Analyticity, unitarity, and non-resonant background scattering

Volume 32B, number 3

PHYSICS LETTERS

22 June 1970

ANALYTICITY, UNITARITY, AND NON-RESONANT BACKGROUND SCATTERING R. C. JOHNSON M a t h e m a t i c s D e p a r t m e n t , D u r h a m U n i v e r s i t y , UK

Received 4 May 1970

Requiring simultaneously analyticity and unitarity of an amplitude with an isolated elastic resonance at low energy leads to predictions of the sign and magnitude of associated non-resonant background phase shifts. The case of isospin-2 F-~ scattering in the p-meson energy region is considered in detail, and r e sults include a useful bound on the two S-wave 7r-Trscattering lengths. It is pointed out that analogous r e a soning may be apphed to lsospm -3/2 ~'-K scattering near the K (890).

A p r a c t i c a l s c h e m e f o r exploiting the u s u a l a s s u m p t i o n s of analyticity, u n i t a r i t y , c r o s s i n g s y m m e t r y and Regge a s y m p t o t i c b e h a v i o u r to c a l c u l a t e hadronic s c a t t e r i n g a m p l i t u d e s has been p r o p o s e d [1], b a s e d on the m a t c h i n g of two v e r s i o n s of the r e a l p a r t of the l o w - e n e r g y a m plitude - the one (D) obtained f r o m a f i x e d - t d i s p e r s i o n r e l a t i o n (FTDR) and the o t h e r (U) f r o m u n i t a r i t y . The l o w - e n e r g y amplitude is to be g u e s s e d as a p a r t i a l - w a v e s e r i e s , and its i m a g i n a r y p a r t (/) u s e d in f i n i t e - e n e r g y sum r u l e s (FESR) to fix p a r a m e t e r s of the c o r r e s p o n d i n g h i g h - e n e r g y (Regge) a s y m p t o t i c a m p li tu d e R. Then D m ay be c a l c u l a t e d f r o m F T D R o v e r I and ImR f o r c o m p a r i s o n as a function of t (within the r e g i o n of c o n v e r g e n c e of both the F T D R and the p a r t i a l - w a v e s e r i e s ) with U d e r i v e d f r o m I v i a p a r t i a l - w a v e u n i t a r i t y . The d e m a n d of D = U, to be a c h i e v e d by adjusting the input g u e s s , has b e e n shown [1] to be capable of yielding i m p o r t a n t d y n a m i c a l c o n s t r a i n t s on the amplitude. T h i s note d i s c u s s e s f u r h t e r s i g n i f i c a n t i m p l i c a tions of r e q u i r i n g such an equality, c o n c e r n i n g the nature of n o n - r e s o n a n t background s c a t t e r i n g in the e n e r g y r e g i o n a r o u n d an i s o l a t e d and p r e d o minantly e l a s t i c r e s o n a n c e . F o r e x a m p l e we a r e to p r e d i c t in a c o n s i s t e n t and s a t i s f a c t o r y way the sign and a p p r o x i m a t e magnitude of the l o w e r p a r t i a l w a v e s in the i s o s p i n - t w o channel of ~-~ e l a s t i c s c a t t e r i n g in the neighbourhood of the rho r e s o n a n c e . A g r e e m e n t with e x p e r i m e n t is good. S i m i l a r p r e d i c t i o n s f o r i s o s p i n - 3 / 2 ~ -K s c a t t e r i n g n e a r the K*(890) can be made. The line of r e a s o n i n g , which follows, r e l i e s on the n o n - l o c a l i t y (in e n e r g y ) of the r e l a t i o n

b et w een D and I, in c o m p a r i s o n with the s i m i l a r l o c a l i t y of the connection b e t w e e n U and L C o n s i d e r a set i s o s p i n - r e l a t e d e l a s t i c s c a t t e r i n g p r o c e s s e s , and f o r m an amplitude of d e finite t - c h a n n e l quantum n u m b e r s and odd s y m m e t r y u n d er s - u c r o s s i n g . Suppose to b eg i n with that the l o w - e n e r g y i m a g i n a r y p a r t I is d o m i n a t e d by a s i n g l e phase shift r i s i n g quickly through ~/2 - i . e . an i s o l a t e d e l a s t i c r e s o n a n c e . L e t the r e s o n a n c e define the p o s i t i v e s e n s e of c o n t r i b u t i o n s to the amplitude° Then F E S R p r e dict in g e n e r a l a n o n - v a n i s h i n g p o s i t i v e h i g h e n e r g y a b s o r p t i v e p a r t ImR. Computing D f r o m an u n s u b t r a c t e d FTD R, ( p e r m i t t e d by c r o s s i n g and the F r o i s s a r t bound f o r t < 0), it is found that w e r e R = 0 then D would be s y m m e t r i c a l about the origin, (apart f r o m e n t i r e l y n e g l i g i b l e e f f e c t s due to t h r e s h o l d b e h a v i o u r and any r e a sonable e n e r g y - d e p e n d e n c e of the r e s o n a n c e width, or n o n - z e r o phase of its r e s i d u e . ) B e c a u s e h o w e v e r ImR > 0, D is lifted u p w ar d s by a s u b s t a n t i a l amount° On the o t h er hand, U c o m puted v i a u n i t a r i t y f r o m I r e m a i n s v e r y c l o s e l y s y m m e t r i c a l about the origin. In the e x a m p l e d e p i c t e d in fig. 1 the d i s c r e p ancy b et w een D and U c l o s e to the r e s o n a n c e is of the o r d e r of 15-30% of the m a x i m u m p e a k height, n e a r t = 0. To obtain a g r e e m e n t b et w een D and U s o m e f u r t h e r co m p o n en t m u s t be p r e s e n t in the am p l i t u d e to c o m p e n s a t e f o r the e f f e c t of the h i g h - e n e r g y R e g g e - e x c h a n g e tail. The m o s t e c o n o m i c a l m e c h a n i s m which can b r i n g about the e q u a l i t y D = U (and t h e r e f o r e the one which we suppose N a t u r e c h o o s e s ) is one which both l e s s e n s I (and hence, through FESR, ImR) tending to pull down D, while at the s a m e 199

Volume

32B, number

3

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M_/D ju Fig. 1. Example of the imaginary part (I), and dispersive (D) and unitary (U) real parts at t - 0 of a crossingodd amplitude for scattering of equal-mass (rn) p a r ticles, with an elastic P-wave resonance of mass 5~n, width 1.0m. For ~S-~ 8rn the amplitude is assumed p r o portional to s ~/2, fitting smoothly on. t i m e it a d d s to U - without g r e a t l y d i s t o r t i n g the amplitude. T h i s m a y be a c h i e v e e x t r e m e l y s i m p l y if t h e r e i s it at low e n e r g y b e s i d e s the r e s o n a n c e a r e l a t i v e l y s m a l l n e g a t i v e p h a s e s h i f t in one o r m o r e p a r t i a l w a v e s w h i c h c o n t r i b u t e in a n e g a t i v e s e n s e to the a m p l i t u d e ° F o r e x a m p l e , the d i f f e r e n c e b e t w e e n D and U of fig. 1 c a n be m a d e v e r y s m a l l n e a r t = 0 by s u b t r a c t i n g f r o m the a m p l i t u d e an S - w a v e w h o s e p h a s e shift f a l l s s m o o t h l y to a c o n s t a n t v a l u e of about -15 ° u n d e r the p e a k of I. C l e a r l y , if an a m p l i t u d e c o n t a i n s s e v e r a l o v e r l a p p i n g and p e r h a p s i n e l a s t i c r e s o n a n c e s , t h e s e can c o m b i n e so that D = U w i t h o u t the s p e c i a l aid of n o n - r e s o n a n t b a c k g r o u n d . T h i s m a y be the c a s e in 7r-N s c a t t e r i n g , f o r i n s t a n c e . H o w e v e r , the p r e s e n c e of o t h e r i m p o r t a n t and p e r h a p s r e s o n a n t p h a s e s h i f t s d o e s not n e c e s s a r i l y a l t e r the s i m p l e a r g u m e n t a b o v e . C o n s i d e r as an e x a m p l e 7r-~ s c a t t e r i n g in the s t a t e of p u r e t - c h a n n e l i s o s p i n one. T h e a m p l i t u d e F f o r t < 0 obeys

LETTERS

i m m e d i a t e l y , with the c o n c l u s i o n that the n a t u r a l s i g n for the ( n o n - r e s o n a n t ) l o w - e n e r g y p h a s e s h i f t s in the i s o s p i n - t w o a m p l i t u d e A 2 is n e g a t i v e . B e c a u s e of B o s e s t a t i s t i c s , A 2 c o n t a i n s no P w a v e , so to b a l a n c e the e f f e c t of the P - w a v e p, n e g a t i v e p h a s e s h i f t s a r e e x p e c t e d in both the S- and D - w a v e s of A 2, ( h i g h e r w a v e s a r e p r e sumably negligible.) Now c o n s i d e r i n g the p r e s e n c e of A o, it is p l a i n that an S - w a v e w i t h a b r o a d m a x i m u m , ( p e r h a p s a r e s o n a n c e ) u n d e r the p only r e i n f o r c e s our c o n c l u s i o n . It adds to I, (and h e n c e R), lifting D f u r t h e r , w h i l e lifting U l e s s , and w i t h o u t c a u s i n g m a j o r d i s t o r t i o n of the b a s i c r e s o n a n c e s h a p e of both r e a l p a r t s . A s m a l l D - w a v e i n A 0 r i s i n g to the f0 (1260), and p e r haps a s m a l l F - w a v e in A 1, r i s i n g to the g(1650), l i k e w i s e can only s t r e n g t h e n the p r e d i c t i o n ° (The r e s o n a n c e s t h e m s e l v e s , b e i n g w e l l a b o v e the p, c a n be e f f e c t i v e l y a b s o r b e d into (R). B e f o r e quoting r e s u l t s of n u m e r i c a l c a l c u l a t i o n s f o r c o m p a r i s o n with e x p e r i m e n t , t h e r e a r e s o m e t h e o r e t i c a l c o m m e n t s to be made° T h e f i r s t c o n c e r n s the p h a s e of F at h i g h e n e r g y . If only the A0 and A 1 a m p l i t u d e s w i t h t h e i r f0, q(?), p and g s t a t e s a r e u s e d to b u i l d F a t l a r g e v and s m a l l t t h r o u g h g e n e r a l i s e d FESR, then resonant imaginary parts reinforce while corresponding real parts cancel - leading to I m F ( v ~ co, t ~ 0) >> R e F ( v - * ~o, t = 0). To o b t a i n on the c o n t r a r y the a p p r o x i m a t e e q u a l i t y r e q u i r e d by the p h a s e e n e r g y r e l a t i o n (or R e g g e p e x c h a n g e ) the o t h e r l o w - e n e r g y p h a s e s h i f t s m u s t be s u c h a s to i n c r e a s e R e F r e l a t i v e to I m F . Negative isospin-two phase shifts are just what is need. S i m i l a r l y one m a y c o n s i d e r the t - c h a n n e l i s o s c a l a r a m p l i t u d e G= ½AO + A 1 + ~ A 2, w h i c h

oo

R e F ( v , t) = 2 ~ p f

ImF(v', t)dv' (v' - v ) ( v ' + v )

=

'

A2

D i s r e g a r d i n g the i s o s c a l a r a m p l i t u d e A 0 f o r the m o m e n t , the p r i n c i p a l f e a t u r e of F b e l o w 1 GeV c . m . e n e r g y (i. e. w i t h i n the known e l a s t i c r e gion), is the r h o r e s o n a n c e in A 1. T h e r e a s o n i n g g i v e n a b o v e c a n be a p p l i e d 200

O

(1)

w h e r e v oc s-u), w i t h c o n v e r g e n c e a s s u r e d b e c a u s e the h i g h - e n e r g y b e h a v i o u r of F i s g o v e r n e d by p - m e s o n R e g g e p o l e e x c h a n g e . T h e s - c h a n n e l i s o s p i n d e c o m p o s i t i o n of F is.

22 J u n e 1 9 7 0

-,°

.S - 2 0 -30

(3 0"4

0"6

0"8

I'0

Centre of Mo~ Energy in C~zV F i g . 2. Range of i s o s p i n - 2 ?T-?T n o n - r e s o n a n t b a c k g r o t m d

S- and D-wave phase-shift solutions. (552 and 5D2 r e spectively), as described in the text.

Volume 32B. number 3

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at l a r g e ~; f o r t ~ 0 h a s a l a r g e p o s i t i v e i m a g i n a r y p a r t f r o m P and P ' e x c h a n g e and s m a l l e r n e g a t i v e r e a l p a r t f r o m the P ' a l o n e . E v i d e n t l y n e g a t i v e p h a s e s in A 2 a r e v e r y c o n v e n i e n t f o r t h i s b e h a v i o u r of G. T h e c o n n e c t i o n of t h e s e p h a s e a r g u m e n t s to t h o s e of O l s s o n [2] is to be noted. S i n c e an F T D R f o r G n e e d s a s u b t r a c t i o n (for t ~ 0) t h e r e s e e m s at f i r s t l i t t l e to be g a i n e d f r o m c o m p a r i n g i t s d i s p e r s i v e and u n i t a r i t y r e a l p a r t s . H o w e v e r , a c c e p t i n g as c o r r e c t the s i g n of the A 2 p h a s e s a l r e a d y d e d u c e d , t h e s e a r e s e e n to w o r s e n the d i s c r e p a n c y b e t w e e n a l t e r n a t i v e r e a l p a r t n e a r the p u n l e s s the s u b t r a c t i o n c o n s t a n t is n e g a t i v e . S u b t r a c t i n g at t h r e s h o l d at t = 0 we t h e n find a c o n s t r a i n t on the s c a t t e r i n g lengths a 0 + 5a 2 < 0 ,

(3)

w h i c h t a k e n t o g e t h e r w i t h the ~ u n i v e r s a l c u r v e " of M o r g a n and Shaw [3] and of O l s s o n [3], g i v e s the u s e f u l u p p e r bounds

%<0.2 a 2<-0.04

(4) p

( w h e r e 2 a 0 - 5a 2 ~ 0.5), a l l in u n i t s of pion C o m p ton w a v e l e n g t h s , S a t i s f a c t o r y c o n s i s t e n c y is evident. In p a s s i n g , we p o i n t out t h a t of D i l l e y s ' s two t y p e s of s o l u t i o n [4] to the ~-~ t h r e s h o l d - r e g i o n crossing-plus-unitarity equation, present cons i d e r a t i o n f a v o u r t h o s e of c l a s s H - w h i c h h a s z e r o s b e l o w t h r e s h o l d - and t e n d to c o n f i r m t h a t t h e y a r e l i n k e d to the e x i s t e n c e of the p [4, 5]. S t i m u l a t e d by the f o r e g o i n g s e m i - q u a n t i t a t i v e a r g u m e n t s , we h a v e c a r r i e d out f u r t h e r n u m e r i c a l c o m p u t a t i o n s f o r the v - ~ s y s t e m of the s o r t d e s c r i b e d in r e f . [1], t a k i n g c a r e f u l a c c o u n t of the i s o s p i n - t w o c h a n n e l . We q u o t e s o m e r e s u l t s obt a i n e d by f i t t i n g t o g e t h e r the q u a n t i t i e s D and U f o r the i s o v e c t o r - e x c h a n g e a m p l i t u d e F in the region-3~2
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22 June 1970

r i s i n g p h a s e shift, i n c l u d i n g a r e s o n a n c e . D o was parametrised rising smoothly from threshold a s if to the f(1260), and in s o m e s o l u t i o n s the p r e s e n c e of a s m a l l r i s i n g l~hase s h i f t in F 1 was considered. A b o v e s = s I = 6 0 + 1 0 ~ 2 w a s a s s u m e d the b e h a v i o u r I m F ( v , t ) = ~ v el(t), w i t h ~ ( t ) = = (0.5+0.05) + t/(50+5t~2). For a given initial c h o i c e of p h a s e - s h i f t p a r a m e t e r s , the z e r o m o m e n t F E S R w a s u s e d to fix y ( t ) , and t h e n the S - w a v e p a r a m e t e r s w e r e v a r i e d to m i n i m i s e the d i f f e r e n c e b e t w e e n D and U o v e r a m e s h of s and t - p o i n t s , f o r a c h o i c e of D - w a v e s . T h e n the v a l u e of y w a s r e - e s t i m a t e d , and the p r o c e s s repeated, satisfactory solutions being those s t a b l e u n d e r s e v e r a l c y c l e s of t h i s p r o c e d u r e and s a t i s f y i n g a p p r o x i m a t e l y a G i l b e r t - t y p e F E S R a s c h e c k on the p h a s e . A s a f u r t h e r c h e c k , the c o m p u t a t i o n s w i t h n o n - z e r o D - w a v e s w e r e r e p e a t e d f o r v a r i o u s t y p e s of S e m e r g i n g f r o m t h e s e f i r s t c a l c u l a t i o n s , t h i s t i ° e fitting D and U by v a r y i n g the S 2 and D 2 p a r a m e t e r s . R e s u l t s f o r the b a c k g r o u n d p h a s e s h i f t s 5S2 and 5/)2 a r e g i v e n in fig. 2. Both phase-shifts are negative as expected, w i t h 15D2] ~ 5 ° , 15S21 ~ 35 °. L a r g e r v a l u e s of ]5S21 c o r r e s p o n d to s o l u t i o n s w i t h no D - w a v e s (nor F1) , and t h e r e a g e n e r a l p r e f e r e n c e f o r a n o n - r e s o n a n t 5S0 , p e a k i n g at about 30 ° at 800 MeV c . m . e n e r g y , w a s found. I n c l u d i n g D - w a v e s , (and F 1, w i t h a p h a s e s h i f t r i s i n g s m o o t h l y to 5 ° at 1 GeV c° m. e n e r g y ) , to i m p r o v e the q u a l i t y of s o l u t i o n s f o r t > 0 ( w h e r e c o s 0 >> 1 is p o s s i b l e ) , a p r e f e r e n c e f o r a r e s o n a n t So p h a s e s h i f t e m e r g e d , a l t h o u g h n o n r e s o n a n t f o r m s g a v e only s l i g h t l y p o o r e r f i t s , (as did r e s o n a n t f o r m s in the a b s e n c e of D w a v e s ) . In a l l c a s e s , a f a i r l y n a r r o w r a n g e of s c a t t e r i n g l e n g t h s w e r e found 0.16 < a 0 < 0.22 -0.07
(5) ,

in r e a s o n a b l e a g r e e m e n t w i t h e x p e c t a t i o n . A l l t h e s e r e s u l t s a r e s t a b l e u n d e r the v a r i a t i o n s of input p a r a m e t e r s a s q u o t e d a b o v e . I n v e s t i g a t i o n s in p r o g r e s s a r e a i m e d at e l i m i n a t i n g the a m b i g u i t y in So by e x p l o i t i n g F T D R f o r the o t h e r two i s o s p i n e x c h n a g e s , t o g e t h e r w i t h m o r e g e n e r a l f o r m s of F E S R , and i n c l u d e a t t e m p t s to c o n t i n u e the p h a s e - s h i f t s o l u t i o n s a b o v e 1 GeV, w h e r e i n e l a t i c i t y i s e x p e c t e d to b e c o m e i m p o r t a n t , e s p e c i a l l y in the S - w a v e s . H o w e v e r the p r e s e n t r e s u l t s f o r the b a c k g r o u n d s c a t t e r i n g (which a g r e e w e l l w i t h e x p e r i m e n t s [6], s h o u l d not be s i g n i f i c a n t l y a l t e r e d . T h e y 201

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strikingly confirm our initial qualitative arguments a n d u n d e r l i n e t h e p o w e r of t h e m e t h o d s of r e f . [1]. L e a v i n g ~ - ~ s c a t t e r i n g , w e now c o n s i d e r a v e r y similar situation in the ~-K process - for the isospin-3/2 partial-wave amplitudes under the K* (890). T h e p r e d i c t i o n s a r e t h a t t h e s e l o w - e n e r g y b a c k g r o u n d S-, P - , a n d D - w a v e p h a s e s h i f t s a r e s m a l l a n d n e g a t i v e . T h e p r e s e n c e of a n S - w a v e k a p p a j u s t a b o v e t h e K* if a n y t h i n g r e i n f o r c e s the argument, and the conclusion agree with h i g h - e n e r g y p h a s e r e q u i r e m e n t s a n d l e a d to t h e scattering length contraint.

al/2 + 2 a 3 / 2 < 0

(6)

Again, agreement with experimental indications [6] i s good a n d c o m p a r i s o n w i t h a t l e a s t one t h e o r e t i c a l m o d e l [7] i s e n c o u r a g i g n g ° As a final remark, we suggest that these res u l t s m e a n t h a t a u n i t a r y t h e o r y of h a d r o n s w i l l find non-resonant background indispensible in at least some non-zero-quantum-number-exchange processes.

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22 June 1970

I a m g r a t e f u l to E. J . S q u i r e s f o r d i s c u s s i o n s , a n d to m e m b e r s of t h e T h e o r y G r o u p a t t h e Rutherford High-Energy Laboratory for their h e l p f u l c o m m e n t s . I a l s o w i s h to t h a n k J . P . D i l l e y f o r s w i f t c o m m u n i c a t i o n of t h e r e s u l t s of of r e f . [4].

References [1] R. C. Johnson, Phys. Rev. L e t t e r s 22 (1969) 1143. [2] M. G. Olsson, ref. [6], p. 759; M. G. Olsson and G. Y. K a i s e r , University of W i s consin r e p o r t C00-222, M a r c h 1969, unpublished. [3] D. Morgan and G. Shaw, Columbia University r e port NYO-1932 (2) - 160, (1970), to be published; M. G. Olsson, University of Wisconsin r e p o r t COO270, (1970), to be published. [4] J. P. Dilley, University of Ohio, to be published. [5] J. E. Bowcock and G. E. John, Nucl. Phys. B i t (1969) 659. [6] Proceedings of the Conference on y-TT and K~ Int e r a c t i o n s (Argonne National L a b o r a t o r y , 1969). [7] C. Lovelace, ref. [6], p. 562