PROCEEDINGS OF THE Vm WARSAW ... - Inspire HEP

INSTITUTE OF THEORETICAL PHYSICS, WARSAW UNIVERSITY, WARSAWINSTITUTE OF EXPERIMENTAL PHYSICS, WARSAW UNIVERSITY, WARSAW

INlS-mf — 1 H 1 4

PROCEEDINGSOF THE Vm WARSAW SYMPOSIUM

ON ELEMENTARY PARTICLE PHYSICSKAZIMIERZ, POLAND, MAY 26 - JUNE 1, 1985

WARSZAWA 1985

I K 3 ? I ? ' J T 2 0 ? THEORETICAL FHYSICC, v;ARSA'.v UNIVERSITY, V/AI-CAV

INSTITUTE GF ZXPEEIKEHTAL P H Y S I C S , V.'ARSAW O r i V i i R S I T T , V.'ARSAW

P R O C E E D I N G S

O F T H E ? ' l l l W A R S A W S Y K F O S I U K

C K 2 L E K S N T A R Y P A R T I C L E P E T S I C S

KAZI1-IIERZ, P O L A n ) , MAY 26 - JU-IE 1 , 1 9 8 5

E d i t e d by 2 . AJDUK

V7 A R S Z A W A 1 9 8 5

The Symposium was supported i n p a r t by S t a t e Counci l for Atomic

Energy, Warsaw, P o l a n d .

O r g a n i z i n g C o m m i t t e e

Z. A.JDUK R. SOSNOWSKI

G. BIAŁKOWSKI A. WRCBLEWSEI

D. KIEŁCZEWSKA J . ZAKRZEWSKI

S. POKORSKI

f r o m

INSTITUTE OP THEORETICAL PHÏSICS, WARSAW UKIVEïSITÏ, WARSAW

INSTITUTE OF EXPERIMENTAL PHYSIOS, WARSAW UNIVERSITY, WARSAW

INSTITUTE POR KUCLEAR STUDIES, WARSAW

This is the eighth volume in the series of Proceedings:

1. Proceedings of the I International Symposium on Hadron Structure

and tëultipàçrtiele Production, Kazimierz, 1977;

2. Proceedings of the II International Symposium on Kadron Structure

and ßultiparticle Production, Kazimierz, 1979;

3. Proceedings of the III Warsaw Symposium on Elenentary Particle

îîhysics, Jodłowy Dwór, '980 /appeared in the journal "N'ukleonika"

|6., 147, 1095 /19S1//;

4. Proceedings of the IV v.:arsaw Symposiwm on Eleaentary Particle

Physics, EaaimłBrz, 19E1;

5. Proceedings of the V Warsaw Symposium on Elementary Particle

Physics, Xazisslerz, 1932;

i>. Proceedings of the VI Warsaw Symposium on Elementary Particle

Physio», Kazimierz, 1983}

7. Proceedings of the VII Warsaw Symposium on Elementary Particle

Physics, Kazimierz, 1984.

L I S T 0 J? P A R T I C I P A N T S

1.H.Abramowicz

2.M.Adamus

3.Z.Ajduk

4.J.Baacke

5.B.Badeiek

6.J.Ealog

T.J.Bartke

8.A.Earut

9.A..Bassetto

lO.R.Battiston

11.A.Boha

12.R.Budzynski

14.E.Chriat0Ta

i5.L.Criegee

16.A.Caechowski

18.D«Dorainici

19.Z.Dziemtoweki

20.A.Pilipkowski

21.J.&ajewski

22.E.Geiiser

23.B.Gre^dkowski

24.L.Hall

-Warsaw

-Warsaw

-Warsaw

-Dortmund

-Warsaw

-Budapest

-Cracow

-Genera

-Padua

-Perugia

-Austin

-Warsaw

-Warsaw

-JINK Dutos

—Hamburg

-Warsaw

-Warsaw

-Genera

-Warsaw

-Warsaw

-Hamburg/Warsaw

-Warsaw

-Warsaw

-Harrard Cambridge

25.E.Hlnni

26.A.Hirsehfeld

27.Riam Quang Bang

28.M.IvanoT

29.B.Jozefini

3O.J.Eenrp&

31.P.Kielanowski

32.D.Kielczeweka

33.R.Kirscbner

34.3.Krasznovssky

35 ,M.iCrawczyk

36.P.Krawczyk

39.D.Krupa

4O.E.Zrye

41.Z.Lai ale

42.A.Ma;jevska

44.B.Kele

45 .R.Nahiihauer

46.J.Namyslo wski

48.W.Van Neerven

-Berne

-Dortmund

-Charlottesrille

-Sofia

-Bratislara

-Icdz

-Warsaw

-Warsaw

-Leipzig

-Budapest

-Warsaw

-Warsaw

-Warsaw

-Irrine

-Bratislava

-Lodz

-Warsaw

-Warsav:

-Varsaw

-Borne

-Berlic-Zeuthen

-Warsaw

-Wcirsaw

-Dortmund

49.R.Nowak -Warsaw ' 74.K.Spaliński -Warsaw

50.A.Okopińska -Białystok 75.I.Stamateecu -Berlin

51.M.0l«ehowski -Warsaw 76.M.Staszel -Warsaw

52.B.O»rut -Philadelphia 77.W.J.Stirling -CERN Geneva

53.G.Fancheri -Harvard Cambridge 78.A.StopozyńEki -Warsaw

54.J.Pawełczyk -Warsaw 79.D.Strom -Hamburg

55.J.Pawlak -Warsaw BO.M.K.Sundareaan -Ottawa

56.K.Paw3xmski -Warsaw 81.J.Szabelski -Łódź

5?.D,Perfcermann -Leipzig 82,R.Szwed -Warsaw

58.T.H.Fhat -Wai-saw/Hanoi 83.A.SzyiBacha -Warsaw

59.0.Pigu«t -Geneva ßi.I.Szymanowski -V.'arsaw

60.S.Pokprski -Warsaw 85.P.Szyaiański -Warsaw

6i.Cfeao Wei-Qin -Beiîn^ 86.K.Tanaka -Coluabus

e2.T.Had©żyokl -Warss».' 87.C.Tao -Paris

63.ß.Roftaniuk -BiałyBto* 88.S.Tatur -Warsaw

64,B.Randio -Siałystok Bg.T.Tymieniecka -HamburgAfarsaw

65.K.Sadzj;Éska -ïââé 9O.P.Vecsernye6 -Budapest

66.J.Sehaeher -B*rne 91.A.Wasilcwski -Łódś

67.H.Sćhienauoi -Saisbitrg 92.J.Wdowczyk -Łódś

68.S.SGhlenstedt-Esflin-Łeuthen 93.A.Wróblewsl<i -Warsaw

6-9.P.£chBÄB*r -Haisburg 94.G.Wroehna -Warsaw

70.G.Senânovic -Upton 95.K.Zabłocki -Warsaur

Ti.D.Siemieiicsuk-Białystok 96.J.Zakrz«wski -Warsaw

72.V.êimak -Pragu« 97.K.Zalewski -Cracow

75.J.Skanâ -Hambarg 98.M.Zrałek -Katowice

5

?. C !• T E y T S *

PP COLLIDER RESULTS

Chairman: K.. Tanaka

C.Tao - RECENT RESULTS FRCK UA1 °

S.BattistOE - '..'* A!3 Z° IK UA2 A!O TESTS 0? THE STANDARD MCDEL 29

J.Schacher - QCD IB CA2 65

H.HSimi - SEARCH ?CP. MGNCJET AKD MULTIJST EVENTS WITH LARGE

KISSING vm IS THE UA2 EZPERIKSi;? S3

*-nE::oi-E:.oicGY CF SLSCTRCVSAK EFFECTS

Chairman: B. Ovrut

P.Q.Kung - THE 3TAI3DAR2 ELECTRC'rfEAE MCIiEL A S3 SEYCIS: A Rr»"E/ i67

S . S c h l e n s t e d t - liSTEP.MIKATIC" CF THE LEPTCI.'IC ITEVTRAL CURRi."!

COUPLINGS 10:

F.Erawczyk - V.TIA.T CA.\' VE LEARi: FRCK PRECISE r-SAS'JREI^NTS

C? ELSCTRC'.v'EAE QUANTITIES?

D.Dominici - AX EFFECTIVE LA-3RAK3IAr 3E30RIi : : :3 :CEV.' VECTOR !2 .3 : :o

I K A STRC:;G I;;TERACTIJVC- E I G C S SECTOR 1-9-

W.R.KroDp - THE KBILE UCI .NTEl'TRU'C EZ?E?.IKE;>T 5 4 "

L .Ha l l - IDEAS FOR A 17 ke7 K2UTRIKG

•HSH ENERGY LEFTOK IN"SERACT1C"S

Chairman: G. Sen janov ic

L .Cr i egee - HAERCS ?RC3UCTICK A : O SEARCH FOR ::E« PARTICLES

AT BEST . • 15"

1 A'e list here the masses of the speakers only.

A complete list of the authors is giver- in the text of the talk.

J.Skard - TWO PHOTON PHYSICS. ELECTROWEAK EFFECTS -—

D.Strom - RECENT RESULTS ON WEAK DECAYS FROM DESY 489

W.L.vaa Neerven- RADIATIVE CORRECTIONS IN TV/C PHOTON PHYSICS,

INCLUDING EFFECTS FROM NON-PERIPHERAL DIAGRAMS 175

A.Wroblewski - CONSTRAINTS ON MULTIPLICITY DISTRIBUTION

OF QUARK PAIRS 661

J.Nassalski - SEARCH FOR HIGHER TWIST EFFECTS IN KAJSONIC

DISTRIBUTIONS FROM p.v INTERACTIONS 517

E.Rondio - QZ AND V DEPENDENCE CF MULTIPLICITIES IN jutp

SCATTERING AT 280 GeV 529

Chao tfei-Qin - FACTORS I!.' EMC EFFECT 187

K.AbranowicE - OVERVIEW OF % INTERACTION'S 543

I.Sch^flser - HERA - TZL PHYSICS AND THE >^CHIN£ 5B3

Chairsan: Z. ZalewsV.i

W.Stirling. - QCI- AT THE COLLIDER

K.Kra-.vczyl-. - DEE? I.'^LASTIC CCKPTON PROCESS

G.Par.cheri - JETS i:; r JIIia'K BIAS PHYSICS

R.Kirscliner - LOGARITHMIC ASYT^TOTICS OF AJCTLITUDSS IK QCD 195

A.Bassetto - CRITICAL QUESTIONS COfiCERNIUG LIGHT C O S

FORMULATION AND POSSIBLE REN0RMALI2ATI0N

^ OF YANG-KILLS THEORIES 201

A.Okopir-sKa - CPTI.MIZSD PERTURBATION SERIES FOR THE

EFFECTIVE POTENTIAL IK 4>4 FIELD THEORY 217

P.Veesernyes - RI3DEK LOCAL SnÊTRIES FROM FLAVOUR

ANOMALIES OF QCD 229

BAG KCDEI. OCD OF THE- LATTICS

Chairmen: L. Criegee, J. EaacUe

J.Baacke - THE PRESENT STATUS OF THE BAG MODEL -

A CURSORY REVIEW 259

K.Zalewski - EARYOK CHARGES OF CEIRAL BAGS 257

M.Romaniuk - METRIC BAG MCDEL OF HADRONS 263

I.Stanatescu - KDHTE CARLO CALCULATIONS VITE SYHAMICAL QUARKS

D.Pertenuann - REGOLAB LATTICSS AS SADDLE POINTS 07

A TWODIKEHSIOHAX RAKDOK LATTICE TKEOR? 271

J.Hamysiowiki- QUART AHD 61D0H C0KDE8SATES IK A HUC1BOK

A.Bohm - POSSIBLE EVIDEKCE ?OR StIPi3lSy>&IETRI IK EAD80R

SPECTRA 28*

A.Barut - SPIK FORCES IK HADROK SPECTROSCOPY: QtJARK MCDEI

TSRStJS ELECTRCKAGHETIC SPIN ISTERACTIOKS 293

HIGH EKERGY HADROK ISTERACTIOSS

Chairman: A. Eassetto

R.Szwed - KOVEL APPROACH TC SCALING IK EUITIPLICITT 301

K.Blasek .- LEADING PARTICLE E?FEC7 AT KISEER EKERGJEE 313

V.Siaak - F0LARI2ATICK EFFECT OF THE ^-KESOK IK pp

EITERACTI01IS AT 5.7, 12, 22.5 GeV/c 321

- KOTE OK TEE CONSTITUENT QUARK CHOSE SECTIONS

AKD AVEKAGED CHARGED MULTIPLICITIES 327

D.Krupa - HIGHER VECTOR RESOKAKCES IK SEDCL50K FORK FACTORS 655

IKTERACTIOKS VflTH tHJCIJEI

Chairman: V. Simak

- RECEHT RESULTS OK COHERENT AKE- II7C0KEREI;?

PARTICLE PRODUCTIOK IK -UTJCLEUS I"TERACTIOE£

BELOW 30 GeV ? 3 5

e

J.Sartke - SIZE OF THE PARTICLE EMISSION REGION IK

RELATIVISTIC NUCLEAR COLLISIONS FROM

TWO-PARTICLF. INTERFEROKETRY 347

J.Kempa - VERIFICATION OF HIGH ENERGY INTERACTION MODEL

USING DATA ON COSMIC RAYS IN THE ATMOSPHERE 603

E.KryS - DEVELOPMENT OF ELBCTROfl-FHOTOK CASCADES

IB LEAD AND LAYERS OF LEAD-SCINTILLATOR 353

J.Szabelski - MULTIPLE HOOHE IN UNDERGROUND EXPERIMENTS

SRAKD USIFICATIOK, SUPERSYMKETRY AND 5UPERSTKIHG THEORY

Chairmen: V. Kropp, A. Barut

G.Eanânovic - GRABD UNIFICATION: QUO VADIS DOMINE? 669

L.Hail - SUPERSYKKETRIC FLAVOR AND C? VIOLATIOK 449

C.Pigruet - ANOMALIES IK GAUGE AKD SUPERSYt 'JITRIC GAUGE

THEORIES 359

E.Christcva - CK PRODUCTION OF SUFERSYKM2TRIC KA.TORA.VA

. FARTICLES IK POLARIZED e%~ COLLISIONS 375

D.KielcEewska- EX?ERIKEJ?TAL LIMITS OK TEE KUCLEON LIFETII-3:

FROM THE 1KB DETECTOR 383

A.Hirschfeld - SUPERSPACE QUANTIZATION OF GAUGE THEORIES:

GAUGE FIXING A3D GKCST TERMS IN K=1 SUPERGRAVITY 397E.Tanaka

i- MAGNETIC KONOFOLES IK EALUZA KLEIK THEORY 617M.SundaresanJ

B.Orrut - 16/16, N»1 SUPERGRAVITY: THE LOW ENERGY LIMIT

CF THE SUPERSTRIBG 633

- Ec SYMMETRY BREAKING IN TKE SUFERSTRING THEORY 643

3.Hunpert - THE0RE>5 PROVING WITH FIRST-ORDER PREDICATE LOGIC 407

RECENT RESULTS FROM UA1

Charting Tao

Laboratoire de Physique Corpusculaire

College de France

P a r i s , France

Foreword : The organizers of the Kazimierz Conference have asked me for astatus report on the UA1 Collaboration. 1 can only select a few topics whichreflect my personal tastes.The reader should also be warned that many numbers presented in the textreflect the present preliminary status of the analysis at the time of writing ;those numbers are likely to change in the next weeks or months due to changesin selection criteria. For final numbers one should refer to the papers whichshould be published in the future and not to conference talks.

As many results are not final yet the presentations are reflecting againpersonal choices and not the opinion of the whole UA1 collaboration.

Writing this talk up, I find it difficult to avoid personal biases. 1 hopethe reader and my collaborators will understand and accept this choice.

totraductwm

Last year out of the analysis of about 100 nb accumulated at / T = 540

CeV by each experiment, both the UA2 and UA1 collaborations have presented

exciting results, which might open new perspectives to physics (ref. 1,2). After

the runs taken in 1934, 'the total integrated luminosity available has tripled.

in this talk I review the few topics of major interest with a (persenad

emphasis on whether new phenomena can still be claimed.

Outline

1. W, Z* decays and production

2. Monojets

3. Dimuons

4. Search for top.

1. W, Z* dtcaiyt awd production

1. Electron and rouon channels

The selection of W—>e V and Z°—> e+ e" is summarized in taMe I and

the corresponding selection for the muon channels is shown in table 2.

Fig.l shows the distribution of the observed e for the VI- —» e »»

candidates.

The expected cross-sections presented in taMe 3 show good agreement of

both UAI and UA2 experiments with the standard V-A theory. Once the

relative efficiencies are taken into account, the electron and muon channels

give similar results in agreet.ient with the lepton universality of weak current.

10

2. Backgrounds to W, 2*

TaWe • fives the number ol expected backgrounds and their source with

the selection criteria given in tabie 1. To estimate the background we use a

mixed event Monte Carlo, using the real W—*ev events. The W four vector is

reconstructed.

We remove the electron track and shower from the event and replace it

with a lake V A decay, with a four momentum equal to the reconstructed one.

The tl, 2° decay products are then passed through the complete detector

simulation and reconstruction procedure and overlayed onto the "electron

substracted" event.

In this approach, the real underlying event is used and there is no

ambiguity about the jet activity and the Pj spectrum of the W.

3. »,. 2" mass

Whtte the mass of the 2* can be obtained by a fit to the reconstructed

effective masses oi the dilepton pairs (e+ e~ or p* u~), the * mass is obtained

from a fit to the mass of the e (JJ) and transverse missing E - .

In order to reduce the errors due to background and improve the

Jacobean peaks, we use an enhanced irar»verse mass distribution and apply

tighter fiducial cuts. (Fig.2)-

The results are given in table 5 together with the newly calculated

values fw sin* | ^ and * which are m extremely good agreement with the

standard model.

0. OCp and

The developments in the resummation of soft gluons allow more precise

calculations in QC0.

However many parameters contribute ( A , the shape of the parton

distribution, non tending terms, and the choice of the scale at which to

normalize ^ ) . which still lead to an error in the normalization.

For example the probability for producing a W with p T y 2% CeV is

between 3% and 6% (ref. 3) and it is difficult to be more precise with the

existing method* of calculation*. However we now have a large sample of W

which can be used to constraint the theoretical predictions.

11

In Fig-3 is shown the distribution of PT (W) for all events compared to

the calculations of Altarelli et a l 3 . For PT<»')>10 GeV, all W's have

reconstructed jets. The number of jets with E-j. y. 5 GeV is compatible with

QCD expectations (ftg.4). Their angular distributions is also in good agreement

with bremsstrahlung of spin 1 objects.

From a total integrated luminosity of ^13 nb~ , there is no evidence for

the production of an object X with mass M x > 170 GeV/c, decaying into * +

jets.

For M > 170 CeV/c', r ' B ( X » - V * ietS-> > 0.02 (90% C Dx «-.B <W-»e V )

5. Z* activity

3et activity in the Z" events is also compatible with QCD expectation

(Fig.5). There is no evidence for anomalous Z* activity.

Also there has been no new Z°—> )* 1 "*• event observed for a total oi

2S Z'. It becomes more likely that the events of this kind reported lest year

have come from internal bremsstrahlunj (ref. *,Ji).

&• "tr production

A further test ot the lepton universality of the weak current, ts trie

observation of * - f t v » er 2 ° - » t * T " .

t decay 41% in 1 charged prong and 16% in 3 charged prongs (ref. 6).

In the analysis of the 136 * " ' of the i'982 and 19S3 runs, a few candidates

were found and reported earlier this year (ref. TV Table 4 jives the jeneral

features of the event selection.

The most important background at lower missing ET comes from 2 jets

events which fluctuate in the caiorirnetry and are reconstructed with missi^

enercy.

The sample jet of five 1 branch candidates and four 3 branch candidates

show distributions in ET, asymmetry, transverse mass and invariant mass

compatible with a X hypothesis -tFif-6 abctf). The CD information is crucial and

effective in reducing the estimated number of background events gf the bijet

type.

W—»e»i can atsc be a source 6f background. None of the five i branch

candidates can be a good electron. Fig.7 fafccde) show the "hadronic" properties

of the tracks compared to a W<-*«*? sample.

12

%e conclude there is evidence for * - ^ t V , t -—>• one charged track.

The expected branching ratio for * — > t V compared to W—»eV is 0.96 -

0.45 in agreement with the universality of weak current. For details, cf. ref. 7.

With increased statistics and an improved trigger we expect to observe

more W-»1rV events from the latest 198<( run.

This is a part of a more general monojet study.

2. Monojctt

Last year UAi reported a few events with very large missing transverse

energy and a single jet which still lack a clear explanation (ref. 2).

An improved missing ET trigger was implemented during the 1984 run.

| "2. ET (left) - J E ? (right) | > 17 GeV and jet >15 GeV

to increase acceptance for missing ET (E'T)physirs especially at lower energy.MA setectiyn similar to the one done in 19S3 for monojets with E ^y't

where tT = 0.7 f £ | E - | was run, yielding 23 candidates after validation cuts

#(X) scanrriF^ ffc*-sample").

%'<tfk ts tn prepress to evaluate the contribution oi alt physics and

background processes it possi&te identifying each event individually. A selection

*i tOVfr E^. { > 5 9*)t K presently performed to help with the understanding of

Asiije irbwi tht "triviBl" QCD bijet background, many "physics" processes

tb the

. W—»iV, >LV where the tepton is raisidentified

"* • •** w-tth a large Ey bremsstranlunj jet

where a large V is produced on

, bg, tt

n!, tE ^

one side (through semlleptonic

decays tor example) and the other

side jives the observed "monojet"

An exact calculation is difficult. Whenever possible we try to use the

oat* itself a* input for Monte Carlo (cf mixed MC method described above).

I cannot jive final numbers yet in this report but :

i) Heavy flavours yield a non negligible contribution to large missing tj

phenomena when they undergo a semileptonic decay.

ii) The signal W—*TV , where -fc—»1 charged branch is confirmed with

10 candidates with a background around ! in the <t sample.

iii) None of the new 1984 candidates is as striking as even: A ((.i

monojet) of 1983 (which largest background was due to W— tV> ,t-+ lft[n.ir\arvl

the charged pion leaks out as a u). We expect <, 0-OU events for the whole

statistics. At the end of 1985, with a new run, we should be able to leli ii

more events like A appear again indicating new physics, or ruing it out.

Difficulties

1) The normalisation is not easy to undertake since factors of 2 cannot

be calculated exactly in QCD but are important in this study. We have to reiy

on other UA1 measured data such as low P_ dimuons, and internal consistency

between the 3V and *•" samples.

2) We don't know if the top exists (althrough there migiit be indications

from the data) and which mass it has. The cross-section for top production

varies rapidly as M" (top) and we don't know well the top lragmentation. This

can yield very large differences in estimates.

3. DJmuons

a) Sources of dirr.uons

1) Z°—> u+ \i , Dreii Yan events, 3/^ and T decays yield opposite signs,

isolated muons, which can be identified and separated by the reconstructed

effective mass M (u+p~).

2) Semileptonic decays of heavy flavours produced either directly by QCD or

via W, Z° decays give opposite sign but also same sign dimuons, and on the

whole the muons are accompanied by hadrons and not isolated.

i) BB mixing analogous to K° - K°;r;xing are an additional source of same sign

cimuons. The observed B lifetime and T1 (b—» u)/ T (b —> c) constraint the

Kobayashi-Maskawa matrix. One expects small B^ and large B° mixing.

b) The- tools which help us in distinguishing between all those processes

are

1. The charge of the muons

2. Hadronic activity around the'muons

3. Mass of the muon pair

it. Mass of pp jet-jet system

5. N(- -)/N {+ -) is a measure of mixing

6. Strange particle content

c) Selection

With the 108 nb atVs~= WO GeV accumulated in 1983, we indeed

Observed dimuons of different possible sources as reported in ref. S. With an

additional 270 nb at Vs~ = 630 GeV in 19S<t, we confirm the observation of

dimuons. TaWc 7 gives the latest event selection of dimuons which yields 221

validated dimuons. F»g.$ shows their reconstructed effective mass. There is a

clean separation between Z° —+v*v pairs (1C events) and lower mass dimuons

(1J50 p V , 62 M : P1)-

d-) background sources of dimuons.

1) Punchthrough. There are about 9 interaction lengths of iron around the

detector, so' the probability of punch through is small, about 10 per hadron.

2) Leakage have bad matching with the CD track; there are also additional bits

in the muon chambers. From test beam re

lity for hadron to leak through as a muon.

in the muon chambers. From test beam results, we expect about 10 probabi-

3) ft, K decays are the largest sources of background. The probability for decay

in flight in the Central Detector is 0.02/ PT per "K1 and 0.11/ PT per K-. *'e

assume 50 % *K- and 25 % K-, therefore about 4 10" /P T per charged hadron.

Low PT K ~ H J V can simulate a high momentum track due to bending by the

magnetic field. This background is estimated, using the sample of single

M Py> 3 GeV. All hadronic tracks with P . > 3 CeV/c are "decayed" into a muon.

The apparatus simulation and the reconstruction programs are used to provide

the expected background which reconstructed effective mass are shown in

FIR.3.

For Muu > 6 GeV/cJ, one would expect 25 events and above 12 GeV/cJ, 8

events. U we use the isolation criteria described below, the numbers are

reduced to 6 and 2, respectively.

e) Isolation criteria

In Figure 10 a, is shown the sum oi the energy deposited in the

calorimeter cells in a cone of £>+. i=\l£^<ix * £>f*") of 0.7 around the direct;on

of the muon. The solid curve is the same sum around a random direction in

minimum bias events.

In Figure 10 b, is shown what we expect for muons from serm-

leptonic bottom or charm decay.Thus we are led to a restrictive definition of an isolated muon. for which(in A * < 0.7)2LET>3 GeV. .For events with two muons, we have defined the quantity

S = (£ET 1)» « (S.ET 2) !

Taking pairs of muons at random trom the 28 *—> uV events, we get thedistribution for S shown in Figure II.82 % of the isolated events would pass a cut at S <,, 9 <GeV)'. This is what weuse to separate isolated dimuons from non isolated ones.In Figure 12, the dimuon masses are shown for the four classes of events.

f) Classification of events (details are given in ref. 9).1) Isolated opposite sign diipuons

These can come from Drell Yan, *T" and heavy flavour processes.We estimate a cross section T D Y (Muu > 11 GeV/c*) - 26*- 1G4 pb, tocompare with the theoretical prediction oi 290 pb (rei. 3).

f * r -\ - I ? ± 4

16

2) Non isolated candidates from heavy flavours.

In Figure 13 is shown the mass of the system of 2 muons + 2 closest jets for

the 161 non isolated events, and expectations for bb, cc. Note that for the

moment the Eurojet Monte Carlo which is used (ref. 10) has no underlying

event, no smearing in the energy deposited by the muon and large uncertainties

in the fragmentation of the heavy quarks.

The bb - cc expectations seem to fit quite well the data, giving

an estimated cross-section V (pp • OQ ) = 1.3 - 0.1 - 0.2 jib for

PT(Q) > 5 GeV/c and | rj (Q)j< 2.

li we include the isolated events, <T (p(T—»Q^) = 1.1 - 0.1 - 0.2 ub.

An additional source of same sign dimuon events is due to B-£ mixing. Muons

are opposite in azimuth, leading particles from^first generation decay.

The u*u* should be accompanied with ft, K°, while the u~u~ events have 7?, "R?

The calculated ratio R = s a m e ** - O.«5 - 0.11 is not incompatible withopposite sign

expectation fpr full mixing. However those are preliminary results.

?) Isolated s.»me si£n dirouons

7 events fall into this category, out of which 3 were reported

previously (ref. $). There are no obvious explanations for those events. We are

studying the effects cri changes in Monte Carlo and contributions from tt .

») Lepton + m%

Last year UA1 reported a clear signal for the production of an isolated

large pT Septon accompanied with 2 or 3 jets. The events clustered around the

W mass and could not be interpretated in terms of expectations from known

quark decays (c, b) but were in agreement with a »' — ^ tb* process with a

semileptonic decay of the top <ref. 11).

The 198ft u sample is under study and still cannot be reported upon atthis time.

1) Selection of e* iets

The samples of 1983 • 19S« wirh an integrated luminosity of 390nb were processed. After a first level selection, asking for

- 2 electromagnetic Cells with E.T > 1 j GeV- 1 high momcnturr: good quality track with p T > 7 GeV., projeciea leng'.i. i-KvJ30 cm, and a number of points above- 2C- CP track well matched in x and ^ with the electromBgnetir duster i < ."> 0" !

- Veto on jei fluctuations. In Aft (r \fZ^pTAA*- i < 0.7. tne electron mustaccount for 90 % of the energy.we are left with a sample of fri7 "e" - jets, which is st;i! dominated b;background due to

- photon conversions which are seen in the detector (96 evts) witr. an estimateof less than 0.1 event unseen.- overlap of one or more IT' with an energetic 7r -.

A tighter selection asKing for

- energy matching | " t "" '£" |^ > *"- strong requirements about the electromagnetic shape of t-he shower

E, . < 200 MeV. matching in space TT 4. 2. electromagnetic shape T* < b'j- a super isolation of the "electron" track : ^ P T - ^ I GeV and S E T ^ i GeVin ^r <. O.fc

yield? only Vi "e" - lets.

Deiining a jet as a cluster of calorimetric ceils with E». > S GeV. there are2 events with 0 jet. 28 with I )et,9 with 2 jets.ft with more titan 2 Jets.

A muon - icts selection is under progress and no report can bt yet given at thissiage-

2) Background estimates

1) A selection of isolated « - ( n. *K* ) - jets was done, yielding 2.3 - 0.5/ l.J -0.2/ I - 0.2 events which could simulate an electron for the 2S/9/^ e • jets

events.2) A selection of isolated 7C° * jets is used to compare the shape of thedistributions with the selected e~ 2 >ets sample. In Figure 1», are shown themass plots M ( % ° or e, V • nearest 3et) versus M ( IT " or e, 9 , both jets).

The probability ior both distributions to be compatible is about S.5 10 .Cos 6* (nearest jet) Vs Ei?UL las defined in Figure 15 are shown in Figure 15 b.

The compatibility is 3.5 10 .

3) Beauty and charm pair production by QCD processes (pp—>bb~ or cc) canprovide high p». iepton • 2 jets.

The Eurojet Monte Carlo is used to calculate the contribution of theseprocesses, since it reproduces well the single rnuon and dinnuon cross-sections.The lepton in a top decay is isolated, whiie it is not in a beauty or charmdeca\. Furthermore the distributions in cos & and E j for the QQg processeswould be more forward peaked, different from the observed events.

3) Interpretation

Both W —>tb and pp —> tt where a top decays semileptonically would be goodcandidates for explanation of the 12 events (electron - 2 jets).The even; rate is consistent and both kinematic behaviours, smirred by theresolution af the detector are very close, and it is difficult to distinguishbetween tne two sources. The large number of e + single jet would favour alower mass top (around 30 GeV). The analysis is still in progress and it is tooearly TO conclude yet on the existence of a top signal and on its mass.

Conclusion

On all the topics presented in this talk, again a word of warning : theanalysis are under progress and nor finai.

However, the 1SS* data do not reproduce the few non-standard eventswhich jumped out from the 1983 analysis. Statistical fluctuations or realphysics ? By the end of 1985, with more statistics to come, we should becapable to answer thai question.

In the mean time we have gained understanding on many fundamentalissues, witnessed a revival of OCD explaining the behaviour of jets associatedto * and Z° production. While »', Z° were a signal in themselves in 1983, theyhave become a background to substract .in the search for new physics.

An understanding of heavy flavour production and fragmentation iscrucial before new physics can be extracted. This is being done.

Final results should be published very soon.

19

Reference list.

1 - Bagnaia et al (UA2 collaboration) PL 139B (198<t) 102.2 - Arnison et a! {UA1 collaboration) PL 139B (198<*) i 1 5.3 - G. Altarelli, R.K. Ellis, G. Martinelli. preprint CflRN TH "015/8* and

Fermilab pub 8<*/107-T (198<.).S. Geer and W.3. Stirling PL 1KB (1985) 373.

it - Arnison et al (UA1 collaboration) PL 126B (198*) 39S.5 - Arnison et al (UA1 collaboration) PL U7B (I98f) 2*1.

6 - Particle Data Booklet (April 19&4).7 - P. Colas et al, UA1 Technical Note TN 80/7S.

For details, see Y. Giraud-Heraud, LPC thesis, 1985 (to be published).S - Arnison et al (UA1 collaboration) PL 155B (1985) W.9 - Details of the selection are to be found in H.G. Moser. UA1 Technical

Note (1985) to be released.10 - The Eurojet Monte-Carlo contains all first and second order QCD

processes.ci. B. Van Eiik, UA1 Technical Note. LAI TN Si/93 (I9S6) and A. All,E. Pietarinen, B. Van Eijk, to be published.

Acknowledgements

It .is always a pleasure to attend Kazirnerz and 1 an grateful to myPolish friends for this invitation and the organizers lor a mosi enjoyfulconference.

The work presented here, is by no means my perbona! work but thecontribution of a whole group. ) v.am to thank all my collaborators, those whohave provided the plots and the information on the analysis, but also those whoare working on perhaps less grataving but fundamental parts pi the large UA1machinery, such as maintenance of the apparatus or data crunching. The resultscome from everyone's effort.

Of course the choice of the topics, the philosophy of the presentationsand the errors are my own full responsibility.

Many thanks to the secretarial staff of the LPC for typing the text anddrawing the figures.

20

FIGURE CAPTIONS

Fig. 1 E-j. (e) for W-»ti> events

2 Mass plot for W, 2° (tight selection)

3 PT (W) Vs Altarelli et al (Ref. 3)

i| Number of jets Vs QCD expectations in ft events

5 E_ of jets Vs QCD expectations in 2° events

6(abcd) E~x, asymetry, M_. M l m for I br and 3 br candidates Vs Monte-

Carlo expectations.

7(abcde)"Hadronic" properties of the 1 branch

8 Effective dimuon mass

9 Effective mas* oi background to dimuons

10(a) Isolation oi u rracKs in \ \ . I " events

(b) Isolation of y from charm, bottom decay

It 5 parameter from ^—»uv' events

12 Dimuon masses for - Classes of events

13 *'ass of di^H-on* - ie!s :or non isolated events

15 (a) Cos0 - , Z°?' doiiriit.or

15 (b) Cos 9* \» CûT

TUPLES

Table 1 I ' - » e V , -'"—» e'f" ••oiecnon

3 • Cross-sections I -M. I \2. Y-.-\

<* Backgrounds to W, Z°

5 M t f . . . S , n ' 9 | .

6 » - » t V 1983 event selection

7 Dimuon selection.

FICUCUUCOIS APPLIED

" "

21

FIT to the W ma**

LIKELMOCD FIT TO THE Z'MASS PEAK

>o

3 -

I

1 1 I'

„ ISevenh

: f li

-

62 96

Z*MASS

FRACTION OF EVENTS©o

a at „

JETS PER 4 GeV-

o> ro

m

1 1 1

1

' \A i f S

1j

€ 2i»M

(if Of

3 5"H. ^

i i t

1

rsjen<—m-H

• o

1

zz

23

I- ;

I i

. -

-i t ,55 i

N

s s s

, r i: i

\?S

ff * •' ' nTi - ^ *** ^ "

//

pn

120,

60-

•SO eo

FIG 6

160

SVJEK3 Y/- -y- - rC

% . 5h

1 • ii

FIG. 10b

FIG. 10a

v 05

• 0.4

g 0.2

01

S fo r W-»»i.v events

9 W 27 36

S. I Em' . l E i a ' CieV2

FIG.11

Z^T]

? ; s= u ^ : i "If

7 Events

20 10

r \' 1• in

r

•.

a

:c.

-

t

ASS

1

1C

161 non-isootsc evens-be, cc'Eurojet!

120

FIG. 13

t *

\<c:

t - r

elel

26

= 5 - K

" g

1!i a.

11 JJII I'IA ' i*^<irt'"Jfc •

, / !

i j s

i

IsC «O «"• •"! " O5> ta • • > - . jn •

3-

I f i O L * t I O N m C L U S T f KS fcV

27

COWISTHIT wilti

I i t i n E«"» l t r

Z. p c 3.0 cev

mm T«!GGE»l < IO«V Z f T e 2 G t V

!fi A r • 0 4

[tECTBOliACWTIC 5HDWCI!PHOFIIC

E l O t G W

3M MotllM «Clwtirgo* m

55 Iron « • S» Ctv113 •• 930

w-

4 Irom VE • 5 « G«V14 ii 630

z •-..-.

TO JET B«CK-TC-»*CK

E^< H E W r { : T 5 Om

in tit = •'• JO'IEM«iV

IS S.V

1453

47

From••

V

IEVEHT5

5«t»v6J0

Table 1 Selection of H, f.Table 2 W->p.V selection

tfs = S«6 Gtv

^ i ; *5OG«V

RATIO

THEORY

0.37* 0 1

0 4 7 : 0 1

17*

OSS

:ooe * o o f

063•005 10 09

1 IS * 0 17

tut?

090

r o oo • «os

0537006 7009

1 • * ! 0.23

Aiwuv

« tVMtK

14

33

Accrotwc*

0 20 ! 004

072 • 003

Cf B (W - O f l

C Bl >•)

067

: o 17 : o is

061: o n : 012

7 r 013

AtUltW

l/i i 63064V

•UTtO

rwonr

• 13

" " - 7

" • - ' *

123

IMf

40

: 20 : 6

79; JI ; i2

2o:os

UA2

101

: 37 ? is

5*! » 59

06 ! 03

TOTAL 10 tvfNTS

O B , SI4 ! 171 - 0 6 pb

T»ble 3 CROSS - SECTIONS

28

1 / T (Gf V>

1 rvFNtf

Sflr torouna*

w - > t VX

morons

TOTAL

SIGNAL

5 0

oo±0?

2 I ±02

J « ± I B

6 4 i t B

526

4 3 0

113 ,

T0SO5

l«!01

IO?iO»

102 3

TOTAL

172

f7±(M

9 t ±05

S3± 19

1? 1 ±10

15*9

CHAMCl

!"-» cc

w-> c?w-j lE

z *—> t T

05

tO t

. 05

- -

< 0 ?

. 0 3

" , .,.-» »0 6tv

0 3

j«to-»

. 0 . 1 0 - '

- -

3.10-1

,3»'0-»

STANDARD MODEL

PARAMETERS

Mw(GeV/cr)

z

S i n 2 O W s i - M 2 / M *

P - « i »ec'Sw/nj

en:;} is

wo i 16 i j

0202 r 00J6

0J I6 *" * 10016

1016 i 0041 ;002l

Table 5

Table* Backgrounds to W, Z°

t36 no-1 Yi

(I98J . IMS)

OATA SAMPit l O B r a ' 1 (19BJI

170 I * " 1 I 19M)

VALIDATION Of HISSING CT I

if; ISGfV or f C j a .-O7VIT7

29 961 IVIMS

of aro OUCTS1 BKVfO-tllClL7 6»«. W >5 SfV

to 6 0 0 Evf«T5

ALIDATION Of TRIGGER JET

GD83 C0-- Clio melcntns

TrtQgcr }«t l r i IS GfV

VRiulttorfnefttty

Dr««)rn ' ; ctisropo track witn pj\ too ntv

Table 6 selection of V~*i

fiUOT. TBIGGII

FILTER PROGHAM

1983 Olt-HM1994 Dn-tint t 168 E )

OtnuON SELCCTiOnj

SCAMMINC OF CVEKT5

72 (u t i t t tro*n 1 / T J C46G»W

289 t«ml« lr»m / 5 E 630 Gtv

37 tMkiyr

tnrpugn rrms

• T| i i matching

T?ble 7 Dinsuons selection

29

W* AM) Z* PRODUCTION IN UA2

TESTS OF THE STANDARD MODEL

The UA2 CollaborationBern - CERN - Copenhagen (NBI) - Heidelberg - Qrsay (LAI,)

Pevia - Perugia - Pisa - Saclay (CEN)

Presented by Roberto BattistonUniversity of Perusia and IMFN

Perugia, Italy

A1STRALTSuriac tba 1984 ran UA2 collected data corraspondins toan intepratad loainosity of 310 nb'] increasing the totalcollactad ltatinosity to 452 nb'1. An analysis of tbaproduction and decay of r aad Z* ia presented, updatingtha rasults on the IVJ properties. Froa aeasaraaents ofthe IVB aassea, ws Mature the weak neutral currentparameters sda'a^ and • ; th?. aeasnreaent of electroweakradiative corrections is also discussed.

30

1. INTRODUCTION

The discovery of tin Intermediate Vector Bosons (IVB), Bade in 1983-by the UA1 and UA2 experiments l)j) at the CERN pp Collider, was a greatsuccess for the Veinberg-SalaB-Glashow unified aodel (GSW) of the weakand electroa*gnetic interactions, SU(2)L«U(1)

>}. The UA2 collaboration

has already reported *"' that the data available at the end of 1983 onthe following processes:

p + p •* If* + anything• •* + v(?) and

p + p •* Z* + anything-• e* + e"

were aestly is agreement with the predictions of SU(3)C*SU(2)L*U(1)(Standard Model). The data were collected in the period 1981-1983(/s « 546 GeV) and corresponded to a total integrated luminosity* - 142 no"1.

lovever, tha data presented SOB* feature* that, with Unitedstatistics. Bight have suggested the existence of unexpected phenaeena.Mamly :

1) a Z* * e'e'T was observed in a kineatatical configurationhaving a low probability for internal breasstrahlung";

ii) . 3 events, interpreted asp + p • V + hari jat(a), » + «v

were found salikely in teraa of a QCD description of the Vproduction Mechanise)*'.

These results, together with the observation of unexpected events byUA1'), have been at the origin of various theoretical speculationssaggestlag maw physics phsnoaens beyoad the Itaadard Modal.

The suksequaat Collider roaning period-, which took place at tha endof 19t4 at higer C M . « U 0 (/a » 630 GeV), was clearly expected toshed sore light on thaae issuaa.

la addition a precise test of the standard model and in particular

of the GSV model, was of great importance and required better

statistical errors on the IVB Basses.

In this paper we present an analysis'1* of the 19S3+1984 data

sample corresponding to a total integrated luminosity £ — 452 nb"1.

Cross-sections are calculated separately for thn 198A data at /s = 630

GeV and the previous data at /s = 546 GeV., The results on f.he IVB's

production cross sections and transverse momentum distributions together

with the VT" charge asymmetry and the 2' width are compared with the GSV

model predictions. The masses of the IVB's, measured usia^ the full

data sample (1983+1984 ), are also reported together with the

determination ot sin'6^., p and of the electroweak radiative corrections.

All the results reported are final unless otherwise is

2. THE UA2 DETECTOK

The UA2 experimental apparatus has been described in detail

elsewhere . Fig. 1 shows a scheicacic view of its longitudinal

cross-section in a plane containing the beam axis. Theie is complete

cylindrical symmetry in azimuth (•), while in polar angle fB) one can

distinguish two regions :

- The central region (40° < 0 < 140°) is covered by a highly segmented,

tower structured electromagnetic and hadionic calorimeter. 'The 240 cells

of the central calorimeter point to the interaction point.

- The two forward regions (20° < B < 37.5° and 142.5° < £ < 160°) havs

been instruaentsd with the ai» of seasuring the forward-backward charge

asymmetry of r decays. They are equipped with 24 (12 on each side)

identical magnetic spectrometers. followed by electromagnetic

calorimeters. In front of both the central and the forward

calorimeters, preshowex counter's guarantee an accurate localisation of

electromagnetic showers to improve the identification of electrons

against background.

32

Stringent efforts were Bade in order to keep the energy resolution

and calibration under control . There are two effects that reduce the

electromagnetic calorimeter response with time:

a) aging ~5% year ;

b) radiations damage (only during the Collider ruiinninj periods)

-15S year "1.

The response is monitored using light Xenon fiashers (to follow

short-term tiase dependences), C o " sources and measurements of energy

flew using minimum bias JSp events. Cell to cell variations of the

calorijnete.r response are known to o=±2.5%. The recalibration of sone

towers on a test bean shows that the uncertainty on the absolute energy

scale is +1.6%.

3. DATA TAKING

Triggers sensitive to high p~ electrons are constructed from

calorimeter phototube (PM) signals with gains proportional to transverse

energy.

Electronagnetic showers nay span across adjacent calorimeter

c«lls. Trigger thresholds were therefore applied to linear sues of

signals from 2 x 2 cell matrices. In the central calorimeter, all

possible 2 x 2 natrices were constructed; in the forward ones only

conbinations in a given sector were considered.

There ar& two triggers sensitive to electrons in UA2 :

- The "tf-trigger", for which the signal from at least one matrix must

exceed a threshold set at 10 GeV,

- the "Z-trigger", for which the signals from at least two matrices,

separated by nore than 60° in azimuth, must exceed 4.5 GeV.

To suppress background from sources other than p"p collisions, a

coincidence with two signals from hodoscopes at 0.47° < 8 < 2.84s on

both sides of the collision point was required. These hodoscopes were

part of an experiment to measure the pp total cross section*^. Their

efficiency to non-diffractive J>p interactions was estimated to be at

least 9S%.

33

approximately - 1.5 • 10* create war* recorded with the V-trlsgerand - 1.1 • 10' with the X-tri*jer is tbe 1M4 COB. Both triggers «ave •total of - 2 • 10* events correspoadiac to an integrated lomlaosity

• * * 310 no'*.

4. TOE H i C I W I ANALYSIS

Figure 2 shows a schematic representation of the signature ofvarious particlas or sjwtamm of particlas in the UA2 apparatus. Shows inthe figure is tha transverse cross section of a quadrant of the centraldetector. The forward detectors present similar features vita theadditional Baasoraaant of the track aoaaattai.

There are distinctive characteristics of different particles la eachcoaponeat of the 0*2 detectoc :

a) is the caloriaster, the small tranaverae aad le«*ltndiaal•xtanciOB elf the electxaawcDatic shower distltypiiihea alectroasfrea particles —rtrraniat asclear ixteractioas ia thacaloriaetex aaterial (isolated charted hadzoas or. hadroa jets).

b) tha aresesce of a track ia the vertex cbaaawrs allows therecoanltioo of am electros fzoa a ahotoa (or a v*) which showerseleetroamgnetieally' ia the caloriacter.

c) a aarticalarly decaerom hart^TmieJ «e alectnaa is giT«a hy thaSaoaetrical overlap of am energetic «• irtth a soft chartedhadroa. The. first pcodBces am electiwi|iitie ahower ia theealoriaeter, while the latter eoatxawtaa a track aad «ey fakeaa electxom. This »scttinem4 ia radweed hy the hits reaolatiAapraahower comstar, which allewe tha precis* localisation of thesheweriax particle.

Therefore, the identification of hiah a,, electcon ia based om thefollowias criteria :

1. the presence of a localised cluster of energy deposition In thefirst compartment of the calorimeters, with mt aost « smallenergy leakage in the hadronic compartment;

2. the presence of a reconstructed. charged particle track whichpoints to the energy cluster. The pattern of energy depositionmost agree with that expected tram an isolated electronincident along the track direction;

3. the presence of a signal in the p'reshover counter, theamplitude of which Mist' be larger than that of a ainiaua<<w<sfag particle. The geometrical Matching of the preshowerhit with the projection of the track mist be consistent withthe space resolution of the counter itself. These features arecharacteristic of a high energy electron starting a shower inthe preshower coaster.

In practice, because of the different instroBentation is the centraland forward regions, these criteria are applied in different ways in thetwo detectors.

•stalls e* the electron identification, together with theirefficiency to detect electrons, are presented In Table la for thecentral region and Table Ib for toe forward ones. An exhaustivedescription of the cuts is given in previous 042 publications**^.

It timid be noted that moat of the applied selection criteria aresatisfied only by Isolated electrons : the detection of high pyelectron* contained in a Jet of high Pj particles is excluded by thepresent analysis.

Electrons Cram photon conversions are r Jawed by requiring a hit is•t least one of the two iamermoet chambers of the vertex detector.Furthermore, la the forward detectors the electron candidate is rejectee!if tt i* f imnisnied by another track of opposite sign at an axiawthalseparation smaller than 90 ar.

35

5. TOE EXECTBOW SAMPLE

In order to reduce the amount of data to be processed in the r

•and Z analyses presented at this Conference, a fast filter has been

applied to the total event saaple.

1. For the W-triggers, it required the presence In the event of an

electron-like energy deposition in the calorimeter (i.e. a

duster of cells with snail leakage in the hadronic coapaxtaent

and saall radios). The transverse energy deposited in the

electroaagnetie coapsrtaent E-"" was required to exceed IS GeV.

In addition the event was required to have a total energy

imbalance in the transverse plane of at least 13 GeV.

2. For the Z-triggars, events were accepted only if at least two

electron-like clusters, vith aziauthal separation of at least

60*, exceeded Ej"" > 4.5 GeV, such that their total uaas was

greater than 30 GeV/c1.

Of the total rabax of - 2 10' V and Z triggers, - 1.2 • 10*

reaain after the filter cuts. These cats, although very effective in

reducing the data seaple to be analysed, introduce biases which Bake the

background estiaate somewhat uncertain for p_e < 17 GcV/c or

H M < 3C GeV/c1. the analysis of the total data saaple is in progress,

but the results presented here are only concerned with events satisfying

pj.e > 17 GeV/c or H ^ > 30 GeV/c*.

After applying the electron cuts we are left with a total of 290

electron candidates with PT* > 17 GeV/c. The Pj* distribution is shown

in Fig. 3. Candidates with Hj.* > 25 SeV/c obtained by the V and 2

analyses are indicated'in the figure.

36

6. TOPOLOGY OF TtE EVENTS KITH AH ELECTROS CANDIDATE

The sample of 290 electron candidates presented in Fig. 3 contains,in addition to real electrons, fake electrons coning froo Bisidentifiedhigh Pj hadrons or jets of hadrons. Depending on whether the electronis real or fake we expect that the event contains either another high pjleptoa (e or v) or another jet of high ptj. hadrons at approximatelyopposite azimuth.

In order to study the topology of the events with an electroncandidate we search for high pj jets using the jet finding algorithm

described in detail elsewhere11^. We consider the jet activity atopposite azimuth to the electron candidate in a 120s wedge. We definethe quantity

where the SOB extends to all jets having an aziouthal separation64 > 120* to the electron candidate and pj. > 3 GeV/c.

Ha tben split the sample of events with an electron candidate in :

a) Events with p < 0.2 : this sample contains W •+ ev.The • cut is estimated to be (91 ± 3)2 efficient, where theerror reflects the uncertainty of the Monte Carlo simulation oflow energy jets. --

b) Events with • > 0.2 : this sample includes the Z* * e'a"sample. It is dominated by two-jet background with one jet•isidcatified am an electron and is used to estimate thebackground contamination to the W-saople.

37

7. THE V •* ev SAMPLE

The total nuaber of V-triggers passing the electron filter cute is

• - 1.1 • 10*. After applying the electron identification criteria the

ssople reduces to 257 events, 130 of which satisfy » < 0.2.

The p-* distribution of these avents is shown in Fig. 4. There are

67 events (71 in the central region and 16 in the forward ones) with

Pj.* > 25 BeV/c. The background for Oy* > 25 GeV/c is estinated to be

9.0 ± 0.B events. The ^'distribution of Fig. 4 shows a clear Jacobiaa

peak at Pj* - 40 GeV/c, a distinctive feature of V -» ev decays. Other

physics processes also contribute to the electron saaple. In additioa to

the background, sham as a dashed line in the figure, we estimate a

contribution of 3.t 1 0.9 events fiesi Z* •» «*«" decays in which one

electron ic not detected and 2.1 1 0.2 events froei W -» tv, i •* ev9, for

the event saaple satisfying Hj* > 25 GeV/e.

The solid curve ia fig. 4 is the Py" distribatiaa expected once all

the contributions to the electrca saaplc are taken into account.

7.1 lacfcgraond to the W electron apactr—

Of the 257 events with an electron caadidaca, 124 (asclodiag the

Z* •* •*•' events) have •e_- > 0.2, nanely aana Je* activity at eayoiite

axianth to the electron candidate Itaalf. Their Pj* distribution is

shorn ia Fig. 5. The steeply falliag aaactnaa sagfeeta that the staple

consists nostly of nisideatified Jets. Althooffc the presence «f raal

electrons with apposite Jets i.iaeir ha enclaasn, we aasane tint the

saaple dee* sot centals electrons b s s V decays. This sasnajK iim is

valid to the extant that the • cat does not reject V't prodncad at

high Py. Should the Jet aanaaita ta the electron candidate he lent

outside the 012 acceptance, thane awanta m U hacaaa « candidates. I*r

this reesea we mse the sanple «• mttaiara the isi>tiiisn< ta V's la the

following way. Froa the - 1.1 • M * flltaaW W-trigaara we anlaet

eveny which ao ant contaia ana ray clMtatrs with p . > 17 BeV/c paaai

38

the electron identification criteria described in Table I . We thanapply tha topology cat in the sane way as was done to the electronsaaple. The distribution of events with » > 0.2, normalized to 124'wrests, is shown as a solid line in Fig. S.

The ratio betweea the two subi —| les resulting frosi the applicationof *** 'opp CBt

(>t + Popp < 0.2) / (jet + Popp> 0.2)

gives the probability that in two-jet events one jet is lost outside theUA2 acceptance. This probability decreases froa ~ 10% to - 21 when pjincreases fro* IS to 25 GeV/c.

The background to the 130 V candidates is estimated froa the pj.distribution of fat events with • < 0.2 scaled by the factor

vfip

("•" + t^ > 0.2) / (jet + Pefp > 0.2)

The resaltiat diatribatioa is shows in fig. 4 as a dashed line.

7.2. The» a—s

To eatrsct tha valna of tha V aass we coabine the data collected byU U ia 1*V> C/s - « M CaV) aad ia previous rans (Vs • 546 GeV).

Tha rsaaltiag saaple caasists of 123 eveats with aa electron havingftp* > 2S CaV/c. The fjj.* distrihntioa is showa Ut Fig. *, toaather withtha aatisMtad hachtrewi (<la»hsa liae) aaa the theoretical distributionat elactroas fna) V aaiays (solid liae).

Tha M M Mlae i» aatzactaa fna this saaale hy i Mjai lag theaistrihatiM •*«/«>,,* * . »i«» tksx Mfsrtsa froa V -* •* decays C«# ia

aalar aa«le).

39

To calculate d^n/dpj* d»t we have usad :

- structure functions froa Gluck «t al.11' to generate the p^distribution

- fixed V width *t tv * 3.0 GeV/c1.

- the decay angular distribution expected for the standard V-A couplingof V to f•rations.The detector energy resolution is taken into account.

- the p_W distribution of Altarelli et «l. l > )

Th« best fit to tfce experimental distribution is obtained with

Hy « »0.S ±1.1 (stat.) GeV/c*

The quoted aass differ froa that quoted pr«li«inarly J) , since a aorerecent and final analysis which is being prepared for publication tras-nsed: is particular the event wit py' » 60 GeV/c has beeo excluded froathe sass fit because does not pass the cuts in the final analysis.

Ve have also fitted the saae two diaensional distribution of asaaple obtained vith stricter electro* cuts, which has such saallerbackground contaBination, and ve find the saae value for the V Bassindicating, as expected, that the fit is not sensitive to the low p^*

part of the spectra. The fit is doalnated, instead, by the high pj.*region, which is sensitive to the topology cut (•__ < 0.2).

To check that the effect of the » o p - cat is correctly taken intoaccount in our Hoote Carlo simulation* we have fitted the transverse•ass distribution of the V saaple. Which has been shown by Honte Carlosimulation to be insensitive to the » cut. Me quote the final value:

My » Sl.l ± 1.1 (stat.) 1 1.3 (syst.) GaV/c*

in perfect agreement with the one obtained with the fit to the (Pj*. 8()distrihotion.

7.3 Forward-Backward Charge Asy—etry

The helicity state of the 5 and q forming the W Is pp interactions

•is defined by the V-A coupling. As a result the W is procuded with full

polarisation in the direction of the p beaa. Similarly the V-A coupling

determines the helicity of the decay products of the V. As a result a

distinctive forward-backward as; jmetry of the charge lepton anst be

observed in the decay V •» ev (Fig. 7). In the V rest frane the angular

distribution of the charged lepton is expected to be of the forst

fr- - (1-q cosB*)1

dcosB*

vhere q is the charga of the lepton and • is the angle between the

charged lepton and the direction of the incident proton. The UA2

experiment is equipped with Magnetic spectrometers in the angular region

20* < 9 < 37.5* and 142.5* < • < 160*. where the sensitivity to the

charge asymmetry is expected to be higher.

There are 30 W candidates with pj," > 20 GeV/c in . the forward

spectrometers, with an estimated background of - 1 event. Their

distribution in the (p"'1, £"*) plane, where p is the electron momentum

with the sign of (q • coce*Iab), is shown in Fig. 8.

The final measured asymmetry is

'V)}/Mtoi: " °"43

where M * (M_~) is the number of positrons (electrons) on the proton

(aatiproton) side, i.e. the right asymmetry, and equivalently for

The measured value of a is in good agreement with the expected

asymmetry

a « 0.53 ± 0.06

where th« effects of the background, of the V * t •• e decs; chain and of

the Z* «v«nt* with only one electron detected have been taken into

account.

6. THE Z' •* e'e' SAMPLE

In the 1984 run - 1.1 • 10' Z-triggers have been recorded, of which

- 1.5 • 10* passed the filter selection described in section 4.

From this sample we keep only the events which contain at least two

energy clusters passing the calorimeter electron cuts described in Table

I.

Although the calorimeter cuts applied are not very strict only

111 events remain with a mass of the two electron-like clusters H g e

greater than 30 GeV/c*. The Mee distribution is shown in Fig. 9a and

shows that 11 events have a M e e > SO GeV/c2 with an estimated background

of — 1 event. After requiring that at least one of the two clusters

satisfies all the electron selection criteria the sample reduces to

IS events. Fig. 9b shows their H M distribution, where one can see that

6 events have Hee > 80 GeV/c* and no event is in the region

42 < Mee < 80 GeV/c*.

Using a sample of two-jet events, we estimate an upper limit of less

than 0.15 events as a background under the Z° peak.

Tox a precise assessment of the sample of the 7 events with

B e e < 42 GeV/c1 it is necessary to study the spectrum et lower mass

values, in order to establish the background level in this region. This

study has not been performed yet, because the event distributions in

this region are biased by the requirement M > 30 GeV/c1.

42

R.I Th« 2' • eeT decay aode

As already Mentioned in the Introduction, one event of the type

•Z* "* sT was observed by UA2 in the data taken in 1983. The probability

thf Internal breasstrahlung can jive rise to such, or less probable

configuration is 1.4%. Therefore we expected a total of N , = 0.11

events of this kind in the 1983 Z' data sample to come from standard

processes.

No such event has been observed in the 1984 data, giving a global

expectation of Neey =0.22 events.

It sbouirl be pointed out that configurations where the opening angle

between the electron and the photon is less than 20° in 8 or 30° in •

are not detected by the UA2 apparatus.

8.2 The Z* mass and width

The total UA2 sample of events with two electrons of invariant mass

H > 50 GeV/c* is shown in Fig. 10. It amounts to a total of 16 events,

8 at a •.. = 546 GeV2' and 8 at /s = 630 GeV, corresponding to an

integrated luminosity 1 = 452 nb~*. The masses of the events taken at

/s = 546 GeV have been very slightly modified with respect to the values

published in Ref. 2 after a recalibration of some calorimeter modules in

a test beam.

All events have M ^ > 80 GeV/c2, where the background is less than

0.Z events.

The final result on the Z' mass is

Mzo = 92.5 ± 1.3(stat.) ± 1.5Csyst.) GeV/c=

where the systematic error accounts for the 1.61 global energy

uncertainty of the calorimeter response. The masses of the individual

events are distributed around M,a as shown in Fig. 11. The 3 events

indicated with an asterisk in the figure are events D, G and H of Ref. 2

which have not been used in the Z* mass evaluation for' the reasons

explained there.

The same event sample has been nsed to fit a value for the width of

the Z", P.. Given the low statistics of the sample available, several

statistical estimators of a Breit-Wigner fit to the data have been

studied. Ve quote

rz < 3.3 GeV/c1 at 90% C.L. (1)

Ve can obtain an independent estimate of T- within the standard

model from the relation *

w & w • ev A it ee

where

R = o_e / Ou = 0.136Z tf -o.o34

as a weighted average of the cross sections at the two centre of mass

energies. The error on R is statistical only, the systematics cancelling

out.

Using the value ru • 2.56 GeV/c2 and th« structure function

paraaetrization from fief. 16, we calculate

l"z » 2.12 ^'^9 (stat.) * 0.21 (syst.) GeV/c2 (2)

and rz < 3.0*40.31 GcV/c* at 90X confidence level, in good agreement

with (1).

Within the fr—work of the Standard Model, we can cooper* tha

•easured value of Tz to the axpactad one to extract the nuaber of

additional light neutrinos expected. Fro* (2) we find:

44

< 2. 5+ 1.7 at 90% C.L.

9. W* AND Z' Pr-DISTRIBUTIQHS

9.1 The V" pf-distribution

± c

The VT" analysis described in the previous sections discarded events

for which a significant amount of transverse energy was detected at

opposite azimuth to the electron candidate (the P o p p cut described in

secton 6).

In the case of the VT pt distribution, a different analysis wasperformed * , releasing the p cut and asking instead for a missing

p v > 25. GeV/c. The results reported in section 9 are still

preliminary. The V transverse momentum p,,, can be evaluated from the

momenta of all other particles or jets of particles observed in the

event in addition to the electron candidate such that

where the sun extends over all the observed jets (pt > 3 GeV/c), P tS p »•

the total transverse momentum carried by the system of all other

particles (spectators) not belonging to jets, and € is a correction

factor which takes into account the incomplete detection of the rest of

the event (? = 2.2±0.5 for the BA2 detector " ^

In the resulting sample of 128 events (83484 data) ve have found

31 events (shown in Fig. 12 with dashed lines), for which the ev pair is

produced in association with a jet or a systas of jets with transverse

energy exceeding 5 G«V. These events have been interpreted in terms of

associated production of Vr+jet(s).

The 1983 data had given hope for sign of new physics beyond the

standard nodal (events A to C 4 O : the 1964 data add three events is

the region ptV>30 GeV ( events E to S ), but the distribution for the

fulj data saaple shows a satisfactory agreement with QCD " ^ (the /s

dependence has been ignored). Therefore the interpretation of events A

to G in terms of standard physics has become more likely.

For the mean value of p t W we find <ptW>=8.5±0.6 GeV/c, including

events A to G which contribute to the highest ptV values; excluding

those events we get a mean value of <Ptu>=6-8±0.4 GeV/c.

9.2 The Z' p^-distribution

The inset of Fig. 12 shows the distribution of the transverse

momentum of the Z°, p t Z > with a mean value of <ptZ>=5.8± 1.0 GeV/c. Also

in this case we have a good agreement with the QCD prediction of Ref 16.

10. IVB PRODUCTION CROSS SECTIONS

10.1 The V* cross-section

The cross section for the process p + p •* V~ + anything followed by

the decay V -* «v is calculated as

otfe » Ny6 / zs n

where Ntfe is the observed number of W -" ev decays, s is the overall

efficiency of th« electron selection criteria, 2 is the integrated

luminosity and n is the detector acceptance.

Ve us* the 87 events with Pj.* > 25 GeV/c for which the acceptance is

H - 0.65. The overall efficiency is t « t • « c u t s = 0.66 ± 0.05.

After subtracting contributions from cha background, from Z' decays, with

on* electron undetected and from the decoy (W •• tw, T -» ev9), we

calculate (final result):

526 ± 6ACstat.) ± 49 (syst.)

for the M~ inclusive production of cross section at /s = 630 GeV times

branching ratio into ev.

An independent analysis based on data from a trigger selecting

events with missing transverse energy gives

owe(630) = 590 + 90(stat.) pb

in good agreement with the result of the electron analysis.

We have recalculated the cross-section for V production at

/s = 546 GeV*', taking into account a new measurement of the total pp

cross-section'' which implies an integrated luminosity of 142 nb"1

instead of 131 nb'1 as used in Ref.2 .

The result is

otf*(546). = 494 ± 90 (stat.) ± 46 (syst.) pb

The cross section ratio at the two collider energies is :

R = o(630) / o(546) = 1.06 ± 0.23

where most systenatic errors cancel.

All the measured values are is agreement with QCD predictions"' :

o (546 GeV) = 360ZX\Q pb

o (630 GeV) = 460 g g pb

R = 1.26

10.2 The Z" cross-section

We use the 8 events with Mee > 80 GeV/c2 to calculate the Z*

production cross-section at Vs = 630 GeV tines branching ratio into

e*e". Five events have both leptons in the central region, where the

efficiency to detect such a pair with the cuts described in the previous

section is 865 and the acceptance is 0.34. For the remaining 3 events,

in central-forward configuration, the efficiency is 89%, while the

acceptance is 0.19.

For en integrated luminosity of 310 nb"1 we find (final result):

O2'(630) = 52 ± 9 (stat.) ± 4 (syst.) pb

This result is in agreement with a theoretical expectation of

As was done for the W, the cross section at •s = 546 GeV has been

recalculated :

02ee(546) = 101 ± 39 (stat.) ± 9 (syst.) pb.

The theoretical calculation from Kef. 15 gives 4 2 ^ | pb.

In Fig. 13 the results on the IVB cross-sections at different v's

are coapared with the predictions of Kef.16.

11. THE WEAK MIXING ANGLE

From the measured values Mu = 81.1 ±1.1 ±1.3 GeV/c* and

•Mz = 92.5 ± 1.3 ± 1.5 ( obtained in the final analysis ) we can extract

a value for sin3By following the definition of Sir1in lj)r

111.1] sin 4^ = sin2ew(qJ=M2

w) = 1-M»W/MS2

This definition is appropriate in the standard nodel with p = 1 (p is

the relative strengbt between the neutral and charged coupling constants

) as inplied by the Higgs doublet mechanism2*'. From an experimental

point of view [11.1] is also appropriate because it depends only on the

IVB physical masses and is insensitive to the theoretical details:

moreover it is also insensitive to the experimental systematic error on

the absolute energy calibration. We obtain:

[11.2] sina8w « 0.23 ± 0.03 (stat.)

Using the standard models predictions for the V~ and Z° masses we

obtain two additional useful relationship for sin J8 u1 1»»»»«) :

[11.3] sin»Bw =(38.65/^)'

[11.4] sialty - <77.30/MI)i

From the experimental point of view the relationships (11.3] and

[11.4] are less sensitive to the statistical error on the mass

determination but are affected by a systematic error. Weighting the

results from [11.3] and [11.4] we obtain the average:

[11.S] sin*6w - 0.227 ± 0.005 ± 0.008 .

Both measurements agree well within the errors. Given the present

statistics the measurement obtained from [11.1] still has a large

statistical uncertainty. On the contrary, the value [11.5] obtained

from the "model dependent" relationships is such more precise, the

uncertainty being already dominated by the systematic error on the

absolute mass scale.

The UA2 measurements of sin*8.. agree with all the previous results

obtained in atomic and neutrino experiments, once one has applied the

appropriate corrections to the low energy data *• . The data are shown

in Fig. 14. The agreement between results obtained in different

reactions and over many order of magnitude of q2 provides a strong

confirmation of the electroweak part of the standard model.

12. THE p PARAMETER

Using the definition:

[12-1] P = H*tf

[12.2] p = 0.99 ± 0.04* 0.01

in agreement with the low energy v(5)N measurements p = 1.01+0.02 li' .

13. THE RADIATIVE CORRECTIONS

The standard aodel with three families and minimal Higgs structure

(one Higgs doublet) predicts for the radiative corrections:

tI3.ll Ar - l- 1

[13.2) Ar = l-ftio

50

where (following Sirlin, Harciano and other "'i''*0)) <x=a(qs=0),

sin28w=sin28uCq2=M2

w), and G is the Fermi constant obtained from the

u lifetime measurement including the electromagnetic corrections of

- order a.

In addition, using the definition [I1 J we obtain a third

relationship:

[13.3] Ar U A 2 = 1-(TO//2G F) * (M*w * (l-H^/M^))" 1

depending only by the IVB physical masses.

Using the relationship [13.3] we obtain:

[13.5] ArUA2 = °"09 ± °'10 ± °"03

Fig. IS illustrates the result [13.5]. Using the relationships [13.1]

and [13.2] tog-ether with sins8K= 0.22O±0.O08 from the low energy v

data" we obtain:

[13.6] Ar = 0.048 ± 0.034 + 0.030

Both determination* of Ar are consistent within the errors with

the prediction of the theory Ar=0.0696±0.0020 *"'

14. CONCLUSIONS

The results of the analysis of the 1984 data (fa » 636 GeV)

do not confirm the indications of new physics of the 1983 data

(<£ - 546 GeV).

The valued of the IVJ masses, of sin w , of the ratio £

between the charged and neutral weak current strengths and the

width of the Z", all agree with the predictions of the ninioal

SU(2)xlI(l) model; also the measurement of the radiative correc-

tion agrees, within the large error, with the theory.

The existing measurements of the IVB production cross-sections

and transverse momentum distributions are as expected from QCD

calculations.

51

REFERENCES

1. Arnison G. et al., Phys. Lett. 122B (1983) 103.

Arnison G. et al., Phys. Lett. 126B (1983) 298.

2. Banner M. et al., Phys. Lett. 122B (1983) 476.

Bagnaia P. et al., Phys. Lett. 129 (1983) 130.

3. S.L.Glashow, Nucl.Phys.22 (1961) 579.

S.Weinberg,Phys.Rev.Lett. 19 (1967) 1264.

A.Salam, Proc.Sth Nobel Synp. (Aspen.-3s garden, 1968) (Alroqvist

and Wilcsell, Stockholm) p.367.

4. Bagnaia P. et al., Z. Pbys. C Part, and Fields 24 (1984) 1.

Bagnaia P. et al., Pbys. Lett. 139B (1964) 105.

5. Arnison G. et al., Phys. Lett. 139B (1984) 115.

6. See also L. Mapelli to be published in the Proceed, of 5th

Workshop on Proton-Antiproton Collider Physics, St. Vincent,

Italy 1985.

?• Mansoulie B-, Proc. of the 3rd Moriond Workshop on pp Physics

(1983) 609.

See also Ref. 8, 9, 12, 13 of Ref. lb).

8' a. Beer et al., Nucl. Instrum. Methods 224 (1980) 360.

9. Boz2O M. et al., Phys. Lett. 147B (1984) 392.

10. see for example Chapter 4 of Ref. lb.

11 • Bagnaia P. et al., Phys. Lett. 138B (1984) 430.

!?• Altarelli G. et al., Hud. Phys. B246 (1984) S76.

13. Gluck M. et al., Z. Phys. C Part, and Fields 13 (1982) 119.

52

14. Hikasa K., Phys. Rev. D 29 (1984) 1939.

15. H. Plothow-Besch to be published in the Proceed, of Sth

Workshop on Proton-Antiproton Collider Physics, St. Vincent,

Italy 1985.

16. Altarelli G. et al., CEKN-TH 4015/84 (1984).

17. H. Hanoi to be published in the Proceed, of 5th Workshop on

Proton-Antiproton Collider Physics, St. Vincent, Italy 1985.

18. A.Sirlin Phys. Rew. D22 (1980) 971.

19. A.Sirlin and W.J.Narciano Phys. Rew. 29D C1984) '.'45.

W.J.Narciano Phys. Rew. 20D (1979) 274.

M.Veltman Phys. Let. 91B (1980) 95.

F.Antonelli, M.Consoli, G.Corbo' Phys. Lett. 91B (1980) 90

M.Consoli, S.Lo Presti, L.Haiani CIFT PP 738.

20. W.J.Marciano BNL 36147 (1^85)

21. K.Winter CEHH-EP/84-137, 1984.

53

Table I - Electron identification criteria

a) Central detector

Physical quantity Desciip:ton CutsEfficiency(isolatedelectron)

Calorimeter energy Radius R», R.

Hadronic leakage

* 0-5 cells

Ho = 0.024 +0.034 In E j

Associated track Track crossing energycluster. •

At least one hit inchambers Ci or Cz.

Preshowor signal j Signal iroin chamber 'Cs! within distance d0 fromtrack intercept.

Associated charge

No additional clusterwithin distance dj ofselected cluster, withcharge larger than Qs

Require energy patternto agree with thatexpected from electronP(X2) > Pc

2 0

3 m . i . p .

60

0.96

0.90

D.9S

0.9C

0.95

Track-energy

cluster match

10" 0.92

Overall efficiency 0.72 ± 0.05

b) Forward detectors

Physical quantity Description Cuts Efficiency(isol/ted electrons)

Calorineter energy Cluster size 2 cellsEnergy fraction (chargedand neutral) in adjacentcells < f0

Energy leakageEleak / E

e m < H»

fn » 0.05

Ho = 0.02,*)

1.0

0.99

Associated track Forward Cr3ck crossingcluster cell. Track

I rainimur,) distance t from

j vertex in transverse

! projection less than to

i! Ai;.-?ocisted transverse

vertex r.r5ck within

lie; ' q>0| At least one hit in

chance.-s Cj or C2. Noideritii:ied co-versior.

tr. - 50

$ 0 = 30 Bir 0.9&

Signal in each MTPCplane within distance60 of track intercept :M < cx, |Ay! < 6y

Associated >!TPC charge

Q > Qc

30 ms

6y = 20 no

'it 6 m. i.p.

0.99

0.93

Preshower-energycluster

Distance of projectedMTPC position andcluster centroid asevaluaced from PM ratio|Axl < Ao

100 mm 0.98

Momentum Momentum p endcalorimeter energy Esatisfy

0.89**)

' Overall efficiency 0.75 • 0.05

*) A cut H • 0.03 i» applied if the enerjy is shared between

two adjacent cells.

**) This value takes into account both internal and external

breatstrahlung.

55

FIGURE CAPTIONS

1. Longitudinal cross section of the UA2 detector in a plane

containing the bean axis.

2. Schematic representation of particle signatures in a quadrant

of the 0A2 central detector.

3. The electron p-, spectrum of the 1984 data (/s = 630 GeV).

4. Electron p^ distribution of V candidates for the 1984 data

sample. The dashed line is the background estimation. The solid

line is tbs fit the distribution when electrons from all

processes are taken into account (see text).

5. t— distribution of electron candidates (histogram) and of jets

(solid line) with p > 0.2.

6. P— distribution of elecrrons from V •* ev in the entire UA2 data

sample (,£ = 452 nb" 1). The dsshed line is the background

estimate, while the solid one is the expectation from W -* ev.

7. The prediction of the theory for the forward-backward charge

asymmetry for V ->* v. ( P. Aurenche and J. Lindfors N.P. B185 )

The dashed regions show the angular coverage of the UA2 spectrome-ters.

8. Plot of (p"1 • q • ~ 'o-fl'i) '*" E'1 for the 3D W •* ev candidate

with p Te > 20 GeV/c detected in the forward regions. 6£ is the

laboratory angle of the electros with respect to the proton

direction.

9. Electron pair mass spectrum of the 1964 data : a) after

application of calorimeter cuts on both electron candidates, b)

after requiring that at least one of the two candidates be a

certified electron. (Preliminary)

10. The Z* mass peak of the entire UA2 data sample. ( Preliminary )

11. Mass values and errors of the individual Z° candidates. The

dashed vertical line correspond to the fitted M ,.(Preliminary)

W12. The transvers momentum distribution p 4 of the 126 W events.

The events with a jet or system of jets with the transverse

energy exceeding 5 GeV are shaded, the events A to G are cross

-batced. The inset shows the transverse momentum distribution of

the Z°.

13. Total production cross-sections of the IVB at the pp-collider

measured at two c m . energies; only the statistical errors are

shown.

2 •>•" • • •

14. Comparison between the low energy sin & measurements ( from:

K.J.Marciano, BNL 36147 1985,f - 1 assumed ) , and the UA1 and

UA2 results. The radiative correction is appliedto the data and

only the statistical errors are plotted.

15. M . versus M - K . The ellipses indicated by a) correspond toz Z* V1 sigma of statistic (solid) and statistic+systematic (dashed)

errors. The predictions of the standard model at the tree level(dashed line) and with the first order correction (solid line-)are also shown (b).

FORWARB CAl«MH£TER

Fig.

57

HADRONICCALORIMETER

ELECTROMAGNETICCALORMETERPRESHOWER

TRA

- CK

N6

BEAMPPE

rig.

58

35 15p? (GeV)

55

Fit. 5

59

25 -

3 "

w

5

UA2

W -

-f~>r

123

L

1913 •

~ev

events

1

f9S4

m25 35 45

pf (CeV)

Tit. 6

55

I1

Fig. 7

60

rip-

a)IM2 » K/«• 430 GcV

Z* SAMPLEC*ler»Mf«r cufl.

HI events

i—i n

alt n m Iq:S tvtirtt

b)

n IL« 5» 74 K 1W

•V (O»VI

F i g . 9

61

s -

10 -| UA2 1963 • 1984

% events

4 -

2 - -n,SO 58 M 74 n

Tit. 10

Fij.

62

UA2 1982*1983*1984135 events

-UA2 1982 4 1983 * 198416 events

I 5 0 data

1 theory

10 2PT

20

Events p!f>26GeV

8 dal-a

.Altarelli. Ellis.Greco, Martinelli

4.2 theory

FEC G

10 20 30 40 50pT corrected IGeV/c)

Tig. 12

60 70

100806040

20

10B64

=2 -

Altarelli et al.

CERN-TH 4015

>

0.3 0.5 10 3.0 5.0 10.0

VT(Tev)Fig. 13

30 50 100

63

10*

10-

3-101

2-10*

8-104

—•—

- —

1 1 1 1

«O liymmitry

^N »C«tt»r inf

UAKI3):

fnw * 80.9 * 1.5m , . t5.6I1.S

UA2(84):

"».« 92.5*13

i

.15 .20 .25 .30 .35

Fig. 14

Fig. 15

65

QCD IN U&2

Tbe UA2 Collaboration

Bern - CERN - Copenhagen (NBI) - Heidelberg -

Orsay (LAL) - Pavia - Perugia - Pisa - Saclay (CEN)

Presented by

JUESG SCHACHER

University of Bern

Bern, Switzerland

ABSTRACT

Data on the jet cross-sections at </s = 630 GeV are

compared with those obtained at /s = 546 GeV. The

agreement between the experimental results and the

QCD prediction is good. Results on a search for an

enhancement in the jet-jet invariant mass around

150 GeV are presented. A preliminary snalvsis on

nulti-jet events is also reported.

66

1. INTRODUCTION

The study of hadron-hadron interactions is pp collisions at very

high energy gave us the opportunity to test predictions of the strong

interaction theory, the Quantum-Chromo-Dynamics (QCD), at short

distances . In fact one of the most outstanding benefits of the very

successful operation of the CERN pp collider1' is the possibility for

detailed measurements of hadronic jet production and their fragmentation

properties in an energy domain where hard processes can be separated

clearly froa the soft hadronic interactions''7'.

During 1984 the SPS Collider operated at an increased energy of

•/s = 630 GeV where jet production cross sections are expected to be

larger. The results are reported in this article which is organized in

the following way: In section 1 we discuss the data taking and the data

reduction. Section 2 is devoted to the study of the jet production

cross-sections and their comparisons with Vs = 546 GeV data and QCD

predictions. In section 3 we present new data on a search for a

possible enhancement in the jet-jet invariant mass distribution around

is.. = 150 GeV while section 4 is devoted to a multi-jet study providing

the possibility of a measurement of the strong coupling constant a ,

followed by the conclusions.

2. DATA TAKING AMD REDUCTION

The UA2 detector has been described in detail elsewhere* . It

consists of a vertex detector for measuring particle trajectories in a

region without magnetic field, surrounded by a highly segmented

electromagnetic and hadronic calorimeter (central calorimeter) which

covers polar angles 40* 5 6 S 140* (-1 £ t £ 1J and the full a2imuthal

angle. The central caloriaeter consists of 240 cells each covering 15*

in i and 10' in 8 and built in a tower structure pointing to the centre

of tl.a interaction region. The cells are mechanically arranged in 24

azimuthal slices (called orange slices) each covering 15*. The forward

regions (20° £ fl S 37.5* and 142.5 £ 8 £ 1601) are each instrumented

67

with twelve toroidal Magnet sectors followed by drift chambers and

electromagnetic calorimeters.

The results presented hera were obtained with the central

caloriaeter only and were recorded during the 1984 CERN pp collider run

at /s = 630 GeV.

Three different triggers were used, all of thea selecting pp

collisions which deposited large transverse energy into the central

caloriaeter:

1. two-jet trigger: the total transverse energies in any four

consecutive azimuthal slices and eight azinuthal slices

opposite in $ had to exceed both a given threshold (typically

18 GeV),

2. single-jet trigger: the total transverse energy deposited in

any eight consecutive aziauthal slices had to ex-.eed a given

threshold (typically 30 GeV),

3. total trarisverse energy trigger: the total tranverse energy

collected is the central calorimeter was required to exceed a

threshold, normally set at about 60 GeV, except for special low

threshold data runs.

Background fros sources other than pp collisions was supprcs'.,•••* at

the trigger level requiring a coincidence of the trigger conditions

previously described with two signals ("aisiaus bias" trigger) obtained

froa scintillatoi arrays covering an angular range 0.44' £ S S 2.84' oa

both sides of the collision region*', the total integrated luminosity

accumulated was If * « 310 nb"1. An uncertainty of ± 6 X t*as estimated

for it dt froa tha observed fluctuations during different naming

conditions and froa tha overall uncertainty in the cross-section

accepted by tha "minimum-bias" trigger.

Ik» events collected include a saall background contamination.

Beaa halo particles cam either satisfy the triggering conditions

68

directly or appear as an accidental overlap with a "minimum-bias" pp

interaction. The background events exhibit a characteristic pattern in

the detector different fron a genuine event. They are rejected from the

data sample if any of the following conditions is satisfied:

1. if they are associated with an early signal in the smtll angle

scintillator arrays,

2. if they have an abnormally large total transverse energy

fraction in the hadronic compartments (more than 60% in the

second one or more than 95% in both together),

3. if they contain only one cluster (see below) with more than 90%

of its energy in the hadronic compartments.

These requirements reduce the background contamination in the data

sample to less than 2% and the loss of good events due to the cuts is

negligible (< 3%).

A straightforward clustering algorithm which takes advantage of the

fine granularity of the calorimeter segmentation has been adopted1* .

All cells which share a common side and have a cell energy

Kce,j > 400 MeV are joined into a cluster. Ihe jet direction is

measured from the centroid of the associated cluster with respect to the

centre of the interaction region. In each event the clusters are

ordered in decreasing transverse energies denoted by

E* T > E*T > E'T > ... The individual clusters are taken massless,

equating therefore p— = E_. Actually by attributing a zero-mass

•oaentua vector (p =•= E) to each cell which participates to a cluster,

one observes a mean "invariant cluster nass" of 9 GeV for a cluster

having tj. > 30 GeV. This value results froa the combined effects of

shower and cell sizes and of actual jet masses, and it is in agreement

with Honte Carlo a isolations").

69

3. INCmSIVE JET PRODUCTION CROSS-SECTIONS

We studied the inclusive jet production cross-sections:

(1) pp -» jet + X at n = 0

(2) Pp • jett + jet» + X within |n| < 0.85

using data from the single-jet and the two-jet triggers, respectively.

In order to partly account for final state gluon radisi.,on, the jet

momentur vector is modified by adding to it the momentum vectors of all

clusters having E— > 3 GeV and separated by an angle n> from it such that

cos u > 0.2 . The evaluation of the cross sections for reactions (1)

and (2) fro* these events requires the knowledge of the integrated

luminosity for the normalization and the determination of the detector

acceptance including the effects of the energy smearing on the steeply

felling fx_ spectra. A Monte Carlo simulation of the detector is used to

calculate the acceptance. The Monte Carlo events are processed through

the sane analysis chain as the data. The comparison of the transverse

momentum (p^) and the two-jet invariant mass (m..) distributions of the

reconstructed Monte Carlo events with those of the initial partons used

as input provides the acceptance functions t (p_) and e (m..) by which

the observed cross-sections must be divided to obtain the corrected jet

and two-jet cross-sections. The uncertainty due to the model dependence

of the acceptance calculation has been studied by using different

independent models*"*^, by changing the details of jet fragmentation and

by varying relevant analysis parameters, such as the accepted rapidity

range, the parameters of the clustering algorithm etc. The systematic

uncertainty on t is estimated to be ± 35$.

Further uncertainties have to be considered in addition to this

systeaatic error: The normalization is also affected by the ± 8%

uncertainty in the knowledge of the integrated luminosity and the

systematic error on the calorimeter energy calibration is contributing

another ± 30V The overall systematic uncertainty oil the cross-sections

for reactions (1) and (2) is estimated to be ± 45% after adding all the

contributions in quadrature.

70

The cross-sections for inclusive jet production d*o / dp— dn at

D = 0 both for •s = 546 GeV (1983 data) and /s = 630 GeV (1984 data)

are shown in fig. la as a function of the jet transverse momentum p_.

The cross-sections for two-jet production do / dm., with both jets in

the interval -0.85 < n < 0.85 are displayed in fig. lb both for

/s = 546 GeV and /s = 630 GeV. The quoted errors include the

statistical errors and the energy-dependent systematic uncertainty on

the acceptance functions.

The cross-sect ions at A = 630 GeV are systematically higher than

those at /s = 546 GeV. In order to examine in more detail the observed

increase of the two cross-sections, the ratios between the values at

/s = 630 GeV and at Y'S = 546 GeV have been evaluated. The results are

presented in figs. 2a and 2b : Only the statistical errors have been

taken into account, since the systematic contributions are expected to

cancel at first order. A p~. (m..) dependent increase of the jet cross

sections is observed.

The data from UA2 can be compared with those from the ISR11' in

terms of scaled invariant jet cross sections. The "naive" parton

model12^ predicts that, as a function of Jtj. = (2 pT/Vs), the invariant

jet cross-section is given by E do/dp3 = f(x-,) p^"*. Fig. 3a shows the

quantity p .* E do/dp* as a function of x^. both for ISR and UA2 data. As

expected from scale invariance violation, the UA2 data show that at the

same x~ the cross sections are lower for higher Js. A phenomenological

fit of the form E do/dp1 « fCXj) p T" where fUj.) = A (l-xT)m/'xT

3 was

performed. A good overall fit is obtained for all data (ISR

at /s = 45 GeV and A = 63 GeV and UA2 data at /s = 546 GeV and

/s = 630 GeV) with the parameters n = 4.74 + 0.06, m = 6.54 ± 0.15 and

A = (8.3 + 0.4) • 10"17 (fig. 3b). The collider data alone are best

described. in this context by n = 4.5 ± 0.3, m = 7.3 ± 0.2 and

A = ft-6 ± 0.2) • 10-".

The measurement of the inclusive jot production cross-sections,

reactions (1) and (2), can be compared directly to QCD predictions.

Several QCD calculations have been reported2*"1' . In the following the

data are compared with a QCD prediction performed using the

71

paraaetrizatioa of Eichten «t a l . 1 " for the structure functions with

Q* » p * and A * 200 MeV. The comparisons between data and QCS are

shown in fig. 1« «nd in fig. lb for <fm <* 546 GeV and •» = 630 GeV. The

agraeaent is rather good and this is also seen in fig. 2 vhere the

ratios between the two different energies are displayed.

It has been suggested that a possible substructure of quarks and

leptons would manifest itself as a new contact interaction visible at

large momentum transfers"). The cross-sections for reactions (1) and

(2) are expected to deviate at large Pj. or •,. fro* the QCD behaviour

depending on the energy scale lc which characterizes the strength of

this new interaction. A detemination of A£ from the data is in

principle possible, but the uncertainties related to the QCD

calculations and the systematic contribution; attached to the data limit

its accuracy. Comparisons between data and predictions lead to a lover

limit of Ac > 370 GeV at 95* C.L. (figs, la and lb).

4. SEARCH FOR STRUCTURE IN THE INVARIANT TWO-JET MASS DISTRIBUTION

Following the indication of a possible structure in the jet-jet

invariant mass around ISO GeV as seen in the 19S3 data, we repeated the

same analysis with higher statistics for the 1984 data. A sample of

clean events with two jets in the central calorimeter and not too much

energy in the forward-backward calorimeter were selected. For each

event, the two-jet invariant mass was computed increasing the energy of

the second jet in order to balance the transverse energy of the first

jet (FyH = 0). The • .. distribution for 1983 and 1984 runs are shown

in figs. 4 and 5, respectively. The best description of the data in

terms of a smooth distribution is obtained with the following

parametriaatioa: dN/dm.j * a e*"jj + I e*"jj. A simultaneous fit to

the data allowing also for the presence of a mass peak was performed.

The peak was parametrized by a gaussian with a width of 13 GeV, as

expected for the mass resolution in this region. The fit for the 1983

data resulted in an enhancement of n * 73-23 events at a mass

•jj m 147±3 GeV using only events with m,. > 80 SeV. The 1984 data

72

shew, on th* contrary, no significant structure around •,. « 150 GeV.

In fact, a fit for a fixed •., • 147 GaV fives a snail dip of

a « -47-so «v«nts. Also the combined distribution froa the 1983 and

1984 runa (fig. 6) shows no significant buan above a smooth background:

Around • ., = 147 GoV , only a "signal" of n = 26 ± 42 events survives.

S. MULTI-JET ANALYSIS

two-jet production is the doainant hadronic process at large

transverse energies at the CERN pj5 Collider'" . However the fine

granularity of the UA2 calorimeter allows for the observation of a clear

signal of events with three or Bore jets. In QCD the main multi-jet

events source is the strong radiative correction to the elastic

parton-parton scattering processes. The relative rate of three-jet to

two-Jet events is proportional to the strong interaction coupling

constant a . A comparison between the two and multi-jet cross-sections

gives therefore a determination of a . Such an analysis is in progress.

We report here only an account of the experimental method together with

some preliminary distributions.

Multi-jet events have been studied in a kinematical region where the

jets are well separated. This is also the region in which the

perturbative QCD calculations"^ give reliable predictions. The

"multi-jettiness" of an event is studied in terms of the variables

P_ou . This variable measures in fact the lack of planarity of a

multi-jet configuration when the collision plane is defined by the

proton line of flight (?) and the highest E^ cluster (p*i):

(pi * b)

73

Fig. 7 shows the distribution of P T°u t separately for two and

multi-jet events in the laboratory frame. The results of the

predictions of ref. 16 for three parton final states oBtained with

A = 200 KeV and normalized to the two-jet sample is displayed in fig. 6.

The systematic uncertainties (luminosity, structure functions,

fragmentation, multiple soft bremsstrahlung, energy scale) are found to

be small (<30%). Calculations with different values of A result in

distributions having the same shape but different absolute

normalizations.

6. CONCLUSIONS

During the 1984 run, the UA2 detector at the CERN pp collider took

data at Vs = 630 GeV. The data collected at this energy, together with

those collected at /s = 546 GeV by the same apparatus during previous

runs, were used to investigate the dependence of the inclusive jet

cross-section and of the two-jet production cross-section on the centre

of mass energy. The QCD predictions were found to agree very well with

the experimental results. A search for structure in the two-jet mass

distribution showed no evidence of a bump around 150 GeV. A preliminary

analysis on multi-jet events indicates a good sensitivity in the

determination of the strong coupling constant a . Work on this subject

is in progress.

REFERENCES

1. R.P. Feynmen, photon Hadron Interactions, Benjamin, New York,

1972;

S.M. Berman and M. Jacob, Phys. Rev. Lett. 25 (1970), 1683;

S.M. Berman, J.D. Bjorken and J.B. Kogut,

Phys. Rev. DA (1971), 3388.

2. The Staff of the CERN pp Project, Phys. Lett. 107B (1981), 231.

3. UA2 Collaboration, M. Banner et al., Phys. Lett. 118B (1982)

203.

4. UA2 Collaboration, P. Bagnaia et al., Z. Phys. C20 (1983) 117.

5. UA2 Collaboration, P. Bagnaia et al., Phys. Lett. 138B (1984)

430.


283.


291.

8. B. Mansoulie, The UA2 apparatus at the CERN pp Collider,

proceedings 3rd Moriond workshop on pp physics, editions

Frontieres, 1983, p. 609;

M. Dialinas et al., The vertex detector of the VA2 experiment,

LAL-RT/83-14, ORSAY, 1983;

C. Conta et al., Nucl. Instr. and Methods 224 (1984) 65;

K. Borer et al., Nucl. Instr. and Methods 227 (1984), 29;

A. Beer et al., Nucl. Instr. and Methods 224 (1984) 360.

9. M. Bozzo et al., Pbys. Lett. 147B (1984), 392.

10. ISAJET programme, written by F.Paige and S. Protopopescu,

BNL report 31987 (1981).

75

11. T. Akesson et al., Phys. Lett. 118B (1982), 1S5 and 193;

T. Akesson et al., Phys. Lett. 123B (19 i), 133.

12. J.D. Bjorken, Phys. Rev. D8 (1973), 4098.

13. R. Horgan and M. Jacob, Nucl. Phys. B179 (1981) 441.

14. B. Humpert, Z. Phys. C27 (1965), 257;

B. Humpert, Phys. Lett. HOB (1984), 105.

15. N.G. Antoniou et al., Hadron and jet production at Collider and

ISR, Me Gill Univ. preprint (1983) and Phys. Lett.l28B (1983)

257.

16. Z. Kunszt and E. Pietarinen, Phys. Lett. 132B (1983) 453.

17. E. Eicnten et al.. Rev. Hod. Phys. 56 (1984), 579.

18. E. Eichten et al., Phys. Rev. Lett. 50 (1983), 811.

FIGURE CAPTIONS

1. Inclusive jet production cross sections. The data points

correspond to the two collision energies /s = 630 GeV (full

circles) and Vs = 546 GeV (open circles). The additional

systematic uncertainty, common to both data sets, is ± 45%. QCD

calculations are shown for Vs = 546 GeV (dashed line) and for

A = 630 GeV with three different values for A£ (see text).

a) PP "* jet + anything as a function of the jet transverse

momentum p_.D) 5P * jet + Jet + anything with both jets in the

pseudo-rapidity range -0.85 < i) < 0.85 as a function of the

two-jet invariant mass m.

2. .Ratios of the inclusive jet production cross sections between

/s = 630 GeV and /s = 546 GeV. The systematic errors

approxiaately cancel in these ratios (see text). The expected

increase from QCD calculation is also shown, a) and b) as in

Fig. 1.

3. Scaled invariant jet cross sections as function of x_. = 2p_/Vs.

The UA2 data from the pp Collider, /s = 630 GeV (full circles)

and Vs •= 546 GeV (open circles), are compared with AFS

measurements1 from pp collisions at /s = 63 GeV (squares) and

/s = 45 GeV (triangles).

a) pij*-scaled invariant jet cross sections.

b) pij.n-scaled invariant jet cross sections using n = 4.74 from

the bast fit (see text).

4. Two-jet invariant nass distribution for 1983 data.

5. TWo-jet invariant mass distribution for 1984 data.

6. Combined two-j«t mats distribution for 1983 and 1984 data.

7. P— distribution for two-jet events (crosses) and multi-jet

•vents (opea dots).

77

8. Theoretical P~out distribution from ref. 16 using

A = 200 MeV for two (dashed line) and three-parton (full line)

final states compared with experimental data.

0 =lJ

79

UA2

j - a c D

o10

1 UA2

• - • ,

a)"1

<V=(â/dprdnlô (/s= 630 GeV)

d^/dpr^!,^ (Vs= 546 GeV)

l ?50 100

Pr (GeV)150

do/dm (/s= 630 GeV)da/dm (/s= 546 GeV) ~

- i 1 1 1 1 i ' _ i ' • • >

100 200m (GeV)

300

Fig. 2

,n-J7-

•

_

10-»«

10 " 2 '

10-30

K)" 3 '

-

•

•

-

•

a)

I

%

i

j 1

i it !

I

t 1

Scaled jet cross sections

UA2 Pp — jet.X• /s= 630 GeVo/sx 546 GeV !

AFS pp — j*t .Xa/s= 63 GeV

. * 4s= 45 GeV

ft

. • • • » :

• f f 1, f i i

0.1 0.2 0.3

2 p,//s

0.4 0.5 06

,o-»

10-"

1 *"

0.1 0.2

Scaled jet cross sections

UA2 pp — jeWX• /ss 630 GeVo/s= 546 GeV

AFS pp — j e t . Xo/s= 63 GeV• /s= 45 GeV

0.3 0.4 OS

Fig. 3

0.6

81

lf<

. • ' ' ' 1 ' •

•

•

u

' • 1 ' •

•

•

• •

' ' ' ! • '

*•

I1

• ' • ! ' - - • \ •

" " *

i .

1111! III. II ! . i . l |2tt 2S«

-

•

-

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I..'we

Fig. 4

3' "•

1£

•••

* '

V•••

-

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• I0.VI

Fig. 6

83

SEARCH FOR MONOJET AND MULTUET EVENTS WITHLARGE MISSING PT IN THE UA2 EXPERIMENT

The UA2 CollaborationBern - CERN - Copenhagen (NBI) - Heidelberg - Orsay (LAL)

Pavia - Perugia - Pisa - Saclay (CEN)

presented by Hans Hanni, Lab. for High Energy Physics,University of Berne, Switzerland

ABSTRACT

Using the full data sample (Ja"dt = 310 hb"1) collected during the 1984pp Collider run (Vs = 630 GeV), the UA2 Collaboration has carried outa search for monojets and monophotoiis, as well as for multijet eventswith large missing transverse momentum ( p j ) . No significant signalcould be isolated from background for either event type. Therelevance of this result with respect to SUSV models is discussed.

• The method has bean checked by measuring the process pp — W+X,W - ev through a study of events with an electromagnetic cluster andlarge | i T -

1. INTRODUCTION

In the 1983 CERN p"p Collider run (v's = 540 GeV) the UA2Collaboration has observed ' electrons produced in association withhard jets and large missing transverse momentum. From the samerunning period the UA1 Collaboration has reported "' events with largemissing transverse energy accompanied by a jet (monojet) or by aphoton type electromagnetic cluster (monophoton). A common feature ofboth event samples is the presence of a large missing transversemomentum C -|0 • The subject of the present talk is a search for eventsof the UA1 type (monojets and monophotons) * and for multijet eventswith large missing p-j- as suggested by recent SUSY model calculations3 " 5 1 , using the data collected with the UA2 detector during the 1984 j5pCollider run (Vs = 630 GeV).

Lacking the highly selective power of the requirement of an electronsignature makes such a search much more vulnerable to backgroundthan was the case in Ref. 1. In particular the accidental coincidence ofa pp collision with the interaction of a beam halo particle in the UA2calorimeter may simulate an event of the type we are searching for. Amajor part of the present study is devoted to rejecting suchbackground events.

In the UA2 detector, the lack of coverage at small angles withrespect to th« beam line is a trivial source of events of thn typeobserved by the UA1 Collaboration: Two-jet events, in which one of thejets is produced at less than 20° to the beam, appear as single-jetevents in the present UA2 detector, thus accompanied by a largemissing transverse momentum. This limitation precludes a significantsearch for events having less than =30 GeV/c missing transversemomentum. Above this value, configurations in which one of the twojets escapes the UA2 acceptance become unlikely. We shall thereforerestrict the present search to missing transverse momenta exceeding30 GeV/c.

In this talk we discard searches for events of the type [pp - W*X,W - tE, t —evb), or (pp — . . . - eTjets(s)*large Pj), which have been

85

reported on by B. Mansoulie 6 ' and H. Plothow-Besch ••, respectively,

elsewhere.

2. APPARATUS AND TRIGGERS

The UA2 detector has been described in detail elsewhere ' . We

briefly recall its main features.

Apart from two narrow cones along the beams, the detector providesfull azimuthal coverage in three distinct regions of polar angles:40° < 9 < 140c, the central region, and 20° < 6 < 40°. 140° < 6 < 160°.the forward regions.

In the centre of the detector a set of coaxial cylindrical dr i f t ardproportional chambers detect the cha:ged particle tracks produced inthe collision and measure the position of the event vertex.

An array of 480 calorimeter cells, each cell covering a similar domainof longitudinal phase space (15° of azimuth and =0.2 units of rapidity},measures the energy density in the final state. Each cell is segmentedlongitudinally, the inner compartment containing electromagneticshowers. While hadron showers are usually contained in the 4 5absorption lengths of the central calorimeter (CC) ', providing ameasurement of jet energies, they only deposit a fraction of theirenergy in the forward calorimeters (FC) which are =1.0 absorptionlength thick. These forward regions are equipped with magneticspectrometers which measure the momenta of charged jet fragments, theenergy of the rr°'s being measured in the calorimeter cells.

Two scintillator arrays (referred to as veto counters in thefollowing)*' located at a distance of 8.5 m from the center of UA2 onboth sides of the detector and covering a polar angular region of2.4° < 6 < 7.0°, are used to reject background events due to beam haloas described in the next section.

' borrowed from the UA5 experiment '

66

The data presented in this talk were recorded during the 1984

running period of the CERN pp Collider with two newly installed

triggers, the pij-trigger and the single jet trigger. The fSy-trigger

required that the modulus of the missing transverse momentum vector

(constructed by hardware from the transverse energies ' measured in

the calorimeter cells) exceeded 30 GeV. The single jet trigger required

that the scalar sum XE-r- of all the transverse energies measured in the

cells of any azimuthal wedge with A * = 120° of the central calorimeter

exceeded the same threshold of 30 GeV.

Two small angle scintillator arrays (covering, an angular range

0.44° '. 8 < 2.84°) on both sides of the collision region are used in

coincidence with both triggers to provide a "minimum bias" signal ' .

A sample of minimum bias" events was recorded simultaneously with the

date presented here to provide a measurement of the integrated

luminosity Jifdt which amounts to 310 nb ' 1 .

3. DATA REDUCTION

In UA2 there exist two major background sources for the type of

'ents looked for in the present analysis:ev

a) Two- or multi-jet events from genuine pp interactions, where > 1

let is at least partially lost in an insensitive region of the

oetector (see sect. 2). This background is referred to as "QCD

background" in the following.

b) Background from beam halo particles, which either satisfy the

triggering condition directly or appear as an accidental overlap

with a "minimum bias" pp interaction. This background is

referred to as ''beam halo background" in the following.

' The gains of the photomultipliers were adjusted such that their

response is proportional to the transverse energy E-j- = E-sin6

87

QCD background events can be recognised if they leave at leastsome trace of energy in the forward calorimeters opposite in azimuth tothe jec (system) seen in CC. Thus the forward regions in spite of theirlacking hadronic calorimetry can be used as a veto region against theQCD background.

Most of the beam halo background events are easily identified whenthey satisfy one of the following conditions:

i) if they are associated with an early signal in the smail angle

scintiliator arrays (minimum Dias counters),

ii) if the event has an abnormally large total transverse energyfraction in the hadronic compartments,

iii) if the event contains only one central calorimeter cluster withmore than 90°6 of its energy in the hadronic compartments.

Events satisfying any one of the above conditions, are rejected. Theloss of good events introduced by these cuts has been shown to be

121

negligible ' . However, while these background rejection criteria arewell suited to the study of two-jet events, they become insufficientwhen the data of interest are required to have a large missingtransverse momentum.

To further reduce the remaining beam halo background we takeadvantage of the fact that a large fraction of these events ischaracterised by an early timing in either the left or the right handside veto counter array, depending on the direction of f l ight of thehalo particles ( f ig . 1).According to that the sample of early timing events was used

1. to study the energy deposition pattern ot the beam halo back-ground in order to establish suitable cuts other than the timingcut itself (to reduce residual beam halo background due to theinefficiency of the veto counters),

88

2. to measure the efficiency of these cuts of rejecting beam halo

background.

The inefficiency on good events of these cuts was measured usingbalanced two-iet events as well as minimum bias events.

20-

20

Fig. 1: Scatter plot of event timing measured with the two vetocounter arrays on both sides of UA2. Abscissa: Timing of veto counteron incident proton side, ordinate: timing of veto counter on incidentanti-proton side. Only the events in tha upper right hand side cornerare genuine pp collisions.

89

4. SEARCH FOR MONOJETS

The search for monojets with the UA2 detector was motivated by theUA1 monojet and monophoton events reported in 1983 ~'.

4 . 1 . Monojet Event Selection

The data sample used for the monojet search was the one recordedby the p"-j- tr igger. For all these events the value of the p'-p wasre-determined by software (using the single cell energies), and a p£ycut was applied at 30 GeV, where the hardware pj tr igger had beenfully efficient. The remaining data sample was subject to the standardUA2 initial selection criteria described in sect. 3 to reject backgroundfrom non pp collisions.

The remaining data sample of 10249 events contains mainly eventsfrom the two background sources described in sect. 3, namely OCDevents with at least one jet in an insensitive region of the detector,and beam halo background (estimated to account for •- 40% of thepresent events with Py > 30 GeV'i.

At this stage all events are fully reconstructed, and the forward jetenergies are recalculated using also the momentum information of thecharged tracks.

In the following we describe the specific cuts applied to furtherreduce the QCD and beam halo background-.

a) Keep only events with a sufficiently centered vertex:

' zver tex ' * ^® m m ' ' T n e probability to loose jets is higher forlargely displaced vertices.)

b) At large distances from the beam the nalo induces showers in theouter calorimeter compartments, resulting in leading clustershaving a large fraction f^ of their energy in the hadroniccompartments. Therefore we require the leading jet ( jetD tohave fH < 90%.

90

c) In addition we require the leading jet to be well contained in CCby selecting only events where less than half of the energy ofjet l is found in edge cells: E e d g e c e | | s / E t o t < 50°c

d) At medium distances the beam halo induces long showersdeveloping in the central calorimeter parallel to the beam line,resulting in characteristic energy patterns. The calorimeter cellarray consists of 240 cells arranged in 10 "r ings" of cells havinga same 9 or 24 "slices of cells having a same <)>. The largestnumber M« of "r ings" hit within a wedge of two adjacent "slices"(Nln ^ 10) has large values for this configuration. We requireN R < 7 .

These cuts leave a sample of 3128 events. About 5% of them are dueto beam haio background, which can be further reduced by requiringgood timino:

(.) We .-eject events with an early timing in the veto counters byrequiring t e v e n t > -23 ns.

At this stage the beam halo background accounts for onfy about1-2%, and the bulk of the events are of the QCD type: 1 jet in CC,> 1 jet in an insensitive region of UA2. As mentioned in sect. 3 the lostjets usually leave some traces of energy in FC. We therefore impose acut on the scalar sum of the energies (JE-r°'">'>) measured in anazimuthai wedge with half opening angle Atp = 60° opposite to the(transverse) direction of t+ie leading jet (OCD events contain mainly 2jets ~' back-to-back in the transverse plane):

f) I E j O p p ( i n FC alone) < 3 GeVXE T

o p p ( i n F C , C c ) < 10 GeV.

The p'-j- spectrum of the remaining events is shown in f ig . 2 togetherwith a Monte Carlo prediction for OCD two-jet background (smooth line)obtained with the ISAJET •• programme. There is good agreementbetween our data and the QCD expectation, but a small non QCD typesignal can of course not been ruled out.

91

Fig. 2: ?-j--sp>ectrum of final monojet sample (known W — ev eventsexcluded). The smooth line is a Monte Carlo prediction by theISAJET ' programme for QCD two-jet background.

Above ^-j- = 65 GeV only one event is present: (t has a fT of100.2 GeV, and no track pointing to the energy cluster. Since it hasenergy in the two inner (i .e. in the electromagnetic and first hadronic)compartments of CC, but none in the outermost (second hadronic), itcan neither be interpreted in terms of isolated photons/TTr's nor ofordinary jets. We assume that the event is a beam halo backgroundsurviving our cuts. Keeping it, however, in our sample we quote anupper limit on the cross section

o = n/tJiPdt / (1)for monojet production in the region pt— > 65 GeV:

n(piT > 65 GeV) < 3.9 (901, c.I.) (2).

92

The total cut efficiency e of the monojet selection has been estimatedusing minimum bias and balanced two-jet events:

c(monojet cuts) = 0.54 (3).This gives for the integrated luminosity of 310 nb"1 of the 1984 run:

23 p b

We can evaluate this upper limit in a somewhat lower e>T~region:There are 40 (218) events with -p in excess of 5G (40) GeV. Usingthe region 40 < -j-< 50 GeV to normalise the Monte-Carlo calculation,the estimated background from two-jet events in which one of the jetsescapes the UA2 acceptance is 46 events for 0>T > 50 GeV. From thisresult the 90% c.l. upper limit for the production of events with a jetplus tT > 50 GeV is

> 50 GeV) < 73 pb (90% c.l.) ( 4 ) .

•4.2. Search For Monophotbns

As a simple extension of the monojet analysis, a search for monophotonshas been carried out using the final monojet sample of sect. 4.1. andlooking for events with isolated electromagnetic clusters. Removingfirst atl known W - ev events the following photon selection cuts wereapplied (for definitions see Ref. 14):

g) Require the energy cluster to show smaJI lateral extension:cluster radius < 0.5 cells, and to have small leakage into thehadronic compartments;

h) ask for no charged particle track coming from the vertex withina 20" cone around the line connecting the vertex with the CCenergy cluster center. Accept only events with at most onepreshower cluster with charge > 3 mip (minimum ionising particleequivalent) in the same 20° cone.

93

The sample of events with no preshower cluster present is called the

"unconverted sample", events with one preshower cluster belong to the

"converted" sample.

non converted

converted

40 50

Fig. 3: P-p-spectrum of themonophoton candidates. For

interpretation see text.

Fig. 3 shows the p-j--spectrumof the remaining convertedand unconverted monopnotoncandidates. A scanning byphysicists on a computerizedgraphics display (MEGATEK)has shown that out of thefive converted events fourare probably W — ev events,where the electron track hasnot been reconstructed due totracking inefficiency.

The conversion probability in the preshower detector is about 70°o. Itseems therefore very unlikely, that the unconverted events are due toisolated photons/n0 s. Their much more probable interpretation is interms of beam halo background.Estimating the overall efficiency i of the monophoton search to be

e(monophoton cuts) = 0.52 (5)and restricting ourselves to the region PyV > 45 GeV. where weobserve no event,

45 GeV, p'T > 30 GeV} (901 c.l ( 6 ) .

we quote the following upper limit for the monophoton cross section:

monophoton. > 4 5 G e V , p y > 3 0 G e V I < 1 4 p b ( 9 0 % c . l . ) 1 7 ) .

4.3. Search For The Proces W — ev Through The Analysis

Using the process pp — W*X, W - ev, a cross check can be establishedbetween the electron and the p1- analysis of UA2. To this end eventscontaining electromagnetic clusters and large &j have been selected byextending the present monojet analysis: Again, the final monojetcandidate sample of sect. 4 . 1 . has been taken and further cuts havebeen imposed to select electromagnetic clusters:

i) cluster radius < 0.5 cells (as in monophoton search);

j) hadronic leakage < 12OD (efficiency for 40 GeV electrons: 92°O).

UA2

, n I30 M) 50

p , IGeV)

Fig. 4: P^ spectrum of electromagneticmono-clusters found through the fijanalysis. The smooth curve represents theestimate for background trom two-jets.

Fig. 4 shows the fiyspectrum of theremaining events: Thereis a distinct Jacobianpeak at around 40 GeV,indicating as ev. ' tsou res the W — evdecay.

This event sample can be c npared to that one found through theelectron search: From the 66 events with pS-r > 30 GeV andP T

e m > 30 GeV, 48 events have been also found in the electron search,the number of non overlapping events thus being 18.The background estimate for the W — ev search through the pi-p analysisgives an expectation of 7.7 : 1.0 events from QCD (two-jet)background and 1.1 i 0.6 events from beam halo background.

In comparison a scan of the non-overlapping events on the MEGATEKgraphics display shows that about 9 events can be ascribed to the OCD(two-jet) and about 4 events to the beam halo background. This isroughly compatible with the above estimates. Two events are probablyW — ev candidates.

Taking on the other hand the event sample found in the electron searchand applying to it the same "f i l ter" cuts as in the monojet analysis,

Eedge < 5 0 °~ Ehad / E tot < 10%; * > / > 3 0 G e V ; ^T > 3 0 G e V 'leaves 50 W - ev events in CC, which are reduced to 42 by applvingthe specific monojet cuts of sect. 4.1 and 4.3. This is in agreementwith the efficiency of these cuts as estimated from minimum bias andbalanced two-jet events.

A comparison of the two values for the cross section,for the process(pp — W*X, W - ev) shows agreement within errors:

alpp - W*X, W - ev) = 590 t 90 pb from p T analysis (8a),a(pp - W*X, W - ev) = 540 t 70 pb from electron analysis (8b).

5. SEARCH FOR MULTIJET EVENTS WITH P-,

Multi-jet events with f5T are expected to provide a signature for2-51

SUSY particle production at the j5p Collider. Recent calculations J" JJindicate that with the present integrated luminosity it is alreadypossible to observe the production of gluinos tg) or squarks («f) withmasses around 50 GeV if they would exist. The dominant sources of 9or <; production m these Model* art Pp - 55 ' X, $$ • X or 3? »Xdepending on the masses. Each of the 5 or ef decays into ordinaryquarks lor gluons) plus a photino (7) * ' . The final state eventtopology consists therefore in general of two (or more) unbalancedhadronic jets and p T due to the undetected"'*.

The main physics background is expected to come from standardQCD production of i 3-jet events. They can fake such eventconfigurations if one of the jets is badly measured or escapes detection.

96

5.1 . Event Selection

The initial data sample for this analysis (72454 events) consists ofevents which have 2 or 3 jets with E-p > 15 GeV inside the fiducialregion of the central calorimeter (|n.l < 0.85). The jets are orderedaccording to decreasing Ey ( E j 1 > Ej* > ^T*^ ' anc* t h e 'ead-ing Je t l s

fulf i l l ing the single jet tr igger threshold: Bj1 > 30 GeV.

Multi-jet events with large fi— have been selected by the criteria&j > 35 GeV and 15° < « i S 020° where 012 is the azimuthal separationbetween the two largest jets. Ordinary QCD two-jet events have 3>ustrongly peaked towards 180° *•••' whereas events with <J>u < 15° arecontaminated by background from beam halo particles. A total of 174events satisfy this selection. They are submitted to the following cuts"which aim at rejecting event configurations where the large pj is dueto the QCD background from multi-jet events with badly measured jets:

i) ZE-p(CC) < 20 GeV, whe>-e IE-p is the transverse energysummed over ail cells of the CC not belonging to the jets;

ii) IE (FC) < 12 GeV, where 2!E 'S the energy summed overall electromagnetic cells of the FC;

i i i j <t(f£j-j,f£j.,-p) > 30°, where p-,., is the p-,-vector of the jetsystem and ^ y p r is the py-vector of ail ceils in the FC.

The last two cuts are illustrated in f igs. 5 and 6. The distributions ofthe present event sample are compared with the ones from well-balancedtwo-j«t event* (defined by p T J < 3 G«V, * l t > 170° and ji-j- < 10 GeV).in general a significant amount of energy is found >n the FC apposite to|£^j in' spite of the large overall 4-j-- This is a clear indication forQCD multijet background as mentioned already in sect. 3. Theadditional FC energy in SUSY events with genuine f*T is expected to besmall and not correlated to p^-j (as for the well-baianced two-jetsample).

97

• ) data samplt of thisanalyst*

40

tin

8 0 0 40

(FC)(G«V)80

Fig. 5: Energy deposition in FC of the events of this analysis (a)compared to that of well-balanced QCD two-jet events (b).

Fig. 6: Angular distribution in the transverse plane of the fT in FCwith respect to the direction of the leading jet system in CC for a) theevents of this analysis, b) well-balanced QCD two-jet events.

96

Residual beam halo background is rejected by requiring that no vetocounter signal be present outside the correct beam-beam interactionwindow.

5.2. Results

Two events survive the selection criteria. One contains an identifiedelectron, appearing here as the leading jet (E-j-1). The event is of thetopology pp - W*jet+X, W - ev and is included in the data samplediscussed in J. There remains only one multi-jet event with &j. Itcan most naturally be interpreted in terms of a QCD multi-jetbackground event because it has * (p^ . j ,^ -p C ) = 42° (which is near thecut) and it has also a central calorimeter cluster of Ey = 7 GeVopposite to p j . .

Nevertheless this event is retained in the evaluation of the upperlimit on the cross section for 2- or 3-jet events with &j > 35 GeV and15° < «is <120° The efficiency of the cuts i) to iii) and of the vetocounter requirement has been determined experimentally using thesample of well-batanced two-jet events mentioned above. The resultinglimit is

This cross section limit can be compared with predictions from SUSYcalculations. The event generator of ' has been used to simulate qifand gg* production at the J5p Collider. The final state jets have beensubmitted to the same very restrictive topological cuts as the data.Cross sections of about 10 pb are predicted for q or g masses in therange 40 to 60 GeV, whereas lower observable cross sections areexpected for smaller or larger masses. The sensitivity to lower massesis suppressed due to the experimental selection criteria. The presentanalysts does not exclude events with such low cross sections and istherefore not in contradiction to the prediction of this specific model.

99

6. SUMMARY AMP CONCLUSIONS

Using the full data sample (Ji?dt = 310 ntT1) collected during the1984 pp Collider run (Vs = 630 GeV), the UA2 Collaboration hassearched for monojet and monophoton type events as suggested by the1983 data of the UA.1 Collaboration '. and for multijet events withlarge missing transverse momentum (pV) as predicted by SUSY modelslarge missing transverse momentum ip-rJ as predicted Dy S U J I moaeis

' . No significant signal could be isolated from background foreither event type. Nevertheless we can quote the following upper limitsfor the production of such events in pp collisions at vs = 630 GeV:

a) monojets:a ( p i T > 6 5 G e V ) < 2 3 p b (90°o c . l . ) o r :aWT > 50 G e V ) < 7 3 p b (90S, c . l . ) ,

b) monophotons:aiPjV > 4 5 G e V , p 'y > 3 0 G e V ) < 14 p b (90% c . l . ) ,

c) multijets • large pS-y-:c(pST > 35 GeV) < 27 pb (90°Ojp.l.).

These results are not in contradiction with the SUSY models ofRefs. 3-5.

The experimental method has been checked by measuring theprocess pp - W+X, W - ev through a • study of events with anelectromagnetic cluster and large pV, and comparing them to the onesfound in the electron analysis '. Agreement within errors has beenfound for the two event samples as well as for the two measurements ofthe cross section:

a(pp - W»X, W - ev) = 590 ± 90 pis from * T analysis,o(pp - W*X, W - ev) = 540 ± 70 pb from electron analysis.

100

REFERENCES

[ 1] P. Bagnaia et a l . , UA2 Collaboration, Phys. Lett. 139B,105 (1984).

[ 2] G. Arnison et a l . , UA1 Collaboration, Phys. Lett. 139B,115 (1984).

[ 3] G. Kane and J.P. Leveille, Phys. Lett. 112B. 227 (1982);P. Harrison and C.H. Llewellyn-Smith, Nucl. Phys. B213,223 (1982); E B223, 542 (1983).

[ 4] J . Ellis and H. Kowalski, Phys. Lett. 142B, 441 (1984);J. Ellis and H. Kowalski, Nucl. Phys. B246 189 (1984).

[ 5] J . Ellis and H. Kowalski, CERN-TH. 4072/84 .[ 6] B. Mansoulie, Proc. of the 5th Topical Workshop on

Prcton-Antiproton Collider Physics, Saint-Vincent, Italy,1985 (to appear).

[ 7] H. Plothow-Besch, Proc. of the 5th Topical Workshop onProton-Antiproton Collider Physics, Saint-Vincent, Italy,1985 (to appear), CERN-EP/85-86.

[ 8] B. Mansoulie: The UA2 apparatus at the CERN pp Collider,Proc. of the 3rd Moriond Workshop on pp Physics (1983), 'p. 609: editions Frontieres 1983.

[ 9] A. Seer et a l . : The central calorimeter of the UA2 experimentat the CERN pp Collider, Nucl. Instrum. Methods 224,360 (1984)

[10] "The UA5 Streamer Chamber Experiment at the SPS pp Collider"The UA5 Collaboration, Phys. Scr. 23, 642 (1981).

[11] R. Battiston et a l . , UA4 Collaboration, Phys. Lett. 117B,126 (1982);G. Samguinetti, Proc. 3rd Moriond Workshop on pp Physics(Editions Frontieres, Dreux, 1983), p. 25.

[12] P. Ba^naia et a l . , UA2 Collaboration, Phys. Lett. 138B,430 (1984); Z. Phys. C20, 117 (1983).

[13] F.E. Piiga and S.D. Protopopescu, BNL report 31987 (1981).[14] L. Mapelli, Proc. of the 5th Topical Workshop on

Proton-Antiproton Collider Physics, Saint-Vincent, Italy,1985 (to appear), CERN-EP/85-84;1983 data: P. Bagnaia et a l . , UA2 Collaboration, Z. Phys.C24, 1 (1984).

101

DETERMINATION OF THE LEPTONIC NEUTRAL CURRENT COUPLINGS

H. Klein. S. Schlenstedt

Institut fur Hochenergiephysik der A * dor DDR, Berlin-Zeuthen , DDR

An analysis has been performed in ordtr to determine the vector end'axial-vector leptonic neutral current coupling constants based on the full setof low energy data ( n , •*•*, atonic parity violation. cO. uO) publishedprior to /I/ or presented »t the Bari conference /2/. Thus the numbers quotedhere represent an update of the analysis described in detail+in fit includingnow also Bhabha scattering and tots) cross-section data in e e~. Calculationsare performed in the on-mass shell renormalization scheme. Systematical andstatistical errors have been added in quadrature. Errors quoted for multidi-mensional fits define the one-standard deviation for a given parameter inde-pendently of the others. . - 2Fig.l displays the result of separate determinations of v»I-+I.+2 sin 6

and a=Ij-Ij for e. u,T using leptonic data only and all data. resp.

The inclusion of hadronic data is seen not to change the results apart fromv which is determined by the BCDMS u- asymmetry measurement. The numbers for

oil data are: ve=-.O47 + .035. v =-.147 + .162, ae«-.*86«.019, a =-.610*.046

and i f : -.468+ .059 at XVD0F=1.0. The x vector coupling is badly fixed by

the second-order tern in A,a only giving v t =2.8 » 1,1, Assuming leptonuniversality one finds (v,o) values as given in tab.l end 90% CL contours asshown in fig.Z. Similarly to these fits one finds j = ,87 + .08. sin Q

= .197 • .015, I* (e) ».O6 • .05. I1? (u>=.10 • .04 end I* (T) =-.03 + .06— 3 — 3 — » J —

based on a l l data considered. Assuming y *1 and sin 6 = '215 we get for ther .h . charges I , (e)*.001 • .017, I , (u) = .071 + .035. I , (x) = - 044 + .057

R " R~and I j « .012 + .014 for e=u=X . Setting I j = 0 we get j = 1.007 • .032.

.215 • .010 with a correlation of .22 and XVOOF«1.0. 90K CL . contours

are displayed in fig.3. For J *1 the data without e*e" yield sin2© = .211+.Oil.Rewriting |he e e" asymmetry formulae one find* sin © * ,iB6 + .021.

Taking sin © * .223 + .006 as the preliminary average from VN scattering /I/

trie mean sin 9 at low energies is .218 • .005 and the average including the

recent M^ measurements /4/ gives sin © = .220 + .004. These numbers areimproving our earlier evaluation based on data published until end of 1984which gave .213 + .008 and .216 t .006. respectively.

IM M. Klein. S. Schlenstedv, PHE 85-01. Z.f.Phys.C to appear/Z/ I. Ouertioth (PETRA). 0. Hedin (=734). Bari conference/3/ R. Pain (CHARM), A. Blondel (CDHS). Bari conf., P.G. Reutens

et al. (CCFRR), PL 152B (85) 404/*/ L.di Lelln, Bari conference

table 1all data

e e . e

e only

V

-.033+

-.023*

-.030+

.034

.037

.045

»-.509+

-.513*

-.499+

.012

.013

.026

corr .-.06

+ .04

-.03

T02

0

-.2

-.6

ve y

_sinV,215

o eV, ve

• all data

v

•1

-.1

Fig;i

ail data

90% CL _

Fig. 2

3

1.2

0.8

e*e",ve

all data

.2

90% CL -

.3

Fig.3

103

SPONTANEOUS PARITY BREAKING BY QUANTUM CORRECTIONS

M. Olechowslci and M. Spaliriski

Institute of Theoretical Physics, Warsaw University,

Warsaw, Poland

Attract

Spontaneous gauge symmetry breaking by quantum corrections

is considered in the context of the left-right symmetric mo-

del of the olectroweak interactions. We find that a realistic

model can be constructed only if the theory includes heavy

gauge singlet farmiona.

1, Introduction

The idea of symmetry being broken by quantum corrections

is a very appealing one for both aesthetic and practical rea-

sons- These latter include the hope that a hierarchy of gauge

symmetry breaking might arise if some of the symmetry breaking

is a perturbative higher order effect. In this note we take

up this question in the context of the left-right symmetric

model of the eloctroweak interactions [1]. This model is parti-

cularly attractive from this point of view because the required

hierarchy (measured by the ratio M,,, /M,,. ) may be relativelyR L

small [2j. Similar ideas have been recently considered by

Cvetic [3j; our conclusions however differ from his. Cvetic's

racial is reexarained in Section 2. We find that the conditions

for the minimum of the effective.potential found by Cvetic

cannot be satisfied. In Section 3 the role of the discreet

left-right symmetry ie considered. We show that the desired

minimum of the effective potential exists when L-R symmetry

is broken explicitly. In Section 4 we examine the idea that

heavy fermions (like the right-handed neutrino) could change

the negative conclusions of Section 2. We find it is possible

and give bounds on the heavy fermion mass. However the simp-

lest model of this kind does not agree with phenomenology.

We end by discussing a way out of this impasse which entails

the presence of heavy gauge singlet fermions in. the theory.

In trie course of this work we also considered a Gildener-

Weinbarg type of approach [4j. In this framework the effecti-

ve potential is. flat in one direction n(4>) . Radiative cor-

105

rections determine the magnitude of <•> and (,$$"} , where

<£$> i s not parallel to n<<!»> .

Formally j ^ = 0(«0 but in fact = 0 . '

The proof i s analogous to that of the. Georgi-Pais theorem [5].

In view of the above we concentrated on the approach of

Cvetic.

2. Cvetic's model

In this Section we analyse in detail the idea put forward

by Cvetic ™. Following him we consider the Higgs fields

necessary to break the gauge symmetry of the left-right sym-

metric model of the electroweak interactions but do not ex-

plicitly introduce the scalars needed to give masses to far-

mions..

Let ^L. and ^R b« scalar fields transfarming under

SUL(2) X SUgCZixUg^Cl) as (|.O.m) and (O.|,m) respectively.

A nonzero vacuum expectation value of < t ( <{>«) breaks SU, (2)

(;SUR(2)} gauge symmetry. The potential

for real vacuum expectation values < •u,n> =Vc,« h a* B" »P"

proximate O(2} symmetry in the space of «pt and ^ , j the

symmetry becoming exact for A2=0 . In that case the quan-

t i ty

106

(2.2)

is fixed at tree level, and there is a circle of minima in

the space of tp,., if* • This infinite degeneracy is removed

when X 2 * O . There are two possibilities:

for yz<o(2.Z)

or <^=0,'.fR*O for >2>0

We have four minima in both cases. For sufficiently small

X j however the potential is still nearly flat along the

circle and one may hope that the one loop corrections will

shift the vacuum expectation values'- •• , such that

(2.4)

We parametrise the vacuum .expectation values by the

"radius" if and "angle" oc :

âtp 'S in* , \pRs\fcoSK (2.5)

For phenomenological reasons we are interested in a solution

with 4 » l « c ) >O since in this case |(pjtl» 1 if>tl ie

M*y 5> M« . Due to the'U-R symmetry of the Lagrangian

we can confine our analysis "o angles satisfying the condi-

tion

107

(2.6)

The effective potential to one loop order is a function

of C S C O S K and if . Correction to tp is ssall and

not important to our problem. We look for a Cj such that

:e*-c**0 , r£r,Va1 >0 (2.7)

where V is the one loop effective potential. The tree

level contribution to the effective potential is given by

V=Vipîp5 + 0(2) -symmetric terms =

(2.8)

= A 2 u>*C2(l-C2) + terms independent of C2

The radiative correction £"V is a sum of contributions

from gauge boson, scalar, and fermion loops [7]

The fermion contribution is dropped because of the assumed

smallness of fermion masses; the scalar contribution is also

unimportant since it respect the 0(2) symmetry. Thus only

the gauge boson contribution is relevant for modifying the

tree level pattern of symmetry breaking. This correction is

given by [7} :

108

(2.10)

where A is the renormalisation point and

_4 cos-fw ,.-2 t, _ , / . 414-ZsWwl VCVR 1_ C2*11/

M w > M j and & are independent of C . We now cal-

culate the quantity (A*a A)

Substituting the gauge boson masses (Mj'fJj^M^

M 3 = M 4 = M W R . M5iC = M ^ ) we find

109

It is easy to calculate that for J> CZ7Q

while

0 te-:15)

Thus for the full one loop potential (the ferrcion contribu-

tion dropped) we have

which shows that the stationary point of V is a local

maximum, not a minimum. It is clear from eq, (2.15) that our

conclusion does not depend on the parameter "j . The magni-

tude of ^2. m aY influence the position of the extremum

but not its character. Thus we conclide that the conditions

written down by Cvetic have no solutions, and the only minima

110

are at c = 0, ^ or 1. This corresponds to three solutions:

unbroken SUL(2)

(2.18)

C2=i yt^y1 ifft'O : unbroken 5UR(2)

Neither of these satisfies the requirement (2. A) which ex-

presses the phenomenologically desired hierarchy of symmetry

breaking,'

In conclusion radiative corrections can modify the tree

level pattern of symmetry breaking. For example the discreet

left-right symmetry may be broken and simultaneously SU,(.2)

syrcnetry nay be restored. However the possible patterns of

symmetry breaking (eq. (2.18)) are the same as in the tree ap-

proximation f eq. (2.3)) , and only the critical value of Ag

is changed.

3. JiscrsDt left-rj-ht symmetry

The gauge syfinetry group in our model is SUL(2) X SUR (2)x

Xt'e , (l) . The scalar fields potential V depends in general

on nany constants \L , AJJ , A and fields $J , >£ ,

4> . Let us define a transformation T:

111

The potential V is a scalar undsr T . In the case of left-

right symmetry we have At = X R and T is equivalent to

$Z ** $k • lt corresponds to C*<"* (4-C*) in the nota-

tion of Section 2. Thus we have

ivhe re V S g«jj V and V

a -F (4 -C*) (3.3)

At the extremun of the potential

From eq.(3.3) it follows that Ffc1) changes its sign at

c = 7J. Thus V (Co) does not change its sign in the neigh-

bourhood of cQ = . In the model proposed by Cvetic F(c )= O

only for c2= and from eq.(3.4) it follows that v"(c2) is

always positive at the extremuro.

There isn't such a conclusion in the case of the left-

right symmetry been broken explicitly. In this case T trans-

112

P 2

formation is not equivalent to exchanging c <r+ fl-c }-

Thus we cannot write eqs (3.2) and (3.3). In general V ( c Q )

changes its sign at c = •= ; minima and maxima of V(c ) are

possible.

There is another problem connected with the left-right

symmetry. U'e will now consider a model with the gauge group

UL(l) * UR(l) for simplicity. The Higgs potential is

Denoting the real v.e.v. of fa_(4>p) by f^ffft) we have

the extremum conditions

(3.6)

A continuum manifold of vacuua (an elipse in the space- of

fu-^R ) appears for

7)

Thus to have a degeneracy (which would be removed by radiati-

ve corrections) we need two conditions on the parameters,

while only one can be naturally imposed by the choice of

renormalii-ation point'-4'8-'. If we had imposed the discreet

113

L-R symmetry, v/e would have only one condition to enforce,

which could presumably be done in a natural fashion by choo-

sing the renormalization scale appropriarely. There is also

another reason for having explicit L-R symmetry: we would

like it to be broken spontaneously rather then explicitly in

ah ad hoc manner. But having exact L-R symmetry we have also

the negative conclusions of Section 2, We propose a solution

of this problem in the following Section.

4. Heavy fermions

We consider now r/hat happens when there are heavy fer-

mions in the theory, that is fermions with masses of order-

Mv, or greater. In this case the fermion loop contribution to

the effective potential is not negligible and can in fact

alter the results in a qualitative way. The fernion couplings

to the scalar field tj> are of the form hvj/cjjij/ , so that •i

the masses of the fermions are given by

mF = h <cf>> (4.1)

so

The fermion loop contribution is given by [7]

1U

- T r

for Dirac fermions and a half of that for Majorana fermions.

Again we compute the expression 2V + (/l-2c*) V which can

then easily be used to check whether a given extremum is a

minimum or not. For one pair of heavy fermions we hav/e:

S =where S = —~~T^T" • ihis can also be written in the forra

Let us assume that there is an extremum at c »c so

This extremum is a minimum when

115

(4.6)

which gives a bound on the ferrnion mass. Thus in the model

under consideration it is always possible to have a minimum

at some c = c such that a hierarchy of symmetry breaking

is created by radiative corrections. *

It is known'- -I that heavy fermions may destabilize the

vacuum. iVe have to analyse our model from this point of view.

It is easy to check that V = V + 5 V G + £ V.. is not bounded

from below when eq. (4.6) is satisfied. However it is not a

serious problem; the effective potential contains the scalar

fieid contribution <5 Vg . This contribution does not de-

pend on c but it depends on the radius tp . The vacuum is

stable when for ip approaching infinity the sum oVg+0V5

dominates oVp which is easy to arrange.

The question now is whether our mechanism would work in

a realistic model of the electroweak interactions. In such e

model a heavy fermion naturally appears - the right-handed

neutrino. However this possibility is not easy to implement

for the following reason. In the model discussed above it

was crucial that the fermion coupled to the scalars via a

Yukawa coupling of the form iLdriL' . However such a coupling

is possible only if the scalar field tr&nsforms in such a

116

way that this term is a singlet under the gauge group. Thus

<j> has to transform as an odd-dimensional representation of

SU(2). For the right-handed Majorana neutrino $ is a trip-

let ( A. in the notation of Mahapatra and Senjanovic'- -y.

However experimental evidence implies that the SU.(2) brea-

king must proceed via a non-zero vacuum expectation value of

a scalar doublet [icQ . This problem can be resolved by in-

troducing a gauge singlet feririon 5 so that couplings like

if(f-s) or J(i}'£J?) are possible. Now <p is a doublet

and the model can be made to agree with phenomenology. The

inclusion of such gauge singlet fermions has been discussed

-for quite different reasons in ref.^ll"] .

V.'e would like to express our appreciation and gratitude

to prof. G. Pokorski for many stimulating discussions.

References

[1] J..C. Pati and A. Solan, Phys.Rev. D10 (1974) 275; Phys.

Rev.Lett. 31 (1973) 661,

i?.N. Mchapatra ar.da.C. Pati, Phys.Rev. Oil (1975) 566,

255S

G. Senjanovi-c and R.N. Mohapotra, Phys.Rev. D12 (1975)

1502

H. Fritzsch and P. Minkowski, Nucl.Phys. B103 (1975) 61

[2] M.A.B. Beg, R.V. Budny, R. Mohapatra and A. airlin,

Phys.Rev.Lett. 38 (l977) 1252,

G, Beall, M. Bunder and A. Soni, Phys.Rev.Lett. A3

(1982) 848

117

[3] M. Cvet ic , Nucl.Phys. B233 (1984 ) 387

[<l] E. Gildener and S. IVeinberg, Phys.fiev. D13 (1976 } 3333

[53 H. ueorgi and A. Pais, Phys.Rev. D1O (1974) 1246

[6] S. V.'einberg, Phys.Lett . 82B ( 1979 ) 387

[7]] S. Coleman and E. Weinberg, Phys.Rev. 07 (1973) 188B

[8 ] £. Gildener,.Phys.Rev. D13 (1976 ) 1025

C93 H.O. Politzer and S. ttot-fram, Phys.Let t . 82B (l979-),242

[ lOl 3.E. Kim, P. Uangacker, M. Levins and H.H. Williams.,

Rev.Mod.Phys. 53 ( l 9 8 l ) '211 .

£11] 0. V.'yler and L. Wol fens fain, NucX.Phys. B216 (1983) 205

•C.N. Leung, R.1V. Robinett and 3.L. Rosner, AIP Conf.

Proc. 99 (19b3 ) 202

C.N. Leung and?3.L. Sosner, Phys.Rev. 028 ^1983) 2205

(12"] R.N. Mohapatra and G. Senjanovic, Phys.Rev.. D23 (I98l)

165 .

119

AN EFFECTIVE LAGRANGIAN DESCRIBIHG MEW VECTOR

MESONS IN A STRONG INTERACTING BIGGS SECTOR *

D. Dominici

Department of Theoretical Physics

University of Geneva

1211 Geneva 4, Switzerland

AbSTBACT

— Using the hidden symmetry formulation of the non

linear <T-model, an effective Lagrangian describing a

new triplet of vector mesons arising from a strong

Higgs sector is presented. Some phenomenological conse-

quences are also discussed.

* Partially supported by the Swiss National Science

Foundation

On leave from "Dipartimento di Fisica, Univ. Firenze,

Italy"

120

1_ INTRODUCTION

Recently there has been a growing interest in investi-

gating what happens in the G-W-S model when the Bass of the

Higgs becomes large and close to the unitary limit (Mg-lTeV)

(1.2,3,4].

It is known from a long time [5,6] that when the mass

of the Higgs Mjj^lTeV partial wave unitarity is violated in

the tree scattering amplitudes of longitudinal vector bosons

W^,Z, and Higgs. Therefore you expect the sector of W^,Z^

and H to acquire the characteristics of a strongly coupled

theory.

This is clearly exemplified by the Cornwall, Levine,

Tiktopoulos theorem [7] which relates, at high energy

is»m^> the amplitude TtW^,WL,ZL,H) for scattering of longi-

tudinal bosons and Higgs to the analogous one T( w+, w~ ,'z, h)

for a scalar^ theory in the ' tHooft-Feynman gauge. Therefore

we can study the behaviour of the sector w£,wr,Z,,H at high

energy simply studying an 0(4) scalar theory. The self cou-

pling of the theory is \-{.G^/fi'iH^, exemplifying why the

theory is expected to become strong when MJJ •» oo .

A J-0 resonance (a scalar Higgs bound state) is

expected. Its width, as the perturbative Higgs mass grows,

becomes larger and larger and its mass saturates

[nj.0<800GeVJ [5,2]. J=l Higgs bound states are also

expected [3,4]. They can be phenooenologically relevant,

because if their mass is low, they can alter the high energy

behaviour of the G-W-S model.

The aim of this talk is to present the construction of

an effective Lagrangian describing these vector bound states

[3]. The aethod is based on the hidden symmetry formulation

of the non linear a-model [8,9].

121

This approach have been recently suggested to describe

p-vector mesons in QCV, together with vector meson dominance

and the KSFR relation (10j.

In our case we apply these ideas to the

SU(2)xSt/(2)/SlH2; nan linear e-model that we obtain from the

Higgs sector just taking, as suggested by Appelquist and

Bernard [11] the formal limit Mp -f co in the generating func-

tional. This non linear a-raodol can be reformulated in a

linear way by introducing vector mesons of the hidden local

symmetry SU(2)loc.

At the classical Jevel these vector bosons are nothing

but auxiliary fields. However in two dimensional a-modsi a

kinetic term is radiatively generated (3j. Here WE assume

that such vector fields become dynamical.

Their masses, mixing with W and Z, their coupling to

fermions and th« phenomenological consequences are dis-

cussed .

2^ STRONG INTERACTING HIGGS BOSONS

The Higgs sector of the G-W-S aodel can be reformulated

in tent of a 2x2 aatrix

in tens of which the Lagrangian becoaes (we neglect for the

•oaent SU(2)LxU(l)y interactions) :,

where f 24SGeV is the vacuum-expectation value of the scalar

122

field. The Lagrangian is invariant under the SU(2)^xSUf2)„

transformations M — V^MV^. The potential has a minimum

located at M*M=f 2 implying <0(M(x)|0>=f. When the gauged

linear c-model is considered, the Hlggs mechanism takes

place and the gauge bosons become massive.

A convenient way to analyze the regime where the mass

ecomes large is

generating functional

Mr, becomes large is to take the formal limit MJJ -* oo in the

In this limit we get a 6 ( M ' M- f *" ) and we pass from the

linear to the non-linear a-model

where now D is the SU( 2 ] r >:U ( 1 , v co variant derivative, and

U-V is a dimensionless field such that

In conclusion if the aass of the Higgs becomes large we

<et as .an effective Lafrangian a non linear fauged cr-nodel.

The HiJJs particle desappeared fro» the theory; however it

is not completely decoupled since the reaainios effective

theory is not renormalizable. Therefore when computing at

one-loop level new divergences appear which require new

counterterms and the mass of the Higgs H» plays the role of

the cut-off.

As it was proved by Veltmsn [5] and Appelquist and col-

laborators {llj the sensitivity of the theory to Mj, is only

logarithmic.

123

3_ HIDDEN SYMMETRY AND NON LINEAR a-MOPEL

The Lagrangian for the not) linear •& .u-model can be con-

structed firstly by considering the following transforma-

tions (see for instance Ref. [9])

(3.1)

(3.2)

where x -» g(x), g(x ) eG.g

Let us denote the set of local transformations (3.1) by

The Maurei—Cartan ^

(3.2) and transforns as

$y. The Maurer-Cartan vector field g^S g is invariant under

(3.3)

( 3' 4 }

under the local ft. After a decomposition of ui along the

unbroken subgroup H and the orthogonal part we get

(3.5)

where Ta6Liejj,X ijj.

Under the local transformation (3.1) we have

(3.6)

(3.7;

and the following, two invariant Lagrangians can be con-

sidered :

125

If we introduce a vector field Vu transfcming as

we can consider one sore invariant tera

(3.85

(3.9,

v 3 . 1 0 i

I 3 . 1 1 i

(3.12)

Since we are interested in an effective Lagrangian we

linit ourselves to terns up to two derivatives of the fields

and we consider the following expression :

126

|_- L «• <* L (3.13)

where a is an arbitrary parameter.

At the classical level V satisfies the equation of

notion V =AU and using it in (3.13) we get 1=L'*' which can

be proved to be equivalent to the kinetic term for the son

linear c-model.

In two-dimensional models [8j a kinetic term for V

radiatively generated and therefore the Lagrangian is

L - L l l > ->

Applications of these ideas to QCD by interpreting the

hidden vector mesons as p.vector mesons have been proposed

[10.;. Here we apply the rae'thod to the SU ( 2 ) xSU ( 2)/SU ( 2 ) non

3inear c model describing a strong interacting Higgs.

4 _ VECTOR MESONS AND THE SU(2) XSO(2)/SU(2) H££ LINEAR o-

MODEL

In our case G = SU(2) LxSU(2)R, H=SU(2)n, the group ele-

•ent is t=(L,R), LeSU(2)L, ReSU(2)R. The Maurer-Cartan fora

is u (g)dx", where

We decompose w as

127

; 4 . 1;

(4.2)

or equivalently _

(4.3)

where

• fc*

Let us introduce a gauge field of it, = s t JC2)' l Q C

(4.4)

\T . < IThe Lagrangian is constructed in tens of the two

invariants, given by Eqs (3.8) and (3.12)

128

, -f (4.5)

where

Note that if we introduce a field U=LH+ we can rewrite L' '

(4.7)

which is the standard kinetic term for the non linear

cr-nodel.

When SU(2)LxU(l)Y gauge interactions are turned on

(4.5) and (4.7) are replaced by

(4.8)

129

( 4 9 )

where

(4.10)

with Y =YuT3/2, w£°^=W^°^T/2, and the effective Lagrangian

is given by

where a is an arbitrary parameter.

Furthernore as anticipated in the previous section we

will assume that a kinetic ten for the vector mesons V is

radiatively generated, so that the final Lagrangian is

L. L" ,- LM •

W.Y.V

The lagrangian (4.11) describee the interactions of

vector bosons V^'.K^.Y and the scalar fields L,R. All the

130

gauge bosons, apart the photon, are expected to acquire

Basses by eating up the scalar fields. This can be expli-

citly seen by perforssing the transformation : .

(4.12)

We get

* \<lCw. +t.»ws (4. 13)

It is known, in fact, that a non linear gauged o-Bodel is

equivalent to a massive Yang-Mills theory.

After the rescaling »-><W, Y-»j*Y. 2V-> f"V we can g"et

the aass aatrix in the charged and in the neutral sector

(Table 1).

Note that in the liait $" •> oo V becoaes an auxiliary

field and we recover the aass foraula's of the G-W-S aodel.

5j. rERMIOW-VECTOB MESOW COUPLIMG AMD tOW-EHEBGY PHEHOMENOL-

OGY

Let us consider the feraions of the G-W-S aodel and

denote thea in a coapact way by *T and ^B.

131

If we assuae <p^, 4^ singlet under the hidden

2), t

with the W's :

, then the feraions couple to V^ via the aixing

(5.1)

froa which we get the couplings

6

(5.2)

where the parameters hw,hy,A,B,C,D. correspond to the parame-

ters of Table 2 when b=0.

However in correspondence of each doublet 4-L we can

define the new fernion XT=L*4'T which is a singlet under

L-^g^L, but which transforas as

when L->th. Therefore we can consider one lore invariant

ten :

132

r X t (5.4)

where b is an arbitrary parameter.

An analogous ter» can be considered starting froa the

right-handed doublets +R (of course only for quarks if there

is no right banded neutrino}. For simplicity we will assme

zero such a term. A liait on the bound of the coupling of

Xjj=R+'<'jj to V can be obtained fron KL-R£ mass difference.

Summing up we consider the coupling (5.1) and (5.4).

After the gauge _ transformation (4. 12),.a rescaling "*••* x+F*

and a redefinition iK •> RL <l>, tie get the couplings of Eqs

<5.2)-(5.3) and Table 2.

We can now construct an. effective charged and neutral

current hani 1 toman. In particular

where a,X,c can be expressed ia tern of A,B,C,D. In order

to understand possible low' energy effects we performed the

following analysis.

He-used the numerical values of Gj.,e, X=-.22 to fix

g,i',f- Then we analyzed how the standard >odel predictions

are altered for different value* of g" and a (equivalently

Mv) .

For instance using (g/js")Z=l/20 we present in Table 3

the prediction* for £M^/H^, J4M|/M|;P-1 and c for different

values of b and Mv. Here p is the neutral to charged current

ratio.

The conclusion is that the aodel predicts relatively

133

small deviations from the S.M. for relatively small Basses

of the new vector bosons V .

In particular the relation

is substituted by :

In Figure 1 the predictions of the node} for differentn

values of the ratio (g/g")~ are reported.

*L. ASYMMETRY IN e+e" •> M~ U' C£CS.S SECTION'

A quantity which is sensitive to the existence of new

neutral vector bosons is the forward-backward asymmetry

which is defined ss [12]

. 0

'\

w h e r e d/o* 6 i s t h e differential cross section for

e e •> n u , © is the center of aass scattering angle, ando

S=(p1+p2i", with Py and p, the momenta of the incoainx par-ticles .

In Figures 2 nna 3 two tsaa;..ies of the- asysmelry for

13 i.

Mv=250GeV are reported.

ACKNOWLEDGEMENTS

It is a great pleasure to -thank the organizers of the

Symposium, in particular Dr. Z. Ajduk and Professors G.

Bialkowski, S. Pokorski and W. Wroblewski for the enjoyable

meeting.

I wouid like to thank R. Casalbuoni, S. De Curtis and

R. Gatto for the stimulating collaboration on this subject.

REFERENCES

J'• R.N. Cahn, S. Dawson, Phys. Lett. 136B, 196 (19R4).

M.S. ChanowiU, M.K. Gaillard, Phys. Lett. 142B , 85

!1984,-. M-K. Gaillard, LBI.-1B590 {October 1984), UCB-

2) .•: B. Einnorn, NucJ . Phys. 246B. 75 (1984). R. Casal-

buoni, D. Dooinici and R. Gatto, Phys. Lett. 147B. 419

(1934). R. Poccei, MPI-PAE Pth 65-84.

3) R. Casalbuoni, S. De Curtis, D. Doainici and R. Gatto,

Phys. Lett. 155B. 95 (1985).

4i P.O. Hung and H.P. Thacker, Phys. Rev. P31. 2866

(19B5). P.O. Hung, Charlottesville preprint (1985).

M. Veitman, Acta Phys. Pol on. B8. 475 (1977).

135

6i B.K, Lee, C. Quigg and H.B. Thacker, Phys. Bev. D16.

1519 (1979).

7) Cornwall, Levine and T. Tiktopoulos, Phy*. Rev. DIP.

1145 (1974).

8) A. D'Adda, P. Di Vecchia and M. Luscher, Nucl. Phys.

B146. 73 (1978), B152. 125 (1979).

9) A.P. Balacbandran, A. Stern and G. Trahern, Phys. Rev.

D19. 2416 (1979).

10) M. Bando, T. Hugo, S. Uehara, K. Yamavaki and T. Vana-

gida, Phys. Rev. Lett. 54_,. 1215 (1985). M. Bando, T.

Kugo and K. Yamavaki, BPNO-45-84.

11) T. Appelquist and C. Bernard, Phys. Rev. D22. 200

(1980). A.C. Longfcitano, Phys. Rev. D22. 1166 (1SB0).

T. Appelquist y M.J. Bowick, E. Cohler and A.I. ilauser,

YTP-84-14 (tiovenber 1984).

12) E.R. De Groot, G.J. Gounaris and D. Schildknecht, Z.

Physik CjLi. 127 (1980).

136

Table I : The physical charged (W^V1) and neutral (A,Z°,V°)

vector boson states and their masses. Masses and mixing

angles depend on the gauge couplings g, g1 of

SU(2) xc|i) and the new "strong" coupling g", on f

which plays the role of the vacuum expectation value and

on the mixing parameter a.

Charged sector

= y ic - (c2-4ag2g"2) ]

c=g2(1+a)*g"2a

limits for large g" : M2 * -g2(1 - (S;) 2) , M 2*^-ag" 2, tp + -^

Neutral sector

5a 'r—

G (1+a'J-G" a1

Z°=A-cos5+A_sin£.sini<'-V,sin5cos^ t g \f =^ 3 g"G

a._ a(g2-g | 2) 2

where

4g-s"

M2Q = | - [ d - (d2-4a'G2G"2)1/2] d = G2(1+al)+G"2oc'

V? = Y ( d + (d2-4a'G2G"2)1/2]

l i m i t s fo.- l a rge g" : M2 ^^ (g 2 tg ' 2 ) [T- (% ~?' } ] ,Z° 4 2 2 2

^ ^ ( g t g ) [ T - % ? ]Z° 4 <g2*gl2)g"2

137

Table 2 : Couplings of the physical vector bosons W", V" and

A°» Z°, V° to the fermions. The couplings are written

in the forms of Eqs (12) and (13). For b = 0 they

coincide with those of Egs (14).

sinip + 2g"bcos(p)

e = *i?- cos iG

A * GcosC d + b l ' ^ i +b2l-(

B = -Sg-cos ?(1 - ptgi sin*)

l-

= -^g- sin S<1 + jtctg5 sin «)

Table 3

138

My « 250GeV

b » 0

.05 .06 -.003 .001

M = 250GeV

b - .1

.002 .001 .0004 .0003

8S0GeV .05 .06 -.0002 .0001

b = 0

My * 850GeV

b « .1

.0001 .0001 < 10~4 < 10~4

139

Figure captions

Figure 1 : tL, as a function of »z for the G-W-S-model {solid

line), for the model of this paper when g/g" =

= .16 (._), g/g" = .22 (...), g/g" = .258 ( ),

ti= 250 Gev.

Figure 2 : The e e • u y asymmetry for M^ = 250GeV t g/g"=.22,

b = 0 ((-.-) is the asymmetry for the GWS model).

Figure 3 : The e e • u V asymmetry for 11 = 250GeV, g/g"=.22,

b = .1 ((-.-) is the asymmetry for the GWS model) .

9C

I . . i . • • , I | . . • i . t • . • . i . . . I . i i . • . • . I

72 ~

70 ^ s ^ . 8 M . a e . 6 8 . s o . s a . 9 1 4 . a n . 9 8 . 1 0 0 . 1 0

Figure 1

1 0 0 . I S O . 2 0 0 . 2 5 0 . 3 C " d . 3 5 U . MJ?

Figure 2

142

1 . C '

vv L

r

- l i D O . - 5 0 . r OV 5 0 . i o o . 1 5 0 . azo. C . 3 0 0 . 3 5 0 . 14

v¥ (Gevj

J-

F i g u r e 3

143

THE PCI HP3I1E NEUTRINO EXPERIMENT

N. Bauraann1, Z. Greenwood, K. Gurr2, W. Kropp, M. Mandeikern L.

Price, P. Seines, and K. Sobel

(1) The Savannah River Plant, Aiken, S.C. , USA(2) The University of South Carolina, Aiken f CSA

Preliminary results are reported from the Irvine Neutrino

Oscillation Experiment. This is an electron-antineutrino

disappearance experiment at a reactor, and is sensitive to mass

squared differences in the range -.02 to ~6. eV

1. Introduction

This report gives preliminary results from the University ofCalifornia, Irvine (UCI! Neutrino Oscillations Experiment. Wehave constructed and installed a detector in our Savannah RiverPlant (SRP) Neutrino Laboratory. The detector is designed to bemovable, and can, with relative ease, be positioned at anydistance from the reactor core between about 18 and 50 meters.The neutrino flux at the closest position is approximately1013/cm2/sec.

a single detector, measuring the neutrino spectra from the samereactor, at various distances from the core, avoids many of theproblems associated with reactor and detector systematics whichwere possible in early experiments.

2. The Reactor as a Neutrino Source

Power reactors, used in most other (reactor) neutrino oscillationexperiments, produce neutrinos from the fission of significantfractions of each of the isotopes 2 3 5 0 , 2 3 8 U , and 2 3 9Pu. The

neutrino spectrum of each component is different. At the SRP

production reactor, 23SPu produces less than 8% of the fissions,

and 2 3 8U less than 4%. As a result, for energies between 2 and 8

MeV, the difference from a pure 2 3 5U neutrino spectrum is less

than 1.5%.

3. A Disappearance Experiment

The detector is designed to observe neutrinos vie the inverse

beta reaction:

v. + p •+ e + ns

Since the neutrino energy is less than 10 MeV, the reaction is

nonrelativistic, and the positron kinetic energy is:

E e + = S-^ - <Mn - Mp + ae) - E w - 1.8 MeV

A measurement of the positron spectrum is thus a measurement of

the neutrino spectrum.

If the vo changes to another neutrino type, the new neutrino will

be below its inverse beta threshold, and thus be unobservable. A

reactor experiment is, therefore, a disappearance experiment: a

deficiency in the v signal is expected if oscillations occur.

4. The Detector

A schematic of the detector is shown in figure 1. The target is

300 liters of NE313, a xylene based scintillator loaded with 0.5%

gadolinium. This scintillator provides pulse shape information,

distinguishing between lightly and heavily ionizing particles

(positrons and protons). The gadolinium provides fast, high

efficiency neutron detection. The target is surrounded by 1100

liters of a mineral oil based scintillator, divided into two

volumes. Each volume is viewed by a large number of

145

photomultiplier tubes (PMT'e). The detector is surrounded by two

inches of low background lead, a three incn thick plastic

scintillator anticoincidence, and finally, eight inches of

additional lead. Room is provided for additional neutron

shielding (B or Cd).

An electron antineutrino from the reactor can interact with a

target proton. The resultant positron stops and promptly

annihilates. Each 0.5 MeV annihilation gamma ray has a high

probability of depositing its energy in the liquid

scintillator. The neutron, with kinetic energy of tens of KeV,

quickly thermalizes and is captured with a characteristic time of

10 us, by a Gd nucleus. The resultant ~ 8 MeV of gamma rays are

detected with high efficiency. The signature cf an inverse beta

event is then Fig. 1.):

a. a pulse in the target with an electron pulse shape and

between 1.8 and 8 MeV, and,

b. A second pulse within ~ 15 us for which the sum of

target and blanket energies exceeds ~ 4 MeV.

With the efficiencies and thresholds discussed below, we expect

an event rate of ~ 500 per day at 18.3 meters. At 50 meters the

expected rate is ~ 70 per day.

5. Detector Characteristics

a. Energy Resolution:

We have measured energy resolution of the Target and Blanket

detectors using various sources. The results can be expressed as

follows:

"target " °'12

and

• 146

• 0 . 20 / E (MeV)

b. Detector Efficiencies:

Monte Carlo simulation of the detector properties have been made

and give the positron and neutron detection efficiencies.

Integrated over the expected energy spectrum, the positron

detection efficiency, with a threshold energy of 1.8 MeV is found

to be ~0.7. The delayed neutron efficiency with a gamma

threshold of 4 MeV is found to also be ~0.7. Experimental

measurements of these quantitites verify the results of the Monte

Carlo.

c. Neutron Capture Time:

The neutron capture time also enters into the overall detection

efficiency. This is measured with various neutron sources and is

found to be ~ 10 us, in agreement with the Monte Carlo, and with

direct calculations.

6. analysis methods

V?e plan to treat our data in two fashions:

a. The experimental results will be compared to combined

theoretical^- and experimental- determinations of the reactor

neutrino spectrum. (This technique suffers from the possibility

that any observed deficiencies can be due to inaccuracies in the

predicted spectra, or to systematic uncertainties in the detector

calibration. Conclusions are thus limited). This analysis will

not be further described in this preliminary report.

b. The spectra measured at different distances from the reactor

core will be compared, giving conclusions relatively free of

detector systematics, and in so far as the reactor fuel content

is (on average) constant, independent of the source, spectra.

7. Preliminary Results

We have taken data at two distances from the reactor core, 18.2

and 23.7 meters:

a. 18.2 Meter Data:

Approximately 40,000 neutrino events were collected at the first

position. Figure 2 shows the observed energy spectrum. The

corresponding positron spectrum is obtained using

EOBS = E e + + 0.75 MeV

(The terir. 0.75 MeV is the Monte Carlo determined fractional

positron annihilation gamma energy collected by the target

detector.)

b. 23.7 Meter Data:

Figure 3 shows the corresponding spectrum for the -10,000 events

collected thus far at the 23.7 meter position. We expect to take

~40,000 at this location, to have equal statistical significance

at both measured postions.

The integrated rates, greater than 1.8 MeV (observed energy),

normalized for power variations and solid angle difference gives

the ratio for position 1 and position 2 the value 1.038 +0.020.

A ratio of unity is expected in the absence of oscillations.

Figure 4 shows the ratio of the data' for the two positions,

normalized for power and solid angle, along with the null

hypothesis (ratio equal to unity) and the ratio value from the2 2

Bugey Experiment (Am =0.20, sin 26 - 0.24),

We test the null hypothesis with the observed ratios for each of

the n energy bins. The result is i » 39 for 29 degrees of

freedom. A similar test for the Bugey prediction gives *2 = 45.

Our preliminary analysis thus offers little support for the Bugey

result, and favors the absence of oscillations. A full 40,000

U8

event sample at position two should greatly improve the test.

The addition of data from other points will, likewise, produce a

more restrictive result.

The conclusions of our experiment can also be presented as an

exclusion region in the parameter space of delta m^ and

sin22e. The results of this analysis are shown in figure 5. The

region to the right of the OCI curve is excluded at the 95%

confidence level (2 sigma). With the larger event sample now

being collected, we can expect the exclusion region to grow to

include smaller values of sin 26.

Also shown in figure 5 are the regions of parameter space allowed

by the Bugey experiment, and the region excluded by the Goesgen

experiment*. Note that this latter group expresses their result

as a 90% confidence limit, i.e., an exclusion at 1.6 sigma, while

the Bugey allowed region has a 3 sigma significance. Neither CJCI

nor Goesgen has as yet ruled out the full range of allowed

solutions based on Bugey.

The Goesgen group has also done the absolute analysis (referred

to i. section 6a above), comparing their spectra with a

prediction derived from the theoretically and experimentally

determined neutrino spectra of the various fuel isotopes in their

reactor. The analysis rules out the allowed Bugey region. The

exclusion however, is again only a 1.6 sigm'a limit, and is

subject to the errors in the predicted spectra and to detector

and reactor systematics, especially normalization.

7. The Future

We expect to continue to operate our detector for several more

years, collecting samples of equal significance at three or more

positions. Selection of these distances is now under study using

Monte Carlo techniques.

A data sample at 50 meters would improve our limiting values of2 0

sin 20 and delta m* by factors of ~ 3 and ~ 2 respectively.

However, because of the reduced flux at such a great distance

from the reactor core, the data sample would require a counting

period of about two years.

This work is supported in part by the U.S. Department of Energy.

150

References:

1. Davis, B.R., et al, Phys Rev C, _19, 2259 (1979)

Vogel, P., et al, Phys Rev C, _2±, 1543 (1981)

2. Schreckenbach, K., et al. Physics Letters, 99B, 251, Feb

(1981) von Feilitzsch, P., et al, Physics Letters, 118B,

162, Dec (1982).

3. Cavaignac, J.F., et al, Physics Letters, 148B, 387, Nov

(1984).

4. Kwon, H., et al, Pbys Rev D, _24_, 1097, Sept (1981)

Vuilleumier, J. L. et al. Physics Letters, 114B, 29B, Nov

(1984) Gabathuler, K., et al. Physics Letters, 138B, 449,

April (1984).

151

Figure 1. Schematic of -he detector. The operation of the

detector is indicated by the superimposed sketch of a neutrino

interacting in the target.

Figure 2. Tne observed energy spectrum at the 18.2 meter

position.

Figure 3. A preliminary observed energy spectrum at the 23.7

meter position.

Figure 4. Plot of the ratio of the rates at the two positions,

versus energy. The rates are normalized for solid angle and

reactor power. Also shown are the expected ratios for the null

hypothesis (ratio eqaal unity) and the Bugey hypothesis.

Figure 5. Preliminary exclusion plot for tne UCI experiment.

Also shown are results from the Goesgen and Bugey experiments.

Regions to the right of the Goesgen and the UCI curves are

excluded at the confidence levels indicated. The Bugey curve

shows a region of parameters allowed by that experiment at 3

s i gma.

152

Neutrino Oscillation Detector

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Fig. 1

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OBSERVED ENERGY (MeV)10.0

30.0

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d

5

Lti

LU

20.0 -

10.0 -

00.0

(PRELIMINARY)

2.0 4.0 6.0 8.0

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156

CM>

CM

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157

APRON PRODUCTION AND SEARCH FOR NEW PARTICLES AT DESY

L. Criegee

DESY, Hamburg. FRG

Abst-c.-t Recent results from e+e~ experiments at DESY in the fields of T decays,inclusive particle production, fragmentation mechanisms, and search for New Particlesare reported.

1. T Decayt

1.1 T(25) -> x* T(1S)New measarements of the decays

T{25) - n r V T(15} (1)+ (2)

have bees performed by the Crystal Ball collaboration / I / . Isospin isvariance predictsa ratio of 0.S3 between the two channels. In theory the reaction is described as theemission of two ghioss which then fragment into pions (see /Z-4/). Because of the lowenergy the first step is treated by a mnltipole expansion of the colour field strengths,while the second one can be derived from PCAC and current algebra.

Out of 190000 T(2S) decays 90 events of (1) were selected, with the T(15) identifiedby its ft+fi~ or e+e~ decay. The extracted branching fraction is

BR ( T(2S) - jf°jr°T{iS) ) = (8.0 ± 1.5) % ,

consistent with an earlier measurement of CUSB / 5 / . The invariant r°r° mass dis-tribution shown in fig. I is distinctly different from phase space (dashed curve), andpeaks towards high mass values as expected from the theoretical descriptions. ( /2 / ./ 3 / , / 4 / all leading to the solid curve, if their'parameters are adjusted as shown inref. / I / . ) The data are consistent with an isotropic production of the ff°sr° system,and with its spin being sero.

The Crystal Ball collaboration has also analysed 169 events of the jr+r~ decay (2).The invariant mass and angular distributions are consistent with those observed inthe jr0*0 mode. Parameters for the TT+T~ invariant mass spectra are also consistentwith those obtained by the LENA, GLEO, CUSB aud ARGUS collaborations /5,G/.

156

The ratio of the branching fractions is [iro*o]/l*+x~j = 0.47 ± 0.11, consistent withthe value 0.53 expected lor an 1=0 assignment of the TJT system, and therefore withisospin invarianee of the reaction.

1.3 T(2S) -> P StatesThe Crystal Ball collaboration has also reported new measurements of the radiativedecays of the T(2S) to the P-states. Inclusive measurements /7 / yield the energiesand branching fractions of the three photon lines, and one combined line for theP -> T(15) cascades, all consistent with those of other experiments. A tentative spinassignment of the P-states can be made by noting that the branching fractions ofEl transitions are ~ (2J+1)- E*. The measurements are indeed consistent with theassignment of J=2,l,0, respectively, to the x" (HO keV), x" (130 keV) and x1 (160keV)-Iine, in analogy to the order of the P-states of charmoainm.

Direct spin assignments were derived from the analysis of the complete 77 decaycascades to the T ground state /8/ . The Crystal Hall collaboration extracted from200000 T(25) decays 66 events of the cascade T(2S) -• ixa - • 7iT(lS) ~+ 77^7",and 71 of the cascade through the xf state. (The x 7 nas too large a total width toshow a liveable radiative decay.) Various spin assignments were then compared usingthe logarithmic likelihood for N events:

A fwhere W ; is the correlated distribution (= probability density) in the angles of thefirst and second photon, and of the recoil T(1S) for the ntH event, as expected for spini. Fig. 2a shows the measured value ^Z^'n(Wo) = —0.17 for the J='O assignmentto the xa state. For comparison the figure also displays the (Gaussian-distributed)expectations for a series of Monte Carlo 'experiments' generated with J=0,l,2, alltested for consistency with J=0. It is evident that the probability for a spin zerostate to produce the observed value is very low (0.2%), meaning that J=0 caa beexduded with hijjh confidence level (99.8%). As shown in fig. 2b, the assignmentof spin terc to toe x" s""-«e sa 2 t i" 'es=I probable (10~'% C.L.). So neither stati: iscompatible v--ith J==0.

Aitboairh Sgs. 2<i, b indicate a preference for the assignments Jo=2, J«=l, theydo not exclude the opposite seo.uence. This can better be done by considering thelogarithmic likelihood for the combined assignment ( J a =i , J/j=2) and comparing its.gsin with a series of Monte Carlo experiments. As shown in fig. 3, the combination(0^=1, J»=2) is improbable, 0.6% confidence level, acd the assignments Jo=2, 3g=lare therefore strongly prefssTed.

It should be noted that the exclusion of the spin assignments Jo=0, JJJ=O doe* notdepend on any specific assumptions about the multipotarity oj the 7 radiation, whilethe assignments J<,=2, J^=l make use of the (natural) assumption of El transitions,and of both spins being different.

159

1.3 T(15) - r+r-Usin& the cascade decay T(2S) -* x+x-T[lS), the ARGUS collaboration / 9 / h*smade a aew measurement of the decay T(1S) -* r + r~ , sad obtained the branchingratio

BR(T(i.S) -* r+r~) = (3.07 ± 0.46 ± 0.22)% ,

consistent with lepton nuiversaEty.

2. Incimlve Particle Production

2.X K\J_The TASSO collaboration has reported new data oo Inclusive K° and A production/10/ . A comparison with earlier measuremeeeEf of *• and p production shows that thefragmentation features do not depend on strangeness: < pr >«-o «s < pj- >» and< Pr > A « < Pr >p' The observe;! rates are of coarse lower, and caa be explainedby a reduced probability of producing it p&irs from the sea, as compared toquarks

2«J / {«0 + iS ) = 0.S6± 0.02 ±0.05.

2.3 pc. K*The JADE coBtboration has reported new measarementa of the inclasrve productionof the vector mesons pz and K* jllj. Together with published dsta on the jr. K, D*and D production they allow an investigation of the pseudoscalar/vector productionratio ia the fragmeatatioa process. This ratio is predicted from spin soantiag to bei . but should be larger if the differences of the messes My, Mps are considered. TheJADE collaboratiou finds afi measurements consistent with

2 ^ 7, *"The JADE collaboration also presented new measuremects on inclusive 7 and x°production /12/ . The invariant ir° spectra agree with the r + sad t~ data over allthe accessible x-range. The observed photon spectrum agrees quantitatively with thatexpected from the dominating r° decay, initial state radiation., n decay, and probablya small contribution of final state radiation. To study this last contribution whichwould be dne to radiating quarks, the JADE collaboration selected a smaD eventsample with well isolated photons which have a low probability of originating- fromdecays (ftp. 4). After subtraction of the estimated decays and initial state radiationthey are left with 56 ± 22 events, consistent with the expected 41 i 5 from quarkbremsstr&hhing.

160

a.4 F, F*Tbe ARGUS collaboration has analysed new data oa F meson production identifiedby the decays into 4>x and <*SJT / I S / , and obtained a more precise F mass value:

mr = 1973.6 ± 2.6 ± 3.0 Af eV .

Tbe fragmentation function for inclusive F production was found to be 'medium soft1:

The ARGUS collaboration also found clear evidence for the vector meson F* in thedecay mode r - • F i •* d>m as shown in fig. 5 /14 / . They report the massmr- = 2109 ± 9 ± 7MeV, and the mass difference mr * —mr — 144± S>i 7 MeV,in agreement with preliminary values from the TPC collaboration /IB/.

i& 5" (1530), n -The ARGUS collaboration used the particle identification potential of their detectorand tbe large accumulated luminosity in the T range for sa extensive study at inclusivehypwon production. The identiScation of the heavier baryons is based on £6000reconstructed A's, o factor of 10 more than previously available in e+e~ experiments.They led to (690 ± 47) 3~ -» Aff~ and, for the first time, a signal of decoupletbaryons, namely (132 ± 26) 3°(1530) — = ~ r + (fig. 6 ) and an indication of (42 ±14) fl~ —»AK~ decays (ail preliminary, analysis in progress).

2.6 AntideuterontBased on dE/dx measurements the ARGuS collaboration found six oncidetiterons andfurther verified them either by a mass determination via TOF, or by their annihilationpattern /16/ . Fig. 7 shows that the invariant cross section is at least two orders ofmagnitude lower than that extrapolated from pions, fcaons or antiprotons. The ratio{AV/A>-) / (Aj/AV-) s is similar to the oae observed in hadron-hadron collisions,suggesting in both cases the formation of the S from p and n which were producedclose in phase space.

S. Fragmentation McchanEgma

S.I Fragmentation aad Branching Model*The three-jet events Srst observed at PETRA aro a convincing indication of an under-lying hard pan on topolog)', like a qusrk-antiquark-giuon system. For a quantitativeanalysis of this underlying- system, and a precise determination of the quark-giuoncoupling constant a, it becomes increasingly important to understand the transitioncf such parton systems into observable hadron topologies.

In addition to the weU-knowi. phenomcnological fragmentation models, like the

I.F. independent fragmentation models /17 / in which each hard parton fragmentsindependently into hadrons, and the

161

LUND string fragmentation model / IS/ ia which the colour strings stretched be-tween the quarks aad the fiaou fragment in th'ir CM system into hadrons,leading to similar observable 3-jet structures.

PAH.TON-BR.ANCHING models have bees developed in which the partous degradeby successive jrfuon emission (which is treated in the le&dinfj-Jog approximation of per-tcrbative QCD). Hadron formation occurs at the end through Shs decsy of iow-mas3(col;>itt~neutrai) parton clusters. Several Monte Carlo modeis have been developed/19/. of which the one far Gottschafc can be readily compared with experiments. Asnoticed by several authors /20/. however, the 3ofi-g!uon ?adiatioa into large sagleshas to be treated coherently, and largely cancels by intsrfereace . This has been siin-aJssed in Webber's Moste Carlo jiij h? cot aUoviaj any giaon omission &ng-!e to befexger than the preceding oce. As shows below, this ordering cf the emissioc sagtesSeads to distinct effects on the hsdrea level.

It is wei! known that tht I.F. aad the LUND model dejeribe rnost inclusive dbrr:-butioos of charged particle* equally weE, provided that suitable (me&siag 10 - 50%different) values of a, are osed. Figs. 8g and 8h demonstrate this for two typi-cal distributions measured wiib high statistics by the TASSO coiiaboration /22/. Adistinction between the models has so far only been achieved through the study ofspecial distributions within 3-jet eveat3, aad the use of charged plus centra! particlesfor improved accuracy.

S.3 Energy Flow Between Jet«The JADE collaboration has stndied the energy and particle flow in selected planarevents. Fijr. 9 shows the 'antenna pattens' of the energy and particle Sov projectedinto the event plane. The events are superimposed such that the most anergretic jetpoints to iero degTees, and the 2ni ead &Tt energetic one foiiow towerds positive an-gles. The remarkable featare is s strong dip between the sc-de3ned first and secondjet. An earlier JADE paper /2S/ had demonstrated that the I.P. models fail to repro-duce this dip, while the LUND model explained ii qualitatively through the absenceof a colour string between the two quirk jets, and also quantitatively. Accordta?to a recent investigation at the JADE collaboratioa /24/ also the Gottschalk model(dotted carve) does sot reproduce the observed depletion, while the Webber modelpredicts it very weB. As s measure of the particle depletion the ratio between thelevels of tbe third (3-1) and the first dip (1-2) is shows is 8g. 10. This depletionratio is seen to be ~ 7/5 for unseiected particles, and even higher for selected particlemasses or momenta p°*! (out of the event plane]. The figure demonstrates that theGottschalk and two typical I.E. models yield hardly any depletion effect at all, whileboth the LUND string model and the Webber branching Monte Carlo predict theobservations very well.

S.S Particle Momenta latide Jet»The dependence of the depletion on the particle masses can he explained, in tbecontext of the LUND mode!, ss aa effect of the Loreatt transformation of the itadronafrom the strings' CM to tbe LAB system. This transformation also affects hadronsinside the jets such that the me&n angles of the relativistk hadrons follow the origins!

162

partoa directions, while those of the slow ones deviate.

The PLUTO collaboration has studied this effect usiag the so-csiled weighted jetsaojnest*. ia which the fcadron momenta # are weighted with 6 factor Iftj""1 beforethe sttinm&tioa over the jet. Fig. 11 shows as RE example the lateral shift Apr of theweighted momentum of jet no. 2 sa a functioa of the power n, with the nubias'-^ jotmomentum (n=ii as reference /2b/. The shift is seen to be distinctly different fromi.sro for larger Hi and well reproduced by the LUND and Webber inodei, while theGottschalk aad particularly the I.F. models fail.

Tbe JADE collaboration has instead studied the mean lateral shift oi tbe hadroBmomenta as a function of the longitudinal momentum, with the jet axis as reference/2</. The observed shifts shown in flp. 22 are significantly different from £ero, and

weli described by tb.1 i'.UND aad Webber models, while the others fail.

The effects which have been established here mean that the jet sad the average hadrondirections differ in genera: from those of the origiaal fast psrtoua, and that one hasto be careful ia deducing the partoas' kinematics from those of the jets, in particularin the case of the steep angular distributions typical of gluon radiation.

3.4 AteewmentSince the Webber mode! is successful in describing these aad also many other hadrondistributions /25/, and 8ince it contains fewer atlitrary parameters than the squallysuccessful LUND model, it seems to be a highly favoured too! for analysing hardpartca reactions.

Unfortunately, however, this model has one particular deficiency which ma&es it un-suitable for this kind of analysis: The basic matrix qqg matrix element is truncatedto allow the LLA summation, and underestimates the (fe-,v) hard giuoss emitted atlarge iugies which determine the number of distinct 3-jet events. As an example, theasymmetry of tbe energy-energy correlation, which for several reasons is consideredone of the least model-depcndect quantities for determining' a, /26/, is reproduceda factor of two lower than expected /24 - 28/. So, until this deficiency can be cured,the more phenomenological LUND model will remain the best choice for correlatingthe jet aad parton kinematics.

4. Search for New Partlclei

4.1 SUSY Particle*The status of the search for SUSY particles was reported by S. Komamiya atWarsaw Symposium, and has sot changed significantly since.

4.2 ^fractional ChargesThe ARGUS collaboration has looked at the specific ionisatioa dE/dx of ~ S-10' neg-ative charged particles produced in > 2-prong events around 10 GeV CM energy (pos-itive tracks are contaminated by p and x + from beam-gas and beam-wall interactions,and therefore less useful) /2S/. Two search regions were defined, as indicated in %.

163

13: One for relativist* fractionally charged particles, 0.4 < {dE/dx}/{dEfdx)mHt <0.6, and one for bearily ionising object*.

•In the first region, no event was observed. The Kcond region contained 35 candi-dates, of which t were identified af i {reported above), 18 as overlapping ee and 1 asoverlapping rir, leaving no exotic objects.Fig. 14 shows t ie resulting limit* on the production cross section at Q » £, Jand J objects expressed a* the ratio R, to the prodsetkm of p+n~. The dashed linesassume a phase-space like production speetmn ~ ir*(E, the solid ones the exponentialdistribution normally observed in jet production by e+e~.

4 J Tb» f (8.8)The Crystal Ballcollaboration bad obserred in 100000 T(1S) decays, collected in 1983,a statistically significant resoaaace aroosd 1070 MeV photos energy, correspondingto a transition to a narrow state at 8.3 GeV fi&j. Daring 1984 nev data with twicethe statistics were collected /30 / , yielding (after the sane cuts) the photon spectrumshown in fig. IS. In contrast to the previous enhancement of 87 ± 21 events, a slightdip is observed at the same energy, corresponding to — 20 ± 20 events. A carefulcomparison of the calibrations and other features showed ao systematic differenceswhich could explain the appearance and disappearance of the effect. Only a sEghtljrhigher cross section was noticed in the 1984 data, indicating a mecbise energy shift o»~ 4 MeV towards the center of the T resonac.ce. Upon this Tye and Rosenleid / S i /conjectured the existence of a second resonance dose to the T, which did sot producethis photon transition, while the T did. The Crystal Ball collaboration considers thisexplanation to be unlikely, and rather attributes the appearance of the photon Useto a statistical ftnctnation. Some erratic behavioor of the enhancement a* a functionof the cats applied supports this view.

The 1984 data result it as upper Emit

BR (T - T € ( 8 . 3 ) ) < 0.08% at 90% CJL..

This a lower than other Ikaits /C/ given on the product

BR (T -» 7*(8.3)) • BR{(~ r + f ) < 0.08% at 80S C.L..

One should note that the predicted braftcbisg ratio for the T decay into a siandwdHiggs boson of 8.J GeV is of order 0.01%. So the givea fimit» do not exclude theexistence of a Higgs as such in this mans range. It appears however qefle unlikelythat, if it exists, it can be discovered with the described experimental technique

4.4 Search for MocoJeUEvents with only one energetic jet have been reported by the UA1 collaboration at theCBRN pp collider /32 / . A possible explanation was given by Glasbov and Manohar/ & / who suggested an extension of the standard model to include four Higgs bosonsof which the first, ha

}, is very light and loap-Eved, so that it escapes detection, thesecond, />§, has a few OeV/c*, while the other ones are very heavy. The monojets

164

could them have been produced by the decay Z° ~* Aj Aj, with the subsequentdecays (1) *S - / / o r (n) AJ -* ft? / / , with / being a fermioa, in particular ther, or a qnark fragmenting into hadtoss.

Since virtual Z° bosons are also produced in e+e~ annihilations, the same kind ofmonojets should appear at PETRA, with a predictable rate depending on the mass ofthe AJ, and the //-branching fraction r = T (AJ -*ff)f{T (Ag -* ff) + T (Ag -/ / AJ)) whkt for a set of given selection cuts affects the detection efficiency.

The JADE collaboration has searched for monojet erentf characterised by a highmissing transverse momentum pr,min< *»d a low energy Euek in the hemisphereopposite to the most energetie jet /34/. Fig. 18a show* the distribution of observedmuKiproag (> 4} events in these variables, and fig. 16b a Monte Carlo simulationfor ao assumed /•£ A* production with masses 0.2 and 8 GeV/c*. No real event wasobserved in the region Pr.mut > 1 GeV, Eatct > 1 GeV, whereas the efficiency fordetecting the conjectured decay Aj -» / / fa seen to be very high. Similar searcheswere done for different assumed A§ masses. For the case of a very light rKass alsomonojets with 2 charged (+ neutral) particles were investigated, and for the case ofa heavy hj the appearance of one separated jet each from the / and / , resulting in atopology of two aeoplanar jets. No single real event was found within suitable cuts,leading to the Bmits

1 OeV/t? < m(Ag) < 21 <?eV/e* (9S% C.L.).

Tkis excludes the whole mass range whkb » compatible with the monojets reportedby UA1 /32/. If the branchicj fraction BR (Z* — k\ Ag) is assumed to be differentfrom the one gives by the Gl&show-Manohar model, the excluded mass range changesaccordingly / /

The same search also excludes supersyizunetric higginos Xj, x§ with the decay xS —If Xt (Xi Kfht and stable} in the typical mess range

2GeV/c» < m(xS) < 28 GeV/c* ,

H the ratio of the 7? branching fractions is f = T(Z° -* x?X») / T (Z° -• uM v^)is aroand O.B.

In summary, the experiments at DESY haw produced new results on:

• the decay T (25) -» *a*° T (IS), demonstrating thai it conserves isospin,and that mass and angular distributions agree with QCD predictions,

• the decays T (25) - • -JX* -» TT T {IS), giving aa unambiguous spinassignment to the P-states,

• the stnage/noustrange and vector/pseudoscalar ratios in the fragmentationprocess, and inclusive t° and photon production with indications of finalstate (= quark) bremsstraklung,

165

t the detection and mas* of the F* meson,

• production of antldenterons and (preEminary) of the decoaplet baryoatE°(1S3O) and £T in e+e",

• kinematic effects in the qjp fragmentation io accordance with the LUNDand Webber models,

• better limits for the production of fractionally charged particles,

• limits excluding the production of the £(8.3) in radiative T decays, and

• limits on monojet masses, excluding their production according tothe GUshow-Manohar model.

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/IS/ TPC Collaboration, W. Roftna&n, LBL-17845 (1984) (uopnbSshed)

fUf ARGUS Collaboration, H. Albrecht et al., DESY 85-634 (1985)

/17/ P. Hoyer ti al., Nuel. Pbys. B1S1 (1979) S49A. Ali et al., Phys. Lett. 938 (1980) 155

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/IS/ B. Aaderwon et al., Phys. Reports 97 (1083) 33T. Sjortraud, Comp. Phys. Comm. 27 (1082) 243, 281 (19S3) 220

/IB/ C.G. Foe and S. Wolfram, Nuel. Phys. BIBS (1980) 285R.D. Held and S. Wolfram, Nucl. Phys. B213 (1983) 66T.D. Gottsehaik, Nucl. Phys. B214 (1083) 201

/20/ AM. Mueller, Nucl. Phys. B213 (1983) 85; B228 (1983) 357G. Marchesini and Bit. Webbers. Nucl. Phys. B238 (1084) 1

/21/ B £ . Webber, Nacl. Phys. B238 (1984) 492

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/24/ JADE Collaboration, W. Bartel et al., DE5Y 85-038 (1985)

/25/ PLUTO Collaboration, H. Maxeiaer, Univ. Wnppertal thesis,internal report DESY.PLUTO 85-02 (1985) (to be published)

/26/ A. Ali and F. Barreiro, Pbys. Lett. 118B (1982) 155; Nucl. Phys. B236(1984) 360MARK 3 Collaboration, B. Adcva et al, Phys. Rev. Lett. 54 (1985) 1750

/27/ PLUTO Collaboration, Cb. Berjer et al., DESY 85-039 (1085), Z. Phys.C2S (in press)

/28/ ARGUS Collaboration, H. Albrecht et al., DESY 86-037 (1985)

/29/ Crystal Ball Collaboration, C.W. Peek et al., DESY 84-064 (1984)

/30/ Crystal Ball Collaboration, S.T. Lowe,SLAC-PUB-3683 (19S5) (Moriond 1985)

/SI/ H. Tyt and G. Rosenfeld, Phys. Rev. Lett. 53 (19S4) 2215

/ « / UA1 CollaboraiioB, G. Araison et al., Pbyt. Lett. 139B (1984) 115

/33/ SX. Glashow and A. Maaohar. Phys. Rev. Lett. 54 (1985) 526

/34/ JADE Collaboration, W. Bartel et al., DESY 65-022 (1985)see also: CELLO Collaboration, H.J. Behrend et al., DESY 85-061 (1985)

probability densitye i g

probability densityntt' I § 8

a

1

probability density8 6 8

AH

1.3-

4000 6000£ Y (MeV)

Fig. 4 Sjwsctwm at bokted photon*

8000

ARGUS

2.0 2.2 2.4

400

SOLO

3X9

10.0

0.0

2.6

...|—4.—i—i,• » • •

t.«S

Fi;. 6 fenariuit sum of the B"r+

169

10-'

Fig. 7 Inrariaat eaerc *pectr» c< T, K, p and i

50

0.01Q5 10

TASSO

. . . . 1 . . . . I

1 • . . . 1 . . . • ,

hi :

•

20

3MO1

Flj. 8 Speeta of ftusvene mooi«st* ta and oat of the erettt plane (solidJinw: LF., dashed liaet: LUND}

170

»r wrarm

0 Energy aad particle distribution in the event plane

| lure) ZQ

§ Mtobcr

Q Goltsclalh

fW ticksK*on 38389

Pip. 10 Ritio of particle depletions between jets

171

o.o;

-o.os

-0.10

-0.15

1—' 1 •—1—«—1—"—1—' 1 ' —— Vim GOTTSCHALK— HOYER WEBBER

. —til

1 . I y |

•

• •

*

1.0 2.0 3.0 «.Q 5.0 6.0 7.0 6.0n

11 L«tctilriiHlotih» wdfhtad naneiiim of Jet • *the powers

ASLund 20KferGottachaft

-01 -

-02QO

Flf. « Awmje tntmne momeata of putklet wrt. tfct jet «odf

172

0,1 0.2 0.5 1

apparent momentum [GeV/c)

Fif. U dE/*tT». d ^ u o t aonestca

<«i d • • • » • • • ! • • . . . . . J — 4 -1 0 V 1 100

i MassfGeV/c2]0,1 to

A .14 Limits (00* CX.) for tke production of fraction.] chartns

t?3

200

I I I t t I I ! ! I I I I I I I I I I » I I I i t

0.75 1 1.25 1.5 1.75Energy in GeV

Fif, IS Photon spectrum from T(IS) -» tX

t » » • » at

ITI

«. III ^T- ^ ^ ^ -i - - *

• s » a » »

16 «al b) rimlatka

HAPIAJIYE COCTBCTIOSS IK TWO-PHOT'OK PB7SICS,

IKOIiUPIS'3 5FFECT? ?ROM NQK-gSRIPHERAL DlJVSRtlWS

W.L. van Neerven

Institut ftir Pnyslfc, Unlv<:rsi.t£t Dortmund

• Dortxiand, ?Hfi

and

J.A.M. Vermaseren

NIKHEF-E

ftmsterdaiDr The Netherlands

We give a review of the ful l virtual ar.a soft radiative corrections tothe two-photon reaction e e~ •• e e~X where X i s a pointl ike pseudo scalarparticle. We find that the effect o i the non-peripheral diagraas i s ne-gl igible so that this class of graphs can be ignored in a l l two-photonreactions.

176

The most general two photon reaction is given by the process (fig. 1)

e. e_ —%• c e.

where X stands for any hadronic or dileptonic state. These processes are

very interesting since they enable us to test some of the predictions of

perturbative and non-perturbative QCD. In particular we want to mention

the measurement of the photon structure function'1 and the possibility

of observing glue balls' in the YV channel. Knowledge of a pure QED re-

action like e.g. e e -* e % u u is useful because it can be used to

calibrate cross sections for those processes containing hadrons in the

final state. At this moment the data for this process' ' are in good agree-

ment with the theoretical predictions obtained from the lowest-order con-

tributions1 ' {fig. 2). This indicates that the radiative corrections ' '

are still smaller than the experimental errors. This situation may change

if the accuracy of the experiments improves. In two photon physics the ra-

diative corrections car. be divided ir. four categories:

(i) Corrections due to the e~e~i vertices, also called peripheral correc-

tions; they include the contribution frota the photon self-energy

diagram (fig. 3a,bi.

;ii.' Photon exchanges between the electron and the positron (fig. 3c,d).

;iii; Corrections inside the yy final state (fig. 3e,f).

(ivj Photon exchanges between the YY final state and the in- or outgoing

electron or positron (fig. 3g/h).

There exist several calculations of category U) in the literature.

However, they were performed via approximation methods like the Weiszacker

177

Williams or the (double) equivalent photon approximation iD.E.P.A.<. This

means that their predictive power is only limited to tJie small t. = q.

2 2

region (-q •* 100 m ) > To remedy this deficiency one has developed alge-

braical and numerical methods to compute matrix elements and phass space .

integrals watch lead to accurate results in all regions of jr.hase space. ' '

Further one has coaput^d the sum cf the real and virtual photor. contribu-

tions in such a way that a minimum cf co!i£"-ter time is needed. v"' ' There-

fore the final expression is suitable for ar. event aenerator. Finally methods19''

have been found to compute higher order one loop N-point functions which

enable us to calculate the non-peripheral contributions in category i: and

iv. In the sequal we shall concentrate on the last two points. Befor*

presenting our results it is very useful to give an order of xtiagmruat esti-

mate of the radiative corrections due to category <i;-(iv). Here We shall

follow the analysis in ref. 10 which is based on the presence or absence

of mass singularities appearing in physical quantities like cross-jections

and decay rates. This enables us to predict the powers j of terms like

lnJ s/m , In - t./» if s, -1. >> m which are typical for radiative cor-

rections.

The rules which determine these logarithmic power corrections car. be

summarized as follows. In the axial gauge a Feymaan graph with 3 pairs of

k "'

identical propagators behaves like In a" if one integrates over k pro-

pagator momenta (provided j-k £ 0). Bence the total and single differential

cross sections corresponding to the lowest order graph behave usymptotxcaliy

C s , - ^ » a^) like

ii. Fig. 2 (2;

respectively.

178

For categories ;i)-(iv) the corresponding results are

Fig. 3c,d (4)

JL Fig. 3g,h (6)

Thfc results above suggest that only the peripheral diagrams belonging to

class (i can provide us with an appreciable correction since we expect

(4 "j -Us -JU. (7)

For LEP with >'s = 100 GeV and t » -100 GeV we obtain io - 5 % and

A(<3c/dt; - 4-5 i. However, these estinates have to be taken with caution

since they only apply for asymptotic values of the kinematical variables.

We will see later that for realistic experimental values of the variables

the results of the above estimates and the exact calculation differ appre-

ciably from each other. This can be traced back to the omission of power

corrections and non-logarithmic constants. Also the coefficient of the

leading logarithms in the corrections can be aoallcr than the one found

179

in the lowest order cross section so that the extra logarithm is partially

compensated. The above analysis can be improved since the coefficients of

the In a2 terms are determined by the anomalous dimensions appearing in the

operator product expansion. In order to compare the above predictions with

the exact results we have chosen as an exasple the siaplest two photon

process where X stands for a pointlifce pseudo-scalar of which the lowest

order contribution is shown in fig. 1.

In the calculation of the radiative corrections we have to split

the cross section a in a hard photon psurt a (A), a soft photon part o <&,X)a a

and the virtual contribution c,,(*>. Here A denotes the upper and lower bound

in the s integration (see fig. 3b) of o and e respectively. X is the infra-

red regulator which appears in c and o . The soft contribution is calculated

in the following way

(8)

where dPS denotes the rwaining integration* and F U V, n are indicated in

fig. 3b. In the last step of eq. 8 the intagration over d», and dys have

been interchanged so that we have neglected the s, dependence in FVV and

in the kinenatical boundaries of d?S. The error cade .in the i n o-o +os+o

is of the order of a power eerie* in (&-m ) / • -It can be miniolzad if we

take the liait & * a . Bowser in this case we' have to cancel large nusfeers.

which are of the order la(4-ii^)/»^ ,betwecn o^ and » s+o y. Therefore it will

take a lot of raejuter tins to achiere numerically accurate results unless

we •Intwire the erroe cadei above doe to the iatercnawje of integration* la

180

eq. 8. In order to compromise between the exact calculation (line 4 in

fig. 4) and the standard eiJconal approximation (line 1 in fig. 4) we have

performed tile interchange of integration for the pole part of ttf only

2 2 •

which has the form W (s =m )/(s, -si > (see line'3 in tig. 4). In this

way we can take a reasonable large value, for A and still achieve an accu-

rate numerical result for e » oH(A)+Js(i,X)+cv(X).

In the case of the peripheral contribution we obtain the following

results. The corrections to the total cross section are given in fig. 5.

They are of the order of 1 % - 2 % whereas the estimate is about 5 %.• This

discrepancy is due to the fact that the vertex and the electron, self energy

contribution cancel the real photon part almost completely leaving the

hadronic and leptonic part of the photon vacuum polarization to account

for the total correction. The contribution of the vacuum polarization gives

even a stronger enhancement to the differential cross section do/dt. (see

fig. 6). Therefore the result at t^ - -100 GeV2 (about 6 %) is in rather

good agreement with our estimate below eq. 7 (about 5 % ) .

From th« estimates in eq. A we cannot expect a noticeable contribution

from the non-peripheral diagrams. Nevertheless we have computed them in

order to see whether there are some regions in phase space where they can

become large. The computation of the diagrams corresponding to this class

is rather awkward and we will give a short sketch of their calculation.{10)

Cancelling the nuoeratar terms the virtual diagram (fig. 3c) can be re-

duced in scalar 5,4,3,2 and 1 point functions. The five point function can

be reduced to five four point functions by either using the procedure given

in ref. 9 or the projective transformation method in rcf. 12. After the re-

181

duction we find 112 different Spence-functions which are evaluated on a

•CYBER 173 coqputer so that one evaluation of the matrix element (single

precision) takes about 40 ms.. In the case of the real photon contribution

we have only included the soft part determined in the frame work of the

eikonal approximation. Introducing a cut off in the soft brcmsstrahlung

integral of s. < (m -Ku) (fc < u and u « 5-m ) and adding the virtual and

soft contributions we obtain the following results. For./s » 100 Gev (LEP

energy) we gat a correction of -O.002 % (M « 0.135 GeV) and -0.004 %

(ML - 1.5 GeV) respectively. Our estinate on the basis of eg.. 4 was about

O.Ol^t. Therefore these .corrections arc extremely snail in comparison with

the non peripheral ones which are 1.55 % and 2.15 % respectively. For'the

4 2

single differential distribution we get Ado/dt » -1 % at t :. - -10 SeV .

Since do/dt, ~ 1.2 pb/GeV this correction will be unobserveble. For smaller

ti this nunber will bacooc even saallar. Fin&lly we obtain for the double

differential, distribution the result 4d2o/dt.*t, = 5-10 % whan t. and t.

are larger than 10J GtV . However for t > 10 Gafir the cross section is

0.4 pb which is about 10~4 of the lowest order total cross section so that

in this region this correction also becaoe* unobservable. The above results

will becooa even — » H e r if X It a milti-particle state or a haaror. (forn

factor).

Snwirisxng our findings we conclude:

i) The contribution of the nonparlgheral diagram (class ii) to the radia-

tive correction can be completely naglacfd. The saae holds for the

class iv graphs. This result iaplies that two-photon processes are facto-

rixable to a high degree of accuracy.

182

ii) The peripheral, graphs are the dominant ones. However, the correction

to the total cross section is small at LEP energies (about 2 %) so

that for future experiments only the corrections to do/dt. (t. - -100 GeV

L do/dt. ~ 6 %) may be relevant.

Acknowledgement

. We want to than.- Prof. J. Smith for reading the manuscript. Further W.L.

van Ne&rven is indebted to the-Sundesministerium fur Forschung und Technologie

for having partially supported this work.

•B2

References

1) W.A. Bardeen, Two-photon physics, Proc. 1981 Int. Syap. ov Lopton and

Photon Interactions ax High Energies, ed. W. i'feil.

2) Ch. Eerger et ai., Phys. Lett. 94B i!980i 254;

K. Brandeiik et ai., 2. Phys. C C_ (1980! 11?.

3) P. Kessler and J. Smith, yv collisions, Proc. tau.er.s :.?SC, ed.

G. cocharfi and P. Kessler (Springer-Verlag, Berlin;.

4j G. cochard and S. Ong, Radiative corrections tc y. processes in e e ,

e e , e e collision rings, in: yy collisions, Proc. Amiens 19SC, ed.

G. Cochard and ?. Kessler (Springer-Verlag, Berlin;;

y. Srivastava, Radiative corrections for photon-photon scattering,

4th Int. Conf. on Photon-Photon Interactions 1381, ed. G.W. London

;World Scientific, Singapore!.

5) K.I.. van Neerven and J.A.M. Vermaseren, Nucl. Phys. B236 < 19841 73.

6) P-A. Berends, P.H. Daverveldt, R. Kleiss, Nucl. Phys. 3Z53 (1^655 421.

7) R. Bhattacharyna, J. Salth/ G. Graaawr, Phys. Rev. Dl_5 (1S77) 3264;

J. Smith, J.A.M. Veramseren and G. Sraaner, Phys. Rev. D15 (1977i 32SG:

J.A.M. VerBBSoren, J. SBith and G. Granaez, PhyE. Rev. Dl£ (1979) 137.

3) J.A.H. Vennaserwi, Kuol. Phys. 3225 (1983) 347.

9) W.L. van Neerven and J.&.H. V u n n r r a , Phys. Lett. 137B (1984) 241.

10) W.L. van Nesrvan and J.A.M. Veroaseraa, Phys. Lett. 142E.(1984) 80.

11} T. XinoShita., J. H»tto. Pfays. 2 (1962) 650;

B. auocwrt and H.L. van Hswrven. Bucl. Phys. 817S (1961) 49O.

12) G. fBooft iai M. WiltMiV, Macl. Phy». B153 (1979) 365.

18A

Figure Captions

+ - + -Fig. 1:- Lowest order graph of the process e o - s e X.

Fig. 2: Lowest order cue graph contributing to the cross section of

•the process e e •» e s X.

rig* 3a,b: Cut. graphs of class i contributing to the cross section.

3c,d: Cue graphs of class ii contributing to the cross section.

3e,f: Cut graphs of class iii contributing to the cross section.

3g,h: Cut graphs of class iv contributing to the cross section.

Fig. 4: First order corrections to the total cross section of

e'e -> e e X computed via methods U ) , !2) , (3) and (4)

mentior.ee in raf. 5, M = 0.2S Gev, »'s - 30 Gev.x

Pig. 5: First order correction to the total cross section as a function

of the pseudo-scalar mass M , (!) >'s = loo Gev, (2) >'s = 30 Gev.

Fig. 6: . First order correction to the differential cross section do/dt

as function of t for /s * 100 GeV, M^ = 0.5 GeV (1) contribution

from do/dt. (2) oontribution froa da/dt.,; contribution from

the vacuum polarisation to

185

Hfig. 2

fig. 3

correction in %

correction in t*

Oi : r

ZX 01

1 °

o<

correction in If.

0>

991

18?

FACTORS IK SWO EFFECT

Chao

Xastitnt* of Mig» toergy H»T»ie». ftcxWia Siniea

V.O.aox U l , BBlJiag, P.R.Oitaa

In 1982 European Ruon Collaboration (QIC) discovered' ' that except

of the known nuclear shadowing effect and the Ferni motion of nucleons

in a nucleus the momentum distribution of quarks in bounded nucleons in

a nucleus is softened comparing with the one in a free nucleon. By

reanalyzing their empty target data of deep inelastic electron scatte-

ring in the early 70 's SIAC group soon confined this discovery. '

Why is the aonentua distribution of quark Moving to the lover region

in nuclei? A coranon explanation is that the quarks in nuclei nove in a

larger confinement size than those in a free nucleon ' '. Based on the

uncertainty principle argument including also the additional gluon

bremsstrahlung due to a lower cut off in the larger confinement size

F.E. Close et al.' ' have related a •oaentun scale change ^ which

gives the softened aoaentuu distribution of quarks to the change of the

confinement size R as follows:

(1)

• The talk is based on work* ia refs. |5| aid |C|

188

where F * and F_N are structure functions for a bounded nucleon in the

nucleus A and for a free nucleon, respectivelyi RA and Ro are the

corresponding confinement sizes.

But, eq.(1) did not answer the^following question: what is. the

physical picture for this enlargement of the confinement size? If a

certain picture is introduced, are there other physical factors

accompanying with the change of the confinement size, which should also

influence the EMC effect, or show other physical results? Now I will

give an analysis for two main pictures.

The first one. Nucleons in a nucleus are very close to each other.

There are certain possibilities that two nucleons collapse into one

cluster which, of course, has a larger quark confinement size. Usually

it is assumed that the 6-q cluster is twice larger and heavier than a

single nucleon. This gives it confinement size fz tines larger. Based on

the nonentuii scale change idea F.E.Close it a L ' have got ?•. is about

2 for iron. For eq.(1) they obtained the corresponding enlargement of

the confinement size of 0.15. This is a quite large change if you think

of the corresponding confinement volume change of 0.52. To give such a

large effect about 58% nucleons in iron nucleus should be changed to 6-q

clusters.

On the other hand, based on the same physical picture C.E.Carlson et

al. and many other* considered other factors which were not included

in Close's description. A «-q cluster has 6 valance quarks and the *ass

of it is twice heavier than the »«s of a nucleon. The change of the

quark number will increase the nuabcr of spectator quarks for each

process froa »# to n'o-a^+ 3. The counting factor in the structure

function then becomes

789

The aass change will change the variable

(3)

where X is the usual B. variable. Considering these two factors Carlson

et al. have fitted EMC data with the enlargement of the confinement

volume of 0.1B which is auch smaller than Close's value 0.52. The proba-

bility of forming 6-q cluster is now 30% also much scalier chan Close's

value 58%. But the correspond™g momentum scale change is not considered

in their work.

According to our opinion ' ' once the physical picture of forming

soae 6-q clusters or any other similar clusters is accepted, the factors

of the aoaentua scale change and the increase of the quark nunber and of

the nass in the cluster should be considered at the same tine. Assusmg

the average confinement volume in a nucleus A, v -(1+ A.iv , where vA A o C

is the voluae of a free nucleon, considering the three factors to : ner,

namely the changes of the momentum scale, of the number of spectator

quarks and of the mass in the changed confinement volume the following

three equations are obtained:

(4)

By adjusting parameters A A a systematic fitting to SIAC data has been

made' and the retrelt is shown in fig.l with parameters listed in

table 1. For comparison Close's result with only the momentum scale

190

change and Carlson's result with only the quark number change and the

mass change are given in the same table. It can be seen clearly, with

all three factors putting together the necessary change of the

confinement volume to explain the EMC effect is considerably smaller. In

fact, for iron our fitting gives A p e » 0.14 and J F e« 1.23 which corres-

ponds to the probability of forming 6-q clusters p « 24% , while Close

et al. Give "$_ * 2.02 corresponding to p - S8t and Carlson et al. have

A = 0.18 with p - 30*.

Now let us turn to the second physical picture. It is assumed that

the volume of each nucleon in a nucleus would become larger due to the

existance of surrounding nucleons. The quarks moving inside the enlarged

nucleon then feel a larger confinement size. We have given an physical

explanation of this r.udeon volume change based on the soli ton bag

model by T.D.Lee. In the eoliton bag model a phenontenological scalar

field (TIE introduced to describe the color dielectric property of the

vacuum. The color dielectric constant K is related to 0* field by

0" — <r»o- *) , (5)The effective potential Vim per unit volume is shown in fig.2.

ff "0~IK«C> corresponds tc the physical vacuum with v(0")" o. Inside the

nucleon bag 0"s0(K*l) and v(ff«0) = Bo. The energy difference B inside

and outside tht bag is tne pressure on the bag surface. Bat, conside-

ring one nucleon bag in a nucleus around it is a distorted vacuum with

(A-l) other nucleon bags embedded in. Sine* there is no precise method

to describe this multiply connected region consisting of domains with

two different Cvaiu*», we introduce an effective dielectric constant

K A > o for' the region surrounding a given nucleon bag.

The ctfectiWt C-fi«ld is 0^ < 0\, • which gives a smaller pressure on

191

the oaa surface: B r- A B . isee fig.2). The corresponding sag size of s

bounded nucieon is therefore enlarged. In fact. Close's worn with only

the momentum scale change included can be explained by this picture

instead of the 6-q cluster model. The corresponding volume change of s

bounded nucieon in iron should be 0.52.

On the other hand, according to the bag model when the size of the

bag is enlarged, the mass of it should become smaller. In the first

order approximation

(6)M «*• '/ft .

The E. variable becomes larger, which makes the plot of the structure

function v.s. original B_. variable shrinked and gives EMC effect.

K.Staszel et al. have done a work along this line ' giving a volume

change of 20% for a bounded nucieon in iron and the mass of it about

0.94 Mo with Me the mass of a free nucieon. But. the momentum scale

change is not included in their work.

Combining the two factors together: the bounded nucieon vith a

larger volume has a smaller mass and the momentum scale should be

changed, the necessary volume change of the nucieon is only 6* to

explain the EMC effect in an iron target. This is again much smaller

than Close's estimate ( 52%).

It is impossible to tell which physical picture is more reasonable

only by EMC effect. Besides, even for the same picture different con-

siderations give quite different physical results. To judge the two

pictures people are looking for other experimental support* for each of

them. There art some strange phenomena in high energy h-A »nd k-h

collisions at high Pt> for example, the cross section of h-A collision

°"hA w r i t t e n •« *" • «"«, "it1* * «F to 1.3 ~ 1.7 for hi8h *t t m t i , the

1S2

jnexpected ISC11 IT-production and the subthreshold Tr-produetion. All

these may be explained by the existance of larger clusters, e.g. 6-q

clusters, in nuclei. Unfortunately they ail correspond to t.he kinematic

region vithXapproaching to one, or even larger than one, which is not

the same region of 0.2 < * < C.6, where typical EMC data are taken. On

the other hand, the picture that bounded nucleons have larger volumes

has been used tc explain some nuclear properties at louer energies. For

example, the form factor of pi and the response function of (e.e'J

quasi-elastic scattering is explained by C M . Shakin et al. based on

this idee ' . To make <» more certain conclusion it would be nice if

any experiment can give the mass M' c£ the effective target in 0.2<1t<

C.6 region. The two pictures we have discussed give different change of

the effective mass from M , the mass of the free nucleon: M' >M for 6-q

picture and M'< M for enlarged nucleon bag.

Even in the region of 0.2<X<0.6 there are still so many uncertain-

r.ies's about physical pictures and included factors. Related with tnese,

it is also difficult to teil how big is the change of the confinement

size in a nucleus. At snail t region data from different groups are not

consistent yet. There are many problems left unsolved around EMC effect.

To understand nuclei at the quark level tnere are many works which

should be done both experimentally and theoretically. Here is one of the

fields where nuclear and particle physics neat.

The author would likt to thank the VIII Warsaw Synpoaiunt on Elementary

Particle Phytics for the kind invitation.

193

References

|1| EMC.J.J.Anbert et al.. Phys.Lett. 123BU983) 123,275. .

|2| A.Bodek et al., Phys.Sev.Lett.50(1983)3431; 51(1983)534.

|3| P.E.Close, R.G. Roberts, C.Ross, Phys.Lett.123BU9331 346.

R.L.Jaffe, F.E.Close, R.G.Roberts, G.ROBS, Phys.Lett.134BU9e4)449O.Nachtaann, H.J.Pirner, Z.f.Phyaik 021(1984)277.

|4| C.B.Carlson, T.J.Havens, Phys.Rev.Lett.51(1983)261.

|5| Chang Chao-hsi, Chao Wei-£in, preprint AS-ITP-84-034,BIKEP-TH-S4-20.

|6| Peng Hung-an, Chao Wei-Qin, Liu Lian-sou, Liu Peng, ChinesePhys.Lett.2(1985)63.

|7| M.Stascel, J.Rorynek, G.Wilk, Phys.ftev.n29(1984)2638.

|8| L.S.Celenza, A.Harindranath, A.Rosenthal, C.M.ShaKin, Phys.R-v.c3i(1985)946; Brooklyn Collage Report Ho.84/111/132, 84/121/134, 1964.

Table 1: McBentuB scala change parameter %.

and confinement vclune change £ •

our: our remit : ref. |5j

JCRR: Close et al. > ref.(3!

Carlson: ref.|4|.

Model » 9Be l2C 1 7 U 4 0 C ^ r . 1 9 7,

.our £ A ,07 .08 .11 .13 ^1* -19

•JA 1.11 1.14 1.18 1.22 1.23 1.32

JOW ( AJ .52

- A 1.40 1.60 l . »» 1.86 2.02 2.46

Cerlaoa Ak îB

IS

11

to

He

0 U M M U

1 1

F c

V«7)

aA <V

ri».J Th« chanf* of t u «n*r«y amglty 4u« to tM elwifa or «M• f f m i ™ IT «iald in tha MCUUS (r*f. |«||

o.e a*

Tt* (Ml. »*|«|-4iCumit ml t i

iMlntb cnu McMaw far

195

LO&ARITHMIC ;.CYK"gOTICS OF AWHJTUIlSS IS QC1-

•i. I - i r sc i iaer

;.ei:tion Ihysiit, Karl-'.arx-iiniversitet Leipzig,

Consider the amplitude of some process in a (riven cr-dei of

uerturbaticn theory and adu a furtaer loop i s ail possible

7;E.VS. In some cases tne sua of tue contriou."cicns v.itn t;ie

additional loop can be expressed approximately i r terss of

a loop integral witn tne original amplitude. IE ter.T.s ;f

graphs this means that vhe aaciitional locr aas Deei: sepa-

rated our of the bloct: representing the original asi.liuade

(Fig. 1). Cne nae to answer 'cue questions aoju" zue ap>ro-

xiaation, where succ an expressioi! is valid, ana about zLt

range of inte£ratioii in zhQ loop inuegral. I'hsr. we arrive

at an equation TJaat allows tc calculate the ar.pli~j;;;fr ir. ;•

definite approximation.

1'he separation of ?. jluon loop proceeds o.' a ? .asis cf

zhe b.-erass-rahluiig tiieorea ,"t/. '.rixe tue sso erTruz c; -i;c-

^l-cxi i r iiuaajicv i-epreses'satioii,

p,• • : - * ' pa t p pi + k

jre TV."C- altosv Ii"h-.-iiê J.O:JSS"W£ cf :_e

If -;:;e transveise aoaeitua K is sî&iier 'uai: t,xie "or.-j.csverse

aoaents in the ether loops, tieL -fle c.o:::ir;ar!t ccntrioutior.

is the cne viizh single-particle interêd.ate states i t tne

cnaniiels (p. - & ' ana ;p^+k. ) • Tiie gluoz: is sevi^rs.zea iroii

•cae saplituae and effectively, i t i s ccuplec to tae external

lines of tue oriîual asolitude. iae Dole factors '.v.-k''1- -k«

and ft. +«;! = ,' 0s) lead tc a logaritnmic loop integral.

Contributions witnout "=aese poles are suppressed sv least by

a logarithm eff s .

The method of separation of uhe particle v. i th the lowest

transverse aosentrum has been applied to calculate scatteriiig

196

tarliûues •.vita quEistua number exchange in double' logarith-

mic approximation /2/.

Consider -he amplitude of the virtual Gompton process

*(pc) wiiicn is relevant for calculating the fragmentationa. UsliKe the scattering amplitudes in the egge

reelon it a&s only one almost ligat-liie erternal momentum p_

jr^ ** J wnereas the virtual photon a&s a large cf »^MZ .

l.-i the amplitude with -the additional gluon loop T~(a p, & )

we separate the contributions with a pole in one of tne

channels pi k': or

lu tae icop integral ever c tae pol^ f.actors (p +

rise ic a ioe.a;'itaai of o* . /. .:o,o £) i s the contributiont,o r.iie CoupTon, amrliTuce zitr: ar. aaditional gluon *?v/ithcu~ jvcles'ir .;.•'_--*1* • '-^ t::is stage we have achieved ararr-ial separation : C;he aiditicnal ^luor couples with oneenc tc. '..z. exrei'i.al _L_r.e rpbut -::".e or^sr end i s coupled tothe irjier ;.=rt cf the block.

li. crier to study t.-je separation of the gluon from theblcci •'£ ccii£ii_er &. rt --.-jjrcicle iitera^diate srate j1 p< /= • rrri.iuLiz-- '.c the J -enaonel iiaîii>-jy pctrt oi' tne3o.Tr:or aapIirMde. The additional gluon. loop couples efi'ec--ively to or.c. of the exzern.al lirês p and to one of their-'ce^.e-iiste iires ,£>• if i t s ~raxsverse aomentua Kt .vithrespect to :p,p.i is e=all. rîs condition can be fulfilledfcr a:v pt- arc all i;:teraediete spates only if" tne ÔEentun

k oi îe adcitioDbl gluon is clsse i t ciirecticn to p .lais i s tiie iU-CA't conai-cion of angular ordering / 3 / .i'he su2 or cor,triDut;j.or!.E witn tae gluon coupled effectivelyto the interffiedia'te lines p,- is given ny

The indexU) reaieds us that we are considering a fixedw -particle intermediate state. T-* is the colour matrix

197

e^sociated uo net fcIucs coupling tc trie lias <" . _iie j.oê

irua; and the colour factors in zae SUE can be iisra tc be

universal if anmlar crdorizic nolas. 'jsis resultE in fuli

separation / V , ?!£•£•

t Jr

S is the energy fraction oi' -ce gluon in tiie rest frame

of <5 .la theories without gauge particles the Hegee asyaptotics

in leading logarithmic perturbation theory is given by ladder

graphs, wnere the momenta of cue s -channel intermediate

s-ate particles obey "cue multi-riegte kinematio£

In gauge theories taere are aadi:ional loops c± 'oremsstrai-

lung type, iiven such a gluoa v.'ita nomentum A (1) tie

ladoer particles are divided into groups wita theix S-cdakov

parameters obeying

U I #L >* f r $•• •

(h * <^t , ? < ft ; (b)

<; ; */ < * , /?/ > F . fi

A'ith respect to the particles (i; vhe giuon # is eoitted

with sn angle that is smaller than the emission an^le of any

particle i, it the luon couples to the block built Dy the

particles i then it is separated by angular ordering.

.JiEilarily the gluon may be separated- from the block of

particles (o). v.ita respect to the block of particles (1)

•cue i.;luOE aas the smallest transverse momentum and it may

oe separated relying; on the brensstrahlung theorem. Graphi-

cally the relation of the sûos s& *ae blocjrs (i;, (1)

and (o) can be represented as in Pig.3«

In the colour singlet channel with positive signatiire

the contributions of the type (1) cancel. In this case the

contribution of all breaisstrahlung gluons can be easily in-

cluded into the sum of ladder grapus. I'he virtual gluon

198

coEtric-tions resul- in the realisation of tile two par-

ticles exciiaat-ed in trie f -caanr.el. j.i'ter separation the

real giuons look like ladder particles. laeir contribution

can be combined witis the one of the ladder particles.

C'r zz.e basis of this result we are able to write down

s. generalised ladder equation, which is most conveniently

expressed in teras of partial waves. 3or the case of gluons

exchanged in ihe t -cnaniiel (racuaa channel) we recover the

equation obtained first by -F'aain, iuraev and Lipatov /?/.

For tne case of two guards in -ne t -cnannel (flavour

tge) zne corresponaing equation reads

f t f - K ' l

1 /9'kl

V

êcifjing tê jorc tera by a r -fux.c:ion the solu.ion isr.?e Sreen function of tnp T.erturtatively generated ..eji--eoacarrying seson cuaîua nuaoers.

/" . / V..".. 3ribov, iadera. ?is . £ (1^6?),35"/£/ .. . iarschner, i.,1'.. Lipatov, j.ucl.Vhys. 3.213 (1$S3),122

/ 3 / V.I. irmolaev, V.S. ?adin, risna i,cSTF 3_ (I981),ê5;;..H. wieller, ihys.Lstt. 1O4J (19Si),'io1

/4/ h.. Lirsenner, in jroceedings of the XXII Conferenceen hign inergy I-aysics, Leipzig 1J£4jYadern.Fiz. to be published

/3/ V.£. ?adin, ii.A. Kuraev, L.i:. Lipatov, rhys.Lett.§03 (1975), 5C

199

Separation of a sluoi: wiuti lov.'est transverse so

Pig.2 : Separation of a r:luon from the Tonipton s.-i.-Iitude.

Fig.3 : Bremss^ral-.lun.;- glues

sep£i"ation in & la.-.aer graph.

201

:RITICAL QUESTIONS CONCERNING LIGHT CONE FORMULATION AND

POSSIBLE RENORMALIZAT1ON OF YANG-MILLS THEORIES

A. BASSETTO

Dipartimento di Fisica "G. Galilei"- Univ. di Padova, Italy

Istituto Nazionale di Fisica Nueleare- Sezione di Padova

202

The light cone gauge choice seems to play a more and more

important (and intriguing ) role in several recent theoreti-

cal developments concerning field theories.

Since its original formulation ' it was realized that the

price to pay for having only independent "physical" degrees

of freedom was a lack of manifest Poincarg covariance and a

very singular infrared behaviour of the vector quantum propa

gator. While the use of the Dirac brackets formalism '

allows to recover the Poincarg algebra in a natural way, the

second problem was either disregarded or faced in a non com-

pletely satisfactory way until recently. It was also realiz-

ed' that it was impossible to get the light cone gauge as

the limiting case of an axial gauge when the square of the

gauge vector n. tends to zero.

In spite of these difficulties most of the perturbative calcu

lations concerning parton amplitudes were performed in this

gauge/ thanks to the triviality of the Faddeev-Popov deter-

minant and to the dominance of planar diagrams in the leading

logarithmic expansion' '.

Finally in recent years this gauge choice seems to be impor-

tant in the formulation of several supersymmetric field

and string' theories, nostly in connection with the possi-

ble finiteness of the so«e of them or with the existence

of a useful Hicolai transformation'

Therefore we think of the utmost importance a correct cano

nical quantization of the theory in this gauge as well as the

understanding of some difficulties arising in its renonnali-

zation. In the following we shall confine ourselves to the

pure Yang-Mills case; the introduction of Dirac fermions does

not entail any essential complication.

Originally the theory was quantized using null-plane brackets ' .

Starting from the usual lagrangian density

203

where

= - | F F ^ • (1)

F3 = 3 A* -3 A2 + g fabC A b A C (2)

we impose the gauge condi t ion

(3)

with constant light-like vectors n = (n ,n) and n = (n ,-n)u o j o

Ther. we have the constraint equations (the "evolution" occurs

along x )

Da beina the usual covariant derivative and A* , F 3 ' £ the•^ " • - a

independent; coordinates and conjugate momenta. Were it possible

to invert the operator 3_, egs. i4; and (5) woulc' allow us to

express A^ in terms of A^ and F '' . Now, in order to invert

3_, one has to impose a boundary condition on the potentials

at a certain value of x > e.g.

This would be acceptable, were the true physical evolution

occurring along x and would make the light-cone gauge some-

what similar to the space like axial gauge A3= O. In parti-

204

cular a natural choice for the inversion of 3_ would be

n.~l 3 U,y) = - sign(x~ - y") (7)

leading to the celebrated principal value prescription for the

spurious singularity in the vector propagator'

As however the physical evolution occurs along xQ, the true ti

me coordinate, we think it is not worth imposing a priori boun

dary conditions which interfere with the time evolution of

our system. One might at this point speculate that a null pla

ne algebra of fundamental commutators is not really equivalent

to the usual equal time commutator algebra when the spectrum of

the system contains nu) I squared inomenta. This will be clear

from cur treatment, in the sequel.

Let us start again from the Lagrangian density (1) to which

we add a gauge fixing term as well as suitable sources ' '

. = - 7 " . F" ' - • HA - JA + K.'. (8/

ere here Lagrange multipliers and K the related sources.

Colour indices are understood. The canonical hamiltonian den-

sity becomes, after ar. integration by parts, for j = K = 0

H = — (F . F . + — F-, F., ) - A D. F . + A nA i9)? oi oi 2 -k 3k o l Oi

leading to the equations of motion

D a b T'"'b= -n1 Aa (10)

and to the. secondary constraints

205

ab b , a ....D. F . = A , • (11)l oi

(12)

We notice from eq.(H) that the Gauss Law is not satisfied.

From eqs.110) and (11) we get the equation for the Lagrange

multipliers

n3Aa= 0 (13)

which would entail i\ = 0, were it possible tc invert the ope-

rator n&.We shall keep instead in the sequel A j* 0 and we

shall work with a redundant number of independent degrees of

freedom, namely A, and P . , on which we impose the equal time

canonical commutation relations

Aa it,x), Fb (t,y0 = i5 ^ . <53<x - y) (14)i 03 J-3

pb (_ -,-, _ 4 ,ab soj

following from the corresponding classical Dirac brackets.

The Gauss Law will be eventually restored by selecting a sui-

table subspace of the Hilbert space. In this "physical" sub-

space the Poincare covariance is also recovered

In order to get the expression of the free vector propagator,

we consider the free (i.e. g =0) field algebra. We denote by U

the radiation potentials, whose Fourier transformed components

satisfy the equations

(k~kv- g'^V) U* (k) = rÂ^lk) , (15)

nÔ* = 0 (16)

Their solution is

206

ua<k) = Ta<k)5(k?) • n - ^ A a ( k ) + i L • Mr*) ua(k)( 1 7 )

where k? = 2 '" ' -" "' . y< and

nWTa « k N a = 0 . (18)

The potentials ~-^ can be expanded on two orthogonal transverse(a ) "

polarization vectors e (k)

vi> (k) ta" !k) _ (19)

and the frequencies t_ quantized in the usual way. The longi-

tudinal-temporal part of the potentials

f20)

can be expressed as

and,according to the support properties, ; and fa cai be decom

posed in frequencies as follows

;a(k) - 9(k3) ;

a(kJ + 6(-k3) ;a {-*), (22)

a(ic) = i6(k3) fa(k) - i6(-k,) f a(^) (23)

The algebra of eq.(14) induces on ca and fa the peculiar commu-

tation relations

207

(24)

ail the other commutators being sero.

Denoting by \O> the Fock vacuum, we ge;

(x) 5 <O| TfuNx) oNo))|0> =

= <O| T(T*(X) T^(O))jO> + (25)

+ <o\ T(ra(x) rb(oi) lo>

and, taking the commutation relations into account,

<o! T -a x ~b<o )!o-> - ± i ! !

r n k + n k •*. n g + n g ,

I-%, + 2 - * - J ^ — T £ - JL-^rJ*^ • (26)

<o| T(ra(x) r^(o))lo> - - Tf^rr }&T-%TZ'£ e±**

n k + B k P.*k "ySjo" n v s

leading to the final expression for the free propagator

""{-g +2££_2sfilL) (2«)

where

Cnk3 ' (nk)

206

the prescription proposed on an empirical basis in refs.|8|

and |11| .

The product of distributions in eq.{28) is now well defined.

The Wick rotation is possible and no singularity arises at

kj.« O in internal lines. The gausson field f behaves as a

ghost.propagating along a generating line of the light cone.

This ghost was killed in the previous treatment based on

null plane commutators.

As we have anticipated the Hilbert space of the Feck states

has an indefinite metric.

We can consistently define a physical subsapce r\ by impo-

sing as the only condition the weak vanishing of the Gauss

operator

0, v;.' eK n <30>/p

: 1o • \JWe have shown ' that J\ , is stable under the usual PoincarS

algebra and has a positive srfmidefinite metric. Moreover the

perturbative /-matrix maps y(r into itself and its restriction

to J- is a unitary operator.

Following a standard procedure it is now possible to intro-

duce the generating functional for the Green functions

WCJM,K3. - J?"1 |dCAu.A3 . i / W x (31)

where L is given by eq.(8) and the boundary conditions on the

potentials are such as to guarantee the prescription (29) for

the free propagator

This functional is the starting point for the study of Green

functions in the "four component " formulation. Before doing

that, we want to show how a "two component" formulation, si-

209

milar to the original one, but with the Gausson included, can

be recovered in t h i s framework.

We notice that in eq.(31) we can immediately perform the t r i v i a l

functional integrations over A and nA , get t ing

W CJ , K3 = ,v~' Id [A , A + ] e x p | i /d"xl ] (32)

where

- 7 F F o + \ (D X- 3 A ) ! + (D K - 3 A ) •4 a 6 a 8 2 + - + a - a

(33 )

At th is point we observe that the integrations over A+ are gausslan

and with a determinant depending only on the external sources K,

Therefore we can make A+ disappear, not inverting the equations

of motion,but just by performing another functional integration

We f ina l ly get

W CJ , KJ = ;V~'cdet D J~l exp Zif d*x I 2 3 •J • • • ( 3 4 )

• \ d [A ] e x p t i / d*x i . )

with

ab rab . acb c2 = 0 3_+ gf K , (35)

T F O F . + 3 ^ A 3 A - J A + K D 3 A -4 a B a 6 + a - a a a a + o\ (36)

~ ID'1 I H A - J A A J K) 32 - U~l (D D A -i a a a a' . a a

210

-g A A3 K) JC3 K + C~' (J - 3±K) 33 a a + *

Iand

£,- J+K + i C9+ KJ2- j CzT(J_- 3aK) + 3+K3

J , (37)

The functional « C ^ , 0 3 provides us with a fcona fide two com-

ponent formulation with only physical degrees of freedom expli.

citly exhibited _

We stress however that the operator \T_ has to be interpreted

in the sense of eq, (29.) and therefore the Gausson is still

creeping into the theory. A quite analogous calculation in-

cluding Dirac fermions could be performed showing that integra_

tion can indeed eliminate half of the fermionic degrees of free

dcre .

Of course both formulations lead to identical Green functions

In particular we observe that WC Jv,0 3 obeys the following basic

identity

f CCT , 0 3 - 0 (38)

Which.allows us to relate Grean functions with unphysical

AKternal lines to Green functions with only physical (i.e. tran£

v«r»e) external lines. The simplest example of such relations

ha* been reported in ref.112 j.

They are different from Ward identities which always involve

Green function* with at least one external line being a Lagrange

j«ultipUer I* II 1 2 I.Osing both kinds of identities one recognises that the entire set

of Green functions can in principle be reconstructed starting

only frost the physical sector.

At variance with the Green functions> the one particle irredu-

211

cible vertices are quite different in the "two component"

and "four component" formalism. This can be immediately un-

derstood if one remembers that the two effective Lagrangians

are different. We do not expect that the irreducible ver-

tices in the two component formulation obey any useful Lee-

type identity as no redundant degrees of freedom are present

anymore. For this reason we find simpler to discuss the re-

normalization problem within the four component formulation

From eg.(34), performing the gauge variation

&AV = D^', 8A = 0 (39)

we get the Ward identities

V U

sJ -» Vis?' ISS] W C^' K ] = 0 (4O)

Then, by introducing the generating functionals for connected

Green functions ZCJ",K3 and for proper vertices rCa'',>. 3 , aJ

and1 being the correponding "classical fields", we derive the

Lee identities

where

"These identities provide us with simple relations among proper

vertices; in particular the two point vector vertex is ortho

gonal to jc ,t .

The peculiar features of' the light-cone gauge, when the pres-

cription (29) is adopted, are the appearance of the vector n,*

which is not present in the original Lagrangiar. as well as the

occurence of divergencies with non local character in the pro-

per vertices.

212

These features are already present in the one loop calculation

of the Yang-Mills self-energy reported in ref.j1i|. However

it is easy to convince oneself that these non local divergen-

cies are harmless, as they are always multiplied by the vector

n, and therefore cannot contribute to the Green functions.

It is not difficult to show that, at the one loop level,

it is possible to make all the Green functions finite by in-

troducing two renormalizaticn constants Z and i , without

spoiling the unitsrity property of the bare theory' '.

We can indeed define renormalized quantities as

A.(o)- B "- A, (43)

with

and

I": n V

R. " * ,'Z ig •-(1-2 "' ) -rr ) (44)nn

o = Z•o

(45)

K .

One can easily verify that by choosing

(46)z = -1 + II

4 « 1 + 2g;N —-1 6 :

the one loop propagator can be made finite.

The Lee identities one can derive from eq.(41) assure the one

loop finiteness of the three and four legged Green functions.

213

Usually, once the theory has been made one loop finite, it

is possible t6 set up a recursive argument to prove renormali-

zability to any »rder. Unfortunately , in facing this program,

we encounter the following difficulty. Owing to the presence o

the vector n* , we have four possible (i.e. orthogonal to n )

u uindependent tensorial structures contributing to the complete

propagator, namelyn k + n k

(47;

(•18)

(1)X- * ™*

t ( 2 ) _ .fcyv -

*

nn

n |

*

k +V

tnk

nV

k,.

i n v n: *

n*k

Cnk3

(50)

At one loop level only the first two among these structures carry

divergent contributions) which can be cured by the prescription

(43-46). However we do not know any general argument preserving

the appearance of all the four kinds of divergent terns at many

loop level. Individual diagrams do indeed exhibit such divergen

cies. The counterpart of this difficulties in the two component

formulation is e.g. the possibility of having two independent

tensorial structures, namely 6agand kpfe for the complete pro

pagator and only one available renormalization constant Z.

If this were the case, to make them finite we should need a more

general renormalization procedure which however would be incom-

patible with the unitary of the renormalized theory.

We remark that a complete two loop calculation of the propagator

could either settle the problem, were the divergencies found, or.

in the case they cancel,perhaps provide a way to understand

this cancellation from a general view point.

In conclusion a correct and unambiguous formulation of Yang-

Mills theories in the light-cone gauge is available both in

the four and in the two component formulations.

Renormalization is feasible at the one loop level, whereas the

inductive proof at any order is still missing.

215

REFERENCES

(1) J.B. Kogut , D.E. Soper , Phys. Rev. D1(1S70) 2901.

(2) P.A.M. Dirac. Lectures on Quantum Mechanics,Belfer Graduate

School of Science, Yeshiva Dniv., New York (1964).

(3) A. Hanson, T. Regge, C. Teitelboira. Constrained Hamiltonian

Systems. Accademia Nazionale del Lincei, Bcma (1976).

(4) w. Rummer. Acta Physica Austr. 4_1_ (1975) 315.

(5) see e.g. G. Curci, W. Furmanski, K. Petronzio. Nucl. Phys .

B175 (1980) 27.

(6) L. Brink, 0. Lindgren, B.E.K. Wilson. Nucl. Phys. B212 (1983)

149.

(7) H.E. Green, J.H. Schwarz. Nucl. Phys. B243 (1984) 475.

(8) S. Mandelstam. liucl. Phys. B213 (1963) 149 and references

therein.

(9) V. de Alfaro, S. Fubini, G. Furlan, G. Veneziano. CERK pre

print TH 4021/84 (1984).

D. Amati, G. Veneziano. CERS preprint TH 4116/85 (1985)

R. Floreanini, J. P. Leroy., J. Hicheli, G. C. Rossi. ISAS

preprint 23/85 EP (1985).

(10) A.Bassetto, H. Dtlbosco, I. Lazzizzera, R. Soldati. Phys. Rev.

031.(1985) 2012.

(11) G. Leibbrandt. Phys. R«v. D29(1984) 1699.

(12) A.Ba«setto, M. Dalbosco, R. Soldati. DFPD 15/85, OTF 120

preprint 1985.(Phys. Lett, to be published).

217

oi-giHizar> Ksa'juBB&'JiQM SSRISS FOR 'JHS BPFECMVE

Pi (fr* FIISLT)

Anna Gkopifiska

Institute of Physics, Warsaw University Branch

Lipowa 41, IS-*?* BlfcuYSTOK. Poland

In this talk I would like to present e non standard method

of perturbation expansion in QKD. I will limit myself to the

self-interacting real scalar field ( x f ) in H dinensioaal

Euclidean space. The method is formulated in path integral

approach and can be easily generalized for other theories, even

at finite temperature•

+ supported in part by Polish Ministry of Science, High Education

-tod Technology, troblea MR.I.?

216

Conventional perturbation expansion

The vacuum persistence amplitude in the presence of linear

interaction with external current J(x) is represented as the

Feynman integral over histories

where the classical Euclidean action is

*L« * faC44»^4W * f <#«) **4to] (2)

The effective action can be defined as

where VUI - In 2[J? is the connected generating functionaland the background field

is th« vacuum expectation value of the scalar field <j)(x) in the

presence of external source J(x). It can be shown that P tpj

is the gene-rating functional for one-particle irreducible Green

functions in the theory with the vacuuaa expectation f(x). In the

physical theory the sources are absent, hence PDfQ satisfies

- %<) . 0 (5)

Only constant solutions f(x)-vf of this equation can describe the

true vacuum, since we dont expect translational invariance to be

broken. Therefore the vacuum should be the stationary point of

the effective potential, defined as an ordinary function of f

Below, for notational simplicity, we shall suppress the space

arguments and the integrations over then.

219

?he generating functional 2EJ2 is usually calculated by thesteepest descent sethod -. _

where ^0 i s chosen to sat isfy the c lass ica l equation of notion

| i s - 3 W (8)

Performing the functional Gaussian integrations we obtain

This expression, ordered according to powers of the i - ' r cconstant, i s knovm as the loop expansion. Tc each order, tileeffective potential can be calculated d i r e c t ! - froa the definingequations (?),(4-) and (6) . 'J?o order £ we

fhere M = f <*^*PHowever i t was pointed out t l " that the loop expacsion i s

valid only if the 3aussian integrals in (?) converse i . e . if

i£ positively definite operator. This is- not satisfied for

y'^'rr i*2 t h e case when we expect the syaaetrj to be broken

spontaneously.(B^ C, the classical potential double well

shaped). Ihe iaaginary part of the effective potential appearing

in the order -6 is a signal of the failure of the loop expansion.

Interpolated loop expansion, which takes into account the ccntrri-

butions of both wells independent?, has been proposed ilj. "lie

interpolated one loop result is real and convex. It differs froa

ordinary oae loop result (10) only in the region vnere this last

is not valid.

220

Optiaiseo. perturbation expansion

:!e can use e siasle trici: that the steepest descent aethod

remains applicable ir. the case when a < 0. .Ve add and subtract

the tera -i-jf^ i n tiie classical iagrangisn and we treat -gty-V+

as R free term and -j-jf a+X )'( as a perturbation. The generatingj

functional 3£J? may be calculated as in (7) >y expanding the

exponential

where the formal parezceter fc has been introduced to identify the

order of approxinsstion. At the end we set 6«1. The Gaussian inte

grals converge as loc- ss we choose an arbitrary function Si(x)

to be positive everywhere.-Intelation of this expression term

by tern givet tae j;6rturbstion series

If we tske ~!T constant through the space, the first order approxi

mation of the effective potential becomes

»e ainiciaff the sensitivity of the approxiaant to ths variations

of the unphysical ptr rjeter SI , as advocated by otevensoa's

of ."iiniaaJ Sensitivity 123. Requiring JlZto fulfil

gives the following equation for

221

It is interesting to note that the first order result or optimized

perturbation expansion (OPE) is the saae as the Gaussian effective

potential (GEP). GEE was obtained froa functional Schroedinger

equation by variational methods with Gaussian trial function C3»43.

J?he Gaussian effective potential has been shown to sua all the

vacuum diagrams without overlapping divergencies, therefore should

be auoh better approximation then the one loop effective potential

which sums only all tadpoles attached to this loop. The systematic

expansion in £ ofers possibilities to inprove this result.

Zero dimensional 4* theory

The simplest illustration of the superiority of OÊ to the

conventional loop expansion (LE) is the case K«C. The generating

functional Z[J3 beeoiaes an ordinary integral

and can be computed nuiaeriealy as well as all its derivatives.

It will be convenient,when discussing numerical results,to intro-

duce a diac-nsionless parameter f&Si • Scaling the variables

>:*/C ^ » 0" "X.v and z«A* Z gives

z(p= fcL*e~ a x" x +^ k (is)

First we will study the value of the connected generatingfunctional as a function of u? for given j. To first order in C?Ewe have

where x0 is a solution of the cuoic eouatios

the aiaiaal sensitivity of the value of w(,i) to the

222

parameter B = a we obtain1%

*_ (21)

?or j>»0 the resulting w(u) is plotted on j?ig.l compared with theone loop (LE), interpolated one loop (ILE) and the "true" resultobtained nuraericaly. For /£=0 bot\ LE as woXl as ILE diverge,contrary to our result (OPE), which is finite and does not differmuch from the numerical one. For jc>> 1 OPE approaches LE and theyare in good agreement with true result. For />•< 0 OPS has twobranches (0PS3 and OPZK), which correspond to the choice of thesyanetric (xo=C) or aos syauetric (x0 »-f) solution of ec.(2C).?or amail IJj the syaaetric solution is better and for -/o> 1 theunsymaetric one, v/hich approaches H E .

HOY/ we calculate the effective potential. To first order in01'E we ha 'e

V(tf> =

where

oo- — "* | L x, /"*"is found from the requirement of the aisiasl sensitivity of 7Cf)to the variations of &> , The results for different values ofare plotted on Fig.2 In comparison with, the classical potential

+ Wk (24)

and tru« effective potential. ?or f.^0 our result agree quitewell with nuae-ical one. For p > C (not sham on the fi-surss) thep ( )agreeoent aaelioxates as p increases. For ft< 0 the asreenentspoils as \i£\ increases, in 0P3 the second ainiaua appears, v/hichfor ft< fif 5 7 becones deeper then the smmtri one

2 X

for ft.< fiff -5.7 becones deeper then the symmetric one.-!fhe effective potential at the miniaua, according to eouations

(5)«(5) and (6) is equal to •• •• • and becosies v(0) in aero

5**

223

dimensional field theory. The v3lue of the effective potential

at the minimum for /<??/<•£. is equal to W Q - ^ C O ) and for ji-<f*a'

can be shown to be only slightly higher then Wyp^.CO). therefore,

the .value of w(0) obtained frou the OPE for the effective poten- .

tial is better compared to the value obtained from direct calcu-

lations of w(0) in OPS.

yield theory in N?l dimensions

The field theory for K=l becomes quantum mechanics. rhe anhar-

nonic oscylator (AO) and double well potential (DaiP) have been

extensively studied. It was proposed by Cashwell t53 t° write the

Haailtonian

as '

( 2 6 )

and to study the expansion in powers of C. For the grouna state

enery of AO (which is equal to V/cC}) the result to first order

gives the error less then 2,o. I?or D'.vT one has to S U E aore orders

if the wells are far apart. It was shown by Jtevanson ;6: that

GiiP is a generalization of Cashwell's first order result, incor-

porating a translation of position operator. Therefore, Stevenson

suggested that the systematic expansion can be found which gives

SXr as a first order result. Gaussian effective potential gives

the error less then lO.-o even for S'JF. Jhe qualitative features

of GEP are very similar to our results for 2ero diaensional case,

only the critical value of diaensicnless parameter which marks

the transition from single well to double well behavior in quan-

tum mechanics has a different numerical value.

For N>1 the necessity of renoranlization appears and has been

studied for the'Gaussia effective potential C*,?J. 2?he renoraaliaed

mass for one particle state has been shown to be equal to the

variational parameter SI*". The theory is shown to be trivial is. <

K-4 dimensions.

224

References

1. Y. yujxmoto, L. O'HaifearLaigr., G. Parevicini, IJuol. ihys. 3212

(1983) 26e.2. P.M.S. Stevenson, Phys.i-iev. D2? (1981) 2^16.

3. J.M. Cornwall, E. Jse'Kiw, E. ioiaboulis, ihys. ."iev. DIG (197*) 2428.4. T. Barnes, G.I. Ghandour, x'hys. iiev. 322 (193C) 924.5. a.î. Cashwell, Acs. of rhys. 123 (1979) 155-6. t .K.S. oteveason, Phys. Hev. D5C (196+) 1712.7. P.K.3. Stevenson, UW-Hadison preprint

Figure captions

Pig. 1: The value of the connected generating functional for j ^c ,as a function of j£? = . The solid line is the numericalresult, the dotted lines correspond to one loop and inter-polated one loop result , the dcshed line to the optimizedperturbation expansion Tor symmetric xQ and the dashcd-dotced line for unsyaasetric xo

Fig. 2: She effective potential as a function of the vacuum expec-tation vclue of the scalar f i e l i . The solid line is thenuaericsl result , tne arshed line corresponds to the opti-mized perturbstion exi'.c:". !or. arid •chs dotted line to theclp.ssicnl potentir.l.

a) for f «Cb) for M--2c)

225

FIG. 1

226

- . , .,:-^.~**'"

- .5

FIG. 2 a

227

?IG. 2 b

228

V f

I : -

FIG 2 c

229

HIDDEN IOC 4 1 STOMETSIES FROM gLATOUB ANOMALIES 0 ? QCD

J. Salog'and P. Vecsernyes

Central Research Institute for Physics

H-1525 Budapest <M4, P.G.B. 49. Hungary

Kaking use of the generating functional of the non-

-Abelian flavour anomalies of QCTD we construct a gauge

invariant pheaonenological Lagrangian of pseudoacalar

and vector mesons, which is equivalent to the extended

Wess-Zumino Lagrangian in the lov energy approximation.

The gauge kinetic term of the hidden local syoaetry is

necessarily present and the gauge coupling constant is

isterained by the equivalence.

<28tv8a University, Budapest , Hungary

230

In a recent paper [1] it was shown that JL% , the usualnon-linear tf -model Lagrangian based on the manifold G/Ewhere G = dUt (2)*SURl2>U(<1} and H = Uyfe) is equivalent toa iagrangian L. posaeeing GJUWI * Hu«»| symmetry. The gaugebosons of the hidden local symmetry Eloo((| .has been success-fully identified with the vector mesons 5 , tu . Ihe equiva-lence holds in the absence of a gauge kinetic term for thevector bosons. The main assumption of \Al Has that this termis somehow generated by the underlying dynamics of QJD.After adding this term to Le by hand, the vector bosons be-came dynamical but the exact equivalence between the twotheories has been lost. However, their equivalence was still,valid as the aeroth order approximation in a low energy ex-pansion.

In the present paper we will show that the emergence ofthe gauge kinetic terms ia the consequence of flavour anom-alies of QCD. Following the sethod of Wees and Zumino [2}but aaking use of both the usual, fiardeen, and the "spurious",non-topologic&l, anomalies the effective Lagrangian £4 hasbeen constructed {3,4). ?he full L-»grangian A.-*-/., describesthe interactions of pseudoscalar mesons in the presence ofexternal electroweak fields up to first order in the lowenergy approximation [5). «e shall construct the locallygauge invariant Lagrangian !>., corresponding to£,. L^ auto-matically contains the gauge kinetic terms and is almostuniquely determined by the equivalence.

?irst'we extend the quadratic Lagrangian Lc of Li] byintroducing in addition to the gauge vector field Y- an axia.lvector field A, as well:

231

Here

where St and Jfc are U(2) valued scalaro satisfying theconstraint det 1^= d e t ^ . "^ and Jy are matrix valuedflavour vector and axial external fields, respectively.Under the G,uk.i« H ) ^ symmetry our fields are transformed*as follows:

7 J. M . toil V rU)£<*y + Lfci^HLV, t*.,,J*!> , u (3)

The gauê is fixed by demanding

J..-U-J, (4)

uhere J caa T»« expressed in terns of the ohiral Goldstonebosons:

Using (4) > (5) and th» identification

of course the presence of V"r ami ^r breaks &jt.t^ . How-ever, part of it can be gauged, supplementing (3) by thecorresponding trans foraat ions on V^r and wf». . GJhough inthe applications we shall consider its UHJ subgroup only,corresponding to the Interactions of mesons and -photons,the general expression (2) proves to be us«ful in whatfollows.

232

the first term in (*) becomes the gauged <£ -model

Lagrangian:

where DÛ= V 1 + *r<A - U Xr, JLr-

The second and the third terss in H) are identically-

zero if we use for Vj. and Ap the classical equations of

notion:

Thus «£, and L. are equivalent, though the latter con-

tains tvo, so far arbitrary, constants «. and V . However,

when the gauge kinetic term

is added, the special value «. * 2 is preferred. As it vas

shown i» £13 this choice yields

C*> J*»ia * \ % universality [$],

W «*•*,«i»^ 4* KiJR?. relation [7],

(c) vector aeson dominance in y*» coupling.

How we turn to the construction of &,j. We recall that

the calculation of the effective Lagrari«rian i 1 proceeds

in two steps £3,8,9j. First one looks for local functionals

2w» CUJ..«*"/»J and Z^-Jv^", J'TJ whose chiral variations are

the BardeenC-to} and the non-topological [11] anomalies, re-

spectively. Then the gauged effective Lagrangian is ^ 1 =

233

where Vp. and J^r are the chiral transforms ofand J?f. with respect to f + = i it :

The 2 functionals are determined only up to chiral invari-ant terms, but -^wj and -|f|- do not depend on this ambi-guity, ie note that Z^y. and Z«t are vector gauge invari-ant. 2 ^ ia given by [3.11]:

Z W I is local only in five dimensions. In the language ofdifferential forms it la given by

lie look for Gjww x H i«^ invariant Lagrangians L M-and Lsn which are equivalent to J^tn and ^-»j up to firstorder in the low energy approximation. That is we have torequire

Apart from uninteresting selfcouplings of the externalfields (-14) is satisfied if we choose

since from (7) and (H) we have

-U =

234

Z\^, and Z denote chiral invariant contributions

to (42) and (13) > respectively. Ihey do not contribute to

JC ., but they are present in L,, in general. The most gen-

eral choice for them is

where 1j-t and Ep, are the field strength tensors of • *.

L r = Vp + A_ and fi_ = V.. - A.. .

Thus cur complete lagrangian

X = L. + L ,. + L^j MS)

contains the arbitrary constants o ,k and "K.. We choose

« = 2 as in [<1 to aaintain the property (9c). Moreover,

we choose X= 1 in order to cancel the strong momentum de-

pendence in the £** coupling thus maintaining properties

(9a) and (9b) as well. The parameter I can be determined

from the axial meson Ease.

With this choice we hava •

- LfrtW* •*-*- IPA)-A+ 2

The physical vector and axial fields are J* , A* , where

/ .*•„ *" • " <»,'',»•!*• <20)

r 2_ r

Here

These fields together with Jin diagonalize the quadratic

part of L.

235

From (19) we see that the vector and axial vector

kinetic teras are necessarily present in 14 3.3 a conse-

quence of the equivalence between I> and the extended

chiral Lagrangian <£j . Moreover, to.? values of the vector

and axial coupling csnstants are also fixed by the equiv-

alence. They can be read off froa (49) :

3 = ZJT %K = 2-<X ^ . 122)

From (4) and (22) we can calculate the $ aas3:

^ , . z J i a f n = »ltHtV . (23)•/e have also calculated 3one decay width using our effec-

tive Lagrangian L. Table I. contains the numerical results

of these simple calculations together with the corresponding

experimental values 2 3 .

Cur results are or-ly alightly different free those of

earlier calculations [43.44.451. The difference is sainly

due to our use of (22) while in other calculations ^ = %utx

was calculated froa the J~*JIJ* width.

Che rathe* good ap:ree~'?!it between our results and e?.-

pericent supports th? ii/pothesis that the rector mesons

5 , u> are indeed the effective jauge bosens of the

hiii.l=»i local syrjDetry of JZD. io far it is not incvn whether

this is a general rr.sr.o"«non, i. «. in all strongly inter-

acting theories if sc-.e global chiral symmetry 5 breaks

iovn to a ssaller or.-=, H with the eêrjer.ce of composite

G/H rassless Goidstor.a bo-sons t.ie.i a^yng t«-e otr.er com-

posite states thsre are alvsys present rassive vector

bosons correspondir.r tj :•>. hi-i-i'sn snd spor."s.-.=-ousIy broker.

.atoc-A gauge sysnetry.

Pinally we would likts 1:0 corner.: or. ar attempt; [14]

to directly identify the 5 ar.4 • A^ aeson fields with

the vector and axial fialis present in the jfauged *ess-

236

2u=ino Lagrangian £• = &- V*- »^f i **"p) • ^ e authors of ref.

£14] concentrated on the Lagrangian JL w% and intro-

duced the interactions between vector, axial and pseudo-

scalar fields by U3ing, instead of (A5), the lagrangian

V/hile it is evident that (24) and (.45) are eatirely dif-

ferent fur.ctionals, there can te effective vertices which

turn out to be the saae, This happens to be the oase with

the 77? vertex which is the nost important one in the phe-

noêr.oi.ogioal applications. It ia given by

L = ^ ^ V ^ ) ^ P 1 = ^TriW7)*p-J L25)i- ccth cases.

However, it has seen r.oted [15] that (24) is incon-

sistent i.T that in general it fails to reproduce the lov

er.er .' "neorers of current algebra incorporated in ^-«j .

Cn the cth=r hand, we constructed our larrangian (^5) so

as to reproduce «^v» (-^2^-w) when (?) and (-15), the

equation cf -otior. of the vector and axial fields in the

low er.erry apprcsi.-ation, a.rs used.

The aush-r? woull ii.-te to thani i. Hras-cd ar.d ...

::ar.-:aritis fsr ni-erour. ir.tsrftssi.--- discussions.

237

Decay

P(W-»JIV)

Calculatedwidth from li

\63 HeV

420 MeV

877 keV

_92 ke7

7.25 HeV

Ziperinent

1541 5 MeV

315 t 4-5 EeV

86-11 60 keV

6.901 C.27 KeV

?abie I. Ihe calculated decay widths of rector andaxial mesons and the corresponding experi-mental values

236

References

[1] M. Bando et al., Hiroshima preprint ev& 84-22 (4984).

[21 J. *ess and B. 2ucino, Phys. Lett. 373(497-1)95.

[5] J. 3alog, Phys. Lett. 14=3 (1984) 197.

[4] .E. 3ao, Gifu preprint G«J3-'1(19S4}.

[5] J. 3alog, EFKI-3udapest preprint EFKI-1985-O6.

[6] J.J. oakurai, Cuxrenta and Mesons (University of .-'hicago

Press, Chicago, 1959).

[7] K. rCawarabayashi and >:. inzuii, Phys. ?.ev. Lett. AS

(1966) 255;

itiazuddin and Fayasuddin, rhys. Rev. 147(1966)1071.

\B) S.K. Pak and r. Kossi, :;ucl. rnys. 3250(1985) 279.

{9] J.L. Petersen, .iiels Bohr preprint UBI-r3-34-25(198iJ.

[10] *.A. 3ardeen, Phys. Hev. 134(1969) 1848.

[11] A. Andrianov and L. 3onora, l<"ucl. rhys. 3233(1984) 232;

w-.-S. Hu, 3.-L. loung and 2.W. /IcKay, rhys. £ev. J30

(4984) 835;

A.P. Salaohandra/j et al., Pays. Aev. 325(1962) 2713.

Particle Data Group, Hev. ."od. Phys. 56 (1984) 34,

) ?.J. C'Sonnell, ?.ev. "oi. 5hys. 53(1961) 673.

[14] 3. Kayaakcalan, o. Xajeev and J. Ssheshter, Fnys. dev.

330 (19S4) 594.

[15] 3. Rudaz, rhys. Lett. 1453 (-1954) 291.

239

THE PRESENT STATUS OF THE BAG MODEL

- A CURSORY BEVIEW -

Jiirgen Baacke

Institut ftir Physik

Universitat Dortmund

4600 Dortmund 50

Federal Republic Germany

1. introduction

This lecture is supposed to be a review of the present status

of the MIT bag model.1"13 I would not think that such a task

could be accomplished by enumerating all miscellaneous recent

results in order to give a complete status report. Rather I

will include also all the well Known "old" reeults and try tc

give a critical estimation of. the relevance of the model for

present day physics. I will let myself be guided by the

following questions:

(i) Which are the essential features cf the bag model that

make it still an interesting field of research (beyond mere

phenomenology)7

(ii) What can we learn from the model about the consequences

of confinement?

(iii) What are the open problems (including mere phenomenology)?

In the following sections I will recall those topics that I

think to be essential and make some coraaents pertaining to the

questions mentioned above.

2A0

2. The Basic Postulates

The bag model is based on the following postulates

<i) A hadron is a cavity in the QCD vacuum with sharp bounda-

ries, filled with an appropriate number of valence par-

ticles. Inside the cavity QCD can be treated perturbatively.

(ii) Volume energy

Besides the energies contributed by the valence (and sea)

particles there is a volume energy

EVll » B V (2.D

where B is the bag constant and V the volume of the cavity.

(iii) Quark boundary condition

The quark fields obey the boundary condition

'. y( \j, « \f (2.2)

This condition insures that the vector currents of the

quarks and thexr energy momentum flow do not penetrate the

bag surface.

dv) Gluon boundary condition

The gluon fields satisfy

or, for a static cavity,

«•?«. * « I, « 0 (2.4)

As a consequence the total colour charge inside the bag must

vanish and there is no flow of colour or gluon energy through

the surface.

241

Some comments:

(i) The gluon boundary leads effectively'to an infrared cut-

off of (size of the bag)" 1. This indicates a possible

way of short-cutting infrared problems, not yet really

explored.

(ii) The boundary conditions modify the Feynman rules of QCD

by modifying the propagators. Explicit propagatorsr 2-4 T

are known for spherical cavities. The gluon propagatoris only defined if the total colour charge Q a = 0. This

entails that tree and 1-loop graphs cannot be uniquely

separated,

(iii) The quark boundary condition could be superfluous. Since

quarks carry colour and colour fields are not allowed to

penetrate the bag surface the gluon boundary condition

alone should be sufficient to colour quarks. For heavy

quarks the classical self energy leads to a mirror charge

potential.1 For light quarks the corresponding potential

is too weak.• J

(iv) The quark boundary condition violates chiral symmetry. Its

restoration by coupling to the pion field leads to chiral

bags, '. ] one of the highlights of bag theory.

(v) A mechanism that is able to explain confinement of quarks

by their gluon field (see (iii)) must at the same time

explain the mechanism of chiral symmetry breaking since

without the explicit quark boundary condition the theory

is symmetric for massless quarks. It would be interesting

to seek a connection to models of the QCD vacuum like

those of refs. [81.

(vi) While the bag model is not a theory of confinement it is

a theory with confinement. It is in my opinion a very use-

ful laboratory for studying tb» consequences of confinement.

242

3. The String Tension, Reqge Poles and the Relation to

Potential ModelsC5>9'10:l

When considering a static quark antiquark pair at sufficient

distance one finds ' ' that the bag becomes an elongated

flux tube of radius (8og/3«BI ' and an energy per length

(string tension)

So the bag model predicts a linear potential V = or and is re-

lated to the string model. Along with this model one predicts

Regge poles with a slope

^ (3.2)

Using a' - .88 GeV one finds the constraint

This constraint is satisfied by the model parameters obtained in

the fits to light ^ 1 1 1

and the ones obtained from fits to heavy guarkonia '

«, *. S "B'l/V * H O K t v (3.5)

Unfortunately both sett disagree, a point discussed further below.

Though the bag model predicts a string "potential" one should be

aware of the fact that this ia by no means a potential of the

same kind as the 1/r potential of QED. For QED (he total electric

energy of a pair of oppositely charged particles is

(3.6)

243

with the self energies

Ej - T~ $ » * ^ 5 ^ * V (3.7)

and the interaction energy

-"'*-* ' (3.8)

in QED most of the self energies is incorporated into the

kinetic quark energy a/Zi-B*. An explicit contribution arises

only to the extent that the charged particles are off shell.

(Lamb shift.)

In a colour confining theory the total field £ cannot be

written as

In the bag model th« boundary condition n2 = 0 cannot be ful-

filled by the colour electric field of a single quark or gluon.

In AJler's bag ^ the effective action is not quadratic in the

fields. As a consequence self and interaction energy cannot be

uniquely separated, as already realized in [11]. The self energy

can no longer be "essentially" absorbed into mass renormalization

even not for heavy (= almost on shell) quarks where it plays an

interesting rSie^14^.

As a consequence the field energy must be "boosted" if the quark

move. The factor (1-8') applied to the mass (incorporating

the self energy) must now be applied to all the elements of the

string (moving at different velocities). The factor 2* in

a' = 1/2TO arises from this fact. If the o • r potential is put

simply into an otherwise correct relativistic wave equation,

wrong Regge slops are found (as e.g. in [15]). This is a very

trivial statement, but usually ignored in potential models.

4. The bag model and the phase transition towards the quark

gluon plasma

The bag model has essentially the same asymptotic level density

as the Vensziano model and Hagedorn's Statistical Bootstrap.16

This was already realized in the first article on the model. -*

More .specifically one finds

where Qn is the n-quark phase space and d the (spin, color and

flavour) degeneracy. One finds that asymptotically

i... - * < 4 * 2 )

where

is the Hagedorn temperature, T » 160 MeV.

Taking d = 3 2 * 40 corresponding to 8 gluons with 2 helicities

and 2 * 3 quark flavours one finds,

which is the right order of magnitude.

Eg. 1,4. t) is based on a virial theorem1" that says

This means that very massive hadrons are also very extended. In

Hagedorn's approach the heavy hadrons are statistically compe-

titive due to the large number of their internal degrees of

freedom. If energy is added to a hadron gas a la Hagedort. the

heavy hadrons are produced abundantly and absorb the energy into

their mass. Therefore the temperature stay* low, always below T .

The fact that the heavy hadrons are extended in the bag model

245

reduces their phase space through a' covolume factor

(V- IV;^ «- (V- I $£) ' (4.5)

and T marks no longer an absolute limit, it rather becomes a

phase transition temperature ' T _ = 1.2 T Q towards a

quark gluon plasma.

Some remarks:

(i) The relation between the real transition temperature T t

and B may be grossly wrong. Zn the simple minded approach,

the pion mass is -800 HeV, because the spin spin splitting

has not been introduced. Now the x is certainly the most

prominent particle in hadron thermodynamics and it should

be treated as correctly as possible. Otherwise one cannot

trust the results quantitatively.

Cii) The model supports Landau's old hydrodynaaical model. -

Since in Landau's model the transition between an ultra-

relativistic "prematter" and the hadron gas explains

the exp (-p /TQ) spectrum observed with high accuracy in

high energy hadron collisions, the real phase transition

temperature should be at T -160 HeV.

(iii) In the model the violation of chiral symmetry is related

to the boundary condition. Above TQ the boundaries gc

away since the hadrons merge. Therefore restoration of

chiral symmetry is most naturally to be expected to occur

at this and the same temperature.

246

5. Light hadrons

The phenomenology of light hadrons was one of the first successes

of the model. Unfortunately this application of the bag nodel

is plagued with a number of problems which still wait for a satis-

factory solution. One should, however, keep in mind that the

fully relativistic bound state problem has not even in QED found

a satisfactory solution. Tackling the bound state problem of con-

find massless quarks is therefore a fierce enterprise.

The mass of a hadron is ob.tained by minimizing with respect to

the bag radius R the energy •*

Here »,, are the dimensionless eigenfrequencies of the occupied

modes, AE(R,ni. , a ) contains the colour electromagnetic inter-1/4

actions. The bag constant is found to be B ' * 145 HeV. a iss

determined essentially by the spin spin interactions, a value

as ~ 2 is required. The term -Z/R was introduced on phenomenolo-

gical grounds and was first supposed to arize from the Casimir

effect.

Now the difficulties:

(i) Translation invariance is obviously broken. Several solutions

have been attempted. The more formal approaches, ' 'F.231

starting from field theoretical bags have been developedr 221

very far by now but still problems in doing the MIT bag

limit. Practical approaches consist essentially in interpre-r 24i

ting the bag as a wave packet or bound state wave func-

tion1 and to subtract the additional energies contained

in the c m . motion of the bag as a whole. Such corrections

go into the right direction, even quantitatively and are

essential1"" also for other static parameters like the mag-

netic moments.

(ii) The -Z/R term could up to now not be explained by the Casimir

effect. Part of it can be attributed^243 to the cm. motion

247

correction which is of the same form. On the other hand

the self energies of the quarks calculated recently J

contribute with the opposite sign and constitute now a

major problem.

(iii) One loop contributions like Casimir energy and self energy

are not yet well understood theoretically, This will be

discussed below.

(itr) The parameters a and B disagree with those found for heavyC27]

quarkonia (see below). Barnes has attempted recently

to reduce the value of o by considering corrections to the

wave function induced by the Coulonb interaction. This in-

creases the effective overlap of the wave functions and

reduces therefore the value of a necessary to explain the

spin spin splitting to a « 1.

6. Heavy Quarkonia

Heavy Quarkonia present the advantage that the quarks become non-

relativistic, or at least accessible to a Breit-Fermi type appro-

ximation (v <. . 5c ). Two approaches have been shown to lead to

a reasonably successful phenomenology: the adiabatic approximation

of Hasenfratz et al. and a self-consistent approach by ourr 2sir 2si

group. The adiabatic approximation leads essentially to a poten-

tial model. Hasenfratz et al. did a detailed numerical calculation

in order to determine the shape of the bag that minimizes the sum

of volume energy V-B and the energy of the static gluon field de-

termined by the quarks as point sources and the boundary conditions

(depending on the skope to be determined). They found:

(i) For elongated bags the energy is proportional to the length

with the usual string tension o = /32nBa/3'.

(ii) For all quarkoniua states except the highly excited +'•'••

the shape of the bag is spherical; it is a good approximation

to use a spherical bag from the outset, in which case the

energy can be evaluated analytically.

248

(iii) The spin-spin forces can be calculated for spherical bags,

(iv) The phenomenological parameters are

to be contrastet with

for light hadrons.

Some comments:

(i) The discrepancy of the model parameters will be found again,

this time within one and the same class of states, for

heavy flavour hadrons; see below.

(ii) A similar approach has been taken, on a different theore-

tical basis by Adler ] who minimizes the action

(6.1)

with - M

in order to determine the force between heavy static quarks.

In the self consistent approach followed by our group one pro-

ceeds as follows

(i) Use the spherical approximation from the outset, fix the

center at the c m . of the quarks.

(ii; The radius is fixed, but used as variational parameter at

the end.

r 2 231

{iii) The knowr. ' static gluon Green's functions in the sphe-

rical cavity fix the nonrelativistic potential of the quarks;

this includes a confinement potential caused by the self

energy (see also [5,103). One calculates the quark wave

functions in this potential and obtains the 0 t h order quark

energy.

249

<iv) Use the Breit-Fermi approximation to calculate all v /c

corrections, SS, 2S and orbital (£ , P r ) . Again the self

energy is included.

(v) Add the volume energy and minimize the total energy byr i 2"1

varying the radius. The phenomenological results " are

quite satisfactory.*

The model parameters are

lie

fUV

Some comments:

(i) The agreement of the model parameters with those of Hasenfratz

et al. elucidates the quality of the too different approxi-

mations.

(ii) In contrast to potential models in the bag model all func-

tions describing the interactions are fixed, only two para-

meters can be adjusted (except for the short range 32 inter-

actions where we have used running a).

(iii) The v /c corrections are complete in the sense that they

include also orbital corrections usually dismissed in poten-

tial models. The confining part of the potential yields

straightforwardly an LS force of the correct sign.

(vi) The self energy plays an important rSle in this (and also

Hasenfrat2' et al.CS'1o:i) approach.

The fact that the T IP-states are somewhat too near to the 2S

states is due to the fact that no running a was used, a running

a rises the S states.

250

7. Hadrons with Heavy Flavour

Hadrons with on heavy and one or two light quarks have first been

considered in refs. [30 ]. For these bags the problems of trans-

lation invariance is minor. One can treat the heavy quark essen-

tially as static, fixed at the center of the bag. For the wave

functions of the light quarks one uses the solutions of the Oirac

equation with boundaries, either without (F-modee) or including

(C modes)L the Coulomb potential.

The wave function at the origin is larger for the C modes, there-

fore the spin spin interaction between heavy and light quarks be-

comes larger for C modes, the value of a required to fit the datar 32i

is therefore smaller « s « .8 for F nodes, oa » .4 for C modes ;

these values are sixilar to those found in the fits to heavy quar-

konia. However, the values of a required to explain the ob-

served X.. A. splitting using the light-light quark spin spin inter-

actions is four tines as large, i.e. equal to the "old" MIT fit

value a * 22 This is a problem that requires to be solved, several

ways out of the dilemma have been indicated in [32], but need to

be worked out in detail. Taking into account Coulomb corrections

to the wave

diff»renc«.

to the wave function seems not to be able to explain the whole

I. Chiral bags

Chiral bays have been developed in refs. [34-37] in order to

restore the chiral invariaace destroyed by the quark boundary con-

dition

which gets replaced by .» _,

\

and by introducing a pion field *(x).

251

The controversy about little bags with a pion field only

outside the bag and cloudy bag admitting pions also inside

it seems to be settled in favour of the latter ones. Ttie cloudy

bag model is theoretically on a more logical footing, especially

when derived as in ref. [37], and phenomenologically favoured.

The most attractive aspect of chiral bags is at present theirr 38 "i r 3 91connection to skyrmions especially the relation of the

baryon number of the sea quarks to the topological baryon

number current in the skyrmion model, a subject covered by

Zalewski's contribution to this conference.

9. 1-Loop Calculations in the JBag Model

The bag model was conceived as a simple phenomenological model,

cutting short the difficulties of a basic theory by simple postu-

lates. It seems therefore somewhat far fetched to calculate 1-loop-

corrections in such a model. For several reasons this is, however,

not so, 1 loop calculations are unavoidable and/or of phenomeno-

logical relevance.

Consider at first the parton distribution. It can be derived from

the graphs

where ^^* represents the quark propagator in the bag. The second

graph is a one loop graph, it is the sea parton distribution,r 41 1which was found to be divergent.1" This is even the case if

represents the guark propagator in a smooth, nonconfining poten-r 42i

tial. The problem still lacks a cure! We have here a 1-loop-

effect which is experimentally accessible, and divergent in the

model and also divergent if milder versions of confinement are

considered. Therefore it is of relevance beyond the model, the

HIT bag model has the advantage that one can do real explicit

252

calculations which facilitate the analysis. This is also the

case of other 1-loop effects. The Casimir energy was introduced

as a phenomenological parameter in the model fit to light ha-

'-11 •'as a -Z/R contribution. A correct analysis shows that

the effect has quadratically, linearly and log divergent contri-

butions. It is yet an unsolved problem how to cure this d i -

sease. Closely related is the calculation of the baryon number

in the chiral bag.1-40-1 This is definitely a 1-loop effect that

is calculated along the same lines as the Casimir energy. While

the baryon number turns out to be what one expects it to be

M6) * ;jjr (6- 4- is*. 2©>

in order to match with the baryon number of the chiral hedgehog

(skyrmion) outside the bag.

L 44"!The 6 dependent Casireir energy is again divergent:

+ f i n i t e p a r t

So if one is pleased with N( 9) one should think about Ec(e) as

well:

The self energy of confined quarks was evaluated recently ,

this was certainly the most involved calculation that has been

done in the model. It was hoped to be negative and found to be

large and positive fbut at least finite!). A common feature

of the one loop effects is that they come out too large or even

infinite. This would be improved by a cutoff in-the loop momen-

tum. Physically this i* plausible since the bag model is anC 45 ~*

effective theory. A parallel situation occurs in dielectrics

where the C«simir energy is likewise divergent, but where the

divergent integrals are cut off by the fact that the dielectric

constant goes to unity if the photon frequency and/or momentum

becoae large. In QCD we know similarly that gluons and quarks

become free if they are far off shell. Zt remains to find a

convincing way to introduce such a cutoff.

253

10. Conclusion

All the technical conclusions have already been presented at

the appropriate place. My general conclusion is that while the

bag model is not in a perfect or even good shape especially

concerning the theoretical aspects it is in an "interesting

shape". I think that all the problems I have evoked will lead

to deeper insights on the way towards their solution. I think

therefore that the model will be also in the future an inte-

resting theoretical tool for gaining an understanding of the

world of hadrons.

It remains to apologize for having omitted to mention a lot cf

further interesting aspects and substantial contributions to

the subject.

References

[ 1] A Chodos, et al., Phys. Rev. D9, 3471 H974);A.Chodoe, et al., Phys. Rev. 5Tc, 2599 9 4

C 21 T.D. Lee, Phys. Rev. D1_9, 1802 (1979).

[ 33 T.H. Hansson and R.L. Jaffe, Phys. Rev. D28, 882 (1983).

[ 43 J. Baacke, y. Igarashi and G. Kasperidus, Z. Phys. C17,161 (1983).

[ 53 P. Hasenfratz and J. Kuti, Phys. Sep. 4OC, 75 (1978).

C 63 A. Chodos and C.B. Thorn, Phys. Rev. D1_2, 2733 (1975);T. Inoue and T. Maskawa, Progr. Theor. Phys. 54, 1833G.E. Brown and M. Rho, Phys. Lett. £2B, 177 (T579).

C 73 A. Szymacha and S. Tatur, 2. PhyE. C7, 311 (1981);A.W. Thomas, J. .Phys. C7, L283 (1981).

C 83 A.Le Yaouanc et al., Phys. Lett. 134B, 249 (1984);S. Adlex and A.C. Davis, Sucl. Phys. B244, 469 (1984).The cancellation of the infrared singularity in this paperis exactly analoguous to the cancellation of the infraredpole of the gluon propagator in the bag.12,263

C 93 K. Johnson and C.B. Thorn, Phys. Rev'. D1_2, 2733 (1975).

£10] .F. Gnadig, P. Hasenfratz, J. Kuti and A.S. Szalay, Phys.Lett. 64B, 62 (1976);P. Hasenrratz, R.R. Horgan, u. Kuti and J.M. Richard,Phys. Lett. 95B, 299 (1980).

25A

[11j T. De Grand et al., Phys. Rev. D12, 2060 (1975);for more recent global fits with various corrections see e.g.C.E. Carlson, T.H. Hansson and C. Peterson, Phys. Rev. D27,1556 (1983);7. Radozycki and S. Tatur, Univ. of Warsaw preprint, March 1984.

C123 J. Baacke, Y. Igarashi and G. Kasperidus, Z. Phys. C1_3, 131(1962).

[13] S. Adler, Phys. Lett. 110B, 302 (1982).;S. Adler andT. Piran, ibid., 113B, 405 (1982).

[14] J. Baacke, Y. Igarashi and G. Kasperidus, Z. Phys.C9,203 (1 981)

[15] D.A. Geffen and H. Suura, Phys. Rev. D16, 3305 (1977) finda1 « 1/8o. The reader may work out this easily for the Diraceguation (1 static + 1 ultrarelativistic quark) with theresult o' = 1/4o.

[16] R. Hagedorn, Suppl. Nuovo C m . 3_' 147 (1965);see R. Hagedorn, CERN preprint CERT-TH.3S18/84 for a recentreview.

C173 J. Baacke, Acta Phys. Polon. B8, 625 (1977); eq. (2.6) of thispaper is incorrect, the correct result can be got by summingeq. (2.5) analytically. Only eq. (2.5) was used subsequently.

[18J N. Cabibbo and G. Parisi, Phys. Lett. S£B, 67 (1975).

C191 L.D. Landau Xn Collected papers of L.D. Landau, Gordon andSreach, N.Y. 1968 and original refs. therein.

['203 J. Baacke, Z. Phys. C2, 63 (1979).

[213 M. Betz and R. Goldflair., Phys. Rev. D2£, 2848 (1983).

[22] H.R. Fiebig and E. Hadjimichael, Phys. Rev. D30, 181, 195 (1984).

[23] R. Friedberg *nd T.D. Lee, Phys. Rev. D15, 1694 (1977),PI 6, 1O96 (1977).

[24] J.F. Donoghue and K. John»on, Phys. Rev. D2J, 1975 (1980).

[253 A. Siymada, Phys. Lett. U6B, 350 (1984);A. Szymada, Phys. Lett. 146B, 350 (1984).

[26] J. Baacke et al., Z. Phys. C21, 127 (1983);S. Goldhaber, T.H. Hansson and R.L. Jaffe, Phys. Lett. 131B,445 (1983).

[27] T. Beurnes, Phys. Rev. D30, 1961 (1984).

[28] J. Baacke and G. Kasperidus, Z. Phys. C5, 259 (1980);J. Baacke, Y. Igarashi and G. Xasperidus, Z. Phys. C&, 257(1981), ibid C9, 203 (1981).

[29] J. B*acker Y. igarasbi and G. Kasperidus, Z- Phys. C U , 131 (1982).

[30] E.V. Shurgek, Phys. Lett. 93B, 134 (1980);D. Izatt, C. De Tar and M. Stephenson, Nucl. Phys. B199, 269(1982).

[31] W. Wilcox, O.V. Maxwell and K.A. Milton, Phys. Rev. DH. Hoge-asen, J.M. Richard and P. Sorba, Phys. Lett. 119B,272 (1982).

255

[32] A.T.M. Aerts, T.H. Hansson and J. Wroldsen, CERN-TH.4068/84

C33] for a recent review seeA.W. Thomas in Progr. in Part, and Nucl. Phys. VI, 325 (1984).

C34] A. Chodos and C.B. Thorn, Phys. Rev. D12, 2733 (1975);

T. Inoue and T. Haskawa, Prog. Theor.TEys. .54, 1833 (1975);

C353 G-E. Brown and H. Rho, Phys. Lett. 82B, 177 (1979).

[36] A.W. Thomas, J. Phys. G, L283 (1981).

[37] A. Szymacha anfi S. Tatur, Z. Phys. C7, 311 (1981).[38] M. Rho, A.S. Goldhaber and 6.E. Brown, Phys. Rev, Lett. 51,

747 (1983);A.D. Jackson and M. Rho, Phys. Rev. Lett. 5J, 751 (1983).

C39] T.H.R. Skyrum, Proe. Roy. Soc. London Ser. A 260, 127 (1961);Nucl. Phye. 31, 556 (1962);E. Witten, Nucl. Phys. B223, 422 (1983).

[403 J. Goldstone and R.L. Jaffe, Phys. Rev. Lett. 51_, 1518 (1983).

[41] J.S. Bell, A.C. Davis and J. RaJelski, Phys. Lett. 78B, 67(1978).

[42] J. Baacke, Phya. Lett. 83B, 103 (1979).

[43] C M . Bender and P. Hays, Phys. Rev. D14, 2622 (1976);P. Candelas, Ann. Phys. (K.Y.) 143, ITT 0 982);J. Baacke and G. Kasperidus, Z.~PEys. C2«, 133 (1985).

[443 L- Vepstas, A.D. Jackson and A.S. Goldhaber, Phys. Lett.B140. 280 (1984);I Zahed, U.G. Meissner and A. Wirzba, Phys. Lett. B145, 117(1984);P.J. Mulders, Phys. Rev. £30, 1073 <1984).

[45] see e.g. p. Candelas, Vacuun Energy in the Ba j Model,University of Texas (Austin) preprint 1982 for a discussion.

257

3ASXQX aHAaGBS OF O J I I U J . SAGS

I. ZtiLmakl, Iwtttut* at JfeciMr Ihpiioa, O M O O W , Folawt

In a popular version of the cftiral bag model the quark wave

functions satisfy the free particle Dirae equation inside a

sphere. Let us denote the radius of this sphere by H. The bound-

ary condition is

f 5 r = R . (l)

Here n is the unit vector along the external normal to the sphere.

The components of vector <£ are Paul! aatrieeB acting on the

isospin indices of \f, whicii is assumed to belong to an isospln

doublet. Parameter © is a constant known as the chiral angle.

Outsi'ie t'ns bag there is a pionio cloud. This niodsZ has oeen

proposed long age 111, but only much more recently i t has been

r-eslized j 2\ that the boundary condition (l) inplies that the

bag hes non^ssi-o bary.oiiic cliarge (usually i'i»actional) even, when

the bag is':sBpty" i.e. ioea not contain valence qusrlta. The

reason is that condition (l) violates chargs conjugation ayE-

metry and oonsaquently the energy spectrum is not invariant

under change of sign of energy E —*• -S.

In order to see how the asymmetry in the energy spsctrun

tjer.arates a baryoiiic number consider the spectrum of single

farnion energies

?• = r- + e n = +1, +2, ... , (2)

where € is a constant. Let. us assume for siaplicity that each

fermion has baryon number one. Ths bsryonic number of the vecuuai

is defined as 131,

258

3 =-| lim 2f(B,t) . ' (5)** t -««"0

Here tha aumsiBtion extends over all the energy eigenvalues.

function f(B,t) tends to B/| 5| for t •*+o and is such that for

t> 0 series (3) converges. Kote that an atteapt to perform term

by term the limiting transition t -• +o would leed to a meaning-

less expression. It is believed that f»r "reasonable" functions

f tne liait (3) decs not depend on f. Let us choose

E, t) = ex?(-JE| t)signE . (4)

Then for speotrus (2) series {'c) deduces to a geometrical pro-

gression an;.

I t i s irnpSrtarit thet for anj' fi::ea '.'. > 0 including in the sue

(3) only the terns v;ith | u | > i ; CTIC-. =2ain obtains (n). Thus this

result iepends only en ttie energy shi r t s of the -ner^y levels

inj»l.

liote that for f = ej.p(-j»-e/aj t)si^nS one would obtain 3 = 0.

This function f is not considered "rearons'bie", hov/ever,"oecause

i t is not charge conj'igation {S •* -?) odd.

A similar cslculation fields for the Jirac eijuaticn -•ith

isassless quarks and bounfiary condition ( l ) : £• = 8/ ,TI in the one

difflensional case (-H<x<R, spin projection sv = - A / 8 ) and

for the three-dimensional case |3—oj

3 = i-(e - sinecose) for |«j<rv'2 . (6)(

259

Result (o) is very nice, because assuming that the pion cloud

outside the bag is a Skyrme soliton |dj truncated by removing

the part of it, where r< a, one rinds that the baryon numbers

of the pion cloud (sic!) exactly cancels ths baryor. number <5)

When the fercion mass m / 0, tvro changes occur;

i. The levels corresponding to |n|*>l shift a little.

This, however, does not affect the result (o) J4,7).

ii. A level with E = 0 may occur. This level can contribute

+1/2 or -l/2 to the baryonic charge B | 8-10J.

Having solved the Dirac equation with boundary condition

(l) one may try to- compute the density of the baryonic charge

inside the oa% from the formula

B(x) = - | lin Zi|f*(x)#-(x)f(Sft). (.7)2 t-» +0 E E .

Por the one-dimensional oes the calculation can be perl'oraed in

the following three steps.

i. One extracts the.part af the serie* divergent for f=S/|Sj.

The summation of this part is identical with the caculation of

the integrated baryonio charge ana yields qf s. To? H = lA-

ii. In the remaining convergent series it is legitimate to

put f(K,t) = sign(E), but- the series converges so slowly that

it would be necessary to SUE thousands of terms to get a reason-

able accuracy. Therefore,one extracts the slowly convergent

(0(l/n)) series and sums it analytically. The resu.lt ie

Bas(x) = - |sinKxsin(aex)ln)3sin(2Kx)j |x|<i/S (6)

It is seen that this result is logarithmically divergent for

|xl —»l/a. This partly explains, why the numerical summation

2'60

is so inefficient.

iii. The remaining rapidly convergent series is summed

numerically. The result obtained for e = ,^/4, m = -I, S = 1/2

is shown in Fig. 1.

The corresponding calculation for three dimensions is much

harder. >Vork to find the mean square baryonic charge radius of

the bpg including the contributions from the valence q.uarks,

the soliton and the vacuum in the bag is, however, in progress.

It seems interesting to study how the baryonic charge of

the fermionic vacuum ai'fscte the phenomenology cf the chiral

bag model. At first sight the infinite spikes iii the density-

look bad, but since they are integrable, it is net clear whether

they lead to an.' real difficulties.

References

\l\ A. Ohodos ani C.3. Thorn, ?hj'8.j.ev, Jlg(l9V5)sJ733.

J Sj M. Hho, A.S. Ooldhsber an: 3.E. 3rown, Pnys.Sev. Letters

13| J. Goldstone and A.L. Jaf .e, Piiys.icev. Letters, 5i(l9SS)lal6

| 4 | T. Jaroszewica, ?:-jys. Letters 31-45(1984)317.

|o | M. Jezabek ani K. 3ale-.vSKi, 2.Phys. 0^511984)365.

|6 | T.H.R. Skyrme, Proa.P.oy.Soc., AaoO(l£-dl)ia7.

17| K. Jezab'eic and K. 3slewsi:i, Phj'S. Letters, 514e(lSS4)354.

J8| R. Jackiw and C. Sebbi, Phys. Rev., 2i3(l97o)3298.

|9| J.S. Bell &vz H. Hajaraman, Kucl. Pftys. 3aao[F38J (1985)l.

JlO| H. Jaciciw et al. iJueLPhys. BaSofFS9] (lg35")a-35.

(ll| K. Heller, JI. Jesabek ani K. Nowak private communication.

261

Figure caption

3aryoiiic charge distribution in an empty one-dimensionsll chiril

bog for « = V4i ai = 1 »nd S = 1/3.

263

METRIC BAG MODEL OF HACRQKS

M.Romaniuk

Institute of Physics *

Warsaw University Division, Bialystofc, Poland

Aba tract.

The new model of hadrons. based on bag approach and "strong gra-

vity" philosophy is given. We have calculated is this nodel

(called h«re the Metric bag model) sane properties of light hadrons.

The agreement with experimental values Is excellent.

*Postal address: 15-424 BialystoX, 41 Lipowa Street, PolaiwS

1. Introgaction

In this short presentation we would like to report some of the

results in the new baglike model of hadrons.

It is well known that the QCD suggests that the nonlinear

color quantum effects inside hadron give confinement of color

charges. But at present no proof exists. In this paper we

suggest that these quantum effects are equivalent to an effective

classical metric. Several authors indicated'the

of 6es.crii>ing hadrons as strongly curved universes:

especially felack holes and neg-atively curved anti - de Sitter space

"to provide feag-like structure fi].

ceietal*tipp«, h0veve<r, givs only ijualitajtive description of

fi 4e*ail«3 Synaijiical theory of "infiuped strops gravity" does not

»:*;i«-t g'p *(£ bujjlt * .wo.del 1» which the tsechanism of confinement is

lp a ssif-c'COi^xiftettt- tnamMir.

is that lngi<S« a hôron «xiets the metric tensor

field ^ ^ tfo«r*iEti'nfl with the usual Minkowski metric n^ .

•*&& fc$ p*8'ht to b* proaucftd toy sc»e source. In our case this

is th* tati-e ka& sfherically symp«tric -classical dust.

i* described by Einsfce,l.n-!iibert action,

it* result is well known

The above wefcric describee the static Einstein »icro-universe.

I»s±4« • h«dr6u ti»er* are two metric The total behaviour of quarks

lan£ gluons) i* datersiined iy "9/**" (Einstein metric) and total

b»haviour of leptons and the foton is fletemiined by "r^j* (Minkowski

**tric).

265

2. The model

The hadrcr. (as extended system) can be fully described by action

t — a •*• it ' ^

•quarks vacuua •

Avacuum ^ e s c r i t > e s t*ie extra Poincaxe term and is given by Hilbert-

Einstein "strong gravity" classical forwula

A ? uV - £ pi (R-ZA) (2,

(the metric is given only by dust).

The first tens describe* the valence quarks which wove in the

effective Einstein Metric

The wave function U-'(x) of a "free" quark satisfies • generalise!

Dirac erpjatioa

where D , = i^ - f^ - the cow*riant <5eriyatives a»d

connections expressed

a •> t J - (-5)

t-uare position - dependent generalized! Dirac matricesit

where the tetrads are given by:

266

L«t us now choose the standard representation for âand Penrose

representation for J""f2J.

Then the positive frequency solution of (4) for the ground state

is 9iven by: . . \

>

* = cos "$ , x « COB § , E •• — , XQ * 2.5.It

Her* M, A - normali«ati-4>n constants.

, /; +f3jT)i* «bov* sclwtien may be transforssed under « similarity transfor-

tefejtioTt to * stanflard rftpreeentjition, but physically meaningful

(jUJuntitl*s So not !*»pend of our choice.

Here we only Roto th*t tine qu-nntizstion «ay be given by a standard

for the f lat *pac«. . .

j. T|>c «l<8K;tfowf«>: ptrawcUri of a nucleon

X foeS t#»t for the gaafk wave function might be the calculation

of the static parameters of a nucXepn.

the generalized formulas for the static parameters are as follows:

- mean square charge radius

• 2*7

- axial - vector coupling constant

H - < N \ (Sî *' * ' V+*- magnetic nooent

The wave function of a three guarV state JH> has the explicit

SU(6) symmetry. The numerical value* in the lowest quark eigennode

are given by (see Table III):

<r 2 > V 2 = 0.75 < K > ,

9i " T

*. The spectrna of the ma«sg» fpg th^ grgnnil yt -be

The total •nergy of the b«9 witb « radivje K is equal to tb*

where

8 , 3 . ' . •

E « T-fatL + B . * 2 1 " 2 / K the total kinetic enftr» (1«i

*• ^ *•** zero ppint energy (i e «aqpt the resultE0 "~i* given i^ Ford f3l)., •

= ff* - = S £ "vscum" energy of the <Just

He note at this poitrt. th«t in the present work we also tatoe

into jtcoount the pertuibative one gluon exchange correction*.

n 8 )

268

. is a calculable function in the color magnetoetatic

approximation.

Th* mass computation, using the normal bag approach, was made in

three steps:

A. First, by ainiaiziug the formula (15) we obtain the bag radius

H L * °- (19)

S.-Second, by eliminating the 2MET from the above equations by

putii>g ZXEt = gr* we simplify their structure.

tiiniwiting aiKS metric cortsctions the bag energy becomes

es »re iedSpendftfit $o we ou^jht to give the

JO**t .<w*-t«e ica», 5he«» is no problew with th« c. ».

bec*u«* ttoe v*cuun part of mass cancels itself.

* ^t.Jjfti'viîc Jhcsd«l tjie »ore appropriate formula reads f 4 J i

o

flttiny pr^c«ûr« i* basad on minlinizatioB of a function

tp conp*riE aiffer**»t fit* »e introûpe quantity

««fin«<S n J4] •

269

The numerical results of our model are given in Table 1 (for

baryons) and Table II (for baryons and Hiesons) .

Table I

Results of the best fit of the metric bag model with c. HU

correction for the ground state baryons

Hadron B a g r a d i u s Fitted mass Exp. mass Difference

[Mevl+ 2+ 3- 4- 4- 4+ 2+ 5

0V,

[Gevj5.9035.7695.7055.5167.0086.7946.3796.365

• [c-ev]0.9371.1131.11971.3221.2371.3.B31.52*1.672

Q.9391.1161.t931.3181.2321 .3851.5331.672

Values of the parameters arc:

• = a.. = 6 MeV, is ' 333

oM * 4

Table II

a^ - 8 .37 , <T= 0.0*87.

Results of the best f i t of the ftetric bag *o&&l With e .

correction and 'with two \racuu» parameters C t l o r mesone

baryons) for this grounfl s ta te hadrcns.

Bag radiusoev

5*72-55.5566.8626.6726.4&06.2917.4S97.1267.<?96.7855.8225.4S0

Fitted massGev

1^19*1 ..3t<1.2381.3*51.50*1 ,«3S6.7*60 . &9$0.75*1.04'20.5250.125

1.31*

1.345S•5 .533

1.0200.4960.146

Mev

- 3-4- 1 *• 5

3712

27- 22- 25

15

Values of the parameters are:

mu = md = 0, » s = 315 UeV, oCs » 0.34^

Stt * 20

= 0.101,

270

Tafele III

Electroweak parameters of the proton

^T > 9R /A (e/2M)

model (case X) 0.8? fin 1.13 2.34model (case ID G.M fro 1.11 2.87

experiment 0.64 £M 1.25 2.79

B&5 radii and quark masses a& in Table I {case I) and Table II

(case II) .

. Conc-lusior.

I t was fownd that the above version o-f the metric bag model

pfoVide-s the dteplaJjation fOl tlie spfectrwi *nd s t a t i c parameters Of

Che iigfrt' ftadron*.. If «s* e«y ô give a real is t ic , interpretat ion to

t»c<inc»3^ Itsrft i*« *#• t ^ t the effective watric se ts l ike soiae

t *

, 4596 (1.978)

111 (1979)

«OC'. London .A33j>, 517 (1974.)

. Rev. ^1_,r975 (1930)

., llucl. Vfeys. M24., *84 < 198-49

6.' ifatsiiheoK, ?}^-*. Rev. ©6, 45*7 (1973)

271

REGULAR LATTICES AS SADDLE -POINTS OF A TWO

RANDOM LATTICE THSOR?

D. Perfcermann

Sektion Fhysik, Karl-Marx-UniversitSt Leipzig, DDE

Abstract

A two dimensional random lattice theory with a free

massless scalar field theory is considered. We ar>alyse the

field theoretic generating functional for any given choice

of positions of the lattice sites. Asking fo-r sadaie-points

of this generating functional with respect to the positions

we find the hexagonal lattice and the triangulated version

of the hypercubic lattice as candidates.

1 - In t roduct ion

The random lattice theories were first introduced by Christ,

Eriedberg and Lee (1). In a word, the random lattice is e

discrete set of N sites in a space-tiaie with a volume £. The

co-ordinates of these sir.es are randomly choser. in Q . The

lattice is characterized by a fundamental length scale L,

where q is the density cf sires and D the dimension of space-

time. This length scale is a generalization of the lattice

spacing in regular lattices.

The average over an ensemble of lattices yields the restora-

tion of rotation iLorentz- ) and translation symmetries for

Lf*C. The continuum l:..r.it corresponds to

272

n -»oo or L -» 0. . • O.2)

The random lattices are constructed by a decomposition of

in D-siraplices. To each realization of a randoa lattice belongs

a dual lattice based pn a Voronoi-construction (1).

The generating functional is given by

n , d D r i zL< ri' - v • • <i-3>

Here, the integration is extended over all possible realizations

of D-dimensicnal co-ordinates x: • of the lattice sites. The

2T lr.,J.) is the generating functional of an Euclidian (or

Hinkovski! field theory in a fixed realization of site co-

ordinates r^ an<5 with external currents J. .

k motivation for the introduction of random lattices was the

conception e-f a discrete approximatisc o€ the flat Euclidian

(or Mirôwski) ^paoe-tirae including the restoration of rota-

tion (Lorents) and translation symmetrie-s. Fri-edberg and

ften (2) ir4 fven (3) have presented computer siai-ulations of U (1) ,

SI.M2}, sna S'J(3) gauge theories on random lattice-s. The average

action anc the specific neat on the random lattice are much

closer tp the expected continuum results than On a regular lattice.

But there is BO drastic change with respect tc the phase struc-

ture between random ana regular lattices.

If w«s suggested by T.E.Le« (4J to introduce the random

lattices, as theories of s-pace-titne it_salf. The random lattice

describes « discrete space-time Including gravity. The co-

ordinates of lattice sites are dynamical variables as the

fields on the sites., links etc. Fo"13.owing Cohen (5) there are

two possible interpretations:

(i) In L«e's version based on the Hegge calculus (6) the cur-

273

vat are resides in (D-2) dimensional "tones." oi the lattice-

di) In Cohen's interpretation (5) the space-time i£ identified

with the co-ordinates of lattice sites. The gravitation

corresponds to derivations from regular lattices. The con-

tinuun' limit (1<—» C) yields no gravitation.

It was shown by Cohen (5) that in D=l dimensions the regular

lattice is a saddle-point of Z^lr^'J^' i n e<3- (1.3) foxr a free

massless scalar- field theory. It is mentioned that a non-zero

mass does not change the result. For dimensions D>1 the situa-

tion should be similar. The possible candidates are Bravais

lattices with at least a D-point symmetry.

Our aisne is to shown, what happens ir> D=2 dimensions

discussing a free massles scalar field theory.

2. The frge scalar field on a flat random lattice

We chose a scalar field theory, because the integrations with

respect to the fields in zL<r i«J i) are Gaussian. This allows

simple analytical estimations. If we are interested on effects

of the fields themselves on the lattice shape, we may restrict

us to the case J^=.O. For simplicity »*e di&cuss a flat Euclidie*

space. The generating functional ifi

The sum in the exponent of eq.<2.t/ runs over all links (ij)

of the lattice. The ^ . are the link coupling*. They are noa-

vanishing only if i and j are ôint by a link. Tc specify ehes«

link couplings we use the roost natural choice

'•v _ °»: "volume" of (£>-p dim.dual of (ij; ,_ ..A±-i ~ ~~~~ ~~~~—————————*————' , \A.Z\

Li; length of (ij)

274

For the properties of this most natural choice of link coup-

lings see ref. (1,4) .

3. Saddle-points ir. two dimensions

Our aijne is to performe a saddle—point -estimation of the

co-ordinate integral in eg.(1.3). We ask for such lattice

configurations concerning the positions of lattice sites which

extirereiae the field theoretic generating functional Z_(r.,J.)

for a free ntassless scalar fiel-d theory in two dimensions.

A.Local extremal configurations

The configurations of lattice sites in the neighbourhood of

• single site- i vtfiicii extremize Z ir.,J.) we call local ex-

tremal configurations. The decimation o-f the site i is defined

as the integration ©f Che site i with respect to the field

i*

/ n dî

Using »<g. X2.1) the resulting l j«

The expression within the es^ponential term is the effective

action after the decimatios

couplings "X are defined by

.action after the decimatlos of the site i . The effective link

lUi, . (3.2)

Dn two dimensions, they have a jr.uch raore complicated structure

ther. in the one-dimensional case in ref. <5) . Due to the second

term on the right-hand side of eq.(3.2) the appearance of a

link crossing in the effective lattice is possible.

275

In connection with the local extremal configurations we dis-

cuss the factor ^

j toio

in eq. (3.1) concerning the co-ordinates of xQ. This factor

contains the sum of couplings A .. of links joining i Q with its

neighbours j. The investigation of the necessary conditions

for an extremuni of the factor yields regular polygons (Pig.1)

and rectangular configurations (Fig.2) as solutions. In the

case of the rectangular configurations there is a freedom in the

choice of triangulation (decomposition in 2-simplices). But

the necessary conditions for th« extrensum are satisfied by the

4-neighbours and 8-neighbours configurations as well as the

6-neighbours configuration in version A only (set Fig.2).

Under the polygoraial configurations we find the hexagon which

is constructed by regular triangles as basic geometrical objects.

B. Global extremal configurations

Now, we are interested on such realizations of th« total set

of co-ordinates which extremize the field theoretic <fener*t.i.nej

functional Z^lr^/J^ with respect to all lattice sitas. Inte-

grating the fields ty, on the lattice site* i and chasing

for one site i^ to avoid a divergence of Z., we obtain

7^VThfc Aî' a r e t h e effective link couplings after the decimation

of the lattice sites i. , I^, , i;. .

There is the suggestion, that a lattice constructed by local

extremal configurations should be a global extrenal configuration.

Using the hexagon as basic element one obtains the hexagonal

lattice (Fig.31. Very sinilar, the 6-neighbours hypercubic lat-

tice shown ir. Fig.4A is constructed by the 6-neighbours rectan-

gles in version A (Fig.2). A combination of the 4-neighbours

ari<5 (it-neighbours rectangles yieldfc the &/4-neighbours hypercubic

lattice presented in Fig.4B. Due to the triangulation this type

of bypereubic lattice tFig.4A,B) is not identical with the usual

one. fhe elsmetstar structure of the usual twp dimensional hyper-

eu-btc lattice with îfferen-t lattice spacing is the rscta.ncle.

Kowever, bfjnierning our v&rsion this rectangle is not eiementar

frut consists of two elemental triangles. If a,Xl fluctuations

with r-eagpe-et te the Cc-orîn^tee are frozen^ the fiiagronal link

^^1s<s. v*Riâ.. ?&£ this Case tfra effective linking

is tfe* 6*J6e ac in toe visual fcyperQiabic lattice.

Tlie a*ialye4.» &f the iiecess.arj' coftditic-ns of **si extreiaum o-f SL

iaclun£in§ tfet isynswf-tfie* o^ the. lattices yields the hexagonei

«ell %e the twp v-ersioSt <>s triapguletefi hypercabic

seS Above ae solu-tloR*.

, the fca«"dle-points c<»vsiiSerea *re aot the only pos-

s i b i l i t i e s . Furthermore, we have to ask Sot the s t a b i l i t y .

We s t r e s s , ttent s maxirra.1 z cGrr*spc>nd« to a set of minimalIs

fc couplings f&i-i' iFox the isost ratur-al {Shoice of th« A's

tft *q. (2.2) th i s o«n be actii-ev*d for r*£ul«r l a t t i c e s ,

Where these \ s are of tire order 0(1) or z&ro. in contradic-

tion to th«t , there i s always the poss ibi l i ty to have very small

link lengths 1 ^ «n<3 l*rg« dual link lengths (5"^ in a pure

277

random configuration. These arguments may confirm our expectation,

that the configurations considered maximize Z . But a definite

answer to this question can be obtained from a detailed' investiga-

tion of the stability only.

References

(1) N.H.Christ, R.Friedberg, and T.D.Lee, Mucl.Phys.B202(1982) ,89

Hucl.Pbys.B£i£,FS€,(1982),31 0,337

(2) R.Friedberg and H.C.Ren, Nucl.Phys.B235,FS11,(1984),310

(3) H.C.Ren, Nucl.Phys. B235,FS11, (1^84), 321

(4) T.D.Lee, Phys.lett. 122B (1963),' 217

Kucl.Phys. B225, PS9, (1963), 1-52

(5) E.Cohen, 1984, Harvard Univ. preprints HUTP-84/AO11, A014

(6) T.Regge, Nuovo Ciroento J_9 (1961), 558

Figure Captions

Fig.1 : The neighbourhood of a lattice site i in the shape of

a regular polygon with K vertices.

Fig.2 : An illustration of triangulated rectangular .configuration*.The numbers 4,S,...ctc, belonging to the picture* labelthe corresponding number pf neighbour* of t&« site 1 .

Fig. 3 : a piece of a hexagonal lattice * whicsfe consists of regular

triangles as basic geometrical objects.

Fi-g.4 : Pieces of a hypercubic lattice, in two versions of triang-ula-tion:

A. All lattice sites have 6 neighbours.

B. The lattice sites have 8 or 4 neighbours in alternateorder.

278

6 version A

6 version 3

FIG. i

279

FIG, 3

280

A 6-neighbours lattice

B 8 /4- neighbours lattice

231

POSSIBLE EVIDENCE TOR St-gERSYMMETRY

IN HADRON SPECTRA

A. Bohni

Physics Department

The University of Texas

Austin, Texas 78712, USA

Abstract

Following the example of nuclear physics a tower

of meson resonances and a tower of baryon resonances

are combined into a supertower described by Osp(1,4).

This is phenomenologically justified by the approximate

equality of the mass splitting for these mesons and

baryons.

282

In a recent f i t of the hadron spectra to the quantum re la t iv i s t i c

osc i l lator model, whose spectrum i s described by SO<3,2) representations, i t

was noticed that the meson and baryon resonances with only u- and d-quark

content ( i . e . hadrons not containing s - , c - or b-quarks) have approximately

the same nass s p l i t t i n g . As mesons and baryons are described by the saae

class of SO(3,2) representations, they have also the saae mass level

structure. la nuclear physics a picture l ike tW,s - In which different

nuclei (even-odd and even-even) have the saae level • structure and level

spacing after the ground state levels nave been adjusted - i s taken as

evidence for super symmetry. '

The best docurnented meson tower, the combined p-ar-tower i s described by

the SO<3,2)-irreps denoted by D { u . «s+l, s»l) and the best documented

baryon tower, the N*-tower, i s described by W l ^ r » s " l / 2 ) . As these two

representations can be combined into one irreducible represcrtation of the

superalgebrs Osp{1,4)3'

(1) Dtj^n - t+X, s-l/Z)

i t i s quite natural co look for evidence of supersyaoetry also in the

low-mass hadron spectra. In the saae way as ue have carried the idea of

collective rotations and oscillations over into the relativistic domain we

shall also try to carry the idea of supersymnetric level structure over to

the aass levels of hadrcms.

The physical interpretation of the Oap(l.A) will be derived fro« the

pV.vslcal interpretation of the S0<3,2) subalgebra1^ >5K According to this

interpretation ;i - ei£enval«e(ro) and j -with j( j+l)-eigenvalue(S2)- are the

discrete quantoa nuabers that label the hadron resonances. In the full

283

representation of the quantum re lat iv i sc ic osc i l lator and rotator away from

rest''5 n - eigenvalue (T^11) and j ( j+l ) » eigenvalue'.-*? pW"), where W »

Juv

' ;generators of physical Poincare transformations, M « (P P1*) ' ; therefore

the physical interpretation of j i s the hadron spin, a * eigenvalue (M*") i s

the hadron mass, u i s a new principal quantum number, which in the

nonralativistic Uni t goes into tbe vibrational quanturc nunber. ' nnln> the

lowest value of (i and s » J a i a the lowest value of } characterize the SO(3,2)

representation* The representations chosen in (1) are both singletons.

This taeaas mathematically that an irreducible representation of the maxlnal

coapact subgroup SOC2)r xS0<3)c characterized by the pair (n , j ) occurs at

most once. Physically this means that u and j are a sufficient set cf

qoauttM nuabers tu label the hadron* in one tower, s ' l I ^ l t c a n i n t h e

non-relativistic correspondence be interpreted as the total quark spin

( i . e . »-l for p; » e t c . and a-1/2 for K).

In Figure 1 we have drawn a level for each pair of lumbers {v*prl , j )

that occur i s D<2,1) and we hawe also drawn another level for each pair

(^(r-1/2, j ) that occur U 0(3/2, 1/2) . If the leiwls would have bee a dram

al l with v as ycoordinate, tben Fig. I would oe the conbinatloa of the

•eight <iiagra« (or K-type) for DO/2,172) and the ueight diagram for D(2,l>.

The y-coordinate of the leve ls i« not exactly v but:

(2 )

vfaere • ia tht aass of the hadron that b«« been aa»ign«d to tne ((t,j)

level ' . The enpirical par«aet«r> I/a', X* and i^2 hava been determined by

to the aetson and nacleon spectru* separately ana i t turned out that the

284

values of I/a' and X. for Che meson and nucleon towers agree within error.

Therefore a joint f i t to al l resonances in the <tr*-p*-and N*-tower was

performed and gives the values

(3) I/a' - (1.03 + 0.036)GeV2, X2-(C.015i0.008)GeV2

vith x2 /n o - 9.9/28.

For the y-coordinate of the level with half .integer j we have chosen

*J — ? t — 1

m -nQ~(baryon) and (m'-m^dseson)) for integer j . In this way the ground

states masses have been aajusted to the same y-coordinate. (For direct

comparison we have also drawn the experimental values of m -»0 with their

error bars). Figure 1 i l lustrates the evidence for the supersymnetry, i t i s

not overwhelming as the values of the quantum number v (or u) can be

arbitrarily assigned to the resonances, and have been assigned such that (2)

gives the test f i t .

Ue shell now give a brief description of the mathematical structure

behind Figure 1:

To eacfi level (v,j) there corresponds an irreducible representation

space '3Cv(a,j) of the Poincare group, which i t obtained from the

representation (mZ,( JFVH, j ) ) of T4gJ (S0(2)r *S0(3)- ) by induction.7* The

cpace of integer spin BCates - related to 1X2,1) of (1) i s thus the direct

sun

where Che values over which the SUB excend* follow fron the weight diagraa

of 0(2,1) . The sâce of half-integer spin states i s the direct sua

285

where the reduction follows fro« the weight diagraa of D(3/2,l/2). To (I)

and therewith to all levels of Fig. 1 corresponds then the direct sua of

(4a) and (4b):

The operators which transform between tte different (•*,j)-levels af the

sane SO(3,2) oultlplet coae fros the components of the operator r which Is

an S -rector operator (and also a 3 -vector operator) that in addition i s

a generator of SO(3,2). As basts of the s)»c*s (4) one chooses the Wigner

basis

(5) fp,J»J3, & with v - eigetnraloe PÎ*

r4 performs the transitions between • the different (v.j) U w l t . i o t W i *

vectors at rest, |p-0, j,J3,n>» lo geaeral these transition* are perfumed

by the operators

(6) I*{f)

where L(P) i s the operator aatrtx which depends upon the center of aass

velocity operator ? »P

L(F) has the property

286

(t)

aad

(«>

The operators that trans fora between the (v , j ) - levels of the two

different SO(3,2) aultiplets are obtained In the ssae way from the

components Q of an S - * pi nor operator chat in addition i s a generator of

SspO,*), i . e . together with S aoi T i t fulf i l ls the defining algebraic

relations of Osp(l,4). Again for basis vectors . at rest the Q Cand q )

ttianaelvec perform these craacltiooc.' la general these transitions are

pefformM V ti*e operators

wh»re %^ ! • 'title Oj»ratoi:

and £p i« t>» oeoter-«)f-«a«» melocity operator tjC*. Tive «atrix «(»),

where p# i* the ci|«n«alue of the operator p , i s the (1/2,0) © (0,1/2)

representation of the inverae boost and L(P) i s the (1/2,1/2) r a presentation

("»e?.f-representation") of the inverse koo»c.

It i* important to choose a basis for toe (pinor operator which i t

adopted to the Kignet basis of tbe repre»ent«tion space. This is achieved

11 one chooses fot y the ossit representation

287

(12) I T = <M,J3,j - 1/21 r | j - 1/2, J3,M>

where | j= l /2 , j-j, |i>, j 3 = ±1/2 u - ±1/2 Is che basis coat has the same

property In the 4-dlmensional representation (1/2,0) © (0,1/2) that the

basis |j ,j3(i> j - 1/2, 3 / 2 . . . , n, )f=l/2, 3 / 2 . . . has in the inf ini te

dimensional representation D(3 /2 , l /2 ) . In this basis one obtains from the

anti-commutation relations of Osp(l,4):

using (9) , (10) and (11) one calculates r.hat

Jin,- 1/2

The physical interpretation of Che generators i s not identical with the

conventional one but resembles i t : Fo i s not the energy but related to i t

by a constraint relation . I"m i» not the momentum bat Is related to the

intrinsic,non-commuting raoraentun of an extended object n by projection into

the plane perpendicular to i t s center-of-aass nomentun P . i^ goes into the

usual oscil lator momentum in the non-relativistic contractior. l imit \

As the Q are on the same footing with che r and the V are not

assumed to commute with the F v (except in the limit when a l l levels In

Fig-. 1 have one and the same mass) one can also not expect that the Qo

coonute with the center-of-taass momentum F . Tor the above constructions i t

Is sufficient to assume that T and thus Q commute with the center-of-raass

velocity P .

288

iie nave thus shown that OspO.4) can be Incorporated into a

relativistic quantum mechanics which combines meson and baryon resonances

into an infinite superaiultiplet. The possibility of such a mathematical

combination of two infinite dimensional SO(3,2) raultlplets was quite

obvious. The hope that physics may also allow this Is based on the

pheTKraenclcglc.il resulc(3), depicted in Fig. 1, which was unexpected, and

quite surprising taking into consideration that the non-relativistic

correspondence for the integer spin case Cvibrati'ng di-quark) and for the

half integer spin case (tri-quarlc) are rather dissimilar.

26S

References and Footnotes

1). A. Bohro, M. Loewe, P. Magnoliay, The Quantum Relativtstic Oscillator I I ,

University of Texas preprint No. DOE-ER-03992-576 (1984).

2) A. B. Batantektn, I. Bars, ?. iachello, Phys. Kev. Letters &7, 19 <1981);

H-Z. Sun, M. Vallieres, D. K. Fer.g, R. Gilsore, fc. 7. Caste*,

Phys. Rev. C 29, 352 0 9 8 4 ) . A similar kind of evidence for

supersymmetry in atonic physics has recently been reported by

V. A. Kostelecky aad M. M. t&eto, Phys. Rev. le t ters J>3, 2285, (1984).

3) C. Fronsdal, Phys. Hev. D2£. 1988, (1982); V. ifci-ienreich, Phys. Letters

HOB 461 (1982) P. Breiceniohner, D. Z. Freediiw-., Ann. of Phys. _l£i> 2 4 ?

(1982); D. Z. Freediaan, R. l&colai, Socl. Phvs. B237, 342, : <>6A.

4) There nay be nore resonance towers with the same level spacing ( e . g . the

£*-tower). There are also other representations uf C>sp(l,4), but most of

there contain S0(3,2) representations with higher multipiicltv ami vouiiJ

thus lead to degenerate mass l eve l s .

5) We use the notation of reference 1: S 0 O , 2 ) r K ~

S0(3,l)c , S0(3)-po ^ S0(3)c x S0(2/r, . Thus T are the operator*(iv " (iv i j ' o ••• i

whose matrix elements in the 4-diaensional representattor. are -y-Y and Sare those whose matrix elements are Yau.v'

6) The construction of the representation and the meaning of the splittingJnv " Muv + Suv h a s b e e n d e s c r l b e d i n f u i l catall for the case r>(l/2,0)

and D(l , l /2) in A. Bohm, M. Laewe, L. C. Btedenham, and H. var Dam,

Phys. Rev. 028, 3032 (1983).

7) The situation i s a l i t t l e bit more complicated due to the constraint

relation which gives (2 ) . One has to induce from the reducible

representation of T^ ® CS0(2)r xS0(3)- ) , do this for every m!>0 and*o i j

then apply the constraint. Another way of describing the sane procedure

i s given in the reference of footnote 6.

8) Equation (2.34) of A. Bobs, M. loeve, P. Magnollay, The Quantum

Relativtstic Oscillator I , Univ. of Texas preprint, DOE-ER-03992-574

(1984) or A. Bohn et a l . Phya. Rev. let ters 53 , 2292 (1984)

9) Such a bosonic syaraietry was callee "Belativlstic Symmetry" when i t was

introduced; P. Budini, C. Fronsdal, Phya. Bev. U-tt. U , 968 (1965).

290

Figure Caption

Fig. 1: Mass level diagram, af obtained from a fit of the nucleon

resonances, and of the Y - 0, CP « +1, j = normal meson

resonances to tne siass formula (2). On the horizontal

axis we have plotted the spin of Che resonances j.

Vertical.";- above it we havs 7letted a level fcr each value

of B ' -' fs£, where the value of S. (baryons) - a~ (neson)

has been fixed such that the ground state levels coincide:

IB"(K) - E I (baryon; « us (t-1)'- ffl- (roeson). v is the nev

vibratioaal quantum number which has been assigned to the

resorance: V = eigenvalue (T, - -j) for haryons v - eipen-

vsJue fT- -1) fOT neED.is. Rot shown in thi£ Figure is the

prediction of m * 2756 MeV for j * 7~ (experimental H(2750)

*nd of ai « 2793 MeV for 3 - -i| iexperinental NC27O0)).

291

2 ~2m -m

V=7

V=6

V=5

v-.«

V = 3

v=2

v=i

— —

«

. ..

- 4 -

• + -

• t - t ;

- t T . T "

» l • t » i »

- T

I. \

~- i J

1

i- ™t"

T

i

—1_^.

3 \ 4 % 5 % 6 j

293

srzi-i 'CRCsa I N HAQRON SPECTROSCOPY;

QUARK KVJZl. \Z::MS gLfcJTRCry.SNETIC SPIN INTERACTIONS

Asia. 0. Barut*

Departement de Physique •Ihe'oriqueUniversi ty de Oenfeve

1211 Seneve <*, Switzerland

ABSTRACT

The sp in -o rb i t and spin-spin p o t e n t i a l s in hadron

spectroscope are usual ly t r ea t ed as a pe r tu rba t ion

to an unperturbed Haniltonian both in n o n r e l a t i v i s t i c

and r e l a t i v i s t i c models. We study these terms non-

pe r tu rba t i ve ly ; they are not uniforaily s c a l l and

indeed .^cdify ic general the unperturbed p o t s n t i a l

in such a way tha t t he o r i g i n a l unperturbed bound

s t a t e completely disappear . Some remedies are d i s -

du3=,ed 2nd a cocparison i s nade with t he spin forcesi

i n QED.

1. ffllBODUCTICIg

One of the cain pillars of quark model and QCD is considered to be

the success of hadron spectroscopy, at least for heavy quarks. It is

generally stated that "apin forces Si'c w«ll understood" * . However.,

the practical hadron speetrosfiopy is based on phenomenological poten-

tials, supposed to be motivated by QCD or by assumed ec>Ei"i enent. We

shall enphssiie here that such potentials have severe fundamental dif-

ficulities both at short and at large distances so that tbe parameters

obtained from fitting with these potentials cannot be reliable. These

considerations imply that one has to use other asethods-tuan potentials

to grasp the nass problems of hadrons based on quarks, e.g. sum rules

294

cr lattice-type calculations.

On the other hacd, the'concept of potential should be applicable

to any r.onrelativistic or relativistic bound s*ate or resonance prob-

lems based on constituents. If the constituents are real physical par-

ticles, no difficulties arise. But if the constituents oust be confined,

there seems to be no adequate nay of forrauleting a1 dynamics based on

potentials. We are thus lead to compare the dynanical aspects of model

of hadrons based on reel particles (in which quarks can be introduced

aaThesatically or group theoretically, but need not be the dynamical

entities) versus the quark raodel.

Innumerable papers are being written on various fits vith various

potentials. In viev of the fundamental difficulties nentioned above,

these fits are questionable. Sooe of these difficulties are known and

have been poirj'vd out. sporadically. Th& purpose of the present work is

to sharpen thcst problems, to formulate then more quantitatively, to

see if the isr.setiLs.zis readies vorkt-t and then cccpare them vith alter-

native dynamical models.

i. CRITIQUE OF THE FHEIiOMSHOLOGICAL POTER'.flALS

In hadron spectroscopy based on qutrk-nodel one starts from a LO-

variant effective interaction between the constituents u and v

£ = ^Hw)S(n2) + (u/uXv-vmq2) , (1)

thus introducing two potentials S{q ) «nd V(q ). In the vave equa-

tion for one of the perticles S and V behave then as a (Lorentz)

sctlar and vector potential, respectively, hence the notation.

Before giving the usual nonrelativistic fora of the potentials,

I discuss firBt the relativistie dynamics. If ve introduce cooposite

fields tlx^fXg) * U(JCJ)V(JL), we can rewrite (1) in terms of * and

i, also the kinetic energy terms. Varying then the total £ with

respect to * (instead of varying X with respect to u. and v sep-

arately) we obtain for * the linear wave equation

295

tYW ( i3 ) - E 1 ) 8 Y 0 + y ° * i Y U i 5 y - n j + 3 + y - V - Y " } * = 0 . (2)

The c o r r e s p o n d i n g H a u i l t c r . i a n i s

H = a,-p1 + 31B1 + u.,-p\ + 3,2, • S^ i r jg , + ( 1 - ^ •Qg)V(r) . (3)

In the center of mass system thus, the mass operator beecoes

K = (ctj-oO'p + B,c1 + i^x.. + B

?:-:te : Eq. (2) is actually a i-tirae equation, if S an3 V depend

enly on r . Then 4 is in3epender.t of the relative t i ze , q°, and

can be written as t{x ,x. ,z). This aeans that £(q"") and V(q ) in

Eq. (l) contain a 5-function, S'x -y ±r), corresponding to the propa- .

gator of exchanged laassless particles. Jor both photons and gluons the

V-part satisfies this condition (in the case of gluon there ere also

nonlinear tenns, of course). 3ut the scalar part S(q ) appears to be

purely phene=er.clcgical. Ana this is cne cf the difficulties about this

confining potential right from the beginning.

I shall now discuss vhat is generally and typically *~ done in

practice at least for heavy quark? (sfoen also for lighx ones). One

deduces from (1) a nonrelativistic Haailtociac, for exasple, toy naXing

a —expansion of (It), and obtains

\(5)

vhere for two equal mass spin 1/2-quarics, y = c/2, and

r - v") , (6)

236

[There are further relativistic ter-.:

which are however usually neglected.]

k± a concrete example, ve quote the rr;oae2s

z.'zz'.ir>£ niÎine^r terr.s i.". glucn f i t ld) , "uhe êc^r.i if the cor.fir.ir.g

r-:-er.-ial s'»egeEtea fro.i a string cciel, tut fc.ioulfi sJ.so ooce frot the

gi>.cr. .-.xchar-g1-; if the therrj' (Qw) is corre=~.

ISe effective r&iial potentIEI "tec sisi;

(9)

The Xert three spin forces ere treated as e j-erv-irbstio:: to the solu-

tions of the jr/perturbed pctential V__ giver, ty the first three terns.

Th=re is a =icisuc of V ar.d one locates sound states.

Hcvei'er, the spir. potentials as they ttsuid are act uniformly

src&li, and one should ari sac .treat the cor.piete potential (6) nonper-

r.urbatively. For exasple for rhe 3F. -nesons, we hare

! • ! « -2 , <s)2> - - i

and the spin-orbit aad spin-spin forces are both attractive. If ve plot

thei. the complete potential vith typical parameters a =0.15 - .6;

i = 5>. 15G«v , it • 300 - SOOGeV, the lininuir cf V disappears j thereV

is no unperturbed bound stsê anj' isore. li; addition the HamiltoniBn

with (8) is not essentially self-adjoint sod heace the probleo is not

297

well for-uiatec:.

Cnt ia.1 thi.iX ;f rcse raireiies. Cne is t c ir.c3--:de the . r e l a t i v i s t i c

corrections. But the r^- tern i s s t i l i an a t t r ac t :vp potentis.! (see "q.-*2

(7)! . The etner ana io r t important fact i r i s to .r.cl 'iie the A -terra in

•_r.e spin-orbit i n f r a c t i o n s . This gives a re ju ls ive hard cure to the;, 3

potential going l ike ",'r , off se t t lag the a t t r ac t ive 1/r -po ten t ia l

at short d is ta rwi i . Uov the problem is w-:.i-defined, and bound s ta tes

occur, but at much shorter distances then i'or V and the aagcitudeE

of the energies are quite different . This neans that i f we wish t.c

identify the hudrons vith su:h a aynacics, t i e paraiieters a , m, K

LTXST. bo chosen ccapietly d i f ferent ly .

There i s a second a^jor problem, in rjiark 3yaaaics with potent ia ls

vf the forn (7) nenely, with the lcng-range oonfi.-.ins po ten t i a l S = k r .

I t is i l ea r that s:ieh a long-range potant ia j results i t very large

van as.- Walls forces betveen two haarons, saj- two mesons or between a

aescr. and a caryon ' . These forces are exper-aentally absent. Also

the sign of the qq- and qq-firces seems t o be the sane so tha t no can-

cellat ion of the very large terras car. occur as in molecular physics .

If indeed t h i s i s the aechar.isn; of haircn binding, then tbe effect of

a l l exchanges of gluons with different colours between t»o colour s in-

glet hadrons oust exactly cancel. This problem adds t o the other basic

one, as to why only colour s ingle ts occur in nature. For t h i s reason

we can conclude that not only the parameters of the quark model ob-

tained fros (8) are unrel iable, but also tha t the spin-forces between

the quarks are not well understood.

3. SFIH FGBCBS IE ELECTBOHAGNBIISl'l ATO EX.ECTROHAG5EIIC KOPEL OF

HADEOHS

In this section I compare the quark spin-forces vith the apin-

forses in electroaagnetisn. The situation here is aueh nore clearcut

and siigile» Assuming the sane vector potentiel V «£ ia Eq. (2) which

in -3ED represents the photon exchange, we arrive of course at the sane

radial potential as (B) which with. U./3a -• a * 1/37, m •* n , k * 0,

298

and with the inclusion of the A"-terx, reads

This potential, again, for exeraple, for 3F.-stai.es '.L-S = - i ,

<S > = -1) has a deer. 2ir.iir.ur. at r » — , in addition to the

Ccuiocb minicu-ti at r w — ). '»"- excect that there are resonance0 ctrr.e

-tates at energies L. ~ E .'a. I t is ren;=;r}:able that we obtain the

same fiual range and energies for hadronic states but vitb entirely

different mass r.sraaeter for coi)£tituent£ (s instead of n j andk e G-

for the coupling constant (o instead of -as I)

But of course vith the leptonic constituents ve nust use fully

re la t iv is t ic kiaeaatics. This is indeed possible. For a Dirac particle

r,ovi.-i£ in the ; :.eld of a fixed magnetic dipole represented by the po-

tent ia l A « v—T > the Kaiiltonian

H * o«(p - eA) + BE (11)

j ives, after sguaring, the equation

{/ * ~ S ' L + ~rt^ * T-T-lta = Ea«P . (12)*•* ar! iEr» 12 io'r11

The effective pstentisl in this equation has the sane deep well as in

(9) snd we expect from uncertainty relation narrov resonances at

energies

A dynamical theory of hadrons and leptons has been put forward on

the basis of electromagnetic dipole inte-actions of the type (11),

Dipole interactions are short ranged, provi<Je_ a deep potential well {a

bagj at short distances for ccnfinenent, but the potential turns over

instead of rising linearly, so thrt. a l l cocposite states are narrov

299

resonances. The tuo cain difficulties of the quark, potential, the spin

forces at abort distances and the van der Vaals forces at large dis-

tances do not arise here

REFEREHCES1)V W. Buchoiiller, International School of Physics of exotic Atoos,

Erice, 193V, CEH*-IH-3S36/&%.2) J.L. Bosner, Spin Dependent Forces in Quark Model, Univ. of

Chicago preprint, 5FI 81t/33 (".98U). <3) A.O. Barut and S. .Keagr, Fortschrittc der Pijysik U , 309 (1985)-li) F. Eichten et aX., Phys. Rer. 221, 3090 U9T8); K 1 , 203 O980).5) A list of references dealing vitn tbe long-range isteracticBS

between badrons is firsn ie O.K. Cj-eenberg and H.J. Lipkin, Buel.Fhys. A370. 3*»9 (19SJ).

€) A.O. Barut and R. Ra^zka (to be published).T) For recent reviewE oo th i s model see A.O. Barut, in CEP of Strong

Fie ias . W. Oreiner (edi tor) , Fieaun Press, t$S* and A.C. Sarat,in tanalen der PbytiX. T965.

Permanent address : Depurtaetst of Physics, UBiTreMrty of Colorado,Botti4er, CO 80309, USA

301

NOVEL APPROACH TO SCALING IN MULTIPLICITY

R. Szwed

Institute of Experimental Physics

University of Warsaw. Warsaw, Poland

ABSTRACT

Problems concerning scaling in multiplicity are briefly reviewed, and the

standard approach fKNO scaling) is compared with the new one (KNO-G

scaling), tt is shown that, with the new approach, the pp multiplicity data

obey multiplicity scaling perfectly within the whole available centre-of-mass

energy range from 2.5 to 62.2 GeV. However, the seating is broken by the

CERN Super Proton Synchrotron Collider pp multiplicity data of experiment

UA5

302

1. INTRODUCTION

The CERN Super Proton Synchrotron (SPS) Collider, because of itsenergy, has created completely new domains cf high-energy physics; but ithas also revived an interest in the traditional part of high-energy physicsdealing with the global characteristics of hadron-hadron interactions -- forexample, multiplicities. Fortunately enough, the new high-precisionIntersecting Storage Rings (ISR) data were published [3] simultaneouslywith the SPS Collider multiplicity data [1 ,2 ] , thus tr iggering off anextensive discussion of old unresolved problems concerning scaling concepts

The aim of this paper is to present our own understanding of theseproblems, and to draw attention to certain concepts [8] which appear to beunknown to the wider physics community. The discussion is limited to thepp multiplicity data measured in a ful l rapidity range (tor references, see[9 ] ) . For most of the numerical details and derivations which are notincluded in the present paper, the reader is again referred to the previousreport on the same subject [9].

The plan of this paper i t the following. Section 2 will give a briefreminder of the present standard understanding of multiplicity scaling.Next (Section 3.5 the fK>n-standard approach to KNO scaling [8] will beintroduced. In Section 4 the validity of this new scaling, calfed hereafterKNO-G, wtfi be demons*rated by making a comoarison with the existingproton-proton multiplicity data. Section 5 contains a comparison betweenthe existing Collider multiplicity data and extrapolation, within KNO-Gscaling, to the SPS Collider energies. The summary and conclusions aregiven HI Section 6.

2. PRESENT STATUS Of THE KNO SCALING

2.1 History

IR 1S72, Kob», Nielsen »r»d Olesen showed that multiplicity distributionsshould scale at asymptotic energies ir. the following way [4 ] :

Pn = 1^<n>*(*3, i = nA«>, <n> = 2 nPR ,

where ' • » = 3 n ' ° " ' •* ^ e Probability to produce n secondaries in the finalstate. The $ function is an energy-independent scaling function normalizedby the following conditions:

= I P.- - I l/<n><(itn/<n>) s J (C(z) = 1It If ft _

and

It should be stressed that the above formulation of KNO scaling hss beendone for the asymptotic energies [4] snd that it is not self-consistent atthe small energies, since the integrals in the normalization equations arebadly approximated by the sums.

303

2.2 KNO sealing violation

Figure 1 shows all the available ps multiplicity data (vs=2.5 to 62.2GeV) plotted in the KNO variables [ a) linear vertical scale and b)logarithmicai vertical scale]. On the one hand, it is seen that the dataform a kind of a common curve. On the other hand, the size of theexperimental errors is comparable at small *=n~n ' '<n

c-| , ' v v i t h t h e Point size,so it is obviously impossible to draw one singTe curve (scaling function y)through all the data points. This clearly demonstrates the well known factthat KNO scaling is violated. The same conclusion is usually reached in smore quantitative way by shov-ing that the normalized statistical moments(e.g. C(,=<nI<>/<n>kl rise with energy [TjC.11.-3j, although the* should s-.ayconstant. Also, D-,An> {D,-[Z(n-<n>)-Pn ] } should be constant withenergy, which implies that" D1=A<n(.^>. In fsct, the data obey theempirical formula of the form D?=A(<.n~h>-e) (the 'Wrobfewski relation' [ 5 ] ] ,where asl for the pp multiplicity data, thus contradicting KNO scaiing. Itis usually claimed thaT the parameter a, being connected to theieadmg-particle effect, has a dynamical origin [12,11.13, 3 ] .

As a short conclusion to this section, one can say that KNO scaling isvery approximate, and is in fact badly violated when applied to the fullenergy range.

2.3 Rescue by the non-diffractive data

There is a common belief that KNO scaling may be rescued by usingonly non-diffractive data [6,3] . (Non-diffractive data means mrltiplicit-,'distributions for all meiastic events, with diffractive events excluded fromthe sample.) This belief is mainly based on two experimental observations:a) people believe that the D versus <r>~h> reistior. is recovered fornon-diffractive data to the forir supporting "ideal KNO scaiing, Dt-A<n(; 1>[14,16]. b) the C., moments plotted for non-diffractive data at ISRenergies [3] show no change with energy within the present experimentalerrors

So the diffractive component in the multiplicity data if believed to beresponsible for the KNO scaling violation.

3. GENERALIZATION OF KNO SCALING (KNO-C)

In 1977, Golokhvastov [8] proposed to reformulate KNO scaling in thefollowing way:

P(n) = l/<n>ii>(z), z = n/<n>, <n> = f nP(n)dr> ,0

where Pin) is a continuous probability distribution o* continuousmultiplicity n. However, we do not observe P(n', experimentally: it is thediscrete probability ?n which is measured end which can easily beevaluated from P(n):

P n = J P ( n ) d n = J t ) d

304

Note, that the ner'nalizatio.-i equations are exactly fulfil led in Golokhvastov'sformulation of KNG seating. As can be seen, KNO-G scaling is in fact aone-to-cne image of the standard KNO, but is applied to the continuousmultiplicity. Obviously KNO-G - KNC when Vs - =>, because

i»*1An>Pn - T *UJdz = l/<n>ii)(n/<r,>) ,

but Asymptopis is far away, since <n> (which is a scale in the problem)rises with energy only logarithmically.

Perhaps it >s easier to understand the whole concept of generalizationusing the simpit picture plotted in Fig. 2- where the solid curverepresents o scaling function i|;(zj plotted with the usual KNO variables.Usinc the defir.itions of Pr within KNO fF,. = 1 '<r>>V[n/Jn>] and KNO-G [PR="j ifffzidz], it is easy to see that whereas Tht P_ ^ are represented bythe ares of rrtt piotted rectsngles, the p^ K N O " ' are represented by thedashed are? under the scaling function.' So KNO-G seems to be thesmallest possible generalization of the KNO formulation, making it applicablefor ar^ scale <n-

4. THE KNO-G TEST

!h this section some features of KÔ-C scaling will be pointed out, andthe scaling will be checked aga'nst the pp multiplicity data.

4.1 Cnar.fle of the scaling function

it seems to be convenient to change a scaling function \p{z) to itsprimitive function [9] defined as follows:

4>(z) = j i t f ( z ) d z - i .t

So,

P = f *(z)dz = *Un-1V<n>]-<Kn/<n:0.

A. change of the scaiir.g function facilitates the numerical evatuations [9 ] ,Dut in no sens* it is compulsory fEj .

4.2 Graphical test of KNG-G

It is easy to see from the relation f8.9J:

S f l - X P, = j (Hz)dl = -<f{z) [assuming 4>(«>l = 0] ,''"• n /<£>

that the data at different energies should overlap if sums S (S =ZP) areplotted versus z. A graphical test of KNO-G sealing is shown in Fi'g. 3.tt can be s«en that all data point* lie on < single curve.

305

4.3 Energy behaviour of the C, (Cu=<nk>An>k) moments

Obviously the C^ moments should be constant with energy,.

Cb = <nk>An>k = 7 zk*(z)dz = const(s) ,

but KNO-G scaling does not require the constancy of the Ck moments. Sothe experimental observation of the rise of the discrete Ck moments withenergy does not contradict KNO-G in any way.

4.4 The Wroblewski relation

Following the KNO-G formalism, we can write

(D2/<n>)2 = <n2>An>2-1 = J l\(z)dz-1 = const(s) .

Hence

D, = constx<n> = A<n> ,

but it can be easily shown [9] that

<n> 3 <n>*0.5 and D, £ D, .

Then

0 2 a D, = A<n> s A(<n>-0.5) ,

where h = 0,1,2,3. . . (e.g. a negative multiplicity in pp collisions) If wechange to ncu (nc^=2,4,6.. . , ncn=2"2n), we immediately get the well-knownWroblewski relation [5]:

D2 = A«n c h>-1) .

Therefore, the Wroblewski «-elation is a simple derivative of KNO-G.

4.5 Generalization of the Wroblewski dependence

It is easy to see that not only the second-order discrete and continuousdispersions are approximately equal, but that this holds for any k-orderdispersion:

\ , Dk, Dk = [I(n-<n»kPn]1/k .

For targe enough <n> we immediately get

O k s A k « n > * 0 . 5 ) .

In generai, instead of the multiplicity measure n ( n = 0 , 1 , 2 . 3 . . . ) , a measurem is used (m=mo*ln), e .g .

PP collisions: m0 = 2, I = 1; m = n . = 2*2n (2 ,4 .6 ,S . . )it p collisions: m0 = 0, I = 2; m = n . = 0~2n CO.2,4,6. ..'>

306

vp collisions: m6 = 1, I = 2; m = n c h = 1*2n (1 ,3 ,5 ,7 . . . )

Therefore, because of the charge conservation, we can write [9] ageneralized Wroblewski relation:

D km s Ak(<m>-mo*l/2) ,

for example:pp collisions: D. s A

k (<n c ( ,>-1) , k = 1,2,3...n'p collisions: Dk = Ak(<n |.

>h>*l) / k = 1.2,3...

vp collisions: Dk 3 ^k l - nc h : > ' k = ^,2,3. . .

In fact, it was observed experimentally that D. ( k= l ,2 ,3 . . . ) moments wereexactly related to <nch"> °V t h e above formulae [10.11,15]. but thisobservation was never understood. Now we can say that this is mostprobably a simple consequence of KNO-G scaling arid the chargeconservation law.

4.6 Quantitative test of KNO-G scaiing

Figure 3 demonstrates in a graphical way that KNO-G scaiinc works forall avvisble pp data t'vs=2.5-62.2 GeV). To do the test in a morequantitative way, the function of th« form

2<t>u(z) = -(1-az)e , a = \ nv'b-2b .

has been fitted to the whole set of pp multiplicity data [9] , The y,"obtained was^ rather good I if systematic errors of the data ere taken intoaccount): x" / f s i D -1 -26 J9]. It should be stressed that although the fittedfunction <fc has only one free parameter b lb=0.6273~0.0CK1!. it is still ableto describe all available pp inelastic multiplicity data with very goodprecision. The best illustration of this precision is the comparison betweenthe behaviour of the different multiplicity moments wrth energy, and thetheoretical predicted' behaviour using the function <J>C . i.e. "

P t h (<n>,v») = •D[<'ri*l)/<n>]-<|>0(n/<n>) .n

where <n> E <n>*0.5 [9] .SUich a' comparison is shown in Figs. 4 and 5 [D=D.,, $=DAn>, Y-J =

i V D , Yo = ny'D - Cj. - <nk>/<n>k, u^ = 2(n-<n>)kP r ] . ~ As can be seen,tne solid line follows the data behaviour exactly, thus again providingproof for the KN'O-G scaling validity in a whole energy range for allinelastic events.

5. EXTRAPOLATION TO THE COLLIDER EMERGIES

It should be stressed that at the most the standard KNO scaling isbelieved to work only at high enough energies, say in the !SR energyrange [6 ,3 ] , which leads one to suspect that the scaling function iy(i) willstill continue to change above the ISR energy range [5] Therefore, anyextrapolation froir !SR energies to the very much higher SPS Collider

3C7

energies may produce a fortuitous result. The situation is completelydifferent if one takes into account the KNO-G scaling. Since KNO-Gscaling is really perfect in the whole energy range, it should be relativelysafe to use it tor extrapolation.

Furthermore, it should be noted that it was not necessary to subtract adiffractive component from the data in order to obtain perfect KNO-Gscaling. So an experiment measuring multiplicity distributions for allinelastic events at the SPS Collider is very much needed. Again it shouldbe stressed that so-calied 'non-diffractive' or 'non-single-ciffractive' datasamples usually have very large uncertainties because of differences causedby the subtraction of a diffractive component. Unfortunately, the onlyexisting SPS Collider multiplicity data that measure secondariesapproximately in a full solid angle are of the non-diffractive type [1]

In order to compare the extrapolation within KNO-G witfc the recentUA5 data [1], the validity of KNO-G should be checked for lower-energynon-diffractive data, which should lead to the scaling function $o

r' , andthen extrapolation could be performed. We use Fei-milab and ISRnon-diffractive data (for references, see [9]) , and get the following bestfit of the scaling function:

<t>o"d(z) = -(T0.11042)e"°-39242 , x ' /NO = 1.42 .

We now only need to know the average multiplicity to be expected at Vs=540GeV. Using the extrapolation within the 'log formula, we get <n>=12.!1[9]. A comparison of the KNO-G predictions with the UAS data is shownin Fig. 6. ft is clearly seen that the experimental data differ from theKNO-G predictions only at the high-multiplicity tail. Also seen are ratherworrying 'snake-type' fluctuations in the data. .Assuming that the UA5data are correct, it can be claimed that on top of the usual particleproduction mechanism responsible for the KNO-G scaling, there is evidenceof an extra process contributing to the high multiplicities.

6. .SUMMARY AND CONCLUSIONS

The scaling derived by Koba, Mielsen and Otesen for asymptoticenergies is very often applied to tht low-energy data, where its formulationis not self-consistent. Golokhvastov proposed a minimal possible extensionof KNO scaling, making it self-consistent for all energies. This resulted ina number of consequences which are now completely changing ourunderstanding of th« multiplicity scaling concept. These are summarizedbelow.

6.t Inelastic data

6.1.1 Validity of the scaling

Standard approach: The scaling is badly violated if one takes into accountthe whole energy range. It may work locally at a given smali energyrange, but only approximately.New approach: The validity of the scaling is confirmed over the whole

energy range with one scaling function (one free parameter).

308

6.1.2 Usual checks of the scaling

Standard approach: The rise with energy o* the normalized moments (e.g.C^) is a direct proof of the scaling violation.

New approach: KNO-G implies the rise cf the normalized moments withenergy (Figs. 4 and 5J, exactly as observed in the experiments.

Standard approach: KNO breaks down because o< the exact validity of theWroblewski relation [D-AUn h>-a), a=1 for the pp data].

New approach: The Wroblewski relation is a simple consequence of KN'O-G.

Standard approach: The parameter a has a dynamical origin connectedwith the leading-particle effect.

New approach: The parameter a. which should be exactly 1 for the ppdata, {for each k-orcier dispersion t ^k = A k ' ! ' n -h > " 1 ' - ' has a o u r e |Vmathematical origin. This is a consequence of going from the continuous tothe discrete multiplicity. The dynamics is hidden in the A^ parameter!

6.2 Non-diffraetive data

An introductory remark: (n our opinion the KN'O tests based onson-diffractive data are of poorer quality than those based on data frominelastic events. Thft is due to the uncertainties caused by thesubtraction »f a singl« diffractive component, especially in low-energy data.

6.2.1 'Ideal KNO scaling'

Standard approach: There i& » belief that in this case the dispersion isrelated to In* average by the formula D=A<n .>.N*w approach: H the rton^diffractive data, were to obey KMO-G scaling,

the relation would he the same as for all meiastrc events [D iA'Knc^>-U],and tfiit seems to be the case if »ne tooks more carefully at the data [91.

Standard approach: There is no ri&e of the Cy. moments with energywithin the ISft energy rancte.

Nei ap-proech: There, fs n.c rise seen within the !SR energy range owingto the large experimental «rrors and the fact that, already at this energyrartge, th* C^ moments rise with energy in a logarithmic fashion (Fig. 5).

$.2.2 Comparison with UA5 data

Standard approach: Clear observation of the scaling violation in the wholemultiplicity range.

New approach: Only the tail of the multiplicity distribution (n , >40) is notconsistent wrtfi extrapolation within KNO-C scaling. However, the tails ofthe multiplicity distributions ar* subject to large systematic errors.Therefore, before arriving at fina) conclusions about the scaling violation,there is need for new multiplicity data measured by a different experiment.

309

AcknowledgementsI warmly thank A.K. Wroblewski and K. Zalewski for many useful

liscussions and remarks on the subject.Furthermore, ! am very grateful for help given by my colleague, C

,,'rochna, with whom I have worked out most of the results presented here.

references

,1] G.J. Alner et a l . , Phys. Lett. 138B (1984) 304.(2] G. Arnison et a l . , Phys. Lett. 123B (1983) 108.[3] A. Breakstone et al . . Phys. Rev. D30 (1934) 528.[A] Z. Koba, H.B. Nielsen and P. Olesen, Nucl. Phys. B40 (1972) 317.[5] A. Wroblewski, Acta Phys. Pol. B4 (1973) 857.(G] K. Boekmann, Bonn University preprint BONN-HE-83-21, and Proc.

Third Int. Conf. on Physics in Collisions, Como, Italy, 1983.[?] G. Pancheri, preprint CERN-EP/84-96 (1984), and Proc. 15 th Int.

Symp. on Multiparticle Dynamics, Lund, 1984.[8] A . I . Golokhvastov, preprint JINR Pl-10591 (1977) and Sov. Jour.

Nucl. Phys. 27 (1978) 430.A . I . Golokhvastov, preprint JINR P1-10871 (1977) and Sov. Jour.Nucl. Phys. 30 (1979) 128.

[9] R. Szwed and G. Wrochna, preprint CERN-EP/85-43 (1985), to bepublished in Z. Phys. C.

[10] E. de Wolf et a l . , Nucl. Phys. B87 (1975) 325.[11] W. Thome et a l . , Nucl. Phys. B129 (1977) 365.[12] A.H. Muller, Phys. Rev. D4 (1971) 150.[131 J. Dumarchez et a l . . N-uovo Cimento 66A (1931) 114.[14] K. Alpgard et a l . , Phys. Lett. 121B (1983) 209.[15] H. Grassier et a l . , Nucl.^Phys. B223 (1983) 269.[16] A. Wroble\v:ki. Proc. 14" int, Symp. on Multiparticle Dynamics,

Granlibakken, Lake Tahoe, USA, 1983.

Figure captions

Fig. I: Graphical test of KNO for inelastic event samples; n , denotes ailchargeo secondaries, z=n u'<n . >.

Fig. 2: Graphical illustration of the difference between the standard KNOand the KNO-G approach.

Fig. 3: A graphical test of KNO-G scalinc-.Fig. 4: Statistical moments: D-[<ns u-1"^ h ' '2 ] ' <t>=<n . >/D, i-.-V-s/D1,

^ j u / D " * [where S^^ in^ - ' r \ _ ^> ) PnJ depencfence on <r,c^> to; pp dat? ina whole energy range. Trie solid line corresponds to the statisticalmoments generatec within KNO-G as described in the text.

Fig. 5: The same dependence as in r ig . 4 but for normalized algebraicmoments (Ci =<n >/-.'n> ) .

Fig. 6: Comparison of UA5 multiplicity dats [1] with extrapolation don?within KNO-G for <n> = 12.11, as discussed in the text-

v>

310

o

o o

Id

d

" . ' • • : ' % ' •

i

A•:.<••

c-J — — — >-' -: 6 O O

311

i t-'v

->- I

tf> iO t frj •*) C4 f< **

90 .oo o b

!

J2

o

312

I "is- 1•ice

(i « a ^ —

u.Ml

313

LEADING PAKJICIE E. FBCT A? HIGHER ENERGIES *

Mikulas BlaSekI n s t i t u t e of Physics, EPHO, Slcv. Acad. Scl .

CS-34 2 28 Bratislava

I t is shown that the presently aveilable data on charges

mul t ip l ic i t ies prefer such a modification of the KNO sca-

l ing function which involves an energy dependent measure

of the leading par t ic le effect .

1. Introduction of the leading par t ic le effect J l j he l -

ped to remove the apparent discrepancy betv/een tne pp 5ata

in the ens energy range Vs frorc about 2 to 24 C-eV ana the

KNO prediction [l]- Especially, for the dispersion ti,, =

= ( <n"> - <C.a> ) , the KNO scaling hypothesis induced the

re la t ion V- = const..<'n^> where < r.'> is the aver-.'ija .':hf;r>£s •?

ni\. ';ipliei'.v while ".he data suggested [ I j the re la t ion

D^ = conat^s'n^ ~*>c ) with oC= 1 and (cons-*.,,) £ » 1/3. I t is

well icr.'Own that tne con.star.ts "const." end "ctcst^" usually

contain the par&aeters specifying the concrete form of the

KTiO scaling d is t r ibu t ion . The change of those pare/Deters with

energy ( i . e . the change of "const," or of "const,," with ener-

gy"1 involves the char;?.?- of the corresponding SNC (or ISO -li-

ke) curve with energy.

Several data have been f i t ted t>y neens of the fcrcu3.Se

containing the measure of the leadinr p s r t i c l ? effect , (com-

pare e.£. L3"l ! f 4 J ' l 5 j ' * J t i s "JS'Jally taten ?.sr grantee that

this parameter «c specifies the col l is ion unCer cocsi

Subaitted to the Synposlam

3K •

but otherwise it does not depend on the energy. However in

the present contribution we went1to emphasise that several

data presently available suggest en energv dependent (more

specifically, an oscillating with energy) leading particle

effect. In this way, it is possible tD preserve ar. energy

independent "const," in the second dispersion T.entionea

shove (of eo;irse, in a smaller energy region the parameter

o< can be considered as acnrcxirately energy independent

ouar.tit.y5.

2- One of the siT-plt>st. realizations of the ?r*."0 set-ling

hypothesis starts in the field cf trie cuar.tu- statistics

the assumption that there are only the stne.-.as tie fields.

The well known rrocedurf leads *.c t.-.e 3ssc—Einsvein :"&r:del-

nice=Plsr*ois.-J:olys, e tc , soTpsre e.£.|_6j'' d is t r ibut ion for

n charged par t i c les ,

where <r.) is tne svertpe ci:Bvg%& «-u2 t ip l io i ty a.ir. ".' cen be

considered af the sverfics nuTter cf cel ls i.i The ccrrespon-

ding pnase spsce (or as tr.e aversre r.i;r:ber ef TiDdes). I t isths t

well knr-v/nV'the appropriate l i ^ ' t i r . r nroceaiire leads froc. (1)

to the faTTie distribLitiar. r_ .

<n>F {z=n/<rn">i = Vl" : 1 ' ^ 1 %""'s / f(V) (2)

revealing the :<X0 scaling property. This distribution is

quite often applied for f i t t i ng the charged multiplici ty da-

ta (co"psre for instance tne desh-dottea curve in Fip.S of

r e f . [ 7 j ) . The Sose-2instein distrisutdeii (1) i t se l f was ap-

plied recently e.g. xv the pp and pp coll isions in TB") and

315

TQ] (even If the Bose-Einatein distribution (1) does not

exhibit the KNO scalir.ff property, i t s t i l l can be considered

as B scaled distribution arising in a corresponding linsit

from a more general distribution which is expl ic i t ly known;

compare e.g. |_6j , [_10j ) •

3 . The increasing- energy stimulated the search for gene-

ralizations of (1) and thereby alao of (2K Especially, the

Perina-McGill distribution £l i \ ,["l2j eppears te generalize

the Bose-Einstein distribution (11 in auch a sense that in

every (phase spacel ce l l e superposition of stochastic (with

the mean occupational number <a^)) and coherent (with <n^>)

field.; is allowed. In the corresponding U n i t , this d i s tr i -

bution again scales (in the variable 8-n/<a> where ^n>=<h^ +

+<fn-i>) and i t s appliaation can be foun-d in several papers

(e.g. fl3]). •

4. the next generalisation in this direction involves

the assumption that besides the cells with superpositions of

stochastic and coherent fields there are przaent also cell*

•with pare coherent fields (and just those fields characteri-

ze the influence of the leading particle). The corresponding

probability Pj^fn) to observe n charged seconaaries eat b*

expressed in the fora of the Pefina-Horak distribution £l*l

which, in the appropriate Halt [15] leads to the following

scaled distribution,

pjj (3)where

In

316

n - « ) / l < r r . > -«), ( 5 a )

£ [n ] R ^ = ( 1 - ^ ) A nT> (5b)

and I (at) represents the sodified Bessel function of the

f irst kind. Moreover, »< , the iseasure of the leading psrticle

effect J_16J specifies the mean occupational number in the pu-

rely coherent field while that nunber m the coherent compo-

nent of the superposition is sTn^ -<*' so thet the full oc-

cupational number in the coherent fields is .x + (<nn>-^} =

=<&„">. Let us ncte that the distribution (4) is seeled in the

verlable z^ , re l . (5e ) . The change of the parameter «>< v:ith

the energif brings only an additional encrg^Sependence into

the sealing variable z^ *hich. anyveajr, already depends on the

energy via the average smltiplicity. Then the f i t of the da-

te ty mean* of the *c*ltng function. <<) in which V. anc" R are

energy independent <in .• giv-en range or'energies> assures the

ftxietenee of a •c«ling. This kind of scaling requires or.ly a

very stal l po-3ificstlofl of the Peynnsaii scaling assuaptions

JlT^which represent an ingredient on the way to the KKO sca-

l ing. \\ie ncte that with<«-»0 rel.(») leads to th? scaling

fancti^wi wJilcfc.JFolljwi* from the Perine-icSill distribution

an4, atwecver, with Re-* 0 the gawna distribution i2~> is ob-

tained. >

4* The charged maltiplielty mooents <n^> ^jLa^ V{a) —»

P(n) d» can be expressed in the fora

I:f^ «x)fvi (6)

where V- = j *" >iz). ds. For tl>e C nonents and for the dis-** C

persions S we obtein,

317

... ^ (7)

and

J"u (8)

•ffe obtain from (7),

< 9 )

and from (6),

? of (Dq) (10)

where

X - T~ /"q\f-l)q+k (V - 1) flit

Considering the scaling function y 8 S i* i" given by

rel.(4) we obtain

V0 =

wnere F is tne confluent hypergeometricml function. The

iiomalization of,-i , re l . (4) is such that V = V = 1. Ta

king into account (12) and (11) we obtain

(13b)

l+R2t?

4] , etc . (IJc)

5. In conclusion we mention at least one application

of the aforementioned relations. ?irsx of a l l , as i t i s seen,

re l s ( l l ) or (13) contain only t:#o parameters, ffl and E» , en-

tering the scaling function (4) . As far* as those two

318

fig.l. The dependence of the para-meter « sess'jring the leading parti-cle effect given by (10*. (for thesecond dispersion, q=2) or. the eve™rage charges aaltiplieity, for theBuonrpPoto.i scattering d*t» publis-hed in per. [?],

ters do not depend on tne energy (in a given energy range),

rel.(lO) can be expressed in the form 3 = const,(<C.T>-o< )_ j H e *

with const^X^ q being independent on the energy.

In the case of the auon-proton scattering the dispersi-

ons D2 have been

published *J] in

the cms energy

range 4 4 >T £ 20

GeV. Inserting

tnea subsequently

in rel.no) and

using rel.(13a)

with W = l ani

R = 3-02 {for the

whole ass energy

range under con-

sideration) -lie

obtain the depen-

dence as it i» seen ir. Fig.I- This result suggests the exis-

tence of ©eeillation* of the parameter measuring the leading

particle effect, with the (cms) energy. At the same time,

the-corresponding KJTO like dependence, compare pel.(3), is

sufficiently scearste. This result aeans that in the present

case the function « ) is 6 good scaling function and the va-

riable zK , rel.<5a), is a good scaling variable (we recall

that ia this case (D?) - (DO ).

Similarly, the moments obtained in pp and pp collisicns

•t ISR and SppS collider energies also suggest an energv de-

pendence of the parameter<T

319

R e f e r e n c e s

1 A. Wrobiewsici, Acts Phys. Pol. B 4 (1973) 857.-2 Z. Xoba, H. B. Nielsen and P. Oiesen, Kucl. Phys. B 40

(1972) 317.3 C. P. Viang, Phys. Rev. 130 (1969) 1463-

4 0. Czyzewsky and E. Rybioki, Nucl. Phys. B 47 (1972) 633»

5 J. Dunarchez et al. N. Cim. 66 A (19811 114.

6 V. Bla2ek, Cae=h. J. Pn.ys. B 34 (1984) 838.

7 3*!C Collab., V,. Arneodo et a l . , Mul t ip l ic i t ies of chargedhadrons in 280 GeV/c muon-proton sca t t e r ing . CSRK pre-pr int E?/85-?6, March l a t , 1935.

8 P. Garruthers and C. C. Shin, Phys. Le t t . B 127 (1983)24 2.

9 UA 5. Collab., G. J. Alner et al., A new empirical regula-

rity for multiplicity distributions in place of KHO

scaling. CEHN preprint EP/S5-62, 30 April 1985.

10 Hi. Blazek, Acta Phys. Slov. 29 (1979) 3.

11 J. Perina, Phys. Lett. 24 A (1967) 333.

12 'il. J. WcGill, J. Math. Psychol. 4 (1967) 351.

13 V;. 3i,yajima, Progr. Theor. Phys. 69 (1933) 966;P. Carr-uthers and C. Z. 3Mh, Phys. Lett. 137 B (1984)

425;M. Hla2e!t, Proc. Int. Conf. "Quarks'84", Tbilisi, Acad.

3ci. USSR, 1984 (to be published"..

14 J. Pefina and H. Horak, J. Phys. A 2 (1969) 702.

15 "i. 31aSek and T. Elaaek, Phys. Lett. 3 (in press).

16 R. \!jiller, Haol. Phys. B 74 (1974) 145.

17 3. P. Feynnan, Phys. Her. Lett. 23 (1969) 1415-

18 W. Blale'K, to be published.

321

POLARISATION SFFE'CT OF THE p-KSSGN IK

IKTSHACTIOKS AT 5.7, 12, 22.5 SeVc

LUDSSILA - collaboration

(Alma Ats-Dubna-Heleinki-KcSiee-Liverpool-Kcscow-Prague-Tbilisi)

Presented by V. §in>ak, Ins t i tu te of Physics, CSA7, Prague

Czechoslovakia

Study of spin dependence of mu l t i pa r t i . l e production

i s of considerable in te res t due to unexpected strong

polar isat ion effect in many experiments [ l l . Our f i r s t

indication of the (5° spin alignment e t 22.4 GeV/c we

observed in a preliminary study C2 3 and confirmed ia ref .

[ 3 l • Our present analysis i s based an inclusive pp date

a t 22.4 and 12 GeV/c and on f i t t ed channels pp -. 2 3T* 2 V~

+ neutrals of 5.7 exclusive data ( - 4 5 000, - 9 0 000 anfl

-35 000 eventt i . e . 1.2, 2.2 and 3.3 e v / ^ b ) . The 5.7 GeV/e

data consist of 574 ev. pp -• 2 T + 2 V , 6 435 ev. pp -•

2 » + 2 S ~ 5T* and 28 540 ev. pp - . 2 3f+ 2 1 " X°.

The effective mass spectrum in the rang* *

is described using standard f i t t i n g formula

where <Sj a*-& 1^ are corresponding cross-aeetions and

normalisation integrals, the relat ivist ie p-wa>e B.-l. function

BW(M)

is the decay aonentusu For the combinatorial background ti»

exponential parametrization BGCB)=q.«xp{- fit K) has been

322

where the factor q takes into account the JT 3T "* phase

•pact. The fixed values T^ = 155 MeV/c2 and K,* 770 HeV/c2

was used, leaving only 3 parameters to be fitted (the

background slope /I , and the croaa-eections Gl » C B Q ) «

The effective oass distributions of 3F 7T * pairs produced

in various cos & £ intervale ( © j, is an angle between

<]?•* and ? ii t "pf. * "p£> J ift the res t system of the

^ -sieson) after background subfraction (as illustrated in

fig. 1) enables the construction of cos © j distribution

(fig.l) of'•*"*" from P0-decay. The cos £L distribution in

Q »• 5T* 3T"" decay has the fora

ci

TThe ^ p eleaient ccrrespoads to the probability of the zero

<9-neson spin projection on the reference axis {for the

iaotropy ^Je = 1/3 >.

Our results (fig.2) point out 5, 4 aad 3 atandard devia-

tions froia the uniform eca &r distribution for pp —* ^>° + X

at 5-7j 12 and 22.4 GeV/c respectively, with a large probability

of zero spin projection on the noroal to the production plane.

SiaiXar effect could be seen also for o* Meson 5.7 QeV/c

data (tig. 2. ).

It is interesting to note that in pp -% Q° * X at

24 GeV/c [4} . the <»^0 » 1/3,corresponding to the ieotropy

distribution fel * t sake possible to connect the observed

&m spin alignoent with the annihilation sechenisa (but we

323

Bust pay a t t e n t i o n to a t t r i b u t e the d i f f erence in pp and pp

to ann ih i la t ion channel only ) . The assumption of

n e g l i g i b l e spin alignment in nonannihi lat ion channels and

roughly equal values c f ^ J o for 5 .7 , 1£ and 2 2 . 4

GeV/c data (with lOOSi, 70%, and 50* a n n i h i l a t i o n

r e s p e c t i v e l y ) however would ind ica te an increase of the f » .

alignment in an n ih i l a t i on channels with increas ing priaary

momenta. I f such an increase sa turates at aoae prinary energy

we can expect a decrease of <? * f o r higher primary

momenta of p [ 5 ] •

in increase of P ^ with increas ing p { ^ )

at 5.7 and 12 GeV/c data ia observed ( f i g . 4 ) . S O M motel* f o r

p o l a r i s a t i o n mechanism in pp i n t e r a c t i o n s could be found

elsewhere £ 6 "J .

REFER2KCES

[ l ] See e . g . kit Oonf .Sroe. , He. 95 Par t , and F i e l d s ,

Substr ies No. 26 , Mew Tork, 1963: High Energy Spin Phyaice

1962.

O.I . BrailoT*. e t a l . , Lud«ila Ce l laborat ion , Hucl.Jfcya.,

B137 (1978) 29.

[3 ) M.1. Batyaaya e t a l . , Laomila Col laborat ion , Csech.J .Phya. ,

B2I (1961) 1353; Trot, mt the Coat, an l u l t i p a r t i c l *

Hrnej»ica, »ctre Baw ,OSl,ABfu*t 19KL, p . 2CU

[4} T. Blobel e t a l . , n s j r a . i e t t . , 4M (1974i 7 3 .

[5 ] V.Y. Babintzev e t a l . f Xad .F iz . , 39 (1984) 1189.

U 3 J . Ftafnik e t a l . , C t e c h . J . R i y s . , B33 (1963) 889 .

R.Lednicky, Czech.J .Phya. , B33 (1983) 1177.

CAPTIOUS

Fig. 1. Effective S » M distributions of ff* 3"" pairs

produced in various cos @j - intervals in transversity

frame ( X H C Px x P< J ) in the reaction

pp •* 2X* M r " • neutrals at 5.7 3eV/c after

substraction.

Fig. 2. £« decay angular distribution* in the transversity

tram* Th* cunrea tn4 j?^ represent the results

of the fits t<y *q (2).

Fif. 3. £>*• decay angular iistribution* in the transversity

frsdM for the data 5.7 GeV/c in different intervale

of Jciaenatical variables.

Tig. 4. The dependence of the ^ ^ on p2 of the o»°-

- I .

2000

-h &

I* CeV/t ft H OeV/c

• J «0.5J10.05

» -'• 0 I. -I.

eas 0 r /

5.7

,

04-1 9 l r ^

-I. 0 I -I 0 -I.

rfT

326

Co1

t

1• J - 1

} % -"*. O •"» -

s s "*i i

i

1 'i : '

' c ?l i i>-7~l

A « V- 6i

*>

%

b

t

if

G> I

327

OS THE CONSTITUENT QU/LBK CROSS SBCTI08S

AND AVERAGE CHARGED MULTIPLICITIES

V. Simak and J. Vavra *

Inst i tute of Rjysi.es of CSiY, fragile

Czechoslovakia

We have revised additive quark aodal for i n e l a s t i c

hadronic cross sect ions with i t s implications to the average

charged m u l t i p l i c i t i e s . She correction to addit ive quark

model (AQK) ia on the l e v e l of 3 ab onljr i f we take into

account a l l possible luark-quark cross s e c t i o n s . We have

attributed average charged m u l t i p l i c i t i e s to the const i tuent

quarks cross sect ions and found soae r e g u l a r i t i e s accroding

to flavour of the quarks.

Total and i n e l a s t i c cross sect ions ia

Several studies of the total cross sect ions has been

devoted to the regular i t ies in to ta l cross sect ions .£*•! •

The most popular among the models appeared to be additive

quark aodel [%] . I t s version [ }} with the two cooponent

poaeron exchange could a lso explain the recently Matured

hyperon cross sections f4 3 • there a l so have been published

aoae atteaps to c lass i fy tbe a u l t i p l i c i t j af secoaftary

part ic les according to flavour of hadrons £ 5 ] and recently

the s ignif icant differences i a average charged m u l t i p l i c i t i e s

for ccson-baryon ana baryon-baiyon interactions have bate

noted [ 6 ] .

l ) On leave froa the Technical University, Praqua, Chechoslovakia

328

In the previous paper we have studied various approaches

to the analysis of the total cross sections in frame of

additive quark codcl £7] • We had solved the system of

linear equations

-ft•here GL^'.

and fe i* & correction factor to AQ£ and 5 t ( « K ' ) s ( ; .

HtF|», &}, 1T*/v1p'lF«',*£**) . The satisfaotoiy comparison

with hyperon cross sections waj gcofi when we assumed the

fii^, * 0 for mitleeis-aucleon interactie.iis and /$*^. * °

witli confiitien Ag^ * Ari*^2- ôr E*&on~RI*clet>n interactions

Ihe solution of •<} (1) for €"«*'anfi Afu could fiepehd

e& the input of 4Sr C « **r) . tn different solution the GOc'

»nd / b ^ differ urithin t*o standard deviatiens giren by

experimentel *rrors auC fittitig procedure, aproximatirf the

€Tr * fl.«>s + * jt*»'5 « Ihe ateraee values of constituent

eroas sectione <*,,' anfi îtw are pre«er.t«d in Fig 2. the

uftused (5" (M**O »nd recalculated fr«a G*.«'and A have

natchet! with experimental values within the errors.

The small velu* of ^ ^ ( F i g ^ ) compared with hadronic

cross section* and i ts decrease with s -» •• support the

•inple additive quark nodel (1). All calculations have been

donne for Sc > 3 QeV.

329

Multiplicities in AQK

If we consider that the observed multiplicities could

be a result of an individual constituent quark-quark inte-

ractions we may adapt the similar additivity as for total

cross sections (eq (1) ) for the integrated inclusive cross

sections:

where <^*Ou.'correspond to the aultiplicity of the charged

particles produced in the individual quark-quark interactions

and the ^<*">/k is aultiplieity correction to the additivity.

Experimental data have been •proxiaattd by ***J •••*•/»*«« Ats.

As we have no complete set of experimental data of multiplicities

with neutron target we use experimental multiplicities with

proton target and solve eg (2) under the assumptions suggested

by results on ^

iZThen we have six different unknown elementary interactions

uu, uu, ud, su, su, /S , for which we can determine both €*9 ? /

and < f ^ f / from £^( »*') and * <•">!,*'. The eorrespondiqg

solution of eq (2) under conditions (3) is presented in Pig.4.

Interesting result seems to be the rapid increase of multiplicity

330

. which ia associates with non additive part of inelastic

cross section /istJ .

From the general rules of additive quark model (2) one

can derive the relations among the average multiplities and

inelastic cross sections, which do not depend on correction

factor fi end *"- t '» with assumption

aao <'<H> .<;.».> we have obtained relations:

(4)

*\

(5)

which ere coapared r i th experiment in Fig.5 .

In all calculations for v««' and *^M\» we have

two types of cinematics : (i) without kineaatie correction,

i.e. s(q,q))=s(H,H^ and (ii) with kineaatic correction »(q,q')=

s(H,H')/lAB) where k,h are the number of constituent quarks

in badrons CA=2,3; B=3). There seeos to be no easentialy

difference ixi the results in respect to the type of used kine-

matics. However the kinematics (ii) leads to the corrections

to A3! which are leas sensitive to the primary energy (Fig.3).

HEFERESCES

[l] K. Johnson, S.B. Freimn, fhys.Bev.Lett. 14 (1965)189;

P.G.O. Preund, Ft1y3.RsT.Lett. 15 (1965) 929.

[2 3 S.X. Levin, L.L.Frankfurt, JET? Lett. 2 (1965) 65;

H.J.Lipkin, F. Scheck, I*j'».Rev.Lett. 16 (1966) 71.

331

[2] E.K. Levin, L.L. Frankfurt, J£TP Lett. 2 (1965) 65;

K.J. Lipkin, F. Scheck, Fhys.Be^.Lett. 16 (1966) 71.

H.J. Lipkin, Rays.Rev.Lett. 16 (1966) 1015;Z.R\ys. 2Û2

(1967) 414; Nucl.Phys. B78 (1974) 381; Fłjys.Lett. B56

(1975) 76; Phys.Rev. Dll (1975) 1827; Itoye.Re*. 017

(1978) 366.

[3] H.J. Lipkin, Kucl.Phys. B214 (1983) 13/6.

[A] S.F. Biagi et a l . , »uel.Hjpa. B186 (1981) 1.

[5j A. Wroblewski, Acta Phys.Pol. B16 (1965) 379.

[ 6 ] A. Wroblewski, Acts Phys.Pol. B15 (1964) 785.

[ 7 ] V. Siœék, J. Vsvra, Czech.J.Rjya. B34 (1964) 635.

FIGURE CAPTIOKS

Fi£ . l Comparision of Ç (*"(») and Cr(?'^) with A3i of the

Q' (2) calculated from €T(H,H') .

Fig.3 The effectirity of tl» A3*' «Vi'•>/*,« * e * fua&tion of

priaary energy. The dashed lines correspond to the

kir.eirgtics ( i i ) (see text) .

Fig.4 Average charged sult ipl ic i t i«« <."*> • and ^**Â calcu-

lated from «9 . (2) for / % # ' * a n d /* l w ,« / * « fa. •

Fig.5 Coaparison of predictions of AQK for <'*>_ awl

•qs. (4), (5) with experiment.

332

•

b

a. JW^ * !

\

in

*

\ • • • *

V 4 < S

333

V \

\ \\ \

\

A

IsV

Xk

\\

A

i\A\vV

V

[—

\

1

[\riii

i

\A\\\

}

.

\1

#

Irtl \t \I \r \»V

-I

I

$

sttil

3

1

334

A

V

T * -r -> ^

fV

335

RECENT RESULTS OK COHERENT Kit INCOHERENT

PARTICLE PRODUCTION IK V* - NUCLEUS

INTERACTIONS 3ELC« 30 GeV

SKA "I" - Oolite, r i t i e r .(IHZF Serlia-Seuthen - IKE? Serpukhov)

presented by R. Nahnhauer, IHEP Berlin-Zeutnen, GDR

1. IntroauJtisaITae data v<& presert in ths- iOllrwing coae i rc~ an exposure

of the heavy freon f i l l e c 'oucbie chanter ;jKA.-I TC t:ie T.ide fcaiid(an t i ) aoatr iac oeca, of the iierpukbcv 70 C-eV prctor. acceiere.-"!.l"ne aver-j.ce neutrino crergy i:i cur £jrperime»t lo a t OUT 6 •- 7 iisVuepisriQsiii on the s i s t i s t i c s used. iT'hc pi*ope^xles of "lie cbaiiLêrliquid (ir.adronio interact^cx. ler.£-tfc fy.m~ 6i- s&, radiat ion lenj ihX* = 1'i ca) provide B COOO iuent i f ics t icn c l auoas, b&arcas BH£

photons ia the rira.1 s t a t e . The c'aanber i s therefore well suitedto stucy the coherent pioE producticr. .:-st.otioiis

inclusive straiite particle production

vA—+*~K*(A)X (4)<inc ounalative proton production reactions

C6)

«, X (7)

™ae statistics available for tfce following iiicrrreapoacs to k.730) S26C1 c"r rge current (ant;'; :: Usvsats.

326

ir. this analysis v?e investigate for the first r>.e a l lthree possible ccr.erent pi on production Dh&imsl;; of •„•£&:: char-ged asc neurral oiirrent interaiticr-s in c~iS &;:p eric int. ?r.isallows T:S tc deterr-inc i t & nods! independent way the iso spin,structure of the e::ial v.-ea*: current.

5?OK tne inclusive strange particle- reaction (4) v;e cal-culate the tc;al strange anc charr; particle crest section isthe unexplored low e-r.ergj- rsgior. near t;:e proauctior. t'irss.:oldc.?or ".'" ezitJ^t '£ v.e "easure also ~he. fj?tt.r~.entation -u:.ci;i&:is.

For curauiative t;rotonE ae trj- to Ein.sv.sr tiJ? questicr. tov.'iiici: e::ter.s -heir productior. is. cue tc £r.ovsr. eficcts 1 i!-:=.•ir.traiiucletr aascadlr^: cr to theoretically treiictse pr&nonsr-Elike nucleor. clusters v;itsin nuclei.

rent vios proauetior.

Sh€ coberest prcduCtic:: of rior.s in nsutriao scatteringo i" nuclei i t slietohpd ir fic« 1. -he c.i:s.j-c:ci cr neutral v sakcurrent covpies tc. tbe pseudoecais-r ;ion rie tie dive~£;€:-c«of the azi&l vfector aaxTent. B;* reasons cf PCAC i t s crostsection ir. proporvional to the ;rcss iti^tiC'ri o. coherentelas t icC - 2C£t-er.L-.i 'VK. t.ie nuclsar targes / I / .

She first f>fcs&ri £yioxi oi neutral current induced sinjileHeutrai pior. production has" feeCE rsportec fron tie ji£cho:;lad ova experiEiKst /2./ and oonf i-rsed r-y 3ar£anelle data / 3 /and reeiatlv '^T tht CKaJCi - e^periaer.t /i/. Oc::s:-sr;t di-.'ii-a:.-ti-s-6 charged oiyrrc-r.t iiEteractions iaducsa by ant insist fineshave ts-es found in t BS3C - esperisaent / ? / .

.-ocseie for ecljerft-nt picifc production based on POAC have.beea d.feveloj;eu l;j-. iacfcaer / c / and P.ein ana Sehsal / ? / . Shelast nod^L decprioes tiie available dats rather ".veil.

fis-6 general feature oT coherent processes i s that tha noi.-.en-tuci transfer /i/ tc the target auclsvs' is .saail so that the effec-tive diaeiifficn o-f space iaveivefi in the iTitsraction is large3OEj>ared tc the diaecsionE of ths tarset. s-na this o:ie c&nderive the coherenoe condition

337

?:2 v.lth -. = i:. • : 3 a : -c, • v i / r . ^ i . i : ;-.;.

Oi-.o -EC c::? fact, thr-t co/:ere:it ittirco'ior.c Talre rlt'.-e en

tho nuclûr .ic z. v.acle J.:-.L tc-32v.ee of t?c sr^.ll "or.ei.tur. trar.s-

for th« ati-uc:: sueleuc re.nci.ir irice-tectoi.

JOT cufi'ieienilr s:x.ll values of /t/1 we s:rpect poner-cn e?:-

i:b:.r;,_'e tc be tbe co;.:i a2at p;-cc££; ir. tlie ".-criaraiel, i . s . the

reaction i ; ecccntiall;.- ui;"fra:tive /"/. It is tr.sr. poesitle ty

cti;cyinr: t;ic properties oi" tbe final state :3eEor. xc ge- inforKa.-

tion about the space tiras ctrustui's {for .cefinitione of variables

3cs i'i,j. 2) aiid quantma nuubfcrs cf the v;ss.i: current. 2he isosrra-

lar a::ial vector coupling is for instsjioe detemined i"

^

* 22.

In the standard model of nea:: interactions one gets p »4

azzvs.Lr.c tiic-.t the a::ial vector conponentc of cijarge;; and neutral

cujr-jntr fern an triplex in isospace.

Coherent pseudoscalar meson production can only taire place

via the longitudinal polarised state of the 7: ~ (2.) &a.d cor-

responding interference tarris /9/. Contributions froa the inter-

ference load to an scj-raetric behaviour of the cos 4 distri-

bution vfhure & ic the acii.iuthsl ans;le between lej-ton anc "ncdron

plane (.^l£. 2). Hscently auc'c an effect has been obse.rved in

ciffrcctive pion production /10/.

?or the analysis of rections (1) snd (2) -;e fcllov.'£ tie

procedure of /5/> ê cevide our data in twe subsaiiples.

i) coherent candidates: two prong events frith nuor. «.nd pion

which have no dov; protons or pions (see fie. 3a)

ii) control sanple: tr/o prenr; events v?ith one or nore stubs

with a naxinal aonentur. of 5C0 MeV/c neglected for cal-

culating the kinejnatical variables. ?or the neutrino

data we add also the quaeei-elastic reaction Vw. ~*f* (•

to the control sample considering the proton to be a T*

(see fie. 3b).

338

I'hc /V diitrisuticnc f~r sample.; i) ani i i) ^rs --'•"-"- --- fic-4.

/ ' t / i 0.15 (-iev/o)". -s ocserve a cictr ooijercr.t oinrâl at•3B.11 / t / . la.raaetrisi.'iir tiia ais~ributioris v.itr. an e?:ponentiaifunction expC-bt) v,'t fins for the siaj-e cv =(17.i-'-2.7)(5sV/s}"'1

and ; • =(i.'l.c-7.i)(ieV/c)™£ in agreeiiisiit '.vith v.nat ".vs err~s;tfroc con&rsnt scatt^rin^ tailing intc account z .„ cffc-cti: ar.-experimental resolution. Iron. fir . - wo oaliulate fas ratios ofcoherent t-o total -_r.ciusive charged current; V - scatterir.- tcbe

v

, , . j 0

at ar. average ener&y of 7 G°V.

In f ig . 5 '-'e show the (iis-tributiin of xhe oosine -if tbiasicutfcal angle cos S for reactioris (1) sa.c (2) . Both V tnaV -d i s t r ibu t ions show aa asyiy&a-jric beu&vicur wcisn le&ds tonoa-zero vE.lyes ?i <ooe^> j = 0.17-O.Oi ax:d < o c s ^ = O.29-O.1-.As shov.x. in / 6 / a posi t ive <»ccs^> origins.ter. from a trar.s.-erie-

a l intsri 'ersnce p a n cf the intermediate vector "OOSOIJ

?or the coherent T"c production reactior. (3) •••'£ select

events: vfith or_L3' photons in the fiaai state (see fig. Jc). '.Vs

aesand tbat the vertex point of t e first if1 convarsicn is -.vithii:

a reduces fiducial volume of 1.5 a" and correct for corresponding

losses ano scancing efficiencies. Fcr i;hs baokgroiuic studies

we use data of the reaction "Op—•*>pT* and laonte Carlo calcu-

lations for The reactionvn-»**»?"*(sec fig. 3d) which is the

sain source of bacigrcKina for coherent 5"* production. To cal-

culate the cress section for reaction (3) we studied ths distribu-

tion of ths angle of the first decay Y v-'i"h the neutrino direc-

tion. As one can see frca fig. 6 a clear peat at srnail angles

appears in contrast to what is erpec.ee! from tie tackgrousfi

339

reactions. (The comparison with the limit's Carle nodel has been

aozie normalizing; octh distribution tc each other for C^*£*s0»*x' )

Caking into account all neccesearing efficiencies we calculate

a ratio of coherent events tc charged currant V - sven-;s

To demonstrate that the observed signal is due t o T ° - produc-

tion we show in fig. 7 the yy - nass distribution. A peal: at

is visible. »'e want tc remark however that it is very

complicated to identify the second decay u from the 7 c under

our experimental conditions, therefore the two particle dietriou-

tions &.re strongly biased. ^

kAssuming a dependence of di^ on tie atonic number A of

we calculate now the coherent cross section for reactions (1) - (

and find:

L T*) = (134^20). 10"40 ca2-/ < nucleus>

r") = (15S±61)«1C-40 an2fi nucleus >

&Z*, (**') = (. 7S-2£).1O-40 ca;2/Vnucleus >

Deriving the azial vector isovector coupling constant jb from

the cross section ratio of reactions (3) to (1) we get

//&/ = 1.09^0.21

in fcOOd agreement with tfitl = 1 as predicted in the standard aofiel.

•Hhe theoretical expectation for the cress sections In tne

Rein-Sehgs.1 model /?/ for our energy and average rajoleus is

again in good agreement with .our ise&sured' values .

3. Eeutral strange particle productioi:

The dais sample -sec' In this anai;-eljs cor.sist£ c:' 6322 charred

current evente in a fiducial volume of 2.1 x-\ Tz*- c:served nuai>ers

ol strange pai-ticles are givei. in table T. ?or the calculation

of the true nunber of 7°'s produced losses fron the fitting proce-

340

iure, the taininua decay length, the chamber geonetry, the V

interaction length and unseen decay modes are taken into account.

i A ufc

1

corrected number

452*45

196^23 ~

^2*37

21*21

ra

7

3.

1.

0.

te /5

1*0.

1*0.

5*0.

3*0.

6

3

se;-tisE

i on iaospir sycraevyj charge conjugation and ccne fc~ps-

facts one aac r&iste ti:e sotal strands particle cross

to vae cross secticr. of neutral strange particles by /11/

Using the values givers in table 1 we calculate

r xhs estr.'-atior. of the total chare cross seciion the relativeiautioas of £.11 strange as,d charn particle production chan-

nels have been calculated for o\xr energjr it-Eicn in the fraae'.vorioi tits ptrtc^ a<>6ei r'cliov;iS5 rsi', /11 / . ?or the calsulctios. v:«hevs used tiie radios *:

B/^Y * O.1£, 3B/S * 3 3Ei the etrangensssiujfrsssic^ ft.;ttr icr ^ragnientatiGr. /^".A . i.bove the ei;ariiThrcsiiola ••"» S GV* we estis»t€ tbsa

In fig. c v;e give the cepead&see of the fractions of producedll°'s and C 'e on •uise av&ils.ole hadrocic- energy .squared ".V .u'hsrees the fraction of K 0 | 3 rises logarithmically v.itij theeaer£3't ?(A\) shows a threshold effect at lew W and than i tbeha-'es approsimttsly coastext.

She fragnentaxion fuactioas for K°'s ana A 's are aonparedir. fie. :i to those of other ne^trino-eEperin-erts/JiA/^/* J«-S or scan see for z ^0.3 already rather good agreeusnt between thedata at different energies has 'Been reached.

341

4. Cumulative protoa production

Protons are called cumulative in the following if they go

backward with respect to the current a ire lit ion in the laboratory

system, ,'ie study their production in the laoaentuc* region"

0.3.6 p * 0.7 G-eV/c where protons can clearly be identified under

our experimental conditions. 2he analysis is based on 3130 (730)

events in the charged current ^( V) nucleus reactions and 500*

nucleus interactions.

We compare our data with an intranuclear cascade nioael /14/

including a special two nueleon mode F T a —*• pp of pi on absorp-

tion. It is known that the cross section of this process has a

rather sharp (oa iaun in the momeEtua range pT •*• 0.25 GeV/c.

furthermore it has been shovm /15» 16/ that the production of

secondary protons by the absorption of * mesons is at least

3 times stronger than oy 5" " aeson.3. To check this we sfcov in

rig. 10 the momentuia dependent ratio of 7T" "to *~+ assort tor

events with and without cumulative protons. She ratio agrees

for both classes of events esept for the region Sj_= O.2-C.3 GeV/c i.e.

the range where the absorption cross section is aaxiaai.

In fig. 11 we give the angular distributions sf-pret-one in

two momentum intervals measured in the v* -t-experimeEt. The data

are compared -with the intranuclear cascade model -with and without

pion absorption. As one can see the contribution of the process

T"(BS) —** M is sro%7ing both with the proton production angle*

and with its momentum. The nodel including this process describes

the data well.

Cumulative protons must be produced independently if their

source is secondary intranuclear rescattering. Then their multi-

plicity distribution (fig. 12) should be described by a ?6isaon

distribution. Such a curve (dashed line in fig. 12) noraalizac

to ITg = 0 does not describe the data. However the noraalisatioas

is not correct because e.g. quasi-fres interactions at the peri-

phery of the nucleus can not produce cumulative protons. There-

fore we fitted only the experimental data wita ST?*^ tc a Pcisscs •

distribution and found good agreement for ali sorts of beais par-

ticles (solid line in fig. 12).

342

Again the cascade model including absorption (full line

histogram) describes the data well in contr&s. to the nodel

without; this process (dashed lijie histograa).

In V A-interactions we investigated the correlation between

the mean multiplicity of cumulative protons £ !;_,> ana tie Eus'cer

of protons H (with p >.3 GeV/e) in the forward hemisphere.

Normally it is assumed that I? is proportional to the nurr.oer

of intranucleer interactions. As ore can see in fig. 13 the

experimental dependence of <ci;=> on E is nearly- linear agair-

supporting the assumption that intranuclear resoatterir^ is

the source cf c maulstire proton production-. Both aodeis of

intranuoleer re scattering, show &,lso s.n increase of C "p> with 1; .

!Ehe one including the two nuclear afcsort:-;ion r.iode ci" pious esf-

oribes the data well (full line).

5. SumK&ryStudying oonerent charge and ;iei.:tral pion producticr. v.i

h&v& measursro for the first tiae in cr.e e::peri~;jar:t a i l tares .states of the isoxriplc-t of the axial -,vsak current.

?or the i seep is 1 axial vector coupling con a taut v.'t derivemodel indeper.isr.tly //>/» 1.0?-C.?1.

»e meesure an asvasieTrit angular distribution fcr ths asi-mutisal engie petvseeji leftonic and naaror.ic plane, indicatingthe pi'esaiiOB ot iaterêrence t-c-ras is tiis weai ourrest.

Our results are ir. good agreement -ith ths 2?C.T.C basea i odelof Bein end Sehgel for coherent tion production.

V.'e exteruSed the inclusive strange particle investigationsto tire lev; energy region and aeter!;:iried the totel strange andcliarsi particle production rates xo be R""1" * 14.4-3-3 <*-• andS • x 7.5-£.c % in agreement with expectations fro;.; other ex-periment s.

Investigating cumulative proton production we SonnS e::peri-mental hints that this process is iue to secondary interactionsin the nucleus instead of scatterii-c en cucieer clusters.

Comparing these results with intranuclear cascade sod elcalculations we found gooc agreeasnt if a tv.-o nuciecn node ofpion absorption has been tai;en into account.

343

/ 1 / i i . i . Acler, Pays. Jtev. 135 (1964) 9c?/2 / K. F&iesaer et . S.I., Pays. Lett. 12?2 (13c3) 23C/ 3 / 2. I s iasa i , 3. 3eix, J. ^crtlzi, II2HA o2 / 23 (15b2/ 4 / ?• Bergsnia ex. a l . , CETJv-SP / S5-4'' (19&5)/ 5 / x. jarage CE. a l . ?hys. Lett. 140B (15S4-) 137/ 6 / K..S. Lsckaer, Sucl. ?hys. 3153 (1??S) 526/ 7 / D. Sein, li.K. Sehgal, IJuoi. Fuys. E223 (1964) S3/ S / O.P. Cbo, IJuol. Phys. 311= (1976) 172/ " / A. 3artla, K. Press ano ViB J^jerct to, Phj/s. Rev. 1/16

<1?77) 2124/1C/ r . Alien et . a l . t CSHK-SI / 85-33 (196=)/ 1 1 / £. Grassier ex. a l . , Ivucl. Pays. 3*94 (15&2) 1/12/ 1". Bosetti et. a l . , liucl. Pvys. E209 (1^?2) 2b

/13/ J.P. .Serge et . a l . , Phye. Rev. Lett. 3fc (1?7S) 127/ K / li.S. he.rs.nov- e t . a l . 2. Phys.. C21 (iyS4) 157/15/ V.ii. .-.sator;iaii e t . a l . , Xaa. Pia. 3^ ("i3&3) £34/16/ G.R. Gulfcarân et . a i .

I *

g

1-i

j

l >

10/ »S03 p

* — ' .

—S &'

sO

•O

u

CO / ^ 503 p / NP

! iA. m

It

* •

T

i •

5^B

o

events/10adegr2

e•fr

•o

iV

!1 a

!i

i ^

1.3•c*

o

g^*—

encombinalions/60 MeV

346

a.a

Fig.10 .

F i g "

V •V:A

*

IT*

\

* « 1 4e t

Fig.13

3i7

SIZE OF 'JKB PAHTICXS BKISSION REGION IK g5IA3?IYISTIC

NUCLSAK COLLISIONS FROM TVO-PAHTICia

Jerzy Bartice

Institute of Buelear RiysicsCracow, Poland

Knowledge of the space-time develojEent of high-energy nuclearc o l l i s i o n s i s e s sen t ia l for understanding the mechanism of these pro-cesses and might help to discriminate between various theore t i ca lmodels. I t has been pointed out by several authors' ' tha t studyingcorrelat ions between ident ica l secondary p a r t i c l e s with email r e l a t i v emomenta can give information on the dimensions ef the endesicn region.This Dethod, analogous t o that proposed by Hanbary-Brovm and Twirs i nac-tronony' ' , i s usually referred t o as t v o - p a r t i c l e interferottetry.I t measures the average separation between the emission s o u r c e s , < 4 r > .I f a certain spat ia l d i s tr ibut ion of sources i a aesumed, then i t al lowsto estiniate the scar, radius of the emission region.

Theoretical formalisz for ident ica l bosons has been developed inr e f s . ~ ^. For the case of ident i ca l pious i t has been shows that two-- p a r t i e l e eraall-angle eorrelat ione are s a t i s f a c t o r i l y described byquantum s t a t i s t i c s a lone . The correlation, function can be written inthe fora

where q - S^Sg , p • ?1+P2 » ^1 2 " fotzr-moeieat* o f p a r t i c l e s ,q0 - lEÊgl . The parameter > ( X i 1 ) depend* on the configuration ofths part ic le eniss ion region, the degree of eofcarence and on s p e c i f i cdynamical correlat ions in the studied process , Th* parameters rand X characterize the space-t ine dimension* of the p a r t i c l e emissionregion.

Pion interferometry was applied t o hadxon-hadxan, hadron-Raeleuaand nucleus-nucleus in terac t ions . In x e f . ' ' the zadiuc of the pi onemission source wae found to be approxisately V.4 fa for three typesof hadron-hadroB interact ions studied (Tp, t p , pp), and references toe a r l i e r papers are a l so glTen. Similar values of the radius of the

Address: ul.Eawiory 26a, 30-05? Krak6*, Poland

346

pic-a. source were obtained for pp and p-Xe collisions . Interac-

tions of pionj- with carbon nuclei were studied in ref.' •''. Results

cf pi as interferometry for collisions of various nuclei have been

given is a number ol papers' . We shall discuss the data on the

Eise cf the pion emission region is collisions involving nuclei as

projectile and target.

In ref..' ' it has been poiatc-d out that in experimental papers

various spatial distributions of the emission sources have been ae-

sused /the surface of a sphere, several forme of Gaussian-type dis-

tributions/, and thus the published values of the "radius" cannot

always be directly compared with results of other authors. In order

to allow such a coaparison, we propose to net the

, which Bean* tlie necessity of correcting some of the published

values of xQ aa showi in Sable 1 of ref. '. Tee conversion factor

to the nas radius ie V3/2 for the Gaussian-type distribution in the

fora exp^r2/*?2) aeeajset! in refc.'4'^', f5 for the distribution

exp(-r /2R ) used in refs./>f , and 1.0 vften the sources are suppo-

sed to be uniformly distributed over the surface of a sphere of ra-

diu« *'1'2'.

The experimental values of th« nws redii of the pien emission

region for collisions of various nuclei have been collected in TA312 1

and plotted vs. A! in FIGUHE 1. Ksey ciearly inareaee with increa-

sing Base of the projectile, A_ • and are dose to the values of the

•effective imclear radius* of the projectile itself, R-« 1.21 /J^5 ,

shown in the Figure with a straight line. The "effective nuclear ra-

dius" it obtained fros seasulcEente ef inelastic /interaction/ cross

sections in collision* of various nuclei, assuming the foncule

/5Vce« e*g» ref.' •"« "Central" collisions i»r«U.y reveal larger radii.

Collisions icvalving nuclei ff.-itt it possible to investigate also

two-proton correlations* It teetss interesting to cccpare them v,lth

tvo-pioc correlations as it is believed that different pertieles are

emitted «t different etages of the inttraction *', end Chue such a

cocparieoE can provide information on the space-tice developEant of

the collision procees.

The ease of identical fencions has bees treated theoretically in

refa,/'7'e''(iioa-relatiTi«tic) ssA in rfcf.^^Crelitivistic). Tor pro-

tcms one should *lso take into account attractive strong and repul-

349

/q/sive Couloab interactions. According to ref. ' , the correlationfunction can be written in the foro

where B0(c,p,r0,T) , Ae(k*) and B i(q,p.rpfT; aeeeribe effect*of quantua statistics, Coulomb and final-state strong inxeractions,respectively, &K « 0.5 V-?*"* Here the paranetere r0 and T wereintroduced assuming a Gaussian-type distribution of particle emis-sion sources. 2he Bean-square raaiue (ras} of the particle emissionregion ie by \/T larger than r0 •

Siae of the proton emission refion in hadron-nucleue collisions/2K_2°'/has been investigated in refs. for carboi*, aeon and zencn

/27/target nuclei, respectively. The xenon data ' are nev - they ccatfros the 180-liter (la long) xencr. bubble charter weposed to thebeam of 5.5 &eV/c negative pione Jroa "ihe ITEP synchrotron inMoscow. Results of protor. interierocetry applied to collision ofnuclei are given in refe. ' ' ~ ' ' . The ros radii of the protoneaiseion region are collected in 2ABLE 2 . Kore data would be neecedin order to snake any systematic behaviour appaxent.

Two-particle interferoaetry allows to estinate not or ly the aearradiue of the particle ecissict region, but also to Sin6 i t s geomet-rical share by a proper selection cf particle paire ia epace angle*- see rvi.1 . I t can ales provide information on the expanaicnvelocity of the particle eaissian rerior.. Ttie has beer, pointed oatrecently t>v 5cctt Pratt , however, a careful look into the litera-ture on the subject revealc that this prc'rlea, wac already treattday Eopylcsv and ?odgorets*y in 1974 . Sctr. antiyses req-jire, how-ever, high statisttee experiments waict xight hopefully be dcr.e inthe near future.

1. G.I. lopyior, K.I.fodgoretsky: Iad.?i«. J^, 392 (i9?2/\ t§, 65CiVblV.', 1£, 434 <?974.;; fs or. J.Kuei. a y s . 21' 215 C1972; i ^gr 5 6(1973); 21, 215 :nn)]

2. 6.1.£opjaov: Ehy6»I«fc. ÔB, 412 (1574,1

3. G.Coccoai: Soys. ie t t . 42E, 459 (19744. F.B.Xano, S.E.KocEir: Jhys.iett. 2§B, 556 (1978.)5. B.Lednitsky, E^l.rodgoxetsiy: J1KR, P2-122O5, Dabaa, 1979

350

£. K.Eiya;;ina: Fr.ys.Lett, 9JB, 195 (1560,'7. S.A.Koctit: Fcys.Lett. 703, 43 (1977)S. K.Sij-a.iisa: Kiys.Lett . 132B,299 '.1933)S. a.ânitBlry, V.i . Iyubcshi tz: Tad.Pis. *£, 1316 (1961) [Sov . J .

Suci.Jbys. 2£> 770 ',1961 ) j10. r..Haaeury-5rows, R.w.Swiss: PhiLKsg. ££, 5S3 (1554,''; Kature

176. 1C4S '.1956;11. K.Dt-.'tscr-rsrji ex &1.: IJuei.Hsys. £?CUt 253 (1982}12. S.LeKarzo e-t a l . : Fr.ys.Hev.B 2£, 3=3 (1984.)13. S.ArigelOT et a l . : 'Iae.?As.x 53, 1257 (19S1}; [Sov.J.Kuel . rhys.

21 , 671 (-i?8i)JU . G.i:.Arakisiu.ftv et a l . : Tac.Fiz. ^ , 543 (1984'15. K.AEhai.a"niar. et £.i.: Z.Jhys.C 26, 2^5 .'1SS4)16. £ .K.Crowe et t l . t i s Prac.7th r'irh r.r.crry Heavy lor. Study,

IS-rBfTadt, ?-1? Ocioter I j e i . GCI-S5-10, !•=.—ttict. 1SrI;;, r - •"-""- I17. *.-.££.,io (.'• a i . : in Ircc.Sth J;irh-rr,e:r£v he£v« i n £-..- ••;• ;'..•..../,

Berkeley, ;&-22 Kay 19&1, L2i~12652, Berkeley, 1?S1, p.35016. i .Seavis et a l . : aye.Rev.C 27, 910 (1983)19. £.£eavis et a l . s Fsys.Rerc.C ££, 2561 (1983)30. S.T.yung e*. a l . : E^ys.IieT.lett. ^ 1 , 159? (197S*21. D.Beavip et a l . : in Proc.7tli Kieh Energy Heavy Ion Study,

Ieras tadt , 6-12 October 1984, GSJ-S5-10, Ifer^ftadt, 1585, p.77122. .T.Bertie, K.r.cwaiski: Phys.Sev.S ^C, 1341 (19B4) ,23. l.'i'enih&ta : ir. I^sc.Tth Eir;h Energy Heavy Ion Study, 2ars=tadt,

8-ii October I9c-i, G£I-8S-tO, I^rr.stadt, 1935, p.41324. E.H&gaaiye : Fcys.Fsev.Lett. £9, 13S3 (1982)25. S.Anpeljr et a i . : i ' td . J ic . ^T, 1357 (1980)2e>. £.i.A2iBW et a l . : Eiys.hev.I) 2£, 130i (19S4J27. J.B&rtfce et a l . : ' preeer.te<: e t tiie Second Iriterrjational Conference

or. 1,'iiCJ.evs-Kucle'jE Ceiiisicr-E., Vieby, 1C-U June 19S5, Abstract2.21 , xc be published

26. E.ialea et a l . : preEected a t th« Second International Conferenceor :;-jcieii«!-irucle«£ Ccllisior.s, Tiscy, 10-U Jure 1935i AbstractE.23 , subaitted tc 2&d.?is. .

25. r.EarbaJci-eh e t a l . : &ys.? .ev. ie t t . 4&, 1268 ('19S1.)30. E.wisnan. e t tl.i in 2rt>c,7tt. High Energy Heavy lea Study,

s t aa t , S-12 Oetefcer 1984, GS1-S5-10, Earastadt, 19S5, p.78331. K.jJeirtrcfcaaiKi et a l . : Muel.Eiy». B103. 198 (1976.)32. Scett FrsL-it : ayE.K«v. le t t . ^2 , 1219 (198*).

351

TABLE 1. Radii (roe) of the pion eaissioa region

E/A,GeV selection ras radius,fm detector Sef.

PPP

d

12C12Ct2C20Ne404040,40404040

56

'AT

Ar

B 200le 200Xe 200

Ta 3.4

Ta

CCTa

2iaF

£C1KC1KC1KC1

Ar Bal,Ar b-iOi'AT PbjO^

J?e

RbSr

3.4

3.43.43.4

1.8

2.11.81.51.21.81.S1.8

1.7

1.2

all inel.all iael.

n> 20

all inel.

all inel.

all inel.

•central*all in«l.all inel.

all inel.all icel.all iael."central41

all iael.all iael.•central"

all iael.

all inel.

1.66 * 0.041.53 * 0.131.45 * 0.11

2.20 i 0.50

2.90 * 0.40

2.753.763.40

2.24

3.S23.536.044.65

4.044.37cs *,

t 0.76t 0.88£ 0.30+ 0.96- 1.95* 0.40* 0.61- "S.10* 0.54* 0.61* 1.35* 1.14S 0.963 **

* 1.47

nstr.c&aia.. 12Stz.cbatt. 12

bub.cha*. 14

but.ohaa. H

bub.ch^cu 1-;

tnib.cbas. 14

magn.apee-tr. 17aapi.spectr. t6str.ehasaiex 1€•tr.chaaber t?

2020

TABiE 2. Radii (xms) of the protaa eaissida r*gia*

.V

i-t

do

•j-i

6

s4He*He

12C12C12C1ZC

40Ar40Ca93H*

k%

C0

0GC

cKC1

taxca

m>

S/A,GeV

3.43.4

3.43.43.43.4

1.81.8

0.4

0.4

•electida *M t*iiv*,£*

aU ia*l,p<30Dall inel,»3QD

all inel.trx.fr.all la«l, otitaiie•cvatral9trg.fr.•eentralfotttaliU

all inel.•ceatal"

all inel.

all ix«l.

ea 6.02.«

5.0 * C34.5 * Q.45.0 * 0.53.5 " 0«-4

2.72.1

ca 4.S a /

c* 6.1 **

tott.chwt.

Wb.cba».buk.cteB.bub.eham.

•tr.cha*.•tx.ebaa.

n.lall

Kl.Eall

«$t5t5

2 |29

30

352

' sr>|pDj SUJJ

Fig. 1

353

'"Institute of~*iu=lsar Stuii'js,' L6iz, I-'oland^'UniversiT cf h&di, .Pisys.i'epart^'srt, £6dzf Poland

?. Introduction

?he present work concerns the investigation cf the electrasasaeticcascades in lead and lead-ssiatillalor calorimeter îiieii art oftenused in e/eulaioa chaaber experiments. The investigations vere per-formed by 'analysis of the cascades using the ^Kite-Carlo method.The simulations were based on the detailed modelling of a three-dimensional propagation of particles allowing for all processes inwhich parviales say participate.

The accurate knowledge of the lateral distributions is indispen-sable ic the analysis of extensive air showers, data tssa lar/re oa-

s, emulsion chambers and other ecscic rsy phenomena at highThe experiments that probe energies significantly beyond

10'- eV aye- r.ade with extensive air showers /SAS/ arrays vhich ovsr-co~5 the very low flux •••itfc very large areas. In the HAS research•ridely applied is tile method cf estimation of funcasental parsne-xers / i . e . a^s parameter and the number of particles/ based on as-sumption ^hat lateral distributions of particles in air Phowc-rs areof the s=ce type as in the electromagnetic cascades. In 3IAS experi-ment i t is difficult to aeas>ire directly the lateral distributionsof particles in individual showers. 'The lateral distributions areusually reconstructed by fitting the theoretical JXu function totne ezpcris(=ntally measured density of particles at several distan-ces from the shower axis. Certain experiseni-al data and the inves-i-i&ations of electroEatnexic cascade in air bj- use of Mcate-Carlcmethod indicate -char .he greneraliy used HZS c-urv-es are pro"baolyinaccurate / the so-called age pasracietar of 5AS obtained on thebasis of the U3<?d lateral distribution. funstiOE. vas surprisinglysnail: saalier tnen 'anity, althouâ the .eoEsides-ed showere v.'erewell belov Eaxisua of thsir dsvelopaent/i

To obtain significant nunber of events above several ffeV, thereare used larê, erouad-based detectors vhere iracleaar ai'.z. 2-raj'filas are exposed for long periods. The inferr.ati c-n about proper-ties of hadronic isreractioas are derived only an tfee basis of de-

354

teeted electromagnetic cascades vitfc visible energies above a thre-shold of .2-2 TeY depending on the defector. The energy of incomingelectron or photon is esti.âtsd fcasinj or. only ssall fraction of allelectrons which are closed in a radius up to feveral hundred nicro-roeters. The proper knowledge about behaviour of the structure func-tions sear the ehover axif ./"cor? approximation"/ i s necessary foraccurate esriisstioa o:. energy. ?;ieEe facts caused that detailed in-vestigations of the electronafr.etie cascades in different absorbersvere -ar.aertaker.,2, Ciutliag of t '.g resalx:- of "core approxir-axior." obtained fror:

-ii-6 FSistir.^ solu-.ions of .îffusicn ecuaticns are obtained u^dersisrplifyiag asru.-rptior.s cDr.cerring the electrcn;a.Tnetic casoade <is-

vftloriner.t. Xr. the eilculs.tio^s the fallowing processes vere taker:is'.o aoco-.it : pair proiuction fcv t.".- ^hctons, bre-Et:-'-hlun-r by c-le-2-trcns /-Jtilis-sc for tae pair r,roijc-vicr» a.-c" bre.".-i:r^lunf crosssecTinjis v.-ere t.-.e expressions valid at e:<vreafi.lv hi?h er.er~ies, theso-called as-—r-stie fcr^^lae/, cclli;icn snerty 1IFS-JE ciffsrei b;-elec-trcr.s arc ~r.ulo=t scat*erir^. 'Jr.c&r srtsh s.ss-j.-3tticr.s the slec-"trcâirr.-etic ?s-z:&cs d: %ficr.-;-r.t do-v.c nor i-r.crid cr. th% absortsr.

Ir. t?;e £3r,l;-ti-=l srlaticna of - iffusicn er.:aticns i t •••;£ frund

ccnbiTiej trciuot Ôr/K£ ani T, but r.ot en £5?sra;E variables Zo >ncir . I t ce?j £.r that if v-5 have r.v.-sr^cs.1 \-;lur-s of the structure f-ii;-ctioE r.ear ^he a::is for s shr--er of certain tr icar- enerr-.' £„, ' ecaj;. prsaict The n-A-srical values fcr a shover c-f different primaryenergy. The cascade curves for scastsnt r?.iii car: be soiled ic t'r.3siirîlar sr.ftr.-e, if th:- mister cf electronc is expressed as a fur-ctic::sof s, vhere s i s the Fh5---er &se of a es.se cd e /s=3t/(t+21r.2c/^.J/. -herealtion ;ai:fe-s i t ciiite sicple to perfcrz the actual cxalysis of theEho--e-r pheacaena near t-he sno .-er aiiis arid i s '.-ideiy used in cali-brntioa snd intfrprstation cf es:êri3sr.ts* This ray be particular?-iaportart i s a cas* of -ccsraio r.ay experimentB '••hea the data are <?::tr£~extrapolated to 6xtreas3.v r.&gh energies.;;. ks&xzvticzi-s. safle in our sisalstjoae

The follo-^'ing processes were taJcen into account in calculations:pair wroductioa and fcraustrahluag - including L?'A effect, ionisationlosses, Ccaptoa effect, single and multiple Cculocb ssatterins, photo-

355electric effect, annihilation of the positron, inelastic scatteringof electrons and positrons with production of delta electrons, "h*landau, Poraeranchuk «nd Kigdai effect makes the cross sections forbremstrahlung aad pair creation decreasing as primary energy increa*sea /?ig.1/« So that, the mean free Bath of shct*er particles i s len-ghened. Hence the cascade curves are deforaed.

7igi1a«?he product of differentialcross-sectioa of breastrahlung aadv, -'her* v=S/2 ;b; The differential oross-ss-srtioa

. of pair production, vraere 3=3/2

The simulation of electromagnetic cascades was saade fortprioaryenergies of particles in the range 50-T00 Gey and 10~1OO TeV#

different primary particles!electron or photon, diff*re«t chaa-bers:pure lead and for lead-scintillator sandwiches.4. Longitudinal and lateral develorasant of the

cascades In pure leadThe longitudinal development of cascades for primary particle

energy of 50-tOO i»eV and 10-100 TeV has P«eE calculated siovrii to athreshold energy 2=1 Me7, The results are presented in ?ig,Z viewcascade curves, for constant radii, ar*-plotted.

fvw \

H

\•

AT

Fig.2 Cascade curves of alectroasradii 50 and 100 .laa.

photon initiate* shovsrs for

Cascade curves present the n-uabers of particiee al>ove fixeathreshold / l KeV/.aa a.function o-f depth .in a » absc-rber. la ?if..3the saaie results as a function of the ag€ parte&tftr are coaparejv;ita the aaaiytieal results. As OB« can sec tpaar results anew- that"core approxiaatioa" /in the early stage Of aevelopnrest/ for. loverenergies i .e . 50-100 5e7 i s rather good wit for higher energies 1PMeffect makes this relation varoag. Tr.e cascade curvee arc defaThe number of particles obtained froa our data change* ironlower than that in the care approxiratien at lovr s to greaterat higher s. These devitîoas bav« significant eon sequence* for th«esti-Eation of the pricary particle eaergy;

356

is 0 m t WWWiS at 01 « s « M «. S

Pig.5 The t raas i t ioa cwrves of electrons obtained In our simulationare compares; with "care apnroxiEation" / t he numbers of electrons for100 fi«V a*i«S TOO ?eT are scaled to 50 GeV and 10 TeV i . e . devided by2 ssA 10 respectively/*

0?i the basis of the presented cascade curves we estimated the energyof The priaary par t ic le tor different depthe of development of cas-cades according1 to core

u t /=»y*

«;6 } *.T

i •«*.* J » . ;

A.6• » » * .

iS.l

e.t

1.3

t . 0

T.3

1.1

S.5

ZM.S

1.2

22*3

22.1370 .0

4*6,0

Tat. 1 The er.*rg^ ,af jsritary -jgio-tan estiiaated according to "coreapprox'i*atlaK" fax rac.i.* TO© JUS ana 50 jam.The reai snscgy. o? pfiagry photon i s : a. i-0 TeV isnd b . 100 SeV

A« oa* cga se», JSie value of energy ie terainated froz our c&taohassrf:.ae frsr: a VaJn*. lO'.-.-er tfasas the energy of priisary tfnoton/•unfles'eEt.iSat,iea pf ajierg.v/ at lc- ' » t s the velut greater thant h i s srt iJi t t*r 9/o?ercstJ)t%tiot: of er.exgy/;

$, lpflasag<i:.o-f th^ in.'&aracts.ecgws structure of .cfaar.ber on the

i* a .sa&d ieh 01 safty aiie«ts of netal pla-tes aftd .auelesa? temasiOE rlatet , bat i t i s difficult to treat thecascaide d«velo.pareat ia ar. iafesecteneou* aedia by theoretical calcu-lations. tTbe structures oif afeaabero iRveetigated in our sinulation' eres eiailar t» that upei Aa ?J»Ai ezp«riaeat / ia the 7XAL experi-ment the enaiisios cha.-r.bsr v;er« exposed to the monoeaergetic e3^c-tror. beajs at energies ap to 300 SeST / , stailar to the big gawa&-bstdroR det-ector U3ed in cosmic, ray experiments , i»e; Paalr aadChacaltaya etiee /aftally tfee costsie ray t lorimi f.er8 have one or

layers o* carbca % such e. ch!arbc has better capability to di-

357

scriainate liadrons from photons and electrons/.

absorber

thickness ofabsorb./cm/

|P»

12

14

Se [Pt (So

• 52

.5 j.5

Pb

1

Sc

.5

90

Pfc 3c

.52

Pb Pb

Tab.2 The structure of chambers investigated in our simulation.So.1:this sandwich i s similar to that used in ?BAi exr>erineat/1/f

No,2:-his one is similar to the big gamaa-hadron detector used incosmic rays research/i.e, Pamir Collaboration /2/ / ;

Comparison cf the transition curves obtained for t'*c different struc-tures of chambers?the lead-ssintillator and the pure lead one isshown in ?ig.4 and 5. As can be seen /i"ig.4/ tne introduced seintil-lator plays an important role, giving sabseoueat reduction on numberof secondary particles /iÂL-and our structure chamber are not exac-tly the sane/. The transition curve obtained in our siaulation for asandwiche structure shows siailar tenders? as in seen in experimenti .e . the smaller electron size than in pure lead chamber.

Pig.4 Cagcade curves of electronsin photon and electron initiatedshov-ers for raiiue 100 Km..

The number of secondary electrons is determined by thickness andkind of absorbers and their localization in sandwiches.. In Fig.5 'pepresent the transition curves for prinary photons of energy of 10 3e7and 100 2eV /for constant radii of 100 Mm/ together vith these obtai-ned in sinilar Paair chamber; The interesting results of tne present

ttcml"ig.5 Cascade curves of electrons in photon initiated shower;

a. for all particles with threshold 2mec2b. for tjae same threshold but in radius 100 Aim.

simulations is that the number of particles after penetration of the

90 cm scintillator and 3 cm lead increases if lead follov this scin-

tillator. ?urthemore the numbers of electrons after first 4-5 cs in

lead are comparable vitn those after 90 cm of scintillator and 5 cs

of lead for primary pnotons oi' energy 100 TeV; One can see taat even

356

priaary photons of energy 10 TeV can penetrate both chambers. Thiseffect i s stronger when ihe prinary energy of pnoton increases and'•'hen tiie "core approxiaation" IS used. I t is important to ciiscussethis problem "because xt xs very fashionable LO look for very stran-ge phenomena. 'xae present results snow the influence of cotn effects:lnnosiogeneous structure of chamber and LK>! effect on the transitioncurves. A detailed simulation would be necessary to quantify this ef-fect with respect to different experiments'. It i s not possible toestimate the energy of primary particle without taking accuratelythese two effects into consideration.

6. Conclusions

ai l t was stated that inhomogeneous structure of chaaber has an ef-fect on the transition curves. It appeared teat titnbers of secon-dary electrons /for constant radius and constant priaary photonenergy/ is determined by thickness and icind of absorbers andtheir position in aasdviefees.

b;It i s not possible to describe the lateral distributions of nar-ticlee with "core apsroxiaation" in^sT-si-ideat sf the prisaryparticle energy because the JJT.'. effect rlays an iarortant ril-? ir.transition curves. The aethod of describing lateral distributions,of electrons by the "core aoproxirati 3n" used in enscber e:-:peri-aents leaaa to underestiiation of the priaarr photon er.er™ atlow values of s and overestinsation at r.ig'-er s.

c.It is interesting res-ilt for sandviche strjct'ore of cr.aabers,si i i iar to these used in oosisic ray research, *iiat the nunfcer ofparticles after penetration of the"thick /90 cc/ saintillator andtnen soae /3 CK/ lead can increase, -his effect is stron-er for"core approxinati-OE11 than for all particles and increases ••••iththe priaary particle energy; It Beans that the interpretation ofSAS experiaental ciata in iisulsion chaaber --.'ould be possible aftertaking into account precise structure of every ehaaber.

Aeknovledgeaent sve «*OU1Q like to express the sincere thanks to Prof. J.vc:o-.-<ezykfor many discussions and suggestions;

5eferencea/ 1 / Hotta K et a l . , 19B0,Phys.aev.3r22,1/2/ Trudy ?IA3J,v.154, Koslcva 19B4

359

ANOMALIES IM GAUGE AND SUPEHSYMMETRIC GAUGE THEORIES

Olivier PiguetDepartement de Physique Ifce'origue

University de Geneve1211 Geneve 4, Switzerland

Gauge anomalies are defined and recent applicationsto N - 1 rigid supt. (•symmetric gauge theories arereviewed.

1. Introduction

The aim of this talk is to present a short review on the anomaliesof gauge theories, in particular of If = 1 supersynnetric gauge theo-ries. The scope being bounded by the space at disposal and by the litpi*«of my own knowledge, too, I shall oranit subjects like the relation ofthe anomalies with topology, index theorems, etc... as veil as the prob-lem of anomalies in (super) gravitation and superstrings. Beginning inSection 2 with a very summarised account on the derivation of the gaugeanomaly In d = 4 usual Yang-Mills theories. Which will allow a* tointroduce the concept of anomaly as a non-trivial solution of the con-sistency conditions, I give in Section 3 sous generalizations to sj-:<ce-tines of arbitrary even dimension : this vill give ae the opportunity tointroduce some useful differential geometrical concepts. Section 4 isdevoted to the (sketchy) proof of the existence «ad uniqueness of thesupersymmetric gauge anomaly in d » 4, Jr * 1 super-lfang-HAlls theo-ries (SYM). I present briefly ir. Section 5 quite recent results on theexplicit computation of the anomaly in STO theories.

2. Yang-Bills theories in 4 dimensions

As this section owes to be an introduction to the subject of gaugeanomalies I shall not follow the historical path (1,2} but instead startdirectly with the pertuxbative derivation of the chiral anomaly as givenin 1975 by Becchi, Soviet and Stora [33, who first prove its uniqueness.More recent developments [4] in the fraaework of differential geometry

Work supported in part by the Swiss National Science Foundation.

360

with applications to space-tines of dimension d ^ 4 are postponed to

Section 3.

The classical Yang-Bille theory Iin 4 dimensions) is defined by aset of gauge fields A*fx! lying in the adjoint representation of acon$>act Lie grot$> G «J<3 a set of natter fields ^ (x) (scalars orxpinors) in some unitary representation of G, with (local) infinites-iioai gauge transformations

S & ix> - i u(s) + i[u(x),A <x! ] = D u(x)\ • * " ,2.1)

* * < x ) •£ i.(x) i* <*>*. *.<x) •i." 1 1 3

The matrices T represent the generators of G ir. the natter fieldrepresentation. For the gauge fields we use the matrix notation

A • A*A* , w = U*A

where the 5. 's can be taken as the generators of G in the fundamentalrepresentation.

The gauge transformations (2.1; leave invariant the action

S !A,v1 <= -4/oxTrF F H V • S (A,v) (2.3)ir.v 4' . \it tatter • <•••>*

with r * 3 >. - > A - ij* ,A ].

In order to extend the classical theory to e quantum theory oneBust replace tk« iccal gaugs invariance 12.1] by the >'rigid) ERS invari-ancc, which involves the new fields

c(«; •= c*-ix;/*-, e'-tx) * c'*!xl>* , B<X) « a*(xi'* . (2.*;

Bcre c* and c' lanticonsuting scalar fields) are the FadSeev-Popov!?-) ghosts [53 ani 1* are Lagranje sultiplier fields 16} used toiqplen&nt a gauge fiKîg coridition. The BBS transformations read

81, 4 ()t)

SCtjt)

»c' (x-»

- ».C(X)

-icaixiTf,.

' iC2tK> - |{<

•= B(x)

and ar* nilpotent i

. !xj

:(x) ,c(x)}

sB{x)

361

s2 = o .

They leave invariant the action

S ' i

U i + TrE^sc;

where we have introduced! external fields E coupled tc those BRS varia-tions which are ccnposite fields. GiA> is the gauge fixing condition:one can take e.g. the Landau gauge

G(ll) • SÂ" . • (2.8)

The local functional S(o< is the g«rarating functional of the treevertices (the Feynman rules I). The corresponding object .in she quantvntheory is the vertex functional r(*,if ,c,c'^S,E) - the generating func-tional of amputating one-particle irreducible Green functions - fromwhich the connected Green function* are defiuced through t Legendre trans-forraation. The BBS invariance it then expresised by the Slevnsv identity

In perturbation theory taken as a foraai expansion in powtts of % -it corresponilE to an expansion in TeynaaD diagrans accordine to th«number of loops - T is a foraai power series i* 1i« its zerotj> oriSertern coinciding with the classic*] action £(o! (2,7;. The ai-a of the"renoroalizatiori prograo" it tc construct a vertex functional r ful-filling the Slavfiov identity {'2.9) at all orders, toêine* with »fixing condition which for the Landau aauge ,'2.S; reads

One can show - at least for cases where «1Z fields inciudinc thefieias are aade aassive through the Eiggt aechatiiSB - that the fulfill-nent of the Slavnov itentity allows for the existence at a yr.ittry S-•atrix I3,7]. Or. the other hajKi, if one cannot fulfill the Slevnpvidentity there is no unitary S-matrix, hence no particle physics inter-pretation of the theory : sne says that there is an anoaaly, sc-re pre-cisely a gauge anomaly.

The construction of r is done recursively on the crSer it 1. litorder 6, where .7 » S<o,; ., the .Slavnov identity' (2.9) is fulfilledsince it siaply expresses the 3SS invariB-r.ee of the action (2.7). nt»gauge condition (2.JO) with G(A; giver; jsj> .2.61 is obvious,, too. Letu* look now at the ordar ft. He first compute froa the Feyreaan rules

defined by the action S, , a vertex iuncticnai(o)

• 362

which is Bade finite with the help of a suitable subtraction procedurelike «.g. SPSS !8] or dimensional regularisation with nininal subtrac-tions (9J. With the latter, ani if there is no v,. inter-cticr., theSlavaev identity iE automatically fulfilled by F ((J, and the problemi£ solved. But there is not always such a BRE invariant regularizationscheme : The first well-know exeaples are the gav*;e theories with Y 5

interactions, where there is definitely no such BRS preserving scheme,just because there is an anomaly as we shal! see. A second exainple isprovided i>y the supersyasactric gauge theories lor which no consistentregularisation preserving both supersynDerry and ESS invariance is knownat the present tia» (see however Ref. llOj Where a heat Jternei regultri-zation coopatible with supersyametry and background gauge invariance is

We thus don't assaae r<(J, to obey the Slavnov identity at theorder f.. but -je shali lock for the possibiiity of fulfill in ; i t byooctifyiu; "(o) through the introcSuction of finite counrertems to theaction S ( o ) .

the Ela*/nov operator i 12.9i to f(Oj we oet t break-in; of order * 'the Slsvncv identity being true at order C) :

i!T ) • fti(A,>,c,c',*,Ei) * CfF.2) . [2.I2?

RE i cor.seijuenp* o€ the guantur. action principle - f i rs t proved in theBPKr scheras by X-an, and Ci«rit and Xi;-.veneteir. {Ij} - tiie breaking i atir.s lowest cr<Jef is * jt»cal functional of the fields «nd, for any powercounting rencira»iisai!le"theary, as the present or.e, the dimension of Ldoes act exc*e£ 4. Ws car. aefiuce sore constraints on l by exploitingth* nilpotency of the SRf t..rar.Efc>raationS', expreEEec in the fiaicticnalforaaliÊ as fsliovs. The functional Slavrscv operator i ti.Si beingnan linear, we asscMriste to *t, for esny functional y, the linear op-erator - tae "d-e-riVfiti- s aapping" at the "pcir.t" -.

whicfc obeys th« iden t i t i e s

! 4CY) >= C f<jr any t (2.K)

Jv* « 0 if J!>) » 3 . (2.15)

For v the classical »ctio»; (2.?; we shall use the notation

t i ie , c2 » 0 . . <2.16)•"(oi

(The silpotenc;- of £ follo-.inc fron (2.15-J and the fact that S(oJ

obeys the Slavnov identity.; We note th*t when applied to the fields

363

A,<.,o,c' and B, 6 coincides with s C . 4 ) .

Let us now apply i y , for t = r ( o ) , to Eq. (2.12). Using theidentity (2.14) we get

0 • h4, t * 0(*2t * S4,. _._, + Offl2) ' (2.17)(o) (a)

which yields by picking out the * tern the consistency condition

SA « 0 . . (2.3ft)

If the general solution of the consistency condition has Che form (whichis obviously a solution due to the nilpotency of 6}

1 = tZfk,<li,c.c',S.Et (2.19)

where S i l l local functional of tire fields, our problem is solved attile order * by defining the new action

S ( 1 ) - S ( o ) -*J . (2.20)

Indeed, s,., yields the new vertex functional

r » r. - fti + Otfi2) <2.21)(1) toj

and then

«<r.) « A(r. ,> - i u r fi + of»2} - * « • , , ) - * « 2 • oc* z ) •=i »o) r , . io)to)

The procodure can be iter«tod at atgr order is », the problem beln? .reduced at «ach step to salving the sane classical field xiuoxj equaticaU.18).

I have neglected is'the discussion tbsw th« role of the ?augc con-dition (2.10). It is obvious., Sl6i fulfillirts it. that it xeuiBE truefor r(o) since no interaction involves the field t. One show* rathereasily ttax. the countertenw E (2.19), (2.201 do aot involve S either,so that r(1, satisfies the gauge condition its wall, tad to OK at e«<?hstep of the iteration.

Mow, if there are solutions of the consistency oeeOittcn (2.18>which cannot be expressed in the foi» (2.19) th* absorbtiOB jprocefiure(2.20) cannot.be performed and we are stuck : these solntionE *xt tnoi-alies. Becchi, Houet and Stora hove shown 112.! rhat there is in fact aunique possible anomaly, naoely the chiml anooily 123. They first, show"hat any tern of L depending on the fields v, C, B, X hac the f a n

J2.19) and thus can be absorbed a? a coui;terte«n. Only the dependencecc, A and c being r.on t x i v i a l , the consistency condition takes thentht forn - the BRS operator 5 being qiven by <2.b) -

sitA.cV - D <2.23)

which is nothing else than the Wess-Zunino consistency condition [13].Then they showed that the general solution is given by

i<A,c) - ra<A,c) * si(A) (2.24)

where A is an arbitrary local functional cf A which can again be ab-sorbed as a countertera, and A xs the chiril anomaly - defined up toa variation ac(A) -

/ d x c y * ^ * |*./0»o} • (2.25)

l*e ccaputation cf the numerical coefficient r in <3.2*) gives, at thecrier * *- -•

where 4 ^ ^ i* toe ayxcs«trXc invariant tensor associated with the gau;eG :

4 ^ = |TrK*{X*.ie} . -. (2.27)

For «pro«ps l ik« Sy:i! which <Sc not possess a d , ^ tensor, r - 0of course, r can also be a&de to vanish at the order 1i by a suitablechtfie« fca: the'rtfxreseJttiti-oB T* of Oie matter f i e l d s , A non-renorma-lizatior. theoro. I i4] st*t«« * « - . that the anosaly stays vanishing ata l l oroers ot p€:r-turfa«cion tiieory.

These results where' inlander! to the general case cf a compact gaugegroup IT. Bsf. (15).

$,

The *s*»9ti of jmoaelies say be *xtended to other renornalizabletheories lint M > < rigid sêrsymetetric âtje theories vhich we shallconsider in the next section. One car. also censidex nos-renorxalizabletheories like eravitation or higher dinettsXsn Yanc-Jtills theories.

For S siojple and Ieft-h»n3c!5 ferciens. For. S serei-siccie there isan anooaly for each siaple facto*.

365

is no harm due to non-renormali2ibility as lone is one restricts one-self to the cases where the gauge fields or gravitational fields areexternal fields coupled to currents of fern;cms.! Let u: consider firstYang-Mills fields [4?. It is convenient to start with the "connection"1-fota

ft = iA dx" = i\ A V . (3.1)

The BRS transformations are then written as

sA = -de - {crks

2 (3-2!

sc = c

For the "curvature" 2-foro :

F = dA + A A A , sF = fF,c] (3.3)

c is a O-forn: (function) but of anticoanmting type. The bracket in(3.2) is therefore an anticonmutator and s is taken as an antidenva-tion like the exterior derivation d. In particular

s*1 = d2 «= ds + sd ' 0 ;3.4)

F, A and c being Lie algebra valued forms, let us consider a symmetricinvariant polynomial Jn of degree r, on hie G, and let us supposethat the diaiensioc of space-time is d »= 2n - 2 (for fl = 4, n * 2 , J 3

is the syssaetric trace which defines the syranetric tensor d,^ (2.27)).One can prove the transgression foraula Il&i

J B(F,...,F) = nd/^dtCB(*,rt,....Ft) (3.S)

a times (n-2) tlam

with Ft « tdft + t2Az (we drop the wedge syntol * ) .

This f oraula is non-sense in the case d = 2a - 2 since it involves(2n>-foras and (2n-l),-fonas which are idencically zero it 2 B - 2 di-mensions. But iet us deflae

ita + a

and, checking frojo the HIS tr*nsfon»ations !3.2! that

' - * • - .3.7)

366

let us apply tht cran*gr«ssion formula to the "connection" a for the

antiaerivation 9 :

J \F,...,F) - nP/ dftJ (&,T. ,. • • ,T.) = $t, . (3.8) 'n ' 'o .1 t t

The left-hanc-siae is zero in 2n - 2 dinensions, but the r.h.s. is asuperposition of forms of different decrees. Me expand the integral Qas

Q = 0° + 01 , + ...+ C2*1"1 (3.9)

where, in * font C_ , p means its degree and q its ghost rumber (bydefinition q = C tor A, d and 3 - 1 for c, s). Inserting this in(i.8l, and icsntifyin^ terms of sane decree we get the set of equations

st' , » -a;" , etc... .

It i-how« in particular that the integral of the a-form

- *».O « is; r_ 2 13.11J

fulfills the ccr.t.stercy condition

tv StOKr-s theorem. "*e efsuae thet toe space i s e i ther closed or thefif"' i . • t.:.ish r&piSly eno.:.7h «-_ i n f i n i t y . ; "his cor.stj-jctior. thus cer-tair.ly y i r las ar.cs&lict ;3 . !I i . I t i s not clear i f a l l anoni-iies csn beafcuu-sii ir. th i s •-•ay. Anyhow, in 4-d-.TaeiiEions (n » j ; , with

>"•-• . v»:- ! « S-.TiTr:- • • " . . ) • d !3.2 3)j » b c • « b c abc

one finds

• -fc.e.' - Jt". « Tr/cdI'ASA » i*'5? [2.14)

which IE exactly the anocsly (7.25j , know, to be unique.

Thi? netjiod say te generaliiefi to gravity {171 and to gauge theo-ries KIUI tijiS supers;-sretty :1S,1S»,2CJ.

* .

In 4-diaensicns ana as for usual gau?« tnecries tht N « 1 rigid-supersyaaietric yeuê theories are renorcalizable t'21J. The role of thegauge fields is played in perturbation, thecrj' by the "prepocer.tial" V,

367

a dinension-zero real superfield

V - va(s,6)Ja (4.1)

and that of the matter fields by chiral superfitlda

a. (x,8) , 5 ^ « 0 . (4.2iu -o

(Da * 3/3eo " i<;ctaS 3u * covariact derivative; see (221 for t&fe nota-

tions .and conventions).

The Fafideev-Popov fields are chiral superfields

c * cau,e>i , c' « c'a(x,e)^ • (o.3j

and the BRS transfozoations readV v - Vst = e c - c e car

sV • c - c + rtV,e+c} + siultiple coasuitators £ 5-v,c)

SA » -c T~JA. , s*. = C T I " «.*>

2 - -2sc » -c , sc « ~c

sc' - 8 , s* » 0 , x' - I . si - e .

The Lagrange-oultiplier B is here a chiral superfielc correspondingto a chiral gauge condition like tcx. (2.101)

-I , 5 2D 2V . • (4.5)

The BRS-invariant action is

s, , • s. <v,*> *(O) iOV

4* 1B62V •- CD2ef • 52c

m 4 .. T.«* •> 3 '+ Jd x& 3a« a (+ * , a -terns)

H . 5 2 ( . - V 0 . V ) . •8 ft

The jajrpoee of the renoeaolizatioe progtwa is the construction of avertex functional

riV,»,c,c',*,««t.lield») » S, , *.OVK} (4.7)

368

fulfilling a Elavnov iSer.tity

*!.") - 0 ' (4.6)

similar to <£.9). Ont has thus to solve a consistency condition as(2.18) and, as there* one checfcs that the non-trivial solutions dependonly or th« gauge fieid V and the ghosts c, c. For these terms theconsistency condition reads

s£.<V,c) * 0 • (<5.9)

with s given by (4.4) and where & is of distension 4 ana ghost lumberI - i.e. linear in eCc). Since BupersvEnsetry eoaaiutes with ERS andsince there exists a xenormclization procedure ccxpatiile with supsrsym-aetry !23: (I siaplify a little by foroettir.g the off-c.Tfil isfrareSdifficulties caused by the Baseless dinension z«ro super!ield' v, butth«y can )>e taken care of [24] i ther. i can be assumed to be surwrsyn-r.etric anS' thus toe written a« the superspace integral of a function of

/ J . (4 .10)

The solution of (4.5) i s then shown 125) to be unisw - up to the BKSvariation of ar. arbitrary local function*! of V. It takes the fore ;>fan infinite power series in V, vapst tarce can be recursively ccaputed

v} 4 t'Bv,t-25v? • I V . D S

iZt PV)tVtfl!l] + QiV*J . (4 .11)

The «osffici«nt r »t or*tr H takec the saac value (2.26) as in or-dinary gaug* theories i1j, sc th*t the aechaniss of anooaly cancellationreadies exactly the cane. One can further check chat the taxos explicitlywritten in (4.12) ccntair. the usual *nomaly 12.2%), with trie idantifica-ticn* -

(4.12)

icit co»put«tiont of the STW anonaiy

Sue to the iij*eneiooie»* of the prepottaitial V one should not betoo suzpri*«A vo obtain an infinite power sarier for the acbisaly, butthis do*i not.appear very aecthetic. They were many atteripts on the re-cent psct to fia.4 s closed fora. In a first series of works 126,27,28;

369

the anomaly is derived by an explicit computation of the chiral aacterfields loops, the prepotential being considered as an external fiald.The results are in She fora of a parametric integral with an integrandof "finite size" (but the perfor* of the integral yields an infiniteseries again). the different results axe difficult to coaâre, as theynay differ by BRS variations of local fonctionals of T. The l*tterpoint may explain the puzzle encountered by the authors of Jtef. [27] -.the anomaly they find contains terns which are not proportional to thesymmetric invariant tensor dj jc. As they themselves argue, there ten*could well be sea* BRS variations. In fact, doe to the uniqueness of theanomaly (see above}, it must be so.

Another set of papers [18,19,20] addresses the probla* of computingthe anomaly by extending to superspace the differential geometry KethodI indicated in Section 4. The task is made difficult by the fact thatsuperspace forma [22] cannot be integrated : forms like dfiÂdBj A ...do not define volume elements since for Qrassaanc variables £ the dif-ferentials d9 coanute (see however 129}, where an integration theoryfor suitably generalized superforas is proposed). This difficulty iscricunvented in the following- way. First, transgression foraula (3.S)are written for the superconnections (l-s

where e , e and e^ .(a * Q,...,3j a,4 =• 1,21 are the rielbein 1-forms (22] and, due to the curvature constraints 122],

The BRS transformations (4.4} for the unconstrained prapctential V andthe coiral ghosts c, c read sow

II> = _ac - {^,c} , sc « -e

s$ • -de - {•,<;) , sc » -c .

The transgression formula (3.8} applied to either of thes* superconntt-tions_ yields identities of the type (3.1O) for a « 3 M .• «>, Ht)*»the Q's are now superforms. (6- and 5-*«p»rfaroB do a«e vanisS .*) Oh*can extract from these identities the part of degrrae 4 ic the space-tlBCvierbein ea, which can then ba integrated over spacc-tiaa- £«t us takee.g.. the equation (3.10) for the 4-superfom fti :

SQ4 + 4B^ - 0 . {J.4)With , . .

.1 • b c d 10. » • Ae » « - A I x • ... (4.Si

37C

where the dots represent terms of lower degree in e , the \'e 1 partof (5.4) yields for the superfield xi the equation

3X • total derivative « 0 (5.6)

the expression

/A1 (5.7)

fulfill* tJw consistency condition si^ = 0. One can check that £G

contains the usual anomaly ;2.25) and one remarks that, contraril^ tothe previous results it is polynomial in the connection * (cr v).Qowever Ag not being the full cuperspace integral of a superfieid isctoviously not supersyaatetric. Is it possible tc add to the anomaly (5.7)a BF.S-variatlon tern m order to render it supersynmetric ? Bonora ets.1. [1 3 first extract ifron the transgression fomula identiries an"anomaly" for th* supergyjaaetry coupled to the gauge anomaly (5.7). Inorder to define what is aeant by such en objact, one introduces super-

»R£ traaef onoatious. They act on any space-time integral of ai »s

The jjaraneters i.J art ooaa^tinq constant sspinors sc that the Wess-tuairw ilgebjr* -'J .ig? * C, etc. takes the fors

The oowutat iv i ty of BRS and scjjersyianetry' transformations reads

' W ' • • • » * C ; (5.10)

"anoaaly*1 i» a local functional i-susy^ >c) •c aJ>0 jpolynoaiJtl in • (or %•>, xhicl-. v e i i f i t s ht> consistency

to <5.9;, (S.:5i :

•Do, •ceorito) :«> » thaorca prcvei io 130), th» goaeral solution of thec«OBi»»;«icy condition (first of fcjs (£.11); i s given by

Ji*| ir *«»e local functional of V. lu contrast with ijusyam! i^ , E ba« no r«a*or. to be a polynciaial in i since the quotedttioore* applies tea local functiontls of qncort»trai..n»a Euperfieies, i . e .• f th« j?reppt«rt4al V In the present case,. Bowsra et »1. have explicit-ijr_tolve«. [•>*> «i* prqtl<» <S.13J for 2 and found an expression i.i thefor* i< a farnftrj.- uita^sal s'ieldiag an infinite power series in v.

371

It is now easy to see that the red&fined gauge anoaaly

i'(V,c,c) = i_(*,c) - sS(V) (5.13)G G

is supersymmetric (use (5.10), (5.11) :

s l'(V,c,c) = 0 (5.14)susy G

and fulfills the gauge consistency condition :

sfi'tV.cc) = 0 . (5. IS)G

Girardi et al. [18] had previously found, by differential geonetricmethods, too, an expression for the same anomaly - which could differfrom that of Bonora et al. at mast by the BK£ variation ~f a local func-tional of V due to the uniqueness [25] of the anomaly.

The conmon feature of all the results I have Dentionned is the r.on-polynomial character of the supersycaietric anosaly. That it aust is. factbe so is the content of theorem proved by Perrara et al. Ill] : the super-svnanetric gauge anomaly cannot - in the case of a semi-simple gauge group- be expressed as a polynomial in the superconnection coefficients (5.2),or even in the oore general expressions

-V V ~ V -VX = « B e , X, * D e e ; A « a,a,a . (5.16!A A A A

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Series in. Physics, ftrinceton University ? i*ss (IS83).[231 T.S. Cl«i*., O. Pigueti * . S i i o ia , *r.n. of PhwB. 109 (]9"7) 438.I24J 0, Pijaet , K. Eiboia, Saei. Phys. S24£ ;i9a<; 33«{ B24S :5965) 396.125.3'O. Have*., X, PiboW., Mud. Pttys. S24.7 (1964) 484.tSfcJ M.K. Ki«;L.Mti, Itucl. ftys. «24< (1984) 499.12-7) E. 0ua£Sgni«i, K. Kor.iehi, K. Jtlntchey, Pis* pj^prict ITUP TH-1O/85

9S'(JCJ «, .frerwici;, ?, Riv», Stony Brook preprint ITP-S&-SS-1Q U9e5) .{29] f ,A. Berakic, Sov. J. iiuci. Phys. .33 (4) <1979) 60S.

» .* . Pic-ien, 1.. SuBderaeytr., fre ie «r.iver»it*t Berlin preprint (!9B5),136} £>. Piauat,'•». Schwed*, X. Sibold, llocl, ?hys. S174. i!9B0) 1£3.{313 S. Fitcrfrfc, i,. CirarSello, C. JPiguet, R. Stom, preprint CERN-TB-

41S4 / IAPP-TB-13J. (196S). ' .

373

Note a&3ed

Aftox completion of ths manuscript I receive; the two follcviropapers dealing with the explicit construction of the sup&rsynmetrir gaugeanomaly :

[32j K. Harada, K. Shizuya, Cniv. of Tohoku preprint TD/35/2S0.[33] R. Garreis, H. Scholl, J. Wess, Hoiv. of Karlsrune preprint tft-THEP-

85-4.

375

o?; PRCSUSTJQV 07 SVFSR:"CW?P:S M&.Tc-Rm ~A"T:CLI:S

;,K.BllesV.y, ii.Uh.Christova, S.P.Eedelcheva

raxory of rhsoretioal Physics, JT.T?,, Dubs a, I'SS".

S

Laboraxary o

Abstract

Production of two -different (sapersymmetric?) Maôrsna fer--

mions in collisions of longitudinally pclarised e"*1 ani &" in

coaaidered on the basis of general inrariaace principles.. ?olari

zation craracteristics of the proe&ss are diseassed IT. aetai.1.

Possible aethods for checking 0? and CFT-iavariance ara s^sgeste

Sose tests which would allov as to distinguish >•*other the *ira2

particles ara if I'iajorar.a or Dirae typ« ars prapesei.

376

1. Recently there is an increasing interest in supersymmet-

ric theories and their experimental signatures. In These theori-

es liatioraaa fermions inevitably appear, for instance, as super-

partners of the photon, Ji and Higgs bosons. So, an observation

of Lajcrane, particles will be of particular importance for super-

synssetry. Calculations of the cross sections for production of

iiaclcrana neutrals have been done in a number of papers' "^'.

Tness calculation have anown that it cay be possible to 3ee

these particles at £ " fr "-colliders ur.c.er construction.

In this paper we shall analyse the production of two diffe-

rent 12s.J.o.r&n& fersuoiis .( and \ in the process

£ + ^ *' ^ /' + / (1)

with p-oi*rised initial particles. We suppose that i) X is hea-

vier tiiaK \ ; ii) tiie p&rticle • is stable and iii) process

(1) is detested by observation of the lepton pair froc tr»<* decay

C =• 5. (2)

So, bsi evidence for process (1) mroulfi be a iepton pair and a

large araount cf nissinf energy (teiceu away by /, ),

-he aim of our investigaxion* is to find possible- signatures

whici, would confirm the liajoxan* nature of / and 1 •

Our considerations are based on tna g«neral principles of

GPT-invariance and unitarit;." of the S-iaetrix and we shall assuae

only that •'. and /•, are not aadrons.. Sic assuEiptions specifying

the type of tbe interaction have been made. ?or unpolarisea ini-

tial beans similar investigations have been carried out in ref. "*'.

377

2. Here ws shall c;c:-3ia=r the case of Icr.gitudinaliy pola-

rised positions ar.a alesirc^s*. Frox tr = O?T-ir.vari=r.Cv ir.d -jni-

tsritv of -he S-~:itriT: (up to tenns of an crder of ..e fiê sxruo-

viira cci:di:a«: .r - ) <ve obtfcin the fcllowing relation fci 'he crosa-

secticn of prcjess ("•):

where f is thr arigle bexv.-eeii the momenta of t" and / .'-1;

c.is.3.; the first {sazcao.) iiidex i.er.D:?s Icv;git^di::ai poierisa-

t i o n of f ' f ' £ ~ ) -

She diiierential cross-seetioii of process (1) ha? the gr=ce

ral form

Obviously, the terms lineex in /*•-; and /';-_ are due to parity

violation, iiany relations for the di?fereKtial as '.veil as for

tne integral polarization characteristics of process {') jei; ce

obtained fror: eqs. (3) a.'-i C4). Here we shall discuss only the

i::ts£.ra! characteristics. If we assign ar. .;pr-er is^'-x F(5' to

the quantities dsfinec i r the forward (ba:.:ws.rd) hsris^here. we

iitve

Ihe polarisation cna.i-acteristics ,'.~ ' , C1 " end J ^ J " m

•Lo.-.cituainailv pola;"i£t'd t5 £ ~ befi-is are tjcoected to teobtained at SIC /o / .

378

be determined from the ir.ce.surable ssynanetries

if )' and } art Ka.iorana par t ic les , eq. (5.5 leads to

Jor the to ta l cross section of the process (1 ; v.e find

Ttitr.i' intc ascou::t eq. (3> w& ^btai- .'~J. = - C - . T ^ i s leads to

?.eiatic:;s (?) aâ (9) fellow fro.1*: the assu-pticr. that ' t-r.d ;

are .Vs.; craiifi par t ic les . iVouia i t be possible tc co^firc. this £ ;££»-

tier, by oseokinc these r&IaTions experimeijtaliy? loi us cc-pcre pro-

cess (*) with the processes

where Y end .4' are aleotrieally neutral 2irac fennions (typical

exeaiple: ; •' ana .'/ ' ssy be e hear;,- arjd a l ight neutrir.o in £ theory

with neutrJTic mixLr.ft). If processes (11) and (1i) are detected by ob-

servation of a leptcr: pair froffi the. decays

379

it would be impossible to distinguish process (1) froc processes (11)

and (12) experimentally. In the latter case the number of events

witb a lepton pair and iixsing energy would allow us to determine the

sum f _

where (T '• ff'anii. /•" "''/Ware the cross sections of processesSit/••- ' i/^('.-!«. ' '

(11) and (12). From CPT-invarisnce and unitarity (up to terms of or-

der d~ ) we have

From eq. (15) me obtain

Thus, if the process of production of two Dlrac particles in fi' » -

collision is detected by observation of the pair ( and r , the

cross section Qf this process obeys the seoe symmetry relation as

the corresponding cross section for production at * peir of Hajoran*

particles. Therefore, this synaaetry relation would not be a sigcatu-

re for iiaôrana particles production*.

Note that all the relations obtained here would be exact ones

if CP-invariauce holds. So, any devie-.iopis fros. then would c« aa evi-

dence for G?-vioiation by the aew Isupersyntartric?) interaction ge-

nerating process (1). Consiaer&cle ( %, «<.) deviations from these re-

lations are possible only if Cpy-theorea is violated.

3. Sow we shall show that study of the energy distributions of

the final leptons nakee it possible to distinguish the production of

Haôrana particles frcan the ce.se when Hirac particles are produced.

•iiote that this is IXOT; so if the processes under considerationare dêcted by observation of a pai:- of different leptons { £ T - A ~or £-_ [i+ etc.). We thesk P.Siederaiayer for this reaark. '

380

For process (1) the probability to observe at an angle b in

cm.3. /'"with energy E and / " wixh energy £ is g-iven by

the fcllov;ing expression

d %A, -; - £/ = ^ o02 ">' & sUeJe'wHere £-£<'*' ,'i^ E /' is the partial decay width of process (2).

Hote that the first (second) argument of function />.'/-/ refers

to the energy of C I 4 / •

Proa CPS-isvariance and unitarity of the S-a&trix it follows

that

It/? ft, E') = Wf'fE', £}.By usirig eqs.("7) aiit (18) we obtain the relations

Ihes« relations do not told for processes (1")»(12) as it is

easy to i>e convinced taking into account that

a'

anc

, . 1 / I *

u > / ^ ••/ ' &F' ik; V " (21)

Sc, relations (15) would allow us to cheac whether the lepton pai:

originates fron tii« decey c! Itejorana or Dirac particle. Prom egs.

(19) one oan easily obtain for the corresponding integral quanti-

ties a number of relations specific for th« production of liajorar.a

particles. We have

381

Here the function G &. ,'"'' /^describes energy diatributicn cf

/i-f /'p~) from the decay oi1 / produced in -ae forward hemisphere,

etc. Let U3 note that equation (23) for /IY = ./\~_ = 0 V«6E ob-

tained in ref."'.

Carrying out integration of aqs.(T9) over the v.-hole solid

angle we have for tne spectra of £'l~h.x£ £~

<{<%% [E) =, cfrf'.^ (5).

To complete, we note the.t an evidence for kajorasa aature of

/ and JC would be the absence of

Here /i/^t-^ ( /V^-^ ; i.B ^h.e ausiber of eveate, i s whici the energy

of / ^ i s more (leas) xaan the energy' of £ ~~ .

We would liire to thank S.T.Petcov for useful discus s i or.s of

the questions considered liare.

3tS2

R e f e r e n c e s

1. L.P< P&yet, Fays.Lett . 117B, 460 (1962).

2. J . E l l i s and J . 5 . Hagelin, ?hys. Let t . 122E, 303 (1362).

3. J.i:. ?rere and G. Kane, Huel. Phys. E223, 331 (*S63>.

4. J. E l l i s et a l . Phys. L e t t . , 1322. 406 (1963).

5. S. Petcov, ?iiys. L e t t . , 139 E, 42" (1964).

i. A.W. Cbao, SLAC-PU3-3061 (1953) (A).

383

The Irvine - j-ichi;ar. - 3roci:haver. Ccllaaoraricr.

R.:.L. Bionta', G. 21e'.vitt", C.3. Britten',^D, Jasper*", 3.G. Cortes: ,

P. Chrysicopoulou , H. Clans", .". Crouch , 3.T. 2ye , 3. £rrede' ',

G.Vi'. FoEter11, '.<. Gajswaii1, K.S. Gsneser', T.l. Goldbaber-5,

T.J. Kaines1, T. •'.'. Jones', 3. Fieioze'-vsas"1 >ci, '.V. 3. I>npp*,

J.G. Learned*5, S. Lah-anr^ , J.T.:. LoSecco1". H.S. Perk'", L.Price ,1 1 2 - ' "- '

P. Reincs i J. Sohults , 3. 3eicel , -• Shuaard , D. Sir.cls.ir~,

H.V/. Sobel1, J.L. Stoas^, L.S. Sulak , K. 3. 5vobcdac, G. T'rjorsrcr/,

J.C. van dar 7elle , end C. iVusst^.

^ Tbe University of California at Irvine

c Kie University of Michigan, Arx .-'.rncr, :iishir;cn

Brookbaven rational Lsboratrrv, "."pton, .*.'ev, i'ori

CaliJci'iiis liiiti"i.".--<= oi* 2bC.^.~.oiî~, , icca-::.i

' Cleveland State University, Cleveland, Ohio

The University of Hawaii, Honolulu, Hawaii7

University College, Lonaon, United Kingdom

'.Varsavv Univsrsitj, '•Varsaw, Poland

' Lav?ranco Livercore national Laboratory, Liverr.crs, Califcrr.is

Univetaity of Illinois, JJrsstia, Illinois1 1 Perailsb EAL, Batavia, Illinois12

Case .Testern Reserve University, Clevelsr.d, Chio

Presented by D.

This woric was supported in part by the U.S. Departs-art of Energy

Lo~er Units of the partial lifetimes for 32 uuoleon decaymodes ai-e obtained free. 417 live da.y e:-.r>c3ure of the water Che-retkc-v 1X2 dstector. During this period 4 01 everts crijjir.atincinside a 3300 tor, fiducial volume •••'ere observed.. Tr.'sv are con-sistent with the expected bacr^rour.d of atmospheric J in ter-a c t i o n . Twenty one of the events can bt interpreted as eitherj.) interact! Mis or nueiecn decays into codes without a neutrino

in the final Btate.

1. ZZB SS7SCTQR

The Irvine - j«icr:igan - 3roc-ki:sven (IUB; detector is locatedir. a sal t rair.c near Cleveland, Chic-, at a depth of E'ocut 600 munderground, v.'hioh cprresponds tc 1570 m of water esuivsleat o-verburden. It ccr.rair^ SOOC toija of purified water ir. a rectar.-gular voiuct !1.~ s 1? x 23 a"). 'The Cherer.kov light eT.i~ted cycharred relativist:.c particles (|3>.73 3 i : detected by 204S pho-toûltipsier tuoes !P:r.TJ, e-ech of 12.5 CE dierjetcr, covering thedetector sides. The distsr.ee btêer. the tubes is about 1 n.

About 25000C cjoas pass through the detector per d&y. 2c ex-tract fro", them atmospheric neutrino interactions: and possible

decays we look for cor.tcined eve.nte with vertices at

lesEt- 'c. - frcs the F;.:T planes. Ihis fiducial vc-lune cor.tairiS3300 tons of w^ttr or 2.C x 10' nucleora.

2hree naar.tities: I I , 12 ar.d v art recorded icr osch PUT.The photon arrival t i - e , T1, asasured vith a resolution of 5-5ns K.'.i-i'., ia ur.e-c for ,;:e one Tries! recojiBtriieticn of events. Pol-lowing a trigger on tne T* scale. a second time 3cale, T2, isactivated for 7 .5 / J J enabling the detection of electrons from/A. deceys with an efficiency of ~ £0/a. The integrated pulseheight, Q, Erasures the Cherenirov light yield.

Hae rav. ?" , T2 and ^ values are calibrated using a pulsednitrogen laser light. She PUT tiae response is âssured by va-rying the time of the laser firms with respect to the trigger.

385

The linht intensity can be varied over five orders of tria

enablir.£ tne relative Q calibration.

The total Cherenfccr li^rt yi.di for an event is calculated

by summing of the calibrated pul3e heights corrected lor li£ht •

attenuation in sater and PUT angular efficiency.

The absolute energy scale is deternined by "t e u 3 6 0:? a n

experimental sanple of ouons passing vertically through the de-

tector. A corresponding sample is simulated in the detector usins

the known energy distribution for atmospheric muens ana taking

into account all electromagnetic pi-ocesses giving rise to knock-

on electrons, pair production and breasstrahlung. By comparing

both samples the Cherenkov light yiela measured by total correc-

ted Q can be transformed into visible energy deposited in the

detector, Er.

31ectro:.:r-c:nstic showers caused by electrons and gsnaas orig-

inating in thr fiducial volume deposit essentially all tiieir

enersj' inside tae detector (radiation length is 3c cm) and there-

fore ttieir total energy, E_ = £ . For stopsing s,uons, £_=-»+ ^15 I-1 C -L C

because of the ffiuon re s t mass and the energy deposited belo'" Ci»e-renkov threshold.

Sncrgy resolution depends on the nuaber of l i t KIT'S, I"?I,and is - 100;; / "SPT - 15& ( sys tena t i c ) . The ffieai, KPT i s '30for nucleon decay nodes with maximal Cherenkov l igh t yields(e.ij- P ->• e + c + e" ) .

3 . I50L.-.TI0" CF COrTAIirSD S\rd;TTS

Ai the f i r s t stage of the search for contained events werequire that HFf is between 40 and 300. Tiiie CUT corresponds tcencr^;; ranê froc 150 to -~1500 i«eV and reduces the dsta cy afactor of 3. Subsequently two independent analysis chains areused to crcas-cheek the r e s u l t s . They consist of f i t s to vertexpositions snd tracL- directions or. the basis of the PIvIT timingand torolo~ieal charac ter i s t ics of Cherenirov rsdict iar i .

The renonstruetics of a point source of laser l i s a" andsimulated samples of ccr.l &ired events shows that vertex positionsare found Tilth accuracy of — 50 CK fcr Hiulti-track topologiesand ~ 100 cir: fcr single track events. The efficiencies of vertex

386

reconstruction in the fiducial vclut.s &re 3C:'> for c:t-.lti-tracJcand TO/o for s ingle- t rack events. 1'te ccz.cined eff ic iencies ofooth analyses; are >. 30-.*.

During L'.l days of l ive ti3:e 401 cor.tair.er. event3 werefouni. Apart of energy, S e , they csr, oe characterised by a pa-rsrsfetfer A ( "anisotropy"):

KF? . i:PT

A - z: v ; i f . -£ , *iwnert rt- i s a unit vector pointing froi. the vertex to a E.I2and ••v, i s e weight taking ir.to account the pulce heit~r.t, liti'htSDsorptior. in wster enc an^ ' I s r correct ions . For eve.-.ts .vithone v is i t i i - t rack, v.hich aoir.ir.fetp the st~:nspherio neutrino i:i~terar.tic.-j; A i s 0.1:, (the cosine of the Choronkov ancle) whilefor isotrcpic rr *'i-.e angle tv.'o body events , v.rnch arc cljrirsc:-t e r i s t i c of tacy of the protor. decs;,' r.oces, A < C, >. "lie e o t i -aeted errors c r. A are - 0.0>,

A sca t t e r plct of 2 va A for 401 ccntsir.tci events is îvtr:in ? ig . 7(a). Kc accusulst icn of events is observed fcr E^ closeCo the nucleus r e s t iiass.

la order to chooe recuire^rnts fcr nuclecr. decay candidates»e Sinruiste cvnnt samples for different decay codes. As an ex-aaple the plot cf I,, VE A for simulated p -»/,C rrr0 decs^s issnov.r. ir. TLr. 2(b) . The picii inversc. ions ir: oxyyen £iv? r i s e tothe srrpad ci points at lower energy ar.d higher Qr.isctroi.v. TheSOJ: irvd-cates the cuts used tc se lec t candidatps for th is rncie.

Cuts on i,, arid A as well as required EUO~ decay sijn&turesfor a l l studied decay codes are giver, ir. colusms 2-4 of Table 1.They were chosen to njninize the neutr.no induced background,Tne samples of simulated decays were analyzed in the same- way asthe date to determine the reconstruction and cut e f f ic iencies .They are given in Table 1. The values in column 6 take into ac-count a l l nuclear interact ions of hadrons in oxygen and surroun-ding water ana depend on the applied intranuclear cecca&c cartel(described m ref. 2 ) . They were uses in the calculat ion cf l i f e -

• 387

time Units jirea ir. colu.v.E 10. ihe efficiencies quoted in

colunn 7 ".ver-? oc:air.ed sh-sr. inxeraetioKa in oxj'gsn were neglec-

ted and are essentially rnctiel independent. Both quoted efficie-

ncies include branching ratss for acr.3idered neson decay mcdf-.a.

In order to isolate candidates for the decay modes:

p —i-e+j~, ? - .•.•+y t p -?e+Jt° and p -* /*+JT° more stringent

requirements could be applied. They are looked among the eveste

with t.vo clearly defined tracks with op^nir.j armies > 140°.

The numbers of candidates for different decay modes are

given in column 8. Sane events are candidates for more than one

node as the requirements overlap. There are many candidates for

sone modes with a neutrino in the final 3tate as they Eimic the

quasielaatic charge current neutrino interactions in cur detec-

tor- The candidates for decays ir;to charged leptons are denoted

by letters. Their details are siren in Table II.

p. BACKGROUND 0? S3UTBI3C

Kie expected baclE^round tc nucleon decays cotaea froai

interactions of neutrinos originating frga decays of bairons

produced by coscio rsya in the ataosphere. ^c estisaate the back-

ground we simulate a sample of i> events in our dete.ctor.

There is unfort'jjiately ao data oa low energy neutrino flux.

Je use the flux and the neutrino eacr,jy spectrum calculated in

rcf. 3 for geoniagr.eiic latitude of the 223 detector.

To include properly very particular and rare topologies of

>' interactions with pions produced at large angles wita respect

to charged leptons we ao net use published exclusive cross sec-

tions but instead we take the accelerator neutrino events found

by the Garânelle collaboration i;: a freon bubble chamber. Jbc

events are weighted in neutrino ener,^y bins taking into account

the flux and the total charged and neutral current cross sections.

The electron neutrino interactions are generated oy char.giEg the

observed cuon to an electron of the same moaeatwa.

The events are subsequently simulated ir. our detector and

analysed with the use of the procedure sppxiea to the data.

388

The scatter pict of £c vs k for the simulates sample issbc'vn iit 7ij*' "(cj . The comparison of i. S.TLZ A - i - t r : Dutior-sobtained for ooth eznericenT&l and simulated s&r.pies is sreser.-ted ir. ?i.s. £ er:i 3. Kc aigr.ificant dii'fcier.ce is observed.

The nuct-er of i>ac:-:«rrcund events expected for each r.ucleor.decay aods ( i .e . fulfil l ing identical requirements) is l istea ir.coluar. S of Table 1.

The errors on tne backsrcund estimates wise fi'orr. uncertain-ties IE tb* flux calcal&t^criS, especially for lev-' r.eutrir.c ener-gies (» 30vi) as well as nuclear effects, possible sosnr.inj: aniparticle ioer.t; fication biases in the Dabble c.-.azber : air.ple. V/ehave al3c processed data froir. wo other experiments usin5; .'"e ar.iD? bub-jle cnan-aers. As a result an overall uncerteinl;; r.f (- 5C»,•r 1OC/i) was sssigried to the bac'iigrcunc esticate for ar.y river.node, oecause oi' this uncertainty, we ssice r.e b&ckr'rcuni sub3trs.c-tiori. Considering ell of the candidate events as iat-;leor. ieoayswe calculate the lifetime limits at the ^Q7c confidence level.

Iftsriisg £17 live day exposure we hsve found 401 containedexeats. Their overall rate and sharacteristics are compatible withthe e-xpeoted i-ecsgrouaa of neutrino interactiors.

Hc-vever, we can not rule out cue Is or. aesey us the source ofSPDE events. Ir. particular there are 21 cs:ididE.tee for differentdecay ccdee v.it::out s neutrino in the final s ta te .

Due zo large uncertainties in roe background estinate-a •;ie.derive the sucleon lifetime l ixi ts without bacicgrcuni sub;- u . i : . - . .Por soce modes with charged lsptor.S the liEit of tne Iife;in;e overbranching rate is on the order of 10 years.

Most of the decay modes have signal at the sarte level as thecaekgrouisd. rfe hope tc increase the sensitivity of our detector

for aome of thea by improving the light collection which wouldprovide aetter energy and track reaolutior. end more effective back-^roi^jd rejection. The recent installation of wave shifters aroundthe E£T*s has increased the light collection by a factor of 2. Afurther improvenent &y a factor of 2 is expected at the end of1965 when the EK?»s of 20 cm diameter will replace the existingones.

389

To set better bacr: rrcuna ssî-a^j^ ';•;- ar? ir. process

;-.ev ao-i-:l cnl julatJiors c. air.-Ie pier, pi-i-î-iir. ar.i stu

neutrir.o data frc- di£'2:re3.z buiale o.;s.:;.ii=j"3.

1. H.3. Par's, a. aiewitt et a l , , Ksya.Iiev.Lett. ^4., 22 (15=5);

B.G. Corlea eT 3 l . , Iîya.Iiev.Le'it* 2£, 1C92 {7?Si}:

R.1U Bicnta et a l , , ?nya.Rev.Lett. ^2, 27 (13S3).

2. T.w. Jones et a l . , ~2hya.3a-r.Zeit, ^2 , 720 {i'j-34).

3. ?. X. CJais3er and I. 3taaev, i s rroaseiings of :he Istsriat-.-.n

Jolloquiuin on BSJTJCTI 3oncoa3ervati02., 5-aIt i a i s Citv , Jer-;ar-" ,

1SS4f edited i?y D. Clise (Univeraitry cf .i'isoorisin ?r?ss, y.a_i8

1934) a. Si; I'./U 3aiss?r ir. ?roo»eii:-t^ si" tine il^~?r.-i Ir.*e

nationaZ Coni"er?r:c9 on Neutrino ar.c --.s-rcphysics, edited bv

K. Kleir.lcnecii'5 and i».A. ?aacijo3 ("Voria Scientific, Singa

19S4), p. 372.

390

?i^. 1. A ssettcr plot cf £„ vs A fcr (a? The 401 contained

events, (b) a simulated sa-plt cf the r.ucleor. dec:.;- aode

r -»•/U."1" jc c , arid ic) a reniac sacple ;f 4C1 ev?r.-£: fror.

sinuiated reutrir.r- i'iteraotior.s. Sventa vith zero, sr.e, and

tvvc ider.tifiei _,*<- decays are indicated by circles, arcss-s

ana asterisks, respectively.

?ig. 2. The visible «r.er^y disir i tut ion. Ttie solid line i»

foi1 the dsta sr.d d£i£r.ed line for the siâiat.ed neJtrir.c

ir.teractiorjs.

?iC. 3- Tr.e ar.isotrcpy dis-vriDut'itr:- The solid ar.a cashed lir:es

as ir. rig. 2.

Cciunr 'i. rî- 'hree-tady decay ir.cies ila* ph&se apace .vae as-

sused.

COJUZS 2-4. i;i;fjrer.- recuirer.est regions ocrrespona to different

CoZ'- r. 4. "uCL-er cf »u:r. ieosy Signals required.

USJUZL-: 5. r:.c::.-er of c-ve-r.rs rejected sy r«quii-ir.£ v-o olstr

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38:

SUPERSPACE QUANTIZATION OF GAUGE THEORIES;

GAUGE-FIXING ASP GHOST TERMS IK N=1 SUPERGRAVITY

Allen C. Hirshfelc

Institut fur Physik

Oniversitat Dortmund

4600 Dortmund SO

Federal Republic Germany

1. Introduction

In many physical problems it takes some time before the

correct "simple" system is found which admits a straightforward

analysis. Sometimes the system at first considered become*

simple only when considered as part of a larger entity. Of

many examples which could be cited, I will consider here two \

which are closely related to the case I am concerned with.

The first example is N=l supergravity. A spin-3/2 field, con-

sidered in isolation, turns out to be a rather complex system,

and its quantization presents great.difficulties. On the other

396

hand, a spin-3/2 field, coupled in the very specific way pre-

scribed by N=: supergravity to a spir.-2 field, is surprisingly

tractable. At the classical level, the combined system admits

a well-posed initial value problem, with causal solutions'" .

Also at the quantum level supergravity is well-behaved; it is

both one- and two-loop finite, whereas pure gravity diverges

at the two-loop level1 '. The second example I wisn to n>.».-r!-

tior. is quantum Yanq-Mills theory, which is by itself rencr-

roalizable, but for which the N=4 supersymmetric extension is

finite^ . In both cases it is of course the increased symmetry

of the extended system, here supersynunetry, which explains the

improved behaviour.

The strategy of the superspace quantization methods is

similar to that used in the above examples. The classical gauge

theory under consideration is extended by the inclusion of

certain auxiliary and ghost fields, coupled to the original

fields in a specific way, such that the eorabinec system is

"supersynrnetric". The supersymmetry referred to here is not

the usual one based on the Poincare superalgebra, but the 8RS-

symmetry based on the Kugo-Ojiir.a algebra-of conserved charcss1 " .

The supersymmetric system is most conveniently described in

a superspaee formalism. The extended system giver rise to s

quantum theory which is unitary by construction, and in the

case of ordinary Jang-Mill* fields, renormalizable.

In the very early literature on the quantisation of non-

Abeiian gauge theories the ghost fields were introduced in a

completely ad hoc manner, in such a way as to preserve units-

rity . Later the Paddeev-Popov mechanism became the con-

ventional framework for quantizing gauge theories. The super-

space quantization approach is, for ordinary gauge theories,

and for supersyjnmetric Yang-Mills theory, a perhaps more ele-

gant alternative to the Faddeev-Popov procedure. For super-

gravity tae usual Faddeev-Popov procedure- breaks down, but the

superspace method allows also here the correct determination

of the quantum Lagrangian.

399

The superspace quantization method in essentially it*

present form was introduced by Bonora and Tor.in foe rhe case1-7-1

of ordinary Yang-Mills fields , and was systematized and

extended by various authors . In the presett paper we

briefly review the method for ordinary and supersymmetric

Yang-Mills theory, and for quantum gravity, before presenting

the case of primary interest, N-1 quantum supergravity.

2. Ordinary Yan^-Kills Theory

We briefly review the usual fonwiisticn of ordinary Yang-

Mills theory. The underlying manifold is 4-dimensiontl space-

time, with coordinates x « (x°,x',». ,x ). Th€ primary dynami-

cal variable is the gauge potenial h (x), describing a spin-1

field, with field strength

{'••>

We are using here the customary matrix notation *...<*) *Ai(x).t'.,

where the T^ are the generators of the gauge group, and simi-

larly for F . A gauge transformation is written as

(2)

a(x) al6O being matrix valued, and the gauge potential snd

liield strength transforms according tc

"1 iv, ou).

The classical Yang-Mills Lagrsngian

is manifestly gauge invariant.

400

When we attempt to quantize the Yang-Mills theory,

starting from Eq. (5), we inur.edlately run into difficulties

related to the singular nature of this Lagrangian. A gauge-

fixing tern is usually introduced at this point, the volume

of the gauge group is divided out of the functional integral

according to the Faddeev-Popov procedure, etc. Instead of

following this route the superspace method extends the ori-

ginal system in the following fashion.

The underlying manifold is taken to be the superspace

with coordinates (x,e,6), where x stands for the specstime

coordinates, and P ,6 are anticonssiuting Grassrcan variables.

The local gauge transformation is now a function of (x,6,P),

and car. be written, up tc an ordinary gauge transformation,

as

Lifco a(x) in Bq. (2) the objects c(xi, c(x) an£ b(x) are all

matrix-valued. Th*y express the degrees of freedom associated

with this general gauge transformation, and correspond in the

usual foirwiisjr. to the Faddeev-Popov ghosts, antighosts, and

auxiliary fields, respectively. The particular combination of

fields in the #f coefficient in Eq. (t) is just chosen for

convenience, so as to make this correspondence exact, other-

wise the fields used here would be mixtures of the usual ones.

Th« primary dynamical object of the theory is a super-

field A (x,t,8!, which results from the usual gauge potential

by a g«r>araliz6d gauge transformation:

'%, e, o) AîxJ if t%, *, ej(7)

401

Inserting the explicit form of U(x,6,9) front Eq. (6) yields

the exp

fields:

the expansion of A, (x,s,S> in terms of ordinary spaceticie

Here D, is the covariant derivative:

etc. We see that the saperfield % (x,e,S) contains the ordinary

gauge field as lowest component, plus the ghost, antighost,

and auxiliary fields in its higher components. The field strength

corresponding to A is

The classical Lagrangian of Eq. {5} is expressed in this for-

malism as

and is obviously independent cf (e,e). Instead ci s':artinc. z'~e

quantization froro the classical action S c l * ,'d x X ;xi we -jse

the "quant'jm action"

Performing the d9d9 integarations according to the rules of

Berezin integration, and usiag the component expz~ sion of the

A. field of Sq. (8), leads to the following expression:

The first tern corresponds, after partial integration and con-

sideration of the Euler-Lagrange eq. for the auxialisry fie." 3 b,

402

to the cov'eriarit gajge-fixing condition 5WA = 0. The second

term corresponds to the Faddeev-Popov tera in the usual for-

mulation, it gives the kinetic tenr. for the ghost fields as

well as the ghoet-gauge-£ield coupling.

3. Supersynunetric Yang-Mills Theory

In supersymraetric Yang-Milis theory the field strength

V is given in terns of the prepcter.tiai V by

u/, = ^2"/*"V.T;._. <cir (13)

Here o is a spinor index, ar.c D% the supersynaietric covariant

derivative. Ir, the super-space quar.tizatios approach the pre-

potential V is exteuaed to a superfield V, related to V by an

extended gauge transformation analogous tc that of Eqs. (6)

and (7):

tf^- t r* f i v t r (14)

wi th , „ #?- *i * ** (&'* tic erf

u • e ' (is)

Here c,c ana B are chiral f ie lds;

1)C- 2> C - "S B * C. (16)In terras of these variables the quantum action takes a similar

form ro that of Sq. '. '-. ") :

Here d z and d z are the chiral and full volume elements in

ordinary superspace. The component form of (17) again results

403

from performing the dede integrations,

Here L,,c ; [V,cj is the Lie derivative. This expression agrees

with tnat obtained for the gaugs-fixing and Faddeev-Pcpov terms

by the Faddeev-Popov method, see, e.g. Ref. [12]. Further de-

tails of tne superspace quantization calculation?, for super-

syaimetrie Vang-Kills theory are given in Xef. [133.

4. Ei.-tStsln Gravity

In the Einstein-Cartan fornralation iravitaticnai theory

involves two symmetry groups, local Lorentz invariance, and

general coordinate invariance. The independent cynaniical va-

riables in the first-order formalism are the Lorentz connection

»u and the 4-bein e". in the superspace quanti3ation method

these1 quantities are extended to superfields in > way similar

to that shown explicitly in the previous cases. The quantum

action is

[tr (Z&

c, re,>

JHere ct, c,, b are ghcst- and auxiliary fields ?.isociatea with

the local Lozentz symmetry, c^ , cV", B are associated with

the general coordinate symmetry. The Euler-Lagrar.je eg. for

the auxiliary field B is

which is the-; familiar gauge condition for harmonic coordinates.r i a i

The expression (19) agrees with that obtained by Nakar.ishi'

by another method. Further details or. the superspace approach

to quantum gravity are given in Ref. "15].

5. $='• Eupergravity

Supergravity is invariant with respect to local Lorentz,

general coordinate, and local supersyrraetry transformations.

The primary dynamical object is the 4-bein e^(x) describing

the graviton. Its suaercpace extension with respect to local

supersymmetry transformations includes the gravitino field

• (x). The action is

In terms of components this yields three gauge-fixinc Lagran-

gians, the first fixes the gauge with respect tc local Lorentz

transformations, the second with respect to general coordinate

transformations, the tnird with respect to local superEymmetry

transformations:

[22)

The S-fields are as usual auxilisry fields associated with

the various symmetries. To lowest order in the field

e^ a s* + h*" the above expressions reduce to

. (23)

r 1 6 *lThese are the expressions usually given in the literature1"

In higher order calculations, however, it would really be

necessary to use the full expressions given in Eqs. (22),

along with the associated ghost terms, in order to preserve

unitarity in the perturbation series. Further details concer-

ning the superspace quantization of supergravity are giver in

Ref. [17]. Compare also Ref. [103.

6. Conclusions

The superspace formulation discussed here provides a na-

tural setting for the various fields involved in guantum gauge

theories. For conventional theories the effective quantum

Lagrangian is easily obtained^ in agreement with the results

of the Faddeev-Popov procedure. In contrast to -chat method,

here the gauge-fixing and ghost terms are produced simulta-

neously. In N=1 supergravity the Faddeev-Pcpov method is not

applicable in its usual form, but the superspace quantization

approach yields also in this case an exactly BRS-invariant

Lagrangian, which agrees with the form known in the literature

in the weak field limit. The full Lagrangian includes quartic

ghost couplings , and other terms which do not arise in

the usual Faddeev-Popov procedure, but which are necessary to

ensure unitarity.

406

Referencss

[13 J. Isenberg, D. 3ac and P.E. Yasskin, in "Mathematical

Aspects cf Superspace", eds. K.J. Seifert, C.J.S. Clarke

and A. Rosenbiuin, C. lifcidel, Dordrecht 19S4.

r 2 J S. Deser, J.E. Kay tr.d K.S. Stelie, Fhys. Rev. Let;. ;&,

527 (1977).

[3] S. Mandelstam, in Proc. 21st International Conference on

High Energy Physics; eds. P. Petiau anc M. Forneuf,

Journal de Physique, Collogue C3 supp. an No. 12 (1982).

[4] T. Kugo and I. Ojirea, Prog. Theor. Phys. Suppl. ££, 1 (1579).

[5] R.P. Feyraaan, Acta. Phys. Polonica 2_6, 697 M963).

[6j L.D. Paddeev and V.K. FOFCV, Phys. Lett. ^SE, 29 (1967>.

[7D L. Bonora ana M. Tonin, Pbys. Lett. 98B, 46 (ISST).

[8] R. Delbourgo, P.D. Jarvis and G. Thompson, Fhys. P.ev. D26,

755 (1962).

[9] L. Alvarez-Gaarafe and L. Baulieu, Sucl. Phys. B212, 255 (1983).

C1O3 F.R. Ore ar.d P. van Nieuwenhuizen, Rucl. Phys. B2O4, 3.17 (19t:j.

[11] A.C. Hirshfeld and H. Lsschice, Phys. Lett. 1O1E, 48 C9S1).

t123 B.A. Ovrut and J. Mess, Phys. Rev. D25, 409 (1982).

[133 U.K. Fslck, A.C. Hirshfeld and J. Kubo, Phys. Lett- USB,

175 (19£3).

£143 K. Bakanishi, Pro^. Theor. Phys. J5f, 77S (1979).

[15] N.K. FaicX and A.C. Hirshfeld, Phys. Lett. *38B, 52 (1984).

[163 A. Das and D.Z. Freedmar., Kucl. Phys. E1U, 271 (1976).

[173 N.K. Falefc and A.C. Hirshfeld, Dortmund preprint DO-TH 84/04.

N.K. Falck, Dortmund thesis 1984.

'183 R.E. Kallosh, Kucl. Phys. BUT, 141 (1978).

i.07

THEOREM PROVING WITH FIRST-ORDER PREDICATE LOGIC

B. Humpert

Artificial Intelligence Group

HASLER AG, Research LaboratoriesCH-3OCO BERN 14, Switzerland

and

Institut de Physique NucleaireUniversite de Lausanne

CH-1000 LAUSANNE, Switzerland

ABSTPACT

We give an introduction into "Theorem Proving1 by presentingthe basic notions of propositicnal logic and predicate logic.The equivalent formula manipulations permit the transforma-tion of a statement to clausal form, and Herbrand's Theorem,supplemented by the Davis-Putnam rules, can give an 'inprinciple' proof of its correctness. Modern theorem provinghowever is based on the resolution principle which uses uni-fication, factorisation and resolution to derive new clauses;they eventually will lead to a contradiction (refutation)proof. The large number of possible clauses is reduced by afirst resolution refinement: the deletion strategy. In thistutorial we skip all proofs but add illustrative examples.

Submitted to the Symposium

1. INTRODUCTION

Over the past decade, theorem proving or more general, auto-

mated reasoning, has developed into a powerful scientific

tool whose potential is becoming gradually visible now. Al-

lowing for automated,' deductive reasoning within the frame-

work of logic, it has become an important discipline of ar-

tificial intelligence. Originally, it grew out of mathemati-

cal research domains which sometimes seem closer to logic or

even philosophy than applied mathematics or (theoretical-)

physics. And yet, the applications of theorem proving sy-

stems in 'practical' problems Of pure and applied research

is impressive. We mention just a few cf them: proving or dis-

proving theorems in group-theory, set-theory and Boolean al-

gebra, real-time systems control, logic circuit design and

validation, puzsle solving, robotics control, advanced pro-

gram analysis and computer science in general, expert sy-

stems, and so on. And the number of applications keeps

growing!

These notes are intended as a rapid introduction into the ba-

sic terminology and the main tools of this field. We conse-

quently limit ourselves to a presentation of results and

their practical use, referring for the detailed proofs to

the specialized literature.

Section 2 gives a short introduction into the history cf lo-

gic and theorem proving. In section 3 and 4 we present the

proposition*} logic and first-order predicate logic which

allow for"equivalent formula manipulations to arrive at the

clausal form of a'statement; this is the starting point of

the successive deductive steps. Section 5 discusses Her-

brand's Theorem and the Davis-Putnam rules which provide, at

least in principle, an algorithm to prove a statement, if

the statement indeed is true. Modern theorem proving is ba-

sed on the resolution principle which we present in section

6 whereby the notions: unification, factorisation and reso-

lution are explained. In section 7 we demonstrate how a first

step towards resolution-refinements can be achieved, and

section 8 is reserved for our conclusions.

403

The second cart of this tutorial will focus on the many re-

solution refinements, and in the third part we discuss the

practical applications of existing theorem proving systems.

2. HISTORY OF THEOREM PROVING

In this section we trace the ir in steps in the development

of present-day theorem proving or automated reasoning.

Theorem proving [lj-[5j is based on 'Symbolic Logic1 which

was influenced, to £ .large extent, by ihree scientists:

Aristotle, Boole and Frege. In ancient Greece, Aristotle

("350 B.C.) was the first to try to describe the laws of lo-

gic, meaning the forms that correct arguments could take.

Note the emphasis on "the forms...", since logicians, in ge-

neral, are not concerned with the content of arguments, but

only with their forms. Unfortunately, Aristotle's list of 19

Syllogisms does not exhaust the correct argument forms. In

1654 G. Boole invented 'Boolean Algebra', a mathematical

theory covering the way in which elementary sentences

(propositions) can be combined with connecting words

(connectives) like 'and', 'or1, "if, etc. G. Frege's 1B79

publication of the 'Begriffschrift' opened up the modern pe-

riod of logic. In his 'Predicate Logic' the building blocks

are object* (nair.ee, variables,...) and relations (predi-

cates) between them. We thus have the possibility, to combi-

ne the tbility of the- propoeitional iogic to construct new

sentences from old ones, vith the syllogistic ability to

delv« ir.tc the internal structure of the sentences. Predica-

te logic thu* includes all the correct forms of both of it*

predecessors, and more besides. Since 1879, the predicate

calculus has been pushed by a series of major mathematical

discoveries to the present highly developed state.

The roots of theorem proving go back to the 17th century

where 7. Hobbes in 1655 advanced the view that logic reaso-

ning might be reduced to and understood as a kind of symbo-

lic calculation or formula manipulation. In 1666, G.W. Leib-

niz put forward * concrete realization by proposing that

every concept should be assigned a code number. Either it

would be a prime-number if the concept was not further ana-

lyzable into simpler ideas (primitives), or else it would be

the 'product' or conjunction of two or more ideae (compo-

sites). The numerical multiplication of cod* numbers would

thus be associated with tb* 'logical multiplication' of the

concept for which the numbers stand. Around the turn of this

century, these ideas were revived by G. Peano, which, howe-

ver, is more known for his axioms concerning the arithmetic

of the natural numbers.

The development of theorem proving in the early 20th century

is almost undissociable from the 'Foundations of Mathematics'

in which notions as 'set', 'funtion', etc. were analysed,

with B. Russel (1902) as one of its well-known exponents.

Around 1920, D. Hiifoert initiated the construction and study

of a logical formalism in which the proof-methods, to esta-

blish the properties of a system, would be unquestionable as

to their trustworthyness. With the security of the logical

tools firmly established, he then hoped to move into forma-

lized mathematical theories and prove there to be consistent.

Whilst formalizing the 'semantics of the system', with in-

tuitive notions as 'truth', 'satisfiability', etc., A. Tarski

(1936) used Cantor's (1895) general set and transfinite num-

ber theory to portray the totality of all models of a pure

predicate calculus.

The work of K. GSdel (v>1930) is by many considered as the

greatest mathematical discoveries of all times. Be showed

that any formalized theory of arithmetics must always be de-

fective in one of two ways: either if would be INCOMPLETE,

meaning that some true formula ('true' in the intended

interpretation) would not be provable, or else it would be

INCONSISTENT in the sense that all formulas, true and false

alike, would be provable; moreover, if the formalised theory

is in fact CONSISTENT, then the proof of its consistency

would of necessity have to transcend by appeal to principles

that have been left out of the theory. The consistency of a

mathematical theory can therefore only be proven in metaia*-

thematics, if at all.

Finding a general decision procedure to verify th« validity/

inconsistency of a formula was considered iong ago. It was

not until A. Church (1936) and A.M. Turing (1936) that this

was proved impossible. Church and Turing independently sho-

wed that there was no general decision procedure to decide .

"a priori' on the true- or falsehood of formulas in first-

order predicate logic. However there are proof procedures

which can verify that a formula i£ 'valid' , i_f indeed it is

valid. For invalid formulas, these procedures in general

will never terminate. lr: view of the result of Church and

Turing this is the best we can expect to get from any proof

procedure.

A concrete realization of the Church/Turing"result was given

already in 1930 by J. Herbrand. By definition, a valid for-

mula is true under all its interpretations. Herbrand there-

fore developed an algorithm to find ar, interpretation that

falsifies a given formula. However, if the civen formula is

indeed valid, no such interpretation can exist and the algo-

rithm will hait after a finite number of trials, -if the for-

mula is not 'valid', meaning that it can be 'invalid', 'con-

sistent ' or 'inconsistent', the algorithm will either ex-

haust the search space in which case we can be sure of the

'invalidity' of the formula, or it will continue the search

for ever without giving a clear answer either way. Hersrands

procedure was very difficult to apply since it was extremely

tirae consuming to carry out by hand. Kith the development of

more and more performing computers, logicians again became

interested in mechanical theorem proving. In 1960, Herbrend's

procedure was implemented on a digital computer by D. Prawitz

and P. Gilmore, followed shortly by • more efficient proce-

dure proposed by M. Davis and H. Putnam. These early attempts

were in that sense disheartening that they required an ex-

plosive number of trials to supply a proof, which rendered

Herbrand's idaa somewhat impractical. Instead of trying all

instantiations of Herbrand Universe, J.A. Robinson proposed

in 1963 the 'Unification Algorithm' which, in some sense,

predicts which trials will produce a "winning combination".

A major breakthrough in mechanical theorem proving was achi-

ved in 1965 by J.A. Robinson Cl]r he developed a single in-

ference rule, the 'Resolution Principle', which was shown to

be highly efficient and quite easy to implement on computers.

413

Since that moment, theorem proving has developed enormously

with resolution-refinements being proposed such as : set-of-

support R., Linear R., Paramodulation R.. and many other re-

solutions 12 2, and with efficient theorem proving programs

being developed [3]. The field of application of these pro-

grams rapidly extended beyond logic and mathematics [4] into

all areas where deductive reasoning is required, thus lea-

ding to the term; automated reasoning £5}.

3. PROPOSITIOKAL LOGIC

Symbolic logic consideres languages whose essential purpose

is to symbolize reasoning encountered not only in mather.a-

tics, but also in practical problems of technical nature,

and even in daily life. In this section we introduce the

principle notions of prepositional logic (or the propositio-

nal calculus) which was invented by George Boole in 1854.

Proposition;

A proposition is any declarative sentence that is either

true or false, but not both. The symbols suph as P, C and P.

that are used to denote-elementary propositions, are alsc

called atoms.

Example 1: P : It is hot. S : The sun shines.

fl : It is humid. H : People go swisaning.

R i It will rain. T : People play tennis.

Logical Connectives;

From propositions, we can build compound propositions try

usind the logical connectives :f(not), V(orK Aland;,

,—— (if ..then), • — (if and only if m iff). These five logi-

cal connectives can be used to build compound propositions

from atoms, and more generally they can be used to build mo-

re complicated compound propositions by applying them repea-

tedly. The binding strength Ot the logical connectives grows

in the above listed order with <*l acting on its closest atoro

and * » reaching farthest.

Example 2:

D^ : If it is hot and humid, then it will rain : (PA Q) —«.R

D 2 : If it is humid, then it is hot .• 0 —* P

Dj : It is humid now : Q

C : It will rain : R

G ^ [(Dj A D2 A. D 3)—*C]

The formulas in Dj are the hypotheses, whereas the conjectu-

re is represented by the formula in C.

The £ -sign assigns a formula to the symbol on its left-hand

s ide.

Formulas:

Formulas in the propositional logic are defined recursively:

(i) An atom is a formula,

(ii) If G and H are formulas, then ( M G ) , (~H),

(GAHJ, (GVH), (G — + R ) , (G •-» H) are formulas,

(iii) All formulas are generated by applying the above ru-

les.

Example 3: G [((SV(PAQ)M (*R)) — * i.HVS)Z

G : If the sun shines , or •

if it is hot and hur-.id, and

if it will not rain ,

then people go swimming , or

then people play tennis . '

Truth Value:

The T(true) or T(false), assigned to an atom, is called

its truth value and denoted as: TV(P) « { T o r T } .

415

Truth Table:

The truth value of a formula is determined by the successive

application of the following truth table

G

T

T

TT

H

T

V

T

T

(not)

T

T

TT

G A H(conjunction)

T

T

T¥

G V H(disjunction)

T

T

T

*"

( i f . . then)

T

T

G — » H(i f and only i f )

TT

T

Interpretation;

Given a prepositional formula G, let A^,...,An be the atoms

occurring in the formula 6. Then an 'interpretion I of G' is

an assignment of truth values to Ai...An in which every kx

is assigned either T(true) or T*(fal»e). but not both.

The truth value of the propositional formula G, as detenni-

ned by repeated application of the truth table, can there-

fore be different for different interpretations:

TVj(G) • T,T* . The following terminology is in use: "G is

satisfied/falsified by I', or "I satisfies/falsifies G*. Ar.

interpretation I that satisfies G is called 'a model':

ItTVjfG) - T .

Exairple 4:I

GCP.

I l »

1 2 - *I 3 *

1Q,

I * .i ,Q* i

k [(PAO)'>} » ^T,1

• {T\

- fr.:

r, r}T,T}

Model

Model

not Model

Model

Note, we frequently use -the short-hand notation (.. it

follows..),4*$ (..it follows in either direction..): these

symbols however are not part of symbolic logics.

Valid. Invalid, Inconsistent, Consistent:

A formula G is said to be 'valid', iff it is true under all

its interpretations: Vl: TVjiG) » T. A valid formula is

also called a 'tautology1 f B } .

A formula G is said to be 'invalid', iff there exists an

interpretation under which the formula is false:

31: TVx(Gj = T .

A formula G is said to be 'inconsistent', iff it is false

under all its interpretations: V I: 7Vj(G} =T* . Ar incon-

sistent formula is also called a 'contradiction' i5* C3 ), and

it is denoted as "unsatisfiable'. A formula G is said to be

'consistent', iff there exists ar. interpretation under which

the formula is true: "51: TVj(G) = T ; it is also denoted

as 'satisfiable'.

Example 5: Valid : G4 [PV(fP)] = J| (Tautology)

Invalid : G * L ( P A Q ) — • Rj

Inconsistent : G & [PA (<JP) j = O{Contradiction)

Consistent : G£ E(PAQ} —«. R]

Logical Consecuence, Equivalent:

A given fornula K is s&id to be a 'logical consequence'

( B Icons' of a given formula G, iff for all interpretations

I in which G is true, H is aiso true:

Vl: TV!(G) - T =%> TVj(H) = T .

Two formulas H and G are said to be 'equivalent' (or K is

equivalent to G ) , iff the truth values of H and G are the

saxe under every interpretation of H end G:

•Vl: TVi(G) - TVi(H).

Example 6: (see below).

"Laws" (Equivalent formulas):

Ar. adequate supply of equivalent formulas, for simplicity we

shall call them "laws", will permit us to carry out formal

manipulations on a given propositional formula without chan-

ging its truth value. They are listed below and referred to

in the text as indicated:

iff Law : G**H » ( G — H ) A (H—»G)

if-then Law : G -* H - («G) v B

Commutative Law : G )! a » H ]( G

Associative Law : (G J[ fl))[ H « G )[(E^ M)

Distributive Law : GvfTi A. "M) = (G v H ) A (G V M),

G*A '{H VTJ) •> (G A H) V (G A M)

• ,B Laws : 6 J " " U ' G J ° "^ y fJB

Negation Law : G Y (e*G) * |S , •»(<VG) = G

Morgan's Law : f»(G } H) « (i»G! X (»H)

Conjunctive/Disjunctive Kormal Form:

A formula G i s said to be in conjunctive/disjunctivt normal

form iff i t has the

Gconi.- [GiAG 2AG 3A...] , G d i 6 j.£ [Gi V G2 V G3 v . . . ] ;

any given propositional formula G can be cast in either of

these two forms:

Deduction Theorem:

Given formulas D},...,Va and a formula C, C is logical .con-

sequence of t>ii..DB iff the formula (Dj*. . . AD n) — • c is

valid:

Theoren

C Lcons. (Di^.-AD,,) 4=^ [{DjA. . .ADn) c ? • • (Tautology)

Ax iom» 1 T Conjecture

Refutation rhccrem:

Given fcrnulas D}.....,Dn and a forrruJa C, C is a logical

consequence of D}...Dn iff the formula{&!A,•,ADnA*>Ci i>

inconsistent: •

C Lcor.s .D

Example 7: (see Example 2)

G £

Di : (PA Oi—• B =

D2 : C - V P '

D3 : Q » 0

C : R = R

A PJ C I

Thus: C Icons

.ADPA ~ C] = Q (Contradiction)

= D (Contradiction)

T~ <?=

419

FIRST-ORDER PREDICATE LOGIC

i this section, we shall introduce the first-order logic

nich has three logical notions called predicate, terms

.nd quantifiers. We shall extend the equivalent calculus of

the propositional logic and introduce the Prenex-, Skolem-

and Clausal- normal forms. Predicate Logic was invented by

Gottlieb Frege in 1879.

Predicate:

A 'predicate' is a relation between objects, as

P(x, f(x),a, ... ), GREATSR(x, y), LOVE(John, Mary) etc., whe-

reby the 'objects' can be constants or names, variables,

functions, etc.. An n-place predicate P(tj,...,tn), also

called 'atom', maps the n symbols tj...tn into the

{T,T*J -space.

Example.8: E[x. y) : x » y (equality)

L(x, y) v : x likes y

MOTHER(x) : Mother Cf X

MOTHER(Peter) : Peter's mother.

Logical Connectives:

The logical connectives as introduced in the prepositional

logic apply here in the same way.

Terms:

The -terms are defined recursively as follows:

(i) A constant is a term,

(ii) A variable is a term,

(iii) If f (...) is an n-place function and tj.-.t., are

terms, then f(tj,..tn) is a term.

(ivj All terms are generated by applying the above rules-

Example 9; s, fx, ffx, fffx, ... : a, fa, ffa, fffa

a,b, f(x,y), fix.a), f(a,b), f ( f U , a), f (a, a) ) , . .

Note, to simplify the notation we frequently omit the func-

tion or predicate brackets; thus f(f (a) 1 = ffa.

V, 3 -C'jartif iers:

Since we have introduced variables, we use two special sym-

bols and to characterize variables. The syirbcls V ana 3

are called, respectively, the 'universal quantifier' and the

'existential quantifier1. If x is a variable, (¥x) reads as

"for all, each, every x", while (3x) is read as "there exists

an x: for some x; for at least one x".

Example 10: P(x), K(x), D(x): x is a patient, quack, doctor.

D, : Sone patients like all doctors : O O I B O A C V J ) {3>(y) —»• L&.yjj JD 2 : No patjent likes any quack : ftfc) [ j w -• (Vy)\ ({(.)}-»• r» L^)}C : Therefore, no doctor is a quack: (ifr)£;tr«) —» rJ k«> ]

Scope of a Quantifier, 3ound Variables, Closed Formulas;

The application range of a quantifier is denoted as the

'scope of trie quantifier'. The occurrence of a variable ir. a

formula is 'bound' iff its occurrence is within the scope of

a quantifier; not bound variables are 'free1. A variable can

be both free and bound in a formula. Any formula containing

free variables cannot be evaluated. We therefore shall

henceforth assume either that all 'formulas are closed1,

meaning that they do not contain ar.y free variables, or that

the free variables are treated as constants.

Quantifier Scope of Q

bound, free variables : (Qx) G(x,y)

bound > .J I free (no Q)

421

Formulas:

Formulas in the first-order predicate logic are defined re*-

cursively:

(i) An atom is a formula.

(ii) If G and H are formulas (depending on terms), then

{e»G), («*H), (G«H), (3VH), (G — H) , ( G — » H) , «C»

formulas.

(iii) If G is a formula with x as its free variable, thPt;

(Yx)G and (3x)G are formulas.

(iv) All formulas are generated only by a finite numbeir

of applications of the above rules.

Interpretation (over a domain):

An interpretation of a formula G in first-order predicate

logic consists of a non-empty domain D with an assignment

of "values" to each constant, function and atom in G, where!

(i) every constant (and variable) is defined in D,

(ii) all (n-place) functions are defined as a mapping:

Dn — £—*• D,

(iii) all (m-place! atoms are defined as a mapping:

Note, (Vx), (3x) are interpreted as: "for all elements x in

D", and "there is an element in D".

Truth Rules;

For every interpretation of a (composite) formula over a do-

main D, its truth value is evaluated by using:

(i) The propositional truth table,

(ii) TV[(Vx)6] - T , if for all xC D: TV[G] « T" ,

»T* . otherwise.

(iii) TV[{3x)G] - T , it for one x€D: TV[G] - "F ,

» T , otherwise.

±22

Valid. Invalid, Inco. -ister.- . Consistent;

The notions 'valid', 'invalid1, 'inconsistent', 'consistent'

are defined as in tne prepositional logic.

"Q-laws" (equivalent ¥ , 3 -fcrn'ulas):

Similar to the "laws" in propositior.al logic there exist

"Q-laws" (Q for quantifier) in predicate logic which allow

fcr formula :nanipuiatior.s without cnariaing their truth-value.

The following Q-laws involve the Q^ {= V,3)-quantifiers:

= (Qx)(HxXG)

= (Vx)(HvAGx)

Ox)HxV (3x}Gx - (3x)(HxVGK)

Hot eqv. : (Vx)HxV CVX)GX / (Vx)(HxvGx)

(3x)HxA (3x)Gx / (3x)(HxAGx)

rename: x —» y

Prenex Normal Form:

A theorem is proven by conjoining its negated conjecture C

with the axioms DjA...ADn, and the subsequent search for a

contradiction • ; such procedure is called a proof by refu-

tation. We thus are always concerned with a for-ula contai-

ning quantifiers, atoms (depending on constants, variables

and functions), and the logical connectives defined earlier:

G ^ tD 1 *. . .AD n

Ax—* Conject .— t I—Contradiction

-FORMULA (Q. A(f . jc .a) , A . . . )

Quantif iers—I i—Atom» t—L-Connectives)

i.23

In the propositional logic, we have introduced the conjunc-

tive/disjunctive normal forms. In the first-order predicate

logic, the analogue is the 'prenex normal form' which will

simplify the proof procedure. It is obtained by equivalent

formula manipulations which leave the truth value of the

original formula unchanged. Using the "Laws" and "Q-Laws", a

set of formal steps permits the separation of all quanti-

fiers into the 'prefix', and all atoms into the 'matrix'.

The matrix consists only of the atoms of the original formu-

la, separated by the A (and) and V(or) logical connectives,

and it is cast into conjunctive normal form:

Prefix G-Matrix

GPrenex * (Qix)(Q2x) C G 1 A G 2 A G 3 ...3

The logical connectives V(or) are now absorbed in the

"clauses" G^. The set of steps is summarized below with

the necessary "(Q-JLswfe" indicated:

Step 0 .• ell formulas closed :

Stap 1 : eliminate — » , m » : L.awi

Step 2 : pull o> to atoms : Q-Laws

Step 3 : rename variables : Q-Laws

Step 4 : allVj3 left most s Q-Laws

Step 5. : conj. normal form : Laws

The application of these steps is easy once the reader has

gone through the details of son* examples.

Example 11: (see ex. 10)

Di : y x ) CPx'A (Vy) { Dy - • Lxyjjt>2 : (¥u) CPu—*(¥») { Kv - *

o.k.

El : Qx)(Vy> CPxA { («Dy) V (Lxy)J]D2 : fVuKV'v) £«PuV i lnKv) V («Luv)}]

C : (\J (Vz ) £ (WDs > V (wKz ) ]

Step 2: M C : » z ) [ D z A Kz]

Step 3: o.k. - carried out already in Step 1 .

•Step 4: GprenexA (3*) (3ss) (Vo) (Vv) (Vy)[Px*|«Dy v LXV}A 1 <v>PuV<v KvVWLuv} A Dz A Kz]

^ Clauses ' Literals

Clause, (positive, negative) Literal:

A 'literal' is either an atom (E 'positive literal') or its

negation ( E 'negative literal'). A 'clause' is a disjunction

of literal*.

Literal : L s [ft. or (<^\)1 . Clause : Gx& [L].V L2V. .. ].

Skolere Normal Form;

We can further siriplify a formula in prertex normal forTi by

disposing of the3-quantifiers and by dropping subsequently

ail V-quantifiersr this procedure is called 'Skolemization'.

We focu« our attention on the prefix: a variable associated

with an 3-quantifier (3-variable) i« replaced by a 'Skoiem

function1, and theS-quantifier in the prefix is dropped.

The Skoltm function depends on all variables which preceed

the9-variable (when goina from left to right). Tf no

V-quantifier preceeds an3-quantifier, the 3-variable is

replaced by a 'Skoletn constant*. Skclew functions/-constants

must be different from all other functions/constants already

in the formula. After all3-quantifiers have in this way

been eliminated, the remaining^-quantifiers in the prefix

are dropped.

Gprenex 4 (3x) (Vy} <Vz) (3u) (Vv) Qw) [x, y, z, u, v, w]

Ic = a

us q(y,z)1

w 2h(y,s,v)

I wea'x eqv. |

Sltolero functions

a- i &G S ) c o l e n ! - C a ' V ' z « g ( y . z ) . v , h ( y , z » v ) ]

The resulting formula Ggfco].em is not (fully) equivalent to

the original formula Gp r e n e x? it is 'weakly-equivalent'

instead, since Gprenex *-s a logical consequence of &skole:n'

but the opposite does not hold true. However, by suitably

constraining the Skolem functions, any model of Gprenex c a n

be extended to be a model of GsjjQiej,,, which turns out to be

sufficient for the refutation proof procedure of a formula.

Example: (fa) (¥b)tVc)(3x)[ax2 + bx + c = 0]

Skolam fct: x*f(a,b,c)=fabc la•f2abc+b-fabc+c«0]

'abc

Clausal For:7i:

The clauses G; i-n t n e Skoierc nd'ss&l forrc of & formula are dis-

junctions of literals which depend en f* '-qusr.ti£ie-d; variables,

constants and the Skolera furictions/constsrit*. The variables are

'standardized apart' meaning that the variable* of each clause

are given different symbols, due to the (fjcuivalent} "O-lavs"

the truth value of the formulas remains unchanged. Wt furthermore

introduce the 'set-notation': a formula in clausal form S there-

fore consists of a set of clauses and each clause is composed of

a set of literals, whereby it is understood that the literals

are connected by the V(or) symbol and the clauses by the A{and!

svnibol.

. . . . K GisU^ y

( x , y , . . . ) ( u , v , . . . )

The 'unit-clause' consists of one literal only, and the truth-

value of the 'empty clause' is defined to be a contradiction

( S D ) . The empty set cf clauses is defined to be a tautology

(=11).

unit-clause S { L } , empty clause s { ! H Q


[Px A («\»Dy VLxy} A{A»PUVWKVV*»LUVJ A Dz A Kz]

SXolera f c t s . : x « a, t * b(»a)

S « { G i , . . . , G 5 } Clauses: C3i A Pa

G2 & [~ Dy V lay J

G3 4 f BlPu Vf« Kv WrtJ

G5 A Ka

427

Theorem :

Let S be a set of clauses that represent the clausal form of a

formula G. The G is unsatisfiable (s inconsistent) iff S is

unsatifiable (= inconsistent):

G unsatisfiable < j> S unsatisfiable

(inconsistent) (inconsistent)

5. HEP3RAHP'THEOREM

In the previous sections we have set up the logic "lan-

guages", in this section we shall consider a first proof

procedure which is essential in that it gives an 'in prin-

ciple' proof of a valid theorem. This method, which is the

basis for most modern automatic proof procedures, was deve-

loped by J. Herbrand in 1930.

Kerbrand Universe:

Suppose we have manipulated a theorem into clausal-form with

the set of clauses denoted by S S { Gi ,63, . . . 3 . The clauses

in S depend on constants, (¥-quantified) variables, and on

members of the 'original plus Skolem) functior.-set

9 35 { f ,g, .. - of S. The Herbrand Universe is defined recur-

sively, H o consists of the set of ill constants in S. Hi

is the union of H o and^CHp). T?'\B.C) stands for the set of

function values resulting from the replacement of all varia-

bles by the Ho-constants, with all possible combinations

being considered. Hj is the unicn of Hi and fHi-Ju etc. This

process is continued up to infinity. K^ is the 'Herbrand

Universe fHU)' which simply consists cf all the variable-

free terms that can be formed from the constants and the

functions in S.

S(c3t6, V ^ S , fcts)

/

H o = ia,b,...}Hi = HOV 7(HO)Hi a Hi U 7 (Hi)

1— 3rd-Terms —,• 1

v „ s Herbrand-Universe [csts, f(csts), fftests),...j

f

Example 13: S * ! P(fx, a, gy, b!

o {Hi = (Ko, fa. fb, ga, gb }H2 = lHl« ff»< ffb- fS£< fSb< 9-a- 9ffc» 59°'

Ground Atoms. -Literals. -Clauses:

An atom (s predicate) with all its variaDies replaced by

members of the Hertrand universe is called a 'ground atom'.

Ground atoms are indicated by an index or an apostropri after

their syn-.bol ,

Ground Atoms : f P, , P } = P( vars € fw,) H P*

L,fIndices: HgQ-elements I—Predicate

Obviously tne set of Grd—atoros can be very larcje if net in-

finite. This terminology is analogously used for the lite-

rals, clauses and terms.

Herbrand Base:

The set of atoms with all their variables replaced by T.e.--

bers of the Herbrand universe is called the 'Herbrani Base

(HB)1 ,

H-3a3e : HB [P1.P2 Ql.Q2 ]

Herbrand Interpretation:

A mapping of the Herbrand base into the {TFJ -space is c&l-

led a 'Herbrand Interpretation (HI)' (see earlier definition

of interpretations). Obviously, there exist many Herbrand

interpretations of a given Herbrand base.

*• ' 0

H-Ir.terDretation : Hi HE a s * lr,T) (Mapping) .

Example 14:

ITx ; E ' x, x )! ; .,

,f y ;

A n a s Conjecture

Q& I ( V x i E ! X, X ) ] A ~ [ rVy) E ( y , y J J

= (SyJv'Vxj ' E U . X J A tv E ! y , > ; ]

failure nodes

S - C l a u s e s = I E ( x , X ) , rv E v a , a ) }

Itt = { a J

KB = { E v a , a ; J

Self-Tree (see below):

Example 15: S-Clausss = { P(f2,a)v p;x,fy), NP(fy.2 ! J

HU = { a, fa, ffa... . ]

HB = f P(fa,a), F(a,fa), P(fa,fa), P(a,f£a),. . }

Sem-Tree (see below):

faxiarenodes

— V ]f ^ ^ -- L_ ry j,

Eer.ant.ic Tree:

In t'-«- preceedinq examples we have explicitly given the

'Herbrar.d base which can consist of an infinity of ground

attrs, In definite a particular interpretation we assigr.

T c r T to each of its elementis: [Pj ,?2 , . . . ,Ci.O2 . • • • j • Tnere

obviously exists a very large if not. infinite number of Her-

crand interpretations. They can be vizualized by the 'seman-

tic cree' as shown in the above examples. Each of its layers

i£ reserved for one ground ator of the Herbrar.d base which

is indicated or the right-hand side of the drawing. The or-

dering of tr,e ground atoms is ir. principle arbitrary. The

arcs going to the left are given the truth-value IT and tho-

se going to the right the valueT.

A more formal definition reads: A 'semantic tree' for a

clause-set. S is a bir.ary tree with the arcs labelled by li-

terals frc;;n the Herbrand base enumeration. Note, in this

slightly different definition the yrcur.d atcr, ano its riega-

tior. are assigned to the arcs of a isyer i.-.t-tead of the cor-

responding tr'jth-values, which means simply a charge in no-

tation. All senantic trees consist of some 'nodes', and the

lines between them are called 'arcs'. The noce at the top is

called :he 'root node'. All nodes, except the root node, ha-

ve a unique 'parent node'; these are the nodes imir'ediately

above them. Some nodes have 'daughter nodes' immediately be-

low them. Those without any daughter nodes cue to the exhau-

stion of the Herbrand-base or due to the occurrence of a

failure node are called 'leaves'. The set cf all daughters,

daughters of daughters, etc. of a node are called its 'des-

cendents'. The subtree consisting of all aescencents of a

node and the arcs between their., is said to be ' ionr.sted' by

the node. Removing some nodes and arcs from a tree to for:?, a

subtree is called 'pruning'. A sequence of consecutive arcs

and nodes is called a 'path', and a path running frorr the

root node to a leave without repetition is called a 'branch'.

The "length of a path' is the number of arcs it contains.

The length of the path frorr. the root node to a particular

node is called it 'depth', and the length of the path from a

node to the farthest leave it dominates is called its

'height'. A semantic tree is called 'complete' if on any

branch all ir.enibers of the Herbrand base occur, either as

Lj orfi»Li (i=l,2,..), but not both.

Whilst going down a particular branch of the semantic tree

we infact select a specific Herbrand interpretation, and we

chec)'. whether S is true or false, la the fornm case all

clauses must evaluate to T meaning that at ieast or.e of

their literals rust b e T . If indeed -> ch ar. interpretation

(called 'ground T.odel ' ) is found, S is satisfiable and a

contradiction proof is excluded. The corr^spc: cir.g r.cde in

the semantic tree is -.arked by Q . In the latter case we can

limit overselves to check whether at least one clause ir S

is falsified, meaning that all the literals in that clause

have the truth-value T. If such a clajse is founj th* cor-

respcadi^g node is called a 'failure node' and Toazi-:e<3 by O

(see exaraples ahcve), ana trie search procedure alona that

oranch is stopped. Along that 'failure branch' the truth-

value of S islF, anc we do not go further. Instead, starting

again from the root node, we analyze a different branch, and

consequently evaluate S for a different Herbrand interpreta-

tion. If or. every branch of a semantic tree a failure node

is found, it is called a 'closed semantic tree'.

Clause = TorTF?

!

Branch: \ [first; failure node.- 'Sclause = T Vl^ =7

specific HI nocei node •• Vclauses =T 3L^ = T

A33

Lemnas:

, A cl.v:se-set S J.S satisfiable iff it has a Herbrand (or

Gra-) model

S satisf. 3 HI satisf.

. A clause-set S is unsatisfiable iff all Herbrand inter-pretations are unsatisfia'cle,

S unsatisf. <$=> V HI

Herbrand Theorem:

unsatisf.

Version I : A clause-set S is unsatisfiable iff, correspon-ding to every complete semantic tree of S, thereis a finite closed semantic subtree.

S unsatisf. Tree: 3 Subtree (finite, closed) .

Version II: A clause-set S is unsatisfiable iff there ex-ists a finite unsatisfiable set S'n of groundclauses of S,

S unsatisf. 4. •) > 3 S'n (£ S') unsatisf.

' finite grd-clause set .


S-Clauses * { Px, («DyV Lay), (<«PuV*> Kv VNLuv), Da.Ka }HU = { a )HB » { Pa.Da.Ka.LaajS •(P.D.K.L}

Sena-Tree :

S - LPA(»»DVL)A(*»PVI*KVI»ILJADAKJ

= [<PAK)A(*>{PAK)Vi»L)AFA(wDPL)3 « [ O V ((PAK)A («L)A (O V (DAL) ) 3- [(PAK)VwL) A (DAL)] = [DALA(«*LA(PAK) ) 1- [DAUA«L)A(LA(PAK)}] - [QADA(LAPAK) ] « qed .

DF-Ri'-les;

The above method of 'truth checking' was implemented on a

computer by P. Gilrcore in I960. It however is inefficient;

as is easily seen, even for a small set. cf ten two-literal

clauses, there are 21Cl conjunctions. To overcome tnis in-

efficiency K. Davis and H. Putnam introduced in thfe sa:ne

year a more efficient method. The input tc their procedure

is a set of ground clauses S1 on which the following rules

are applied:

1) Tautology Role

One-Literal Rule

: Renove all tautologies, { LVLCJ = •

where the notation L C = N L is used.

: I* a one-literal clause {!.} is

present, delete it together with

all other clauses containing L and

remove in the remaining clauses

each occurrence of Lc,

3(L)-clause

3) Pure-Literal Rule : If L appears in some clause, where-

as L c is absent from any clause,

delete all clauses containing L,

4) Splitting Rule : Choose a literal L such that bc.th,,

L and Lc, are present in the clause

set. Replace the set by two new

sets, set 1(2): drop all clauses

containing L (Lc) and delete LC(L)

in the remaining clauses,

5) Subsumption Rule : Remove every clause G^ if there ex-

ists a correspondino "sub-clause" with

each of its literals also occuring in

VG, ^ . . . L - j l

At the end cf this analysis either the ground clause-set is

empty S'=tf which means that S' is satjsfiable (afc. 3 ground

model), or the ground clause-set is unsatisfiable S'=

(=*• contradiction),

<p sÊ' satisfiable

IClausesj * j

»=#S' unsatisf iable .

DP-Theorem:

Each DP-rule preserves the (un)satisfiability of the original

ground clause-set S'.

Example 17:

1) S' * { BVAVCVWftVD } a* { BVCVD )

— ^ Tautology Rule

2) S' »{ A,AvBVC, t*AvD»E j " ^ { DrE }

—• One-L Rule

3) S" « { A V B , AVC, AVE } = ^ 4

—• Pure-I< Rule

4) S' » { A V B , NkvC, ~AWE ) =*> { al , {C, EJ

—• Splitting Rule

5) S ' »{AVBI?C, ArEyPyCvB J => t AVB»C

' t • t

Subsumrcion Rule

6. RESOLUTION" PRINCIPLE

The major drawback of Herbrand's procedure is its Generation

of an exponentially growing sequence of ground clauses. The

resolution principle, which was proposed by J.A. Robinson in

1965, can be applied directly to any clause-set S (not neces-

sarily ground clauses) to test its satisfiability. However,

we first need to understand the notions: substitution, uni-

fication, resolution-factorisation, and PR-Deduction.

Substitution:

A 'substitution' is a finite set of the iorir. {t^/xi... tn/xn \

where every x^ is a variable (to be replaced), every t^ is

a terir. different from x$_, and no two elements in the set have

tne same variable after the stroke-symbol. When tj...tr are

ground terms, the substitution is called a "ground substitu-

tion1. The substitution which consists of no elements is cal-

led the "enpty substitution' ana is denoted by £,

\

Terms tj

w- S {ti/xi tn/*n) S {t/xj4

I Variable: x^ f XJ

Example 18: V & {a/x, v/y, gu/z} (vars =p terms) .

Composition of Substitutions:

Let 5*2it/xi and X = is^'J be two substitutions. The 'composi-

tion' of V and A . denoted by S"»A > is obtained frorr

{ t^X/xx....tnX/xn.si/yi....sR/yn\ by deleting tjX/xj for

which tji-x3 and by dropping any element »i/fi if y^_ is anong

delete; 1) (

a) («t/yi) if vi€{?)•

137

Composition Rules:

Example 19: JT = { a/x, v/y, gu/zj , A*{b /v , u/x, z/wj

ff-.X =faX/x,- vA/y, (gu)A/z}u f b/v, j ^ x , z/w J= {a/x, b/y, gu/z, b/v, z/w}

Xor = [br/v, u<r/x, zoVw Jv { <£?^ v/y, gu/z }= £b/v, u/x, gu/w, v/y, gu/z } .

Instance:

An 'instance' of a literal (or clause) is obtained by the ,substitution &" which replaces simultaneously its variablesby terms,

LS" as L({"x*1 "> {$ ) (simultaneouslyi)

Substitution i Variables^**" Terms

Example 20: L(fx, y, z, w) = L{fa, v, gu, w) .

Unification;

The procedure of finding a substitution which will make twoor more propositions (literals except probably for the sign)identical is called 'unification',

r via Substitutions

U...) * L(...)

Unifier 3";

A substitution &" is called a'unifier1 for a set {L2...Ln} iffLjF « L2T* ...*Ln(T. The set {iLi...Lnl is said to be 'unifiable'if there exists a unifier for it,

{Li , .. . ,Ln) ur.ifiable

438

Host General Unifier (MGU):

A unifier « M G U f° r a s e t iLl-«-Ln' -s cailed a 'most generalunifier' iff for each unifier C there exists a substitutionX such that: C = S M G U ° ^ > • e generality of the substitu-tion, meaning the maintenance of as many variables as possi-ble (instead of constants, etc.), counts for it to be a MGU,

•Substitution

3X •• C *0MGl! ° *

MGU (vars.)Unifier (csts)

Example 21:

^ S Q(x,b,y) , L 2 as Q{w,b,z)

Unifier :{T= {x/w, c/y, c/z} =^

MGU :&= {x/w, z/y J =^

Qs'x.b.c)

Q(x,b,z)

MGU Construction:

A number cf unification algorithms exist. Perhaps the easiest

to understand is that which proceeds left to right comparing

the first, second,... symbols in the two (or more) proposi-

tions. At each step define the disagreement set D^ = (t)j, xî

which must consist of a term t^ and a variable not occuring

in tfc, otherwise no unification is possible. 3 y = {t-K/xfc}

is the corresponding substitution. After all symbols have

been compared, the MGU is obtained by the composition:

0MGU * VVV"""'

ILfx. , L ( f ,> ' , . . . ) ] Disagreement Set-: D> M

if tJ* I Unifier : | k •= i

MGU : GMQU • ^QO'I !

( is unique I)V

Unification Theorem: If two (or more) propositions are uni-

fiable, the above sketched unification

algorithm will determine the MGU which

is unique.

439

Resolution:

The general inference technique to derive new clauses from

known ones (not necessarily ground clauses) and their suc-

cessive use to derive eventually • , is called the 'Resolu-

tion Principle'. The technique is simple: If the original

claus'e-set S contains • , then S is unsatisfiable. If S how-

ever does not contain • , the next thing to check is whether

Dean be derived from S by generating new clauses. By Her-

brand's Theorem, S is unsatisfiable iff there is a closed

semantic tree T for S. Clearly S contains D i f T consists of

the root-node only. If S does not contain Q , T must contain

more than one node. However, the generation of new clauses

will change some T nodes to failure nodes and it thus will

reduce the number of T nodes until the empty clause O is

eventually derived.

Factor:

If two (or more) literals with the same sign of a clause G

have a mest general unifier "& S MGU, then GG is called a

'factor1. If at is a unit clause, it is called a 'unit fac-

tor'of Gj

Factorf : 1) G(..L1...L2) : same-sign L's.

2) 30 £ MGO(L!,L2)

3)F(G; - G» : (includes G) .

Example 22:

Gj k PavpxyPy r ^ ^ G i ) - G or (PavPx) or (PaVPy)or Pa

Gift PxV PfyVNQx anfrT(Gi) « Gt or (Pfy V Qfy)

Binary Resolvent;

Let G and H be two clauses with no variables in common. Let

I<1 and L2 be two literals in G and H respectively. If Li and

l«2 have a most general unifier 6 s MGU, then the clause:

,H) 5 {GC-L^j U IHS-L^BJ is called a 'binary resolvent'

of G and K. The literals L3 and Lj are called 'literals re-

solved upon1, and G,H are Ehe 'parent clauses' ofJBJt,.

B i n a r y R e s o l v e n t B S . : 1) G {.. I^TT. ) , H ( . . L 2 " > s d i f f - s i g r . L ' s .

2) 3 9 = MGUCLj.,**) L2)

3) JfcH. !G,H) = {c-6 -LxSi U {n8 -Ltfl .23:

G&PxVQx, K ^ PaVRx , %% = J G S - L ^ t u{tig-L^3\ = QaVRa

, BR - |Qxv RxV-« O>w Sai or

Resolvent:

A 'resolvent' of the (parent) clauses G and H is one of the

following binary resolvents; (i) alt. of G and K, (ii) a

ISlL of clause G and a r- ctor of H, (iii) a WR. of a factor

of G and the clause H, (iv) a SK. of a factor G and a factor

of K,

Resolvent % s % (G,H) «= H ( "V.Q), T(H) ) .

Clause or Factor

Theorem: The resolvent fc of two given clauses G and H is a

logical consequence of these clauses,

1 (G.H) Lconi {G,HJ

Lifting Leama:

Most theorems on resolution are first proven for ground

clauses and then 'lifted' on the general level of clauses

with (iT-quantified) variables by the following lemma;

H subsumes G 4=£ 3 9" •* H 0"£ G

Substitution ' »— subsurced clause

Example 26:

Px subsums Pa, PaVQb ? PaxVPyb subsumes Pab.

Deletions;

Delete any clause in the clause-sets S^S2,... of the level

saturation method which : (i) is a tautology, !ii) is sub-

sumed by another clause.

Deletions: 1.) if Ja. (Tautology)

2.) if G^F < Gj (Subs, clause)

for all sets Sl, S*,...

Deletion Strategy;

The K-deduction supplemented by the level-saturation method

and the deletions, is called 'deletion

strategy'. It is a resolution refinement and preserves

soundness and completeness,

"Rp • R + Level Saturation + Deletions «^ sound + conplete.

Example 27: (see ex. 10}

(1) Pa

(2) Ca

(3) Ka

(4) o)Dy v Lay

(5)

442

If G' and H' are instances of the clauses G and H respecti-

vely, and if R' is their resolvent, then there is a resol-

vent K of the clauses G and H such that P.1 is an instance of

R.

3t(G.H)Instance

Lifting Lemma : G, •*+

(Robinson) G',H'

R - Deduction (Refutation):

Given a clause-set S and the target clause A. An 'R-deduc-

tion of A*, denoted by s (rrA. is a list of clauses {Hlf . . ,Hna

in which each clause is either: (i) a member of S, or (ii) a

factor on an earlier clause, or (iii) a resolvent of two

earlier clauses. If A s O . we speak of an "R-refutation'.

Hote, R-deduction. needs both, resolvent and factor.


5-Clauses

(1) ?a

(2) »#&y VLay

( 3 ) NPu Vet Kv VeO Ku v

(4) Da

(5) Ka

R-Clauses

(1+3) =^ (6) «»Kv

(5+6) (7)(»Laa

(2+4) •• (8) Laa

(7+8) =*>• (9) •

R - Deduction Tree:

An R-deduction can be represented by an 'R-Deduction Tree1

where the clauses are indicated (possibly by numbers) on

the nodes and the literals resolved upon ar« shown on the

connecting arcs. The R-deduction tree "grows" from the lea-

ves to the root-node.


(1) (3) (5) (2) (4) S-Clauses

R-Clauses

R-Soundness:

The R-refutation derives new clauses from the original ones,

in an attempt to derive the empty clauseQ. A conjunction

which contains TF1 is unsatisfiable, since it will be false

whatever assignments are made to the other clauses. For this

to imply the unsatisfiability of the original clauses we

must show that the resolution R cannot derive a clause which

would make a satisfiable conjunction into an unsatisfiable

one*, that is, any new clause must be a logical consequence

of the old ones. This fact is called 'Soundness', because it

shows that resolution is not a faulty rule.

Soundness Theorem: If there exists a R-refutation of a

given clause-set S, then S is unsatis-

fiable,

S unsatisf. <—— S i — |*~l

R - Completeness:

That the inference of new clauses via R-deduction will al-

ways lead to the empty clause O , provided the original

clause-set S is unsatisfiable, is guaranteed by its 'cow- ,

pleteness'.

Completeness Theorem:' If a clause-set S is unsatisfiable,

there is an R-deduction of the empty

clause D .

S unsatisf.

eventually found!

7. DELETION STRATEGY

Resolution is inore efficient than the earlier methods such as

Herbrand's procedure used by Gilmore. However, unlimited ap-

plications cf resolution may cause many irrelevant and re-

dundant clauses to be generated. Thus there is room for fur-

ther improvements by the 'resolution-refinements1 which are

considered in detail in part II of this tutorial. A first

step in this direction is the 'deletion strategy' which was

considered by R. Kowalski in 1970.

Level Saturation;

The R-deduction does not specify which clauses are used to

generate factors and resolvents in order to find • . One

straightforward way is the 'level-saturation method': com-

pute all resolvents of pairs of S clauses, add the resolvents

to the set S, coj^>ute all further resolvents, and repeat this

process until the empty clause D is found. We thus generate

the sequence of clause-sets:

S° « S TRfSj.SzJa {"RfGj Gjjj Gj€Slf Gj€S2,

s2 * Tt(s°.s°)s2 - m.(sou s1, s1)

= , S2)

sn+l » "R (S°w SXU...Sn. Sn)

This prescription generates redundant and irrelevant clauses

which, by using the subsequent definition, nay be dropped

whereby the soundness and completeness property remains va-

lid.

Subsumption:

A clause R 'subsumes' a clause G i f f there is a substitutionCT such that HtTSG. G i s called the 'subsumed' clause.

H subsumes G 4=£ 3 «• : H VS. G

Substitution i '— subsumed clause

Example 26:

Px subsums Pa, PaV Qb ; PaxVPyb subsumes Pab.

Dele t ions:

Delete any clause in the clause-sets S^.S^,... of the level

saturation method which : (i) is a tautology, (ii) is sub-

sumed by another clause.

Deletions: 1. ) if /EL (Tautology)

2.) if Gj«-£ Gj (Subs, clause)

for all sets S1, s2,...

Deletion Strategy:

The R-deduction supplemented by the level-saturation method

and the deletions, is called 'deletion strategy'. It is a

resolution refinenent and preserves soundness and conolete-

ness,

•Rp m R + Level Saturation + Deletions *>• sound + complete.


(1) Pa

(2) Da

(3) Ka

(4) o>Dy V

(5)

4 4 6

r(l+5) ->

(2+4) *»

(3+55 •>

L>(4+5) c^

(7)

(8)

(9)

Laa

wPuVi%»Lua

(1+8) =>

(7+10) =S>

(10)

(11)

Laa

8. SUMMARY

In this tutorial we have given a condensed introduction intothe basic notions and fundamental reasoning techniques oftheorem proving. Since we aim at reader* which mainly areinterested in the application of the mathematical theory, weleft aside all proofs and illustrated the definitions andtheorems by examples. After some historical remarks, we in-troduced the basics of the prospositional and predicatelogic and we showed how the equivalent formula manipulationsare used to arrive at the clausal form of a formula. Her-brar.d's theorem gives a first technique to certify the vali-dity of a formula. We here introduced the Herbrand -universe,-base, -interpretation and showed how the semantic tree helpsin the truth checking; the Davis-Putnam rules simplify thelatter step. Robinson's resolution principle is at the baseof all modern proof techniqu >.# in theorem proving. The reso-lution-deduction allows for tne derivation of new clausesuntil a contradiction is found. We explained the notions:substitution, unification, resolution and factorisation, andwe defined the R-deduction which is sound and complete. Inordfer to reduce tne huge number of deduceable clauses manyresolution refinements have been proposed, *>e here limitedourselves to the deletion strategy as the mot:!1, obvious andsimple technique.

Part II of this tutorial is reserved for the resolution-resolution refinements, and part III will focus on the appli-cations of theorem proving/automated reasoning in science,technology and every-day life.

ACKHOMUEDCEMEWTS s

The author thanks E. Oubuis, E. Peter and Dr. D. Ryter fortheir reading of the nanuscript and helpful suggestions, andDrs. P. Vogel, K.S. Kolbig and Prof. C. Joseph for theirinterest and encouragements*

The author al«o thanks the CESS Theoretical Physics Division,where part of this work was done, for its kind hospitality.

REFERFifCES

[1] J.A. Pobin«on

L04UC1 Form and Function

Edinburgh University l»re»«, Edinburgh, 1379.

[2] C.L. Chung, R.C.-T. Lee

Symbolic Logic ttnfl Mechanical Theorem Proving

Academic Pre««, H«w York, X973.

131 A. BundyThe Coaiputer Modelling of Math«matieel ReasoningAcademic Press, New York, 1983.

[4] D.W. LovelandAut:omatecl Theorem Proving: A Logical Basi«North Holland Publ. Co. , Air.stetdara, 197S.

[5] L. Wos, R. Overbeeck, E. Lusk, J. BoyleAutomated Pêasoning; Introdoetion and ApplicaticrisPrentice Hell, inc.. Hew Jersey, 1934

S!.T"£R.SYMMETRIC FUTOR AND CP VIOLATION

Lawrence J . Hall'

Lyman Laboratory of Physics , Harvard Cr.iversityCambridge, Massachusetts 0213E

U.S.A.

Abstract

The flavor and CP properties of the supersyxmetric standard

SU(3> xSU(2) x D(l) model (SSM) are briefly reviewed, and several

important results are stated. A new mechanism for flavor and CP

violation is discussed. It is interesting and important since it

0 -0occurs whenever the SSM is embedded in a larger scheme. B -B

mixing and B -*• (hard y+x) are good places to search for such non-

standard flavor violations. A new scheme for CP violation in super-

syaeetric models is sketched.

A general broken supersymmetric SU(3) xSU(2) xU(l) model contains

an enormous number of soft supersymraetry breaking parameters. With s

Jew assunptioas (see Sef. [1] and references therein) about the origin

of supersymmetry breaking, Chese can be reduced to jast x few paraneters.

The resulting standard model <SSM) has become canonical, and its

phenomenology has been studied close to exhaustion [2]. The flavor [3,-i'

and CP [5,6j violating properties of the theory have received particularly

close attention and will be reviewed below.

The bulk of the paper will deal with the possible effects of flavor

violation beyond the SSM. Such effects are important because they ars

ubiquitous, and interesting because they have been so little studied and

have surprising results. For example, all supersynmetric jrand unified

theories known to me have u •+ ey with a branching ratio of at least 10~~ .

450

Such facts surprise even the expert. This work was dor.e in collaboration

vith Alar. Kostelecky and Stuart Raby. For details Lhe reader is referred

to 17].

The paper ends by sketching a new origin for CF. The idea is to

solve the strong CF problem by breaking CF spontaneously in a heavy

sector. Fermion masses are preserved from acquiring a phase by the aor.-

rer.orEalization theorem; thus 6=0. Scalar masses, however, ars unprotected

and it is these which are the source of £. Models are currently uri.iez

study [8] and allow the phase responsible for E to be the sane as that -»>

which is responsible for the baryon asymmetry of the universe.

The SSM is described by the superpotential:

f = D'VqH' + LA ECH' -Hi'"Xr.qH + uHK1, . (1)

where Q,L are left-handed doublet quark and lepton superrieids, Uc, D1"

and E* are left-handed singlet antiquark and antilepton superfields and

H,H' are Higgs superfields. At the Planck scale supersynmetry is broken

ir. hidden sector, resulting in the following soft supersymaietry breaking

operators at tree level:

re " CSAA » BAA, (2)

•«gaugino

vhere E is the apparent scale of supersyar.etry breaking, which we consider

to be of order the weak sâle. A, B, and C are dimensionlest constants

of 0(1), $ represents all the scalar fields, X all the gaugino fields and

f-'i,) i» =*>e trilinear (bilinear) part of f.

At tree level the only flavor changing interactions are those of the

Kobayashi-Maskawa (KM) aatrix, K, in the charged current couplings of W~

and w~ to quark mass eigenstates. At the loop level the squark and quark mass

matrices can only be diagcnalized by different rotations on squark

and quark fields. This introduces further flavor changing Interactions,

for example, in the gluino couplings. The origin of this flavor violation

is the anooalous dimension £or the squark mass operator shown in Figure 1.

It leads to a radiative squark mass:

* 5 " & > + A£v (3)

As we will not be interested in processes involving external real squarks,

we will work in a "super KM" basis [6] in which tIC^D and D C A D D are

diagonalized by superfield rotations, so that the gluino couplings are

flavor diagonal but the left-handed squark mass matrices have flavor

changing radiative contributions

(4)

where SL. _ are the real and diagonal quark aass matrices. It is important

co notice that the SSM has no flavor nixing angles beyond those in K.

However, K appears is squark aass matrices at veil as in weak charged

currents, the notation is sinilar to that of [6], where a detailed dis-

cussion of C? violation is the SSM is also given.

The following results on the effects of K in the SSK have been

obtained 13,4,5,6]:

(1) The KT~K, nass difference, An. is probably doainated by the

usual W box diagrams. The supersymnetric (gluino) box diagraa is sup-

pressed by (o ,a 6.)/a .

(2) The supersyoaetric box (gluino) will dominate c_ if the tcp

quark is heavier than the squarks, since it is enhanced by » /n2 con-

pared to the usual V box diagrae. Ic this ease ». is smaller than ino •

the standard model.

452

(3) If *• 1* reduced, Chen so is t'/z: the penguin diagram is

proportional to »., and chc superpenguins are small as chey do not

involve large logarithms.

(4) Tor a /« i 1/2, the flavor changing and C? violation properties

in the K system of tfe« SSM are not greatly different from those o£ the

standard model.

(5) In the B -B system, the supersymmetric box diagram is a

2 2factor (ot./ou) (m /n) f of the conventional box result, f varies

0 -0between 1/10 and 1 depending on the parameter C. Hence B -B mixing is

probably (but not necessarily) enhanced in tne SSM.

The SSM conserves e, p, and T numbers. This is easily seen by

starting at the Planck scale in a basis in which X is diagonal. Thus,

scaling of slepton masses cannot introduce leptonic flavor violations.

It is a very general result chat trilinear terms in the super-

pocencial, which contain quark superfields, lead to flavor changing

squark masses.

Let L.(i •].,...5) be the five varieties of light quart and lepton

superfield: Q,U ,D ,1.,E . Suppose that the SS.M is extended by

where x.y.z are additional suoerfields with ths correct gauge rapreseata-

tions to couple :o quarks and leptons. As usual, I have not vrirter.

explicit generation indices. The diagrams of Figure 2 generate squark

and slepton masses, for example, for i»Q

where each n-. is a 3x3 aatrix in generation space,and £ o is a column

vector in generation space. If the fields x,y,z are light compared to

the Planck scale, then these diagrams contribute to the anomalous

dimensions si Che squark masses, and so the terms in (6) are proportional

to a large logarithm In M /M . If x> y and z masses are comparablep x, y, z

to M > then (6) should be considered as a heavy mass threshold effect.P

In either case the induced flavor changing effects ia the lov energy

theory can be large. It should be stressed that it does not really

matter if x,y,z are heavy (~M ) or light (-M ,) • It should now be clear

that it is almost impossible to incorporate the SSM in a larger unified

scheme without inducing such flavor changing interactions.

It is possible, that the couplings r,,c, are small and only give

unimportant corrections to the SSM. This is the case of the quark and

lepr.on Tukawa couplings in minimal supersyranetric SU(5). Ir. this

theory x « H.,5 the tvo color triplet partners of Che Higgs doublets,

and n is X or X.T (which is symmetric). Scaling between M and the grand

unification scale yields scalar masses:

where T(Q,L'C,£C) and i(DC,I.) are the 10 and 5 superfields. The most

interesting feature of these aasses is that the E scalars have picked

up a mass squared proportional to \.A • The lepton mass matrix is

proportional to X£ • X . Since siepton and lepton oass aatriceE cannot

be diagonalized by a superfield rotation, these slept;?, casses induce

2 + 2e, [i, and T nunber breaking effects: iB_c

a K m,.K. Unfortunately,

these effects are too small to be important; foT example, the branching

-15 A'ratios for \i •*• ey is -10 {100 GeV/m) ,. Nevertheless this illustrates

how superheavy particles (in this case B ) can alter the low energy

theory via souark and siepton masses.

454

The ,'SM, describee by (1) and (2), has chiral symmetries very similar

to that of the standard aodel. X_ can be untie real and diagonal fay rota-

tions OB L and t superfields so that e,u, and T are conserved. In the

limit the: A. has • ftassless eigenvalue the chiral symetry which rotates

c i° cthe left-handed anti-dewr. superfield, T> - e 'D, , becomes exact. Hence in

this lijait n.^0 to al l ordeis.a

Additional t&ncs in the superpctential, (5), chanee this chiral

symmetry structure. For example, they tray rorcj the C* to have the san;e

chiral charge, so tha: even X_ has a zero eigenvalue the down quari1. ir.ass

is not protected from raaiative corrections. At low energies this chirsl

syramezry breaking is manifested via the radiative scuark aass matrix:: ^ O -

If V is tha rotation on the Ec superfields necessary to diagonalize ? ,

c — 2

then the chirai synnnetry en l> is broken if (Vi« CV) f 0 for ar.y i f 1.

In general iM j wiil have non-zero otf diagonal elesifcnts. However, even

if i t is diagonal, non-cegenerate scuarks are sufficient for the flavor

r.ixint; angles of the rignt-handed quar'n. sector to be observable.

Similar censiderations apply to the leptor. sector. Radiative2 2

contributions to AM c and J2i~ will in general break e, u, and t number,

although total 2ep-on number is conserved. Ihe neutrinos regain nassless.

Ihe dowr; quark receives « radiative ssss. fros the diagrar. i~ Tiiuie 3,

cf! ,,.2 l f , w 2 )

^ !

where we have taken equal scuark and gluino cesses and d' is a heavy

flavor of down type quark of aess c , , .

These weak scale radiative corrections allow us to use the tret

level SU(5) oess relation:

455

•-SSL -v -£ L ^ i. Mev.K J S K j S U

He staply need E " ' 4 I . » I , , which can be arranged by having A=l,

m , = 200 GeV and the d iaens ionless f lavor mixings - 1 / 5 .d

There may a lso be radiat ive correct ions to m and a . Those ofeC u 0

a are probably small, while Am » m al lows an easy understanding

of the ra t io m /it , in a four generation model [ 7 ] .u d

In (8) d' cannot be s because the re su l t ing flavor inser t ions give

too large a contribution to An_. However, ar. exc i t ing prospect i s opened

by considering the case that d1 i s b. This requires the dimensionless

f lavor changing inser t ions , 3, » AM / « ' and <5 " Axicgc/m", to be ~1

for A - 1.

One important consequence of 5T „ - 1 i s in TS (bd)-B (bd) mixing.

The r a t i o of prompt same sign to opposite sign d i l eptons seen in the

0-0production of B B pairs i s given by

(9)

where x ' AM /I" i s a measure of the mixing, expected to be -1/20 in

the standard model, and probably s l ight ly enhanced in the SSM [A]. For

more general mixings £. ~ c [7 ] :L ft

Clearly i f 5, . are responsible for m. - im, , then e i ther B.-B mixingL>R o d c c

i s maximal, or m i s g r e a t e r than s e v e r a l TeV. Such a a x i a a l mixing i s

not yet ruled out i f the E seraiieptonic branching ratio i s l e s s than

that of the B+ [ 9 ] .

The diagrams of Figure 5 give a non-charm hadronic filial srate,

predominantly B -* o7, with branching rat io:

456

In order that the secileptonic branching ratios of B mesons are notB E >

diluted by more than 10%, R(B •nil) <0.1, thus for oL - 5R - 1, to - 150 GeV.

' The process B -* (bard y +x) could be observed directly. The diagrams

of Figure 6 lead to

RCE ->-hard y+x) - -fz — k r*1— !— , (12)60 2 i si j

which could be as large as 10 . This is 1-2 crders of magnitude above

the standard model background [10], and the hard y would provide a

particularly good signal if the B mesons uere produced near rest.

The electric'dipble noment of the neutron, d • places important

R B •

constraints oa Im c , and also on st even if o are real providing

they are large.

In the SSK diagrams such as those of Figure 7 give

For m- t> m 100 GeV, and Re A 1, this constrains the phase of A to be

- 5.10 " [5]. If this phase were real due to CP conservation of the

hidden sector, then in the SSM sufficient squark insertions of the form

(4) tnust be placed on the scalar line so that s^ appears is the result,

la this case the equivalent phase i s [6] (i>a.i»*/tL)sS,s s ; "v 10~ giving

A ^ 10 ecm. With additional flavor violations a contribution = it.

is possible as shcvn in Figure 8:

^ 5 b ^ M !9

For the case of Re 5 rw 0(1), the constraint on Che effective phape isLtfR

457

l-aproved by three orders of magnitude. Hence even ii A is real, we are

led to

In $1$*, - 5.1C"6. (14)

3We conclude that if 5, „ are large, then they must also be essentially

L,R

real.

It may be that the source of A and of 5 . is CP conserving. InLtf A

this case the dominant contribution to d again comes from Figure 8, but

now the cross oa the left-handed squarks is the SSM insertion of Eq. (4)

while that on the right handed squarks is again 6 . The experimental

limit on d gives

, B Q ) 2 < 5.10-6_

Large values of 6 then imply that s, is less than in the standard model.R ' 0

One might worry that this is a problem. If s^ is so small where

does C coae from? It is at this point that one is driven to a new theory

of CP violation which I describe below. In my view, within the context

of supersymmetry it if certainly as good, and potentially ouch better

than the KM origin of CP.

My discussion of d., suggests several things about CP violation inh

an underlying theory. Firstly, that the trilinear terms in the super-

potential responsible for 6 , and also \ , are real. This certainly

suggests that CP should be imposed as a symmetry on the dimension A terms

of the Lagrangian. This also removes the strong C? problem: 6 vanishes.

In this case where does CP violation come froa. One possibility is

simply to introduce it softly in terms of dimension 2 as was done in a

section of Sef. [6]. There :.t was shown that a phase in ^ c g c o r ^ s

was sufficient to give £ at the one loop level. In the present case,

453

adding arbitrary soft CP violating squark masses falls far short of our

aims, since we would no longer understand why c. are real. Instead we

impose CP on the entire Lagrangian and break it spontaneously in a super-

heavy color singlet sector. For example, consider an SU(5) theory with

generations of gauge singlets N, in addition to 5 a-10 (F+T), whose super-

heavy mass matrix M has beccrae complex by spontaneous CP breaking. In

this case, the trilinear interaction A FK H (where H is the Higgs 5-pitt;

allows the phases of M to be fed radiatively, via the diagrams r.f Flours 9,

into the D squarks mass matrix:

2 + •*-

I find this oechanisa of CP violation very exciting- Consider the

following interesting features:

(1) The strong CF problem is solved. Potentially troublesotse

radiative contributions to A are absent in supersymtetry because of the

non-rencnnalization theoreo. One simply has to check that the phase does

not feed reaaily into the eluino mass matrix.

(I) One say understand why c is small. Tha phase is in (16),

whereas the conventional insertion (A) is real. It is very reasonable

for (4), which yields in,, to be larger than (16), which leads to e.

MThis is because the iogarit.Tn in (4) is In -^-, while that in (161 is

% 0 -0 * V

lr. ;-*-. For large j"j> B -B mixing and b •*• dy one r.eeds a large* G

trilinear of the fona (5) which does not involve X.

(3) l'It is likely to be scalier than in the standard nodel. This:

is because the penguin diagram is proportional to s. which vanishes, ana

the superpenguins have no large logarithn.

(«) Finally, it is very easy to see how the phases associated with

S are the origin for the CP violation necessary to cause a baryon

asvnroetry in the universe, iB.

459

From the diagram of Figure 10 iB = XVM A ^ / . ^ - A t race i s avoided by

having some N heavier than thfe Biggs t r i p l e t s , and seme l i g h t e r . The

phase of (J, XM~*"A * occuring in Figures 9 and 10 lead to t and AB respec-

s"" stively. It is hard to imagine a siapler and acre direct link between

these two quantities.

A more detailed analysis of this CP breaking mechanism, is under

study [8].

In this paper I have argued that even if the SSM if substantially

correcc, its unification will inevitably lead to new flavor and CP

violations. Flavor violations may allow us to see the effects of super-

partners long before they are observed directly. The aechanisa for CP

violation in supersytsaerric aodeis is still a wide open question. I

have argued for a particularly simple and elegant mechanism as an

alternative to the conventional KM mechanism.

Acknowladaements

I have done much of this work in collaboration with the authors of

References [7] and [8]. I thank nany colleagues at Ka;i=ierz. CERX.

Crete, and the Gordon Research Conference for helpful discussions and

conversations. I would like particularly to thank cy host* wniie in

Poland for making my visit stimulating and enjoyable.

This research is supported ix» part by the National Science Foundation

under Grant So. FHY-82-15249 and also by the Sloan Foundation.

References

[1] L. J. Kill, Proceedings cf the Winrer Schojl ir. Trôvsriaal r'^sies,

Mahabale3hwar, India (1984), Springer Verl«g Lecture Sotes in Physics

208, 197.

A6C

[2J E. Haber and G. Kane, Phys. Rep. 1T7 (1985) 75.

[3] J. Ellis and D. V. Jtenopouios, Phys. Lett. JOOB, 44 (1982).

R. Barbieri and R. Gatto, Phys. Lett. 11 OB, 211 U9S2).

M. Suzuki, Berkeley preprint DCE-PTH-82/8 (1S82).

T. Inami and C. S. Lim, Kucl. Phys. B207, 533 (1982).

B. A. Campbell, Phys. Rev. D28, 209 (1983...

M. J. Duncar., Nucl. Phys. B221, 285 (1983).

A. 6. Lahanas and L. V. Hanopoulos, ?hys. Lett. 129E. 461 (1983).

E. Franco and M. Mangano, Phys. Lett. 135B, 443 (1964).

3. A. Campbell and j. A. Scott, Phys. Lett. 135B, 423 (1984).

0. Shanker, Nucl. Phys. 3?04, 375 (19S2).

J. F. Dcnoghue, H. ?. Nilles and D. Uyler, Phys. Lett. 12SE. 55 U?83>.

M. J. Duncan and J. Trampetic, Phys. Lett. 134£, 439 (1984).

J. M. Gerard et el., Ftys. Lett. 14QB, 3*9 (1984).

A. Bouquet, J. Kaplan, and C. A. Savoy, Phys. Lett. l4gB, 69 (19S-).

[4] J. M. Gerard et ai., CEKX preprint TH-392" (1984).

M. B. Gavela et si., Phys. Lett. 154B. 147 (1965).

[5] J. Ellis, S. Furrara, and Z. V. Sar.opoulos, Phys. Lett. 1143. 231 (1?82).

J. PoictiiT-.ski and M. B. Wise, Phys. Lett. 1255, 393 (1983).

[6] M. Dugar., B. Grinsteii., ana t. Kail. Sucl. rhy=. B255, 413 (1965).

[7] L. J. HJII, V. A. Eostelecky, dr.d S. Saby, Harvard University preprint

HLTF-35/*063.

i8j A. Dannecberg, L. J. Hail, S. Raay, and L. Randall, Harvard University

preprint Hl'TP-aS/AOei.

[93 T. Ferguson, trocee.c.iîas of XXXI Zntsmatiir^C Conference on High

Svevcit Physics, Leipzig (1984), Vol. I, p. 191.

[10) E. Caajfcell and F. O'Donnell, Phys. Rev. D25, 19S9 (1482).

46?

Fi,;ura Captions

R,- liat.Vi contr ibut ion to i!M" in the SSM.

2. Radiative contr ibut ion to iM" fro-a the operators of Eq. (5) .

3. Radiative cor t r ibu t ion to the do.jn quark mass.

B4. A "ypicaJ diagram involving o n which contributes to AM .

5. Diagrams contributing to b -» dg.

6. Diagrams contributing to b -• dy.

7. Typical contribution to d in the SSM.

8. Ivplcal contribution to d a"n models vith 6>: .e c'ea flavor vioi.ation

in ?<riark masses.

2 J. +9. R i J i ? t ive contr ibut ion to LZL.C = ' . . ^ l . ; -

-•* • { •

10. Radianive contribution to AB = A'.MM'?-„.N N

462

Figure 1

UC,DC,H,H'

sFigure 2

/

Y

X / \Q 0 0 Q

Figure 3

• &4 VD

, r •«# ^*d C Q Q

U S3

Figure 4

B

*D t

bv

-9—

B

; D C

d c

A 64

Figure 5

4I ^

b g d

VV\

S tX 9

g

Figure 6

8

9

Figure 7

\ u

I

g

Figure 8

3» i if-^—K > 3- «

Figure 9

mM

As

\. \

\

ACS

Figure 10

N• \

N

4S?

THE STANDARD ZLZCTP.OKZh? Mr DEI AND 5EY0SD: P. KEVI2K

Phaa Quang Hung

Department of PhysicsUniversity of Virginia

C h a r i o t t e s v i l l e , Virginia 22901U.S.A.

Abstract

A pedagogical review of the standard eleccrcweak model is followed by

a phenoraenological analysis of eiectroweak. interactions which led to a

partial ccr.f ircation of the standard model. The phenomenology of the »

and Z bosons is briefly reviewed with the purpose 01 shovisg the scrikir.g

agree-ent between experiment and the staidard Taodel. The pre-eciirg reviev

is followed by speculations on the unexplored Higgs sector of rr.e siodel.

Aspects of strongly coupled Kiggs nodels are discussed in this section.

The raview is ended vith a discussion of what could lid 'nsyonc the stsr.carc

model.

468

With the txrerisentai discovery of tj-.c '«" and Z bcsuiis, a chapter hc.=

beei: closed on the nature cf the slectrowesW interactions, specifically on

the forrc of the interactions anc the ider.cicy ;•: thtir carriers. Vijthaut any

doubt, s'ji;!) achievements were the result." of the incredible predic*..ir.£ po"..x-r

of the standarc model." It is, hgvever, not the- tn; of :h~ story. Ar.

inportanr ingredient of the stsr.d&rc nodel is btill missing: it is t'-.e

elc«ive Higgs bosori which plays surh a crucial roie ir; tht- cheerv. Is i;

there or isn't it there is a question wnose answer we wculc lii.t tc '••:ic-« in

the future. If the Kijgs boson aere to exist, would it be eic.-en:;r-- or

cor-posite? Are the scalar fields of the thecry stror.giy or wea'..Iy couple:?

The discovery of the Kijgs bosun itself would ciearly settle tftis i5S-ê.

In the neantire. any indirect clue >;s to its naf.:re will certainly be vslccr.e.

Another issue which has importar.t ra~i:ic2tior:s for bet':, theory 2-c fc::ptri-.cnt

is the cuestior. o: w'r.az lies "ir_-;tCi2:el;.'" beyor.d the stf.r.darc scdei. Arc- •

tr.erc new g--~e i::ter2cric:;s vith i>::ra \<'s a::c 2's'1 In sh'jrt, is there

ir.ythir.g elt-e? Ar.ctr.er question o: r.c :is= irrortar.ee it uhcther or not

the V>" an.i Z bcscr.s tre as icrcar-.ertal as one- -ight nave thought cr tl.ey

just aprear tc be sc at c-jr pres-sr.i energies. ThesE ars ar.crg the kind cf

fcr.din-.ental cur-stijr.s he;efully to be ar.£»ert: by future accelerators.

Ir. this lecture-, 1 vculc lir.e zo revie:. first the i-pcrtar.r irgrecie-.-ts

cf z'r.i s-tzr.zt.-i electrc-c.il-: zodel. :;e::t, a sh;rt cis:.usEior of the ph«r.c-

ocr.jlcry •-•>.•;'- led tc the oetemir.atirn of tha veai: neutral c:z:n~ts is

prc'sentec. The strii;ir.; asr-seêr.t btt-.ter; e:-:v-;rir.er:t anc the sLar.card

-eciil is further er.r.aticed :y the disce'verv of r.he ;•." sne Z bcecrs whose

prcperties are sj^-crizec next. The preceding -ere or less star.darc

ciscussior. is ther. followed by a sore spetu-lstive one on the nature of the

Hi;;:; boson, especially or. the possibility of having a strongly couplta

Higgs Jâc] v;;.n vt» ?'•••• '..Mi: 0lOj,lczl con&eqaeaces. TW i ' . ; U ' f '-• ~.'ea

wich t:'rê speculations or. what flight be beyond the standard modfci.

I . The Standard "lectroweak Model

s2

Some of th? main ingredients of th;:- standard electraweak modal a re

flUDrw.rized in th i s sect ion. Firs: . , the gautie symaetrv of the mode.' i .>

SU{?,\ x L"(l) whose gauge bisons are {V"-'°; and {Bi rr;=pect-iVelv. Ttiis*- y

iy-.r.-._-;v i.s spontaneo'jsly broken by the non-zero vacuut expectatir-r. value

(VEV) oi a Higgs field 4. In principle, <t c;.n be an arbitrary lcprebCtteatior

or SlU;),, the .simplest being a doublet, naaely if - (4O). As we shaU see

belou, nac:«re appear- to choose the doublet r6"--'tcr,tation. It turns om

that the Higgs potential for the doublet representation has a global

symmetry which is larger than the gauge symmetry itself.

The discussion of the Eiggs sector vitn a single doublet parallels that

of the old C-model. In fact, one can write a 2x2 matrix

= — (o+lri-t") - ( ° ? ) >where the vertical direction represents St'CI).

transformations while the horizontal one refers to global SUC2V transformations.

. in other words, l3' + gL*S^ ur.dc-r SU(2)L x SU<2)R with gL RtSU(2) p. The

Higgs potential which is invariant under SU(2), x SU<2)_ 2 0(4; is then clve*

by

•a-/1 °\Ke learn ne<t chat this potential has a aininun: st <$> - -*-J I . arid

v'I\& 1/

consequently SU(2)j x Su(2)j, is broken down to SU(2)V (its diagonal subgroup).

As for the gauge svmnetry, S0f2), x U(l),, -»• LV{1) since the U'j)L i e.m. e.m.

I., t T V « j •. U'VI 0 /

oi J) aT.'nU.ilaces the vacaizr. i.fc.

Now, t h i s global symmetry SU(2) has an important consequence en cUe

mass re l a t ionsh ip sezween the gauge bosons. Iv. f ac t , irom t!ie k i n e t i c ten.

•v IT CD i^")(D"1), t/tife obtains the following gauije boscr mass Tr.ctri?.

•ne r.otic«s that tlic diaeorial e lescn ts of TfJ preserve glob i SVl'2).. invar ian;e

Frr cicarapj- , V" and 17° navt the same îrss T g v ~. I t ;» the si.Miir, uetwc^:.

W° and 5 ii'-.e off-diagonal elenancs of1"l) which bre^KS Sl'(2},. - Diajjonsiisir.^

^ . one finer. X'\ • v g 2v 2 , 1^ • -j (g 2 +g '?)v 2 , and M" = 0 where

sad

- >

with tanA, - s'7-g. It is also c»sv £e see that

- 1 (5)

These two relations, ( if ) *nd ( JT ) , are very important, predictions of

the standard model. Equation ( H ), which Is sometimes referred to as the

471

electroweak unification relation, provides a definite prediction for the "J-nasr

in terns of experimentally measurable quantities, namely

Fwhere BJis Che e.nt. fine structure constant and G_ is the usual Fermi constant.

Equation ( *> ), which is a consequence of the residual global SU(2),, symmetry,

is Vi.ll. tested at low energies and provides a prediction for the Z-nass,

namely

The structure of the interactions between W~, Z and quarks and lepcone

cau be directly determined at low energies. Without repeating veil-known

2 2 2derivations, it suffices to state that, at q «»y» n^.

where T? - T(T, I it,), T,, » -=- , Qt the charge operator, p - 1, ani

~ » -tj (this relation is used in conjunction with Efl. (e^) to give

Eq. ( 6 1 ) . ^tch-r-od describes the well-known charged current interactions

like beta decay for exanple. There is nov an extensive literature on

subject so let me concentrate instead on reviewing what we knov about •

The left-handed neutral current couplings for the tensions deternined from

"^neutral *r* 8iver) b y CL * "^SL***1*2*?' w h l l e t h e right-handed one* are

E - -Qsin Sj. (0"l). The standard moflel predictions for quarks and leptons

are then given by

472

s i . --*

r a \

From a pure phenomenologicai point of view, it is preferable to deturcinc

C,'s and E_'s separately fron experiment without prejudice since a priori one

cannot know which aoo*l is right. A systematic pher.oiBenologic.il determination

o£ neutral-current couplings was undertaker, with, as a result, the confirmation

of the standard model at low energies (or any model which looks like the

standard model at low energies). Let rae recast the procedure used to

determine neutral-current couplings.

II. Confrontation with Low-gnergy Experiments

The reactions used in this approach are v+E-> V + X , ve-»-Je,

•*!-•• eq (SLAC awd atomic physics experiments), and c e •*• ja y~• Let me go

over briefly each of these reactions. For lack of *pece, the readeT Is

invited to consult Kef. (3) for sore details.

1) y-hgdran scattering:

A&swsing only lcft-hand«d neutrinos, the following couplings can, in

principle, be d«terEin«d froB tfcis reaction: e,(u), ET(<i), 6R(u'). and

e.{d). Without going into too many details, it can he shown that

v 4 S(I«G, "isoscalar taraet") •• v'-t-X determines z. (u) +£, Cd>" and

e («;•+e^Cii)" while V +< \* Xvy + X determines £ L(u)2, e,(d)2, C R(u)

2,;hile *v' +< V* V

-ately. Notice that

2the quark-carton model only e, , is determined and not c. .

f"

?.£ c,(d) separately, tfotice that because of the incoherence assumption of

£73

Experi.nentallv, from the CHAW-i experiment for exa:n;>ie on isoscal-r target

f. >••*•) + £ (J.J = o-"SoS i o-oi5 ,

£ (ft

while froa the BEBC experiment on neutron and proton targets, one obtains

It is customary to rewrite e in te-as of four other independentLi jK i

constants: a (isovector vector coupling), g (iscrvector aKial-vector),

Y (isoscalar vector), and £ (isoscalar axi«l-v«ctor), naaely, £,fu> «

•| (a+B + Y + «), ^(d) - j (-a- 8 + Y + 4 ) . eRfu) « | (a- 6 + y- «). an<S

^(d) - j (-a+3 + Y - 4 ) - the sign* of e^ R can be determined by knowing

the relative sign* an* aagnitudes of tt, S, Y, ana £.

Pro* exclusive reactions such as v + H -* v+iS(123*), v* + p * v +f>,

v + D •* v + trt t it was found chart the couplings ace pEe4cainaBtl\ i»ovectos-

i.e. |S! > \a\ > |Y|> 1*1- The signs of a and $ can be fis«d by

the reaults of SLAC (•' j + • •«~ j + X) a»n4 atomic physics «xperiaants,

ve scattering, and factorizaticxi (one Z-bosos txcluna* hypothesis),

a, 0>O. CaBint Vack to &L R, latomimt a.S > 0, one deduces £L(u) > 0,

Ej^d) < 6S Cj(«) <0, and ^ W > 0 - ^h< coaparison betwes experiment and

toe standard model is present^ in the taMc which follows. One notices

die rtsurkable agrecantt bctuecn Che standard node!, which Is described

by a single p*r*«ec«r sia'S^, an* experiment. One can also try tc ds a

tHo-paraaetar fit with p and sin 3V> Sucn fit »11I be given further below.

Lat UE turn to the ochec reactiona.

Experitaent 3.3^10.026 -0.153 t 0.02i -0.419i O.c;: 0.07610.0*3

standard nodeI,_-?o _ « ,,v 0.35 -0.15 -0.42 0.08

Table 1

2) v-s scattering

This type of reaction determines the vector and axial-vector couplings of

the electron, na»ely £v and g^. The reactions v^ e * lv,. e which are mediatt-d

only by neutral-current interactions are represented bv ellipses in the 6y~g,

plane. T~e intersections of these ellipses give four soiuticris for g,, ati-d g

(the sign ar.e x,, *g arebiguities). The ve~e reaction, which has both charge i

and neutral-c1.:! rent Interactions, contains a £V~8A inrarferer.ee tern whose

sign is knowr. The intersection of the V -e ellipse with those ol v. -e and . -

v, -e therefore resolves the sign ambiguity. The results ar« g*xp « OiO.lS

and st**P » -O.Sfci Q.14 with still the 8^**8» ambiguity. The standard model

predictions witti »m*S B • 0-23 are g'U « - -|<l-s«ir.2Sv) - -0.04 and g^" " " T-

Ajain, anc notices che »gree»*nt between the standard model asd experiment

once the fifSF*-*gfK^ aeeiguit;-' is resolved. As we shall see below, it turns

3) «-<i scattprln;

This type of reaction include* the SLAC experixcat (eT «+?:--er _+X) and

certsir. types of atoxic physics experiments. There are four coupling constants

invelvaj in the parity-vielating parts of the interactions (the parity-conserving

neutral-current interactions are completely masked by the nuch stronger electro-

wagaetic interactions). They ate denote<!3by Û^vJ* 1), S C V ^ " 1 ) , Y(AevJ*°),

and ':.(V A J ) , vhere V and A stand for v«ctor an* axial-vector couplings

respectively.

475

The SLAC experiment measures the asymaezry A - (0(eR)-C(eL))/(c(eR)+a(«:L))

C p , 3 - ***• 2 * 3 ~which is expressed in terras of a = (— e )(8a+-r y) and a, * (— e )(9fc + T 4).

vi • fl

Measurements gave a^ * (-9.7*2.6) x l(f5 GeV"2 and a, - (4.9 18.1) x 10"S GeV~2,

v - ~Zin terms of which one gets: <*+•£ » -0.61 0.16 and £ + - "= 0.31x0.51.

Atonic physics experiments on heavy atoats (e.g. Bi, Til, Cs) oessure parity

violation which arises from A_V interactions (a and y) since there is coherencee q

for the vector part (V ) (the "charges" are additive). Because the nucleon spins

tend to cancel each other, the V A interactions are relatively unimportant. The

parity-violating Ramiltonian is proportional to the "weak, charge" CW(Z»*O "

-[a(Z-N) + 3Y(Z+N)] where Z and 5J stand for the numbers of protons and neutrons

respectively. For example, <i,(Bi) « A32-627Y and Qw(Ti) • 42&-612Y. Experinentally,

the Seattle group gave Qjj(Bi) » -11SS19 v*iile the Berkeley '

- -170 +559r

4)

Two paraoeters wbieh have been measured so f«r are h ^ and h which represent

the deviation from QED and the forvard-btckwird asymmetry respectively. In th«

standard model, h ^ « TC1-4 sin 6^) and h ^ » . The JADE grouj obtainedj

h ^ - 0.01*0.08 and h ^ « 0.18*0.16 to be coopared with the standard model

2predictions (sin 9 « 0.23): 0.0016 and 0.25 for hy.. and h^ respectively.

S) Factorization tests

The hypothesis of a single Z-boson gives relations between different neutral-

curreot couplings. For eximpie, one can have: i « ~ ; •%•

r — " —s anong others. A look at these relations reveal that one h«s a

g -dominant solution f or \>e reactions and that a, U > 0 with e given by

Table ljin remarkable agreement vith the standard aodel.

It is fair to say thac all "low" energy tests performed so far arc *

consistent with the standard model i.e. the neutral-current interactions are

given by

H.C.;

with p«l. The one-parameter fit (p"l) to the standard model gives sin"8 =•

O.233 1G.009 (£0.005: theoretical uncertainty). Ihe two-parameter fit gives

p * 1.002*0.015 (±0.0U)j sin26w - 0.234 ± O.013(±O.OO9) with the most recent

one (from *v'K reactions) being p « 1.02 * 0.02: sin2ey - 0.238*0.03.

notice that the vorld average for sin'S., is:

III. Confrontation with High-Energy Experlmeat*

Kith the observation of V and Z by the UA1 and L'A.2 groups, the second- -

crucial cast of the »tafi£ard model was provided for the first tine. Knowing

Che vsaJt angle S^, one can use Eqs. < & ,7 5 to predict the H and Z masses

im Che lowest erdar of perturbation theory, i.e.

TV

the&e predictions *r« ho.v«v«tr awdified bj- radiative corrections which affect

the iefinitieti of Wg and Mj as v*ell as elvat of a, Cj., and sin^.. There has

>eer. «xtei)slvc discussions on this subject in the literature and I would

strongly rcccmmetid their reading. Instead, let roe sunaiarize the theoretical

predictions for sis 3^, M., and V~.

the value of sin 6^ as jives by (4.3 ) is measured at Q «K,- In order

tc> estiaace My and X^, one has to "run" cin 5 , to a wass scale cf order X,,.

2 "The "htist" value for sin B,, obtained so far is

As for M^ and >U. radiative corrections to O(ci) give

2 1/2^ = (7Ta//2 G sin 6W(1-Ar)) , * y^/cosQ^, where ir is the 0(o) corrections,

a = 1/137.035963 and G is the rnuon decay constant obtaineo from the mucn

lifetime computed to 0(oc), namely G = 1.16638* 0.00002 x 10~5 GeV"2. To

estimate ir, which includes vacuum polarization effects, one needs to knov

the mass of the t-quark as well as that of the Ki^gs boson. For example,

taking DL, = m and m.. s 36 GeV, one obtains,

M -. u.es ±caH

M - 77"3° t

where br * 0.0696 ± 0.0O20. The predictions for sir,2^ - 0.22 4 0.00b are

(My) . - 82.4+1.1 GeV ai.d (M z) t h - «.3 ± 0.9 CeV. Experimentally

^ V e x * 80.9 ±1.5 ±2.4 GeV(UAl), 81.2* 0.8 ±1.5 GeV (UA2) while for il^,

one obtains (M.) - 96.5± 1.2(±3%) GeV <UA1), 92.4t 1.1-t 1.4 GeV (UA2).Z exp

Needless to say. there is good agreement between the standard node]

predictions and experiment. More precise asasureaents of M , and M, will

certainly throw more light on any possible deviation froo the standard

model. Also, if m is known and M, is oeasured with a good accuracy then

one car. use sinS^ - Tl r to determine M^ accurately, naaely

476

V % I ^-^v^ v

where

Notice that A » 38.65 GeV for m * 36 GeV s:id a^ = M_.

Other iar^rtant quantities to be measured accurately are tile total widths

of W ana Z> With only three generations, they are estimated to he

r (W) - 2.8 GeV, rtor(Z) s 2.9 GeV, assuming mt = 40 GeV and sin25w = C.^2.

rl deviation fr-':a these values could signal many things, tor instance, if

rat >^ 40 G«V, :', t(2) could be smaller than 2.9 GeV. If : tor(Z> > 2.9 GeV

however, it is: than possible that a 'ourth type of neutrino is present.

Remember that I"(vv> ~ 132 MeV and T(tt) i 76 MeV Cor sir. 8^ * 0.22 and tnt 40 Gt-V.

I would like to close this partion by mentioning that so far there, seems

to be good agreement between experiment and the standard'iaodel in the decay

properties of V and Z. Exotic events (monojets, etc..) have not been

definitely oonfinr.ed or discounted.

As we have seen afcove thfcre i« a renarkeble agreement between th«

stanriard nodel and experiaent, both at the "low" energy end as i eli as the

high energy end. Tn«re is however a "cioud" hanging over the success of

the standard model whicfc i£ nothing bat the existence or non-existence of

the Higgi, b^son.

479

IV. The Rices Sector

There has been ample discussion in Che literature on various methods

for detecting a "light" Higgs boson. By "light" 1 mean s^ « 1 TeV. In

fact the background exceeds signal when HL, > 630 GeV. Let me, however,

concentrate on an unorthodox possibility, namely HL, > 1 TeV.

Why 1 TeV? Hell, it has to do with the question of whether or not the

scalar fields in the standard model are strongly coupled, from ( 4 ) one

finds that, in the minimal standard model with one Higgs doublet, the

physical Higgs boson has a mass HL, " JTk 0 <J"s 250 GeV). Since X, the

Higgs self-coupling, is arbitrary there is no way to predict how large nu.

is. Within the framework of perturbation theory, one can however set

limits on .V,.

If one first assumes that A is amall and compute the one-loop radiative

12correction to V(<£) (Eq. (l)) t one finds %h«t in order for the true

ground state to be an absolute local minimum, it, has to satisfy c lower

bound o?> 4<<» 2K. Here, tc - * . {3(2ft4.+'S*)-4rM* } * = i — T

** ' *4n*<#>* * Z i f I 64-Z«p>4

4 4 4{3(2MH+Mz)-4Mt ...}, where the contribution from the Hig#s loop it neglected

in K. Since it is »till unclear how large mt i« and whether or not there

are other heavy feraionst one cannot put a definite lower bound on a^. For

exanple. if there are only three generations and if • f 40 GeV, one wonla

obtain m^> 7.1 GeV.

On the other hand. If X (or m^) ie too large, tree ucitarity vlll be

Tiolated. Consider, for example, the s-wave scattering of longitudinaj

V s . The s-uave amplitude for H -WL•* 'V+W, grows with s ana becomes a

constant for «»•?. In fact, <l^(W W - 4 W W j - > - 3 V ""tL

•480

Partial-wave unitarity requires that la^j < 1 which implies that aij, < 1 Tey

(i.e. \<~). If the Higgs mass txtieeds 1 TeV, one ther. has a stctor which

is strongly coupisd. It raises the possibility that, for such a systen,

there nay be interesting bound state formation.

As we have discussed at the beginning of the text, the fiiggs potential

has an 0(4) synisetry, similar to the linear 0-model. There, it is customary

to work with the non-linear c-modei (for large nT) to describe n-ir scattering.

Also, it was found that in IT-IT scattering, one expects- the following beur.d

states to be formed:

i) I - J * 0 bound state ("5"). This is not a well-defined resonance

ii) I * J » 1 bound states (P1s). Those are proisir.ent resonances with

in - 775 M«V and T s 130 MeV. Also, a combination of current

algebra »ni vector meson dominance gives us a relationship between

f * f , the scale of chiral siwaetry breaking f and the

pm ts ' " ir16

jr. » namely the KSRF relat ion

A strongly coupled fiigjs model of the type used for the standard aodel

would be expected to behave similarly to the a-raodel described above. This

expectation is born* oiit by detailed calculations.

A naive estimate of the mass scales to be expected for these objects

iv. the strongly cottpled Higgs model can be obtained by rescaling, namely

•u.,,.1 s mriv'^) = 1 > 9 T-«^ f e r the I • J » 0 bousd state and * s » (J£-) s 2.1

for the I » J « 2 bound state, where n,, „ s 700 lieV, kf% 250 GeV and £_ a 93 MeV.

These naive estimate* ju6t serve as guiding signs foe the search of these

.objects.

A 81

The .above numbers are given for the one-Higgs doublet case, for the

multi-Higgs doublet case - P « 1 for any number of Higgs doublets - one I13*

the constraint :tfTs (250 GeV)' where U. is the vacuum expectation value for

t L

the ich doublet. The constraint comes from U^ « -j % IJ^ - - g (i/I G p ) ~ *

-7 g (250 GeV)". . For example, if there are two Higgs doublets, one could have

tf, » 50 GeV and Jl « 245 GeV. It also turns out that after symmetry breaking

there are a triplet (I » 1) of Nambu-Goldstone bosons (absorbed by V and Z).

a triplet of pseudo-Nambu-Goldstone bosons and two physical Higgs bosons. In

fact, for N doublets> there arises one triplet of S-C bosons, S-I triplets of

pseudo's and N physical Higgs bosons.

In a strongly coupled K Higgs doublet model, there would be HÎ " J = Oj

and N{l - J » l\bosons of the types discussed above. Since

,...,N) < 250 GeV, it is clear that there would be bound states which

can be lighter than 2 TeV and therefore more accessible to near-future

colliders. Detailed analyses revealed that scattering of longitudinal li's

produces bound states in the TeV mass range while bound states of peeudo N-G

bosons generally have lover masses.

How does one produce these bound stscex? So far, the best plsce to

produce them is in pp or pp collisions. It is also conceivable that if

e e machines (especially th« linear colliders) can get to high enough

(e.g. several hundreds of GeV cm.) energy, the lower mass bound states could

also be produced. Since th* coupling of K, and pseudo K-G bosons to ferraions

is of 0((m-/v)«l) the production of the I « J « 0 bound states by direct

collisions of W, or pseudo's is extremely inefficient at CERN coiiider and

Tevatron energies (0^10 nb typically).

The I « J « 1 bound states (referred tc as V j ) can however ne produced

by mixing with V~ and Z because of the likeness in quantux nucbers. ' The

is obtiin*d in analogy with the old vector aesoo 3ominar.ee, namely

4822

ii IT ~ 8 'oT~ c "*" va vi»ure f v is the aniiiaj t " 5 ar.ti where the KSK7 r^latioM-V " 2f^ ij IT V

f « has been used. Armed with these considerations, one ca'n relate the

cross-section for producing V ' s to the one for producing H's. In fact,if rj

x o(pp-*-W+X) where R^ = ( ( T x r ) ' * n y / ' s ) /? ^ ,

((t f)'t"!11y/s) t'1e ratio o£ respective luminosities. Typically, in order

to obtain a non-negligible number of /-events, one should have

^ "-IT*i - — <0.3. It means that the CERK colider can only produce if's with

Bij.- 170-190 GeV while for Tevatron I (2 Tev cm. energy) , mtf< 600 GeV. For

heavier 17's, one should probably have to wait for the SSC.

Possible signals of a strongly coupled Higgs sector were discussed exten-

19sivelv in tt>e 1'tcrature, for example multi-W production and exaLic

1J-de-.->ys,' ' Of interest is the possibility of rinding low rasg V s

Stace that wou .i indirectly ioply that there may be ncie than one Hiygs doublor.

(remember that By > 2 Tev for one doublet).

Certainly; strongly coupled Higgs models provide us with interestin;.-

phenomena worth being er.plored Eince it is one aspect of the standard model

which still remains e mystery.

V. Beyonc t'nc Standard ?todel

tftvao one it fac«l with the prospect of trying to look for what may go

.on "beyond the standard model, there are three immediate questions one might

ask. The first one is whether or not one can obtain the standard made!

results without the standard nodel. If yes, how natural is it? The second

question involves tfte nature of the Higgs boson, naisely: is it elementary or

composite? In other word's> can one understand the standard model without

as elementary Higgs boson? The third" question has to do with possible

ex tens lens of the standard model. Let ae try to address thos<= 'ssues or.e

by one.

483

First, let us look at a scenario in which the scar.dard-model Higgs

boson is not found at any future attainable energy. There are two possibility

either the Higgs boson is very heavy and strongly coupled or it is not there

at. all. The first possibility was discussed in an earlier section. Let me

instead discuss the second unorthodox possibility. Phenomenologicaily, it is

easy to produce the standard model results without a Higgs boson. Following

Ref. <20,2]), all one needs is a global SU(2) symmetry and a triplet of

massive vector bosons V~' which may or may not be gauge bosons and which

couple to the isospin current. To reproduce the correct neutral—current

structure, this global SU(2) symmetry is explicitly broken by the mixing of

the photon field with W0,' in the form: - -jIV F W^ , where A w is arbitrary.

With a W-raass term of the form -j MJJ W "if (global SU(2)-invariant), one can

use the abov. y-W mixing and diagonalize the mass matrix to find

M^ - M £ / U - X * V ) and o£KC - k~ (J3"Xft| e.m.*2 whlch iE Precisel>' the

standard model result10*21 if one identifies A — • "sin23w".

GF z 2

As usual, — - -*-r . Bovever, to obtain the standard model prediction72 ®£

' 21for W and Z masses, the Y-tf nixing has to satisfy the so-called "unification"

condition: X^, « — « sin6u. The question is now which dynamics produce*th«

low-energy effective L»grangian " ( r f f (- ^ c h a r g e d + * ^ K . C . ) a n d 'Afi " f"

One attractive possibility is a model in which W's, quarks and leptcms are

22 e

composite. In,order for X^. * — , one then assunes a complete M-dominance of

the electromagnetic forn factor for the composite quarks and leptons (in

complete analogy with vector meson dominance). With these assumptions, one

can then reproduce the entire standard model results. There are however

many disturbing questions concerning the validity of this class of aodels.

Notice that, in this case, the weak coupling g plays an analogous role to the

p-N-N coupling f. First of all, it is not clear that universality holds

484

i.e. the couplings of "»' to quarks and iepton;; are not necessarily the same- in

this scheme. Secondly, g = —: e 0.65 is a fairly vcak coupling in contrast

slntfW

with f £ 5.6. Whst is the origir. of the dynamics that makes £ so much less

than fc,™? Why is the isoscalar neutral current so much suppressed experiinon-

tally in this scheme? These and other questions force us to look for na-urnl

answers if composite models are to be taken seriously. Another point worth

studying experimentally and theoretically is whether or not composite V's and

Z can have ^fang-Mills structure.

The answer co the second question of whether or not one can understand

the standard model without an elementary Higgs boson is yes. Technicolor

models were developed precisely to provide an alternative to an elementary

Riggs boson because of problems such as the naturalness problem for example.

In these theories• technifennions belonging to a technicolor gauge group con-

dense to break SU{2), x U(l) u spontaneously at a mass scale ^250 GeV. The

techni KG bosons are then absorbed by H and Z to give then their masses. With

the appropriate choice of transformations for the technifermions under various

gauge groups, it is possible to obtain the usual standard model results.

However, if one tries to extend these nodels so as to give masses to the

fennions as well, one encounters serious difficulties such as the flavor-

changing neutral currents for example. Despite these difficulties, the ide2

of technicolor models is still very attractive. Feraion masses are probably

generated in a different way than extended technicolor. Notice that the

low-energy effective theory for technicolor is precisely a strongly coupled

HijgE model. There is however a slight difference. In technicolor theories,

the raass of the physical Higgs boson is expected to be around one or two Tevs

(the scale of techni-strong interactions) while in a strongly coupled Higgs

aodel, it could be anywhere above that value.

The third question deals with the possibility of finding structures

between 100 CeV and a few hundreds of Tevs, in particular those which come

435

froiu extensions of O.:s. standard snooei. It aesns Chat SU(2) x 0(1).. has

to be parr, of a larger gauge group. Furthermore, such an extension woui:: imply

tht; t-xifat.'.nee of extra heavy gauge bosons and likely also new fernior.s. A24

particularly attractive model of this class is the so-called left-right symmetric

nodal (g, • g ), e.g. SU(2)L x SU(2) x L'(l) (of course, there could be nore

than one U(l) factor). The new gauge bosons here are the ones which couple to

the V+A currents. They are the so-called right-handed bosons, tff and Z_.

There are various ways to find the limits on the mass of W . For example,

25 1bu-decay gave Mu > 380 GeV while the KL-K.. mass difference gave R. > I lev.

R ~ T- s "R ~ •

Of course, W£ and Zfi may be detected directly by their decays e.g. K -»eH.

Among the interesting aspects of this class of model to be searched for in

the future are the mass range of W,, and the chi'ality nature of its coupling

to fermions. Also, one would like to know whether there is a right-handed

partner of f.he left-handed neutrino sad whether it is a Majorana particle.

Certainly, there is a rich structure to be explored in the future. Theoreti-

cally, left-right symmetric models are interesting since they provide a aore

natural explanation for parity violation (from the breakdown of SC(2). x SIK2)O).

There is an alternative way to look at what may lie beyond 100 GeV. This

28is the so-called petite unification approach. One looks for a gaugegroup

GZ>S(I(3)C x SU(2)L x U(1) Y «ith the requirements that there is some structure

in the Te* region, sin 5^ has the correct experimental value at lov energies,

and constraints fro« rare decays have to be satisfied. A ainimal example

which satisfies these requirements is Che group C • SU(4)D x SU(2). x

S0(2)R x S0(2)£ x SU(2)£, where S U W ) p . S - is the Pati-Salan gauge group'9

which breaks down to SU(3)C x U(l)g at a mass scale M = rcass of gauge boson?

connecting quarks and leptons (with the simplest Higgt representation, there

is no proton decay;. The other SU(2)'s combined with u(lj arc broken sown

to SU(2), x U(l)y at a cass scale H~ My (thers: could, of LOU; :e, be nore

steps in between). The results are given in terms of the following bounds:

4 86

K > 300 Tev (frr.r. K + ye; and 380 GeV < >l - .\'hV < 10 Tev .'the lowci- iu,u:)d

comes frco. t-aecay and the upper bcund corcts iron: the sin'-:,, cons-crainc).n

It it certainly an interesting mass range, to explore.

The above exanples are only a few among a multitude of acdc?r which jive

some structure beyond 100 GeV. Whether or not anything will be- there, car. only

be determined by the next generation of macr.ines. I have not discussed the

interesting topic of the phenonenology of supersynraetry in these Jecfjiet.

The interested reader can look for it in the nice lectures given by Larry Hsj.

(to appear in the same proceedings).

Conclusion

We nave se»n bow well the low-energy date agree with the standard model.

We also saw t'r.r crewniny achievements of the standard rodei wher V~ and Z

were discovered near the predicted aasses. Despite all oi thesif successes,

one important ingredient - the HiggE boson - ts still aissing. It is ncv

more important than ever to search actively for this elusive object and to

contemplate on wh£t might be beyond the standard model.t

I would like to thank Dr. Z. Ajduk, Professors G. EiaJkowski, S. Pokorski,

K. Wrftblewshi and the orgirisers of. the Symposium for a stimulating raeetir^.

Ky tJiankt also gc to prof essots K. Cbadan, J. Iran thanh Van, and L. Oliver

£or the hospitali:y at Orsay, Professors T.K. Trucng and T.X. Fh^r. for the

hospitality et Ecole Polytechnique, and Professors M. Chanowitz, M.K. Gailiard,

and M. Suzuki for the liospitality at L5L. This work was supported ir part

by :tsr.

487

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the XlXth Rencontre de Moriond (1984), ed. by J. Tran Thanh Van, and

other reference therein.

23. See the review of E. Tarhi .and L. Susskind, Phys. -Rep. C M (1981) 277.

24. For a review, see R.K. Mohapatra, lectures delievered at the NATO Simmer

School on Particle PfcysicB, Sept. 4-18: Munich, West Genaany; Maryland

preprint SD BP-PP-84-0012 (1963); C. Senjanovic, in "Phenomenology of

C-nifiad theories", edited by H. Galic, B. Cuterina and D. Tsdic, p. 133

(World Scientific, Slnaaôre, 1984).

75. J. Carr et *1., 3%ys. *ev. tett. 1 (1983) 627.

25. G. Refill, H. Bauder «nd A. Soni, Phys. Xev. Lett. 48 (1982) 848; F.J.

Gilo£n »na K.E. Reno, Pbys. Rev. 929 (1984) 937. t

27. See, f»r exansple, E. Kayser. Phys. R«v. D30 (1984} 1023.

28. P.<3- Huns. A.J. £ura», and J.B. Bjorken, Phys. Rev. B25 (1982) 805.

29. J. Psti and A. Salaa, Phys. Rev. 010 (1974) 275.

489

RECENT RESULTS ON WEAK DECAYS PROM DESY

D. Stron

Department of Physics, University of Wisconsin, Madison

USA*

abstract

Recent results from e e experiments at EES'i on the following top-ict. are

reported: observation of the decay D° -» Ko«, D° - 5° mixing, D° lifetime,

bottom hadron lifetime, and the limit en tne tau neutrino nas.-.

* Supported by the U. 5. D-jpartm«nt of Energy, Contract DE-ACO2-"i5ERCOaSi,travel support provided by tfte U. S. National Science Pou.-MiBt.ic.is. qrantnumber UTI-B31399*.

490

The el«ctron-po$itrOE storage rings PETRA and DORIS at DESY afford

a good opportunity to study weak decays. The electrons and positrons

can annihilate via a single photon into pairs of quarks or leptons.

Weak decays have be*r. studied in inclusive final states containing charm

and bottom mesons and in exclusive final states containing pairs of T

leptens. The contribution of the experi.-nents at DESY to the understand-

ing of these decays has been large. Here, only recent results are

reported.

Ther

brief, only the particle state will be named. The antiparticle state

should be understood.) ARGUS reports the observation of the decay

D° •» K° it and an update whicn improves their previous limit for D c - 5°

nixing. TASSO has recently measured the D° lifetime. Concerning bottom

maasurenents, both TASSO and JADE report values tor the B lifetime.

Finally, ARGUS uses the decay T » it i t v tc set a nesi upper limit on

the ness of the tar, neutrino.

Theoretical Motivation

Be'ore going any further let us repeat an eld story and consider

tht theoretical motivation for the raeasurenients on mesons contaxr.xrg c

and b quarks. The simplest model to describe the decay of mesons

containing .'leavy quarxs is the spectator model. Here the heavy quark is

treated wLznc:>t regard to .the lighter quarx witn wnicn ;.t is brund. Tl'.e

r.eav. i uark, q, decays tc a lj.enter quark, q', plus a virriial W, w!:i:h

in turn decays t.1 a pair of leptoas or zo lighter qusrks. 'Zse Figure

491

I.) The amplitude for q to decay to q' of a different flavor depends on

tne weak mixing anon? generations- This mixing is described by the K-M

(Robayashi-Maskawa ) matrix:

ucS us ub

"cd Ucs ucb

IT I' Iftd ts tb_

Since the mixing among generations is small, the off diagonal elements

of the K-M matrix are small, while the diagonal ones are nearly unity.

This has interesting consaquences for B decay, which proceeds via the

off diagonal elements U . and 0 . . Thus, the B lifetime can be a

sensitive measure of these elements. A calculation, which was based on

the spectator aodal and the B neson santilcptonic decay spectrum fros

CLEO and CUSB, of the relationship between tne KM elements and the B

lifetime gives2

\192 Tf'

For ad^flf 5 GeV/c2, the factor in front is about 1014sec."1. if Ucb

or U. were near unity, a$ is U , the B lifetime would be very short

indeed.

Charm decay is interesting because of the failures of the spectator

model. These failures are by now well known; for example, the lifetime

of the D and D° mesons are very different. A recent compilation at

neutral and charged D lifetime? gives

= 2.4 ± 0.3,

w /u

q4

H spectator

Figure 1. The spectator decay for the decay of a mesonwith one haavy quark.

D<

u

wd d,s

AnnihilationFigure 2. (•) The exchange diagram for the D end tfie

tira diagran for the D* and F+.

w

dIfc Interference in the decay of the D

Oetween identical d quarks in the *i.naistate.

492

at odds with the spectator model which would have them equal. The

semileptonic branching ratios are also different by about the sane

factor. The MARK III collaborations finds

Br(D* - e v X) +0.5 + 0.1

A number of explanations have been put forward for this failure of

the spectator model. One is the existence of ether diagrams for weak

decay. The exchange diagram,'' for the neutral D meson, and the

annihilation diagram, for the charged D and T mesons, are shown in

Figure 2a. The annihilation diagram for fhe D has a Csbbibc suppressed

W vertex so it could add littls to the decay width of D , while the

exchange diagram could give a substantial contribution to the D c decay

width, giving it a shorter lifetime and .= smaller semileptonic

branching ratio.

toother possible effect is illustrated in Figure 2b. In badronir.

decays of the D , the finai state can contain two identical d quarns.

These quarks could interfere to suppress the decay of E mesons. Both

of these effects are difficult tc calculate, so input froir experiment is

important. Presently we shall see how ARGUS has produced the first

evidence for the existence of Dc decays via the exchange diagram.

Results from Charm Meson Decay

i. Observation of D •» K >t>

This decay is a good place to look for the contribution of the a

exchange diagram, as shown in Figure 3. The only spectator diagram

Milch car. give such a finsi state is an OZ1 violating process which is

ezj'«cred to have a blanching ratio of less than 10 ". The observrtior.

or a r.ui.5taPtial branching ratio tor the dec^y D° •» K° t woeld be the

first direct evidtn-e for M exchange in the charm system.

The ~-RC;.7S collaboration searched Cor this decay by exuinirunc the

final state :r, wr.ich the K° decay t; two pions, and the 4> decays to

two chargad kaons. The D° mesons were first reconstructed without

constraining the two kaons to haê the $ mass. A clear signal of 69 i

1£ events is seen at the D° mass when x = P(K°K K )/P. > C.3, When

ehe two kaons are restricted to the mass interval, 1.01 < m(K*K ; < 1.03

C-eV/c , s signal of 37.7 i 8.0 e\?ents is seen as shown in Figure 4a. In

the lower * side-band (mCK^K ) < 1.01 GeV/c ), a small D° signal of 14 2

i 4.6 events is observed, while in the upper side band, (m<K K ) > 1.C3

GeV/c ;•, 10 - 16 events are observed- The upper side band is shevr. in

ficure 4b. Extrapolating the contribution from the upper ":ide band, the

nonresonar.t contribution to the signal region is estimated to be I ; 2

To obtain a branching ratio, ARGUS compares the decay D° -» K° S*T

with the ona above- The branching ratio can then be calculated from

u

495

w + 1III

s *

Figure 3. The exchange diagran for 0* *

) • I? IB 1« 20 2.1

Fig. 4b Mass K»K-fK- [GeV/e2]

Pigura 4. (a) The resss of K° K*K" combinations for

M(K K ) in the • region (1.01-1.03 Gev/c2)and X >0 .3 .

(b) The mass at J ° S K combinations lor

M(K K ) above the > region (>1.03 GeV/c2}and X >0-3.

496

Br<D°

Bt(D°

K K

+ -

)

• )

ES(D° s K K )

>

where c is the ratio of efficiencies for the two processes which has

been determined by a Monte Carlo calculation to be e * 1.23 i 0.08. The

decay D° * K* T*V is seen only for x > .5, so this cut must also be

applied to the D° • K° • analysis, yieldine 25.7 s 5.8 events. Using the

known branching ratio for • * K K~ of 19.3%, and the branching ratio for

B c • K' v* T~- ot 5.3 i 0.9 t 0.9*. as recently measured by MASK III,9

ARGUS obtains

•r<D<> * ic0 • > - ( 0.99 i 0.32 (stat.) t 0.17 (sys.) > %

where the statistical error contains contributions from both the ARGUS

measurement and the KftRK lit Masuraoent, and the systematic error is

due tc the KfcRX III measurcatent.

C° - D e mixinj occurs ir. the standard osdel via toe box diagras

show, in Figure 5. This mixing in the context of the Standard Model is

thought to be very snail, - 10 . A significant mixing between D° and

D would indicate that new physics is involved.

ARGUS has given an update to their previous D° - 5 C nixing limit.

This limit is obtained by examining the decay of D •» D°it with the

D* decaying to TTT*, K"T*TT IT , and to K'TT'TI0, Hhere the r° is not

reconstructed. If mixing occurred, then the final state would contain

the decay products of a 5°, giving the charge of kaon in the final state

the same sign as that of the transition pzon, rather than the opposite

497

w

d.s,b d,s.b

u W

d.s,b u

wi w;

S. Dlagrao* for B* - B* ni«1ng in tfc*

498

sign which it would have for a normal decay. Experimentally, one plots.

ass

reconstructed D * a&ss for right sign and wrong sign D° mesons as shown

in Figures 6. Because of the low Q value of the D * D IT decay, the

mass difference provides a distinctive signature for true decays, ever,

if the y° has not been reconstructed. To reduce the background fron

wrongly assigned particle identities, full use of the tine of' flight

system and the dE/dx measurements was made. Particle combinations were

only Earned among particles whose hypothesized identity was consistent

with the tiae of flight measurement and the dE/dx measurement.

Futhermore, at least one particle in the K~* and K s v° mode, and the

kaon in the K*T~**IT node was retjuire to have momentum less than .6

GeV/c and to 2i»ve a dE/ax probability for the hypothesized identity of

greater than 0.1. In addition only those combinations Kith >E *

2E(X) >//* > 0.7 were accepted. Ate signal plot contains 88 events

from the modes where all particles are seen, and 22 events in the IS JT it°

MOO*. The backgrounds are estimated at 12 and $ events, respectively.

The wrong sign plots contain $ and 2 events in the signal region, where

the backgrounds are estimated at 5 and 2 events, respectively. This

fives a liait ot ».*% on D c - 5° aisbtg at the 90% confidence level.

this is the fcwt liait thet has been obtained at an e •" storage

ring. The BELCO collaboration has given a limit of 8.1% at the 90%

confidence level. k fixed target prompt nion experiment has found D 6

- B° nixing tc be less than 4.4% at the 90% confidence level. However,

the results of such a fixed target experiment depend on some assumptions

499

Events/0.5 MeV

25

0.13 0.14 0.15 0.16 0.17

m(D**)-m(D°) ( GeV/e* )

Events/0.5 MeV25

0.13 0.14 0.15 0.16 0.17

m(D**)-ni(D») ( GeV/e* )

Figure 6. (a) Correct sign and wrong sign massdifference distributioni Cor the decay 0° *

K * and I n n ,

Events/2.0 MeV15

10

0.13 0.15 0.17 0.19

( CeV/c« )

lvents/2.0 KeV

10

.. . n.i0.13 B.I5 0.17 0.19

m(D'*)-m(D0) ( G«V/e« )38662

(fa) Correct sign and wrong sign aassdifference distibutions for the decay S* *

K t~ic where the v° is not detected.

500

about the production ratios of charmed particles; these assumptions are

not needed Cor the ARGCTS analysis.

The neutral D nesons are identified in the decay D " • D v~ using

•+ °the aass difference between the D and D mass combinations, in a

aecner similar to that described in the last section. The details ofo

the analysis can be found in TASSO publications where signals from D

i Smesons were reported in the following decay nodes:'

This S aeson selection procedure, applied to the data taken since

tfie installation of the TASSO vertex detector, and the requirement that

tJ»c decays be well r«c«istructa£ in the vertex detector yielded 6 events

in fib* acde B e » STT" and 2 events in the node B** t."t*iTt*. Bo events

with valid v«rt«x detector tracts were fcund in the node D° + K ir ir°

apd*. the tracks from the decay products of each C* meson from the

selected events were fit to a cotuacn vertex in 3 dimensions and then

constrained to the D e aass.

Once the vertex position has been found, the most likely decay

distance for the D° decay nay be calculated from the relative position

to the decay vertex tc the bean center. The most.likely decay distance

is given by the expression:

501

Xy V« " 2VxV * Vywhere cr . is.forraed by adding the error matrix determined in the vertex

tit to that from the beam spot, t and t are the direction cosines of

the D momentum, and x 2nd y are the coordinates of the decay vertex,

as found by the vertex fit, relative to the beam center. The decay

distance is calculated by projecting I along the 3 dimensional

proper time by dividing by YB = PpO / MgC.

The proper decay times for the eight events and their errors are

shown in Figure 7. Having determined the proper time for each decay,

the D° lifetime was extracted using a maximum likelihood fit, assuming

that the decay distribution was describad by a Gaussian convoluted with

an exponential where the width of the Gaussian was taken from the

calculated error on theaost likely decay distance. In addition, a

Gaussian term was added to take into account a 15% background in the

•ode D° * \~i , and 30% background ic the aode D° •* K~ir*ir~r*. The fit

gave xDo = («.6 *_ \'* ) • lCf13**:.

The systematic error was estimated by varying the assuned detector

resolution, the background fraction, the fraction of b -» c cascades, and

the detector positions within their errors. The contributions fron

units ofthese sources were respectively, ± 0.7 , ± 0.3, * °*°, s 0.3 in uni

10~ sec. Th* lifetime of a background sample was found to be (-0.07 i

0.6 )-10~ sec, indicating that tbe detector is understood to the level

50: TA5S0

D°-KTX <

«— 1I

i f

— -

I 1 i

0.5 1.0-1.0 -0.5 0

tD U0

Figure 7. The distribution of proper decay tines forthe D° candidates and their errors.

Figure e. 1 1M definition of tito lopact paranctar d.for particlM with * finite decay F»th, *,U M ispaet paraeieter, d, is norutaro.

503

of 0.6 • 10~Usec. ftdding all of these effects in quadrature tc obtain

the systematic error, the preliminary TASSO result is:

This result is in good agreement with the world average as given by the

Particle nata Group o£ (4.4 * 0*e)-10~ sac.

B Lifetiae Results

All B lifetime neasureaents froa e*e~ collisions follow the sane

strategy. it has not been possible to reconstruct > aesons in

the same manner as P aesons have been reconstructed. To obtain the »

lifetime, a sas*>ie of events and tracks enriched in e e~ • fab is

selected. The iapact parameter in the plane perpendicular to the bean,

t, is given a sign depending on which side of the interaction point the

track crosses the sphericity axis. She distribution of the observed

signed iapact paraneters is then used to obtain a lifetiae. The

principle of the signed iapact parameter netbod is shown iti Figure I.

Tracks which coo* fron decays of B hadroBi tend to cross the sphericity

axis ia front of the interaction point, and to have large iapact

paraoeters.

the JADE collaboration hat used senileptonic decays of £ hadrons to

1.9aeasure the B lifetiae. Leptons froa B decay tend to.have large

transverse momenta with respect to the sphericity axis. Selecting

•uons (electrons) with P > 1.8 GeV/c < 1.5 Gev/c) and F > 0.9 GeV/c,

JADE obtains a sastple of tracks which ptiaarily originate from B

with:

election aruon

b * 1VK 60* 65%

b * c + 1\K 8% 6%

C * lvK 10% . 6»

•isidentified 3% 21%

To ensure that 6 is well measured, the angle cf the leptor. momentUE to

the sphericity axis is required to be greater than 6° and less than 57°.

The distributions of the signed impact parameters are shown in Figure 9.

The Euoa 6 distribution has a mean of 282 ± 78 tin and the electron i

distribution has a M a n of 457 i 107 um., To extract a lifetime, a

likelihood fit is perforsed which taker into account the various

backgrounds Listed above. The f i t yields:

*, - i - 7 * I o l « ** <1BUOa«> •"•Tt « 1.13 _ * ' ^ ps (electrons).

In a second method, the C fron al l tracics in the aoun events which

pftssec) the same cuts as the l*pton« were usad. These hadrons gave:

T • 1,7 I 0 ^ pS.

This hadron saaple was used primarily as a check and should not be

averaged with the lectan results as the l«rton and hadron results are

not statistically i.itiepeadan-t.

The 5yst«oatic «rror en these results is estimated "to be i 0.4, and

results primarily fro uncertainty in the B fraction in the final

sesple. Averaging the electron and nucn results, JADE obtains

*» * 1'£ - I'.l t °-* «*•This value 1E considered preliminary.

505JADE

03 0.56 in cm

Flgur* 9. H W distribution of tbe iapact paraaetcrs ofinclaciv* rnxmi, (a) , and inclusive

ft, (b) r frac the b Micichad snapl*.

506

Last year, the TftSSO collaboration present a E lifetime based on

data taken with and without their vertex chamber. Ir. the mean tiir.e

the amount of data with the vertex chamber has been increased by a

factor of nearly two and the value of t has been brought up to datt

using the new data. This update is based on a totai of 2Spb " with the

vertex detector, taken at W = 44 GeV.

The TASSO collaboration uses ar< event shape variable tc produce a

sample of hadrons enriched in bb events. Each hadronic event is

separated into two jets using the sphericity axis. All charged tracks

falling into a cone of half angle 40° about the sphericity axis are

boosted towards the B meson rest frame with a T of ~ 1-5. The

sphericity O.C the boosted jets, S, and S,, is then computed. Those

events which have a boosted sphericity product, S.S,, greater than 0.10

are taken to form the B enriched sample. Monte Carlo studies show that

this saraple contains 32% be, 35\cc, and 33% mi, dd and ss and that about

*C % of all bb events pass this cut. A sairple depleted in S mesons is

foraed fron these events with SJSJ < 0.04. This sample has 6% bb, 37%

CG, and 57* uti, dd, and ss.

The average i is calculated from all tracks in the event which have

momentum > 1.0 GcV/c, which pass strict track quality cuts, and which

have |S] < 0.5 ca. Ihe t distributions fron the data with the vertex

detector for the B enriched saaple and for the I depleted sanpXe are

shewn in figure 10. Table 1 shows the scan ( for two independent sets

of data. Part a) shows the result* obtained since tbe installation of

tbe vertex detector. Part b) shows the results for the data taken

before the Installation of the vertex detector. Figure 11 shows tbe

Sample

all events

b enriched

b depleted

Part (a)

data

56 t

91 t

38 i

50?

Table 1

Vertex Detector <S> (im)

7

17

e

h - :

56

115

47

Monte Carlo

I - IO^B , B

1 2 33

r 6 32

1 2 33

•

= 0

i 2

£ 6

i 2

Sample

Part (b) Drift Chamber <S> (um>

data Monte Carlo

T B * 2'1Q

all events

b enriched

b depleted

63 ± 7 58 i 1

105 i 17 107 ± S

58 ± 8 48 i 2

41 ± 2

40 t 5

40 ± 2

0 33

0.15

0.10

005

0.0

Vertex •» Drift Chambers

-O.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 OJJ 0J 0.4 0.S«{ctn)

Figure 10. The distribution of impact parameters fromTASSO, th» shadeo histograic shows the b•nriched data, the dotted histogram showsthe .'i depleted data.

250

200

E 150

oiDnli Cbomber

W=3£6GeV

Detector* Drift Chamoer

W-«GeV

Figure 11. The averaged I for the range-|<l < 0.5 cmas predicted by * Monte Carlo calculationas a function of B l i fe t ime. The shadedregion shovs tha one standard deviationl imits obtained froa the data. The datawithout vertex detector information areshown in (a ) , (b) shovt the date withvertex detector information.

6.C

MAC

MARKD

DELCO

JADE(preiim.)

TASSO

MARK I

JADE(preta }

Figure 12. The value of the 8 lifetime obtained byvarious experlEents. The statistical andsystematic errors h&ve been added inquadrature.

relation between the mean value of 6 and the B lifetime, as determined

from a Monte Carlo calculation. The B lifetime from data with the

vertex detector was found to be 1.36 i O.tZ ps. This is combined with

the data taken prior to installation of the vertex detector, - = 1.85

0 49

n.o PS, to obtain the final result, T = 1.57 : 0.32 D S .

The systematic errors in such an analysis are not negligible. The

estimated error due to instrumental biases, including beam spot position

and size, nonuniformity of the magnetic fielcJ, and any offset in the 5

distribution, is t 0.16 ps. The systematics from the method, such as

the range of 5 averaged over to obtain the mean value, and the require-

ments on track quality give an error o£ t 0.14 ps. Finally,

uncertainties in the Monte Carlo parameters, such as the particle

multiplicity and the charm lifetime, result in uncertainties in the

fraction of B tracks in the B enriched sample and in uncertainties in

the lifetime of the background, giving a systematic error of tn"r? Ps-

Adding these in quadrature, TflSSO finds,

TB » 1.57 i 0.32 ! I'll ps.

The results of JADE and TASSO, together with other recant results18

are shown in Figure 12.

As was pointed oat in the introduction, these lifetime results can

be used to determine the element 0 . of the X-H matrix. This is in part

possible because CUSB ami CLEO have shown that

^ < .11 (90% confidence level)21

lucb'

The values for \li. | obtained using relation (1) given earlier are

> 0.042 t 0.005 (stat.) t 0.006 (sys.) (TASSO)

|U J = 0 .039 t 0 .005 ( S l a t . ) t 0 .005 t s y s . ) (JAPE;,CO

where the considerable uncertainties ir. the parameters in relation < i'

have not been included in the error (see references 2 and 211. These

introduce an uncertainty on the order of 5%.

Mass Limits on v_

A number of experiments have used the process e e * t T to give

limits on the v^ mass. Ignoring initial and final state brensstrahlung,

each tau will have the beam energy. By reconstructing the decay of the

tau, the er.argy of the neutrino can be calculated as tne difference of

the beam energy and the energy visible in the tau decay. The tau can

decay into E variety of final states. Previous upper limits exist from

the study of the Bode t" * sr~ji+5r~i!ovT, T~ • 5T"(it°)vT,22'23 and

T •• K K IT v . The key point of these analyses is to avoid background

from decays of the tau uhere sotce of the decay particles escape

detection; these unseen particles would distort the neutrino energy

— — + - 25spectrum. ARGUS uses the mod* x •» v ir t u , and avoids background

fron tau decay to unseen neutral pions by requiring that there be no

energy deposited in the shower counters.

lo ensure a high purity in the final sample, a number of cuts are

nade to eliminate background. The events are required to be of a one

against three topology. The event is rejected if there are any photons

with energy > 50 MeV, unless the photon can be combined with another

photon to fora a n" which in turn can be combined with the lone prong to

fora a «. The acser.tux su=, " .p, '. , is required to be nor° than 2.7

Gev"-c, w sjrjrass bear ?as and :we pnotrn events, but ".ass thin .92.E

to suppress exclusive decays, e. 5. 7' • - - j 1 . To reduce the

background from radiative 3ra£'"-as s.-id LUI events, where the pnctcr. has

converted C3 ar. alaccron posuror. pair, the cper.ir.g ar.eies between the

particles in the 3 pror.g side are required c~ satisfy cos 6 ; 0.392. In

addition the energy dsscsited ir. the eiectrcTagr.etic cslor-setar by trie

lone pror.g, ana tnat dapasited by the 3 prcngs is required re i - less

thar. 4 GeV. Finally the tiae ei flight Eeasurec.er.ts and drdx aaasure-

ne-nts are used to reject events wnere any of :Ke particles in the 3

prong decay are .-.ot cortsister.t with a picn hypothesis, and where the

lone pror.g is not consistent with an electron, men, pier, or kaca.

The resulting neutrino energy spectruc froas about 150C tau decays

is shown in Figure 13a. The shape of the spectrus near t, = 0 is

sensitive to rhe 3ass d the neutrino and is shown in Figure 13b. The

two curves show the pradicticr. of two Mcnte Carlo calculations, one for

a masslffss tau neutrino, the other for a tau r.eutrir.o of 3ass 140

MeV'c . The lint is obtained by using a caxinux likelihood function

of various tau neutrino casses. The aaxissua likelihood lonctiar. gives

a limit of at v. 1 < 56 Mev.'c a* the 95% corîsence level.

Consideration of systematic effects increases this I1rr.1t sosewnac.

The following checks were sade: the region of the fit was variec 3j-

tlOO MeV, the assuaed aomentun resolution was degrades by 10 %, and the

assumed background was increased by a factor of 4. Tr.is gave ar; upper

BOMeV

7!

20

ji f n Tj r r j |

L

2 3E v IBeVj

Figure 13. (a The tac neutrino e-.eryjr speetrua ior^ •* — *

the decay T •• t t r v , no correctioris fcr

acceptance have bear, applied.

K rWMeV

30

t 1 - — v n*n"nJ

ARGUS

— mv=0

The tau neutrlnp anerTy spectrua in thereefer; ser.sative to the mass of the taune-jtri^c, the scid line shows a Monte Carlocajxuitaior. for a C taass neutrino, thedashed line sho» a Honte Carlo calculation

for a tau neutrino with tcass 1*0 MeV/c .

n m i w incljdj.r.g tne systematic errors, of 70 Mev/c at the 95%

confidence level.

This value is better than any which have so far beer: published.

Mark II" and DELCO" have published uppe- limits or. the tau neutrino

mass of 143 MaV/c2 and 125 JfeV/c2, and 157 MeV/c" (all at the 95%

23confidence level). The HRS has recently reported a value coraparacle

to that from ARGUS using the decay mode ; ' ? : r r 3 v . They obtained

m(Tu) < 76 MeV/c .

Acknowledgements

I wish to thank D. B. MacFarlane of the ARGUS collaboration for

several useful discussions. The DESY directorate is acknowledged far

their continuing support and hospitality. I would particularly like to

thank the organizers of the VIII Warsaw Symposium on Elementary Particle

Physics for their kind hospitality.

1. «. Kebayashi and K. Maskawa, Progr. Thecr. Phys. £9(1973!652.

2. K. Kleinknecht, Europhysics Study Conference on Flavour Mixing ir.Weak Interactions, Erice, Karch 1984.

3. W. T. Ford, Contribution to Aspen Winter Physics Conference,1985,University of Colorado preprint COLO-HEP-87,(1985).

4. R. M. Baltrusaitis, et al., SLAC-PUB-3532, (1984!.

5. M. Sernreuther, 0. Nachtmann and B. Stech, Z. Phys. C4(19BO}257.

6. S. Guberina, S. Hussinov, R. D. Peccei and R. Ruck], Phys. Lett.89b(1979) 111; See also P.. Ruckl, Weak Decays of Heavy Flavours,Habilitationsschnft, University of Munich, 1983.

7. .1. Bigi and K. Fukugita, Phys. Lett. 91b{1980)121.

8. The ARGUS collaboration, H. Albrecht et al.,, preprint, DESY 85-048,(1985).

S. R. K. Schindler, Proceedings of the XXIJnd International Confernceon High Energy Physics, Leip=ig(1984),Voluae I, p. 171, ed. A. Meyerand E. Wieczorek.

10. L.-L. Chau, Phys. Rap. 9jl(19e3i,l.

11. The Argus collaboration, B. Altaracht, at a l . , Phys. Lett.150aq98t)23S.

12. S. Kussinov, Pfiys. Rev. l«tt . 35<19?6)167«.

13. The DELCO collaboration, Phys. Rev. Lett- 54(1965)522.

14. A. Bod*ten at al., Biys. t«t. 1138(1982)82. Since the conference aa lower limit -of 1.9% (90% confidence level) has been given in apreprint by X. Argvrto, et al., which has been suinitted to PhysicsLetters i.

15. The TASSQ Collaboration, H, Althoff et al., Phys. Lett.I2SSU9S3)493; fhyi. L«tt. 1J8J(19«*)317.

16. G. E. Pordan, S. H. Saxon, RAL-65-037; D. K. Saxon, RAL-84-044.

17. The Particle Data Group, C. C. Dotil at al., Rev. of Modern Phys.56(1984)52*.

18. The JA3E Collaboration, R. Bartel «t al., Phys. Lett. 114B(1982)71;The MkC CDilai»r«tion, K. Pernandez at al., Phys. Rev. Lett.51(1983)1022; R. Prepott, Contribution to the XXth Rencontres de

515

Moriond: Eiectroweak Interactions, Les Arcs, Savoie, France, March1985.The Mark II Collaboration, K. S. Lockyer et al., Phys. Rev. Lett.5^(1983)1316; Contribution to the Physics in Collision Conference,Santa Cruz, California, August 1984.The DELCO Collaboration, D. E. Klem, et al., Phys. Rev. Lett.53(1984)1873.

19. The JADE Collaboration, P. Steffen, Contribution to the Int. Conf.High Energy Physics, Leipzig, July 19-25, 1984; Priv. Com. from J.Spitzer; See also report by G. Wolf, SLAC-PUB-3446, Sept. 1984.

20. The TASSO Collaboration, M. Althoff et al., Phys. Lett.149B( 1984) 524.

21. J. Lee-Franzini, Europhysics Study Conference on Flavour Mixing inWeak interactions, Erice, March 1984.

22. The HARK II Collaboration, G. Hatteuzzi, et al., SLAC - PUB - 3560(1985)(Submited to Phys. Rev. D.); P. R. Burchat, et al., Phys.Rev. Lett. 54(1985)2489.

23. The HRS Collaboration, D. S. Koltick, Contribution to theWashington Meeting of the Anerican Physical Society, April 24-27,1985; See also "Observation of T Decay to Five ChargedParticles", I. Beltrais ct al., Phys. Rev. Lett. 54 (1965)1175.

24. The DELCO Collaboration, 6. B. Mill at al., Phys. Rev. Lett.54(1985)624.

25. The ARGUS Collaboration, preprint DESY 85-054 (19S5).

51?

SEARCH FOR HIGHER TWIST EFFECTS IK HADROWIC DISTRIBUTIONS

FROM MP INTERACT IONS

'l'h.e 2uxopein Kuon Coll».bornticn

Presented by

J. llassalslci, Institute for Huclear Studies, Warsaw, Poland

Abstract •

The hiîer twist effects predicted by Berger [1]

have been searched for in the hsdronic distributions from

fip interactions. The data nre not compatible

with the predicted magnitude of the effects.

516

It iip£ beer; note: ['< ] tJ-.-T the »:".-Cert3 of -jo-ctituente

influence -;i;e 5sep inelastic crosr section, "."•i-h tns ir . i t i^l

pp.rton on the TIIP.SE shell nr.s iisvir,-- no intr: r.sio rior.ierstur. th

following; prediction i s given:

where - Qc is the four momentum transfer in lepton scattering,

y = V-/E ,V" = E »- B is the virtual photon energy,

4 ** - 4/p j, is the trnnsvevse monentma of the pion relative

•to the virtual photon,

4 is the azimiithal angle of the pion (aee Pig.2),

q{x) is the psrton distribution in the proton.

The last two terns in the ebove expression represent higher

twist (HT) effects predicted by Berger. These effects are large

and the;' induce correlations between leptcriic and hadronic

variables. Some of them are similar to those expected from

hard QCD processes.

Results

The data come from the experiments H.A2 and 1M9

performed by the IMC at the CER1I SP3. The description of the

apparatus as well ae the event selection criteria and

519

corrections to the data from each experiment can be found

elsewhere ([l'J, [3]).

J In the V.A2 exparinant only current jst frpgaents have

been ne^stired (I:, fii a > 0.2) The 'beam energies v'ere 120, 200

and 280 GeV. .In the region Q 2 > 2 GeV2, -J43 k events have

been used.

In the I.AS experiment the bep.m energy ".VP.S 230 Ge7 and

in the ™egion Q4 > 4 G-eV", 19 k events were analysed. The

experiment covered full x», region.

In the analysis only charged hndrons have been used and

they were assumed to be pioris.

a. (y, z) correlation

In Pig. 3 we compare the z - distributions of h«drons

produced with different bean energies '.vhere the only !"inen:p-

tics variable which changes significantly between She data sets

is y.

The HT term from (1) is

ff «* (1 - y) ?°r z — > 1 «* fixed Q2. (1a)

The prediction for the chsnge in the z - distributions due to

different y is shewn in Pig. 3b with the solid line. She

prediction is vsithin the errors or. the data points.

b* (Q » P T ) correlation

In Pig.4 we compare the z -dependence o± the average

Pm for the events from different ^ - intervals.

The KT tern in (1) is

2ff*c" (1 - y)-El- for z — > i. (ib)

Q2

Sinilsrily as for the QOD hp.rd glyon emission it gives rise

to the jet broadening at lsrge z but it has stronger Q^ -

- dependence.

This effect is not seen in the data for any of the Q2

intervals.

52-0

c, (j;, z) correlation

The H? tens froir. (1) gives the follov.-ins "vsragcvalue of ccs ij1 :

.'A .

where f(y) is a kinenmtic factor, "he v.#>lue i s positive nndstrongly z - dependent, On ths.. other h=.nd a sini; I~:-- dcpt.-nderi s expected to arise from ths intrinsic no.~15r.tum of pnrtons(kj) tut i t is of the opposite sign [4 ] :

the QCL) predicts negative values of < cos > for z — > 1

([4], [-5]).

This is ilustrated in fig,5 where we conpare the xf. -

- fiependence of <- cos > with the prefiiotion [6] which includes

kn and QC3 effects only> The solid line in the figure shows

the expected additional contribution from the KT tern. The

ejected rise of < cos > at high x_, is not seen in the d*st--=.

The &p.ta in the current Set region is consistent

mainly with kj, effectst the QGD effects being saall. On the

other hand in such a case the assymnetry in the target region

is expected to be eiaiilsr in magnitude but opposite in sign.

The observed effect; is opposite in sign but smaller in rsa£ni-

tuae,

correlation

Pros the KT prediction (lei) one can also expect the

value of < cos $ > to increp.se with Pj. 3?ig.6 shows < cos

as a function of p^ for different Xp ranges. For xp > 0.1

the observed trend is opposite and it agrees with the

expectations of Qjrii with k , (1C.2). For smaller Xp values

there ie no strong variation of < cos > v/ith the p^.

521

Conclusions

The predictions for higher twist effects in the

nadronic final states [1J have been compared to the dats from

the European tluon Collaboration. Ho effects of the predicted

magnitude have been found:

- (y, z) correlations are not conclusive,

~ CPipf Q ) " ' are not observed,

- ($, z) and ( , p~) correlations are of the opposite

sign to those predicted.

Small HT effects are perhaps not excluded.

The distributions of <ct»a 4 > for Xj, >O.1 seem to be

dominated by i T effects with s possible contribution of herd

processes. This indicates that non perturbative effects have

to be included in HJ predictions.

References

1. E. Berger, 2. Phys. C.4(7980)289.

2. EMC, O.C. Allkofer et a l . , Hncl . Ins t r . mnd t 'e th. 179(1981)445;SKC, J . J . Aubert et a l , , Fhys . Ie t t . 1143(1962)373.

3 . HiC, J , P . Albsnese et a l . , Muel. I n e t r . and Eeth. 2,12(1983)111iEEIC, J .P . Albanese et a l . , Phys. Le t t . i44B(1964)302.

4. R.K. Caha, Ehys.. Le t t . 783(1978)269.

5. H. GeorgL and H.D. .Politzer, Hay*. Eev. Lett. 4O(i9T3)3.

6. A. KBnig and t. Kroll, Z. Phys. C16(19B2)89.

522

Figure captions

Pig. 1: The origin of l-.i.~her tv.ist -terns in hadroproducti or

from deep inelastic scattering

Pig. 2: The definition of the hadronic szinuthnl angle ^.

Pig. 3: The comparision of the z - distributions at different

values of y: a) energy scaled a - distributions,

b) the ratio of the distributions at 250 and 120 GeV

beam energy. The data comes from the experiment KA2.

The line is Berger prediction.

Pig, 4: The a - dependence of the ratio of average transverse

momenta from different Q" - intervals. The data cones

frora the experiment UA2. The lines are Berger

predictions.

Pig. 5 s The z - dependence of <cos iji >. The broken line is

"the prediction [63 based, on k^ and QCD effects alone.

The solid line shows additional contribution from

Berger prediction.

Fig. 6 : The Pm~ dependence of <cbs ij > in different x^ - inter-

vals. The data comes from the experiment EA9.'

52:

524

N

0.1

T3| "O

A *

Z H•DJ-O

U

1.0

0.6

525

70<v *100 GeV5<02< 20 GeV2

• 120 GeV , <y> « 0.68fi 200 GeV . <y> = 0.<1

o 280 GeV . <y> = 032

a)

b)

0.2. 0.4 0.5 0*

ig- 3

526

5ooCO

111

CMO

C

rvll

itUJ

527

o

0.1

0

-0.1

-0.2

i i i

1 1T 1 '

* *hW• NA9 DATA• NA2 DATA

— - Modtl of Konig t

—— Contribution ofBers/tr tffecf.

i t i

/ \

/ \

LI 4

1

-1.0 -0.5 0.5

i-e. 5

0.1

0

0.1

0.2

i

•

*-0.VKf

t

i

!cosO)

•8.1 *

I

I

(

I I

}I

0,2 0.5 1.0 2.0

PT OF PARTICLES

529

Q2 AND W DEPENDENCE OF MULTIPLICITIES IK //p SCATTERING

AT 280 GeV

The European Muon Collaboration

presented by

Ewa Rondio, Institute of Physics

Warsaw University Branch, BiaJystok, Poland

Abstract

Properties of hadron multiplicities in 280 GeV /t+p

interactions have been investigated. The c. m. energy

dependence in the range from 4 to 20 GeV as well as

variation with Q are presented. Ho variation faster

than logarithmic is seen in the energy range of this

experiment. Increases in both the mean charged hadron

multiplicity and the K yield are observed with2

increasing Q for fixed c m . energy W. (The detailed

study of fragmentation function shows that extra

particles are produced mainly in the central region

Of Xp.

530

1. Introduction

The proces of deep inelastic scattering can be described

in Qoark-Parton Model as one photon exchanget It is two step

process: firstly the hard scattering when proton constituent

/quark or antiquark/ absorbsvirtual photon and secondly -

both, stuck object and the spectator fragment into hadrons*

hadrons: . .current jet

hadrons .-target jet

The hard scattering is described by the following quantities:

total energy available in the center of mass system /W/, • four-

momentum transfer to the ha&ronic system Afv, and the fraction

of proton monentmn carried, by the stuck quark /Xg./. These three

variables are related by: • - •.

£-i),where SL is the proton mass.

That neaas two of them are independent and needed to described

the hard scattering process con^letly. Because of that in deep

inelastic scattering there is a possibility to study separately

phace space dependence and QCD evolution. Ihis is different

from e+e~ scattering where only one variable E 2 = Q2 = s = 4-E

is needed for a complete description.

The final state hadron multiplicity is an interesting quan-

tity to look at in the context of V and Q2 dependence.

5312. Bxperiaental results

To study total nmltiplicity of hadrons the most important

experimental requirements are high, statistics and high accep-

tance for hadron detection and reconstruction. These were meet

in RA9 experiment ~ade by .European Uion Collaboration at CSES.

This experiment consisted of a vertex aagnet containing a strea-

mer chamber Bith a 1 meter long liquid hydrogen target, large

angle track chambers ana a forward spectroaeter which T.vas used

to detect and measure the scattered nucn and fast charged hadrons.

The complete system detected tracks from energies of 280 GeV

down to less than 200 UsV and gave almost 4 37 coverage in the

hadronic center of mass system. In addition, time-of-flight

hodoscopes and four Cerenkov counters gave good particle iden-

tification over a wide momentum range. ISose details of the

experimental set-up can be found elsewhere {i}.

The analysed data sample consists of 25000 events in the

kinematic range for 11 fxom 4 to 20 GeV and fox Qf above 4- GeV^.

The multiplicities of charged hadrons as well as neutral £°

yields have beea studied.

As track selection and event reconstruction are not r>erfect

one has to unfold the true /corrected/ charged hadron multi-

plicity from the results obtained from reconstuction.

The probability F(m) to observe a given multiplicity n is rela-

ted to the true aultiplicity n by

where C(n,m) is a matrix describing probability to measure real

multiplicity n as m. This is calculated from the J.tonte Carlo

simulation taking into account radiative corrEctions, detector

acceptance, reconstruction efficiency -j.nd probatility of v;rcri~

tr^ck association to the interaction vertex. ?:.e total correction

for all these effects is not ticker thari "iZ'-l. rlaving the ~.crcrix

C(^,c) the trae tnultiplicities hare be-r. fitted by solving the

~etrix eaua-icn derived from 21iri13u.n1 OC' condition, rie fit v:as

scssibls since charge conservation restricts n to oil? odd -"eluis«

The calculated values of X ~ per derree of freedom allcv.ei s. tsct

of the reliability cf the method as well as correct estimate of

statistical errors. Systematic errors were studied using different

track selection criteria and treating the high aultiplicity tails

in different ways. r2he estimated systematic error (.-shich includes

the uncertainties in correction for E°,A, A decays) on the total

multiplicity is m&xinuE ~0.4 *

The results obtained fros using the procedure described above

fcr each energy interval are presented on fig.1. Ibe average char-

ged saltiplicity can be correctly described as linear f-onction

of ln:.7 . Ibe results are also compared with results from 3333 for

^ and 7 interactions with protons ( 2). To conrpsre with nulti-

plicities for e+e"" annihilation one has to use only results froc

forward henisphere where single quark fragceEtation doainates.

This comparison is shov;n on fig.2 and also here the agreement is

good within quoted errors.

The dependence of aultiplicity on total available energy YJ, seen

for all data sanples is clearly the dominating effect but a que-

stion can be asked if it is the only quantity defining hsdran

nultiplicity. She candidate for fche second quantity is tf or x3-.

One can not distinguish what the correct choice is because, as they

are related, any effect existing for one of then v;ill be reflected

in the other. For this study Q, was chcosen.

Pirrure 3 presents the average charged icu.ltiplicity as a func-

533

tion of \" for four (£*" ranges* A systematic increese of the 5-10%

is obc3rv-?.3 between lowest and highest Q,~ internal and i t seems

to be independent of VJ. 'Jhe same data is ploted in fig.A- as a func-

tion of Q2 for different ?V bins. Eie comparison vfith. Lund model

(3) shows cleariv different behaviour /dashed curve/. For the

data an aproximately linear dependence in lnQ^ i s observed and

in the model no variation Tith Qr at low W and a small decrease

gfc high W is predicted.

The relation a+b ln(j2 has Jaeen fitted to the data. The slope

parameter b is positive in a l l W ranges and i t s valuse are ploted

in fig.5. "The mean value for b is 0.23 + 0.03 and is significantly

different from Lund model prediction /-0.0??+O.Oi3/«

The average yield of 2° as a function of fl for different 5J,

ranges is shown in fig.6. C?he analysis siaiilax as for charged

hadrons leads to a mean value of the slope parameter equal to

0.065 + 0.027. Thus the E° yield also tends to increase with Q,

in a similar way as aean charged hadron multiplicity.

Pinaly a question where these esctra particles corns from liave

been asked. Pigure ? a and b shows I/TZ^ <H*'/vxTt as a function

of H for different x_ bins in two ft2 intervals. It seeds tha-t

increases of charged hadron yield are observed in tiie forvjari

central and bacto?ard central regions.

The 5f dependence of total multiplicity is expected in QCD

and could be explain as beeing due to the giuon emission by the

struck quark. ISbrsover, as the mass of the scattered quark is

equal to Jjf after absorption of the virtual photon, one expects

that this highly excited (juark will eait soft gluons and that gizck

bremsstrahlung becomes more violent with increasing Q*%

These possibilities have been widely debated in perturbative QCD

in the past few years (^,5,6). However the quantitative predic-

tions from these thecii es ars difficult to exfceact and compare

•with experimental observations discussed above.

53-i

1. ZI.C, J.P.Albeiiesa et s i . , ITuci. Ir.=:tr. t n i I.k-t'noar 2/Yc (p95^) 111

2. H.Grassier et e l . , Iftici. ?r:ys. E22^ (193J)26V.

D.Zieidnslia et a l . , ?hj-s. fiev, D2J (1953)^7.

3 . G-lngelnar. et s i . , iTucl. Phys. E206 (1582)239.

4 . A.Bassetto et a l . , Jtays. Le t t . 62B(T9?9)2O7; ITacl, Fh^-s. B i ^

(1930)^77j.Ehys. Reports 100(1983)205; IfacX. Fays. 3207.(1952)153-

5 . W.Jurxanski et a l . , rfacl. Ehys. ^ 5 5 ("979) 253.

S. A.H.libeller, Hrv-s, Le t t . 104B(i96i) '6'; .

535

e•oatent_raJZ

en

9.0

8.0

7.0

6.0

5.0

4.0

io

2.0

-

-

- •

- '

o-0

• EMC p*p-

*• v p [ 8 ]

o vp [81

t • .

( ., , , , , , , ,10 20 40 100 200 400

: T-'W2 {GeV2)1000

Tit. 1

Av«r»t« ehirjed hadron sultiplicity at a fur.c.ioR of V2.d € a c e « * j ? c c i i * to the «sticiatcd systeaaiic errors.

536

2 ( n P )

15

10

5

0

i i i i i i ( i i < • i t i i 11

• EMC n+p

o e*e~ <n) corrected for

KJ, A ,, A decays

PLUTOJADE^TASSO A

4 "

i < t i i ? i i r i i l i i t i i

2 . 5 10 . 20 • .50 100

- - .- . VUGeV)

Fit. 2

twiec th« avenge forward y*P multiplicity tnd itscotnpu-isOB with e+e" d«.ta.

537

7.0 -

6.0 -

5.0 -

1.0

' EMC y*p (NA9)

preliminary

1 V

•

1

» } J ••

.<» 21• K• OL

1

••

• . . .

•

<Ct2<20l«t0

10

i.v«r%s* chmrtwl k»dron multiplicity ts » f laietioa ofV* Car <UK«rtat Q*

53S

preliminary—LUND W

* 4-6n 6-8A 8-10• 10-12o 12-14r 14-16« 16-18O 18-20

i l i I t t t t i i l

10 10

-G* (GeV2)«», *

h«droa Multiplicity cs « fiutctioa of«* for d1ff*ir«i»t V bin*.

539

A

V

0.8

0.6

0.4

0.2

0

-0.2

EMC |i+p preliminary (NA9)

Slopes •

Mean 0.23±0.03

LUNDMean-0.077i0.013

i i i i 1 1 1 1 1 t I T I t t l l

10 102 103

(GeV2)

rie. SThe slope p*r«met«r in <Be>,> • « • J> la Q* function

for differcat V bias.

U./

0.6'

0.4

0.2

i . i

t

EMC \i+p (NA9)

preliminary

—

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1

1 <i

1

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1

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W (CeV)

Fit. 6

iverasc K» yield as a function of W for different Q* ranges

-a

=2

30

20

10

0

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a

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542

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<?•

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rrajnecttticn functioa. Ch(»^, w*; ') « J-

0/ ch»rj»4 htdrons witk x_<8, for different x bine.

OVERVIEW OF VX IKTSSACTIOKE

H. Abramowics

Institute of Experimental Physics

University of Warsawt Warsaw, Poland

This talk is meant to be a short review of our knowledge

on neutrino nucleon interactions, given that existing results

may be looked at with some thought and new ones are expected

any time. This review is by no means complete. The topics

which are covered concern mainly our understanding of weal;

interactions and the nucleon structure. Exceptionally the

like sign darnuon production is mentioned, since some new data

have been lately available and the otherwise obscure picture

in this domain seems to clarify, at least at the experimental

level.

A lot of data exist on neutrino nucleon interactions, thus

the choice of results can hardly be objective, even less so

in this talk where they are used -for illustration only.

The main interest lies in the study of the energy

dependence of totsl cross sections. A constant slcpe in the

6(E> dependence is a measure of the point like structure of

the nucieon.

In the range of available energies (up to 200 SeV>. it

seems that for E>10 GeV the slope E/E is fairly ccnstant.

Fig.l is an example of the energy dependence of S/E as

measured by CDHS (1) and CCFRR (2). Although tnere is no

raasor t c e.:psct v<?t snv p;-oc.rO?tc- £--M&::t, one w:»,!c eas i l y

accept e unsll Drop 01 tr,e si or? w:t!" ene-ra-,, -c.f's: s ten*, w.*.>-

3CD oreci ict : on. Tns'-e i s PC riS.i-n- wen--,- t ' i ' j c i - ! oe: = .:>;e t.'-.f-

dots s t i l l ?i!HE'- rrciTi s * a t : 5 t c * i *Juc :n i t iD - . s i ? r r t e n ; . i n

t * cover an> QCC—like aspsr.eence. T*ble 1 contains c : ; i C-T CILI-

i ncwieage or, cress sectior-.c -for V s.ns V1 CC 1 rn.sre.ct; ons Dr sn

sveregs nucleari . Tt-,& lack ot scr eerte* t i r> iri= ^&;r:ti.:of' Q-f

cross =ect;DnE ; = no ! oncer ? wor 'y . Tne E""ecJ=e

(iieasuremsr~ = of CDK£w +ron> s NPB rup - ^ tr- c..n i c p r t v e i

mona tc r i r .g svstec. o-r me CEFf" neutrjnc;. C'triti: l i n e , gsi L i t : ? '

tc- tne value OEter»i:ned t>- CCFC:R. F ig.2- , ' i ?u f i : r es \aj..-s-s u*

CC cross sect icns as =. r u n t t i o n o-f tifT'& sno in : iudt i= =.; =c 'r,c

rfSrt rssLi l ts o-f CDHSW as pr&senteo at Pan ( 4 J .

Tfie CC z-ase secrt; ons. -fo>- V »nd V1 i n t e r s t l i o n E $E r.es.zwec

or. pi-cto"S =v~a neutrons sec^rMel V ^"e presentea i n ta t ' i e I I .

ArsJ ysi E c-t tr,s neasu'ec C D E S sect ions i n t e r c r s- t "e

Q'jsri- p t ' t c r . fr.oaei ms. ; : ve s i reao- s-'ine lr.s-gnt :n*;c- t~e?

n*.ir.ienn HtrLicti '. 'e.

The ai •*- ere-.ti 5.. :-.•"£ = £ sert-icr. e1.pressed in te- i is o*

wr-.ere : srd •• =re trie u»u&j Bjorter. v s r i f D i s s . £ js f . e

;n:op,inc V sne'-qi, 5 i s the •oui"-mofferitair t>"&r.s-c6f tc trie

•"•sjcleon a->c f.-the PJCISD-I r;? = s.

Fc'- a-; isoscciar ta rget ar.c -for- s•:•".'.'tur6 *unc*..ins

6vsrs;ec ever k-' anc ? tfte r c i i ow jng "-eiet icns r.oio

where q and q are the quark and antiquark contents of the

nucleon and R=^/5-r is the measure of the violation of

CalIan-Gross relation.

If one denotes by «<** and o^ the slopes E/E and SK/E

respectively then

and

Table III contains the integrals of F, and xF. and the

corresponding amount of momentum carried by quarks and

antiquarks if a value of .1*.i is assumed for R. The rest of

the nucleon momentum is carried by gluons.

Structurg_functisns_sf_the_nucl.gan

Structure functions can be determined from the

differential cross sections in * way anaiagous to the one

presented in the previous chapter. One may also determine

the antiquark structure function for »/ — ^ frorc art

appropriate linear combination erf differential cross sections

for y and P interactions, for high y.

+• corr.

The x dependence of various structure functions can oe

seen in fig.3.

Since KFJ Describes the valence part of the nucleon

structure, the integral of Fl should &e equal to 3. Actual Zy

QCD predicts!

CCFRR i.2> has measured s value of 2. £~+. itj+. 0r+. 10 at

1 1 1 n

G =3 GeV /c , compatible with ths scale of OCDA .525 GeV/c.

F I G S . ' - 6 show a f u l l x and Q dependence of structure

functions as measured fcy various experiments. The agreement

among experiments is quite good. There is some d3ssgreement

at IOK x, part of which ie due to dif ferent values ot cro=s

sections used, ft di f ferent treatment of corrections winch

have to be applied for the determination o+ structure

functions could probably account for the rest of tne

discrepancies. A l i s t of those corrections can-be fauno in

table IV. The second column of th is table proposes

(•following F.Dydak '3)) a way of analysis, and the th i rd

column points to those corrections ths di f ferent experiments

agree upon (radiative corrections ! ! : > . The agreement

between î-ffsrent ex Deri men ts w i l l become even better i f one

considers the latest preliminary structure functions

determined fcy CDHSW with high precision in a WBB run, and

a i l correct!or.s ss proposed in tabie IV (5).

QCD predicts logarithmic scaling violat ion in the Q

dependence of structure functions described by the

AJtarel l i -Par is i set of equations.

with of j—the etrong coupling const An .

There are di f ferent methods to f i t GCZ> predictions

547

available on trie market. An analytical method has been

proposed by Furmanski and Petronzio <6J based on -fits o-f

Laguerre polynomials in ln ( l /x ) . A numerical integration has

been proposed by Abott and Eurnett <6i . Devoto et al . (7)

and Lopez and Yndursin (S>. ft fixed parametrization of

structure functions at a given 0 i s then assumed . different

in different programs. Qdorico (10) performs the QCD

evolution by Monte-Carlo simulation. All those methods give

similar results, wi.th /^contained in the range .40- .40 OtV/c.

It i s quite remarkable that the f i t s obtained at

relatively low energies (Gr<200 6eV / c 1 ) , extrapolated to

high energies give such a good agreement with what i s

measured in pp interactions at the CERN-Co11ider Csee f ig .7

and ref.11).

With the new measurements expected soon in their final

version one should expect a new round of QCD f i t s , with

hopefully some more stringent limits or> A .

The longitudinal structure function Fj_ »F -2*F^ i s expected

to be non zrro, ouing to the transverse momentu* of auarks

with respect to the nucleon direction <OCD>. Any quark mass

effect i s also expected to contribute to the non zero valu*

of Ft . Eventually spin -0 constituents ' diquerl;s> may also

contribute to the violation of Callan-Sross relation.

The knowledge of Fu i s needed for the determination of

structure functions, A non sero value o-f Fu introduces a

change in the y dependence of differential cross sections, as

compared to the naivt quark p*rton model predictions.

A way to determine F is thus by -f i t t ing the y dependence

i .e . at fixed values of >••. There is even a better way to

estimate FL cat high x - from the V data at high y (say y>.5>.

The results of CDHS '1> end CHARl*. (23) are preEented on

f ig .6 in terms of R=Fj_ /2x^| ae a -function o-f x. rilthcugh the

s ta t i s t i ca l significance is poor, the agreement with OCD far

A = .£ GeV/c i s quite good. The upper liir.i't obtained by CCHS

for .4<x<.7 gives R<. 039±. 014*.025 at <D2 >=3S Sev*/c*".

The only ««y to lesrn «iore About the flavour content of

the nucleon is by using the data on v and v> scattering en

hydrogen and deuterium.

I f for »i i»pl ici ty one neglects the strange"»no charmed

<?u*rks in th-ie nuclear* then from the quark parton

representation ai the structure functions one can easily

obtain t f i * following relat ions:

I t has been Known for » while th-at the ra t io a4 o^/v.^ as a

function o# >: was rtot constant as predicted by the naive

quart: parton model. Fig. 9 shows the CDHS measurement of

dy/uv together with the contour formed by s l l other <jat«. ft

clear drop with ;: is oDEerved, but unfortunately there are no

549

date available at high x , and thue i t i s quite di-f-f i c u l t t o

d i f f erent ia t e among di f ferent models which seem to predict

such a drop.

New data are now avai lable -from the deuterium -f i l led BEBC

for a 400 GeV WB-bsaro <12>. It has been shown for the f i r s t

time that the r a t i o £V /uv i E constant as a function o-f Q

(f ig . iO) and that the antiquark content of the sea i s the

same for protons and neutrons, a feature whicn i s commonly

assumed in a l l the analyses < f i g . l l ) .

By using the Fj structure functions for neutrons and

protons separate ly , one can try t c t e s t the ftdler sum rules

0

This has been done with the new BEBC data, and one can see

indeed that the Adler SUM ru le i s va l id within s wide range

of Oa ( f i g . 1 2 ) .

The EMC effect (13), confirmees by tne BLAC-date (14j ar«J

the BCDftS-dat* (15) h*% been with us for a while. There is a

general agreement -for larae x <sav x.;.3), wnile tne situation

is ouite unclear at low x (fig. 13).

It seems that one of the reasons for the difference

observed at low n i s the different range of Q*" covered by

different experiments. The relation between the ratio of

cross sections and the ratio of structure functions tor a

given nuclear target A and deuterium i s the following

where 6 i s the degree of longitutinal polarisation of the

550

virtual photon. For the mjon aats which cover a high Q

range 6. tenss tc 1. and the ratio of cross sections 1= equc.'

to the ratio of structure functions. In iht case of e--dsta,

for small values of R:

If for some reasons R i s not equal to R (not excluded By

the new SLAC data !16U ther. the lack of Q* dependence for

e-dat« end tne disagreement withj*-dats <fig.i3o> couid be

ft Dunderstood. Actually if the ratio F /F£ for e-date istfcterminad assuming:

the agreement between EMC and SLHC dftta becomes quite good

(fig.14).

While tns EhC ef fect , i f 0 -ino«p«r>dcr,t, i s quite

unimcartant for any kine of GCt> analysis, i t becomes

imoortmnt -for tn*.d»t*r»iination of tti« nucleon structure

functions if we w*nt to go beyond the notion of an average

nucleon within a, {i ven nucjeir t*rg«t.y

The EMC effect Beams tc be present in the v? oate. too

(f ie . IS) . Although much ims* significant th> V d*ta may be

used to test SCMW of the i d u s proposec to explain the EMC

effect. Ttve V data ctm be UPBO to investigate * pessibie ser.

increase at low r. in he*vy nuclei. Th« study of y dependence

of V and f data *«v be oc some help to invetticate the A

dependence of R. An extensive study has tesn presented by

<rwf . 17).

In terms of the ouark parton modell

551

where a=qV(-(-Qse(t is the quark content of the nucleon

(valence and sea) ,a=cjg6a is the antiquarl; content 0+ the

nucleon and q, is the longitudinal component 0+ the nuclean.

Assuming the following parametrizstion for q^ and 0.3 4

one may t ry to f i t the data assuming that for heavy

targets ft and © can. varyj

fi > /i + A/5

The EMC data would lead to *£=.2 and Rj^l.35. It can be

seen from fig. 16 that even if thes* values co.Jld account for

the x dependence of th* cross sect;or ratio, th« y dependence

is MronQ. The best fit to the d#t* is caotained with

Rt=.914.07, thui no increase of the »»a content at seen.

The exclusion of the low ;: region and/or the low 0 region

(to get rid of any possible shadowing effect) does not cnange

this conclusion (Rj«.?a±.O8). Fig.17 show* the prediction of

this simple model .tuned to fit the EMC oata, if different

assumptions are made about the fi dependence of R. It is

clear from that comparison that it is extremely nard tc

accomodate both the y dependence of C'.Ne//£ date suffer frosi large statistical

errors, it is at least clear- that the»-e is no significant

increase in the see content of neavy nuclei.

552

Neutral currents

The nu: 1 eor! i nterect icns y ie ld ons ot tne most prec:=e>

measurement o^ s i r /G , sna trius fiiev give more insight jn to the

highei- o'"der eleztrowesi^ correct ic-ris. The ".fir. argument

against the determi riotion ot sirj*4»w 'rom nucieor.

interact ions is that trie whole pic-ure i s obscured by nsoran

structure. Liewel1yn-Smitn (IB) has shown that i c r an

isoscaJar target most o-f the re le t ion between neutral «nd

charged current cross sections-comes from isospin invarienca

alone, and tne quark parton model i s needed only tor minor

correct ions. H strancje and charmed quarks are riegiecteo ^r.d

no Cabibbe mixing is assumed then:

Two metnods have been us-eo to determine sin riw •from

C> / S" <CC) and i-* (T<CC> / S <<CC> / S <CC> with

giv*s sin v-^ with a theoretical uncertainty c*

4.CiOS:from the Paschos-Woi + enstein raiation w t h

wi t r s theoret ical uncertainty o+ +.003. While the • f i rst

method has a Digger tneoret ical er ror , the second one

reouires #. very good knowledge ai the absolute -fluxes ot

y and V beams.

Table V contains resul ts puplisned recent ly. The «iorid

average hss been calculated by C.Gewemger i l9) ior

raa ia t ive ly corrected data and amounts to :

"I

the lsrt error being due to theoretical uncertainties.

553

The y distribution d<>/dy -for neutral currents has Seen

studied bv the CHHRH Collaboration <2O>. The dif ferential

and of , where

i?*.

A common f i t to NC and CC y Distributions yieio=:

P >

wnileo! - li =.O«>±.O4 is in good agreement with the same

quantity measured in charm changing reactions (opposite sign

dilution production):

Thu« the y distributiort o* NC is in good agreement with

the expectations of the standard moael with * conncnly

accepted value of sin «w.

The diffarential cross sectiori dp/d« can be tspresswa ;n

terms of F*',x> whicti are appropriate comtsi nations o-f F. fiO

xF. where:

where u. <0 >(uB<de) are NC caaplmg constants -for l e t t

tisndsei and r ight hanoed u and d quarks, Results oi the CKAF ft

col lsbaratian are presented in f ig .13 . The CC osts have beei

used in order to check the unfolding procedure necessary fo"

the determination OT dSV'dx tor NC against the st«nda'-d method

used for CC analysis. A •f it in terms o* quart* par ton model

and NC couplings o-f the standard model to the CC and HZ data

yields t h * came parameters -tor both <ful l l ine on -fig. IE)

which indicates that the nucieon viewed By CC and NC locks

the same.

The FMP! collaboration <2i) hes 5=tudie-i the ratao

CC

as a -function c*f ss < + ig . l9> . A quark parton model -f it to

>c > yields-

which shows ag*in that tnere is rto signi-ficant di-f•'er

in the nueiton structure *» deterniined in CC and NC

reactions.

New r-fsi.ilts en neutral current interact ions are expected

soon se well *c Farmiiab as at CEfiN with better s t a t i s t i c s

end »vstem«tice.

A orompt Xlka sign dilepton production has been ooserved

in counter experiments lji.il) ».s well as in Bubble Chamber

experiments tff'JA"). A prompt sisnal of 2-4 6* has been

observed in nost experiment* with a re la t i ve promot rate

(T <f*."fi > /€"< ft • OT ID"" - 10" while known processes, as cc pair

555

production via gluon bremstrahlung or b proouction could

hardly account -for more than a rate of few * 10 a . Since, as

can be seen •from tig.20 (a compilation from ref.22), there

was no'agreement among different experiments especially at

high energy, new data were expected eagerly. With new CPHSW

data, and the revised CCFFR aata the situation seems tc

clarify as can be seen in fig,21 (the CHARM data will Be

probably also revised). There is a clear prompt signal, witn

an energy dependence which looks like a possible threshold

effect. There are some hints (from CCFRP:) tnat the structure

of those prompt events is compatible with cnarro production,

although the observed rates are much highar than predicted by

perturbative QCD.

One should expect soon some -final data on like sign

dimuons, while one will probably have to wait longer for some

theoretical understanding.

556

REFERENCE?

1. H. Abriinrwi cr. at si . , Z. ''hvs. C17 :1 r-, ;; 7 ? 2S3

2. D.MacFarlanfc et el..Z.Fhys.C2o <I9a<;) 1

3. F. Dydal , CERN-EP/B"—171 and PrGcesoinc,? o* the

Internationa! Svmoosiuir. on Leptorc and Photon

Interactions at High Energies

4. w.lcrasnv. contr ibut ion t » the International Contersnc?

on High Energ> F'hysi C£. Bar i 1935

f.. F-.Euchhois.ccntriCutisn to The iritsrr,;.tianal Center en:

an Hi ah E-isroy Physics, har: 1985

t . W, Fu"-f.ansl;i sna R. Petrpn: 10, Noel. Phvs. 5! 75 il^SZ) 237

7. ^.F.MtDCtt sr.ci F;.H. Farnett,HRn;ls Phy.s. 135 tiÊ1:'.1 " o

£. A.Devote and al•.PhvE.ftev.E27 fl979> 5C8

y. C.Lo&s: and F.J./nGjrsin,Nucl.Phvs. BIS" (1?61> i5~

!•>. R.0(»oricc.CERN-TH.'2?e:.

11. F.Berjsms et ai. .Phvs.Lett. i53B <>°35' 111

\Z. V. A l l ass s et j L ^ r e o n n t 1NFN/PD February l»55.to ce

DUCi:shec jn Z.P!-»s.

:7. vi.J. Albert et * 1 . .Phys.Lett. 123B (19£-? 272

et al . . Phys. Rev. i_ett. 50 a ° s : i 1J3I.51 '1=8?:

15. fcCi'r'S 'NAO- Col I s&orst i an, ccntr i sut: en to the inte

t :or . f i Con-ersnce on Kei tr inc Phvsics and Astroony =

".MeL-fi.-iD 94'•. rortR'Uid J=S4

J4. r.C-.H'-nolc 6t ei . . P^vs.. Pev. Let t . E.? '195^; 727. sno

J.5cê;.SLAC-PUS-T5S2

!7, A. M. Sarkor-Cccper .CEP.N. EP S^- i2 l . ano Frorssoi ngs n

"K ei.'t..-i-c i1^' C3r,< e'en;6, Dort.T,-r,£ 1=1"

J5. C.H.-ieweii •.•n-Sinitr..Nucl .Phvs.5r.lt: •. l = Sr-.' 2vt"

19. C. Bewenicjer .Proceedings o-f "Neutrino 84" conference,

Dortmund 1?S4

20. M.Oonker et al.,Phvs.Lett.102B (1981) 67

21. O.&oaert et al..Proceedings o-f "Neutrino 84" conference,

Dortmund 19B4

22. H.E.Fisk. Proceedings o-f the International Symposium on

Lepton and Photon Interactions at High Energies,

Bonn 1<?S1, and re-ferencas therein

23. F.Bergsma et al.,Phys.Lett.141B (1984) 129

Table I: Compilation o-f most recently published charge*}current total cress sections

Experiment

CDHS

CHARM

BEBC<WA47>

CCFRft

BCP

BE8C<WA25)

Average

.62 i .04

.604 i .030

.65? + .030

.669 ± .024

.62 * .05

.60 * .04

.638 + .014

.3C> t -02

.301 ± .01B

.3<5? + .016

.340 * .020

.34 t .029

.30 s .02

.512 ± .008 j

I

558

Tab! g XI: Inte-grsls D-f structure -functions obtained -fromCC V and $ cross secticns

Experiment

i

i

CCFRR

.479 2

.3i2 ±

• 1 ±

.374 +

.061 +

.015

. 030

— J

. 026

. 025

CDH3

.437

. 304

L

. 1

. 350

.047

i

*

*

i

.021

. 043

. 1

.033

.025

Tablg I I I : CC totsi cross sections for V and V on~ protons and

E : peri ment Targ»t

EC£iC(WA21> p i H) ; .56 i .04 .34. t .04 j

I

CDHS p <K) I ."<? I . 0 3 .40 * .03

&EECMWA24! \p (H+Ne) ' .fit, ± .03 .36 ± .03

tEEBC(WA2S)jp (0: ° ± .04 .40 * . OZ

tI

.81 * .06 .20 ± .02

559

TablglV :List of corrections which have to be appliedto cross sections to obtain structure -functions.Column two contains propositions about the wav tomake those corrections, column three summarizesthe actual status al agreement among differentexperiments (with an obvious notation).

Propagator term

CC cross sections

Strange sea

Charmed sea

Charm threshold

Fermi motion

Radiative corrections

_ J

ves

world average•

s-.2<u+d+-s,

neglect

slow rescaling, m=1.5BeV

«.„no

De ftujula et al.

-

-

-

-

-

-

5S0

Tafrle Vt sin*<S>y, as determined -from precise in2ctii.trEir:€?ntof NC and CC cross sections

Expt.

CDHS

CCFSfl

rm

CHARM

Method

Ry . r

P. (••:)

+ 00e>

II. 240*. 0101 +.005 !1

,243±.03 4t l .014)

.220+.014 i

1

56i

Figure Captions

Fig. l -Energy dependence of V (upper points) anci p (loner

points) nucleon CC cross sections as measued by

CCFRft (open points)-and CDHS (-full points). Tne

straight lines represents the averaged values and the

arrows show the allowed scale shi f t .

Fig.2 -CC total cross sections -for >/ (upper part) and ]7

(lower part) nucleon interactions as measured by

different experiments.

Fig.3 -F, ,>;F« and p structure functions and their x depend-

ence as measured by CDHS, compared to the ji and e

data.

Fig. 4 -The K and Q* dependence of xFj for yN interactions.

Fig.5 -The x and S dependence of Fj ior j/N interactions.

Fig.6 -The x and Gr dependence of p for vH interactions.

Fig.7 -The rat io R»FL/2KF,, am a function of x. The cui-ve ia

the BCD prediction wi th A».2E*v.

Fig.8 -x dependence of F(X)«G(K>+-4/9((5(N > *5<>:)) at

Qa»2000eeV2/c2 »s determir»d by the UA1 ( ful l points)

and UA2 (open points) Collaborations an-d extrapolated

fro<» CHARM structure functions (continous iine^.

Fig.9 -The ratiD A,/uv a* a function of :;. The data points

come from th» CDHS hydrgen data. The dashed lines are

contours formed by nany existing data. Also shovtn are

predictions of the quart, psrton model, the diquark

•odel. OCE> and Field ant} Feynoian model.

Fig.10 -The « and Q dependence of the rat io roa^/uy.

Fig.11 The x and Q dependence of the proton and neutron sea

distributions.

562

Fig. 12 -Check of tne Adler sum rule as a function o-f Q .

Fig. 23 The rat io of dif f ©rent j si cross sect ions on nuclear

targets and deuterium as a -function of x and various

values o+ Q as measured in different experiments.

Fig. 14 -The structure •functions rat io -for iron and deuterium

JIS a function o+ :•; and Q .

Fig.15 -The EMC ef fect as observed ir. various neutrino data

isas ref .17 -for d e t a i l e ) .

Fia.Xh -The rat io of the di f ferent ia l cross section as a

overlaid on the prediction of a quark pertor. model

involving a 35% <-fuli l ine) and 9% (dashea l ine) sea

Increase in Ne. (b)-the same for the y dependence.

Fig. 17 -The r.»tio of differentifil cross sect ions as a

function of >• as measured in ^Ne and ?D scattering,

overlaid on the predictions of a quark parton mooel

involving a 357. sea increase in He and q^=0 (full ,Hi ~ W —

1 in») , q «. 06A <Q+qj (dashed l i n e ) , C|J_*.1A ia+q)

(dotted l i n e ) . (b)-the s»«e but -for vNe and vx> data.

Fig. 18 -Structure function* F*. ~ dS/d:: fors a) CC v events,

b) CC P events, c) NC y events, d) !«: v events.

The ful l l ine i» the result o-f a coatmon f i t to CC and

NC data with the same quark dis tr ibut ions performed

by CHARM.

Fig.l1? -Ratio of d i f f e r e n t i a l cross s e c t i o n s -for NC and CC as

a function o-f x for V -t'a) and ( c ) , and V - and (d)

*t d i f ferent beam energ ies .

F ia .20 -Ccunpiletion of the energy dependence ot prompt rates

for like—sign dirouon production. <s*e rm-f.22 -for

553

details)

Fig.21 -New results on like sign dimuon production.

* V * V ^ * <> A

50 100 150 200 250 300

Fig.

564

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583

HERA : THE PHYSICS AND THE MACHINE

Peter SchraiiserI I . Institut flir Experimentalphysik der Universitat Hamburg, Hairiburg, FRG

Abstract:

The physics programme of the large electron-proton colliding beam fac i l i tyHERA is reviewed and a description of the accelerator project is given.

1. Introduction

The electron-proton colliding beam facility HERA now under construction at

DESr offers unique opportunities' to study extremely energetic interactions

between the two basic building blocks of matter - leptons and quarks. The

available center-of-mass energies of more than 300 SeV extend far beyond the

range of present-day fixed target experiments and would in fact require

leptons beams of 50 TeV Impinging on stationary nucleons. The average squared

momentum transfer Q at HERA will be two orders of magnitude larger than that

of the SPS or Tevatron lepton scattering experiments (Fig. 1). Consequently,

the spatial resolution is increased more than tenfold and substructures in the

proton can be investigated down to a scale of 10 cm.

In the energy range of HERA, electromagnetic and weak processes are of

comparable magnitude. While in the study of neutral current fclectro-weak

reactions HERA will have to compete with the new generation of e+e* and pp fpp)

colliders, it will be a unique tool to investigate charged current weak

interactions.

An important quantity to characterize the physics potential of a new accelera-

tor is the number of events which can be collected in a reasonable amount cf

running time. The design luminosity of HERA ir, the bead-on collision scheme is31 • - 2 -1

2-3-10 cm s but in the following 1 shall use a conservative figure of

1-10 cm" s* corresponding to an integrated luminosity of 1 pb"' per day.

Fig. 2 shows the number of charged and neutral current events with Q 2 values

larger than a given QQ. About ten events of each type are expected per day

with momentum transfers Q above the W mass.

r 1 q. 1:

The average Qc in charged

current reactions as a func-

tion of the ep center-of-

mass energy (from Ref.l)

WOO

O.I r

0.0!

Fig. Z:

Counting rates for charged

and neutral current reac-

tions with q above a given

Q . The curves have been

computed -. Mattig (DESY)

using the G1Uck, Reya,

Hoffmann parametrization of

the proton structure func-

tions and A0C[| = 100 HeV.

0 10000 20000 30000 40000 SOOOO

585

In the quark-parton model the reaction ep - eX and ep * vX are mediated by

gauge bosons (y, Z° or W ) which are exchanged between the lepton and one of

the quark or antiquarks in the nucleon (Fig. 3). The infinite momentum frame

of the parton rao.del is actually almost identical to the laboratory frame at

HERA and one can therefore consider the high energy protons as a stream of

nearly free quarks.

proton(820 GeV)

(30GeV)

Fig. 3:

Diagram for deep inelast ic lepton nucleon scattering at KERA. The four-momenta

of the incoming and outgoing lepton, the proton and the exchanged f i e l d

quantum are denoted by k, k \ p and q, respectively; xp Is the four-momentum

Of the struck quark. The variables used are q = k-fc •, Q* = - q 2 , x = Q2 /(2p.q!,

y ' <P-q)/(k- p). For q2 values above 103 6eV2, the lepton is scattered by

wore than 90°.

The scattered lepton appears on one side of the beam and the struck quark,

fragmenting into a "current j e t " of hadrons, on the other side. The remainders

of the proton produce a "proton j e t " of hadrons moving along the beam

direct ion. HERA is thus ideally suited to study Ifepton-quark col l is ions

without "background" from spectator quarks.'According to Monte Carlo simulati-

ons the events have a very clear signature and can hardly be confused with

proton reactions on rest gas molecules or other background.

5S6

To compute the proton structure functions the kinematics! quantities 0 and x

have to be determined. The precise knowledge of the incident electron energy

is extremely valuable but in addition the energy and angle of the current jet

have to be measured with good accuracy. In the case of neutral current events

also the kinematical quantities of the scattered lepton ere quite useful. The

HERA detectors will be equipped with highly segmented calorimeters with good

energy resolution for electromagnetic and hadronic showers. One important

design goal is to achieve equal pulse heights for electrons {photor.s/ and

hadrons of the same energy in order to be able to measure total jet energies

with good precision.

One of the unique features of the HERA collider is the possibility to study ep

interactions with longitudinally polarized electrons or positrons. The

emission of synchrotron radiation in the arcs of the electron ring leads to a

natural transverse polarization of up to 32 % if depolarizing resonances are

avoided. By a combination of horizontally and vertically deflecting magnets

the spin can be rotated into the longitudinal direction in front of an

Interaction region and back into the transverse orientation behind it. With

left- or right-handed leptons the standard mode! of electro-weak interactions

can be tested in great detail and the he!icity couplings of any new particles

to known or new currents can be determined.

2. Physics Opportunities at HERA

2.1 Proton Structure and Test of QCD

The standard physics programme at an electron-proton collider is of course the

investigation of deep Inelastic lepton nudeon scattering which up to now has

contributed so greatly to our present understanding of particle physics,

providing much of the experimental foundation of the standard model and of

QCD. At HERA a vastly extended range in Q and v is accessible and charged and

neutral current reactions can be studied at the same time. Within one year of

running time the proton structure function F,(x, Q ) can be measured with

statistical errors below 10 % for Q values up to 10 GeV whereas a typical

SPS neutrino experiment with similar running time and statistical accuracy2 2

would cover only the Q range up to about 100 GeV (Perkins, Ref.2). At such2

large Q the logarithmic scaling violations predicted by QCG should be cleariyvisible. Mass corrections and higher twist effects which obscure the QCD beha-

587

viour in the presently accessible Q range should not play any role. The deta

yield at the same time an accurate determination o' zhe querk and qluon

momentum distribution functions which are interesting by themselves but are

also very useful-as an input to calculations for pp (pp) collider physics. The

accuracy with which one can nope to measure the QCD scale parameter is about

- 40 MeV for Anrn = 200 MeV (Ref.3). The value of the strong coupling a is

predicted [1] to be ac (10 GeV-) = C.I? - 0.01 for a wide range of t,nrns ? H ? ^ us ? H ?

Q = 10 GeV? H ? ^

(100-5fl0 MeV). A measurement of a at Q = 10 GeV could clearly establish

whether the concept of a "running" coupling constant is correct. Going beyond

the quark-parton picture and standard OCC there is great interest to find out

whether there are additional constituents in the proton like scalar quarks or

whether the quarks themselves are made up of ever smaller building blocks. If

quarks are extended objects the structure functions will show power law

deviations from the values predicted by QCD. Eichten et ai [ 4] and Ruck!

f5] have considered composite quarks and leptons containing the same

constituents which are bound together by the exchange of heavy vector bosons.

Frota the apparent "pointiike" behaviour of electrons, muons and quarks at

PETRA and PEP energies one can infer that the mass scale A^ of these bosons

is at least tn the several 100 GeV to few TeV range. A direct production of

these bosons of even a measurement of propagator effects are therefore not

possible at HERA but the new interactions will lead to a residual four-ferraioa

point coupling similar to the well-known Fermi interaction in 6 decay. These

point interactions can be measured through their interference with the y, Z°

and W exchange diagrams. Depending on the value of the scale parameter hu the

effect on the proton structure function F^x,. Q } may be quite large (see

Fig. 4a). Measurements at p > 10* GeV2 could place limits on >.u of more than

2-3 TeV. Even more sensitive than the structure functions are asymmetries

Fig. 4b shows the asymmetry A between right- and left-handed electrons. A U

values up to 5 TeV seem accessible. The asymmetry has the advantage than most

normalization errors and the not well-known quark momentum distribution at

large Q drop out.

2-2 Test of the standard model, search for new currents

Before going into the discussion of detailed processes 1 would like f.o make a

few general remarks.

- Within the standard model only a single lepton (e or v) can emerge from'th«

lepton vertex of an electnn quark coiiision. Any observation of s haoronic

588

S.O - F i g . 4a ;The oretorc structure function

F-,(>:.Q ; at r. = 0.5 for various

values of the scale narameter / ,.

The dasned ana dotted curves are

the standard model prediction and

the one-photon exchange resul t ,

respectively.

-0.

10

polarization asymmetry

in the standard rode": (dotted

curve) anc for various values of

the substructure scale A^. Tne

label "LL" ("RR") denotes left-

r.anded (right-haioed) coiiplingc

for both the leotons and the

ouari:s in the resioual four-

fermioii interaction.

5

5BS

j e t and/or adcitional leptons coming from the lepton vertex are a clear

indication of new physics. Experimentally, the lepton vertex is well

separated from the quark vertex (see Fig. 3).

- Only left-handed electrons and right-handed positron: couple to the weak

charged current. Polarized beams are a sensitive tool to search for

right-handed W bosons.

- The parity v io lat ion in neutral current processes is large and again

polarized lepton beams are useful to look for new ZJ bosons.

The charged current reaction ep -« vX is mediated by w exchange with purely

left-handed couplings. Additional left-handed W bosons can be detected at HERA

even i f their mass should be beyond the cer.ter-of-mass energy of 300 GeV. The

reason is thai the interference with the standard W exchange leads to a

substantial increase in counting rate at large values sf 0 (see Fig. 5). W,

masses of more than 500 GeV can be determined. The situation is quite

di f ferent i f the second W couples only to right-handeo electrons. In that case

there is no interference an<J the rate for e^p -> vX may be Quite low i f the

mass of the Up is high. A lower l im i t of 400 GeV for the mass of a W boson

with right-handed couplings can be derived from the decay of polarizes mutms

[6 ] . While i t w i l l be d i f f i c u l t to improve such i l im i t at H£RA one has to

keep in mind that the experiments at low Q *re. only sensitive to right-handed

currents i f the neutral partner \*, of the right-handed charged lepton is

massless or very l i g h t , «H^;) < 10 HaV. Since there is no compelling

theoretical reason why right-handed neutrinos should be l i g h t , measureswnts at

large Q are unavoidable to either establish or rule out right-naneed

currents.

To get an Uea on the sensi t iv i ty which might be achieved at HE3A we consider

the rat io R = O D / ° I ° f t n * r igh t - and left-handed cross sections. Assuming

that the kL couples to the normal quarks am! leptons with the same strength as

the standard W,, this rat io TS given by

F i g . 5;

The c o o s sec t ion f o r the charged cur ren t reac t i on e + p - v - t x in a rco

w i t h two l e^ t -ha rded W bosons, ncraiel ized to t.he standaro model cross sec t i on .

the coup l ing constants of the W. and W~. are assumed to be the sair>e and are2 i t

acjiiisted such that at Q * 0 the Fermi coupling is obtained, "he counting rate

«t Q ^. 3 x iO'' GeV is about IOC per year at a luminosity of lj.-10"1cm"2s~1,

so W,, masses well above the HERA center-of-raass energy are accessible.

i F iS- 6:

The cross section for eo-eX

I for left- and rignt-handed

! electrons.

591

2 2

At Q 5000 GeV a rat io fi = 0.05 corresponds to a WR mass cf 214 GeV. For a

degree of polarization of P = 0.80 t 0.01 and an integrated luraonosuy of

300 pb" , shared equally between left-handed and right-handed electron beams,

one obtains a s ta t i s t i ca l significance for the cress section rat io of

R/o(R) = 5.8 .

The s ta t i s t i ca l significance depends both on t •• degree of polarization P and

on the uncertainty o(P). For P = 0.60 s O.OJ. the s ta t i s t i ca l significance

obtained with the same total luminosity is only 3.0.

In the neutral current reaction ep - eX, the photon 2° interference term is

large for Q above 104 GeV2 and leads to remarkable differences between

left-handed and right-handed electrons (Fig. 6) and also between electrons and

positrons.

Polarized lepton beams allow sensitive searches for additional Z° Bosons.

Fig. 7 shows the polarization asymmetry between left-handed electrons and

right-handed positrons. Ki th in s l i gh t l y iiore than one year's running time Z-

masses up to 450 GeV seen accessible.

model

Fig. 7:

Asyrantetry between left-funded

electrons *nd right-iwr>d*d po-

sitrons in the standard aodel

a-nel tn a raMel with s second

Z°. The quantuiy r.^ is the

•nixing parameter for the 2_.

Also shown are "data points"

which might be obtained in

raeasurement with a total inte-

grated lum'ncsity of SCO pb"1

end a degree of polarization

P = 080 ± C.C2.

(Curves from R.,j. Cashrcore,

ftef. 7).

592

2.3 Search for new particles

Electron-proton collisions are ideally suited to search for new lepttis which

coutle to the electron as well as new quarks wnich couple to the u and d

quarks ir> the proton.

A neutral heavy lepton I" and a heavy quark Q could be produced in the

rtaction e q -* I' 0. Provided new charged currents exist wnv-h do not

suppress this process througn small mixing angles ths rates are fairly high

even for large messes of the lepton and quark (e.g. 1 event per day for M. =

M_ - 50 GeV and a mass of the exchanged W boson of 200 GeV). The decay of the '

heavy lepton Issds to pronounced event signatures: ieotons and hadronfc jets

will emerge ~roni the lepton vertex instead of the single outgoing neutrino in

normal charged currents events. If such heavy leptons should be detected,

measurements with longitudinally polarized electrons wsuld be extremely

valuable. Suppose the I" was found to couple to right-handed electrons only

then it would be natural to assume that right-handed electrons belong into

weak isospin doublets rather than into singlets as in the standard model.

"mother class of particles which couple directly to either the electron or the

normal quarks in the proton are their supersymstetric partners. The associated

production o* sca?er ?epton$ and quarks

t + <j-»--e + 5 or e + q •* v + 5

proceeds through the exchange of a spin 1/2 f i e l d quantum [y, Z or W). The

salar e lect ron decays in to an electron and a phot ino, the scalar quark in to a

puar.k and i gluine or photino.

S1n.ce both "f and Q f t i n v i s i b l e the observed f i n a l state in the react ion

e •» t| - e * Q is the same «s in normal neutral current events: a s ingle

electron t r *ck and » j e t of hacrons. Due to the momentum and energy car r ied ri

aviay .by the pnotino and gluino these supersyronet>-ic events w i l l have 6 large :%

irabalanre in transverse momentum and the e lec t ion and the hadron j e t w i l l not ;:jf|

appear back-to-bacb wneu viewetl along the beera d i r ec t i on . By sui table cuts in •• ;§•

coDlanarity anc transverse nomer.tum these exotic events car be very well V |

separa'.ed frGrp conventional neutral current interactions. ' J,;*

5S3

The reaction e 4 q -» v + q is much more difficult to identify experimentally

since the nuino will in general decay into invisible particles. The final

state therefore contains just a single jet of hadrons and cannot be distin-

guished from the-much more numerous charged-current events e + q-> v *q' at

lower Q . There is, however, the possibility of more complicated decay modes

of the nuino leading to multilepton or mixed lepton and naaror. final states.

In that case the events e + q - ^ - v + q w i l l be unmistakable.

The rate for associated production of scalar electrons and quarks has been

computed by Jones and Llewellyn-Smith [8] and by Altarelli, Mele and Ruck!

[1]-Fig. 8 shows the cross section at /? = 314 GeV as a function of the e and q

masses. Almost 10 events per day are expected for m~ = m~ * 40 GeV.

Fig. 8:The cross section forthe associated productionof scalar electrons andquarks t t •? * 314 GeVas a function of the eana $ masses.

20 40 60 80 100 120 !40 160

3. The HERA Machine

3.1 Layout and general description

The HERA collider wi1" be mounted in an underground tunne! of 6236 m circum-

ference and 5.2 m diameter. !T. consists of four 90° arcs and four 360 m icr.a

straigtit sections with the exaerimenial areas. The electron ring oas a nominal

energy of 30 GeV and is equipped with conventional iron magnets. T>ie tiipcle

magnets are excited by a single aluminum current bar. The vacuum chamber is

made from 6 copper alloy and reinforced with lead to contain the synchrotron

radiation. lor, cetter sumps are integrated in the beam pipe of the dipoie and

quadrupole magnets. The electron Rf system will be taken over from PETRA. For

energies above 26 GeV e system with 500 MHz superconducting cavities is

envisaged.

The proton storage ring consists cf superconducting dipole 2no quadrupole

magnets connected in se'iss. The nominal proton energy of 320 Gev' requires o

dipole field of A.64 T and a quadrupole gradient of 90 T/m. The beam pipe is

at liquid helium temperature and will be copper plated on the inside to reduce

the ohmic heating due te image currents accompenyino the proto.i bunches. 7he

quadrupole *mj sextupole correction magnets as well as the-correction diroles

are also Superconducting. The superconducting coils are cooled.!?/ single phase

fceliuw wfcich traverses one octant, is then'expanded through a Joule-Tr.omscn

valve and is returned to the refrigerator as a two-phase mixture. In the

stipole magnets there is heat exchange Between the single and two-phase

systems. The radiation shields in the cryostats «re cooled by helium gas cf

40-80 K.

The 8F in HERA will De 52 MH2 while the 2i0 bunches a>-e injectec fron FETRA.

After injection the voltage of a 208 MHz system is raised and the protons are

accelerated. The bunch length at 820 GeV is 35 cm.

The injection scheme is shown in Fig. 9. Seve-al stages of acceleration are ' .

ne«d«! before the electrons or protons can be injected into HiRA: separate ;

linear accelerators and synchrotrons and the modified PETRA storage ring to j

accelerate electrons to K GeV and protons to 4C GeV. The proton injection I

chain starts with negative hydrogen ions which are produced in an 18 keV ion |

source. A radio freouency ouao'upole accelerates the h" beam to 7?C KeV an- ','•

focuses and benches it at the same time. The next step is a 50 KeV linear ac-

595

celerator. From here the H" ions are injectsa into the rebuilt synchrotron

(OESY III), stripped, and the protons are then accelerated to 7 GeV/c. Due to

the statistical nature" of the stripping process, Lioui/ille's theorem does pot

apply and the proton intensity in the synchrotron is nnt limited by the phase

space of the preaccelerators. The bunch spacing in DESY III and PETRA is

28.8 m like the final spacing in HERA. The estimated filling tirr.s is 20 rain.

3.2 Interaction regions and polarization

In the present design the electron and proton bearcs collide head-on because

particle tracking calculations [9] have shown that witn a crossing angle of

more than a few milliradians synchro-betatron oscillations would be excited

when the electron bunches penetrate the much longer proton bunches. The zero

crossing angle geometry requires that the electron ana proton machines have a

number of dipole and quadrupole magnets in common. The electrons are bent away

from the protons by an angle of 10 mrad in the horizontal plane by means of

displaced quadruples and a dipole magnet. The following proton quadrupoles

and dipoles in this region have to be normal Iron magnets because the

synchrotron radiation generated in the 10 mrad bends woulfl put an intolerable

head load on the cryogenic system of superconducting magnets. This radistion

may also cause a severe background 1n the experiments. It seems possible,

however, to arrange colliraators and shielding in such » vtv that the

synchrotron radiation can enter the centra! tracking chambers tyf the detectors

only after double scattering. The resulting flux of photons with energies

above 50 keV has been estimated [10 J tc be below 10 per second which 1s a

factor of 10 lower than the flux observed in the TASSO drift chamber at PETRA.

Considerable efforts are needed to achieve longitudinal electron polarization

in the interaction regions. The emission of synchrotron radiation leads to a

natural transverse polarization which is huilt up witn a time constant of

25 min at 30 GeV ana may reach a theoretical maximum of 92.4 %. 'he machine

has to be very well aligned for this purpose and depclarizing resonances have

to be avoided. The spin can be rotated into the longitudinal direction cy a

magnetic deflection. For ultrarelativistic electrons, fr.e spin precession

frequency can be much higher than the cyclotron frecuency, the difference

frequency being given by Aw - l/2y (g-21^.. r, with v = E/ir c aid g the Lands

factor of the electron. At 30 GtV a deflection by i.32" is sufficient to

Fig. 9:

Tne HERA

infection

38

verticaf plane 0.66 negative heiicity

H q . 10: Schematic view of the spin rotator in fr-jjnt nf the mtsract icn point.Ths graphs 61. the top and in the middle indicate ttie deflections of theelectron Srotr. in the vertir.s" and horizontal Diane, ftf trip ttozxo^ the c ie r i ta -t ion of the electron spin r t i a tu 'e tr- tne horiztmtnl plane is snowr:. ; similararrangement oehind the interaction point restores tne transve-se nrientatior,in t.Ne iTci.

597

rotate the spin by yo°. A "spin rotator" has been developed [11] wnich b.y

successive vertical and horizontal deflections allows to obtain electrons with

either positive or negative heTirity in aT 1 interaction regions. The

principle is explained in Fig. 10. Particle tracking calculations [12] with a

linear optics program indicate that a degree of longitudinal polarization cf

80 S seems feasible, kith only slight mechanical adjustments of the magnets

the spin rotator can cover the energy range front 27.5 to 35 GeV. The 10 mrcd

bend near the interaction point is part of the spin rotator.

3.3 Superconducting magnets

When the superconducting magnet program was in i t ia ted at DESY several >ears

ago i t was decided to follow closely the successful design of the FMAL

Tevatron magnets. A dipole magnet with warm iron yoke has been developed at

DESY and ten 6 m long units have been bu i l t with toe? ing suitable for series

production. Al l magnets tested so far have exceeded the nominal f i e l d of

4.53 T and have showR good f i e l d homogeneity [13] . Parallel to the DESY

program an industr ial company {BBC Mannheim) has designed and constructed

three 6 m long dipoles with a cold iron yoke which d i rect ly clamps the

,.,. superccnducting co i l s . The f i r s t magnet has been tested and has reached a

f i e l d of 5.75 T. Both magnet types nave thei r r e s t i v e virtues and drawbacks.

The warm iron dipole offers good f i e la qual i ty without any iron saturation but

has a f a i r l y large heat f lux into the cryogenic system because the coi l has to

be centered inside the yoke by many supports. The stat ic heat load in tne cold

iron dipole can be made much smaller because only fe\» supports are needed artd

in addition about one tn i re of the superconductor can be saved but this

magnets suffers from large f i e l d distort ions due to iron saturation afcove 4.5

Tesla.

Stimulated by an analysis [ l 4 j of quench safety aspects in tne HERA -ing i t

s has been proposed [ l b ] to consider a magnet which combines the positive[I

'i features of both concepts and avoids roost of tiie drawbacks. In this new type

| of dipole the coil of the DESY magnet, collared with aluminum clastps, is

I surrounded with a cold iron yoke and a cryostat similar to thot of the BBC

'.} . magnet. In the meantime, the design of this "HERA flipoie" has been completed,

jj two 1 m long units have been built and tester! and four full size magnets of

!, 9 m magnetic length are close to completion. A cross section of the magnet 1s

Si shown in Fig. 11. The coil consists of two current shea's with longitudinal

I wedges for improved field homogeneity. It is wounc from a Ruther'orc type

596

cable witK Ik stranas and copoer to superconcuctor ratio of 1.3. Trie aiaiwter

of the -NbTi filansntL. is 14 jm ano the critical current st 5.5 Tes'a anc

4.0 K exceeds 0000 A. "he coil ij clamper by p^tc -. sic-blanked alumnum

collars which define the e*acT geoTit".ry ?nd tans VIP the bus? Lorertz forces.

The collars have four noses which slide into notches in the iron yoi.e and

thereby provide an accurate alignment. The yo'e laminations jrf surrjurcec b.v

a stainless steel tube wnich serves as the liauic! heliun, container anc at tr.e

same time defines the curvature of the magnet with a radius of 530 IT. The roid

part has to be supported only at th"-ee location: over the length of 9 r. The

computed static heat flux is 2.4 W into tne liquid nelium vessel sr.d c\ 'A "nto

the radiation shield.

The new magnet concept has several obvious advantage;:

- compared to the warm iron dipole one gains 12 % in the central field but

saturation effects are very small even at 6 T

- the static head load is low

- a passive quench protection by means of parallel diodes is possible.

One disadvantage which is present for any type of cold iron magnet is of

course tne large cold mass. Detailed calculations [l?j have shown, however,

that the cool down or warm up to « whole HERA octant could be done in about 40

hours.

The test results il6j obtained with the 1 m long prototype oagnets are

extremely encouring. Botte magnets exceeded tne nominal current of 4990 A at

the first excitation and reouirea just one training step to arrive at the

critical current of the superconductor. The higher harmonics measured et trie

nominal field are plotted in Fig. 12. They ai-e in general lesi than lxlC""1 ar.d

thus well within the allowed limits. Fig. 13 shows the current dependence ot

the normal sextupole coefficient by Besides the well-known hysteresis at low

current which is causeo by persistent eddy currents in the NbTi îiarrenii a

definite curvature is observed at la"ge excitation. This indicates s slight

saturation of the T~or yo*e. These systematic sextupole fielos will be

minimized in future magnets by some additional shimming of the coils and the

remaining contributions can be compensates using the sextuple correction

coils.

Dipoie cross section. 1 super-in-

suUling 2 forward and return

bus - : two pnase nei'uit. - 4

collars - 5 coil - 6 Mffl tube

with correction coiU - 7

radiation shield - 6 vacuum

container - 9 shield cooling

tube - 10 one phase helium - 11

yoke

39042

Fig. 11: Cross section of the superconducting HERA dipoie nwgnet. The coi l is

clamped by an aluminum col lar and then surrounded t>x a cold iron yoke. A de-

ta i led description Is given In Ref. 16.

Superconducting quadruped e magnets have been designed end bu i l t at Saclay

[18]. Two warn iron quadrupoles have reached a gradient of 120 T/H>, well

above the design value ot 90 T/n. Recently, quadrupole magnets with stainless

steel col lars and a cold iron yoke have been constructed and have been excited

to 120 T/ra without quench. These quadrupoi«s are well matched to the H£RA

cold iron dipoles.

The quadrupole and sextupole correction coi ls are 6 m lon$ and are mounted

on the cold beam pipe inside the main dipcles. Several prototypes have been

bu i l t by Dutch industry. The measured quencn currents in an external f i e l d

of 5T were a factor of 2.5 above the operating current. The dipoie for orb i t

correction is a 60 cm long "superterric" C3gnet with iaminatec irof. yol:e

and two saddle-shaped superconducting coi'.s. The f i r s t prototype reached

2.3 T compared to the required vs)ue of 1.4 T.

60C

•10'* t a n

A

3

2 -

1 -

0

-1

-2 \-

3

5 6 7 X 8 9

•KP

A

3

2

1

0

• b n

•2 -

-3

-i,

allowed limits

T i * ¥ r-2 2 4V 5 6 7 8'

Fig. 12: The measured higher multipole fields at a radius of 25 ma, normalizedto the efipole f ie id. 'of the two i m long prototype d'poles. The allowed limits»re 1 6*10-4 for the sextupo1* coefficient b, and i 2.5x10 -1 for all othermuHipote coefficients. The normal (skew) muUipoles are denoted by b (a }.

30

28

-'0

20

30

1•4

1

i

*

\

\

1000>

- /

71

2000 3000

*ar v—40C0 5000 6000

820 GeV

7000 A

I

Fig. 13:

The sextupole coefficent of

a 1 ft ionj prototype dipole

*i a function of the

current in the coil. The

continous curve is obtained

tffien the current in the

magnet is increased, the

dashed curve when the

current is decreased. The

hysteresis is caused by

eddy currents in the

superconducting NbTi

filaments.

601

A detailed report on the present status of the HERA collider can be found in

Ref. 19. The work reported here is being carried out by many people, both at

DESY and at other institutions. I want to thank my colleagues for many inter-

esting and stimulating discussions.

References

[ l j G. Al tare l l i , B. Meie and R. Riickl, CERN-TH.3932/81 (1984)

[Z] Discussion Meeting "Physics with ep colliders in vie* of HERA",

Wuppertal (1981)

[3] G. Wolf, DESY 85-052 (1985)

[4] E.J. Eichten, K.O. Lane and M.E. Peskin, Phys. Rev. Lett. 50 (19831,811

[5] R. Riickl, Nucl. Phys. B234 (1984), 91

[6] H. Steiner, XXII International Conference on High Energy Physics,

Leipzig (1934)

[7] Proceedings of the Workshop: Experimentation at HERA, Amsterdam (1983)

DESY HERA 83-20 (1983)

[8] S.K. Jones and C.H. Llewellyn-Smith, Hucl. Ptiys. B217 (1983), 145

[9] A. Piwinski, Uth Particle Accelerator Conference, Vancouver (1985)

[10] B. Foster, private comnunication

W. Bartel et a l . , DESY HERA 85-15 (1985)

[11] 0. Buon and K. Steffen, DEST HERA 85-09 (1985)

[12] 0. Barber, private communication

[13] S. Horlitz, H. Kaiser, S. Knust, K.-H. Bess, S. Wolff, P. SchnUser an«)

B.H. Wiik, Journal d* Physique Cl-255 and Cl-?59 (1984)

[14] K.H. Hess and P. SchmJser, DESY internal note (Oan. 1984)

[15] K. Balewski, H. Kaiser and P. Schfuiiser, BESY internal note (Feb. 1984)

[16] S. Wolff, Invited Talk, 9th International Conference on Magnet

Technology, Zurich (1985)

[17] H. Lierl and P. Schmiiser, DESY HERA 84-16 (1984)

and G. Horlitz, H. Lierl am) P. Schmiiser, contribution to the Cryogenic

Engineering Conference, Cambridge, Mass. (1985)

[18] R. Auzolle, A. Patoux, J. Perot and J.M. Rif f let , J. de Phys. Cl-?63

(1984)

[19] B.H. Wiik, Invited Talk, 11th Particle Accelerator Conference, Vzncouver

(1985), OESY HERA 85-16 (1965).

603

cv ;;:':n'-iCTio:> MC-'-V-JL USI::G

ccs;:ic ravs i:.'

University of Lodz,ljeyxirtineiit of Physics,

2j6 Lcdz, ul Xovotki 1^9-153,

J.jTdovczyk

Insti tute of Nuclear Scicace

90-950 LoJs,ul Uniwersytcclsa

Poland

Abstjact» Tha snsrjy spectra of high enerjy

rsuoTis and hadrons are compared with the px-e-

UictionE base:-! on the ti>-o coinoonent noiiel oi"

the Galactic cosrîc rays. Sxceient agresfflcnt

of" ih«s •TC'-iot.ions and <3Speri;.:ontaI ' ' i u is

f cmnd.

Tni: reduction

Inforsia.ion on high, energ;' interactions und properties of

cosmic rays at very "tiijjft orarnius are derived comparing c;:-

norimcni.'il data cor.cerriinj cosmic rays in atrao.feoboi-c '.;i*.h

predict.xojis. The predictione i:avc to be oassd on det-iiicci

calculations of cosmic ray propaj-aticn in s.tr.osp:\crc for LZ-.O

assuiscd .-nodcl of hijh saorgy intorS-cticTr. In ronorcl those

pi-e<Iictions iiavo to be obtained for various asEirnotior.t? ab-

out tlie properties of hisii encrsj' intcraotiojis anJ about the

prirjary cosmic ra;*s. In particular verj- important jar iMier

of the prir.sry cosmic rays is their fcass composition.

In order to reduce tho possible nodels of high er»3rt- ' in-

teractions tire sensible ar>pro.-ich is to construct a model

describiri^ -- s veil as possible the availatic experir->cn;;nl

data at the accelerator energies and then to use various

hypothasss to estrapolate the mcdol to bic'ior coc-iic .-ay

ensrjies.

SCK

In jon-:?:';-2 there c::ist ?ov'r'ir.I cosrac j'ciy T/hcvjontnri j :i

Et.:icsthc;c •••hioh can be ; r.v< £ c i •;-_-. too rnr. C3r.;j:ivcl '..'i th the

predictions < Here can be indioaiod cs ±~i <• j-.-.wnt such ;i"C -

uoniena c--3 extensive a i r shov-crs,^' - ray i'an-iilvcs at iv-ious

depths in the atmosphere and the unc-rcT spectra of si;:^lo

p&rticlos .

In ths present vork we have conco!rt,:-a-!-ori on vc-r;' C..i,r,i.lcd

calculation of the spectra of nrjons and soccndc-.iy jfi;5:-cns

in the atmosphere. There exist a vo;ijth of the cicta or. hi^h

enc-ro' iDuoriS at various zenithal angles and z-ouily cotailod

ccir.narison vith prt-clictions can be r.:.-.c!c. TIic dzts. >v hae'r-

ens, on the oilinr hand are relat ively directly r"latic! to

ibe prin^jn- nucl&ons.

In dsrivinj-: of vht niocol in addition io various ir,."or.i.".-

iioiis or, l;a;iron nucli'oa interaction ve have to tf»\e imo

account also the fact riicr in the atno3r.*;orr. the rar;-i!t is

rit>t a proton but l i^ht iv^cr. eus (K,o). -""it- pvcsoni od here

TGIHI2ZS : r» the f i r s t esiontive calcul2.ti.cns «or romco to.-

;;;:^£ into account the G:"i"eci.£ of the nuclear ti>r;;et» This

effeci, .rc-iei-ally foi nn,-or:5 calc-Cations i s not van- s t r -

onr., but a? ic v i l l be £6en zurtber this 2ca<£ to zltjt-.r ir.—

provc.-jtsnv ai tiie ct rsc-r.e".* of tlio cJiporinc-rx.al c!~ ts triti

prodic:tio"s» Fnr ^rcdicrior.s bnscd on the s« .£ r..oclwl 'cui'

without r-jciear tci-'.-j set- i;e—.tia cr.cl a<io'..;:yk v •'"•-r'."• j • i'--e

7-e^scn fcr rather s--.:ill efteci af the ru-clc&r ta^-jci co;..c-r

rroni xjic f^ct tjist the r.uca flu:-:es to isr.^c- zr.ivr.i r.rp do-

te I-.-.incc by the- tr.csx enor2Ciic sccordaries vhich arc le^st

efXi.'Ctod ty vl:t auclsar tar.jot.

I t h f- been ciwtr.;:fraT;od ty .io'.-ciy;: ar:c' Volfc::-.'..ilc', l^rZU)

ji - p i-t^rcc:io:-J vitli c-.cr^-ies uii to u re-..- ti:::cs 10^ C-sV

c.-n tie de&erib'-'C ^-cll by c: scaling vic la t in" formila ol tbo

X o ; " at w _ . _ ^ . ^ ^ ^ _ ^ % k::-t,2C5i

*- r"

^ hero r.he raraîvtor £\ , vhich is « r.:c:.s^r5 of iih.e rco?.i.r.c:

605

violation, i s s^dually increasing with energy. TJie coef-

ficient IJ(E ,2C5) = *v(E ) / K ( 2 O 5 ) describes tne variation

of inelasticity coefficient for charged pion production.

The energy spectra of the others secondary particles can

bo obtained by similar transformations.

For purpose of numerical calculations th« function q(at)

is usually talcen from Fermilab data at 205 GeV *nd appro—

xiirated by the following formula { for IT + n )

qU) = .692<1-x)7'4 exp(2.IS5x) + , S 0 7 ( 1 - X ) ~ * 2 axp{2.%3x),

(2)

wliich is obtained by integration ovor p. of the more ge -

rjcral expression fitting 3imultaoeously the x and p . va -

riation for tb« same Fermilab data at 205 GoV ( EalUsa et

al 1977 ). The expression has the fora

^ . (3)

As a. reasonable cljoice for o< and '.c^ variation i t i s ta'ketn

c\ a .: :15((iio/-05)*'35 - 1) lnf£o/2C5)-, for ii^".? 10* O«V

c<= .072 + .0163 ln(Ba/2C3), for SQ>7.7 to'* GeV. («)

'*B±(BO,2O5) = (B

It i s also assauiad that forntula ( i ) ana (2) oaft be uset!

for describing the energy spectra of Iza-ons any baryoac

( antiba-ryons } . All those ars tahen tofie-tlier but rela-

tive role of 'baryo-ns and antiburyons increases with pri-

mary energy according to the expression

606

r = i - .~:i,:j:^r-n:i:i . (O

I t i s assumed thr.f. the avjvcije energy of Icrior.s and bary -

ons { antibaryons ) increases wii;i fnci~- fas tor t'mra thct

of 'jior.s and the relat ive in e last ic i ty cc.rf "tcicnt for ;.rj>

ctuetion of those also increases viiî ranrjy. The p-:>res',ion

giving the ra t io of the part icles Everu.ju oncrjj- and their

ine las t ic i ty coefficient to ti-.osc or ;;ions arc respectively

.233

'Clio interactions of neoons ;-.'ith protons are also asswod

to be described by tfco sane o:;rreFs:.ors fsccnt that the

tota l inelas t ic i ty coefTicient is talcen to l>e 0.(5 i.nstod

of 0.3 with total multiplicity far, &atr:o anc avcra.-o rntr 'T

inercu:c<c" Ivi- facior of I . i .

T-akin<* iiito iecownt a l l t'jvso assunij-jiioijs forrrjlr- (1) is

-'-c into tile fajlowitsj; e?:r;possions for '::aon r.nd

plyE antilfraryqn production in pfoton intc-roctioia

pip" j:r*.cii»otion in pion interactions

K', l.(~ ,20'}) ) 2

xp:'o;:s.ion for the

tjie "b£ij"yoi«i plus a-ntibarj-ens .produced in

the rpr;*fen i.nt&csotiwjK ,

ienf ficseri pt'i oft &f ;;ie nuciciir t.;xrjrft vc have f.fsu: eO

f-:-at Hyp int.Kraetion <oi~ Pacsron with nuci«n.is can b« dose -

rV^CC as a syiorj-ositioji pf tjTC sul>'s^nucnt interaction cf

tfee ledein^ 'naur*6n isoc.lilias ot ;.-.l 1?oO). The nucleon

int<$-imi;±tig vltii TU*CJ.BWB i* aS-avmrn? to TjacJra -jto ••• r] ccr--

toitt J5j-o.ba't?ilit.i*i£ en* ,tvo 6r a r c intori>ctio::s with

:iiiel.tf«ne in *}i<? riwclp!*?. 'Aie probabili t ies for tjvp n«n -

>r oX infer&CTi0n* usefi py x»s aro c i v e " in tht.- tc'.-ic I .

feijorjjj- af-«»ctnjni o-f t'J:e socortSarios and the inc las t i -

607

Table I

probability of t.n inciciam protor. .nx-vi.nz n collisions

within, a carbon nucleus

.013

city coefficient d i s t r ibu t ion the case of one in te rac t ion

are the sane as for interactions with protons. *.,"Uen ttit

:iurcbor of interactions is highor those distributions aw

obtained by sunning the contribution from subsequent int-

eractions.

l.'hen: Kj is tlie i n e l a s t i c i t y ccvefficicnt in t"r.£ i - th

sabscqucnt i n i c r ac t ioa .

In s i v i i a r l.-ay che inelasiti c i ty coefficient fpr r, -

nucleus ;."lsr!iCticE ea=n b# obtained. Overall S'lccrrsi of

socondar;' yo.rticlss ea.i: be &r.sily o-i^taincd adding the

contributions fror i-arious mi.fter o*~ ittxeractio-n^ with

probabi l i t ies given in table X. T?»a dictributj.f>7.' of i.e.

i n e l a s t i c i t y cocrf^cient Tor 7 - A tollxai.vr car. bo •»: -

ajn.-d by 'lento Carlo oliovirip t;-i.<- s.ubf osi'i-c: . r-.^cla-st . 'ci-

ti.os ir. p - nuclear: i.Ttornctiorit to f J ue -.; t:a t o i M c W ; ^ -

to tbr iissur.',ec fôet-uotiorss a." '.Yu> ;.aelast;eiv,v 5. nc-rri.-

eicnt ir. ;-. - p intercici Sens.

spectra oT tlio soconciari GE can be well rcy.i'.:Qe:rt ^(' bv

vp.r" sim-ilc not i f ica t ion o." the «^;3rossioTj ( 1 ; ano cils

the expressions i.6) a:i<i (?, ' . Vtoe raa-Ji f i'-"U fr.rault ( ! i

'acconos

608

The inoiriStirity coefficient --istribuvion calculc; ed oy

the outlined abovs .-r.et'icc! is conoisi-c-rit vith en e;:".rcnsion

v'-'ich i.'o have dwvfclopoC \ ;i.c:.v.- and :rdni:c::vk TjSj ) on the

basis of oxperin'jial ci.;ta corciulet; by OoncE (I9"3j«

The expression is

f or K

T]-to prig*:-,1 rgxrticlc snectra end mr.s co-:>cs:ricr.

Xhje spectra and the pass conrorition oi" tlio ;j,-i nar;1 par-

ticles for the present calculations vore tr.V.pn acccrdin-

tc< the two component raodol of t?ie Galactic coer.ic rr.j's •

/tccording to that motlol the intej^r^l onrrj-/ s-cctra of the

prirtsrv particles oi1 differect nasses are described by

s" sr »nd crtcrc:-- in G'>.'.'/ niicleus

•A C I).

P 1.72 10'* 3.23 TO"'

c:x -:.r. to- n.:7 io t ;

is inte-rcl sj".uctrar' oi" 'h? ncdit! onsl proton com;-oncrt

609

• Ij.th ciurr?1 iT.ua.i." i " the

The energy spectra of cosnic ray tnuons are .-noasured by

several different tr.ef.iods. The most dvrect method is that

based on tho use of raa^notic spectroscopes. That method

hovevcr ccinnot be extended to very bich energies where the

nuon intensities are low. At those energies the muon spo-

ctra are obtained either from the depth intensity relation

or on the basis of the nmon interaction in the burst dete-

ctors. The most poverful method is that based on the conv-

ortin^ the depth intensity curve into the energy spectrum.

That convortion is however rather complicated, in particu-

lar due to the fluctuations in energy losses.

In the present vorl: the coapErisoii is nude separately

for the sy.octra derived at sea level ( mri-cnotic spectrosc-

opes anci the burst measurements } and for the spectrum ob-

taiuoci from the depth intensity relation.

The soa level measurements are presented and compared

vlth the- jiredictions in fi-c* * .The cdm-.iariison is tnrsdc- hnt'n

for vertical spectrum and Tor two msar horizontal angles

79° and 3&,SC . Tlie Spectra at dirfercrit an^los differs

significantly for variety of reasons. First of a l l the

produced muon intensities are Sir-nificantly bici'»or at ).ap;a

anjlcs. The increas-t? at BKVderete zenithal aaâos Sis j-.rpjj-.

ortional to sec 3 . At tlte angles close to ?0e triis <Jc;i-.crrtv;-

ence breaks doiv-n siftce the aagj? at v'i'.ich tlje nnipn pnters

atnosphei-e is smaller that th<i ohsort'atian aR-ls at sea 1c1-

vol due to the curvature of itteeiyherc, On the o-tTier h: h*i

the muo;i losses Whicb are quitp iftportant-for th« lcvor

nmon f-ncrcicE 100 - JOOO GeV strongly derieilti on the zeni*-

hc-1 ar»gle as the thieluioss of S32-& atiaos'j>Ijer traversed str-

ongly changes. Tit* losses ar« drae both t« $Jie decrease of

the auons enersic-s and to tJieir decays* The calculations

are performed, for two different assumptions about priniaiy

:SSB composition. As we can see there exist dxpelont sgre—

ecictit of tjje observed won fluxes with predictions ">a&ed

on the assumptions about mass composition and high energy

interaction model preferred in the present paper. Cn the

310

other. n&nU xh; -rc-clistior. basac- on the sr.nc hi;;;, Ir.torac -

tion roocica but aisuninn i'-cuvy c-ni-iclaou ir/sss .-.omrositian (

Tan 9t a l 19S2 ) are in c l tar dlaa-roerecnt v:i ;1. tho exueri-

10

TO*

Ti;-. 1 thi 4v-on trscrg;-- S-r'ctrfc. SM! variou-s" angles. 'JScfrnfisuos li.*jfa a-r«. *:«• ppsuiictioas baSod en tbc tvoesi---&rcn* Toi-ei, i"r <:ot,t«fi 3,ip-o*'are tijc ;s;ro2icfcio;

c ^ u T O ^ S t i y i * of «1 (t^J:-^) } x J4e-2t - J3csj-. JcIcisVi aei i'ii'T9i jm *;cn ...;.I*Kofer r t a l v1pSi) {* Ka.eI,Ali:<o.rcr ct a i ( i-97?) ; *. ;;oscov,ia-.J-oTiov (157T) i - iurra-a

). c r ( 9 7 )Tlifr : . - . ; .s t i r . t--.: a l ( > 5 7 T ) .

611

ste •wfab u s o .Deptti in hg cm"*K6f ruck

Fig, 2 Depth intensity curve { oa^fri.T'.orJtai vorie <3*>.tarecalculated so the lioXar Golf" Field reel- after Kriahs-asvoiny igsi j foi- shallow depths . iixe eofttinuoB 'li.no*3 are the pretlictioija based osft tlae tve — c crnj.emeE-tol» the cot tod 1-inps A a*-e slip prcSic:-ci'ons -fVr th«vy enriched spectrum { tan et ai YS33 ) ,

The procticted aad observed d«pt% j-Btonsity depeiioon<;es fpr

Fh-o dii'Ierowt ranges of tUe ApptlM **•© SII«WB tn fijpii-e« 2 anfi

3. Figure 2 concerns sij»illo*' aepilt laverRKS trtt figure 5 *l*o

great dopth up te highest a* ftur eb*erv«d. 3En figiirc 3 tt*er*

are indicated rauon eaergie* carrBSjwridiag to ««• -wajri iiS d«-

Vths, I t again caii be seen titat the agr<»«ir.c'n-t under tb« ass-

Umptiona preferred by *bc present autbore la vcrs' sood.

The goncral consistency of ta* data with the predictions

indicates that tiio assurajstians ab«ut prisar;- OTISS c.oaposi -

tion and properties of high energy ir.tcrstctiena are close to

the real i ty .

o12

10

'Dwirth it\ hg ,c«" 2 K©F reek

.":.£. D Tlit- sa.no as in figure 3 but for ifct-depttfi.

I t iics been uCiiWn at rated by KS ( 'Cerapa and Vdovcsvl:

that 'he fluxes of Ijadrons in the £tn©sj>horfc can well be li

rib-ed usinc tl» ciscussefi here siass coir.pcsition und I»ic5i ener-

gy interacti.ua nocei. Here we vould like to restr ic t our e t to-

rTior) tp cortEin class of hacroris observed ir:- the atmosphere .

Vl;o£p ar t iiadrens ( prcttat ) ".shich can be- reasonably called

-urvivSnc from tive top of atnospiicro t;o fjivon cc.-5t3- vii;icxut

inicracticij. itosii bodrosi-E eST>erir.^t.si:iy or; irccrî.ricd by

ti'c f i-i- -'::'-J- vw'V a r * '-f14 aceor-j-nrsieci bv try sc-onti.-ry v^rta-

S13

I'i-r. ' TUe intosral oner;.;:' spectra of the.T-otens at different XevoXs of th« o6s«rvafio:i:n ;:t Cliacal i;aya 5200nfian<?Jco e>t al {T-971) j»otiliim 3j30o,G.\iss«r *t al (lJ77},:'«o et a l ' (';•%•;. J.^mct al \157>} ; « Aracsse 3200s;,ftri-orov rt. r;l ri.^rttrull linns - predictiecJ spectra of the aurvix-ia~ pro-tone for considereii altitvutes \ noto that th« Cliaca-ltaya ir.tonsitica are ratr-Ttiwlicd ^yy factor 100 Einrithe Tien £;KIR by 1C for t.ae bct-ior see- ).

clos.The observed at various cicr-ths the fliusos of the

ing nucloons are given in fisurp il. At that fi.ûrc there is

also sUovn tUe primary jiroton s-pnetrau and the sijoctal Tl-

uxes o/ the surviving protons ca lcula te aiwier the ssourtp -

tioTi thet the protoij Mean froe path in atmosphere varies

with racrsy as

614

Vor2' jjooo ajroencnt' of the predictions with observationsis seen.

Conclusions

;.e havn dononstratod here that the flux«s of mions ir. atrao-spherc can be well described a-ssusning- ti.e scaling violatingmodal of liipfc energy interaction and the tvo component jiictu-r« of the Galactic cosmic ray ©ricin, *t nay be s t i i l possj. -ble to giat aa aereeaient for different combination of tjj; ass-uaptioriE about the hisli energy invcrractions and primary part-icle mass composition "oat i t appears that that -..'ould lend tocontradiction with observations of different ntirEr.caers ofcosmic rays.

For instance if ve a.£.sv?Uiut the neeoed intensitiesof titc priniario* vitfa various passes woulc bo in c3.ca.r contra-

various ciroct

0 C trt a l tS!?1 i*roc»12tfc IS:.C,K©fcart,7i p . ' 3

1' S et. u i TJSi* lV<»c.Xnt.iyiBp. on Cos-nic Hcys andi-ar-îclcp Mr); s . , •'•olryo,p.*»<><?

i-'Jias, ^ L e* a l 15fiO J«iys ri.av.D, f?. 13Gj.is.ser T fit a l H^T? J.r-hyS.(if', ' 'j>cl.;hys. ,3,1.242G.'icordv I." i e t a l Ig63 'iusfjeent G o r f . v o l . 1 "o 1 p .?2JoUisl: K c t a i IS"? I'Viys.rcV.ij, 15, ;:. 13it-.ione's L •.' 1i?^3 l-rc.&cntcd so tho lot;- lCKCâi'ifralorcivafi:a T Of a l 1977 l'h;-s.. ev.D, I 6 , p . V-'t 1ii;ir>s'.;o T e t e l Jg74 I ' roc.12tb ICIiC.I-oin&rt, 7,p.2759I'CTBi-A J and ;;<;cr.-ezyk J 1C6T J . rhys .ClNucl . : -hys . ,«? , j>, 1271

_>—^_»_— ___ 19S5 11. NlioVO ZX.^KTV i n p r e s s

j j irciwv B A 1577 Froe,i5t>h I C C ,1 lovUiv, 10,p.2i '2

asi.-a!i^' ;-i — l<?51 1T-.&,Thesis ,Universi ty of Corabaysi et cil 1^54 i 'T^c^ .ov . i , ?^ , p . 1

615

N.int i: A t?t a l 1975 Iroc.i^lth Icr.C,.":unchen, ~, 0.225S

J933 Iroc.JSth ICiiC,i:angalore,5,p.336

Tan V K ct al IPS." ..eport of tb.c Univorsity of Tokyo 99822

Thompson G ot -.-.I ?97~ i^rnc. 15th ICKC,Plovdiv,S,o.2 1

'..dovcsvk J and Volfondale A '., tg£4 J.Fhys.G:-uci.F!r."s,, 10,

P.25T

617

MAGNETIC rtON0POL.ES IN KALUZA KLEIN THEORY

H. I. Sundaresan

Deportment of Physics

Carleton University

Ottawa, Canada K1S5B6

and

K. Tanaka

Department of Physics

tbe Ohio State ttniv»r«itj

Colwabttc, Ohio 432U

Attract

We s t a r t wltt) an introduction t» gH^netic acmapca^s tm4 th^n d i s c u s s the

majnetlc aoaopoXes l a 5-dlasnslon*, £tw * u * m t y at s o l u t i o n wijjj rcapact t o

small change* In tae a a t r l c , meA f l n a l l j eat wttk

615

I. Introduction

Schvarzschild found the solution to Einstein's equation

R » 0, (l)

for a metric around a fixed spherically symetr lcal center of mass

<Js2 = - V"1 a t 2 + Vdr2 • r 2 (U62 * sinZ9di|jZ> • g d x ' W

with (2)

X° « t x1 « r x 2 « S K3 « » v""1 < 1 - (2«/r)

HiCB) « 1.3 X IO"52

Is the solution stas>l« with respect t© K»»ll first order departures froa the

SetwarassJiiM ustrlc? To examine thlj 4UAttlon, the Iwckgrouna me trie is

lwticated by ( Jtni the ssall iierturtatlan to i t with h £ . The contracted

fticcl tensor i* e«lled 9 If s«icui«t#« frpn i ^ and I u + 4B If

Oelcul»ts< frc* £ • ft where iK is

«t • - W* . • *r* (3)

^* V;*" V;»>

*»o,o

eie

In order to check the stability of t?ie solution, the metric tensor h

is separated into four factors each a function of a single variable or

coordinate- The separation Is obtained by flegge and Wheeler by generalizing

the spherical harmonics (known for vectors, scalar*, and splnors) to

tensors). The stability of the Schwarzsehiid solution Is studied by

taking " 4R = 0 and looking for diverging solution with a factor of the

form F (t) = e" . They found for various angular momentum states that

IB* = 0 and concluded that the solutions are stable.

II. Magnetic Honopole In Five Dimensions

The magnetic monopole solution ' Is a generalization of the Euclidean

Taut - HUT solution descrlDed Djr the aeCrlc g in 5 dimensions

A,B « 0, 1, 2, S, 5

(5)

vt»re X* » U . r, », «, Jf5)

e c » e-1 ^ \ * /g + Afttr ftV\ (6!

8 r V 02 2 2 1

S e {r s •* « )/V

Here V~ » 1. + (JtM/r), s * sine, and AR . «HU - eos»> In the northern

healspnere and A^ = -4M{1 < co*0) In the southern hemisphere. When at = U,

the resulting subspace Is the Taub-HUT lastanton.

A magnetic nonopcle at the origin has a. magnetic flelfi

6 20

where g is the Biagiv •..ic charge with a string on the negative z axis.

iz

0

In spherical coordinates and a suitable gauge tjie only nonvanishing component

of the vector potential is

The string is ursdetect&Dle If

exp [ - \ef A»0r] = esp [ - ie J A <U0 = exp [ - 1 M ireg] - 1, (7)

(where » chargaa particle Is transported along the closed path enclosing the

»trlng). Kcnce, we g«t «e « *, the Dlr»c quantization condition. These

solution* w* rojular, ststi-c, .topologlctilly stable, and approach the vacuum

solution « « • _ *t tpatltl Infinity.i,0The XaiuiB-Klein theory 1K-K) unlfleE jravlty am) electroinagnetiaB in a

fl»c c!li*sn*lo<t«l oranlfoK! «*lcfc is composed of a four dlaenslorval epnce time

tl ana ar, extr* <Jl*»nEioti 5 , which Is eocnpectlfiad to give rise to an UU)

gauge synotetry. the action in tlw K-K. ttitory is

R. • 5 dimensional curvature scalar of metric t..3 Ac

-= 5 dlnenflcnel coupling constant

Signature of jAB

621

the effective low energy theory is oDtained By integrating over x

where

G - G /2*R

r - a A - a A .liv 11 v v v

To see how the radius R is deteminsd by the electric charge ' consider a

complex field + coupled to g with the action

• »„ •*>«** (10)

and expand

Tttac,

*«•**** • t «> * t f *„>

From Eq». (9) mnd (11), we not*

a . (16«O)*/B,

mas* % ~ I ' ( 1 Z )

In other words both tbe charge and the aats of tKe coaplex field * is

proportional to It" .

622

12We next turn our attention to the singularities of the metric The

apparent singularity at r*0 does not lead to a singularity of the mcnopole as

V » 0 at r - 0. In order to study the singularities In the metric of the

monopole (which are not associated with V » 0) we take dt « 0 and consider

as2 « V(dx5 + 8H sin2* d*) 2 + (dr2 • r2d62 + r2siii2e d *2> (13)

lake r = constant and the tetrad

u**. /V(dxS + 8S sin2 | d •)

dr

The SUl wrtrle is

c takat on th» value n . ( 1 1 1 1 ) which is

equivalent to the statement that the bate vectors v* ore orthogonal.

IP study those singularities not associated trttn v«= 0 It Is sufficient

to assuac r* const, and stud; only

</ a VVl&i * 8W sin2 * 4 •>. « 3 - sit* d* ,

where «e set x = t j - t since tj, is a tlae coordinate regular at the north

pole © « 0, although it Is singular at • • »J.

623

Then,

dt, - £ - * tan f f

which is regular at e - 0 but singular at 6 = «. If we put

t = t, - BM$, then we obtain

which is regular at the south pole 6 * * but singular at 6 - 0. Let us

consider

t in o < « < « where It 1* regular

t. in •• < • < 2* where it 1* regular

The union of these two patehes couU Be a •enlfoW on which the aatrie i«

averywhere noo-alngular. Wewewer. t » t_ - m%, and 4 Is not

dlfferantlabl* o» S2. Ike fuoctloo » C M be

if its aultipie valuednoss 4 « « < 2« ii racpected.

dlfferantiable cm SZ. Ike function « can as treated as a regular function

Thus, tbe eoordinate x or t^ caa only be tfeougtrt of as angl*s with

tj, = t ^ U v K and siallarlf for t$. Wits this undarstanainj the overlap of

coordinate ptttehes 1* proper and metric is free trvm •ingularles.

It is assuaed in C-K theory that the esXra diaeoslcoi is coapcctlfleif i>

~ K

which case the metric t..(x , x ) eae tie expaadad In a Fourier series

624

> n

so x = t_ has a period 2«R. Then the condition that the metric is free ofII

12slngularles can be identified as 16vM = 2*R or

M = B/8 . (15)

Mass of the nonopele K is obtained by evaluating a five-dimensional

energy momentum pseudo tensor.

v2 l* V

Magnetic charge is given fro» the »ctlon In four dlaensions S and A as

. . J8L-.

Therefore g it quantise In units of the Dirac ehftrge. Both the nass Ha

Mid Magnetic charge J arc proportional to 8, ttte radius of the circle of the

fifth dimension, itone*, th* ausgnetic ana gravitational interactions aroonj

monopol** «nd ether field* are weak in M » reglac R « /Q.

the nuacrlcal v+luet ar*

•C « i*^ - l.e x 10~ cm (Bsvton's const.)c

R - |j /*| - 3.7 Jt 10"32 ca

d.r x l<T33ca

625

= = 3

G 4/a

III. Stability of Solution

Ue first discuss the general angular dependence of the perturbative

metric h.-. Take the nonbpole harmonic ID R (northern hemisphere) an

Ao aoperate with the parity operator P which ha* the effect of taking £H*-e,

( 1 W * V . (x) «i(B+q!* x-cose (16)

Now note that

and also slullarly P f -j^X * t-1'**2*"1'1-^, "ber* subscript 6 lndlcttac

southern hemlspltere. Bance we can fora states of definite parity I -1 and

T q U "tllCft

(Parity <-

(2) 1_

(Parity (-:

626

Thus here we have two scalar monopole harmonics of parity (-1) + °*(i and

(-1) * * respectively eacb of which can t>e used to generate vector and

tensor monopole hamonlcs. For example, for parity (-1) + o+q iwe have with

_(1) v(l) r(2) _(2)qla * l ' qlm " x

Scalar: r ( 1 )

-!_rll\ €v -£_ I2Vector: \ax" »

Tensor: V ' v^rlvaUve5' ' / ' " ' ^ ' ^(19)

X « ^

2 3 „ ^ 2 , a ^ 3 !where e « e « 0, €, « sine, e, ' - -ri,,2 3 s z sinv

x2 « 0, and x3 « *.

Any of trite tin* in (9) can be multiplied by an arbitrary function of

r and t, without chancing its transformation property under a rotation.

He new Obtain an •xteiuion of ftcgge-VteelM- h v to flv«

dlisensloni.1* ract^rine oat the tins dependence and Kaluza-Clein

coordinate depwMntc*: tîxît^x*), wit* ri(x°)

F,<x ) - e 1 0* '* we have for parity (-X)*****':

hu

v r ) r • Si- Hilr)I • h«: v

H3

(1) 2 (2) (1)

62?

(20)

h33 " r 2 ^ a s 1 ' " * L*33 * GX33

hAB

H3P ( r )

The R is the radius of the Kaluza-Klein circle. Note the number of radial

functions are reduced when the conditions of tracelessnees and

transversality given below are isposeg,

***„ ' 0, h**;* * tg* 0! 8 0^);B - 8. 121)

Ue consider waves with tt*O «nd n-0 in P,(t) and F (x ) ana first discuss

the c&se q*0, 1=0, and s=0, «o that ls_ has no angular <J*pentSence. Since

q*0, implies tn» nonopole charge is a«ro, the vector potential A vAfilife*:

also V~ • 1 • jj -»1. J» this limit we consider the statollltj of U »

residual flat space resulting fro* the vanishing of the nonopole charge.

We verlfj that the nonopole solution in five dinenslons in the limit of

flat space M=0 reduces to the corresponding llnit Ir. the Sehwarzschild

solution. Equation ('3) can tie written as an eigenvalue equation6

CD

where Oh^- S^/a-cj)-

€26

Positive (negative) values of >- will correspond to stable (unstable)

fluctuations aDout the monopole solution. In the present case of fist space

CDthere Is no contribution from the R._Qr.h term of (22).

In solving (22), we impose the conditions (21) that h is transverse and

tracefree. We obtain from (20)-(21)

*"*- a r * - * K * * B 8 ~°- (23)

2K - -

Substituting for K in (23), we have

H e F - ' rZT * 3H2 + V •

We obtain the following eleenv&lue equations for the components

fro» equations (3) and (22):

(26)

4 H cSH-4 - 1 J. - -4- (K - H } = X H . (A-B.l) <27)ar r «r -' z *-'

-d_| - | H , -ij iK - H2) - XK , (A=B=2,3) (28)ar r

- " H^P - 1 f^2£ • m 2 p- (A«B=!)) (29)drdr >

From Eqs. (25) and (2V), one obtains

629

& \ . 4 d H2 - Jl H-. (30)7 f ~5r '

dr

1 j-The solution of. (30) is H2 - |(i| + -|) e

i X r.

It is clear from Equations (26), (29), and (30) that Hg, H and H2

possess solutions which vanish as r*» for all positive values of X.

For negative values of A., the solutions go to infinity exponentially and

hence do not satisfy the boundary conditions. Equations (25) and (30)

are identical to those of Perry ir. the «=0 limit. Thus, we conclude

that a stable solution Is obtained for the case q=0, 11=0, m=o.

IV.

We outline the procedure to study the stability of the case q=l=-m=l

and ever, parity, fro* (16) we obtain

T| ^ 1 = const, (independent of • and •> (31)

Substituting in (20), we jet for »„, - h ^

Ho ai °

0 -v2

rJH

The traceless condition becomes

63 C

vV m3PK- + H, • 2K i K> • — J - 5 IK • H, c ) + — 5 — A = 0 (33)

0 2 ip i s 2F r '

For terns independent of A we project out the P part by

r.2T I sin 6 <3 6, whereas for terms F U ) that depend on A, we use

0

(F(A))Q • 2» l*a/Z FfA^) sin 6(16 + 2* J^/2 FIA^) sin 6d9

We then note that,

7- (V2A2/r2s2). - 2a J <ix ^ " X. = 2a (2 K 2 - 1),

whert a

Ue start with ten amplitudes iip, Hj, Hj, K, H , , H3, H3p, H^ and

K2 given in (32). the amplitudes H , R^, H3 and K decouple and we

obtain equations that couple the amplitudes K,, K and H . U(. are

interested In the solutions to these equations in order to fir.d out about

the stability for the case q=l. Work is In progress for the solutions of

these equations.

631

References

1. I. Begge and J.A. Uneeler, Phys. Rev. 108, 1063 (1957).

2. C.V. Vlshveshwara, Phys. Rev. Di, 2870 (1970).

3. L.A. Edelsteln and C.V. Vishveshwara, Phys. Rev. HI, 3514 (1970).

4. F.J. Zerilll, Phys. Rev. Lett. 24, 737 (1970).

5. M.J. Perry In Superspace and Supergravlty. edited by S.W. Hawking and

H. Rocek (Cambridge University Press, London 1981).

6. D.J. Gross, M.J. Perry, and L.G. Yaffe, Phys. Rev. D. 25_, 330 (1982),

7. D.J. Gross and M.J. Perry, Nucl. Phys. B126, 29 (1983).

8. R. Sorkin, Phys. Rev. Lett. 51, 87 (1983).

9. E.T. Newman, L. Tamburino and I. Untl, J. Hath. Phys. 4, 915 (1963)

S.V. Hawking, Phys. Lett. 60A, 81 (1977).

10. Th. Kaluza, Sltsunsber. Press Akad. Vlss. Hath. Phys. Kl. 966 (1921),

0. Klein, Z. Phys. 37, 895 (1926).

11. A. Salam and J.D. Strathdee, Ann. of Phys. 141., 316 (1982).

12. C.V. Hisner, J. Math. Phys. 4, 924 (1963).

13. A. Iwazaki, Prog. Theor. Phys. 72, 834 (1984).

14. T.T. Wu and C.K. Tang, Nucl. Phys. B107. 365 (1976).

15. M.K. Sundaresan and K. Tanaka, Carleton University and Ohio State

University preprint (1985).

632

We thank S. Drell and R. Blankenbecler for their hospitality at SLAC

ana Z. AJduk and S. Pokorskl for their hospitality at the Warsaw

Symposium. One of us (MRS) would like to thank C.K. Hargrove for

allowing access to the VAX coroputuer and the other (KT) wmld also like

to thank A. Salam and A. Halprln for their hospitality at the

International Center for Theoretical Physics, Trieste and Lewes Center

for Physics, Delaware, respectively. This work was supported in part by

the Canadian NSEHC Grant Number 1574, and the U.S. Department of Energy

under Contract Humber E5T-76-C-02-1415-00.

633

16/16, S « 1 SUPERGRAVITY:

THE LOW EKERCT LIMIT OF THE SHPERSTRIHG

Burt A. Ovrutfcepartment of Physics, Univarsity of Pennsylvania

Philadelphia, Pennsylvania 19104

ABSTRACT

He analyze irreducible, N - 1 supergravity theories with 16 boson-

ic and 16 fermionlc degrees of freedom. The Lagrangians for pure 16/16

supergravity, and for 16/16 supergravity coupled to arbitrary chiral

superfields are constructed. These theories are shown to have natural

SU(l,l) non-compact symmetry. The low energy field theory limit of the

superstring is conjectured to be of this type.

The superfields of non-minimal (20/20), N • 1 supergravity [1],

along with their lowest components are listed in Table 1. Fields e a

a • m

and i>a are the graviton and gravitlno respectively. Non-minimal super-

gravity is reduced to 16/16 supergravity by imposing the constraint [2]

where is a real superfield. Eqn. (1), with the Bianchi identities,

implies that

where

W • « 2e (G • + — [iZ>#T - 3 T'] - ^(n-l)T ?•) (3)

and n («• 0, - -j) is a real number. It follows from Eqns. (1) and (2)

that

3 n*ca + ca - - b a - j e Wa + fenaions (4)

These equations, along with Eqn. (2), reduce the bosonic degrees of

On leave of absence from The Rockefeller University, Hew York, N.Y.10021.

634

freedom from 20 to 16. A related equation eliminates four fermionicdegrees of freedon from the non-minimal supemmltiplet. The superfleldsof 16/16 supergravity,along with their lowest components, are listed inTable 2. The superdenslty for 16/16 supergravity is [3]

5-0 (5)where

£2 - 2(3n + D - f — (6)S

S(a) - 2(3n + l)(a-l) (7)and

V + 92f°)

"a

f° - - •|<3n-l)S-4n(3n-l)TaT0i

Parameter cs is an arbitrary real number, and e - det e a. First we de-lsrive the component field Lagrangian for pure 16/16 supergravity. In

terms of superfields this Lagrangian is given by

J / d2e £<a>S + h c (9)

Expanding <£(a) and S into component fields, ve find that the bosonic

part of ^ « is

^ g ° - e e [- j R + a(n,a)SS + y b^(l + y

^ | b Aa(i(3n+l)(l-3a))3 Z

where

and R is the scalar curvature. We implement the constraint in Eqn. (2)

by adding a Lagrange multiplier,

2 a (12)

635

where A Is a real scalar field, to Eqn. (10). The Lagrangian can now

be written asB\ji -2(4n+B)i/J

a (13)

where

y(n,a) . . ( W ) d - 3 « ) {14)

Weyl rescale the fields to eliminate the factor in front of the term

- - R . T;ie Lagrangian becomes

* (15)

Finally, make the field redefinitions

X - - i e (16)

z - X + tX (17)

The bosonic part of the pure 16/16 Lagrangian is then given by

(18)

The Kahler potential (up to arbitrary Kahler transformation) is

K(z,i) - - | In (z + I) 2 (19)

Also, the 2-field kinetic energy term is Invariant under the SU(1,1)

transformations

(20)

where d, S, C> $ are real, and af+SC - 1. The potential energy of the

z-field vanishes.

We now derive the component field Lagrangian for 16/16 super-

gravity coupled to arbitrary chiral superfields $ . We denote the

scalar and auxiliary component fields of * by A and F1 respectively.

In terms of superfields, the super kinetic energy part of the Lagrangian

636

is given by

^ S K " ~ £ /<>Z8 aiiCe ~ " 'fif*^.*1)) + h.c. (21)

where

4 . 3JSa - SCn+DT^S01- 2(n+l)S (22)

parameters a, b, and c are any real numbers, and £ is an arbitrary func-

tion. For simplicity we take £ to be real. Expand Eqn. (21) into com-

ponent fields, and rewrite F in terms of a new field & . Then the

bosonic part of ^ S K is

2 -c E

Jtf" - e e [. . . -t- (3n+l)E-2- _§_] (23)oK S S

whereE - 4(-a+a)bf - (-a+b+a)2f;1*(f*1) it

ii± • (24)

^__ will be singular in S and S unless we set E - 0. This can be

achieved by taking

*- f1 v1) Aa - a + b (25)

Weyl rescale the fields so as to eliminate the factor in front of the

term - jR, redefine field T)> as in Eqn. (16), and let

a - X + iX (26)

Also, take

(27)

(28)

637

- Lr—*-7 (O »36A* (a+iV a

where

A2 - 1 - |f (29)

o a - £13*Ai (30)

^ , or - - ~O-3b(3irt-l)) (31)

Note that the last two tarna in Eqn. (23) are not in hermitian form.

Henceforth, for simplicity, assuae that f is honogeneoua of degree p.

For |f|<*l (an excellent approximation for \Al\ $ 1/10), we find that,

to leading order, the field redefinitions

z " a

Z 1 - |a + a| A1 (32)

put the entire Lagrangian into hernitian form. Defining

vfzSz^) - f(zi,z1Jr) , (33)

the Lagrangian in Eqn. (28) becomes

-* 1*3.«3'Z 1.-C 1J*3 aI J. 3'Z

1 (34)where ,

.-«••=>•"•&&

3Z 35SZ1 (35)

and J

-(Y/3-1)K - - YJutflz + i| - i |z + S| v(zx,ziit)) (36)

He emphasize that the form of Kahler potential K is completely

determined from the theory.

In terms of superfielda, the superpotential part of the Lagrangian

638

Is given by

«£** - /d2e£(a) WCt1) +h.c. (37)

where W ia an arbitrary function. Expand Eqn. (37) into component

fields. Then the bosonic part of *^sp i*

Bib **

^ " - e e [. . . + 2(3n+l)E' 4-+ h.c] (38)

whereE' » aW - i (a - a + b) f ^ C f " 1 ) ^ ^ (39)

j£gp will be singular in S and S unless we set E' • 0. This can be

done by taking

f ^ ( f " 1 ) ^ ivi - | w (40)

The auxiliary fields >/( - 2aS) and & are found to be

• -|<^)S (41)

S*1 - -U3*(f"1) ± (42)

It follows that the potential energy is given by_ -2Y/3 ,* _ . _2 _ -V/3 2

V/e - YY | l + z| (Wl (v )1JkJWj +%- iz + z| |wr} (43)

wherep > 3b-l (ii)

i 1 9b

and W - W(Z ). For b - - -7 superpotential W • c, where c is an arbi-0

trary constant. In this case, Eqn. (43) becomes

V/e -^" !z + zT Yic| 2 (45)and the gravitino mass is given by

Y/2 , -,-Y/2 ,.,

Hence, supergravity is spontaneously broken for c # 0. It is clear

from Eqn. (43) that there are two possibilities for the vacuum state.

The first is that the minimum of the function inside the brackets occurs

for a non-vanishing value of this function. The [z + z\ factor

then assures that z -*- °°, and the vacuum is unstable. The second possi-

bility is that the minimum occurs for vanishing value of this function.

639

In this case, at tree level, z is undetermined, the vacuum is stable,

and the theory has naturally vanishing cosmological constant.

For the remainder of this lecture I'll discuss the tree-level

low energy effective Lagrangian of the superstring. The Kahler poten-

tial and superpotential are given by [3,4]

K - M2 [-in(^-~) - 3 H.ni(-~~)- 2 -Ap)] (47)

W ' Wg(S) + dAB(, A V (48)

where M - M_. , /JfR and d,__ are proportional to the invariant ten-rlanck ABC

sors of the gauge group. Fields S and T are gauge group singlets.

W. is such Chat the associated potential energy has a minimum for

finite S, and that W is non-vanishing at this minimum. A specific

superpotential-3S/2b0M

«s =• M3(a + be ) (49)

has been derived from superstring theory assuming that gluino condensa-

tion in the shadow Eg sector is responsible for supersymmecrv breaking

[ 5]. In this lecture we do not require that U,. be restricted to the

form (49). Given K and W ve can calculate the tree-level potential

energy. Define

S - S + s , T - T + f

- . C.CA

Q - T - 2-Sj- (50)and le t I be the Lie algebra generators of the low energy gauge group.

Then the potential energy i s given by

M4 -S2 2 0 >3Wi2, IB s V 1 - aA B 2V/e - ~ {Zj ;DsWr + jfc \~^\ > + S , " Re j(CA(Ta) gC ) (51)

where'

D W « ^ | - 4 W (52)

s sS s

and g is the gauge coupling constant. Mote that V is non-negative.

This is due to an exact cancellation of the -3|W| /M part of the

supergravity potential function. This cancellation is easily traced to

two properties of the effective theory induced from the superstring:

1) the -3 £n (Q/M) term in the Kahler potential K, and

2) the T-independence of the superpotentlal W.

The tree-level vacuum state is determined by minimizing V and is given

by

<D W> - 0

1 (53)<(T> - 0

The VEV <T> is undetermined at tree level. Note that V « 0 at the

minimum. The gravitino mass is given by

3 / 2 (<S><T>3

Since W is non-vanishing at the minimum, it follows that m, ., is non-

zero and that supersymmetry is spontaneously broken. The value of

m,, is not fixed at tree level since <T> is undetermined. We con-

elude that the low energy effective theory of the superstring has, at

tree-level, a "stable" vacuum state (<S>, <T>, and <C > are finite)

with a "naturally" vanishing cosaological constant (no fine tuning of

paraoeters). Furthermore, this vacuum spontaneously breaks supersyn-

metry but does not set the scale of this breaking (».,, is non-zero

but arbitrary). Theories with Kahler potentials similar to (47) have

been discussed in the context of "No-Scale" supergravity models in

Ref. [6j.

References

1. F. Breitenlohner, Nucl. Phys. B124 (1977) 500; W. Siegal and J.Gates, Sucl. Phys. B147 (1979) 77.

2. G. Girardi, S.. Grinm, M. Muller and J. Kess, 2. Phys. C26, (1964)123; R. Grimm, M. Muller, and J. Kess, Z. Phys. C26 (1984) 427.

3. W. Lang, J. Louis, and B. Ovrut, Phys. Lett. B (1985) to appear.

U. E. Witten, "dimensional Reduction of Superstring Models"Princeton Preprint (198S5.

5. K. Dine, R. Rohm, S. Seiberg, and E. Uitten, "Gluino Condensationin Superstring Models", I.A.S. preprint (1985).

6. J. Ellis, C. Kounnas, and D. Nanopoulos, Nuc. Phys. B24? (1984)373.

641

TABLE 1

Super fie Id Loweat Coaponent

Cai " 3 bai

Table I. Superfields of non-minimal supsrgravity along with theirlowest components.

TABLE 2

Superfield Lowest Component

\ -cTa

Table 2. Superfields of 16/16 supergravity, along with their lowestcomponents. B is the antisymmetric tensor superfield

amassociated with W .

E, SYMMETRY BREAKING IH THE SUPERSTRING THEORY

Burt A. OvrutDepartment of Physics, University of Pennsylvania

Philadelphia, Pennsylvania 19104

ABSTRACT

We derive two methods for determining the symmetry breaking of E, in

the low energy superstring theory, and classify all breaking patterns. A

method for calculating the effective Higgs vacuum expectation values is

presented. We show that there are theories with naturally light SU,

Higgs doublets, and classify all theories in which this occurs.

1. Introduction

The theory of superstrings' , which evolved from the string theories of

the early 1970'a , has recently undergone a great revival of interest,

spurred by the work of Green and Schwarz. . An anomaly free, d * 10 super-

string theory is possible with gauge groups 0(32) or E x E . The sub-o o

sequent compactification of the ten dimensional space to M x K, with M, being

Minkowski space and K a compact six-dimensional manifold, places further re-

strictions on the theory. In particular, the requirement that the corapictifica-

tion leave an unbroken N » 1 local supersymmetry in d « 4 implies that K has

SL'3 holonomy^ . The existence of spaces with SI' holonomy was conjectured by

Calabi and proved by Yau . On such spaces one is naturally led to a four

dimensional gauge theory with reduced gauge group EQ x E,/ . Below the Planck

scale the supergauge multiplet gives rise to an adjoint 496 of gluons and gluinos

corresponding to Eg, and an adjoint 78 of gluons and gluinos associated with E,.

In addition there are n_7 27'3 and n.. 27's of E , each containing left

handed fermlons and their scalar superpartners. The number of generations,

N " n,7 - n27> *s restricted on topological grounds. For simply connected

Calabi-Yau spaces K , N turned out to be hopelessly large, H > 36. This ledo ™ — •

the authors of Kef. 5 to consider multiply connected spaces K - K/H, where

B la a discrete group which acts freely on K . For a specific choice of K ,o o

and H « Z x Z_, the number of generations Is reduced to K • 4. There is an

additional benefit to having K be a multiply connected manifold. Define the

VilBon lcop

YU " ? e Y (1)

where Aa are the vacuum state E gauge fields, T a are the group generators,m o

and y is a contour in K. Then, as pointed out in Ref. 5, one can have U 4 1

• (8)even though the vacuum state gauge field strength F° vanishes everywhere.

The reason for this is that, because of the "holes" in K, A cannot necessarilym

be globally gauged to zero, even when F8^ vanishes globally. Therefore, as

long as contour Y is non-contractable, U is not necessarily unity. It follows

from Eqn. (1) that U is an element of E,. For a given vacuum <-onflguration,

A there can be many inequivalent U matrices. Loosely speaking, there is one I'

for every "hole" In manifold K. If we define # • {U}, then, with respect to

matrix multiplication, M is a discrete subgroup of E . As an abstract group

3^ C H. For example, if Aa vanishes everywhere then 7^ - {1}. For non-

trivial Aa, however, one can find that ^ - H. Let # {.' H) and M-'i" H)

m

correspond to vacua A and A respectively. Then 4? and 3^' may be two in-

equivalent embeddings of H Into E,. The possible existence of non-trivial

discrete group M has important ramifications. If Aa is a fixed vacuum stateis

and M the associated discrete subgroup, then denote by >¥ the subgroup of E

that commutes with 4^ • Then, as discussed in Ref. S, at energies belov the

645

Planck scale, E, will be spontaneously broken to >%/'. This result is of funda-b

mental importance in model building. For practical purposes the symmetry

breaking is due to effective Higgs vacuum expectation values (VEV's)

/ YTa A* dxm (2)

in the adjoint 78 representation of E . In this talk we derive methods foro

calculating patterns of E symmetry breaking. For concreteness, we focusb

our discussion on the multiply connected manifold K » K /H with H - Z x Z^ and

four generations. Many of our results, however, are valid for other manifolds

and discrete groups.

2. E, Symmetry Breaking—o

He want to study the symmetry breaking patterns of E on multiply connectedo

Calabi-Yau manifolds K - K /H. In general, H can be any abelian or non-abelian

discrete group that acts freely on K . The problem of E breaking is simpli-o o

fled, however, if we restrict H to be an abelian group. For concreteness, we

focus on the group H • Zj x Z,, (and manifold K - K^(Z x 2 ) with four genera-

tions) although our discussion is valid for any abelian group. Let Aa be a

fixed vacuum configuration, j^( * {U}) the discrete group of Vilson loops

associated with it, and-^ the subgroup of E that commutes with J/ . Also,

denote the SU,C x SU," x U gauge group of the standard model by & . The

following observations will be helpful in deriving £, symmetry breaking pat-

terns. First, the two Wilson loop generators of Z x Z, commute and, hence,

the associated effective Higgs VEV's can be simultaneously diagonalized. These

VEV's can be extended to form a basis for the E Cartan subalgebra. All the

elements of this basis commute and, therefore, m u s t have the same rank as

E&, namely six. Second, & must be at least as large as & in order to success-

fully describe kuown phenomenology. Therefore, we must find the embeddings of

^ in Efi. This problem has already been studied extensively, in particular

646

(9)by Slansky^ , who has listed all the possible symmetry breaking patterns

that preserve & and given the vector boson masses in terms of the symmetry

breaking direction in weight space. Combined with our first observation it

follows that J^must be at least SU3C x SU^" x Uj x V . Third, note that if

U is a Wilson loop in Z^ then U « 1. This follows from the fact that a

non-contractable path repeated five times is contractable. Finally, note

that if 4^" {IK E, remains unbroken. We now discuss our first method ofo

determining E, breaking patterns,o

A) Decomposition into Maximal Subgroups

E, has three maximal subgroups with rack 6: SU,C x SU,W x SU,,b J j j

SU2 x SUg> and SU x V . Let M - Z$ x Z^ and consider SU C x SU w x SU .

If U e j^ a possible form for U is

»k

a*

( (3)

where j, k are integers, a' • B • 1, and, hence, V3 » 1. Letting j and k vary

from 0 to 4, we generate 25 distinct U's, which form a representation of the

group Z. x Z . The group 5 ^ can be read off from the form of Eqn. (3). It is

#'- SU,C x SU W x U x SU, x U (4)

A second possible form for U is

\ I

(5)

Again, j, k are integers, o* • S - 1, 0 " 1, and, letting j and k vary from

0 to 4, we generate 25 distinct U's which form a representation of Z, x Z,.

The group * can be read off from the form of Eqn. (5). It is

647

SU3C Jt SU2

W x Uj I Jj I IIj (6)

Eqns. (3) and (5) are the only possible embeddings of J/- Z^ x Z5 into

SU c it SU.W x SU3 that preserve jf. The embeddings of JT - Z$ into

SU C x SU.W x SU, can be worked out in the same manner. The results are that3 3 J

j#"can be SV.C x SU,w x SU,, SU,C x SU," x SO,, and SU, x V,. The remaining

breaking patterns can be found by embedding Z, x Z, and Z_ into maximal sub-

groups SU. x SU, and S01n xll,. It is clearly preferable to have a more general

method, particularly one that allows a simple determination of the associated

effective Higgs VEV's. Such a method is most easily found by using the Dynkin

formalism*105.

B) Method of Weyl Weights

He can write the most general Wilson loop U as exp{l£A H }, where the H

are the six generators of the Cartan subalgebra of E, and the X are six real

parameters. Arrange the \ into a vector X S (a,b,c,d,e,f). Let a be a root

vector of E . The mass of the vector boson corresponding to a is proportional

to the inner product (X,a). Since the vector bosons corresponding to & must

be massleas, X - (-c,c,a,b,c,O). He can then write U as a 27-diaensional dia-

gonal matrix that depends on the three real parameters a, b, and c. The dia-

gonal elements are given in Table 1, together with the transformation properties

under SU.jC x SU2", S01Q, and SUj of the elements of the 27-plet on which they

act. Throughout this paper we label elements of the 27-plets that transform as

an A under S01(J and * B under SDj by [A.Bj.

To find the group & , we simply embed the discrete group in the above

form for U. For B • Z ; s Zj, «. could let e ^ - a , ei b - g, and e l c - aH*,

where a » g « 1 and j and k arc integers. # • Zj t 2 corresponds to two

of the parameters a, b, and c being independent, while for ¥ » Z there is

only one independent parameter. Furthermore, by using Table 20 in Slansky's

(9)report , the relevant part of which is reproduced here as Table 2, we can

1 lad which ^corresponds to a given & .

As an example, let us find all the eabeddings of jr • Z5 x 1 that break

E& to SU4 % SU2W x L'x x U (Pati-Salam).<U) From Table 1 we see that if

the (e,v) and (u,d) are to lie in a & of SU^ we must have b»-c. (We have

used the fact that the (e,v) corresponds to the weak doublet in the {16,5;]

and the (u,d) to the doublet in the £16,10).) We can then either let a be

Independent or fix a • 0,c,2c,3c, or \c. (Because of the Z symmetries

5c = 0, 6c = c, etc.) For a icuependeut ue consult Table 2 and find that

(X,a) is zero only for a - +(000001), ±(0100-10), ±(0-10011), ±(10001-1),

±(100100), ±(-1101-1-1), ±(-10010-1), and the six roots (000000). £0 we

have 20 massless vector bosons and these roots span a representation of

SU x SU_" x 0, x U . Similarly for a»0 we have 46 massiess vector bosons

and the gauge symmetry is S01Q x C ; for a » c we have SU, x SU, x U , x U_;

for a » 2c, SC. x SU x I' ; for a • 3c, SU. x SU, c V^; acd for a - 4c,

SU, x S u " x U, x U,.' We break to the Pati-Salam group in three cases:

i) a independent, J/ • Zj x Z , e l c - aJ, e l b - a"j, e 1 3 - gk; ii) a - c,

¥' Z5, eic - a

j, eib - a"*. eia • aj; iii) a - -c, M - Zy <ic - a^,

e i b « a - J , ei a - a ^ .

By examining all the values for a, b, and c allowed by the discrete

symmetry, one can exhaust all the symmetry breaking induced by Jf . In

particular we recover the eobeddings found by the first method. In addition,

ve can find for which symmetries the effective Higgs VEV's are zero in some

directions. We use this method in the next section to generate naturally

light Higgs doublets.

3. Saturally Light

Ve would like '.0 have light Higgs doublets to set the electroweak scale

and give masses through Yukawa couplings to the ordinary fermions. In par-

ticular, it is necessary that at leaat one of the light doublets be in the

649

[10,5] or [10,5] representations under [SO^SU^]. At the same tine, super-

symmetric E theories contain extra color triplets that can mediate nucleon

decay" via dimension 4 or 5 baryon number violating, J& Invariant, operators.

These triplets Bust be given very large masses. It is very difficult, even

with fine tuning, ta keep the doublet light while making the triplet heavy.

Fortunately, the same mechanism that breaks the gauge symmetry gives us a

natural method for splitting the doublets from the triplets. Although the

method does not depend on which discrete symmetry we use, for simplicity we

assume that I • 2, x Z,.

The nontrivlal gauge fields that give rise to Wilson loops different from

unity can also lead to a mass term through the coupling of the four-space part

of the chiral superfields to the gauge fields in the compactified dimensions.

In other words, one of the 27's can couple to the 27 through a tern Involving

an effective Higgs 78 VEV (27 % 27 x 78 contain* a singlet). These fields thus

acquire a mass of order the Inverse radius of the compactified dimensions, pre-

sumably the Planck mass, while the other four 27's remain massless. (Note that

we cannot have a 27 x 27 bare mass tern since, until E, is broken, the chiralo

superfields are all massless zero modes.) Let us call the 27 and 27 that

pair off X and X, and the remaining 27's tp. He then expect all the components

of X and X to gain huge muses and disappear from the spectrum. As we now show,

however, it is possible for some of these components to remain naturally mass-

less. This occurs when the diagonal entries of the U's that multiply these

components are unity. That Is, the associated effective Higgs VEV's are zero

in these directions.

As an example, let us use Table 1 to find which components of X and X are

left massless for the breaking to SU^ x SV^1 x Uj x l)j given in Section 2:i) for b • -e, a. independent, none of the U 8 is one, the effective Higgs

650

VEV then has no zeros, and all the components of X and X are massive;

ii) for b - -c, a - c, u d l a g - 1 for the color triplet in the [16,5] and

the singlet in the [16,10], and hence those components of X and X remain

mas»les«; ill) for b - -c, a « -c, the color triplet weak singlet in the

[16,10] and the [16,1] remain masaless.

Using Tables 1 and 2, we can find for which values of the parameters a,b,

and c we obtain light doublets and what the resulting gauge symmetries are.

The light doublets in X can be used as Higgs fields to break SU_" x U and

to give masses to quarks and leptons. The corresponding light doublets in

cannot couple to ordinary natter, and hence we ignore then. He list below

all the cases in which at least one weak, doublet in X is light vhile the color

triplets are all heavy: •

i) b-a+c, a and c arbitrary

masslesst doublet in [10,5]

gauge symmetry: SU3° x SU2" x Dj i Oj x llj

ii) b-4c, a-3c

massless: doublet in [10,5]

gauge symmetry: SU x SU x V

lii) b-0, a-4c

massless: doublets In [10,5] and [16,5], the a ingle t [16,1]

gauge symmetry: SU3C x SU2" x V1 X l^

iv) a-2c, b arbitrary

masaless: doublet in [10,5]

gauge symmetry: SU.C x SU,V x U. x U

v) a-2c, b-0

maaslesa: doublets in [10,5) and [16,5], singlet in [16,10]

gauge symmetry: SU3° x SU2W x Oj x tj

G51

vi) a-2c, b=3c

massless: doublets In [10,5] and [10,5], the singlet Jl,l]

gauge symmetry: SU3° x SU2" x SU2 x l^ x l^

vil) a-2c, b»4c


gauge sysmetry: SU x SU. x Ui

viii) b«0, a and c arbitrary

raassless: doublet in [16,5]

gauge symmetry: SU c x SO," x t^ x U r. V

ix) b»0, a-3c


gauge symsetry: SUj x SU2 x L'j

The embedding* of J/ that give rise to these values for a, b, and c can be

readily found. For example, in case i) let e a » aJ, e c • a , and

e » a^ , a • 1. Thi» corresponds to ^ » Zj x J . In case ii) let

e i c - a-3, e1* - a3i, « i b - a4j, rfilch corresponds to JK- Z .

It is worth re-emphasizing that our light Higgs doublets were not obtained

by Cine tuning. Setting, for example, b - 3c, a • 2c as in case vi) is a choice

of parameters (optiaistically a minimum for the vacuum configuration), but not

a fine tuning, since e *, •" , and e c are restricted by the discrete symmetry

to be fifth roots of unity. These Higgs doublets are totally massless until

supersyzmetry is broken spontaneously.

ACKNOWLEDGMENTS:

The results discussed In this talk were derived in collaboration with

J. Breit and G. Segre. These results were also obtained independently by

E. Kitten and K. Sen.

652

References

1. J. H. Schwarz, Phys. Rep. 89. (1982) 223; M. B. Green "Surveys in High

Energy Physics 3_ (1983), 127 are good reviews of the subject.

2. P. Ranond, Phya. Rev. D2. (1971), 2415;

A. Neveu and J. H. Schwarr, Nucl. Phys. B31_ (1971), 86, Phys. Rev. D4

(1971), 1109; L. Brink, D. I. Olive, C. Rebbi and J. Scherk, Phys.

Lett. ASB (1973), 198 describe early models with fennions and bosons.

3. M. Green and J. Schwarz, Phys. Lett. 149B (1984), 117 for N-l super-

symmetry with open or closed strings. L. Alvarez-Gaume and E. Witten,

Nucl. Phys. 3234 (1983) 269 for N-2 supersyraoetry with closed strings.

4. For recent discussions see also E. Witten, Phys. Lett. 149B (1984) 351;

D. J. Gross, J. Harvey, E. Martinec and R. Rohm, Phys. Rev. Lett. S4

(198S) 46.

5. We are following here the discussion of P. Candelas, G. I. Horwitz,

A. Strominger and E. Witten, "Vacuum Configuration for Superstrings",

XSF-TTP-84-170 preprint (1984).

6. E. Calabi and Algebraic Geoaetry and Topology: A Symposium in Memory

of S. Lefschetz (Princeton University Press, 1957), p. 58.

7. S. T. Yau, Proc. Nat. Acad. Sr.i. 74_ (1977), 1798.

8. Similar mechanisms had been proposed for gauge symmetry breaking in

Kaiuza-Klein theories by Y. Hosotani, Phys. Lett. 129B (1984), 193.

9. R. Slansky, Phys. Rep. 79 (1981), 1.

10. E. B. Dynkia, Am. Math. Soc. Trans. Ser: I (1957), 111; £ (1957), 245.

11. J. C. Pati and A. Salam, Phys. Rev. D10 (1974), 275.

12. \. Hosotani, Phys. Lett. 126B (1983), 309;

D, J. Gross, R. Pisarski and L. Yaffe, Rev. Mod. Phys. .53 (1981), 43.

653

Table 1. Diagonal elements of the Wilson loops V

exp{i(b-3'c)>

exp{i(c+a-b)}

exp{2ic}

exp{i(2c-a}

exp{i(c-b)}

expf-Ka+c)}

exp {ib}

exp{i(a-c)}

exp{l(a-b-2c)}

exp{i(b-a)}

exp(-ic)}

d.i)

(2.1)

(1.3)

(2.1)

(1.3)

(1,1)

(2,1)

(1,3)

(1,1)

(1.3)

(2,3)

S 010

1

10

10

16

16

16

SU5

1

5

5

1

5

10

654

Table 2. Nonzero E. Roots

Root a (A,a) Root a ,a)

(000001)

(0-10011)

(-11001-1)

(0-111-1-1)

(0-12-10-1)

(101-10-1)

(0-H-IOt)

(0-11-100)

(0-1101-1)

(-1010-10)

(-1010-1-1)

(-1101-1-1)

(010-110)

(010-11-1)

(-100-111)

(-101-110)

(-101-11-1)

(-12-1000)

0

0

3c

a+b-2c

2a-b-c

a-b-c

a-b-c

a-b-c

a

a

a *

b+c

2c-b.

2c-b

2c-b

a-b+2c

a-b+2e

3c-a

(0100-10)

(10001-1)

(-210000)

(00-12-10)

(1-11-110)

(1-11-11-1)

(001-1-10)

(00100-1)

(00100-2)

(-1-11000)

(-100100)

(-10010-1)

(000-120)

(-110-101)

(-110-100)

(-111-10-1)

(-11-1011)

(-11-1010)

0

0

3c

2b-a-c

a-b-c

a-b-c

a-b-c

a

a

a

b+c

b+c

2c-b

2c-b

2c-b

a-b+2c

3c-a

3c-a

65 S

HIGHER VECTOR MESONS IN NUCLEON FORM FACTORS

Dalifaor Krupa

Institute of Physics, The Electro-Physical Research

Centre, Slovr.k Academy of Sciences, Bratislava

Vo.itech Kundrat

Institute of Physics, Czechoslovak Academy of

Sciences, Prague

Pavel Masiar

Polymer Institute, The Centre of Chemical Research,

Slovak Academy of Sciences, Bratislava

Abstract

The consistency of higher vector meson states

existence with the isovector electric nucleon form factor

data has been checked. Besides the well established c(77O),

J>(1600) and the not so well established j (12SO) the con-

sistency is confirmed for yet another vector meson with the

mass just above 2fteV so far sinestablished but possibly al-

ready observed.

Presented by D.Krupa

656

According to the vector meson dominance model1' the isovector

part of the electric nucleon form factor 0' (t) can be expressed ap-b

proximetely as the sum of pole contributions due to the appearance

of the o meson and the so called higher vector mesons with the

same quantum numbers as the £i'(i25O), £(1600) etc.

Here t is the square of the four-momentum transfer, m is the

mass of the vector meson, fy is the vector meson-photon coupling

constant and sfLf? i s t nS vector meson-nucleon an'tinucleon coupling

constant.

In the generalised vector meson dominance model this sum runs

to infinity' . Due to the hadronic character of the photon''1' the

masses m and the widths I of these vector mesons are supposedv v •—•»

to be related to those of the p meson (m» and |» ) and form a Vene

ziano-like spectrum

(2)

This is the meson spectrum obtained in the chiral limit of

the dual resonance model, where daughters appear as radial exci-

tations. These relations were determined also by imposing gauge

invariance on the vector meson dominance model' / . More recently

they were derived by using the QCD duality properties' ' and by

using the finite-energy sum rules technique in QCD ' .

By using (2) with the input m. = 769 MeV, [Z = T54 MeV 7/

we have the spectrum of vector meson states with the following

first five terms:

State

Mass

Width

(MeV)(MeV)

S769

154

3'1332

267

Table l.

f"1720344

f'"2035

407

f"2307

462

657

//Of these meson's only the second radial excitation P is well

established as the so called C (I6OO). By comparing the m a s s i f "

with the mass of P(l600) from fief./7/, one can expect that the

mass spectrum of Table 1 deviates from the "true" spectrum for

In order to check the consistency of such a spectrum of higher

meson states with the data on nucleon form factors we use the same

130 experimental data points on the isovector part of the electric

nucleon form factor Gg(t) and t.he same analytical expression to

describe Gp(t) as in Ref./8/:fc

£ = 1

Here

1 <•

where

/ ; . i

(5)

and A. are the coefficients to be suitably determined. The pairs

of complex conjugate poles at w«, and vie. are there due to the

complex conjugate poles at in - (/tvLfi^)'" and -iJ', and similarly

for the higher fc -like vector mesons. By changing N in (3) we can

thus take into account as many higher vector mesons as we wish.

The case with N £ 3 has been studied extensively earlier' , here

we are Interested in N2 3.

VThis description is based on a model of analyticity of G"(t)

in which the multi-sheeted Riemann surface structure is approxi-

mated by the four sheets resulting from the two pion cut from

t=4m£ to infinity and the second effective cut with the branch

points at t=t and at infinity, approximating thus all the other

possible intermediate states as the 43T, KK, NN etc. The short

cut on the second sheet just below the two pion cut branch point

is also taken into account'9' by a pole at w=w .

658

So she model satisfies the following basic properties:

i, analyticity

ii, reality condition G^(t*) = C^*(t)v

iii, normalization Gg(t) = 1

iv, asymptotics Gg(t) > t-i

The consequence of ii, is that the coefficients A^ in the

numerator of (3) are all real and the vector meson poles appear

in complex conjugate positions. When fitting the data by means

of (3) the free parameters are only the w and the coefficients

A' (n=1,2,...L). The coefficient A is not free, it is deter-

mined by the normalization condition iii,. The complex conju-

gate pole positions are fixed at the values corresponding to

the Table 1. The first pair of poles is on the second t—sheet,

while all the others are placed on the third sheet since their

mass is higher than the mass corresponding to the effective

branch point at t.=4mi .

The % values per number of degrees of freedom, as the

preliminary result of the fitting procedure, are shown in

Table 2. Here N and L have the same meaning as in Eq. (3).

«DF

3

4

T

1

3

. 3 0

. 2 9

1

t

N

4

. 6 9

. 2 4

5

10

7

. 6

. 8

Table 2.

By increasing L the number of free parameters is increased,

however, by increasing N more stable poles are introduced but

the rmmber of free parameters remains the same. It is by no

means obvious that by incresing the number of fixed poles one

would get better description of the data, unless the data is

not in agreement with the presence of such poles.

The best fit is obtained for the first four vector mesons

indicating thus that besides the a (770), p (1250) and p(ISOO)

the data prefer yet one higher vector meson with the mass above

2000 MeV. This might be the indication of the f (2150) which

has been se«sn in some formation experiments'' ' ' but omitted from

the meson table of elementary particles.

659

References

/ I / N.Zovko, Fortschrit te der Physik 23., 185-209 (1975)/ 2 / E.Etim, E.Masso', CERN Preprint Ref.TH 3557-CERN,

March 1983/ S / G.Preparata, Invited talk at the EPS I n t . Conference on

High Energy Physics, Palermo, 23-28 June 1975/ 4 / E.Etim, L.Schiilke, Ze i t s chr i f t fur Physik C; P a r t i c l e s

and Fields 6, 303-308 (1980)/ 5 / S.S.Grigorian, Soviet Journal of Nuclear Physics 39.

468-473 (1984) ( in Russian)/ 6 / N.V.Krasnikov, A.A.Pivovarov, Phys.Lett . 112B. 397 (J982)/ 7 / Part ic le Data Group, Review of Modern Physics 56., No 2,

Part 2 (1984)/&/ D.Krupa, S.Dubnicka, V.Kundrat, V.A.Meshcheryakov,

J.Phys.G: Nuclear Physics JJO, 455-469 (1984)/ 9 / S.Dubnicka, D.Krupa, V.A.Meshcheryakov, Acta Physica

Slovaca 3_i, 2O5-213 (1981)/ 1 0 / As the ref . No 7 . , page S176

661

o GIT V.'JLTIILICITY PUT31 BUT IP" 0? ClVuiK PAIRS

Andrzej Y/roblewski iInstitute of Experimental Physics

university of './arsav/Warsaw, Poland

•••cstract

iiorstraints have bsen obtained on quark-antiquark multiplicity

distribution in quark jets by adjusting parameters of Monte-

Carlo jets to fit the experimental relation between the average

multiplicity of charged final state hadrcu s and the dispersion

of their multiplicity distribution.

Hadron production in lepton-hadron deep inelastic scattering

or e e annihilation is a complex process which may be considered

to occur in three steps!

1. Creation of quark-antiquark (or diquark-diantiquark) pairs

in the color field,

2. Hadronization, i.e. the process of quarks Cor diquerks)

dressing up into color singlets, »

3. Strong decays of resonances giving rise to observed "stable"

particles (which eventually undergo further decays on much

longer scale than that of strong processes).

It must be added, however, that the three steps are not

necessarily separated in space-time.

Hadrons observed in our detectors are products of step 3

(or possibly step 2 - in case of the lowest mass states).

Thus they carry little direct information about the initial

breaking of the color field. Step 2 is not accessible experimen-

tally and little understood theoretically. This has pronpted

the construction of phenomenological models like th# Field-

Feynnian [1] or LUIJ2 [2] to describe the evolution of quark jets.

Of course any quantity related to step 1 is of interest for

our understanding of strong processes.

667

It it.- relatively aLr.r.x* to euti::at& t'.io averse- r.vj-.ber

< -*'iq> oT -.i-in.-ii';' euark-m-.tiqus-r1.: c-r.ira (cc-o [3] for rlov.-.il::

of the procedure). In the- preoor.t j'cp&i* sii r.ttetnpt is described

to obtai:: certain oonsxrair.ta on ^ho q? n.nltiplicity tisiri-

bution beyond its first -ior:<ent.

It ic "cnovsi by now that for nuari: jets the- tiver&.-e ciicr c-d

multiplicity <r.-,> end the dispersion D,,= (<n£> - < »-,>")1/2

2..T2 rslated "oy a linear function

D_, = A + 3< r.w>

with 3=0.35^0.01 sa:d A=0--'t7-|-3.0". This fit [it], br-.ced or. data

from [5]j was recently nicely cor.firrr.ed by the SLIC Collaboration

[6j. The parameters A and B (see ?ig.1) seea to be indQpsndtnt

of the quark flavor, since the data for vp pertain mostly

to d quarks, those for' vp and ^ p - to u quarl:3, whereas

the e'e data represent averages over five quark flavors with

u and o quarks entering with ths largest weight.

The method used here to obtain constraints on quark-antiquark

multiplicity distribution consisted of generating llonte-Carlo

quarks jets and adjusting parameters of the model to fit the

above experimental relation between < r.w> and Dp.

The codel used for generating particle contents in quarl:

jets involves the follov/ins assumptions:

1. Only three flavors (u, d, s) are produced with probabilities

uu:dd:ss = 1:1: X .

2. Only qq pairs are produced. Possible production of diquar:-:-

diantiquark pairs is neglected-

3. Ths primary mesons belong to the three nonets! J^= 0

(pseudoscalar), 1 (vector) and 2 T (tensor) with probabilities

r, V end T, respectively (P+V+T = 1 ) .

if. There is no correlation between the numbers of primary

pseudoscalar, vector and tensor r.esons in a jet.

Thus the model ha3 three independent adjustable parameters,

ths strange quark suppression factor X , and two of the three

probabilities ?, T, V.

Composition of particles in quark jets was generated in the

following v/ay. The number Kqq of primary qq pairs in eaoh

jet v;as determined according tc sorae assumed ^.v.ltiplioity

distribution (see below). The nature of each meson was fixed

randomly according to a given choice of parameters X , Ps V, T.

663

The decay chains of unstable mesons v,-ere traced down to

"stable" particles. Then charged "stable" hadrona from each

jet were counted and stored for calculating < n^> and Dp

in a £ivun sample of jots.

.is a first step three simple one-parameter distributions

•;.•••:re t r i e d :

1) Sero-width, i.e. llqq = const.

2) roisson, or rather a "truncated" Poissor. without II qq = 0,

2) Flat distribution between i.'riq = 1 and Kqq = K „, with

••_,,,.. allov/ed to vary.

Tho exclusion of Kqq = 0 is obvious, because in a quari: jet

•one requires at least one particle, charged or neutral, to be

present. •

The results of Monte-Carlo calculations are shoraa in Fig.2.

It appears that v/ith the three distributions for Bqq one can

not obtain Dp and < n^> to agree with experiment, no matter

what parameters P, V, T and X are chosen. In particular,

the Poisson distribution for Uqq yielded Dj, always larger

than that observed experimentally for a given < n ^ .

This result served as an indication that an agreement with

experiment may be obtained by starting from a distribution of

Bqq narrower than the Poisson distribution. Hence the nex;t step

was to try a simple two-parameter distribution, a Gaussian,

for which the width <J was varied for a given <Kqq> and a given

set of parameters P, V, T and X .

The.- surprising result of these calculations was that again

the parameters P, V, T and X had little influence on the relation

between Dp and < n-^, in contrast with the width O to which

the results were rather sensitive. Fig.3 shows an example of

results for < Uqq >=!(., in which case Cf =O.65V<fTqq> = 1.3

v;as found to' fit the experimental relation.

Finally, it was found that the best-fit Gaussian width

increases linearly with <Kqq> as shown in Pig.4. This seems to

be independent of the flavor of the quark which initiates a jet.

One may conclude that, given the assumptions of the model

of quark jets, the multiplicity distribution of qq pairs seems

to be narrower than the Poisson distribution at a given

V/ith the Gaussian ansata for the multiplicity distribution

of qq pairs one finds the wi^th CT increasing linearly with

664

[1] R.D.?iol& ru-;d n.P.Feynnujjn, Kucl.liiys. : ; i ; : , 1 (n7c ;•[2] B.Anderson, S.Gustafson, C.Peterson, Zs.-itschrift. f.Physic

C1.105(1973).[3] A.Y.'ro'clewcki, Acta Phys. Polonica 3io,370i 1923).[4] A.'i7r<iblev/ski, Proceedings of the XV-th Internat ional

3;anpociun on Llult ipartiole Synanicc (Lurid 19-34),ed. G.C-ustafson and O.Peterson (Singapore-, '.Vorld uCior.tir103^) p.30.

[5] H.Grassier et a l . , Huol.Phys. 3223,269(1983),U.Althoff et a l . , Zeitachrift f.Physik 022,307(1984).

[62 M.Arneodo et a l . , Hucl.Phya. B25S,249(198p).

665

Pig.1. Dispersion Dp as a function of the average charged

multiplicity < np> for quark jets. Data are taken from

Ref.5 and 6. The straight line represents the fit

mentioned in the text.

Pig.2. Monte-Carlo results for Dp and < n-p> for three different

multiplicity distributions of qq pairs! flat (•),

Poiason (*) and zero-width (x). The straight line shows

the fit to experimental data as in ?ig.1.

?ig.3. icanples of ilonte-Carlo results for Dp and < a-^>

obtained for Gaussian multiplicity distributions of

qq pairs with <ITqq> = 4 and three different choices

of the width a . The upper set of triangles (A)

is for a =0.8^1 qq>, and the lower ( V) for a =O.5V<3:fqq>«

Black circles ( • ) which agree with the experimental

data (represented by straight line from Pig.1) are for

CT =0.65 V^qq^". The letters denote corresponding groups

of results obtained with the following parameters!

A:.P=0.7, V=0.3, T=0, X =0.22,

B : P=0.5, V=0.4, T=0.1, A =0.27,

C : P=0.4, V=0.43, T=O.12, X =0.38,

D : P=0.25, V=O.S, T=0.15, X=0.33.

In all cases the u quark was taken as the one initiating

the jet.

Pig.4. Dispersion Dqq{=<r) of the qq Gaussian multiplicity

distribution as a function of <Uqq>. For comparison is

shown the relation between Dqq and <Tqq> for two qq

distributions which clo not fit the data: the Poisson

distribution (continuous line) and "truncated" Poisson

without Kqq=O (dashed line).

666

1 2

OF

2 3 i 5 6 7 8

Fig. 2

667

OF

2

1

A

B C D

Fig. 3

POISSON

GAUSSIANFITTED TO

DATA

8<Nqq>

669

GRAND UNIFICATION: QUO VAPIS DOtgNET

Goran Senjanovic

Physics Department*Brookhaven National Laboratory

Upton, New York 11973

and

ICTP, Trieste, Italy

In memory of my brother Pavle Senjanovic

ABSTRACT

An attempt is made to summarize the present theoretical

and experimental situation with grand unification.

•Peraanent address

The submitted manuscript has been authored under contractDE-ACO2-76CH0OO16 with the U.S. Department of Energy.Accordingly, the U.S. Government retains a nonexclusive,royalty-free license to publish or reproduce the published formof this contribution, or allow others to do so, for U.S.Government purposes.

670

It is interesting to observe the reversal of our attitude towards the Idea

of grand unification: it was first considered rather immodest by many people for

trying too much too soon, and now it faces the opposite problem - of being not

ambitious enough, since the gravitational force is completely left out of

consideration*

Still, after more than a decade GUTS have stimulated a large amount of work

by a lot of people, stirred quite a few polemics, become an important link with

modern cosmology and played a crucial role in the world-wide dedicated search

for proton decay. Why? Sometimes I am not sure whether it's due to the actual

progress or the peculiar soclo-psychology of our field that makes fashions come

and go. I hope that this short review will help make up your own mind on this

question.

1 will address the issues of proton decay and sin2ew at length. Me shall

go through the predictions of both the standard 8U(5) theory and its

supersymoetrlc extension. As you know, the SO(5) theory suffers from a serious

defect: it fails to unify even a single family of tensions. It is only natural

then to study its extension, the S0(10) theory which provides a minimal one

family model. SO(10) has rather interesting properties characterized by the

existence of domain walls, strings and raonopoles. The amusing gravitational

characteristics of domain walls and strings are discussed. Finally, I argue

that we have to go beyond S0(10) in order tc incorporate a unified picture of

families. This leads to the prediction of mirror fermions, whose physics is

carefully analyzed.

All of this is incorporated in the following table of contents:

I. The SU(5) theory: sin29jj and tp

II. Supercyanetry

til. S0(10)

671

IV. Domain walls and strings

V. Family unification and mirrors - the S0(18) theory

VI. Outlook

It is only fair to enumerate the topics I haven't discussed here, either

due to the lack of space or my ignorance. These are: monopoles, baryon

asymmetry, strings and galaxy formation, and most important, there is no mention

of gravity.

Before we go on, let me remind you briefly of the reasons we got excited

about grand unification. Besides the usual story of sin26jj, proton decay,

charge quantization, etc., I will also show you how GUTS imply purely massless

photon-on phenomenological grounds.

(i) Charge quantization1

Since GUTS unify quarks and leptons into a simple (or semisimple) group,

charge operator is one of the generators of the group and so

Tr Q • 0

en

This relates Che charges of quarks and leptons* For example, Sl)(5) contains

five dimensional fermionic representation

v° R

and so

3 <4 + < £ . P.(it) sin29w prediction2

Related to (i), which implies the connection between SU(2)L and U ( 1 ) Y

couplings.

(lit) Proton decay*

Unifying quarks and leptons leads to exotic gauge bosons X tnat couple to

both qq and qi states, which obviously leads to the violation of baryon number.

The mass of these particles is the unification scale, and is predicted by means

of the renormalization group2. This in turn determines proton lifetime.

(iv) MasslessneBB of the photon3

How come Oy » 0? Since in gauge theories m . - e <Vy>, where V^ is

a v.e.v. of some charged scalar, the question is how come <My> * 0? In an

Abelian case, you could take the charge arbitrarily small, and so small

m (<10 eV) is perfectly OK. In the non-Abelian case, however, the charge is

quantized as we have just seen. This implies the existence of a physical Higgs

scalar with the charge of order 1 and a mass of order V^ (as in the usual

Higgs mechanism). Such scalars, almost tnassless, are ruled out experimentally

e.g. for their large contribution to g-2 at the two-loop level. Grand

unification forbids the photon mass, no matter hog small! Of course, it doesn't

offer any deep reason as to why this is so.

(v) Baryon asymmetry'*

I will not go into an; details. At high temperature, t - Mx, baryon

number violating interactions are fast enough to create a nonvanishing baryon

number, even if the universe started symmetric in baryons and antibaryons.

Except for the very minimal SU(5) theory, one can accommodate the observed ratio

of matter and radiation nj/Oy = 10~9.

(vi) Monopoles5

The breaking G » 8 » 11(1) produces magnetic monopoles (it's not the only

such breaking) with mass -I/a Mj. Amazingly enough, these monopoles can

catalyse the proton deeey with the strong interaction cross sections.

673

I. The SU(5) Theory: sinJ9y and tp

Probably the main original source of excitement about GUTS comes frois the

predictive power of the minimal SU(5) theory6. As you know, the gauge structure

and the minimal Higgs sector determine both the proton lifetime and sin2e(f.-7

sin29 (M ) - 0.216 ± 0.004H W

t (p + e+it°) = (0.06 - 240) x 1029 yr. (1.1)

«Hs " 1 0° * so* "•*•Whereas sin28w certainly agrees with the experiment, the present M B bound8

T (p + e+n°) > 2.5 x 10 3 2 yr. (1.2)P exp

puts SU(5) theory in trouble. There are still ways out of this impasse.

(i) Increased Higgs sector3

This is likely to keep the low energy physics intact, since the survival

principle tells us that except for the usual light SU(2) x U(l) doublet, the

rest of Higgs scalars become superheavy. Their masses may not be precisely

equal to Mx, however. For example, add _45 (considered necessary by some

people) with the spectrum under 5 U ( 2 ) L X U(l) X S U ( 3 ) C

•l a2 m3

4 ^ - (2, 1, lc) + (1, -2/3, 3c) + (3, -2/3, 3c) +

+ (1, 8/3, 3c) + (2, -7/3, 3c) + (I, -2/3, 6c) + (2, 1, 8c) (1.3)

mi, m ^ mg ID^

This is rich enough to modify predictions substantially.

Assume conservatively 1/10 < mi/tnj < 10; this is still sufficient to

increase rp by two orders of magnitude, while keeping sin2^ essentiall)' intact.

(ii) Number of families

The predictions in (1.1) are fairly insensitive to the number of families

Np, except in the H_»8 case. Since then 6 " o o p=0, the two-loop effects

actually become dominant. Bagger et al 1 0 find Tp(Nf«8) * 1032 yr. Me shall

see in the context of family unification (section V) that Np»8 may not be so

crazy.

(iii) Mew fermlons

Not very appealing; I won't discuss it here,

(iv) Supersymmetry

This, of course, modifies the low energy world, and therefore it goes under

the departures from the minimal theory. The whole next section is devoted to

it.

675

II. SPPERSTHKETKT

In the context of perturbative unification, supersymmetry appears almost as

a must. The reason, of course, Is the infamous hierarchy problem. Super-

symmetry simply prevents the miracle Mjj/Mx £ 10"13 from being renormalized in

perturbation theory. What about its implications for Tp and sin Syj?

(i) d-6 operators

The first contribution is the usual exchange of superheavy gauge bosons.

The results are summarized below11

(MeV)

100

200

0

0

.239

.235

2

4

T P(YD

x 10

x 10 3 5 ± 1

The theory appears completely safe; actually if this was the only source of

proton decay there would be very little hope of observing it in the near

future. Now, sin2ey is slightly pushed up, but still OK — an Important

distinguishing feature from the standard nodel.

(ii) d-5 operators

Obviously, in a supersysmetry theory besides X bosons we are bound to deal

with their supersynaetric partners, the fermionic superheavies X. They also

mediate proton decay, at even faster rates12. The typical proton decay diagram

would be

gluino (W-ino, ...)

676 8

The predictions are not precise; they depend on glutno or V-ino mass and the

•quark masses.

Gluino exchange is particularly Interesting, since it is expected to

dominate13. It leads to a model Independent prediction

p • K*"vp (vT)

other decay modes being forbidden. I wish I could tell you what the lifetime

for this process is; it Is not hopeless, however, and could be seen.

Supersymoetry is plausible and important; don't rule out SU(5) yet.

677

III. S0(10)

Whether or not Slf(5) works we don't know. In any case, I feel that it

couldn't be a final theory. Its asymmetric treatment of up and down quarks and

left and right components is in By opinion unforgivable for a grand theory. The

aininal extension of SU(5) that unifies at least one fanily of feralons is based

on the by now popular alternative SO(IO)111. This theory has a plethora of

rather exciting features that I would like to review now.

Probably the most Interesting aspect of orthogonal groups are its splnor

representations. They are nothing but the Euclidian version of the Lorentz

group. Imagine a Dlrac algebra in Euclidian space (Clifford algebra15)

{y±, Tj} - 2 Sjj! i,J - 1 2N . (3.1)

Then in the usual manner the commutators

Tij s 7 "ij * 4T tv *j] <3'2>

generate the Lie algebra of SO(2N) group (why 2N will be obvious later). The

dimension of the complex spinor representation of (3.2) would be 2", if not

for

^FIVE - lN n *z • • • *ars YFIVE ' 1 (3'3)

which commute with all the generators

WIVE' V " ° • F W E ' .V 0 ' (3'4)

This allows the notion of internal "helicity"

"L" ("R") j ^ ^ - (3.5)

splitting the splnor into 2N~l dimensional L(R) components.

The rank of S0(2N) is N. It is convenient to choose N diagonal generators

to be

T2k-1, 2k = h . k • 1, • • ., » (3.6)

678

since their eigenvalues are ±1. Each physical state can be represented by the

ket:

1ej . . .cK> (3.7)

with

TFIVE - HH • • • S • (3'8)

We shall fix v - 1.

He oust decide now which of SO(2N) theories allow chiral fermlons, i.e.

which forbid bare mass terms. Take a gauge rotation

U « e * 1 * E N • 1 Ej • • • £ { ] (3.9)

so that

0* - iNT (3.10)

The bare mass term transforms then in the following manner:

1"f + i 2 N It • (3.11)

Demanding i2N « -1 requires 2N « 4k+2, where k is an integer. In other words,

S0(4k) are bad theories — they are vectorlike. You can similarly show the same

for SO(odd) groups. Of physical interest are only SO(4k+2) groups

S0(10), S0(14), S0(18) ; (3.12)

S0(10) is the minimal orthogonal GUT.

If you feel that you don't need all this to discuss S0(10) — we shall need

spinors in section V again. Let me just illustrate to you how simple and useful

e-notatlon is; you can list physical states by shuffling signs

!+ + + + + > VR Stf(5) singlet

«R

C '- -*• - -•- •"- v ~ 10 of SU(5)

c

679

I- - + - - > d1K )

I- + > d2R

1+ > d3R 2 of SU<5)

' *• - > eE

I + > Vg

It will help you to know that my neutral generators are

1T3L * 4 ( £ " " C 5 )

B-L * - -j ( e j + e2 + E 3 )

1 iQ • x eii - ~7 ( e i + e 7 + e O • ( 3 . 1 3 )yem 2 ^ 6 1 2 3

Let us now discuss the physics of S0(10).

Proton decay and slr.23w

Me cannot predict either of these: S0(10) is just too large a group.

It contains both SU(5) x U(l) and the Pati-Salam SU(2)T x SU(2)D x SU(4).

u K \j

subgroup; the predictions depend on the route of symmetry breaking. In general

Intermediate scales increase Tp, while having enough freedom to keep sln29(f

consistent with the experiment.

Charge conjugation and parity

The lb_ dimensional spinor of S0(10) contains all the left-handed feraions

c — Tf, and their charge conjugates f, = C(f_) (one family). It is perhaps not too

surprising that one of S0(10) rotations changes each feralon Into its charge

conjugate16

S0(10)(3.14)

680

A remark Is in ordar. Notice that the above is a perfectly OK definition

of charge conjugation; under it jy(e.m.) + -jy (e.m.). Furthermore, you can

show the saae for the rest of the scalar fields affected by this transforma-

tion. In general it also changes every left-handed field Into its right-harmed

counterpart (as in (3.14)], In a sense this operation can be viewed as a proto-

type of left-right symmetry. In S0(10) left-right symmetry is the autoraatic

consequence of gauge symmetry. For real couplings <CP conservation) this

coincides vith the usual definition of parity ?L • TR.

This is extremely important. This otherwise discrete symmetry becomes

gauged and, therefore, has profound cosmological consequence* The usual

disastrous domain walls associated with different value now disappear, or at

least are not stable*

Furthermore, C can be broken independent of other symmetries17. You can

still preserve SU(2)R after breaking C. A typical scenario could be

MX

S0(10) • SU(2)L x SU(2)R x SU(4)C x C

» SU(2)L x SU(2)R x U(1)B-L * SU(3)C

+R S U ( 2 ) L x U(1)R x U(1)B-L x SU(3) c

5L S U ( 2 ) L x U(l) x Sl)(3)c

or any variation that you may find appealing. It is important to know that

there are consistent scenarios with M . down to its lower limit - 105 GeV and

MR as low as 1-10 TeV. There could be a lot of new phyBics in S0C10): rare K

decays such as KL * i>e. n-n oscillations, right-handed currents, right-handed

neutrinos, etc. I will not go into details here; we obviously need more

experimental input before anything concrete can be said about SO(10) breaking.

68?

CP violation (weak)

You may or may not keep Yukawa couplings real. If you don't, you will end

up with the usual Kobayashl-Maskawa mechanism. If you do, however, you have aa.

interesting possibility of breaking CP and F spontaneously. You can even do it

at high energies (Mx). There is no decoupling In this case and CP propagates

down to low energies. Imagine

<<f,H> - Hy. ei 9 (3.15)

where I! denotes heavy Hlggs; this could be the only complex quantity. This

could provide hope for establishing the connection between baryon asymmetry and

CP violation in kaon decays , missing in the standard scenario. However, in

realistic models we find additional (real) parameters which obscure this

connection. More work Is needed.

'CP violation (strong)

It is worth mentioning that in S0(10) Peccei-Quinn symmetry becomes just a

fermlonlc number19

where *16 is the basic splnor (one family of fermlons). More important, since

the scale of U(l)pQ is limited to the interval

10s GeV < M < 1012 Gev (3.17)

it is natural to tie MpQ to the scale of B-L breaking (or SU(2)j). Since

the mass of right-handed neutrinos

M ^ - M B L . (3.18)

This leads to the connection between axion and neutrino itass20

682

V H>Q (3.19)

for each flavor.

Feralon masses and jgospln breaking

One of Che central issues in weak Interaction physics is the relationship

between a^ and sy- Can S0(10) shed any light on this question? At first

glance this appears hopeless. Namely, the only way to relate n^ and ny is

take real 10, with the Yukawa coupling

I* - «> T16 +10 *16 + h'c- ^3-2°)

This leads to niu • ay- Nonsense? Hell, not necessarily. Recently, Chang

et al z l point out isospin breaking depends (obviously) on SU(2)R breaking, and

therefore may be achieved spontaneously and induced into the tree-level

relations* Unfortunately, as in the case of CF violation there are unknown

parameters that obscure the predictions• Still, the idea is quite interesting

and should be pursued.

663

IV. DOMMB HAILS AMP STBIBCS

In my mind, the most Interesting aspect of S0(10) theory are possible

topological deformations: domain vails, strings, and monopoles* As an example,

imagine the symmetry breaking pattern

MvS0(10) • S U ( 2 ) L X.SU(2)R X D(1)B-L * SD(3)cx C

• R SU(2)L x D(1)R x n(l)B-L x SU(3)C x C

+L SU(2)L x 0(1) x SU(3)C

Notice that at the MR stage you get monopolee, since SU(2)R is broken to

0(1)R. These monopoles could be fairly light (- 100 TeV?)! At M8L stage,

we should get strings (breaking of U(l)). Actually, since C is broken

spontaneously (but was in S0(10) to start with), one gets domain walls bounded

by strings — rather amazing objects with possible cosmological relevance22.

But we don't know the pattern of symmetry breaking; it's senseless to discuss

precise properties of these objects. Rather, I would like to describe here some

recent work on the spectacular gravitational properties of domain walls and

strings. There are now new exact solutions to Einstein's equations: it is,

however, enough to study Newtonian approxlmae in order to discuss the relevant

physics.

Domain walls23

The simplest example of discrete symmetry is $** theory without a cubic term

. L - j ( S / ) 2 -J U 2 - " 2 ) 2 (4.1)

with symmetry D: t * ->.

The two minima of the. theory are at <•> • ±v. There ought to be a domain

•*11 connecting the vacua. It ia the famous "kink" solution

$(z) « v th z/S

6 - v-1 /7/X

describing infinite, static domain wall in x-y plane.

(4.2)

The energy of this wall is determined by v, and the thickness is given by 5. It

Is easy to write Che fcaergy-monentun tensor of the kink

diag (1, 1, 1, 0) (4.3)

with f(z) • J T * eh'" j and my aetric being (1, -1, -1, -1). For a very thin

wall we could take f(s) - «<*). Actually you can write down T " for a thin wall

just by using Lorentz invariance.

The important feature of (4.3) is tha: its form is equivalent to negative

pressure in x ana y directions: Px - Py - "P. Pz " Oi T h l s lt n o t B 0 s t r a n 8 5 ~"

remenber the T" for the relativistic vacuum: T" - A«" — again negative pressure

(this is the cause of exponential expansion of the inflationary uaiversu). In

any case, the gravitational field in the Newtonian approximation for slowly

ooving objects (like us) i»2>t

- 4*0 f(z) (4.4)

685

Domain walls are "antigravitatlonal" — they repel ordinary natter. Science

fiction dream coming true? I find this fascinating.

We have studied Infinitely large domain walls; such objects are not allowed

to exist. It is sufficient, of course, that the wall 1B large compared Co the

material body It affects. However, for fast-moving walls this analysis breaksv

down.

Strings

We can perform the sane analysis for strings. Imagine a static, infinitely

long, thin string along the z-axis. Its energy-oomeatun teaser is

T|J - Wi(3c) «(y)diag (1,. 0, 0, 1) (4.5)

where u is the linear energy density.

Now we have a miraculous cancellation (again 'negative pressure" in the z

direction)

'*• " 0 (4.6)

No gravitational field for slowly noving objects!

However, light does bend as Vilenkin shows. He found the exact solution to

be (In "cylindrical" coordinates)2*1

ds2 - -it2 + (dz2 + dp2 + p2dfl) (4.7)

where a varies in Interval [0, (l-4Gu) 2*]. The space has "conical" form, and

if you can travel a large distance you would know it.

686

V. oHOTCATIOB OF FAMILIES

S0(10) Is a beautiful theory, and yet It mistreats the families: they are

just copies of each other. If perturbatlve unification is to shed light on the

faally problem, we mist go beyond S0(10), sticking for the reasons of chirality

to S0(10+4V). Before we do the counting of state, we mist analyze the

properties of the spectrum. He take

Y F I V E " «I «a • • • «5 • • • e s + 2 k " l <-5-»

which implies two possibilities

(1) E l . . . 5 - 1 (ii) ex . . . e5 - -1 (5.2)

By definition case (i) denotes usual fermions, lb_ dimension spinors, and (11)

describes T6's. Hence, the major prediction: for every ordinary feralon f with

V - A currents there is a mirror fermlon F with V + A weak interactions25'26.

Recall that mirror symmetry, as defined In (3.9), forbids the bare nass terms of

the fora fF.

This implies that S0(14) is not large enough. Its spinor has dimension 64,

leading to only two ordinary families and two mirror families. The minimal

group la then S0(18).

S0(18)

The S0(18) spinor is 256 dimensional: it contains eight families and eight

mirror families. This is far too aany if we are to maintain asymptotic freedom

needed for perturbative unification. The strategy is to break SO(1S) down to

S0(10) x S0{8), S0(8) being the fatally symmetry. The physics depends on the

next stage of symmetry breaking.

(a) Bagger and Dioopoulos break S0(8) to U{1)H, needed co protect some

families. They end up (demanding asymptotic freedoo) with four families (and

four mirror families).

687

(b) Chang, Huboch and Mohapatra28 use Sp(4) x SU(2) as a protective

symmetry and end up with three families.

(c) With Wilczek and Zee we demand Peccei-Quinn symmetry to solve the

strong CP problem. We get three families.

Although the number of families is not fixed, in this context it can only

be three or four. In any case, there are mirrors. Where are they? Let us,

rather quickly, verify the consistency of their existence29.

(i) They barely mix with ordinary fermlons. The leading mixing term must

be dimension 5,

L = i- f ^ F , (5.3)mix Hx

where $2 symbolically denotes the presence of two SU(2) x 0(1) doublets (to fora

a. singlet). Therefore, the mixing is

9fF * 5 " 10~13 (5-4>induced through weak breaking. You can produce mirrors only in pairs. The

reason we don't see them is that they are heavy (why?).

(ii) The mixing has Important cosoological implications. The lifetime of

the lightest mirror fermlon becomes

Tp . _ . ! sec

for My = 100 GeV, % * 10ls GeV . (5.5)

This is why geologists cannot find them. Presumably, they all decayed prior to

nucleosynthesis.

(ill) Mirror fermion masses are bounded from above

MF < 250 GeV (5.6)

or else the couplings would blow up before Mx- You ought to see mirrors in

666

supercolliders* I don't know, however, why they are heavier than ordinary

fermions.

(lv) Neutrinos: there i6 a danger of too many neutrinos. The hope is that

SOUK of the SU(2) x U{1) singlet components survive the "see-saw" mechanism,

find their doublet partners and form neutral Dirac leptons, with masses on the

order of the weak scale. This reduces the number of light neutrinos. In the

pure S0(18) case, Chang, Kumar and Mohepatra30 claim the necessity of

intermediate B-L scale being less than 1011 TeV.

It is maybe worthwhile to be cautious at this point. The number of

neutrino species will soon be counted (or at least limited) In Z° decays. For

all that we know this number may surpass the cosmologlcal limit. For example,

there is Linde's point that large neutrino degeneracy cen eliminate the helium

abundance limit. And a large lepton number may be even consistent with GL'TS

such as S0(10) (or SO(16))31.

(v) We should find some deeper explanation of this strange fact that all

mirror fermions are very heavy; possibly heavier than all the ordinary fermions.

I hope you are convinced Chat conventional GUTS, if large emush, can offer

a solution to the family puzzle. Families can be naturally unified in

orthogonal groups. If so, we shall definitely find mirror matter with unusual,

right-handed weak interactions. The fate of this, as is only appropriate, lies

now in the hands of experimentalists. Do not be afraid if the message is

somewhat narcissistic: we ought to look more into mirrors.

689

VI. OUTLOOK

I guess, after all is said, the face remains that the proton is still

stable. It is not surprising that the enthusiasm for GUTS (at least with the

usual Higgs mechanism) has somewhat diminished. A lot of people feel that the

minimal SU(5), the theory that caused most of the excitement, is ruled out. It

is not so easy to make the case for grand unification in such an atmosphere.

However, the problem is not with these theories in general; the problem is that

we got carried away mainly due to the sin 2^ prediction. But, as I have shown

here, this prediction could still be OK, and proton could still be unstable,

just long-lived due to, say, supersymmetry or a larger group, such as S0(10) or

. . . . A good many among us believe in supersymmetry, anyway, so let us w>t

worry yet. Let's keep in mind that verification of GUTS will not come easy, if

it comes at all. This is somewhat ironic. The more we study them, the oore of

their interesting properties seem to be uncovered. These theories can contain

monopoles, domain walls and strings, with profound implications. Monopoles may

catalize baryon decay. Strings may be responsible for galaxy formations.

Strings and dooain walls have fascinating gravitational properties. Even more

important, GUTS nay explain the genesis of matter. They are beautiful theories,

rich in physical and cosmological phenomena. X do hope they do not end up being

another victim of the ugly facts of nature.

Their only serious setback: they leave gravity completely out of the

picture. If we could judge by this simmer's excitement, the string theories may

achieve this monumental task. And if the quarks and leptons are elementary

particles, the true GUT should then be based on an E(6) model, resulting from

E(8) x E(8) heterotic superstring theory. If correct, this theory would explain

the hierarchy problem and determine the number cf families. My ignorance

prevented me from summarizing the situation at the moment, and, in any case, it

is done by Ovrut32, in these Proceedings.

690

ACKNOWLEDGEMEKT

I am deeply grateful to the organizers of this excellent meeting, Z. Ajduk,

S. Pokorski, A. Uroblevski and others for the beautiful week In Kazimierz.

Thanks are also due Prof. Abdus Salam for the warm hospitality at ICTP, Trieste,

where most of this talk was written.

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Vile) M B . C.W. ua&nMiM2.

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