Some consequences of exchange degeneracy and the triple-Regge-hypothesis

Volume36B, number5

SOME

PHYSIXS

LETTERS

CONSEQUENCES OF EXCHANGE DEGENERACY TRIPLE-REGGE-HYPOTHESIS

4 O c t o b e r 1971

AND

THE

J. F I N K E L S T E I N *

Department of Physics, Stanford University, Stanford, Calif., U.S.A. and R. R A J A R A M A N

Department of Physics, Delhi University, Delhi-7, India Received 4 August 1971

We extract the proton-Reggeised rho total c r o s s - s e c t i o n from inclusive data for K + + p -~K ° + (anything) in the triple-Regge domain and find that it is of the same order as physical meson-proton c r o s s sections for a range of Reggeon mass. We also relate other processes such as pp ~ pX and K+p ~ K + X to K+p --~K°X, and get good agreement with data. We suggest methods for calculating baryon exchange p r o c e s s e s as well. Amplitudes involving triple-Reggeon (T-R) c o u p l i n g s h a v e r e c e i v e d m u c h a t t e n t i o n l a t e l y [1] f o r i n c l u s i v e r e a c t i o n s a + b --, c + X, in the l a r g e M, l a r g e s / M 2 and f i x e d t d o m a i n . (See fig. 1 f o r the n o t a t i o n . ) D a t a f o r s e v e r a l i n c l u s i v e r e a c t i o n s a r e a v a i l a b l e now [2,3], in w h i c h a s i g n i f i c a n t f r a c t i o n of the e v e n t s c a n be q u a l i t a t i v e l y d e s c r i b e d [4,5] by s u c h T - R a m p l i t u d e s . U s i n g t h e f u l l p o w e r of e x c h a n g e d e g e n e r a c y , w e p r e s e n t h e r e the f o l l o w i n g c o n s e q u e n c e s of the T - R f o r m u l a , w h i c h g i v e s t r o n g s u p p o r t to the m o d e l : (i) W e e x t r a c t f r o m the i n c l u s i v e d a t a f o r K + p ~ K ° + X (anything) the v a l u e of an a p p r o - priately defined Reggeon-proton total cross-section. It is found, f o r a f a i r l y w i d e r a n g e of the R e g g e o n m a s s ( t , to be c o m p a r a b l e to m e s o n -

p r o t o n t o t a l c r o s s - s e c t i o n s on the m a s s s h e l l . W e p o i n t out why t h i s is both u s e f u l and in s u p p o r t of the v a l i d i t y of the m o d e l . (ii) We c a l c u l a t e r e l a t i o n s h i p s in the T - R region between different inclusive cross-sections, n a m e l y K+p - - K o x , K+p ~ K+X and pp - - pX. (iii) W e a l s o s u g g e s t a m e t h o d f o r e s t i m a t i n g baryon-exchange inclusive reactions such as pp ~ 7r+X. F i n a l l y we s p e c u l a t e on p o s s i b i l i t i e s of o b t a i n i n g b o o t s t r a p c o n d i t i o n s f r o m o u r e q u a tions. In the i n t e r e s t s of b r e v i t y , we p r e s e n t in what f o l l o w s only the e s s e n t i a l s t e p s of the d e r i v a t i o n and the r e s u l t s . D e t a i l s w i l l be p r e s e n t e d elsewhere. Reggeon-proton total cross-section. F o r the T - R r e g i o n of a + b ~ c + X, if a s i n g l e R e g g e p o l e w e r e to be e x c h a n g e d in the t - c h a n n e l ,

* A. P. Sloan Foundation Fellow. M2

Z X

(/4

s

b

a

b

Fig. 1. The triple Reggeon {T-R) region of inclusive cross-secttons and definition of variables. The symbols (p) and (M) stand for masses. 459

VoLume 36B, number 5

PHYSICS

d2a = 6:s2~acR(t)ID(°tR)]2 dtdM 2

•,M,-, / where

C

Xis observed o-T°tai PP

!

-0.5

w i t h A(c~) d e f i n e d by P~(z) ~-~-.~%(~)z ~, and C~R(t) = s o + a ' t . H e r e ~ is the c o m m o n m a s s of a and c and M is the m i s s i n g m a s s of X. T h e n o r m a l i z a t i o n in eq. (1) is s u c h that w h e n o~(t) a p p r o a c h e s a p o s i t i v e i n t e g e r , a T b ( t , M2) is the p h y s i c a l R - b t o t a l c r o s s - s e c t i o n on the m a s s s h e l l . F o r l a r g e M 2, a T b ( t , M 2) i s r e l a t e d to the Reggeon-Reggeon-Pomeron vertex. For comp l e t e n e s s we r e a s s e r t the n o r m a l i z a t i o n a b o v e by noting t h a t a+c~c+a (2)

L e t us now t a k e a s p e c i f i c r e a c t i o n K+p --. K°X. H e r e two l e a d i n g t r a j e c t o r i e s , the p and the A2, c a n c o u p l e to the k a o n s . H o w e v e r , on a p p l y i n g e x c h a n g e d e g e n e r a c y to the p r o c e s s K + + p ~ K ° + X1, w h e r e X 1 is any g i v e n c h a n n e l in X, it c a n be shown that (a) the p+ e x c h a n g e and t h e A + e x c h a n g e a m p l i t u d e s do not i n t e r f e r e , (b) ~t~D~t)is the s a m e w h e t h e r the R e g g e o n R i s the p + b r the A~ and (c) the p r e s e n c e of the two t r a j e c t o r i e s is t a n t a m o u n t to u s i n g an e f f e c t i v e s i g n a t u r e f a c t o r of 2, and a c o m m o n r e s i d u e f u n c t i o n ~K+KO^+(t). F i n a l l y , a r g u m e n t s of e x c h a n g e d e g e n e r a c y [6] a l s o g i v e flK+KOp+ = 2flK+K+po = 213K+K+ w = 213K+K+p,. T h e v a l u e o f S K + K + w ( 0 ) c a n be o b t a i n e d by u s i n g f a c t o r i z a t i o n on the a p p r o p r i a t e c o m b i n a t i o n s of ~ T o t a l f o r I ~ N , NN and NN r e a c t i o n s a v a i l a b l e f r o m e x p e r i m e n t [7]. W e e v a l u a t e the t - d e p e n d e n c e of ~(t) f r o m a V e n e z i a n o f o r m u l a [8], w h i c h e s s e n t i a l l y a g r e e s w i t h d a / d t f o r the A 2 e x c h a n g e p r o c e s s ~ - p ~ 7 / n [9]. N o t e that a l l the a b o v e a r g u m e n t s and o u r e v a l u a t i o n of a T a r e v a l i d e v e n w h e n M 2 is not l a r g e , but K o u r c ~ l c u l a t i o n s a r e new and of p a r t i c u l a r i n t e r e s t w h e n X is o u t s i d e the r e s o n a n c e r e g i o n , i . e . in the T - R d o m a i n . W e p r e s e n t h e r e only the e x a m p l e of M 2 = 4, a l t h o u g h the a n a l y s i s c o u l d be d o n e a s a f u n c t i o n of M 2. U s i n g a l l the 460

lo=.

(1)

D ( a ) = (2a + 1) [ e x p ( - i y a ) ± 1] A ( a ) sin~

2 t2 2~R(t) -~ ;3acR(t) ID(°tR) I (2s) 64s2 (t_4;~2) 2°LR(t)

4 O c t o b e r 1971

._=

~u

-t_4ot-~ ~

LETTERS

i

i

0.0

0.5 t in GeVa

Fig. 2. A plot of ~(t,M 2) defined in the text versus t, for M 2 = 4 GeV 2. The c r o s s denotes the measured value of aTotal. PP a b o v e i n f o r m a t i o n , the i n c l u s i v e d a t a at 8.2 G e V / c f o r t h i s r e a c t i o n [3], and t a k i n g M 2 = 4 G e V 2, w e c a l c u l a t e the q u a n t i t y ~ ( t ) =2/c~(t) (t/to)Or(t) a T (t, M 2 =4), w h e r e t o is 1 G e V 2, and p l o t it in fig. "2" a s a f u n c t i o n of t u s i n g ~ (t) = I + t . N o t e that a n a l y t i c p r o p e r t i e s of eq. (1) i n d i c a t e t h a t ~(t) i s f r e e of z e r o e s and s i n g u l a r i t i e s ** in the r e , o n of o u r i n t e r e s t and i s e q u a l t o a T+o t a l at t = m - 2~ - ~ T h e g r a p h s h o w s that P P • a (t) d ~ e s not v a r y r a p i d l y w i t h t and i t s v a l u e at t ~ 0 is c l o s e to the ~+p t o t a l c r o s s - s e c t i o n [10] at ( E c . m . ) 2 = M 2 ~- 4 G e V 2, w h i c h is about 35 mb. A r e a s o n a b l e e x t r a p o l a t i o n to t = m 2 w o u l d g i v e a r e s u l t not f a r f r o m the o b s e r v e d v a l u e [11] of a~p-~ 27 ± 2rob. T h e f a c t t h a t the a p p r o p r i a t e s i n g u l a r i t y - f r e e ~(t) is w e a k l y d e p e n d e n t on t, and, f u r t h e r , is of the s a m e o r d e r a s o b s e r v e d meson-proton cross-sections, lends added cred e n c e to t r e a t i n g the R e g g e o n - p a r t i c l e c r o s s s e c t i o n s on the s a m e f o o t i n g a s p a r t i c l e - p a r t i c l e c r o s s - s e c t i o n s a s is done in the T - R m o d e l .

Relationship between different inclusive reactions. It is known that c e r t a i n r e l a t i o n s h i p s e x i s t b e t w e e n d i f f e r e n t r e a c t i o n s o b t a i n e d by v a r y i n g p a r t i c l e b and u s i n g f a c t o r i z a t i o n [12]. T h e s e a r e v a l i d in the e n t i r e f r a g m e n t a t i o n r e g i o n . W e now p r e s e n t o t h e r r e l a t i o n s h i p s b e t w e e n K + + p --*K° + X, p + p ~ p + X and K+ + p ~ p ~ K + + X, w h i c h a r e v a l i d in t h e T - R d o m a i n only. L e t u s b e g i n w i t h the i n c l u s i v e r e a c t i o n pp --* pX in t h e T-R domain. The leading trajectories coupling to the p - p v e r t e x a r e P, P ' a n d w ; t h e p andA 2 c o u p l i n g s to the p r o t o n a r e s m a l l . T h e t r i p l e P o m e r o n c o u p l i n g is a l s o r e l a t i v e l y s m a l l [13,5], although we will discuss its contribution below. ** Note that eq. (1) seems to have a double pole at ot = = 0. One pole is absorbed by the ghost-killing zero of fl(t) ; the other has to be absorbed into ~'(t), assuming the nonsese-choosing mechanism.

Volume 36B, number 5

PHYSICS LETTERS

Let us begin with the dominant w and P ' exchanges, and u s e eq. (1) as before. Exchange d e g e n e r a c y , as applied to e v e r y given exclusive p r o c e s s pp ~ p X 1 leads to the fact that ~,,,, ,(t) = = ~ p p p , ( t ) and(~T (t,M 2 ) = ~Tp , p ( t , M 2 ). ~,vWe wp ~Total ~ ,,Total f u r t h e r a s s u m e that ~¢oP .pp for any given t and M2, which is suppozted by e x p e r i m e n t [14] on the m a s s shell. The t r a j e c t o r y function a(t) is the s a m e for w, P ' , p and A2, so that the ratio of (d2a)/dtdM 2 for pp -- pX and K+p K°X is r e l a t e d to the ratio of the v e r t e x functions (~nuw(t))/(~K+K+w(t)). The exact r e l a t i o n upon inv'Sking all t h e a r g u m e n t s p r e s e n t e d e a r l i e r is

(

d2~ dt----~/pp__~X

= g1

f3ppw(t)

(Sl ~ 2 a R ( t ) - 2 ,

~K+K+w (t) \~2/ dtdM 2

~K ° X

(3)

where we have kept in m i n d that data for pp ~ pX is available [2] at s 1 = 38 GeV 2, while the K+p K°X is at [3] s 2 = 16.6 GeV 2. At t = 0, the ratio ~ppw/~K+K+w may be obtained as before, u s i n g I ~ N , NN and NN total c r o s s - s e c t i o n s . Near t = 0 then, upon i n s e r t i n g the data [3] for d2a/dtdM 2 for K+p - - K ° X eq. (3) p r e d i c t s that (d2a)/dtdM 2 at t ~ 0 and ~ = 4 GeV 2 will be 2.06 m b / G e V 4 for the Plab = 19.2 G e V / c data of pp ~ p X . The o b s e r v e d value [2] is about 2.82 m b / G e V 4. This is a l r e a d y a v e r y r e a s o n a b l e a g r e e m e n t . The s m a l l amount of 0.76 m b / G e V 4 by which the exp e r i m e n t a l value exceeds our p r e d i c t i o n could be a t t r i b u t e d to s m a l l e r effects such as the t r i p l e - P o m e r o n coupling, and the s m a l l p and A s coupling. Note that an e s t i m a t e of the t r i p l e P o m e r o n c o n t r i b u t i o n b a s e d on the e x p e r i m e n t a l ly deduced value [13] of the t r i p l e - P o m e r o n coupling ~ p p p ( t ) (which a g r e e s with c e r t a i n t h e o r e t ical e s t i m a t e s rl]) adds about 1.6 m b / G e V 4 to our r e s u l t for ((d'~a)/dtdM2)pp--pX ' t=O - s o m e what l a r g e r than the above d i s c r e p a n c y . However, the e x p e r i m e n t a l l y deduced 7?ppp(t), b a s e d on the H a r a r i - F r e u n d c o n j e c t u r e is probably an o v e r e s t i m a t e . The a n a l y s i s of Pignotti and P e c c e i [5] i n d i c a t e s n e a r a b s e n c e of the t r i p l e P o m e r o n effect for pp ~ pX. T h e s e f i n e r points aside, the a g r e e m e n t to within 25% of just the w - P ' c o n t r i b u t i o n to pp --'pX data is itself r e m a r k a b l e and e m b a r r a s s ingly close in our i l l u s t r a t i o n at t ~ 0, M2 = 4. Eq. (3) can in fact be used to p r e d i c t d(r/dtdM 2 as a function of t and M'2. T u r n i n g to K+ + p

4 October 1971

K+ + X, one can once again r e - a p p l y all the a r g u m e n t s u s e d above, and r e l a t e it quite s i m p l y to K+ + p -- K o + X. T h e r e a r e four t r a j e c t o r i e s , ¢o, P ' , p and A 2 (neglecting the t r i p l e P o m e r o n ) coupling to the K+K+ vertex, as c o m p a r e d to only p and A 2 coupling to K+KO. This fact is exactly offset in eq. (1) by the r e l a t i v e i s o s p i n ClebschGordan factor of ½ in 9K+K+po as c o m p a r e d to ~K+KO,,+. Thus, we p r e d i c t that in the T - R r e gion d'~a/dtdM 2 for K+p -~ Kox is equal to that for K+p --K+X. Unfortunately, not much data a r e a v a i l a b l e for the i n c l u s i v e r e a c t i o n K+p -" K+X. However, it is r e a s o n a b l e to say at Plab = 8.2 GeV/c, that most events in K+p s c a t t e r i n g contain either one K+ o r one Ko and not both. In the T - R region, we have shown that each p o s s i b i l i t y c o n t r i b u t e s equally, and t h e r e f o r e to half the n u m b e r of events. Now the ratio of the i n t e g r a t e d K+p ~ K°X to the total K+p inelastic c r o s s - s e c tion is a v a i l a b l e [2] and is about 40%. This lends support to our r e s u l t although only i n d i r e c t l y . Once again, the d i s c r e p a n c y is in favour of slight. ly l a r g e r K+p -- K+X c r o s s - s e c t i o n as c o m p a r e d to K+p ~ K°X, and could be a t t r i b u t e d to f i n e r effects like the t r i p l e - P o m e r o n coupling.

Possibility of predicting inclusive distributions. F i n a l l y , one can f u r t h e r exploit the fact that aTotal e x t r a c t e d e a r l i e r is roughly of the Rp s a m e o r d e r as the p h y s i c a l m e s o n - p r o t o n total c r o s s - s e c t i o n . A s s u m i n g that the s a m e happens when the Reggeon is a b a r y o n , one can attempt to p r e d i c t the i n c l u s i v e c r o s s - s e c t i o n d2(~/dtdM2 for r e a c t i o n s like pp --v+X. Here, a Reggeised proton is exchanged. I n s e r t i n g a Reggeised p r o t o n - p h y s i c a l proton total c r o s s - s e c t i o n ~ a(pp) 40 mb into eq. (1) modified for b a r y o n exchange, one can calculate (d2a/dtdM2)pp_u+ X. The r e s u l t should, judging from the s u c c e s s of the m e s o n exchange calculation, give c o r r e c t values to within a factor of 2. We know of no m o r e p r e c i s e a way of p r e d i c t i n g such i n c l u s i v e reactions. The v e r y r e a s o n a b l e value of ~(t) we e x t r a c t ed e a r l i e r leads us to speculate about b o o t s t r a p p o s s i b i l i t i e s . Using a total c r o s s - s e c t i o n as i n put, one can get the i n c l u s i v e d i s t r i b u t i o n in the T - R r e g i o n in t e r m s of f3(t). To the extent that the T - R f o r m u l a works on the a v e r a g e for all the events, one can i n t e g r a t e it to r e p r o d u c e aTotal, t h e r e b y e v a l u a t i n g the coupling f~(t). It has b e e n shown [15] that such a b o o t s t r a p gives the s a m e conditions as those obtained in m u l t i - R e g g e models without, however, a s s u m i n g a m u l t i - R e g g e p r o d u c t i o n m e c h a n i s m . To e s t i m a t e these p o s s i b i l i t i e s quantitatively, we i n s e r t a ~wp ~ 30 mb 461

Volume 36]3, number 5

PHYSICS

i n eq. (1) f o r t h e p p - - pX c a s e , a n d i n t e g r a t e o v e r a l l M 2 a n d t b e t w e e n 0 a n d -1 G e V 2. W e g e t a -pp ~inelastic ~ 10mb, i.e. about a third of the observed value. The possibility of improving this quantitatively deserves further study. T h e a u t h o r s w o u l d l i k e to t h a n k P r o f e s s o r Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.

References [1] C.S. DeTar, C . E . J o n e s , F . E . L o w , J . H . Weis, J. E. Young and C. I. Tan, Phys. Rev. L e t t e r s 26 (1971) 675; H. D. I. AbarbaneI, G . F . Chew, M. L. Go[dberger and L. M. Saunders, Phys. Rev. L e t t e r s 26 (1971) 937. [2] J . V . Allaby e t a [ . CERN r e p o r t 70-11 (1970). [3] J. V. Beaupr6 et al. p r e p r i n t CERN/D.PH. I I / 7 0 - 5 2 / Rev (1971).

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[4] P. Chliapnikov, O. Czyzewski, J. Finke[stein and M. Jacob, CERN p r e p r i n t T H 1336 (1971). [5] R. D. P e c c e i and A. Pignotti, Phys. Rev. L e t t e r s 26 (1971) 1076. [6] C.B. Chiu and J. Finkelstein, Phys. L e t t e r s 27B (1968) 510. [7] W. Galbraith et al. Phys. Rev. 138 (1965) B913. [8] G. Veneziano, Nuovo Cimento 57A (1968) 190; 'C. Lovelace, Phys. L e t t e r s 28B (1968) 264; J. Shapiro and J. Yellin, p r e p r i n t UCRL-18500 (1968). [9] O. Guisan et al., Phys. L e t t e r s 18 {1965) 200. [10] G. Giacomelli, P. Pini and S. Stagni, CERN-HERA 69-1 (1969). [11] S. C.C. Ting, Proc. Seminar on Interactions of e l e m e n t a r y p a r t i c l e s with nuclei, T r i e s t e , 1970, p. 171. [12] H. M. Chan, C. S. Hsue, C. Quigg and J. M. Wang, Phys. Rev. L e t t e r s 26 (1971) 672. [131 R. R a j a m a r a n , ICTP, T r i e s t e , p r e p r i n t I C / 7 1 / 2 5 (1971). [14] L. Foa, P r o e . Seminar on I n t e r a c t i o n s of e l e m e n tary p a r t i c l e s with nuclei, T r i e s t e , 1970, p. 221. [15] J. Finkelstein, Phys. Rev. D2 (1970) 1591.