Volume 32B, number 3 P H Y S I C S L E T T E R S 22 June 1970

A N A L Y T I C I T Y , U N I T A R I T Y , A N D N O N - R E S O N A N T B A C K G R O U N D S C A T T E R I N G

R. C. JOHNSON M a t h e m a t i c s D e p a r t m e n t , Durham 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-Tr scattering lengths. It is pointed out that analogous rea- soning may be apphed to lsospm -3/2 ~'-K scattering near the K (890).

A p r a c t i c a l scheme for exploi t ing the usual assumpt ions of analyt ic i ty , uni tar i ty , c ro s s ing s y m m e t r y and Regge asympto t ic behaviour to ca lcula te hadronic sca t t e r ing ampl i tudes has been proposed [1], based on the matching of two v e r s i o n s of the r e a l pa r t of the l ow-ene rgy a m - pli tude - the one (D) obtained f rom a f ixed- t d i s - pe r s ion re la t ion (FTDR) and the o ther (U) f rom uni tar i ty . The l ow-ene rgy ampli tude is to be guessed as a pa r t i a l -wave s e r i e s , and i ts imag- inary par t (/) used in f i n i t e - ene rgy sum ru le s (FESR) to fix p a r a m e t e r s of the cor respond ing h igh-energy (Regge) asymptot ic ampli tude R. Then D may be ca lcu la ted f rom FTDR over I and ImR for compar i son as a function of t (within the reg ion of convergence of both the FTDR and the pa r t i a l -wave s e r i e s ) with U de r ived f rom I v ia p a r t i a l - w a v e uni tar i ty . The demand of D = U, to be ach ieved by adjust ing the input guess , has been shown [1] to be capable of yielding impor tant dynamical cons t ra in t s on the ampli tude.

This note d i s c u s s e s fu rh te r s ignif icant imp l i ca - t ions of r equ i r ing such an equali ty, concern ing the nature of non- resonan t background sca t t e r ing in the energy region around an i so la ted and p redo - minant ly e l a s t i c resonance .

Fo r example we a re to p red ic t in a cons is ten t and sa t i s f ac to ry way the sign and approximate magnitude of the lower pa r t i a l waves in the i so - spin- two channel of ~-~ e las t i c sca t t e r ing in the neighbourhood of the rho resonance . A g r e e m e n t with expe r imen t is good. S imi l a r p red ic t ions for i sospin -3 /2 ~-K sca t t e r ing near the K*(890) can be made.

The line of reasoning, which fol lows, r e l i e s on the non- loca l i ty (in energy) of the re la t ion

between D and I, in compar i son with the s i m i l a r local i ty of the connection between U and L

Consider a set i s o s p i n - r e l a t e d e l a s t i c s ca t - t e r ing p r o c e s s e s , and fo rm an ampli tude of de - f ini te t - c h a n n e l quantum numbers and odd s y m - m e t r y under s -u c ross ing . Suppose to begin with that the low-ene rgy imag ina ry pa r t I is dominated by a s ingle phase shift r i s ing quickly through ~/2 - i . e . an i so la ted e las t i c resonance . Let the resonance define the pos i t ive sense of contr ibut ions to the amplitude° Then FESR p r e - dict in genera l a non-vanishing pos i t ive high- energy absorp t ive par t ImR. Computing D f rom an unsubt rac ted FTDR, (permi t ted by c r o s s i n g and the F r o i s s a r t bound for t < 0), i t 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 rom en t i r e ly negl ig ible effects due to th resho ld behaviour and any r e a - sonable ene rgy-dependence of the resonance width, or n o n - z e r o phase of i ts res idue . ) Be - cause however ImR > 0, D is l i f ted upwards by a substant ia l amount° On the other hand, U c o m - puted v ia uni tar i ty f rom I r em a ins v e r y c lose ly s y m m e t r i c a l about the origin.

In the example depic ted in fig. 1 the d i s c r e p - ancy between D and U close to the resonance is of the o rde r of 15-30% of the max imum peak- height, near t = 0. To obtain a g r e e m e n t between D and U some fur ther component must be p r e sen t in the ampli tude to compensa te for the effect of the h igh-ene rgy Regge-exchange tai l .

The mos t economica l mechan i sm which can br ing about the equal i ty D = U (and the re fo re the one which we suppose Nature chooses) is one which both l e s sens I (and hence, through FESR, ImR) tending to pull down D, while at the same

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M_/D j u

Fig. 1. Example of the imaginary part (I), and d isper- sive (D) and unitary (U) real parts at t - 0 of a cross ing- odd amplitude for scattering of equal-mass (rn) par - t ic les , with an elast ic P-wave resonance of mass 5~n, width 1.0m. For ~S-~ 8rn the amplitude is assumed pro-

portional to s ~/2, fitting smoothly on.

t i m e i t adds to U - wi thout g r e a t l y d i s t o r t i n g the ampl i tude .

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 i t a t 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 ve ly s m a l l n e g a t i v e phase sh i f t in one o r m o r e p a r t i a l w a v e s which con t r i bu t e in a n e g a t i v e s e n s e to the ampl i tude° 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 f ig. 1 can 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 ampl i t ude an S - w a v e whose phase shi f t f a l l s s m o o t h l y to a con- s tan t va lue of about -15 ° unde r the peak of I.

C l e a r l y , i f an ampl i t ude con ta ins 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 combine so that D = U wi thou t the s p e c i a l a id of n o n - r e s o n a n t background . T h i s may be the c a s e in 7r-N s c a t t e r i n g , fo r i n s t ance . 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 phase sh i f t s does 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 above . 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. The a m p l i t u d e F fo r t < 0 obeys

i m m e d i a t e l y , wi th the conc lu s ion that the na tu r a l s ign for the ( n o n - r e s o n a n t ) l o w - e n e r g y phase sh i f t s in the i s o s p i n - t w o ampl i t ude A 2 is nega t ive . B e c a u s e of B o s e s t a t i s t i c s , A 2 con ta ins no P - wave , so to ba l ance the e f f ec t of the P - w a v e p , nega t i ve phase sh 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 igher w a v e s a r e p r e - s u m a b l y neg l ig ib le . )

Now c o n s i d e r i n g the p r e s e n c e of A o, i t i s p la in that an S - w a v e wi th a b r o a d m a x i m u m , (pe rhaps a r e s o n a n c e ) unde r the p only r e i n - f o r c e s our conc lus ion . It adds to I, (and hence R), l i f t ing D f u r t h e r , wh i l e l i f t ing U l e s s , and wi thout c aus ing 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 shape 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 red ic t ion° (The r e s o n a n c e s t h e m s e l v e s , be ing w e l l above the p, can 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 quot ing 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 ions fo 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°

The f i r s t c o n c e r n s the phase of F at h igh ene rgy . If only the A0 and A 1 a m p l i t u d e s wi th t h e i r f0, q(?), p and g s t a t e s a r e u s e d to bui ld F a t l a r g e v and s m a l l t t h rough g e n e r a l i s e d FESR, then r e s o n a n t i m a g i n a r y p a r t s r e i n f o r c e whi le c o r r e s p o n d i n g r e a l p a r t s c a n c e l - l ead ing to I m F ( v ~ co, t ~ 0) >> R e F ( v - * ~o, t = 0). To obta in on the c o n t r a r y the a p p r o x i m a t e equa l i t y r e q u i r e d by the phase e n e r g y r e l a t i o n (or Regge p exchange) the o t h e r l o w - e n e r g y phase sh i f t s m u s t be such 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 . N e g a t i v e i s o s p i n - t w o phase sh i f t s a r e jus t what i s 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, which

o o

R e F ( v , t) = 2 ~ p f ImF(v ' , t )dv ' (1) (v ' - v ) ( v ' + v ) '

w h e r e v oc s-u), with c o n v e r g e n c e a s s u r e d b e - c ause 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 Regge po le exchange . The 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 .

= A 2

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 fo 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 be low 1 GeV c . m . e n e r g y (i. e. wi th in the known e l a s t i c r e - gion), i s the rho r e s o n a n c e in A 1.

The r e a s o n i n g g iven above can be app l i ed

O

- , °

.S - 2 0

- 3 0

(3

0"4 0"6 0"8 I'0

Centre of Mo~ Energy in C~zV

Fig. 2. Range of i s o s p i n - 2 ?T-?T n o n - r e s o n a n t backgro tmd S- and D-wave phase-shift solutions. (552 and 5D2 r e -

spectively), as described in the text.

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at l a r g e ~; fo r t ~ 0 has 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 lone . 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 fo r th is b e h a v i o u r of G.

The 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 hose of O l s s o n [2] i s to be noted.

S ince an F T D R for G needs 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 ga ined 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 ign of the A 2 p h a s e s a l r e a d y deduced , 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 tan t is nega t ive . 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 then f ind a c o n s t r a i n t on the s c a t t e r i n g l eng ths

a 0 + 5a 2 < 0 , (3)

which t aken t o g e t h e r wi th 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 ive s the u se fu l uppe r bounds

%<0.2 a 2 < - 0 . 0 4 p

(4)

(where 2 a 0 - 5a 2 ~ 0.5), a l l in uni t s of p ion 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 ev iden t .

In p a s s i n g , we po in t out tha t of D i l l e y s ' s two types of so lu t ion [4] to the ~-~ t h r e s h o l d - r e g i o n c r o s s i n g - p l u s - u n i t a r i t y equa t ion , p r e s e n t c o n - s i d e r a t i o n f a v o u r those of c l a s s H - w h i c h has z e r o s be low t h r e s h o l d - and tend to c o n f i r m tha t they a r e l inked 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 have 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 fo 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 re f . [1], t ak ing c a r e f u l a ccoun t of the i s o s p i n - t w o channe l . We quote s o m e r e s u l t s ob- t a i n e d by f i t t ing t o g e t h e r the q u a n t i t i e s D and U for 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 r e g i o n - 3 ~ 2 < t < 3 t t 2 , 4 t 1 2 < s < 4 0 ~ 2, (~ = p i o n mass).

Solutions were sought for the elastic S 2 and D 2 waves (notation: (angular momentum)isosPi N) as 3-parameter power series (with appropriate threshold behaviours) for the tangents of the phase shifts. P1 was assumed given as a 2-param- eter effective-range expression containing a purely elastic p of mass 765 + 5 MeV, width 120±20 MeV, and S o was inserted as a 3-pa- rameter effective-range formula capable of reproducing a wide range of possible types of

r i s i n g phase shif t , inc lud ing a r e s o n a n c e . D o was p a r a m e t r i s e d r i s i n g s m o o t h l y f r o m t h r e s h - o ld a s if to the f(1260), and in s o m e so lu t i ons the p r e s e n c e of a s m a l l r i s i n g l~hase sh i f t in F 1 was c o n s i d e r e d .

Above s = s I = 6 0 + 1 0 ~ 2 was 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), with ~ ( t ) = = (0 .5+0 .05) + t / ( 5 0 + 5 t ~ 2 ) . F o r a g iven in i t i a l 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 f ix y ( t ) , and then 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 , fo r a cho ice of D - w a v e s . T h e n the va lue of y was r e - e s t i m a t e d , and the p r o c e s s r e p e a t e d , s a t i s f a c t o r y so lu t ions be ing t hose s t ab l e u n d e r s e v e r a l c y c l e s of th 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 check on the phase . As a f u r t h e r check , the c o m p u t a t i o n s wi th 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 fo r v a r i o u s t ypes 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 , th is t i ° e f i t t ing 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 fo r the b a c k g r o u n d phase sh i f t s 5S2 and 5/)2 a r e g iven in fig. 2.

Both p h a s e - s h i f t s a r e n e g a t i v e as e x p e c t e d , wi th 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 so lu t i ons wi th 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 fo r a n o n - r e s o n a n t 5S0 , peak ing at about 30 ° a t 800 MeV c . m . e n e r g y , was found. Inc lud ing D - w a v e s , (and F 1, wi th a phase sh i f t r i s i n g s m o o t h l y to 5 ° a t 1 GeV c° m. e n e r g y ) , to i m - p r o v e the qua l i ty of so lu t i ons fo r t > 0 (where cos 0 >> 1 is p o s s i b l e ) , a p r e f e r e n c e fo r a r e - sonan t S o p h a s e sh i f t e m e r g e d , a l though non- r e s o n a n t f o r m s gave only s l i gh t ly 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- waves ) . 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 eng ths w e r e found

0.16 < a 0 < 0.22 (5)

- 0 . 0 7 < a 2 < -0 .05 ,

in r e a s o n a b l e a g r e e m e n t wi th expec t a t i on . A l l t h e s e r e s u l t s a r e s t ab 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 quoted above .

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 a t e l i m - ina t ing the a m b i g u i t y in S o by exp lo i t i ng 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 th m o r e g e n e r a l f o r m s of F E S R , and i nc lude a t t e m p t s to con t inue the p h a s e - s h i f t s o l u t i o n s above 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 fo 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 wi th e x p e r i m e n t s [6], shou ld 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

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s t r i k i n g l y c o n f i r m o u r i n i t i a l q u a l i t a t i v e a r g u m e n t s a n d u n d e r l i n e the p o w e r of the 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 , we now c o n s i d e r a v e r y s i m i l a r s i t u a t i o n in the ~ - K p r o c e s s - f o r the i s o - s p i n - 3 / 2 p a r t i a l - w a v e a m p l i t u d e s u n d e r 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 and 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 the K* if a n y t h i n g r e i n f o r c e s the a r g u m e n t , and the c o n c l u s i o n a g r e e w i t h 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 the s c a t t e r i n g l e n g t h c o n t r a i n t .

al/2 + 2a3 /2 < 0 (6)

A g a i n , a g r e e m e n t w i t h e x p e r i m e n t a l i n d i c a t i o n s [6] i s good and 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 °

A s a f i n a l r e m a r k , we s u g g e s t t h a t t h e s e r e - s 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 f ind n o n - r e s o n a n t b a c k g r o u n d i n d i s p e n s i b l e in a t l e a s t s o m e n o n - z e r o - q u a n t u m - n u m b e r - e x c h a n g e p r o c e s s e s .

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 the T h e o r y G r o u p a t the R u t h e r f o r d H i g h - E n e r g y L a b o r a t o r y fo r t h e i r he lp fu 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 sw i f t c o m m u n i c a t i o n of the r e s u l t s of of re f . [4].

References [1] R. C. Johnson, Phys. Rev. Le t te r s 22 (1969) 1143. [2] M. G. Olsson, ref. [6], p. 759;

M. G. Olsson and G. Y. Kaiser , Univers i ty of Wis- consin repor t C00-222, March 1969, unpublished.

[3] D. Morgan and G. Shaw, Columbia Univers i ty r e - port NYO-1932 (2) - 160, (1970), to be published; M. G. Olsson, Univers i ty of Wisconsin repor t COO- 270, (1970), to be published.

[4] J. P. Dilley, Univers i ty 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~ In-

te rac t ions (Argonne National Labora tory , 1969). [7] C. Lovelace, ref. [6], p. 562

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