Volume 28B, number 4 PHYSICS LETTERS 9 December 1968

A NEW EXAMPLE OF

A CROSSING SYMMETRIC REGGE BEHAVED AMPLITUDE

R. JENGO CERN, Geneva, Switzerland

Received 21 October 1968

An example is shown of a Regge behaved, crossing symmetric amplitude, with linearly rising trajectories? such that it has the right branch points in s and t, and the residue of a pole, say in t, is a polynomial in s . The example does not show duality, it is rather like a sum of direct and crossed channel resonances.

There has been recently a great interest in the construction of examples of crossing symmetric scattering amplitudes showing Regge behaviour with linearly rising trajectories [l-4]. We want to present here a new example which, improving the other attempts known to us, satisfies the fol- lowing requirements.

The crossing symmetric amplitude A(s ,t) = =A(t,s) (later on we will also consider crossing * symmetry with respect to the third variable U) has to be such that:

i) for 1.~1 - 00 at fixed t it goes as (-s ) o(t) , no

matter the value of Rea(t), which in our case of linearly rising and decreasing trajectory can be made arbitrarily high positive or negative. This, together with dispersion relations, will ensure the automatic fulfilment of the finite energy sum rules;

ii) it must have a cut in s from s = so to + 00, and a cut in t from t = to to + m, the position of the branch point in s being independent on t and vice versa. Poles must appear only in the second sheet, apart from bound states;

iii) the poles in the second sheet, say in t, near the real axis (resonances) must have a re- sidue which is a polynomial in s. This ensures that at the value of t in question only a finite num- ber of partial waves will resonate, and this is what we would expect from a theory of the Regge

type; iv) the partial waves, say for the t channel,

have a good asymptotic behaviour in t. This means that the asymptotic behaviour of our amplitude for t, s -) m at fixed cos 8 must not be worse than a power behaviour.

All previous models fail in one of the points i) to iv). For instance, the model by Khuri [l] is a Function which has a complex branch point in t

Fig. 1.

for s complex and it does not satisfy ii). The model by Veneziano [2] either satisfies i) and ii) or iii), but it cannot work for all of them. Indeed the function F( -a(s ))r( -a(t ))/r( -a(s)-a(t )) has no singularities in a(s) and a(t) except for poles, so that if a(s) is a polynomail in s it cannot have a Regge behaviour with the proper signature for a(s) + m along the real axis, the proper signature requiring a cut. If a(s) has an imaginary part for s real positive, the function can have a Regge behaviour in s, but then the residue at the pole in t is a polynomial in (Y(S) and cannot be a polyno- mial in s. In such a case the formula by Venezi- ano shows poles at all the partial waves for a value oft (not only the daughter poles).

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V o l u m e 2 8 B . n u m b e r 4 P H Y S I C S L E T T E R S 9 D e c e m b e r 1968

T h e s a m e c r i t e i s m , b a s e d on po in t i i i ) , a p p l i e s to t h e m o d e I s by A t k i n s o n a n d D i e t z [3] and b y K u p s h [4]. In o r d e r to c o n s t r u c t o u r e x - a m p l e le t u s f i r s t s t u d y t h e f u n c t i o n d e f i n e d b y t h e i n t e g r a l r e p r e s e n t a t i o n

1 1 (So - s) n G ( s , t ) ='2-~ni f dn n - a ( - i ) F ( n + l ) s i n n n ×

r (1)

1 1 >,( r(n -c )sin ~(n- c) r(n - d ) s i n ~ ( n - d)

w h e r e c and d a r e r e a l p o s i t i v e n o n - i n t e g e r n u m - b e r s . T h e i n t e g r a l i s done a l o n g t he c o n t o u r s h o w n in t he f i g u r e . I t c o n s i s t s of a p a r t p a r a l l e l to t h e i m a g i n a r y a x i s a t t h e l e f t of t h e p o l e s of t he i n t e g r a n d a t n = 0 , 1, 2 , . . . a n d n = c + 1, c + 2 , . . . a n d n = d + l , d + 2 , . . , p l u s a p a t h r o u n d t h e m o v - i ng p o l e a t n = a( t ) , i f e v e n t u a l l y i t i s to t h e r i g h t . T h e r e f o r e we h a v e a c o n t r i b u t i o n f r o m t he m o v - ing po l e of t h e f o r m

1 (So -S )a ( t ) 1 ×

F ( a ( t ) + l ) s i n n a ( t ) F ( a ( t ) - c ) s i n n ( a ( t ) - c )

1 (2) r(a(t) - d ) s i n n(a( t ) - d)

p l u s t h e c o n t r i b u t i o n f r o m a " b a c k g r o u n d " i n t e - g r a l of t h e f o r m of (1), i n t e g r a t e d a l o n g a p a r a l - l e l to t he i m a g i n a r y a x i s . F o r n -~ R e n + i~o, a t f i x e d Re n, t h e i n t e g r a n d in ( 1 ) b e h a v e s a s e x p ( - n / 2 1 n I), so t h a t t h e i n t e g r a l c o n v e r g e s f o r e v e r y Re n.

It i s i m m e d i a t e l y s e e n t h a t G ( s , t) h a s a b r a n c h in s a t So, a n d we c h o o s e t h e cu t f r o m s o to + oo. A s we c a n p e r f o r m p a r a l l e l d i s p l a c e m e n t s on t he l i n e of i n t e g r a t i o n of t he b a c k g r o u n d , i t i s i m - m e d i a t e l y a p p a r e n t t h a t G ( s , t ) h a s a R e g g e b e h a - v i o u r in s :

G ( s , t ) l s I -~oo ( - s ) a ( t ) "

F r o m eq. (2) i t i s s e e n t h a t , a p a r t f r o m t h e cu t of a( t ) , G ( s , t ) h a s p o l e s in t f o r a(t ) p o s i t i v e i n t e g e r , w h o s e r e s i d u e i s a p o l y n o m i a l in s . T h i s c o r r e s p o n d s in g e n e r a l to a f a m i l y of p a r a l - l e l d a u g h t e r s . G ( s , t ) h a s a l s o p o l e s f o r a(t ) = = c + m and a( t ) = d + m , w h o s e r e s i d u e i s no t a p o l y n o m i a l in s . We can c h o o s e h o w e v e r c a n d d v e r y l a r g e in s u c h a way t h a t t h e s e p o l e s a r e r e a c h e d f o r l a r g e R e / a n d a l s o , p r e s u m a b l y , I m t , so t h a t t h e y wi l l no t h a v e to b e i d e n t i f i e d w i t h r e s o n a n c e s . Anyhow , ff we do no t l ike t h e s e e x t r a p o l e s , we can e l i m i n a t e t h e m b y m u l t i p l y - i ng G ( s , t ) b y a f u n c t i o n of t a s i t w i l l b e s h o w n l a t e r .

L e t u s now c o n s i d e r t h e b e h a v i o u r of G ( s , t ) f o r ] t [-~ co. If ]a(t)[ -~ ¢o in s u c h a way t h a t R e a < N w e can a l w a y s c h o o s e t h e l i n e of i n t e g r a - t i o n so a s to l e a v e to t h e l e f t t h e m o v i n g p o l e , e v e n t u a l l y p i c k i n g a c o n t r i b u t i o n f r o m s o m e p o l e s a t n = l , n = c + m , n = d + m (l , m p o s i t i v e i n t e - g e r s ) . I t i s c l e a r t h a t t h e c o n t r i b u t i o n f r o m t h e s e p o l e s wi l l d e c r e a s e a s ]a( t)] ~ oo l i ke 1/(] a ( t ) ] ) , w h e r e a s t he i n t e g r a l a l o n g t h e l ine w i l l b e h a v e a t m o s t a s a c o n s t a n t . If R e a( t ) -~ + ~o we a r e f o r c e d to t a k e in to a c c o u n t t he c o n t r i b u t i o n f r o m t h e m o v i n g p o l e of t h e f o r m (2), a n d i t i s c l e a r t h a t f o r a(t ) g o i n g to i n f i n i t y l i n e a r l y w i t h t t h i s t e r m g o e s to z e r o f a s t e r t h a n t he e x p o n e n t i a l . L e t u s n o t e a l s o t h a t if s a n d t go to i n f i n i t y t o - g e t h e r , b e i n g f i x e d s / t , t h e t e r m (2) a g a i n g o e s to z e r o w h e r e a s t he l i n e i n t e g r a l b e h a v e s s u r e l y no t w o r s e t h a n a p o w e r in s .

In o r d e r to e l i m i n a t e t h e e x t r a p o l e s a t a( t ) = c + m , a( t ) = d + m we m u l t i p l y G ( s , t ) b y g ( t , c ) g ( t , d ) w h e r e

g ( t , c ) : (3)

1 = r ( c + 1 - a(t)) exp {- a(t )} exp{(a( t ) - c - ½ )lg b( to - t)}

a n d b i s d e t e r m i n e d by t he a s y m p t o t i c b e h a v i o u r of a ( t ) :

a(t ) , b t + h + O ( ( l g t ) L)

g ( t , c ) h a s a cu t in t f r o m to to + 0% i s z e r o f o r a(t ) = c + 1 , c + 2 , . . . and b e h a v e s a t m o s t l i ke e x p ( l o g t ) ~ w h e n ]t] ~o.

W e t h e n c o n s t r u c t a f u n c t i o n f ( t ) h a v i n g a cu t f r o m t o to + oo a n d s u c h a s to k i l l t h i s a s y m p t o t i c b e h a v i o u r , f o r i n s t a n c e

f ( t ) = e x p - [ l g ( t o - t)] 2L

o r

f ( t ) = e x p - ( t o - t )¼

T h e n we c o n s i d e r

F ( s , t ) = f ( t ) g ( t ) g ( t , c ) g ( t c i ) G ( s , t ) (4)

T h i s f u n c t i o n h a s a cu t in s f r o m So to + 0% a cu t in t f r o m t o to +~o, p o l e s i n t in t h e s e c o n d s h e e t w h o s e r e s i d u e i s a p o l y n o m i a l in s , i t b e h a v e s a s ( - s ) a(t) f o r s ~ ~oat f i x e d t , i t g o e s to z e r o f a s t e r t h a n p o w e r w h e n t -~ ~o a t f i x e d s o r a t f i x e d t / s ( f ixed c o s 0). T h e a s y m p t o t i c p r o p e r t i e s in t a r e i n d e e d d e t e r m i n e d b y f ( t ) . A t t h i s p o i n t t h e job i s done . We c o n s t r u c t

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B(s , t ) = F(s , t ) + F ( t , s ) (5)

and then

A ( s , t , u ) = B(s , t ) +B( t ,u) +B(s ,u ) (6)

[Note tha t f o r s ~ ~o at f ixed t B(s ,u) g o e s to z e r o f a s t e r than p o w e r , a s e x p l a i n e d b e f o r e . ] T h e func t ion A s a t i s f i e s t he p r o p e r t i e s i) to iv). As a f i r s t c o m m e n t we no te tha t e v e n if the r e p r e s e n - t a t i o n (1) m a y r e c a l l the Khur i r e p r e s e n t a t i o n [5], i t i s not an i n t e g r a l o v e r K h u r i a m p l i t u d e s . In - d e e d we cannot expand a func t ion in p o w e r s of (So-S) i f i t h a s a b r a n c h at s = s o. We want a l s o to po in t out tha t e v e n if the i n t e g r a n d of eq. (1) shows f ixed s i n g u l a r i t i e s at n = c + 1 , c + 2 , . . . , t h e Khur i a m p l i t u d e (the c o e f f i c i e n t in the s r d e - v e l o p m e n t ) shows on ly m o v i n g p o l e s a t r = a(t) , a( t ) - 1 , . . .

Second c o m m e n t . T h e way in which the f in i t e e n e r g y s u m r u l e s a r e s a t i s f i e d d o e s not show, by the v e r y c o n s t r u c t i o n of t he a m p l i t u d e [a l so 4], any dua l i ty of t he D o l e n - H o r n - S c h m i d [6] type . R a t h e r i t r e s e m b l e s t he i n t e r f e r e n c e m o d e l , o r b e t t e r the m o d e l p r o p o s e d by A l e s s a n d r i n i e t al . [7] to which we r e f e r f o r q u e s t i o n s c o n c e r n i n g double count ing. N a m e l y B(s , t ) i s c o m p o s e d of two p a r t s ( s e e eq. (4)): I) F(s , t ) which h a s r e s - o n a n c e s in t he t channe l and i s s m o o t h in the s c h a n n e l g i v i n g the R e g g e b e h a v i o u r in s , and II) F(t ,s ) which h a s r e s o n a n c e s in t he s channe l but g o e s to z e r o fo r s -~ ~o f a s t e r than any p o w e r and t h e n i t i s s t r i c t l y s u p e r c o n v e r g e n t (a l l the m o - men t ) and d o e s not c o n t r i b u t e any th ing to the R e g g e b e h a v i o u r .

As a t h i r d c o m m e n t the f in i t e e n e r g y s u m r u l e s do not g ive any b o o t s t r a p - l i k e cond i t ion on the p a r a m e t e r s of o u r amp l i t ude .

We want a l s o to no te that i t is l e s s c o m - p l i c a t e d , and in f ac t can be done r a t h e r e a s i l y , to c o n s t r u c t m o d e l s s a t i s f y i n g i) to iv) if the t r a j e c t o r y ~(t) g o e s to a f in i t e l i m i t fo r Re t -~+oo a n d / o r Re t ~ - ~ o . T h e s e m o d e l s a r e aga in e a s i l y c o n s t r u c t e d if we do not d e m a n d dua l i ty of the D o l e n - H o r n - S c h m i d type.

I would l ike to thank fo r d i s c u s s i o n s and c r i - t i c i s m P r o f e s s o r D. A m a t i and P r o f e s s o r A. Mar t in .

Re fe rences

1. N.N.Khuri , UCLR 18426 (1968). 2. G. Veneziano, Nuovo Cimento 57 (1968) 190. 3. D. Atkinson and K. Dietz, Rutherford Laboratory

preprint RPP/A 42 (1968). 4. J. Kupsch, Bonn University preprint (1968}. 5. N.N.Khuri , Phys. Rev. 132 (1963) 914. 6. R. Dolen, D. Horn and C. Sehmidt, Phys. Rev. 166

(1968) 1768. 7. V.A. Alessandrini. D. Amati and E, J. Squires,

CERN preprint TH.922 (1968).

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