Crossing symmetric, Regge-behaved amplitude for a general trajectory

~77~

Nuclear Physics B10 (1969)319-322. North-Holhmd Publ. Comp.. A m s t e r d a m

CROSSING SYMMETRIC, REGGE-BEHAVED AMPLITUDE FOR A GENERAL TRAJECTORY J. B O G U T A Phy.s'ikalischcs I n s t i t u l . Universitiil Bonn. Ge~nzany

Received 5 February 1969

Abstract: F o r any polynomially bounded Regge t r a j e c t o r y a , an amplitude is cons t r u t t e d which is c r o s s i n g - s y m m e t r i c , Regge-behaved, has resonance poles to which only a finite number of partial waves contribute and is pol~lomially bounded. The amplitude is analytic in its variables with the usual unitarity cuts. The ex am pie is based on the generalized interference model of Kupseh.

It i s a l r e a d y c l e a r f r o m a v a i l a b l e d a t a that the R e g g e t r a j e c t o r i e s do not t u r n o v e r a s s i m p l e p o t e n t i a l t h e o r y w o u l d s u g g e s t , but i n c r e a s e a l m o s t l i n e a r l y with e n e r g y s q u a r e d . A s e r i o u s b o o t s t r a p m o d e l m u s t r e p r o d u c e t h i s f a c t . Two a p p r o a c h e s a r e open to t h i s p r o b l e m . One can m a k e s p e c i f i c d y n a m i c a l a s s u m p t i o n s a b o u t the i n t e r a c t i o n s and then by aid of s o m e u n i tarizing multichannel dynamical equation explicitly demonstrate that the trajectories are rising; or one can look for general conditions that the scattering amplitude must satisfy and from these restrictions derive information about the Regge trajectory. The scattering amplitude proposed by Veneziano, which satisfies crossing symmetry has Regge asymptotic behaviour and zero width resonance properties, demands a linear trajectory [I]. It has also been shown by Atkinson and Dietz that if one demands s - and t channel asymptotic crossing symmetry and Regge behaviour, then under some reasonable conditions the residue function and the Regge trajectory are asymptotically determined [2]. Thus it is hoped that a collection of conditions, such as uniformity condition of above mentioned work, will yield a handle in determining the properties of the Regge trajectory and its residue. One can combine the Regge asymptotic behaviour with crossing symmetry and analyticity properties and summarize them into the finite energy sum rules and use this as a starting point. In order to make the finite energy sum rules a solid technique, we must learn how to make practical generalizations of the narrow resonance approximation, since it has been shown that there can be no exact solution in terms of finite numbers of towers of zero width resonances. Our aim in this note is to see what information about the Regge trajectory is contained in the usual restrictions on the scattering amplitude (uni-

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t a r i t y e x c l u d e d ) . We a s s u m e that the R e g g e t r a j e c t o r y a ( s ) is an a n a l y t i c f u n c t i o n o f s with r i g h t hand cut only. It is r a t h e r e a s y to c o n s t r u c t d e s i r e d a m p l i t u d e s f o r t r a j e c t o r i e s that t u r n o v e r , so we s h a l l d e a l with i n f i n i t e l y r i s i n g t r a j e c t o r i e s s u c h that R e a ( s ) ~ = ~ as R e s - - = ~ . The usual condit i o n s i m p o s e d on the c r o s s i n g s y m m e t r i c s p i n l e s s p a r t i c l e s c a t t e r i n g a m plitude A(s,l,lt) are: (i) f o r / fi x e d and s ~ + ~ we h a v e R e g g e b e h a v i o u r . F o r l f i x ed and !s I ~ ~ , the a m p l i t u d e is p o l y n o m i a l l y bounded. (ii) A ( s , / , t t ) m u s t be an a n a l y t i c f u n c t i o n of i t s v a r i a b l e s with the u s u a l u n i t a r i t y c u t s in s , / , a w h i c h b e g in at S o , l o , t ~ o r e s p e c t i v e l y . We do not d e m a n d s q u a r e r o o t b r a n c h p o i n t s r e q u i r e d by u n i t a r i t y . P o l e s , e x c l u d i n g bound s t a t e s , m u s t a p p e a r on the s e c o n d s h e e t . (iii) n e a r a r e s o n a n c e of spin J , s a y a ( l ) = J , the a m p l i t u d e m u s t b e h a v e as A ~ (pole in 1) / ( p o l y n o m i a l in s of o r d e r J ) . T h i s i n s u r e s that only a f i n i t e n u m b e r of p a r t i a l w a v e s w i l l c o n t r i b u t e to the r e s o n a n c e . (iv) f o r f i x ed p h y s i c a l c . n l. a n g l e , s a y in the s - c h a n n e l , c o s ( ) s = Z , -1 . Z 1, A ( s c o s 0 s) is p o l y n o m i a l l y b o u nd ed as s - ~ ~. As a l r e a d y n o t e d by J e n g o , V e n e z i a n o ' s e x a m p l e f a i l s to s a t i s f y al l of the a b o v e c o n d i t i o n s s i m u l t a n e o u s l y [3]. One s h o u l d f u r t h e r note that the c o n s t r u c t e d a m p l i t u d e d o e s not s a t i s f y FF.SR. T h i s is so b e c a u s e S t i r l i n g ' s a p p r o x i m a t i o n f o r the g a m m a f u n c t i o n F ( Z ) is only v a l i d in the r e g i o n i a r g Z i , ;;-~. H e r e c is s m a l l , but m u s t be fixed. F o r R e Z < 0 one m u s t u s e the S t i f l i n g a p p r o x i m a t i o n o u t s i d e a w e d g e of a p e r t u r e e, and thus it is not s u f f i c i e n t to be j u s t off the r e a l a x i s w h e n ,Z! - ~ o . J e n g o ' s e x a m p l e s a t i s f i e s al l of the a b o v e c o n d i t i o n s e x p l i c i t l y f o r a l i n e a r t r a j e c t o r y , w h i l e K u p s c h ' s e x a m p l e f a i l s to h a v e p o l y n o m i a l b e h a v i o u r n e a r the r e s o n a n c e [4]. In the m o d e l c o n s i d e r e d by A t k i n s o n and D t e t z , they did not n e e d to c o n s i d e r the a b o v e c o n d i t i o n s . O u r c o n s t r u c t i o n is b a s e d on K u p s c h ' s w o r k , but the point (iii) is i n c l u d e d f o r p o l y n o m i a l l y b o u n d e d t r a j e c t o r i e s . H i s n o t a t i o n w i l l be f o l l o w e d h e r e . C o n s t r u c t the a m p l i t u d e a s a g e n e r a l i z e d i n t e r f e r e n c e m o d e l A (s ,/ ,a ) = F ( s ,tt ,I ) + F ( s , l , t t ) + F ( I ,a , s ) .

(1)

T h e f u n c t i o n that p r o d u c e s the r e s o n a n c e p o l e s is taken to be the p si f u n c tion ~,(Z)-

F '(Z) r'(z)"

(2)

It w i l l h a v e s i m p l e p o l e s at n e g a t i v e i n t e g e r s , be a n a l y t i c e v e r y w h e r e e l s e and tend to l o g ( Z ) a s y m p t o t i c a l l y . Now d e f i n e F(s././t)

: ~(-a(/))tt(/)l(So-S)

c~(s'/) + (It o - I t ) cg(H'/)l~ ,

w h e r e st) - /o : Ho is the e l a s t i c t h r e s h o l d *

• The, (.ul ()1 ] t l l / ¢ ' [ i ~ ) l l ~

;IS I > ( )

,'4'1 C4

i~ tz/kc, to run t() flat? h'I(.

(3)

REGGE AMPLITUI)E

321

T h e f u n c t i o n H(t) i s an a n a l y t i c f u n c t i o n of t with r i g h t - h a n d cut only and i s c h o s e n so t h a t the R e g g e t e r m will be the l e a d i n g t e r m c o n t r i b u t i n g to A ( s , t , u ) . A p r o p e r e x p o n e n t i a l w i l l do the job. Now l e t

(p(s,t) = a(t) {1

log(-a(/)) expg(s,t)}g,

~(-~(t))

(4)

where 1

g(s,t) = -(So-S)e(to-t) 5-~

0 < e < 5< ~.

(5)

S i n c e we a s s u m e d that a(t) i s p o l y n o m i a l l y b o u n 0 e d , we c h o o s e N to be a p o s i t i v e i n t e g e r s u c h * that ¢o(s,t) o 0 as Ill ~ o~. S i n c e a(t) i s a H e r g l o t z f u n c t i o n , the l o g a r i t h m w i l l not give t r o u b l e on the f i r s t s h e e t [5], and s i n c e N is an i n t e g e r , the z e r o of the b r a c k e t s y i e l d s no b r a n c h cut. T h e c l a i m i s that the c o n s t r u c t e d A ( s , t,u) s a t i s f i e s all the c o n d i t i o n s ( i ) - ( i v ) for any p o l y n o m i a l l y b o u n d e d Regge t r a j e c t o r y . We c h e c k w h e t h e r t h i s i s i n d e e d so. A n a l y t i c i t y p r o p e r t i e s [ c o n d i t i o n (ii)] a r e s a t i s f i e d by s i m p l e i n s p e c t i o n . S i n c e q~(s,t)=>a(t) as Is [ ~ 0% and ¢ o ( s , t ) ~ 0 as It l ~ 0% f o r one of the q u a n t i t i e s s, u, Z s, Z t o r Z u fixed c o n d i t i o n (i) f o l l o w s ,

F(s, t,u)=> ~(-a(t)) H(/) {(s o - s) a(t) + (u o -u)a(t)} ,

(6)

and the full a m p l i t u d e can be w r i t t e n as

A ( s , t , u ) = ~V(-a(t))H(t) {(s o - s ) a(t) + (u o -u) a(t)} + B ( s , t , u ) , with B ( s , t , u ) ~

(7)

0 if Is ] ~ o o and t fixed. On the o t h e r hand

if

IsL--

,

(8)

f o r Z s , Z t o r Z u fixed. T h u s c o n d i t i o n (iv) i s s a t i s f i e d . F o r c o n d i t i o n (iii), we note that if a(t) = J t h e n ~ ( _ j ) - i = 0 and

¢°(s't)l a(t)=J = a(t) = J ,

(9)

and then one h a s a r e s o n a n c e type s t r u c t u r e d e m a n d e d by (iii). T h e a b o v e c o n s t r u c t i o n i s v a l i d f o r a l l p o l y n o m i a l l y b o u n d e d Regge t r a j e c t o r i e s . T h e a m p l i t u d e s a t i s f i e s f i n i t e e n e r g y s u m r u l e s f o r all p o l y n o m i a l l y b o u n d e d R e g g e t r a j e c t o r i e s . T h i s l e a d s one to the c o n c l u s i o n that F E S R by t h e m s e l v e s y i e l d no b o o t s t r a p c o n d i t i o n s . By c o n s t r u c t i o n o u r e x a m p l e does n o t s a t i s f y n e i t h e r the H o r n - S c h m i d d u a l i t y n o r the u n i f o r m i t y c o n d i t i o n s of A t k i n s o n and Dietz. In the g e n e r a l i z e d i n t e r f e r e n c e m o d e l u s e d h e r e , the R e g g e a s y m p t o t i c b e h a v i o u r and r e s o n a n c e p o l e s c o m e f r o m d i f f e r e n t t e r m s . We s h o u l d n o t e h e r e that as an e m p i r i c a l r e s u l t , the a p p a r e n t d i f f i c u l t y in c o n s t r u c t i n g V e n e z i a n o type a m p l i t u d e s d o e s s u g g e s t that a p a r t * The symbol ~ s t a n d s for exponential approximation.

322

J. BOGUTA

from u n i t a r i t y , H o r n - S c h m i d duality or u n i f o r m i t y conditions are quite r e strictive. I would l i k e to t h a n k D r . K. D i e t z and Dr. H. R o l l n i k f o r d i s c u s s i o n s and h o s p i t a l i t y a t the I n s t i t u t e .

RE FE REN CE S [1] G. Veneziano, Nuovo Cimento 57A (1968) 190. [2] D. Atkinson and K. Dietz, Infinite rising Regge trajectories and crossing s y m m e try, Rutherford Laboratory p r e p r i n t (1968). [3] R. Jengo, Physics Letters 28B (1968) 1262. [4] J. Kupsch, A generalized interference with indefinitely rising Regge t r a j e c t o r i e s , Bonn, Univ. preprint (August 1968). [5] N. N. Khuri, Sums of direct-channel Regge-pole contributions and crossing s y m metry, UCRL-18426.

NOTE ADDED ~ PROOF In r e f e r e n c e to eq. (4), one s h o u l d note t h a t if t h e R e g g e t r a j e c t o r y t a k e s a z e r o , we s h o u l d u s e l o g ( - ~ ( t ) + c ( t ) ) . T h e f u n c t i o n c(t) c a n b e so a r r a n g e d t h a t t h e r e i s no b r a n c h p o i n t a n d w i l l s t i l l h a v e a l l the r e q u i r e d p r o p e r t i e s . T h i s f o l l o w s f r o m t h e f a c t t h a t a(t) i s a H e r g l o t z f u n c t i o n . T h e q u a n t i t i e s So, to, u o c a n b e c h o s e n to b e i n e l a s t i c t h r e s h o l d s i n o r d e r to a v o i d l o g a r i t h m i c b r a n c h points at the elastic t h r e s h o l d s .