IL 1WUOVO CIMENTO VoL. 15A, N. 2 21 Maggio 1973

Zeros of the Triple Regge Vertex (*)(**).

A. PATI~ASCIOIIY

.Laboratory /or .Nuclear Science and Department o/ Physics Massachusetts Institq~te o] Technology - Cambridge, Mass.

(rieevuto il 25 Settembre 1972)

Summary. - - The triple Regge vertex, which appears in the helicity- pole-Regge-pole (HPRP) limit of 1-particle inclusive cross-sections, is shown to have certain zeros. The proof uses the upper bound imposed by unitarity and analytieity on the 3-to-3 amplitude in the HPRP limit.

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

A recent pape r (1) gave an interes t ing discussion of the triple l~egge vertex~

specifically the coupling of one reggeon with v a c u u m q u a n t u m numbers wi th two reggeons carrying the same invar i an t m o m e n t u m t. This object appears in the hel ie i ty-pole-Regge-pole (HP R P ) l imit of 1-particle inclusive cross-

sections. The authors showed t h a t the absence of spurious poles in the 3-to-3 ampl i tude for cer ta in values of t could be achieved b y zeros in the tr iple ver tex . Computa t ions done with some F e y n m a n diagrams (1) and the dual model (2)

did find such zeros. All a t t e m p t s to expla in on general grounds the existence of such zeros

t r ied to ident i fy t h e m as nonsense zeros (2.3), bu t no conclusive proof has been

(*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction. (**) This work is supported in part through funds provided by the Atomic Energy Commission under Contract (11-1)-3069. (1) S . J . CHANG, D. GORDON, •. E. LOW and S. B. TREIMAN: Phys. Rev. D, 4, 3055 (1971). (e) C. E. D]~ TA~ and J. H. W]~IS: Phys. Rev. D, 4, 3141 (1971). (3) H . D . I . ABA.RBAN:EL and ]~[. B. GREEN: The vanishing o] triple-reggeon vertices, Institute for Advanced Study, Princeton, preprint (1971).

274

Z E R O S O F T H E T R I P L E R E G G E V E R T E X 275

given. Moreover la ter computa t ions (4) p roved this to be false for wrong-

s ignature nonsense zeros. We will a t t a ck the p rob lem f rom a different direction. I n a previous

pape r (5), employing ana ly t i c i ty and uni tar i ty , we der ived an upper bound on the 3-to-3 ampl i t ude in the H P R P limit. We will re ly on this bound to prove

t h a t indeed the tr iple Regge ve r t ex mus t have cer ta in zeros. To m a k e the p resen ta t ion self-contained, in Sect. 2 we give a brief review

of the der iva t ion of the a sympto t i c fo rm of a cer ta in d iscont inui ty in the 3-to-3

ampl i tude in the H P R P l imit b y use of a 02,1 expans ion (~). Section 3 contains

¢he discussion of zeros in the tr iple Regge ve r t ex based upon the upper bound

on the 3-to-3 ~mpl i tude. Some commen t s regarding the significance of our resul ts for l -pa r t i c le inclusive cross-sections are made in Sect. 4.

2. - The H P R P l imi t o f the 3 - t o - 3 ampl i tude .

The 3-to-3 ampl i tude can be exhib i ted as a funct ion of the 02.1 group varia-

bles by clustering the par t ic les as in Fig. 1 and choosing special f rames of ref- erence F~ and F z (6). FA is defined b y requir ing t h a t P1 and p'~ each have

3'r 2 r 1 y

3 2 1

Fig. 1. - Diagram used in exhibiting the amplitude as a function over 0~,1.

vanishing x and y components , while Q has only a posi t ive z-component . This

specifies F~ up to a z-rotation. FB is chosen such t h a t Q has only a posi t ive

x -componen t and vanish ing y and t components and p~ has vanish ing y-com-

ponent . The f rames F and F B are re la ted b y an 02.1 t r ans fo rma t ion g which we

choose to pa ramt r i ze as

(2.l) g(e, ~) = B~(a)B~(~).

(4) A. H. MUELLER and T. A. TRVEMAN: BNL preprint 16350 (1971); D. Go~Do~: ]~AL preprint THY-23 (1971). (5) A. PAT~ASeXOZV: -~uovo Cimento, 15A, 249 (1973). (6) C. E. JONES, F. E. Low and J. E. Yov~cG: Phys. Rev. D, 4, 2358 (1971).

9'76 A. PATRASCIOIU

Invar ian t s formed out of the momenta are defined in Fig. 2. The ampl i tude depends upon 8 independent scalars which we choose to be tn,, t~3,, t~2,, P2~,

! I p2~, p ~ , ~, ~. The helici ty-pole-Regge-pole (HPRP) limit is defined b y the

$I23P

t3Z Y S 12

/3 r X2 I1

t23f $12

Fig. 2. - Some of the invariants used in characterizing the HPRP limit.

k inemat ica l configuration in which a - + 0% ~--> c~ while ull the other inde- penden t invar iants are finite. The asymptot ic form of some of the invariants is

[ s12 "~a coshct cosh ~,

I (2.2) s12 ~ a' cosh a cosh ~,

s12~,~ b sinh ~.

The 3-t0-3 ampl i tude F~ cannot be expanded in 02.1, since it has poles in the variables a and ~ in the physical region. However us suggested b y MVELLE~ (7) and subsequent ly more rigorously justified (8), the following ele- m en t a ry discont inui ty formula holds:

1+ ]r

(2.3) dise~,, , z - - % + "~2---2'= + + . . . . 1 ~ - J ~ 3' 2 2

1 3' ( 3 r 1

As implied by the Ste inman relat ion (9) this discontinui ty does not possess

any singularities due to poles of normal thresholds in crossed channels.

(7) A. H. MVELT.]~R: Phys. /~ev. D, 2, 2963 (]970). (s) C. I. TAN: Brown University preprint NYO-2262TA-240; H. P. STAPP: Phys. /~ev. D, 3, 3177 (1971); J. C. POLKINGttORNE: University of Cambridge preprint DAMTP 71/36. (9) H. ARAKI: Journ. Math. Phys., 2, 163 (1960). (10) C. E. Jo~v.s, lv. E. Low and J. E. YounG: Phys. t~ev. D, 2, 640 (1972). (11) A. PATRASCIOZU: MIT preprint, No. 285 (1972).

ZEROS OF THE T R I P L E R E G G E V E R T E X 277

The first term on the r.h.s, of (2.3) contains 6(s12a,--m~). Subtracting this term from the discontinuity gives a function which can be expanded in 0~.~,

In the HPRP limit the expansion of an asymptotically growing function of the group variables A(e, ~) takes the form (6)

(2.4) A(:¢, ~) ~ _ ~ exp [-- aT](sinh $)L. 1~1--~'~

Z and 7 are the leading singularities in the 0~.1 expansion. The position of J5 can be determined by analytic continuation to the crossed channel as being the leading Regge trajectory coupled to 1i '

(2.5) L = ao(tn,) •

As recent work has shown (10"11)7 it is difficult to find the position of 7 if we expand the 3-to-3 amplitude. However presently we are dealing only with its discontinuity and, as we shall see, this simplifies things. Indeed from (2.3)

1 ~ 1 r

(2.6) disc~,,~, 2 ~ + " , . . ~ 7 2' - = + . . . . 1 ~ 3 ' 2 2

!

If we keep s ~ , fixed and take s12 ~ @12 ~ oo, Regge arguments for the following ~mplitudes:

(2.7) 2 ~ s~23' and 2 ' ~ sT23' 1 ~ 3' 1' 3

allow the identification of 7 as the sum of the leading trajectories in the 23' channel and 32' channel. Finally assuming that

(2.s) lim lim = lim lira

one obtains the following asymptotic form in the HPRP limit:

(2.9) ~ ~ ~ , 1 disc~l:~, ~ ~ ~ , ~/~(tll,, t2~,, t32,) sin ~1(t~3,) sin 7~zt~(t3~,)

Here f l ( tn, , tss,, ts2,) is the triple-Regge vertex. In (2.9) we have dropped the term containing ($(slz3,--m s) which does not contribute when s12~,--~ c~.

278 A. PATRASCIOIU

%ntroducing the signature factors we obtain the asymptotic expression for

(2.~o) 3 ~ 1 ' d is%. , 2 - - ~ +' , . .~----2' , ' ~ f l ( t x t , , t23, , t3~,)

1 ~ 3 '

1 + vt exp [in~(t~3,)].

siu ~(t2~,)

sm u~2( t~, ) \ s~, /

where T 1 and T2 are the s ignature factors. Using the S te inmann relat ion

(2.11) disc,, ~, = disc~=. 2----~+ [-)-----2' • 1 ~ 3 ' 1 " ' ' " % - - - - ~ / - - 3'

t f ,' The nota t ion in the bubble on the r ight means all subenergies st~ , s23 , 813 are

below the i r cuts. I t is necessary to work with ---{+~} because this is the

funct ion for which we could der ive an upper bound.

3. - Zeros in the triple Regge vertex.

The conjecture t h a t cer ta in zeros mus t occur in the triple Regge ve r t ex made in ref. (1) is based upon the assumpt ion t h a t the same power behaviour

will domina te above and below the cut in s~23,. Then f rom (2.10) one can re- cons t ruc t the 3-to-3 ampl i tude and using (2.11) one obtains

(3.1) 3 ~ 3 '

Fe = 2---%+ [-~---2' ,~fl( tw, t23., t32.)" 1 ~ 1 '

1 + ~0~1% e x p [iY~(~0(tH, ) - - ~t(t~3, ) - - ~2( t32, ) ) ] .

sin zE~0( tw)- ~( t23 , ) - ~(t3~,)]

812 1 ~l(t2s')+c~a(tss') 1 ~- ~'1 exp [izg1(t23,)] 1 ~- ~2 exp [-- i~(t32,)] (s123,)~o(tw) _ _ + F ' sin ~( t23,) sin ux2(t32,) k8123'/ '

where ~v, is some funct ion of the invar ian t s having no a s y m p t o t i c d iscont inui ty

in st23,. Whereas the poles in F due to sin ~1(t~3,) and sin 7~%(t3~,) are expected

*o occur, those produced when ao(tn,)--a,(t~,)--~(t~2,) is an integer are spu-

rious. They mus t be cancelled ei ther b y zeros in fl(t~,, t~a,, t32,) or b y subtrac- t ion ~erms conta ined in F . We will de te rmine which cancellation occurs using

the upper bound on F.

Z E R O S O F T H E T R I P L E R E G G E V E R T E X 279

! Subs t i tu t ing the a sympto t i c fo rm of s~, s~ and s~a, (2.2), we can write

the ampl i tude in (3.1) as

(3.2) 1 ÷ % v, v~ exp [i~[0~o(t11,)- ~(t23,) -- ~(t~,)]]

~ o ~ fl(t,~,, t~ , , t~,) - s in ~r[ZCo(t~,) - - zq(t~,) - - zc(t~,)]

1 ÷ ~ exp [iz~(t~a,)]. 1 ~ v ~ exp [-- i z~ ( t3~ , ) ]

sin ~r~, (t~,) s in z ~ (t~,)

• exp [--iz~[Zto(t~v)- ~ l ( t ~ , ) - ~(t3~,)]] b ~°(t,,')+~m~')+~(~'~ •

. a~{t , , ,) a,~,~(t~,,)(cosh ~)~,,(t,~,)(cosh ~)~,,(t~,)+~,~(t~,) ÷ F ' .

F i r s t let us consider the spurious poles which occur when

(3.3) { - - ~o(tll,) + ~x(t~,) ÷ ~ ( t ~ , ) = ~ > 2 ,

T o ~ = exp [ i z n ] .

I f these poles were cancelled b y subtract ion, then F ' would have to contain

t e rms involving (cosh ~)~ (cosh ~ cosh ~),,(t,1,) or (cosh ~ cosh ~)~lct,,)+~m~,)(cosh ~)-".

As r e m a r k e d b y G0~A)~ER~ER the l a t t e r type of subt rac t ion t e rms are not consis tent wi th the genera l Regge behav iour for fixed :¢. l~egge a rguments

show t h a t for fixed ~ we expec t the ampl i tude to have the a sympto t i c fo rm (cosh ~)~'(t~')G(~, t~v , t23,, t32,). We assume Regge behaviour and thus reject the

possibi l i ty of sub t rac t ion t e rms containing (cosh ~ cosh $)~,(t~,)+~(t,~,)(eosh ~)-". The dual model obeys this res t r ic t ion and indeed no such t e rms are present .

The other t ype of sub t rac t ion t e rms (eosh a)'~(eosh ~ cosh ~)~0(tl~,) cannot

exis t for n > 2 if ~o(0)> 0. This follows f rom the existence of an upper bound

on F der ived in ref. if). There we showed t h a t in the H P R P l imit F cannot conta in cosh ~ to a power larger t h a n two wi thout violat ing u n i t a r i t y - - t h i s

resul t is val id only in the physica l region t~j, < 0. I t follows t h a t the tr iple l~egge ver tex fl(tu,i~3,t3~,) mus t have zeros whenever t~,, t32, and t23, are such t h a t (3.3) holds.

4 . - T h e 1 - p a r t i c l e i n c l u s i v e c r o s s - s e c t i o n .

Let us de te rmine some of the implicat ions of the previous result for 1-par- ticle inclusive cross sections:

(4.1) tu,---- 0 , t 2 3 , = t32,---- t < 0 .

As leading t ra jector ies we t ake three pomerons if the 25' channel has v a c u u m

q u a n t u m numbers and a pomeron and two reggeons otherwise. Thus the con-

280 ~. PATRASCIOIU

dition (3.3) used in locating zeros becomes

(4.2) - - 1 q- 2~(t) = n > ~ 2 , 1 : exp [ i zn ] .

:Notice t h a t (4.2) cannot be used to prove the vanishing of the triple-

pomeron ver tex a t t----0 (~s). This is understandable since our constraint on

the ver tex is a consequence of only ~mitarity, whereas the above-ment ioned

result is one of the numerous constraints (1~) which follow if one consistently reqnires the leading l~egge singulari ty to be a pole a t 1.

An interest ing question left unanswered by our approach is the way in

which poles produced ~t values n < 2 are cancelled. I t m a y be tha t , enforcing

maitari ty and analyt ie i ty via the complex angular momentum, one can obtain

more information than we did. So far, though, such a t tempts have not been

very successful.

The present work s temmed from discussions with F. L o w to whom I am indebted.

(12) H . D . I . A B A R B A N E L , J. F. C H E W , iYl. L. GOLDBEI~GEt¢ and L. M. S&U~DX~S: Phys. l~eo..Left., 26, 937 (1971); C. E. DE TAR, D. Z. F~EEDMAN and G. VENEZlANO: Phys. /~ev. D, 4, 906 (1971). (13) C. E. JONES, F. E. Low, S. H. H. TIE, G. VENEZlXrrO and 5. E. YouNg: MIT preprint, No. 264 (1972).

• R I A S S U N T O (*)

Si dimostra ehe il vertice di Regge triplo, the compare nel limite polo di elicitS-polo di Regge (HPRP) delle sezioni d'urto inclusive di 1 partieella, ha certi zeri. La dimo- strazione fa uso dei limiti superiori imposti dall'unitariet~ e dall'analitieit~ all'am- piez~.~ 3 a 3 nel limite HPRP.

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