LETTERE AL NUOVO CIMENTO VOL. 3, -'~. 8 19 Febbraio 1972

Anomalies of the Triple Regge Vertex.

R. J ~ G O

Is t i tu to di F i s i ca Teorica dell' Universi t~ - Trieste I s t i t~ to Nazionale di F i s i ca Nucleare - Sottosezione di Trieste

(ricevuto il 10 Diccmbre 1971)

The triple Reggc vertex appears to be of relevance in the analysis of the hmlusive reactions (~). We refer for clarity to Fig. 1, where A and B are the incident particles and C is the final particle detected in the experiments (we take A, B and C spinless). The distribution of C is related to a suitable defined discontinuity (s) of the amplitude represented in the Figm'e.

Fig. 1.

t. C C t

Recently, it has been pointed out that the triple Reggo vertex has not the expected Regge behaviour, if there is a pomeron with ~p = 1 coupled with B. The expected be- haviour would have been (s/sl) 2~R~Q'~ s~P (~ fl(Q2), where s = (PA d- pB) s, sl = (Px + PB-- PC) s, Q2 = (PA - - P c ) s, and ct R is the trajectory coupled to AC. The authors of ref. (8) claim that the behaviour is instead g(QS)s -4- ](Q2)sl. The bulk of their argument is simple: since the pomeron represents (from the J-plane point of view) a vector exchange, the whole expression must be linear ill PB- But the other independent vectors are PA and Pc, therefore the result follows. A question arises natural ly: what happens if the trajec- tory ~ exchanged in the BB channel is not equal to 1? In this note we will t ry to obtain a general formula, valid for all the cases.

(1) L. CA~ESOHI and A. PIG~OTTt: Phys. Rev. Left., 22, 1219 (1969); C. E. DATAR, C. E. JONES, F. E. LOW, J . H. W]~IS, J . E. YOU~'G and GHUHG-I. TAN: Phys. Rev. Lett., 26, 675 (1971); C. RISK and J . H. FRIED~L~.~: Phys. Rev. Left., 27, 353 (1971). (2) A. H. MIYEI,I,ER: Phys. Rev. D, 2, 2963 (1970). (q J . 2 ~ , Y. HARA and H. SUOAWA~: Phys. Left., 37 B, 92 (1971).

335

336 m J ~ G O

I t is convenient to dis t inguish the two (AC) channels calling them (AC) and (AC)'. We suppose first Q2 > 0 and we write a 03 expansion in the (AC) channels , which we will cont inue as in the usual Regge theory to Q~ ~ 0. I f aBw is integer we cannot couple the reggeon exchanged in (BB) to par t ia l waves in (AC) and (A~)', when the helici ty differs for more t han a ~ . In other words, there are the so-called nonsense factors, which kill the forbidden t rans i t ion . A simple and in tu i t ive picture of the s i tua t ion is obta ined, for instance, by the use of the formalism of the der ivat ive ampl i tudes (~). In general, a definite angular m o m e n t u m J , for ins tance, in the (AC,) channel , is described by a ten- sor //~t...~+.~r.., ~ (symmetric , traceless, or thogunal to Q = p ~ + p~), and the whole ampl i tude for the t r ans i t ion J in (A~) to J" in (AC)' is wr i t t en as

(1) A~,x....4 n : ~x..."./.~i...r "~' A

where A = PA--P~e, 1~,' is a funct ion of Q and Pn . The tensor indices of ]j+, ~',' are con- s t ruc ted by (~,, or by (~/SQ,)F, (D/OQ,,,)F, where F is a scalar funct ion of Q'pn, because those are the only tensors at our disposal at the upper ver tex for the t rans i t ion J + B -+ J ' + B (the vector Q does not couple to H~.,). If a Regge-pole ~ is exchanged in (BB), then F ~ (Q'pB):Bg. Therefore the form of ]~5~, ' will be (for b rev i ty we call here and later ~ ~-gB~)

(2) 1 8 8 8 8

(K + K')~ 80, ~ "'" 8Q, g 0 Q , "'" cO,k, - - - F(Q "PB) ~'x+~.'~'+l "'" ~+,,~,

~ ( ~ - - 1) ... ( a - - - K - - K ' + 1) (K + K') ! P~'I "'" PB~'KPB"'I "'" Pm,'~,(Q "PB) a-~-K' ~a+l.v~,+1 "'" 5vj.,"j, "

The factor r162 ( o t - - K - - K ' + 1) is the nonsense factor and i t is zero for inte- ger ~ when k + k" > ~ ( the factor 1/(K + K') ! is a rb i t r a ry and has been wr i t ten for convenience). Inse r t ing eq. (2) in to cq. (1) we obta in for the ver tex the cont r ibut ion

a ( ~ - - 1) ... ( z t - - K - - K ' + 1) , (K + K ' ) ! - (A "pB)X(/J "pR)Z'(Q "ps) ~-X-R' C(J, J ' , K, K , (2)

and other te rms of lower degree in 3 "PB (*)" The whole ampl i tude T is ob ta ined summing over J , J' , K, K' . We write i t as a sum over K, K ' since we are in teres ted into the behaviour for A.pB large:

~ / A ' p B ) J r + ~ ' ~ ( a - - 1 ) . . . ( a - - K - - K ' + 1) , 2 (3) T ~ ~(~)(Q-pn) ~' ~_, C(K, K , Q ) ,

~=o ~'=o \Q 'PBI (K + K')!

where ~(~) is the s ignature factor of the pole a.

(~) V. DE ALF&RO, S. FUBI~rl, G. F U R I ~ r a n d C. I~OSSE2~I~: Ann. ol Phys., 44, 165 (1967); C. REBBX: Ann. o! Phys., 49, 106 (1968). (*) The f o r m is a l w a y s (Q.pB)a(A.PB/Q.pB) n, for Q . p B la rge because i) t he Whoole express ion is homo- geneous of degree a in PB a n d ii) t he t cn so r s ~g . r axe o r t h o g o n a l to Q, the re fo re c o n t r a c t i o n wi th PB'S m a y g ive A . p B or (QpB)2/Q 2 - ~ 2 (Q.pB)2.

A N t ~ M A I , I E S O1" 'l'III-', ' [ ' R I P I , E R I 2 1 ; ( ; E V E R T E X 337

Tim ampli tudes C(K, K',Q '~) arc (.xpr(.saed as a sum ,)ver the angular m , m t e l m ~

,gr(,at~.r tlla}l K and Js

co

1 -0 l ' - , 0

and a ( J - - 1 ( :- I) de~cribes the properti(~s of the angular , nonmntum J in the channel (AC), for instance, a Regge-polc gives a(J)~d/ ( , l -c~(Q'~) ) . Therefore C(K. K ' ,Q 2) eonta.ins the Reg~e singula.rit.ies (~md da.ughLers) in the (AC) "rod (At;)' eh~nne, ls.

We make now the hypothesis tha i i1 is possible to cont inue to eomplex K and I f ' and to perform the usual Sommcrfeld-Wa~son machinery. The crucial problem for our inw,sl i~ation is the eont i , luat ion in K, K' of the factor

~ ( ~ - 1 ) . . . ( ~ - K - - K ' - : - 1)

(K+ K')!

which becomes zero for a integer for all the integers value of K and K ' sufficiently large. In this case no Carlsonian con t inua t ion is possible. We must therefore factorize out the zero which occurs for integer a. I towevcr, we mus t also compensate t ha t factorized zero by in t roduc ing poles for those values of K and K ' which give a finite result . A eon- 0 n u a t i o n of this kind is the following:

a ( c r 1) ... (cr + 1) s i n ~ a F(K + K ' - - ~) (4) - - ( - - 1) K + # - - - - F ( a + 1) - -

(K + K')! z~ F(K + K')

We obta in therefore

(5) T ~ 6:(%-fi)(Q .p,,)~ns s i n ~ t a ~ P ( % g + 1 ).

�9 J sTI~-K \-~-B] J ~ n - ~ 7 ~ n , I F(K "4- g ' + l) C(K, K', Q2) .

The contours of in tegra t ion lie, as usual, between the s ingular points K, K'----- O, 1, 2 .... and the other singulari t ies in the complex plane. There arc the singulari t ies of C(K, K' , @), e.g. Regge-poles, b u t also the poles of the nonsense factor F ( K + K ' - - o~). Suppose first we integrate in K ' : the pinches of the poles at K ' = c , - - K - - l with the poles of 1/sin ztK' for K ' = 0, l . . . . . give in t u rn poles in K for K = a - - m, m = 0, 1 . . . . . which cont r ibute a behaviour like (d'pn/Q'pB)X+K'(1/sinztK) for K ' = ~ - - K - - I and K = ~ - - m (due to the prescr ipt ion on She in tegra t ion contours, 1 / s i n a i ( a - - m ) g i v e s ac tual ly a s ingular i ty only when a - - m = 0 or a posit ive integer).

We may therefore write in general

(3 .p,/Q .p ,O~-" (9.,.) +

§ $(%h)(Q'p , , )~ '~s inz~F(aB~- , - D R ( Q : ; : , Q2).

The first te rm represents the cont r ibut ion from the nonsense factor, the second one from the other singularities.

338 ~:..IEN(;o

\Ve see tha t fro" ~ ,5 - - integer only lhe iir~l te rm survive~, which in p ' t r t ieular for xmi ~ 1 ~zives the result of rcf. (~). Thal result is valid in ~eneral for the f r a / m c n t a t i o n re~ion, and not only for the triple [ l e ~ c l imit . If w~, arc interested in part icular in the tr iple tb,.g~e region, we obtain, call inz ~-i~ and ~ ~lw t ra jce t . r ies exchanged in (AC) and (Ai.~')'.

. - , ~ I ~ i i u , ~ ) - , , ~ , [ , l : p ~ , ' ~ ~,,,((?"-)

[ l �9 n \ a ~ ( ~ % + ~ i { ~ t)

which is a modification of the expected triple Regge behaviour.