Energy-momentum transfers in high-energy nuclear collisions

IL NUOVO CIMENTO YoL. XXX, N. 3 1 o Novembre 1963

Energy-Momentum Transfers in High-Energy Nuclear Collisions.

I~. I~A(;ItAV.~X, T. N. R~-c~.~swa3rr a nd A. ~um~A~rA:q~:<

Tata L~.~'titttte of l h tndamen ta l Research . Bomba.q

(ricevuto il 27 Maggi() 1963)

Sum m ary . It is pointed out that tile form of the distribution of energy- momentum transfers in the (( diffraction region ,~ of high-energy elastic i)roton-protol~ and pion-nucleon scattering is applicable to the (~longitudinal ,) enei'gy-molnentum transfers in high-energy ( ~< 10 ~2 eV) inelastic nucleon-nucleon collisions. Except for a change in the value of a para- meter of the distribution, the same is also applicable to the inomentum- transfer distribution in the collision of the virtual particles into which the nucleon-nucleon collisi()ns can be decomposed on the basis of an assumed model. The implication ()f applying these energy-momentum- transfer distributions to the ease of pion-nucleon collisions is examined briefly.

1. - I n t r o d u c t i o n .

Tile t r an sve r se -n lomen t~mt d i s t r i b u t i o n of secondar ies p roduced in hig'h-

energy ine las t ic nuc lea r collisions is s imi la r to the m o m e n t u m - t r a n s f e r d is t r i -

b u t i o n in h igh-ene rgy elas t ic p-p and ~-A" collisions (Fig. 1). This figure shows

t h a t errors in the da t a al low a l a t i t u d e in the choice of the form of the t r ansve r se -

m o m e n t u m d i s t r i b u t i o n , and to a first a p p r o x i m a t i o n i t is no t i n c o n s i s t e n t to

assume t h a t a t least up to abou t 0.7 (}eV/e (*) i t is the same as the m o n m n t u m -

t r ans fe r d i s t r i b u t i o n in elast ic p-p collisions. This fea ture has led us to e xa mi ne

the consequences of the a s s u m p t i o n t h a t th is d i s t r i b u t i o n applies also to ~, l on -

(') I t is possible that the transverse-momentuln distribution for the secondaries produced in high-energy nuclear collisions can be expressed as the sum of two expo- nentiM distributions, i.e., F (pa) 2p • ~ 2 = [---P• + b/p~ .exp [--p~,/p~]}, the con- stunts a, b, Pc and Pl being chosen to fit the data (a -- 0.97, b = 0.03, p0 ~ : : 0.06 (GeV/c} 2 and p~ = 0.56 (GeV/c)2). See Section 2.

1.0

01

0.01

It', l~,,o

\

Fig. 1. - A comparison of the transverse momentum distr ibution of the secondaries resulting from high-energy inelastic nuclear collisions and

the momentum-transfer distr ibution in elastic p-p scattering. The momentum transfer in the la t te r type of collision is approx-

imately the square of the transverse momentum generated in the collision. The other lines drawn for comparison

are the empirical formula suggested by Coccoxi et al. (1) and the formula derived by M~LEClN and Ro-

\ SENTAL(2) on thermo-dynamic considerations ~ ( temperature of pion gas ~m~ which has often

\ k been found to agree with the exper imenta l �9 x ~ 3 , _ data). - - - elastic sca t te r ing( ) p-p:

x exp [--9.4p~_]; . . . . . CoccoNz et \~\. ~, T al. (1): exp [ - 5.5p• �9 . . MIL]~-

\.K k ~ , ! ~ CIN and ROSENTAL (2); @ and \

�9 \ \~ \ ' , ~ , @ positive, and negative se- 1 ,d", eondarles from 24GeV/c

" x x T 4 +Z T\ , I \ , , , p-p();OG--SEvAet I " \ ' ~ \ ", + aL (5), ~300 GeV

\ / ".,,\ " ' , ] ZY'-~; �9 Awu-

- x ! _~ \ "', / HANSEI~" (7); [] �9 / r ' ~ , . . ,< quasi-elastic p-p

1 "~ x 3

01.1 i i J 0 0.2 0.3 0.4 0.5

p~ (GeVlc) 2

j \ , \ �9

0.6 0.0025

(1) Car. COCCONI, L. J. KOESTER and D. H. PEP~KINS: High-Energy Physics Study Seminars, No. 28 (Part 2), UCID-1444 (1961).

(2) G. A. MIL~ClN and I. L. ROS~NTAL: Suppl. Nuovo Cimento, 8, 770 (1958). (3) G. Coccom: Proc. o] the 1962 International Co~]erenee on High-Energy Physics

at CER~V, p. 884. (4) p . DODD, M. JOBES, J. KINSON, B. TALLINI, B. R. FRENCH, H. J. SHERMAN'.

I. 0. SKILLICORN, W. T. DAVIES, ~[. DERRICK and D. RADOJICIC: Proc. o] the Aix-en- Provence Con]erence (1961), p. 433.

(5) V. V. GVSEVA, N. A. DOBROTIN, N. G. ZELEVINSKAYA, K. A. KOTELNIKOV, A. M. LEBEDEV and S. A. SLAVlTINSKY: Proe. o] the I~ternational Conference on Cosmic Bays, Kyoto (1961), p. 375.

(6) E . R . T . AWUNOR-R~NNER, L. BLASKOVITCH, B. R. FRENCH, C. GHESQUI]~RE, I. B. DE I~[INVIELLE-DEvAux, W. W. NEALE, C. PELLETIER, P. RIVET, A. B. SAHIAR and I. 0. SKILLICORS: Nuovo Cimento, 17, 134 (1960).

(7) L. F. HASSle\ and W. B. FRETTER: Phys. Rev., 118, 812 (1960).

E N E R G Y - M O M E N T U M TRANSFERS IN H I G H - E N E R G Y NUCLEAR COLLISIONS 7 9 3

gi tudinal )> e n e r g y - m o m e n t u m transfers in inelastic collisions. I t is shown in Sect ion 2"1 t h a t in nucleon-nucleon collisions, the observed dis t r ibut ion of

inelast iei t ies is in rough agreement wi th wha t follows f rom this assumpt ion.

I n Section 2"2 an a t t e m p t has been made to re la te the above four -momentum-

t rans fe r d is t r ibut ion to the features of the emission of secondaries in the col-

lisions. I t is found t h a t a re la t ively f la t ter d is t r ibut ion of momen tum- t r ans fe r s

t han the above is needed to account for the observed mult ipl ici t ies in nucleon- nucleon collisions. The case of inelastic pion-nueleon collisions is briefly

discussed in Section 3 in the l ight of the above t r e a t m e n t of nucleon-nucleon collisions. Final ly , in Section 4, a recent work of FRAUTSCHI (s) o n somewhat

similar lines is compared wi th the resul ts obta ined in the present paper.

2 . - .N'- .W c o l l i s i o n s .

2"1. I n e l a s t i c i t y d i s t r ibu t ion . - Since ba ryons have to be conserved in the collision and the exper imenta l resul ts indicate t h a t the dominan t inelastic

collision is the one in which the nucleons persis t a f te r the collision with a consid-

erable f rac t ion of thei r ini t ial energies, we can wri te the ene rgy -momen tum t ransfer to the nucleon in an inelast ic collision as (*) (**)

(1) q~ = 2 q L + q~ ,

where ql and q~ represent the t ransverse and longitudinM components and

q~-~ p~; p z is the t ransverse m o m e n t u m acquired by the nucleon in tile col-

lision. The dis t r ibut ion of ql is wha t is essential ly observed in small-angle elastic sca t te r ing at high energies and the d is t r ibut ion of this is given by (1)

(~) F(qkL) = 2q• exp [ - - q~L/q~] ,

where qo ~ 0.l (GeV/c) 2. Since the p• d is t r ibut ion is more or less the same

for both elastic and inelastic collisions, we assume tha t the q~ dis tr ibut ion is

tile same as t ha t of q i and independent of the l a t t e r (***). The form of F(q)

(8) S. C. FRAUTSCHI: NUOVO Cimento, 29, 409 (1933). (*) q2 is taken to be positive.

(*') Even if the nucleon emerges in one of its isobaric states, it is possible to write the relation (1) which will then apply to the nucleon resulting from the decay of the isobaric state; in this respect q2 will be a phenomenological quantity applicable to the nucleon (see Section 2"3).

(***) In this connection it is interesting to note that Cocco~i et al. (2) have suggested that the transverse-momentum distribution is independent of the longitudinal-momentum distribution for protons and pions in inelastic p-.N' collisions.

,~ 51 - 11 N u o v o C i m e n t o .

794 R . R A G H A V A N ~ T . N . R A N G A S W A M Y a n d A . S U B R A M A N I A N

in (2) would then imply tha t the q distr ibution will be governed by the same

exponential as applies to qz in (2). Unfor tunate ly , we do not have reliable

and stat ist ically significant experimental da ta to check this assumption regard-

ing the dis tr ibut ion of q. Since q2__ 2 M T , where M is the mass of the nucleon

and T is the kinetic energy of the recoil nucleon in the inelastic collision~ a

comparison of the p• and T distributions for the relatively low-energy recoil

nucleon is very valuable and should be easy to obtain experimentally.

Once the assumption regarding the distr ibution of q~ and q~ is mad% it is

s t raightforward to calculate the distr ibution in inelasticity, or the momentum

distr ibution of the outgoing nucleons in the c.m. system, since

(3)

=

A P * : - - AE*: ,

(( z ~) represents the axis coincident with t he line of flight of the nucleons before

v

30

20

9 -

IO

/=,

i i i

t

r

J

0.5 10

col l i s ion; A P*, AE* are the momen tum

and energy loss of the nucleon consid-

ered in the c.m. system and k is the

inelastici ty coefficient in the c.m. system

Fig. 2. - The distribution of inelasticity coefficients expected on the basi s of the assumption in the text. Histogram is taken from the work of Gusv.vA et al. (s) at ~300 GeV. Solid line represents the distribution of inelasticity coefficients calculated assuming a distribution for qJl as given by (2) and the broken line assuming a distribution for q, which follows more closely the distribution as given in the foot- note on page 791. The; broken line may be taken to be a better fit to the experimental data since there is likely to be a bias for high-ine-

k lasticity events Jn the sample of cosmic-ray

events (i.e. very-low-multiplicity events are not efficiently observed).

[ _ k ~ M ~] M 2 (2--k) exp , qo2-- - 0.1 (GeV)2; ~(k)= q-~ k (1--- k~)-~ ( l - -k)

q~(k)= M2~ ( l - -k ) 2 (q-~ exp [(l_k)q2oj § q~ exp 1 - - k ql

a = 0.97, b ~ 0.03, qo 2 = 0.06 (GeV) ~, q[ = 0.56 (GeV) ~.

E N E R G Y - M O M E N T U M T R A N S F E R S I N I | I G I I - E N E R G Y N U C L E A R C O L L I S I O N S 795

def ined to be A E * = kE*; E* = ene rgy of t h e nucleoY~ in t h e c .m.s , a n d M

is t h e m a s s of t h e nuc l eon (*). The a b o v e a p p r o x i m a t i o n s are v a l i d in t h e

case of t h e sma l l t r a n s v e r s e m o m e n t u m g e n e r a t e d in t h e col l is ion as sugges t ed

b y t h e s t e e p l y f a l l i ng e x p o n e n t i a l d i s t r i b u t i o n (2). The i n e l a s t i c i t y d i s t r i -

b u t i o n in ,N'-3~ col l i s ions a t --, 300 GeV a n d t h e c.m. m o m e n t u m s p e c t r u m

of p r o t o n s in 24 GeV p - p col l i s ions a re c o m p a r e d in F ig . 2 ~nd 3, a n d these

c o m p n r i s o n s show t h a t t h e m a j o r p o r t i o n of t h e i n e l a s t i c col l i s ions is g o v e r n e d

b y n (( d i f f r a c t i v e - t y p e ~ of m o m e n t u m - t r a n s f e r d i s t r i b u t i o n for q , . I n d e e d

t h e a v e r a g e i n e l a s t i c i t y e x p e c t e d is 0.2 G r o u g h l y in a g r e e m e n t w i t h exper i -

m e n t a l d a t a a t a l l ene rg ies (9). F u r t h e r , i t is poss ib le to ~ccoun t for t h e u sym-

m e t r i e s in n u c l e o n - n u c l e o n col l is ions

b y a s s u m i n g t h a t t h e q, d i s t r i b u t i o n s

for t h e two nuc leons a re i n d e p e n d e n t .

I f k~ a n d k~ d e n o t e t h e i n e b t s t M t i e s of

0.6~- . . . . . . . . . . . . . . .

0.4 Fig. 3. - Dis t r ibut ion of momenta of outgoing nucleons in the c.m.s. Histo- gram is from the work of DODDet al. (4) ~ Q, at 24 GeV and the calculated curve Q- (normalized to the area of histogram) 0 2 according to the dis t r ibut ion in q, assumed to be the same as (2). The peak in the exper imental dis t r ibut ion very near the elastic l imit is not accounted for by the calculated curve. I t has been explained (lo) tha t this peak is due to 0 quasi-elastic scat ter ing of nucleons and we have not specifically taken into account this process which is supposed target nucleon (1).

L 1 2 2.4

p; (G eV/c)

L

\ 3 6

to go through T = � 8 9 isobar excitat ion of

1

$ $2 EoV~PI + 21I e

2}] (*) See Section 2"3 for a discussion of the effect of isobaric excitat ion of the nucleons

in the collisions. (9) D. It . P~RKI~S: Proc. oj the International Con]erenee on Theoretical Aspects o]

Very-High-Energy Phenomena, CERN, (1961), p. 99. (lo) D. R. O. MORRI~O~N: CERN 61-22, p. 153; Air-en.Prov nc~ Con]ere~we Report,

vo|. 1, 407 (1961).

796 R. RAGHAVAN, T. N. RANGASWAMY and A. 8UBRAMANIAN

the nucleons in the c.m. system, and if Q ~ 2 = (q,,)a/(q~ )~ = k~(1 ~ k2)/k~(1 - - k~), the

Q d is t r ibut ion derived f rom the above assumpt ion agrees well wi th exper imenta l da ta as shown in Fig. 4.

20'

to (D tn

u l O

\ \

' ~ 3 cases , [----],

3 5 7 9 Q

Fig. 4. - Distribution of the ratio Q=q~/q~ where q~= (k~/(1--kl)) M S and q~= (k2/(1 - -k2))M 2 taken from the work of Gcs~vx et al. (s) compared with the expected distribution of the same given by F(Q)= eonst/(1 +Q)2 suitably normalized to the histogram. (k 1 and k~ have been taken to be k~b and km~ .... as given in ref. (5)). ____ F(Q)= const/(l+Q) 2.

2"2. F e a t u r e s of the e m i s s i o n o/ the secondaries . - Thus far no th ing has been said abou t details which re la te to the inelastic channel, viz., how the

field surrounding the colliding nucleons radia tes the energy released by the

nucleons. I f the inelastici t ies of the nucleons in the c.m. sys tem are specified, we need a t leas t one more var iable to make a fu r the r analysis of the resul ts

of the collision. Fo r this purpose, we consider t h a t the d iagram in Fig. 5 is

the dominan t one (*) where the v i r tua l par t ic les car ry ing ene rgy -momen tum ql and q2 have to exchange space-like m o m e n t u m ql to sca t te r real par t ic les at

the two ver t ices I and I I . The question arises as to the d is t r ibut ion in ql.

We have examined the consequences of the assumpt ion t h a t q~ is also distri-

bu t ed as qx in (2). I t is then found, as shown below, t ha t the d is t r ibut ion in qj

which is needed, has to have a value of q~ ~ 0.6 (GeV/c) 2.

(') I t is necessary to insist on this diagram at all energies (i.e., ~ 10 GeV) in order to preserve the constancy of the small mean value of k in 2~'-~ collisions; the one. pion- exchange diagram considered by SALZMA~ and SALZ~AN (11) for example, in which the outgoing baryon and the meson lump are clubbed together, leads to difficulties in accounting for the angular distribution of the secondaries and the inelasticity coefficient in high-energy ~ ' - ~ ' collisions. However, it cannot be excluded that the outgoing baryon in the diagram is in an excited state (Section 2"3).

(n) F. SALZ~AS and G. SALZMAN: Phys . Rev. , 120, 599 (1960).

E N E R G Y - M O M E N T U M T R A N S F E R S I N H I G H - E N E R G Y N U C L E A R COLLISIOINS 797

For the sake of simplicity it is assumed tha t qs has

in the e.m. system of the vir tual particles (*)(~2).

Then M n and Mm, the effective masses of the created

particles at the two vertices are predicted. To obtain

the actual multiplicities of secondary particles pro-

duced, we have fur ther assumed tha t due to strong

final-state interact ions an isotropic distr ibution of

momen tum components p• and p, result in the c.m.

system of Ms~ and M m for the secondaries; the

distributions of p . and p, are given again as per

equation (2). The la t ter assumption brings us close

to the (~ fireball ,~ model of meson product ion pro-

posed earlier by several authors (~3). The momen tum

distr ibut ion in the (( fire-ball ~), resulting from the

(~ diffractive type ~ of m o m e n t u m distr ibution (**),

compares well with the experimentally determined

no component in energy

inelasticity r ~:,P.M ., E,.P,:M

, ~)

E:,Po',M

l q, I~ >'~

lq,

I% 4> k, E ; ~ t M

Fig. 5. - The <( internal ~ collision vertices in 3~-,W

collision.

spectrum of pions in ~N)-W annihilations (~4) (Fig. 6a) and in high-

energy nuclear collisions involving 24 GeV/c protons (Fig. 6b). in Fig. 7 the

observed mult ipl ici ty distributions at various energies are compared with

those calculated on the assumption tha t k ( = k~= k2) has a distr ibution as

u... o

,:5 s

0 200 400 600 8001000 p (MeV/c)

Fig. 6a. - Distribution of the momenta of pions emerging from nucleon-antinucleon annihilation events observed in a hydrogen bubble chamber (1~). This is compared with the momentum spectrum deduced from the transverse.momentum distribution assumed according to relation (2) in text and also assuming that Pu is distributed similar to p• but independent of it. Cur- ves: F(p) = (2p3/p'o) exp [_p2/p~]:

- - p~ = 0.06 (GeV/c)2;

- - - p~ = O.lO (GeV/cp .

(') This situation is analogous to the elastic scattering of real particles. (12) E. G. BUBELEV: Proc. o] the Moscow Cosmic Ray Con]erence, vol. 1 (1960), p. 285. (la) p. CIOK, T. COGHEN, J. GERULA, R. HOLYNSKY, A. JURAK, M. MI]~SOWICZ,

T. SANIEWSKA, 0. STANISZ and J. PERNEGR: Nuovo Cimento, 8, 166 (1958); K. NIu: Nuovo Cimento, 10, 994 (1958); G. Cocco~I: Phys. Rev., i i i , 1699 (1958).

(**) It is interesting to note that the (( diffractive type ,~ of momentum distribution more or less corresponds to the energy spectrum of pions deduced from thermodynamical considerations for a temperature of T = m~ for the <(fire-ball~)(3).

(14) N. HOI~WITZ, D. MILLER, J. MURRAY and R. TRIP: Phys. Rev., 115, 472 (1959).

798 R. R A G H A V A N , T . N . R A N G A S W A M Y and A. S U B R A M A N I A N

~40

0 1.0 P* for 1"(- (G eV/c)

2.0

Fig. 6b. - A comparison of the momentum spectrum of negative pions observed i n 2 4 GeV p .p collisions by DODDet aL (4) and the expect . ed dis tr ibut ion for the same on the basis of isotropic dis tr ibut ions of momentum in the c.m.s., i.e., distr ibutions in p• and Pit are the same and independent of each other. A large fraction of the pions are accounted for by this spectrum. The observed ta i l of the spectrum is due probably to low-mult ipl ic i ty events where there is no equilibrium between P~L and Pl~ in the c.m.s, for the pions. Curve: normalized distr ibution i

2 P *a [__P*21 /(P*) = ~ exp p~ j , p~ = 0.1 (GeV/e)2.

Fig. 7. - A comparison of the dis tr ibut ion in the mult ipl ici t ies of the secondaries in J~-3~ collisions a t different p r imary energies: (a) 24 GeV (4), (b) ~ 300 GeV (s) and (c) ,-~ 10 TeV (15). These distr ibutions have been compared with the calculations made as out- l ined in the text . The broken lines represent choice Qf same form as (2) for q and qj with q~= 0.1 (GeV/c) 2 and the full lines for the choice of dis t r ibut ion (2) w i th qo~l= 0.1 (GeV/c) 2 f o r q and q~2= 0.6 (GeV/c) ~ for ql, A mean value of 400MeV per pion in the c.m.s, of the (~ f i re -ba l l s , has been used in the calculations to obtain the multiplicit ies. I t is diffi- cult to assess the bias involved in the experi- menta l mul t ip l ic i ty dis tr ibut ion in (b) and therefore i t cannot be concluded tha t there is lack of agreement with the calculated curve. The good agreement of the dis tr ibut ions at 10 TeV must be considered fortuitous because of the uncertaint ies involved in a t t r ibu t ing the his togram to an average energy of 10 TeV and possible biasses involved in the collection of the exper imental data . Also the region of 10 TeV belongs to t h e mul t ip le fire-ball region (16) and our calculations may not be valid.

200 a)

0 5 10 15 multiplicity of pions

20~ L b)

10 / ' //I

0 5 10 15 2O

l:t/ Ln r

0 10, 20 30 40 rnult/pl/c/ty of charged p/ons

(z5) p . K. MALHOTRA: preprint , Ta ta Ins t i tu te of Fundamenta l Research (1962). (16) :S. HASF, GAWA: Progr. l'heor. Phys., 29, 128 (1963).

ENERGY-MOMENTUM TRANSFEI~8 IN I f I G I I - E N E R ( ] Y NUCLEAI% COLLISIONS 799

shown by t h e solid lille in Fig. 2 and qs has ~ dis t r ibut ion identical to (2). The calculated multiplicdties are clearly too low. Much be t t e r agreement

resul ts if q~ in (2) is taken as 0.6 (GeV/c) 2 for the d is t r ibut ion in ql only. The average mul t ip l i c i ty is expected to rise app rox ima te ly as /~*~ for the fixed

(energy-independent) qr dis t r ibut ion which is assumed. I t should also be poin ted out t h a t Fig. 5, which is t aken to be the dominan t

d iagram for inelast ic ..h'-iT collisions, has the a t t r a c t i v e fea ture t ha t i t predic ts

t ha t low-mult ipl ic i ty events , resul t ing f rom smaller t han average qr, will have

an angular d is t r ibut ion of the secondaries peaked in the forward and backward

directions in the c.m. system, due to the re la t ively high longitudinal m o m e n t a

of the secondaries produced in such cases. Such a correlat ion be tween mult i -

pl ici ty and angular d is t r ibut ion in the c.m. sys tem exists even a t 24 GeV (4).

A crude predic t ion t ha t can be made on the basis of the independence of k

and q~ for the re la t ionship between the mean energy of the secondaries in the

c.m. sys tem and the nml t ip l ie i ty of the secondaries is E ~ . n ~ = const; E~ is the mean energy of pions in the c.m. sys tem and n~ the mul t ip l ic i ty of pions

for a group of events at a given p r i m a r y energy. This predict ion roughly

agrees wi th the exper imenta l dat.~ (4). Recen t work by HASEGAWA (16) suggests t h a t the f o u r - m o m e n t u m transfers

be tween fire-balls in high-energy nuclear collisions is peaked around a value of ~-~ 1 (GeV/c) ~. I t m a y be of in teres t to point out t ha t the d is t r ibut ion of q:

given by Hasegawa has a form similar to (1) for q 2 > 0.5 (GeV/c) ~ but with

q~ ~ 1.4 (GeV/c) 2. I f such large q~ apply, then the average inelast ic i ty will

shoot up to ~ 0 . 7 . I f the inelas t ic i ty for the nucleons is found to be small

even in the mul t ip le fireball region, we have the in teres t ing s i tuat ion t ha t the d is t r ibut ion of q for the nucleons is different f rom the q-distr ibution for the <~ in ternal lines ,> in the collision. In this paper i t has a l ready been poin ted

out t ha t the dis t r ibut ions for q and qs are character ized by different values of the cons tant q0. Whe the r the d is t r ibut ion in ql changes wi th the energy of the in te rac t ing nucleons is a m a t t e r for futm'e invest igat ion.

2"3. Isobaric excitation o] nucleons. - The model t h a t has been assumed

for nucleon-nucleon collisions, as p ic tured in Fig. 5, leaves no room for isobaric

exc i ta t ion of the nucleons which process is suspected to be i m p o r t a n t a t high

energies (:7). I t is not ye t establ ished if in a large propor t ion of the collisions

actual isobaric exci ta t ion takes place. For example , any collision process in

which q~ is small (Fig. 5) cont r ibutes to forward and backward going <~ fire-

balls >> in the c.m. sys tem and leads to collision k inemat ics similar to t h a t of

isoba~:ic exci ta t ion of the nucleons. Therefore we have made no specific assump-

t ion regarding the product ion of isobars in the collisions. Even if isobaric exci-

(tT) B. PETEI~S: f'roc, o/ the 1962 International Con/erence on High.Energy Physics, C+ER~Y, p. 623.

800 R. RAGHAVAN, T. N. RANGASWAMY and A. SUBRAMANIAN

ra t ion of the nucleons takes place in almost every collision, our conclusions regarding the magni tudes of q and qf will remain practical ly unchanged, i.e., the dis t r ibut ion of the former will be characterized by a q~l ~ 0.1 (GeV/c) 2 and tha t of qf by a q~2 ~ 0.6 (GeV/c)2>>q~. The above conclusion is arr ived at by an approximate analysis in which the e//ective inelast ici ty in the collision

is kept small ~ 0 . 3 .

3. - Pion-nucleon collisions.

Since i t is found tha t elastic scattering of pions on nucleons shows the same dependence on momen tum transfers as elastic p-p scat ter ing (1), i t is of interest to examine whether the distr ibution of q~ in (1) for inelastic pion-nucleon collisions will be similar to tha t of q~ given by (2). According to (3), the inelasti- c i ty in such collisions will be near ly un i ty since q~t is now reduced in the ra t io m~/M s for a given inelasticity. An appreciable fract ion Of the inelastic cross-section of pions is found to be (~ peripheral ~) (18) and i t m ay be tha t these collisions arise f rom forces of a longer range than those tha t are responsible for the h i ther to observed elastic p-p and ~-2~ scattering. Some of the inter- pre ta t ions of the peripheral interact ions of pions are based on ~-= in teract ions (19); such an approach would imply tha t a l though the mo- m e n t u m transfers in elastic 7:-7: scatterings are quite small and distri- bu ted probably ve ry steeply with qo s in (2) approximate ly equal to 0.1 ( G e V / c ) ~ . m ~ / M ~ 0.002 (GeV/c) s, i t is relat ively easy to de tec t these very small m o m e n t u m transfers in (~ peripheral ~) inelastic collisions of pions ra ther t h a n in elastic scatterings of pions off nucleons.

4. - Comparison with the recent work by Frautschi.

While the present work was near ing completion, a recent work by FttAUTSCHI (8) WaS brought to our a t t en t ion where ideas somewhat similar to ours have been expressed. We should like to point out some differences

in the two approaches. The type of diagrams, dis t inct f rom th a t in Fig. 5,

which have been considered by Frautschi and which were proposed earlier

(,s) D. R. O. ~r CERN 61-22, p. 153; Aix-en-Provenee Con]erence Report, vol. l , 407 (1961); Proc. o] the 1962 International Con]erence on High-Energy Physics, CERN, p. 606; G. BELLINI, E. FIORI~I, A. J. H]~RZ, P. NEC, RI and S. RATTI: Proc. o] the 1962 International Con]erenee on High-Energy Physics, CERN, p. 613; SIDDHE- SHWAR LAL, R. RAGHAVAN, B. V. SREEKANTAN, T. N. RANGASWAMY, A. SUBRAMANIAN and S. D. VERMA." Proc. o] the 1962 International Gon]erence on High.Energy Physics, CEBN, p. 641.

(19) S. D. DRELL and K. ItII~)A, quoted by D. R. O. MORRISON: Proc. o] the 1962 International Con]erence on High-Energy Physics, GEtCN, p. 606.

ENERGY-MOMENTUM TRANSFERS IN H I G H - E N E R G Y NUCLEAR COLLISIONS 8 0 1

by SALZ~IAN and SALZMAN (11) encounter difficulties in the explanat ion of the

angular distr ibution of the secondaries and the small inelasticity coefficients

(especially at E o --~ 300 GeV), as pointed out earlier in this paper. Frautschi ' s

work is oriented towards explaining multiple (~ fire-balls ~) at extremely-high energies (,,o) whereas our model is applicable to pr imary energies ~ 1012 eV.

I t shows essentially how at high c.m. energies the (( fire-balls ~) emerging from

vertices I and I I (in Fig. 5) should be factored into multiple (( fire-balls ~),

(once ((damping)) of m o m e n t u m transfers is assumed), and this leads to the

interest ing result tha t there will be a logarithmic rise in the mult ipl ici ty of the

number of (( fire-balls ~) with increasing energy of the collision in the c.m.

Frautschi assumes tha t the size of a (( fire-ball )) at a ver tex is always near

the max imum size (rest energy of Mn, M1~---2 to 3 GeV) so as to be in

conformity with experimental data (1,); it is not clear how far the group-

ing of secondaries into (( fire-balls )) vi t iates the true mul t ip ic i ty distr ibution

of (( fire-balls ~> at each vertex. We have allowed a snlooth var iat ion of the

average mult ipl ic i ty of the secondaries at each of the (( fire-ball ~) vertices.

I t is not easy to see how the actual mult ipl ici ty will grow with increasing

energy in the e.m. system once the demand for a part icular mass for the

(~ fire-balls )) is not satisfied; there may be a smooth growth of the size of a

(( fire-ball ,~ up to a certain limiting size before it breaks up. Qualitatively

one can say tha t a fur ther factoring of the (~ fire-balls )) at vertices I and I I

should lead to an overall dependence of the secondary mult ipl ici ty on the c.m

energy (2Eo) which is sonlewhat slower than the E *�89 law found for a fixed q~

distr ibution and only two vertices introduced for the (( fire-balls ~).

Another interest ing feature of Frautsehi ' s work is the difference between

~-A ~ and 5 ' -A ~ collisions. Pions are expected to produce more secondaries

in a collision as compared to nucleons and their inelasticity coefficients are

also expected to be relatively higher. I t should be pointed out tha t at present

there is no definite experimental indication for either of these two predictions. If at all, mult ipl ici ty in pion-nueleon collisions is equal (21) to or probably

less than tha t in nucleon-nucleon collisions (a5,~2) and a sizeable fraction of the

inelastic collision cross-section of pions is ((peripheral ~)(~8) which takes us

to the remarks made in Section 3. I t is impor tan t to determine experimentally

the mult ipl ici ty of secondaries produced in ~:-~" collisions and also the effective

inelasticity coefficient for these collisions.

(20) S. HASEGAWA: Progr. Theor. Phys., 26, 150 (1961). (21) S. J. GOLDSACK, L. RIDDIFORD, B. TALLINI, B. R. FRENCH, W. W. NEALE,

J. R. N O R B U R Y , I. O. SKILLICORN, ~V. T. DAVIES, M. DERRICK, J. H. MULVEY and D. RADOJICIC: Nuovo Cime~do, 23, 941 (1962).

(.z2) F. A. BRISBOUT, C. GAULD, G. B. A. ~IcCuSKER, J. MALOS, N. N1SHIKAWA, L. S. PEAK and L. G. VAN LOON: Nucl. Phys., 26, 217 (1961).

~ 0 2 R. RAGHAVAN~ T. N. I~ANGASWAMY a n d A. SUBRAMANIAN

5 . - C o n c l u s i o n s .

From the analysis made in this paper , i t is possible to conclude t ha t m a n y features of inelast ic JV-A" collisions Can be unders tood in te rms of small-energy-

m o m e n t u m transfers , the d is t r ibut ion of which is similar in form to the energy-

m o m e n t u m t ransfer :d is t r ibut ions in the <~ diffraction region ~> of elastic p~ and ~ - S collisions. For the m a j o r i t y of ~ - A ~ collisions, Up to energies of

~-10 ~2 eV, the observed dis t r ibut ion of inelast ic i ty coefficients, a s y m m e t r i e s

in collisions, and the energy spect ra of the secondaries can be accounted for,

to a first approximat ion , in te rms of the a b o v e ene rgy -momen tum t ransfer

dis t r ibut ion. However , the mul t ip l ic i ty dis t r ibut ion of the secondaries requires

On the average a higher e n e r g y - m o m e n t u m t ransfer in the collisions between

v i r tua l par t ic les into which the nucleon-nucleon collisions are decomposed.

In collisions produced by pr imar ies of a higher energy than this, i . e . , in the

region of mul t ip le ((fire-balls))(~0), the fou r -momen tum transfers between

the (( fire-balls ~> (16) seem to be above t h e (( diffraction region ~). The possible increase in the average m o m e n t u m transfers with increasing energy of the p r i m a r y has to be inves t iga ted in g rea te r detail.

In pion-nucleon collisions i t appears t ha t (( 10ngitudin~'l ~)-momentum trans-

fers in <( per ipheral ~) coliisions are small compared to the cases of per ipheral

collisions involving nucleons; if r:-r~ in teract ions are assumed to domina te the per ipheral collisions of pions (~9), the (( diffract ive ~) m o m e n t u m transfers in r~-r:

in terac t ions appear to be more easily detec ted in per ipheral inelast ic collisions

r a the r t h a n in elastic collisions of pions off nucleons.

We wish to express our sincere thanks to Prof. M. G. K. M E ~ o N and Dr. B. V. SnEE~(A~TA~ whose perusals of a p re l iminary draf t of this paper

were beneficial to us.

R I A S S U N T 0 (')

Si mette in rilievo che la formula della dist~'ibuzione dell'energia-impulso trasfcrita nella (~ regione della diffrazione ~) dello scattering elastico pione-nucleone e protone- protone di alta energia si pus applicare all'energia.impulso (~ longitudinale ~) trasferita alle collisioni anelastiche nucleone-nucleone di alta energia (g 1012 eV). Salvo un cam- biamento nel valore di an parametro della distribuzione, la stessa formula si pub anche applicare alia distribuzione dell'impulso trasferito alle collisioni delle particelle virtuali in cui possono essere decomposte le collisioni nucleone-nucleone in base al modello adottato. Si esaminano brevemente le conseguenze dell'applicazione di queste distri- buzioni dell'energia-impulso trasferita al caso delle collisioni pione-nucleone.

(*) Traduzioae a cura della l~eda~ion~.

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Energy-momentum transfers in high-energy nuclear collisions