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A soluble model for the πΛ interaction
-~
Nuclear Physics B10 (1969) 89-98. North-Holland Publ. Comp., Amsterdam
A SOLUBLE MODEL FOR THE ~rA I N T E R A C T I O N N. M. QUEEN
Department of Mathematical Physics, University of Birmingham, England Received 13 December 1968 Abstract: A simple model for the ~A interaction, based on a forward dispersion r e lation for the inverse amplitude, is formulated. The assumption that the amplitude is dominated in the appropriate energy range by the Y{ (1385) resonance leads to definite restrictions on the possible behaviour of the amplitude at lower and at higher energies. The asymptotic total cross section calculated from the model is consistent with the quark-model prediction. However, a negative S-wave scattering length is required, in disagreement with each of three alternative (mutually incompatible) predictions.
1. INTRODUCTION T h e o r e t i c a l s t u d i e s of the ~ i n t e r a c t i o n with the aid of f o r w a r d d i s p e r s i o n r e l a t i o n s have b e e n m a d e by many a u t h o r s [1-5]. The S - w a v e s c a t t e r ing l e n g t h s have b e e n p r e d i c t e d [1] on the b a s i s of e x p e r i m e n t a l and t h e o r e t i c a l i n f o r m a t i o n about the p h a s e s h i f t s at h i g h e r e n e r g i e s . Even in the a b s e n c e of d e t a i l e d ~ s c a t t e r i n g data, the a s s u m p t i o n that the a m p l i t u d e is d o m i n a t e d in the i n t e r m e d i a t e e n e r g y r e g i o n by one or two r e s o n a n c e s , but is r e l a t i v e l y s t r u c t u r e l e s s e l s e w h e r e , l e a d s to definite r e s t r i c t i o n s [3, 4] on the a m p l i t u d e o v e r the e n t i r e e n e r g y r a n g e . In t h i s p a p e r , we f o r m u l a t e a m o d e l f o r the ~A i n t e r a c t i o n b a s e d on a f o r w a r d d i s p e r s i o n r e l a t i o n f o r the i n v e r s e a m p l i t u d e . Our a p p r o a c h is s i m i l a r in s p i r i t to that of the m o d e l f o r ~ s c a t t e r i n g p r o p o s e d by S h i r k o v [2] and e x t e n d e d by Antoniou [3]. A m a j o r a d v a n t a g e of w o r k i n g with the i n v e r s e a m p l i t u d e is that both the t h r e s h o l d and the h i g h - e n e r g y b e h a v i o u r of i t s i m a g i n a r y p a r t a r e t h e o r e t i c a l l y d e t e r m i n e d a l m o s t uniquely without p r i o r k n o w l e d g e of the s c a t t e r i n g length or c r o s s s e c t i o n . M o r e o v e r , the d i s p e r s i o n r e l a t i o n f o r the i n v e r s e a m p l i t u d e does not r e q u i r e s u b t r a c t i o n s . If a s p e c i f i c m o d e l is a d o p t e d f o r the i m a g i n a r y p a r t of the i n v e r s e a m p l i tude at a l l e n e r g i e s , the d i s p e r s i o n r e l a t i o n then m a k e s it p o s s i b l e to c o n s t r u c t the e n t i r e a m p l i t u d e . The v a l i d i t y of the m o d e l f o r the ~A i n t e r a c t i o n p r o p o s e d below d e p e n d s on the f o l l o w i n g a s s u m p t i o n s : (i) the p r o c e s s is p u r e l y e l a s t i c and p r e d o m i n a n t l y S - w a v e n e a r i t s t h r e s h o l d ; (ii) the Y~ (1385) r e s o n a n c e d o m i n a t e s the a m p l i t u d e in the a p p r o p r i a t e e n e r g y r a n g e ; and (iii) above t h i s r e s o nance, the a m p l i t u d e is r e l a t i v e l y s t r u c t u r e l e s s , r a p i d l y b e c o m i n g p u r e l y i m a g i n a r y and tending to its P o m e r a n c h u k l i m i t .
90
N.M.QUEEN
Hypothesis (i) r e q u i r e s no special justification, since the inelastic t h r e s h o l d and the tail of the Y~ r e s o n a n c e a r e both sufficiently f a r above the elastic threshold. Hypothesis (ii) is supported by the fact that the Y{ is a P~ ~A r e s o n a n c e with a l a r g e b r a n c h i n g r a t i o to this channel and that it o c c u r s at a sufficiently low energy for all the higher p a r t i a l - w a v e a m p l i tudes to be negligible. The S-wave b a c k g r o u n d at this energy is expected to be small, p a r t i c u l a r l y if the c u r r e n t - a l g e b r a sum rule p r e d i c t i n g a v a n i s h ing S-wave ~A s c a t t e r i n g length [6] is a c c u r a t e . A s s u m p t i o n (iii) is supp o r t e d by the a p p a r e n t a b s e n c e of higher Y{ r e s o n a n c e s with l a r g e ~A b r a n c h i n g ratios. M o r e o v e r , f o r w a r d d i s p e r s i o n r e l a t i o n s for the coupled KN channel [7] a r e c o n s i s t e n t with a purely i m a g i n a r y and relatively s t r u c t u r e l e s s ~ amplitude above the r e s o n a n c e - d o m i n a t e d energy region.
2. FORMULATION OF THE MODEL Let F(¢o) be the 7rA f o r w a r d s c a t t e r i n g amplitude at pion energy ¢o, w h e r e both quantities a r e evaluated in the A r e s t f r a m e . Unless specified o t h e r wise, we use units in w h i c h ~ = c =mTr = 1. In the complex w plane, F i s analytic except for the unitarity cuts along the r e a l axis for I w l > 1 and the E poles at e n e r g i e s w = + w E with r e s i d u e s + R E given by n
= c2[1
- (m
-
(1)
w h e r e G is the u n r a t i o n a l i z e d vA~ coupling constant. C r o s s i n g s y m m e t r y and the Schwarz r e f l e c t i o n principle r e q u i r e that F(~o) = F(-¢o) ,
F(¢o*) = F*(w) .
(2)
Since F(~o) is c r o s s i n g s y m m e t r i c , it is convenient to introduce the v a r iable z = 0)2. In the z-plane, F is cut along the real axis only for z > 1 and has one pole at z = zE = 0)2; m o r e o v e r , F(z*) = F*(z)
(3)
.
We n o r m a l i z e the amplitude so that the optical t h e o r e m r e a d s k a(z) = 47 Im F(z) ,
(4)
w h e r e k2 = z - 1. F r o m the usual a s s u m p t i o n that the total c r o s s s e c t i o n if(z) t e n d s to a n o n - z e r o l i m i t as z -~ o% it follows that the i n v e r s e a m p l i tude h(z) =- 1/F(z) s a t i s f i e s an u n s u b t r a c t e d d i s p e r s i o n r e l a t i o n . Hence, for r e a l z,
1 h(z) = ~ P
S 1
p(z')dz' z' - z
+ ip(z) + ~ n
an Z - Zn '
where p ( z ) ~ Imh(z), z n a r e the z e r o s of the amplitude F ( z ) , and a n the r e s i d u e s at the c o r r e s p o n d i n g poles in h ( z ) . We now t r a n s c r i b e the t h r e e basic h y p o t h e s e s listed in sect. 1 into s t a t e m e n t s about the function p ( z ) . Condition (i) implies that
(5)
~rA MODEL
91
F(z) ~ ( k / q 2) e x p (i 50) s i n 5o
(6)
n e a r t h e e l a s t i c t h r e s h o l d , w h e r e q i s t h e c . m . m o m e n t u m a n d 5o t h e S w a v e p h a s e s h i f t . T h e p h a s e s h i f t s a r e r e a l b e l o w the i n e l a s t i c t h r e s h o l d at z = 2.75. H e n c e ,
p(z) ~ K(z)
for
z ~ 1 ,
(7)
where
m2~-z_ 1 K(z)
(8)
=- -
2m A f z + rn2 + 1 Similarly,
s i n c e t h e Y1* (1385) i s a P_~2 r e s o n a n c e ,
p(z) ~ ½ ( r / r ~ A ) K ( z )
for
c o n d i t i o n (ii) i m p l i e s t h a t z ~ zy ,
(9)
w h e r e F i s t h e t o t a l w i d t h , Fir A t h e e l a s t i c p a r t i a l w i d t h , a n d Z y = 4.65 t h e p o s i t i o n of t h e r e s o n a n c e (in g e n e r a l , t h e s u b s c r i p t Y d e n o t e s t h e v a l u e of a k i n e m a t i c v a r i a b l e at t h e r e s o n a n c e ) . A s s u m p t i o n (iii) l e a d s to t h e a s y m p totic behaviour 4~
p(z)
for
~
z >> 1 ,
(10)
w h e r e ~ - cr(oo). S i n c e it i s a s s u m e d t h a t c o n d i t i o n s (7), (9) and (10) d e s c r i b e a l l t h e e s s e n t i a l s t r u c t u r e of t h e f u n c t i o n p(z), w e s e t p(z) e q u a l to a s i m p l e s m o o t h function which interpolates between these three conditions. Such a function is
po(Z) = K(z)
1+
8~
-C
,
(11)
w h e r e t h e c o n s t a n t C d e p e n d s on t h e v a l u e of ~ a n d i s c h o s e n s o t h a t t h e c o n d i t i o n (9) i s e x a c t l y s a t i s f i e d a t z = z y . F o r s m a l l z , [po(z) l i s s o m e w h a t l e s s t h a n t h e v a l u e g i v e n by t h e c o n d i t i o n (7). T h e a p p r o x i m a t i o n (7) d o e s , in f a c t , o v e r e s t i m a t e I P(z)] if t h e r e is constructive interference between the S-wave and P-wave amplitudes. S i n c e only t h e r e s o n a n t P~ w a v e i s a s s u m e d to b e i m p o r t a n t , c o n s t r u c t i v e i n t e r f e r e n c e d o e s o c c u r in t h e i n t e r v a l b e t w e e n t h e e l a s t i c t h r e s h o l d and t h e r e s o n a n c e e n e r g y if t h e S - w a v e s c a t t e r i n g l e n g t h i s p o s i t i v e . H o w e v e r , a s n o t e d b y A n t o n i o u [3], t h e r e m a y b e a s u b s t a n t i a l p e a k in Ip(z) l if t h e i n t e r f e r e n c e i s d e s t r u c t i v e . F o r t h e c a s e of a n e g a t i v e S - w a v e s c a t t e r i n g l e n g t h , we t h e r e f o r e r e p l a c e e x p r e s s i o n (11) b y P2(z) = Po(Z) + P l ( Z ) ,
(12)
where pl(z)
=S;~(z-p)(Z-Zy)2
1
o
Here, ~ is a parameter
for for
~ < z < Zy, z < p or
z > zy .
(13)
to b e d e t e r m i n e d l a t e r , a n d t~ i s c h o s e n in t h e
92
N.M. QUE Eli
r a n g e 1 --< g < z y . W i t h t h i s m o d e l f o r p ( z ) , t h e f u n c t i o n h(z) i s c o n t i n u o u s a n d d i f f e r e n t i a b l e at z = z y . M o r e o v e r , t h e c o n t r i b u t i o n to h(z) f r o m t h e t e r m p l ( z ) c a n b e e v a l u a t e d a n a l y t i c a l l y . In p a r t i c u l a r , d e f i n i n g h l ( z ) _ 1_~rI) the following results,
f ~ Ol(Z')dz' z' -z ' 1
to b e u s e d l a t e r ,
(14)
are obtained:
h 1 (Zy) = - ~ ( Z y - tz)3/67r ,
(15)
and, f o r z < 1,
zl h i ( z ) - - ~ [z 3 - z 2 ( 2 Z y + ~) + Z Z y ( 2 ~ + Zy) - tz z 2 ] l o g L - ~ : - F ]
+~_[z2(zy_tZ)+z(½,2+,Zy_~z2l+~z3+½pz2_~2Zv+~p3] . 7T
(16)
To evaluate the dispersion relation (5), it is necessary to specify the zeros of the amplitude. The number of zeros can be deduced from a theorem of Sugawara and Tubis [8], which asserts that a function which is meromorphic in the cut plane and obeys the reflection principle (3) has the asymptotic behaviour
F(z)
F(z) ~ Nzn-rn(-z)-dP(°°)/~ (z) d?(-°°)/~
as
]z I -~ oo ,
(17)
f o r s o m e r e a l c o n s t a n t N, w h e r e n a n d m a r e r e s p e c t i v e l y t h e n u m b e r s of z e r o s a n d p o l e s of F ( z ) ; ~b(z) i s t h e p h a s e of F(z), d e f i n e d f o r r e a l z s o t h a t
F(z +i~) = ~ IF(z +ic) lexp[i~(z)] ,
(18)
#)(z) = 0 f o r z in t h e c u t - f r e e r e g i o n , a n d qS(z) h a s no d i s c o n t i n u i t i e s g r e a t e r t h a n o r e q u a l to ~ in m a g n i t u d e ; it i s a s s u m e d t h a t n, m a n d qS(~:~) a r e a l l f i n i t e . In t h e p r e s e n t c a s e , m = 1, q S ( - ~ ) = O, a n d
~(~) { -21½v
if A < 0 , ifA>0,
(19)
where A -=F(1) is the S-wave ~A scattering length. Comparing the forms (10) and (17), we find that n = 1 if A < 0, while n = 2 if A > 0. From eq. (I), we see that has a positive residue at the ~ pole. Thus, for the case A < 0, we deduce from the mean-value theorem that the single z e r o z o l i e s on the real axis in the i n t e r v a l z ~ < zo< 1. In the caseA > 0, the reflection principle (3) implies that the two zeros z o and z I are either both on the real axis or are complex conjugates of one another. If these zeros lie on the real axis, then they are either both in the range z~ < z < 1 or both in the range z < z~.
F(z)
3. D E T A I L S O F T H E C A L C U L A T I O N We c o n s i d e r f i r s t t h e c a s e w h e n t h e s c a t t e r i n g l e n g t h i s n e g a t i v e , A < 0.
7rA M O D E L
93
T h e m o d e l t h e n e n t a i l s f i v e p a r a m e t e r s : t h e z e r o of t h e a m p l i t u d e Zo, the c o r r e s p o n d i n g r e s i d u e a o i n the i n v e r s e a m p l i t u d e , the p a r a m e t e r s X and /z of eq. (13), a n d the c r o s s s e c t i o n a. T w o of t h e s e p a r a m e t e r s c a n b e d e t e r m i n e d by r e q u i r i n g t h e m o d e l to give t h e e x p e r i m e n t a l l y o b s e r v e d z y a n d w i d t h F of t h e Y~ (1385) r e s o n a n c e . A t h i r d p a r a m e t e r c a n b e d e t e r m i n e d by r e q u i r i n g the e x i s t e n c e of the ~ pole i n the a m p l i t u d e F(z) at z = z E. T h e Y1 c o n t r i b u t i o n to the a m p h t u d e F is r e p r e s e n t e d by a r e l a t i v i s t i c Breit-Wigner resonance formula ~k
.
.
(2k/q)xvq2
F(E)-
(20)
E y - E - i7 q3 ' w h e r e E i s the t o t a l c . m . e n e r g y , x = F ~ A / F , a n d ~ = ½F/q2. We r e q u i r e t h a t , at z = z y , R e h ( z ) a n d d[Reh(z)]/dz c a l c u l a t e d f r o m eq. (5) a g r e e with the c o r r e s p o n d i n g v a l u e s o b t a i n e d f r o m eq. (20). T h u s , we d e r i v e the c o n ditions X ( z y - ~)3 ao 6~ + zy-z o
h°(zY) dh o
ao
X(zy - p)2
[--dz-z] z=zy +
2~
(Zy - Zo)2 = -
0
m3ky 2xFqycoyE3
(21)
'
(22)
where
ho(z)
-= £ p
J2 O o~(rz :' )~d z ' •
(23)
1 T h e c o n d i t i o n f o r the ~ p o l e i n the a m p l i t u d e i s
ho(z~) +h l ( Z ~ ) +
ao
z~- zo
- 0 .
(24)
In o u r n u m e r i c a l c a l c u l a t i o n s , eqs. (21), (22) a n d (24) w e r e s o l v e d f o r ao, ~ a n d ~ i n t e r m s of z o a n d ~ . S i n c e t h e r e a r e no f u r t h e r c o n d i t i o n s by w h i c h to e l i m i n a t e z o a n d u , the e n t i r e c a l c u l a t i o n w a s m a d e a s a f u n c t i o n of t h e s e p a r a m e t e r s . We r e c a l l that the f u n c t i o n h o d e p e n d s on the p a r a m e t e r ~, w h i l e h I d e p e n d s on ~ a n d u ; m o r e o v e r , e a c h of the r e l a t i o n s (21), (22) a n d (24) d e p e n d s l i n e a r l y on both ~ a n d a o. O u r p r o c e d u r e w a s t h e r e f o r e to s o l v e eqs. (21) a n d (22) a l g e b r a i c a l l y f o r ~ and a o in t e r m s of ~, for f i x e d v a l u e s of the r e m a i n i n g p a r a m e t e r s z o a n d g , a n d t h e n to f i n d the v a l u e of if, if a n y , which s a t i s f i e s eq. (24). F i n a l l y , z o and u w e r e v a r i e d w i t h i n t h e i r a l l o w e d r a n g e s z E < z o < 1 and 1 --< u < z y . T h e m e t h o d is s i m i l a r f o r p o s i t i v e s c a t t e r i n g l e n g t h s , A > 0, e x c e p t t h a t the s e c o n d t e r m i s a b s e n t and a n a d d i t i o n a l pole t e r m o c c u r s i n e a c h of the eqs. (21), (22) a n d (24). T h e two p o l e s i n h(z) a r e e i t h e r both on the r e a l a x i s o r a r e c o m p l e x c o n j u g a t e s of one a n o t h e r . In the s e c o n d c a s e , the r e f l e c t i o n p r i n c i p l e (3) i m p l i e s that z I = z~ and a 1 = ao, so that the n u m b e r s of r e a l p a r a m e t e r s r e q u i r e d to s p e c i f y the pole p o s i t i o n s and r e s i d u e s a r e
N.
94
M. ~UEEN
t h e s a m e a s in t h e f i r s t c a s e . In e i t h e r c a s e , we u s e t h e t h r e e c o n d i t i o n s (21), (22) and (24) to e v a l u a t e t h e r e s i d u e s a n d t h e p a r a m e t e r a a s f u n c t i o n s of t h e two p a r a m e t e r s w h i c h s p e c i f y t h e p o l e p o s i t i o n s . F o r g i v e n p a r a m e t e r v a l u e s , t h e s c a t t e r i n g l e n g t h A i s d e t e r m i n e d by e v a l u a t i n g eq. (5) a t t h r e s h o l d . T h e ~ A ~ c o u p l i n g c o n s t a n t G i s o b t a i n e d f r o m e q s . (1) a n d (5), u s i n g t h e r e l a t i o n 1
_
dI~z]
2¢oZ
R~
(25)
z=z~
U n f o r t u n a t e l y , w i d e l y d i f f e r i n g e x p e r i m e n t a l v a l u e s of t h e w i d t h of t h e Y~ (1385) r e s o n a n c e h a v e b e e n r e p o r t e d , v a r y i n g f r o m F = (17.1 + 4.4) MeV ( r e f . [9]) t o (88+ 10) M e V ( r e f . [10]). T h e a v e r a g e v a l u e 3 7 ~ 4 M e V i s q u o t e d in t h e d a t a c o m p i l a t i o n of t h e P a r t i c l e D a t a G r o u p [11]. S i n c e t h e n u m e r i c a l r e s u l t s w e r e found to b e r a t h e r s e n s i t i v e to F , t h e c a l c u l a t i o n s were repeated for several different values. For the elastic branching ratio, we t a k e t h e v a l u e F ~ A / F = 0.91 ( r e f . [11]).
4. R E S U L T S AND C O M P A R I S O N W I T H O T H E R P R E D I C T I O N S T h e output v a l u e s of G 2, cr a n d A a s f u n c t i o n s of t h e p a r a m e t e r Zo, f o r s e l e c t e d v a l u e s of n a n d F , a r e s h o w n in f i g s . 1, 2 a n d 3, r e s p e c t i v e l y , f o r t h e c a s e A < 0. F o r p o s i t i v e s c a t t e r i n g l e n g t h s , h o w e v e r , no a c c e p t a b l e s o l u t i o n s w e r e found, a l t h o u g h an e x h a u s t i v e s e a r c h w a s m a d e in p a r a m e t e r s p a c e . If t h e two z e r o s z o a n d z 1 a r e a s s u m e d to l i e on t h e r e a l a x i s , e q s . (21), (22) and (24) p o s s e s s no s o l u t i o n s f o r any ~ > 0. F o r two c o m p l e x c o n j u g a t e z e r o s , on t h e o t h e r h a n d , t h e s e e q u a t i o n s h a v e s o l u t i o n s only f o r G 2 ~ 100. T h e r e f o r e t h e m o d e l r e q u i r e s a n e g a t i v e ~A s c a t t e r i n g l e n g t h . We s e e f r o m fig. 1 t h a t G2 i s an e x t r e m e l y s e n s i t i v e f u n c t i o n of Zo, r i s i n g m o n o t o n i c a l l y f r o m G 2 = 0 at z o = z ~ t o G 2 > 100 at z o = 1. T h u s , it i s not p o s s i b l e to d e t e r m i n e G 2 f r o m t h e m o d e l . A c c o r d i n g to SU(3) s y m m e t r y [12], G 2 i s r e l a t e d to t h e ~NN c o u p l i n g c o n s t a n t b y
20C
G2
I
I
I
0.4
0.6
O.O
15C
IOO
50
O Zr
t.O ZO
Fig. 1. The square of the lrAE coupling consLant as a function of z o. The dashed curves are for F = 25 MeV and the solid curves for F = 37 MeV. Curves (a) are for ~= 1.0 and curves (b) for ~ = 1.5.
~A MODEL
95
O~ 8 0 t
t.b) o|
'
i'
40 1
'
"
20
O zf
0.4
0.6
0.8
I.O Zo
Fig. 2. The asymptotic total 7rA cross section as a function of zo. The significance of the curves is the same as in fig. 1.
,'~
0
Zr
0,4
0.6
1.0
0.8 ZO
Fig. 3. The S-wave ~A scattering length as a function of zo. The significance of the curves is the same as in fig, 1.
c2 :
(i-
C2NN,
(26)
w h e r e (~ = f / i f + d ) i s the f r a c t i o n of R - t y p e c o u p l i n g . A s s u m i n g G2NN = = 14.6 [13] a n d ~ ~ 0.4 [14], we o b t a i n G2 ~ 7. M a n y i n d e p e n d e n t e x p e r i m e n t a l a n d t h e o r e t i c a l e s t i m a t e s of G 2, s u m m a r i z e d in ref. [15], a g r e e c l o s e l y with t h i s SU(3) p r e d i c t i o n . We t h e r e f o r e a s s u m e t h a t G2- = 101 10. E v e n with t h i s c o n s e r v a t i v e l y l a r g e u n c e r t a i n t y , it is c l e a r f r o m fig. i t h a t the p a r a m e t e r z o i s r e s t r i c t e d to a r a t h e r n a r r o w r a n g e ; in fact, we find Zo < 0.45. F o r z o w i t h i n t h e p e r m i s s i b l e r a n g e , r e s t r i c t i o n s on the a s y m p t o t i c t o t a l ~A c r o s s s e c t i o n ~ c a n now b e f o u n d f r o m the r e s u l t s s h o w n i n fig. 2. If F i s a l l o w e d to b e a s s m a l l a s 25 MeV (ref. [16]), we o b t a i n the l o w e r b o u n d > 15 m b . M u c h l a r g e r v a l u e s of ~ a r e p o s s i b l e if F e x c e e d s i t s l o w e r l i m it, a l t h o u g h ~ i s f a i r l y i n s e n s i t i v e to p . A c c o r d i n g to the q u a r k m o d e l [17], i
2
= ~ % N +~ ~KN •
(27)
U s i n g the a s y m p t o t i c e s t i m a t e s (r~N ~ 23 m b and aKN ~ 17.5 m b [18], eq. (27) g i v e s a ~ 19 rob, a v a l u e which i s c o n s i s t e n t with o u r l o w e r b o u n d f o r (r T u r n i n g f i n a l l y to the s c a t t e r i n g l e n g t h c a l c u l a t i o n s s h o w n i n fig. 3, we find t h a t A < - 0 . 2 0 fro. T h i s l i m i t is a t t a i n e d when u a n d I~ t a k e t h e i r m i n i -
96
N.M.QUEEN
m u m v a l u e s , and m u c h m o r e n e g a t i v e v a l u e s of A o c c u r f o r v a r i a t i o n s in the p a r a m e t e r s . In t a b l e 1, we c o m p a r e o u r e s t i m a t e of the s c a t t e r i n g l e n g t h with t h r e e o t h e r independent p r e d i c t i o n s . F i r s t l y , s i n c e the A is an i s o s c a l a r p a r t i c l e , W e i n b e r g ' s c u r r e n t - a l g e b r a s u m r u l e [6] g i v e s A = 0. Although this s u m r u l e w o r k s r e m a r k a b l y well f o r vN s c a t t e r i n g , doubts have a l r e a d y b e e n r a i s e d about its v a l i d i t y f o r the vE s y s t e m [19]. Table 1 Four predictions of the 7TA S-wave scattering length A.
Source
A (fm)
Present model
< -0.20 0
Current-algebra sum rule [6] Analysis of --~- decay [21]
0.34 + 0.33 - 0.50
Multi-channel KN analysis [24]
1.7 +3.0 -0.7
A n o t h e r i n d e p e n d e n t e s t i m a t e of A is p r o v i d e d by the a n a l y s i s of the w e a k d e c a y p r o c e s s Z - ~ ~-A. T h e c h a r a c t e r i s t i c s of the d e c a y a r e u s u a l l y s p e c i f i e d by the p a r a m e t e r s [20] a
=
2 Re(AoA1)
/3 = 2 I m ( A o n 1) ,
,
~ = [nol 2 - In1{ 2 , (28)
w h e r e A L is the d e c a y a m p l i t u d e f o r a final ~A s t a t e of o r b i t a l a n g u l a r m o m e n t u m L, n o r m a l i z e d so that l A o l 2 + {All = 1. Since a 2 + f i 2 + y ' 2 = 1, it is c u s t o m a r y to u s e the p a r a m e t r i z a t i o n 1
= ( 1 - a2)~ sin~b ,
y = ( 1 - ~2)½ c o s ~b.
(29)
A r e c e n t c o m p i l a t i o n of ~ - d e c a y d a t a g a v e the a v e r a g e v a l u e s [21] = -0.42 + 0.04 , It can be shown [22] that, if t i m e - r e v e r s a l the d e c a y p r o c e s s ,
q5 = 13 o + 7 o .
(30)
i n v a r i a n c e is a s s u m e d to hold in
t g ( 5 1 - 50) = f i / ~ ,
(31)
w h e r e 5 o and 51 a r e , r e s p e c t i v e l y , the S½ and P,_2 ~A s c a t t e r i n g p h a s e shifts at the d e c a y e n e r g y . F r o m a d y n a m i c a l m o d e l of ~A s c a t t e r i n g b a s e d on p a r t i a l - w a v e d i s p e r s i o n r e l a t i o n , M a r t i n [23] o b t a i n e d 51 ~ -1.2 °. U s i n g this e s t i m a t e , t o g e t h e r with eqs. (29) to (31), we c a n c a l c u l a t e 5 o. M a r t i n [23] d e t e r m i n e d the c o n t r i b u t i o n s to the r e a l p a r t of the S - w a v e a m p l i t u d e f r o m the n e a r b y s i n g u l a r i t i e s in the p a r t i a l - w a v e d i s p e r s i o n r e l a t i o n . If we a s s u m e that the e n e r g y d e p e n d e n c e of t h i s a m p l i t u d e a r i s e s m a i n l y f r o m t h e s e n e a r b y s i n g u l a r i t i e s , his r e s u l t s can be u s e d to e x t r a p o l a t e the a m plitude to t h r e s h o l d . In this way, we obtain the v a l u e f o r bhe s c a t t e r i n g l e n g t h given in t a b l e 1.
7TA MODEL
97
A f u r t h e r i n d e p e n d e n t p r e d i c t i o n i s o b t a i n e d f r o m an a n a l y s i s of l o w e n e r g y K--N s c a t t e r i n g d a t a by K i m [24], a c c o r d i n g to a m u l t i - c h a n n e l e f f e c t i v e - r a n g e e x p a n s i o n of t h e i n v e r s e K - m a t r i x . If K i m ' s p a r a m e t e r s a r e e m p l o y e d to d e s c r i b e ~A s c a t t e r i n g , we f i n d a s u r p r i s i n g l y l a r g e v a l u e f o r A ( s e e t a b l e 1). T h i s v a l u e s h o u l d p e r h a p s not b e t a k e n t o o s e r i o u s l y , s i n c e K i m ' s d e s c r i p t i o n of e l a s t i c ~A s c a t t e r i n g i s a r r i v e d at i n d i r e c t l y b y a n a l y z i n g t h e e n e r g y d e p e n d e n c e of v a r i o u s KN p r o c e s s e s . M o r e o v e r , t h e r e i s s t r o n g e v i d e n c e f r o m f o r w a r d d i s p e r s i o n s u m r u l e s [25] t h a t h i s p a r a m e t e r s g i v e an i n a d e q u a t e d e s c r i p t i o n of t h e l o w - e n e r g y ~[N i n t e r a c t i o n . It i s a m u s i n g to n o t e t h a t t h e f o u r a v a i l a b l e e s t i m a t e s of A g i v e n in t a b l e 1 a r e m u t u a l l y i n c o n s i s t e n t a n d c o v e r a v e r y w i d e r a n g e of p o s s i b l e v a l u e s .
5. CONCLUSIONS W e h a v e s h o w n in a s p e c i f i c m o d e l t h a t a n a l y t i c i t y i m p o s e s s t r o n g c o r r e l a t i o n s a m o n g t h e f e a t u r e s of nA s c a t t e r i n g at low, i n t e r m e d i a t e a n d high e n e r g i e s . T h e m o d e l p r e d i c t s an a s y m p t o t i c t o t a l c r o s s s e c t i o n ~ > 15 rob, which is consistent with the quark model. However, the resulting S-wave s c a t t e r i n g l e n g t h , A < - 0 . 2 0 f m , d i s a g r e e s w i t h e a c h of t h r e e o t h e r , m u t u ally incompatible predictions. F u r t h e r c a l c u l a t i o n s s u g g e s t e d t h a t t h e output v a l u e s of t h e s e q u a n t i t i e s a r e f a i r l y i n s e n s i t i v e to t h e p r e c i s e f o r m a s s u m e d f o r t h e f u n c t i o n p(z), p r o v i d e d t h a t t h i s f u n c t i o n s a t i s f i e s t h e t h r e e b a s i c h y p o t h e s e s of t h e m o d e l . S i n c e o u r c a l c u l a t i o n s a r e e x a c t , f o r a g i v e n input f u n c t i o n p ( z ) , we s u s p e c t t h a t v a l u e s of ~ o r A s u b s t a n t i a l l y d i f f e r e n t f r o m t h e e s t i m a t e s g i v e n a b o v e a r e p o s s i b l e only if t h e ~A s c a t t e r i n g a m p l i t u d e h a s c o n s i d e r a b l y m o r e s t r u c t u r e t h a n a s s u m e d in t h e m o d e l . A m o r e r e f i n e d a n a l y s i s of t h i s t y p e , f o r ~ a s w e l l a s ~A s c a t t e r i n g , m a y b e c o m e p o s s i b l e w h e n m o r e r e l i a b l e m u l t i - c h a n n e l p a r a m e t r i z a t i o n s of t h e KN i n t e r a c t i o n h a v e b e e n m a d e . I a m g r a t e f u l to D r . C. H. C h a n f o r p o i n t i n g out an e r r o r in t h e o r i g i n a l manuscript.
RE F E R E N C E S [1] N. G. Antontou and C. Palev, Phys. L e t t e r s 26B (1968) 301; H. Yabuki, Prog. Theor. Phys. 39 (1968) 118. [2] D. V. Shirkov, Yad. Fiz. 3 (1965) 1149 [English translation: Sov. J. Nucl. Phys. 3 (1966) 834]. [3] N. G. Antoniou, Nuel. Phys. B2 (1967) 648. [4] F. T. Meiere and M. Sugawara, Phys. Rev. 144 (1966) 1308; 153 (1967) 1702. [5] V. V. Serebryakov and D. V. Shirkov, F o r t s c h r . Physik 13 (1965) 227; N. G. Antoniou and C. H. L. Smith, Nucl. Phys. B3 (1967) 277. [6] S. Weinberg, Phys. Rev. L e t t e r s 17 (1966) 616. [7] N. M. Queen, Nucl. Phys. B1 (1967) 207; M. Lusignoli, M. Restignoli, G. Violini and G. A. Snow, Nuovo Cimento 45A (1966) 792; 46A (1966) 430; 49A (1967) 705; A. D. Martip and F. Poole, Nucl. Phys. B4 (1968) 467; A.A. C a r t e r , Cambridge p r e p r i n t HEP 68-10 (1968).
98 [8] [9] [10] [11] [12] [13] [14]
[15] [16] [17]
[18] [19] [20] [21] [22] [23] [24] [25]
N.M. QUEEN M. Sugawara and A. Tubis, Phys. Rev. 130 (1963) 2127. L. T. Smith, U.C.L.A. thesis (1965). W. A. Cooper et al., Phys. L e tt e r s 8 (1964) 365. P a r t i c l e Data Group, Revs. Mod. Phys. 40 (1968) 77. J. J. De Swart, Revs. Mod. Phys. 35 (1963) 916. J. Hamilton and W. S. Woolcock, Revs. Mod. Phys. 35 (1963) 737; V. K. Samaranayake and W. S. Woolcock, Phys. Rev. Let t er s 15 (1965) 936. A. Pals, Phys. Rev. L e tt e r s 13 (1964) 175; W. Willis et al., Phys. Rev. L e t te r s 13 (1964) 291; M. E. Ebel and P. B. J a m e s , Phys. Rev. Letters 15 (1965) 805; Phys. Rev. 153 (1967) 1694; C. J a r l s k o g and H. Pilkuhn, Phys. L e tt e r s 20 (1966) 428; T. Kohmura, Prog. Theor. Phys. 35 (1966) 416. A. J. Martin, Phys. Rev. 164 (1967) 1868. Birmingham-Glasgow-London (I.C.)-Oxford-Rutherford Collaboration, Phys. Rev. 152 (1966) 1148. E. M. Levin and L. L. Frankfurt, ZhETF Pis 'ma (USSR) 2 (1965) 105 [JETP Lett e r s (Soy. Phys.) 2 (1965) 65]; H. J. Lipkin and F. Scheck, Phys. Rev. L e tt e r s 16 (1966) 71; J. J. J. Kokkedee and L. Van Hove, Nuovo Cimento 42A (1966) 711. W. Galbraith et al., Phys. Rev. 138B (1965) 913. G. Shaw, Phys. Rev. L e t te r s 18 (1967) 1025. T.D. Lee and C.N. Yang, Phys. Rev. 108 (1957) 1645; D . D . C a r m o n y et al., Phys. Rev. L e t t e r s 12 (1964) 482. D.W. M e r r i l l and J. Button-Sharer, Phys. Rev. 167 (1968) 1202. T.D. Lee, J. Steinberger, G. Feinberg, P . K . Kabir and C. N. Yang, Phys. Rev. 106 (1957) 1367. B. R. Martin, Phys. Rev. 138B (1965) 1136. J . K . K i m , Phys. Rev. L e t te r s 19 (1967) 1074. N. M. Queen, S. Leeman and F. E. Yeomans, Nucl. Phys., to be published; A. D. Martin and G. G. Ross, Phys. L e tt e r s 26B (1968) 527; N. M. Queen, M. Restignoli and G. Violini, F o r t s c h r . Physik, to be published.
Nuclear Physics B10 (1969) 89-98. North-Holland Publ. Comp., Amsterdam
A SOLUBLE MODEL FOR THE ~rA I N T E R A C T I O N N. M. QUEEN
Department of Mathematical Physics, University of Birmingham, England Received 13 December 1968 Abstract: A simple model for the ~A interaction, based on a forward dispersion r e lation for the inverse amplitude, is formulated. The assumption that the amplitude is dominated in the appropriate energy range by the Y{ (1385) resonance leads to definite restrictions on the possible behaviour of the amplitude at lower and at higher energies. The asymptotic total cross section calculated from the model is consistent with the quark-model prediction. However, a negative S-wave scattering length is required, in disagreement with each of three alternative (mutually incompatible) predictions.
1. INTRODUCTION T h e o r e t i c a l s t u d i e s of the ~ i n t e r a c t i o n with the aid of f o r w a r d d i s p e r s i o n r e l a t i o n s have b e e n m a d e by many a u t h o r s [1-5]. The S - w a v e s c a t t e r ing l e n g t h s have b e e n p r e d i c t e d [1] on the b a s i s of e x p e r i m e n t a l and t h e o r e t i c a l i n f o r m a t i o n about the p h a s e s h i f t s at h i g h e r e n e r g i e s . Even in the a b s e n c e of d e t a i l e d ~ s c a t t e r i n g data, the a s s u m p t i o n that the a m p l i t u d e is d o m i n a t e d in the i n t e r m e d i a t e e n e r g y r e g i o n by one or two r e s o n a n c e s , but is r e l a t i v e l y s t r u c t u r e l e s s e l s e w h e r e , l e a d s to definite r e s t r i c t i o n s [3, 4] on the a m p l i t u d e o v e r the e n t i r e e n e r g y r a n g e . In t h i s p a p e r , we f o r m u l a t e a m o d e l f o r the ~A i n t e r a c t i o n b a s e d on a f o r w a r d d i s p e r s i o n r e l a t i o n f o r the i n v e r s e a m p l i t u d e . Our a p p r o a c h is s i m i l a r in s p i r i t to that of the m o d e l f o r ~ s c a t t e r i n g p r o p o s e d by S h i r k o v [2] and e x t e n d e d by Antoniou [3]. A m a j o r a d v a n t a g e of w o r k i n g with the i n v e r s e a m p l i t u d e is that both the t h r e s h o l d and the h i g h - e n e r g y b e h a v i o u r of i t s i m a g i n a r y p a r t a r e t h e o r e t i c a l l y d e t e r m i n e d a l m o s t uniquely without p r i o r k n o w l e d g e of the s c a t t e r i n g length or c r o s s s e c t i o n . M o r e o v e r , the d i s p e r s i o n r e l a t i o n f o r the i n v e r s e a m p l i t u d e does not r e q u i r e s u b t r a c t i o n s . If a s p e c i f i c m o d e l is a d o p t e d f o r the i m a g i n a r y p a r t of the i n v e r s e a m p l i tude at a l l e n e r g i e s , the d i s p e r s i o n r e l a t i o n then m a k e s it p o s s i b l e to c o n s t r u c t the e n t i r e a m p l i t u d e . The v a l i d i t y of the m o d e l f o r the ~A i n t e r a c t i o n p r o p o s e d below d e p e n d s on the f o l l o w i n g a s s u m p t i o n s : (i) the p r o c e s s is p u r e l y e l a s t i c and p r e d o m i n a n t l y S - w a v e n e a r i t s t h r e s h o l d ; (ii) the Y~ (1385) r e s o n a n c e d o m i n a t e s the a m p l i t u d e in the a p p r o p r i a t e e n e r g y r a n g e ; and (iii) above t h i s r e s o nance, the a m p l i t u d e is r e l a t i v e l y s t r u c t u r e l e s s , r a p i d l y b e c o m i n g p u r e l y i m a g i n a r y and tending to its P o m e r a n c h u k l i m i t .
90
N.M.QUEEN
Hypothesis (i) r e q u i r e s no special justification, since the inelastic t h r e s h o l d and the tail of the Y~ r e s o n a n c e a r e both sufficiently f a r above the elastic threshold. Hypothesis (ii) is supported by the fact that the Y{ is a P~ ~A r e s o n a n c e with a l a r g e b r a n c h i n g r a t i o to this channel and that it o c c u r s at a sufficiently low energy for all the higher p a r t i a l - w a v e a m p l i tudes to be negligible. The S-wave b a c k g r o u n d at this energy is expected to be small, p a r t i c u l a r l y if the c u r r e n t - a l g e b r a sum rule p r e d i c t i n g a v a n i s h ing S-wave ~A s c a t t e r i n g length [6] is a c c u r a t e . A s s u m p t i o n (iii) is supp o r t e d by the a p p a r e n t a b s e n c e of higher Y{ r e s o n a n c e s with l a r g e ~A b r a n c h i n g ratios. M o r e o v e r , f o r w a r d d i s p e r s i o n r e l a t i o n s for the coupled KN channel [7] a r e c o n s i s t e n t with a purely i m a g i n a r y and relatively s t r u c t u r e l e s s ~ amplitude above the r e s o n a n c e - d o m i n a t e d energy region.
2. FORMULATION OF THE MODEL Let F(¢o) be the 7rA f o r w a r d s c a t t e r i n g amplitude at pion energy ¢o, w h e r e both quantities a r e evaluated in the A r e s t f r a m e . Unless specified o t h e r wise, we use units in w h i c h ~ = c =mTr = 1. In the complex w plane, F i s analytic except for the unitarity cuts along the r e a l axis for I w l > 1 and the E poles at e n e r g i e s w = + w E with r e s i d u e s + R E given by n
= c2[1
- (m
-
(1)
w h e r e G is the u n r a t i o n a l i z e d vA~ coupling constant. C r o s s i n g s y m m e t r y and the Schwarz r e f l e c t i o n principle r e q u i r e that F(~o) = F(-¢o) ,
F(¢o*) = F*(w) .
(2)
Since F(~o) is c r o s s i n g s y m m e t r i c , it is convenient to introduce the v a r iable z = 0)2. In the z-plane, F is cut along the real axis only for z > 1 and has one pole at z = zE = 0)2; m o r e o v e r , F(z*) = F*(z)
(3)
.
We n o r m a l i z e the amplitude so that the optical t h e o r e m r e a d s k a(z) = 47 Im F(z) ,
(4)
w h e r e k2 = z - 1. F r o m the usual a s s u m p t i o n that the total c r o s s s e c t i o n if(z) t e n d s to a n o n - z e r o l i m i t as z -~ o% it follows that the i n v e r s e a m p l i tude h(z) =- 1/F(z) s a t i s f i e s an u n s u b t r a c t e d d i s p e r s i o n r e l a t i o n . Hence, for r e a l z,
1 h(z) = ~ P
S 1
p(z')dz' z' - z
+ ip(z) + ~ n
an Z - Zn '
where p ( z ) ~ Imh(z), z n a r e the z e r o s of the amplitude F ( z ) , and a n the r e s i d u e s at the c o r r e s p o n d i n g poles in h ( z ) . We now t r a n s c r i b e the t h r e e basic h y p o t h e s e s listed in sect. 1 into s t a t e m e n t s about the function p ( z ) . Condition (i) implies that
(5)
~rA MODEL
91
F(z) ~ ( k / q 2) e x p (i 50) s i n 5o
(6)
n e a r t h e e l a s t i c t h r e s h o l d , w h e r e q i s t h e c . m . m o m e n t u m a n d 5o t h e S w a v e p h a s e s h i f t . T h e p h a s e s h i f t s a r e r e a l b e l o w the i n e l a s t i c t h r e s h o l d at z = 2.75. H e n c e ,
p(z) ~ K(z)
for
z ~ 1 ,
(7)
where
m2~-z_ 1 K(z)
(8)
=- -
2m A f z + rn2 + 1 Similarly,
s i n c e t h e Y1* (1385) i s a P_~2 r e s o n a n c e ,
p(z) ~ ½ ( r / r ~ A ) K ( z )
for
c o n d i t i o n (ii) i m p l i e s t h a t z ~ zy ,
(9)
w h e r e F i s t h e t o t a l w i d t h , Fir A t h e e l a s t i c p a r t i a l w i d t h , a n d Z y = 4.65 t h e p o s i t i o n of t h e r e s o n a n c e (in g e n e r a l , t h e s u b s c r i p t Y d e n o t e s t h e v a l u e of a k i n e m a t i c v a r i a b l e at t h e r e s o n a n c e ) . A s s u m p t i o n (iii) l e a d s to t h e a s y m p totic behaviour 4~
p(z)
for
~
z >> 1 ,
(10)
w h e r e ~ - cr(oo). S i n c e it i s a s s u m e d t h a t c o n d i t i o n s (7), (9) and (10) d e s c r i b e a l l t h e e s s e n t i a l s t r u c t u r e of t h e f u n c t i o n p(z), w e s e t p(z) e q u a l to a s i m p l e s m o o t h function which interpolates between these three conditions. Such a function is
po(Z) = K(z)
1+
8~
-C
,
(11)
w h e r e t h e c o n s t a n t C d e p e n d s on t h e v a l u e of ~ a n d i s c h o s e n s o t h a t t h e c o n d i t i o n (9) i s e x a c t l y s a t i s f i e d a t z = z y . F o r s m a l l z , [po(z) l i s s o m e w h a t l e s s t h a n t h e v a l u e g i v e n by t h e c o n d i t i o n (7). T h e a p p r o x i m a t i o n (7) d o e s , in f a c t , o v e r e s t i m a t e I P(z)] if t h e r e is constructive interference between the S-wave and P-wave amplitudes. S i n c e only t h e r e s o n a n t P~ w a v e i s a s s u m e d to b e i m p o r t a n t , c o n s t r u c t i v e i n t e r f e r e n c e d o e s o c c u r in t h e i n t e r v a l b e t w e e n t h e e l a s t i c t h r e s h o l d and t h e r e s o n a n c e e n e r g y if t h e S - w a v e s c a t t e r i n g l e n g t h i s p o s i t i v e . H o w e v e r , a s n o t e d b y A n t o n i o u [3], t h e r e m a y b e a s u b s t a n t i a l p e a k in Ip(z) l if t h e i n t e r f e r e n c e i s d e s t r u c t i v e . F o r t h e c a s e of a n e g a t i v e S - w a v e s c a t t e r i n g l e n g t h , we t h e r e f o r e r e p l a c e e x p r e s s i o n (11) b y P2(z) = Po(Z) + P l ( Z ) ,
(12)
where pl(z)
=S;~(z-p)(Z-Zy)2
1
o
Here, ~ is a parameter
for for
~ < z < Zy, z < p or
z > zy .
(13)
to b e d e t e r m i n e d l a t e r , a n d t~ i s c h o s e n in t h e
92
N.M. QUE Eli
r a n g e 1 --< g < z y . W i t h t h i s m o d e l f o r p ( z ) , t h e f u n c t i o n h(z) i s c o n t i n u o u s a n d d i f f e r e n t i a b l e at z = z y . M o r e o v e r , t h e c o n t r i b u t i o n to h(z) f r o m t h e t e r m p l ( z ) c a n b e e v a l u a t e d a n a l y t i c a l l y . In p a r t i c u l a r , d e f i n i n g h l ( z ) _ 1_~rI) the following results,
f ~ Ol(Z')dz' z' -z ' 1
to b e u s e d l a t e r ,
(14)
are obtained:
h 1 (Zy) = - ~ ( Z y - tz)3/67r ,
(15)
and, f o r z < 1,
zl h i ( z ) - - ~ [z 3 - z 2 ( 2 Z y + ~) + Z Z y ( 2 ~ + Zy) - tz z 2 ] l o g L - ~ : - F ]
+~_[z2(zy_tZ)+z(½,2+,Zy_~z2l+~z3+½pz2_~2Zv+~p3] . 7T
(16)
To evaluate the dispersion relation (5), it is necessary to specify the zeros of the amplitude. The number of zeros can be deduced from a theorem of Sugawara and Tubis [8], which asserts that a function which is meromorphic in the cut plane and obeys the reflection principle (3) has the asymptotic behaviour
F(z)
F(z) ~ Nzn-rn(-z)-dP(°°)/~ (z) d?(-°°)/~
as
]z I -~ oo ,
(17)
f o r s o m e r e a l c o n s t a n t N, w h e r e n a n d m a r e r e s p e c t i v e l y t h e n u m b e r s of z e r o s a n d p o l e s of F ( z ) ; ~b(z) i s t h e p h a s e of F(z), d e f i n e d f o r r e a l z s o t h a t
F(z +i~) = ~ IF(z +ic) lexp[i~(z)] ,
(18)
#)(z) = 0 f o r z in t h e c u t - f r e e r e g i o n , a n d qS(z) h a s no d i s c o n t i n u i t i e s g r e a t e r t h a n o r e q u a l to ~ in m a g n i t u d e ; it i s a s s u m e d t h a t n, m a n d qS(~:~) a r e a l l f i n i t e . In t h e p r e s e n t c a s e , m = 1, q S ( - ~ ) = O, a n d
~(~) { -21½v
if A < 0 , ifA>0,
(19)
where A -=F(1) is the S-wave ~A scattering length. Comparing the forms (10) and (17), we find that n = 1 if A < 0, while n = 2 if A > 0. From eq. (I), we see that has a positive residue at the ~ pole. Thus, for the case A < 0, we deduce from the mean-value theorem that the single z e r o z o l i e s on the real axis in the i n t e r v a l z ~ < zo< 1. In the caseA > 0, the reflection principle (3) implies that the two zeros z o and z I are either both on the real axis or are complex conjugates of one another. If these zeros lie on the real axis, then they are either both in the range z~ < z < 1 or both in the range z < z~.
F(z)
3. D E T A I L S O F T H E C A L C U L A T I O N We c o n s i d e r f i r s t t h e c a s e w h e n t h e s c a t t e r i n g l e n g t h i s n e g a t i v e , A < 0.
7rA M O D E L
93
T h e m o d e l t h e n e n t a i l s f i v e p a r a m e t e r s : t h e z e r o of t h e a m p l i t u d e Zo, the c o r r e s p o n d i n g r e s i d u e a o i n the i n v e r s e a m p l i t u d e , the p a r a m e t e r s X and /z of eq. (13), a n d the c r o s s s e c t i o n a. T w o of t h e s e p a r a m e t e r s c a n b e d e t e r m i n e d by r e q u i r i n g t h e m o d e l to give t h e e x p e r i m e n t a l l y o b s e r v e d z y a n d w i d t h F of t h e Y~ (1385) r e s o n a n c e . A t h i r d p a r a m e t e r c a n b e d e t e r m i n e d by r e q u i r i n g the e x i s t e n c e of the ~ pole i n the a m p l i t u d e F(z) at z = z E. T h e Y1 c o n t r i b u t i o n to the a m p h t u d e F is r e p r e s e n t e d by a r e l a t i v i s t i c Breit-Wigner resonance formula ~k
.
.
(2k/q)xvq2
F(E)-
(20)
E y - E - i7 q3 ' w h e r e E i s the t o t a l c . m . e n e r g y , x = F ~ A / F , a n d ~ = ½F/q2. We r e q u i r e t h a t , at z = z y , R e h ( z ) a n d d[Reh(z)]/dz c a l c u l a t e d f r o m eq. (5) a g r e e with the c o r r e s p o n d i n g v a l u e s o b t a i n e d f r o m eq. (20). T h u s , we d e r i v e the c o n ditions X ( z y - ~)3 ao 6~ + zy-z o
h°(zY) dh o
ao
X(zy - p)2
[--dz-z] z=zy +
2~
(Zy - Zo)2 = -
0
m3ky 2xFqycoyE3
(21)
'
(22)
where
ho(z)
-= £ p
J2 O o~(rz :' )~d z ' •
(23)
1 T h e c o n d i t i o n f o r the ~ p o l e i n the a m p l i t u d e i s
ho(z~) +h l ( Z ~ ) +
ao
z~- zo
- 0 .
(24)
In o u r n u m e r i c a l c a l c u l a t i o n s , eqs. (21), (22) a n d (24) w e r e s o l v e d f o r ao, ~ a n d ~ i n t e r m s of z o a n d ~ . S i n c e t h e r e a r e no f u r t h e r c o n d i t i o n s by w h i c h to e l i m i n a t e z o a n d u , the e n t i r e c a l c u l a t i o n w a s m a d e a s a f u n c t i o n of t h e s e p a r a m e t e r s . We r e c a l l that the f u n c t i o n h o d e p e n d s on the p a r a m e t e r ~, w h i l e h I d e p e n d s on ~ a n d u ; m o r e o v e r , e a c h of the r e l a t i o n s (21), (22) a n d (24) d e p e n d s l i n e a r l y on both ~ a n d a o. O u r p r o c e d u r e w a s t h e r e f o r e to s o l v e eqs. (21) a n d (22) a l g e b r a i c a l l y f o r ~ and a o in t e r m s of ~, for f i x e d v a l u e s of the r e m a i n i n g p a r a m e t e r s z o a n d g , a n d t h e n to f i n d the v a l u e of if, if a n y , which s a t i s f i e s eq. (24). F i n a l l y , z o and u w e r e v a r i e d w i t h i n t h e i r a l l o w e d r a n g e s z E < z o < 1 and 1 --< u < z y . T h e m e t h o d is s i m i l a r f o r p o s i t i v e s c a t t e r i n g l e n g t h s , A > 0, e x c e p t t h a t the s e c o n d t e r m i s a b s e n t and a n a d d i t i o n a l pole t e r m o c c u r s i n e a c h of the eqs. (21), (22) a n d (24). T h e two p o l e s i n h(z) a r e e i t h e r both on the r e a l a x i s o r a r e c o m p l e x c o n j u g a t e s of one a n o t h e r . In the s e c o n d c a s e , the r e f l e c t i o n p r i n c i p l e (3) i m p l i e s that z I = z~ and a 1 = ao, so that the n u m b e r s of r e a l p a r a m e t e r s r e q u i r e d to s p e c i f y the pole p o s i t i o n s and r e s i d u e s a r e
N.
94
M. ~UEEN
t h e s a m e a s in t h e f i r s t c a s e . In e i t h e r c a s e , we u s e t h e t h r e e c o n d i t i o n s (21), (22) and (24) to e v a l u a t e t h e r e s i d u e s a n d t h e p a r a m e t e r a a s f u n c t i o n s of t h e two p a r a m e t e r s w h i c h s p e c i f y t h e p o l e p o s i t i o n s . F o r g i v e n p a r a m e t e r v a l u e s , t h e s c a t t e r i n g l e n g t h A i s d e t e r m i n e d by e v a l u a t i n g eq. (5) a t t h r e s h o l d . T h e ~ A ~ c o u p l i n g c o n s t a n t G i s o b t a i n e d f r o m e q s . (1) a n d (5), u s i n g t h e r e l a t i o n 1
_
dI~z]
2¢oZ
R~
(25)
z=z~
U n f o r t u n a t e l y , w i d e l y d i f f e r i n g e x p e r i m e n t a l v a l u e s of t h e w i d t h of t h e Y~ (1385) r e s o n a n c e h a v e b e e n r e p o r t e d , v a r y i n g f r o m F = (17.1 + 4.4) MeV ( r e f . [9]) t o (88+ 10) M e V ( r e f . [10]). T h e a v e r a g e v a l u e 3 7 ~ 4 M e V i s q u o t e d in t h e d a t a c o m p i l a t i o n of t h e P a r t i c l e D a t a G r o u p [11]. S i n c e t h e n u m e r i c a l r e s u l t s w e r e found to b e r a t h e r s e n s i t i v e to F , t h e c a l c u l a t i o n s were repeated for several different values. For the elastic branching ratio, we t a k e t h e v a l u e F ~ A / F = 0.91 ( r e f . [11]).
4. R E S U L T S AND C O M P A R I S O N W I T H O T H E R P R E D I C T I O N S T h e output v a l u e s of G 2, cr a n d A a s f u n c t i o n s of t h e p a r a m e t e r Zo, f o r s e l e c t e d v a l u e s of n a n d F , a r e s h o w n in f i g s . 1, 2 a n d 3, r e s p e c t i v e l y , f o r t h e c a s e A < 0. F o r p o s i t i v e s c a t t e r i n g l e n g t h s , h o w e v e r , no a c c e p t a b l e s o l u t i o n s w e r e found, a l t h o u g h an e x h a u s t i v e s e a r c h w a s m a d e in p a r a m e t e r s p a c e . If t h e two z e r o s z o a n d z 1 a r e a s s u m e d to l i e on t h e r e a l a x i s , e q s . (21), (22) and (24) p o s s e s s no s o l u t i o n s f o r any ~ > 0. F o r two c o m p l e x c o n j u g a t e z e r o s , on t h e o t h e r h a n d , t h e s e e q u a t i o n s h a v e s o l u t i o n s only f o r G 2 ~ 100. T h e r e f o r e t h e m o d e l r e q u i r e s a n e g a t i v e ~A s c a t t e r i n g l e n g t h . We s e e f r o m fig. 1 t h a t G2 i s an e x t r e m e l y s e n s i t i v e f u n c t i o n of Zo, r i s i n g m o n o t o n i c a l l y f r o m G 2 = 0 at z o = z ~ t o G 2 > 100 at z o = 1. T h u s , it i s not p o s s i b l e to d e t e r m i n e G 2 f r o m t h e m o d e l . A c c o r d i n g to SU(3) s y m m e t r y [12], G 2 i s r e l a t e d to t h e ~NN c o u p l i n g c o n s t a n t b y
20C
G2
I
I
I
0.4
0.6
O.O
15C
IOO
50
O Zr
t.O ZO
Fig. 1. The square of the lrAE coupling consLant as a function of z o. The dashed curves are for F = 25 MeV and the solid curves for F = 37 MeV. Curves (a) are for ~= 1.0 and curves (b) for ~ = 1.5.
~A MODEL
95
O~ 8 0 t
t.b) o|
'
i'
40 1
'
"
20
O zf
0.4
0.6
0.8
I.O Zo
Fig. 2. The asymptotic total 7rA cross section as a function of zo. The significance of the curves is the same as in fig. 1.
,'~
0
Zr
0,4
0.6
1.0
0.8 ZO
Fig. 3. The S-wave ~A scattering length as a function of zo. The significance of the curves is the same as in fig, 1.
c2 :
(i-
C2NN,
(26)
w h e r e (~ = f / i f + d ) i s the f r a c t i o n of R - t y p e c o u p l i n g . A s s u m i n g G2NN = = 14.6 [13] a n d ~ ~ 0.4 [14], we o b t a i n G2 ~ 7. M a n y i n d e p e n d e n t e x p e r i m e n t a l a n d t h e o r e t i c a l e s t i m a t e s of G 2, s u m m a r i z e d in ref. [15], a g r e e c l o s e l y with t h i s SU(3) p r e d i c t i o n . We t h e r e f o r e a s s u m e t h a t G2- = 101 10. E v e n with t h i s c o n s e r v a t i v e l y l a r g e u n c e r t a i n t y , it is c l e a r f r o m fig. i t h a t the p a r a m e t e r z o i s r e s t r i c t e d to a r a t h e r n a r r o w r a n g e ; in fact, we find Zo < 0.45. F o r z o w i t h i n t h e p e r m i s s i b l e r a n g e , r e s t r i c t i o n s on the a s y m p t o t i c t o t a l ~A c r o s s s e c t i o n ~ c a n now b e f o u n d f r o m the r e s u l t s s h o w n i n fig. 2. If F i s a l l o w e d to b e a s s m a l l a s 25 MeV (ref. [16]), we o b t a i n the l o w e r b o u n d > 15 m b . M u c h l a r g e r v a l u e s of ~ a r e p o s s i b l e if F e x c e e d s i t s l o w e r l i m it, a l t h o u g h ~ i s f a i r l y i n s e n s i t i v e to p . A c c o r d i n g to the q u a r k m o d e l [17], i
2
= ~ % N +~ ~KN •
(27)
U s i n g the a s y m p t o t i c e s t i m a t e s (r~N ~ 23 m b and aKN ~ 17.5 m b [18], eq. (27) g i v e s a ~ 19 rob, a v a l u e which i s c o n s i s t e n t with o u r l o w e r b o u n d f o r (r T u r n i n g f i n a l l y to the s c a t t e r i n g l e n g t h c a l c u l a t i o n s s h o w n i n fig. 3, we find t h a t A < - 0 . 2 0 fro. T h i s l i m i t is a t t a i n e d when u a n d I~ t a k e t h e i r m i n i -
96
N.M.QUEEN
m u m v a l u e s , and m u c h m o r e n e g a t i v e v a l u e s of A o c c u r f o r v a r i a t i o n s in the p a r a m e t e r s . In t a b l e 1, we c o m p a r e o u r e s t i m a t e of the s c a t t e r i n g l e n g t h with t h r e e o t h e r independent p r e d i c t i o n s . F i r s t l y , s i n c e the A is an i s o s c a l a r p a r t i c l e , W e i n b e r g ' s c u r r e n t - a l g e b r a s u m r u l e [6] g i v e s A = 0. Although this s u m r u l e w o r k s r e m a r k a b l y well f o r vN s c a t t e r i n g , doubts have a l r e a d y b e e n r a i s e d about its v a l i d i t y f o r the vE s y s t e m [19]. Table 1 Four predictions of the 7TA S-wave scattering length A.
Source
A (fm)
Present model
< -0.20 0
Current-algebra sum rule [6] Analysis of --~- decay [21]
0.34 + 0.33 - 0.50
Multi-channel KN analysis [24]
1.7 +3.0 -0.7
A n o t h e r i n d e p e n d e n t e s t i m a t e of A is p r o v i d e d by the a n a l y s i s of the w e a k d e c a y p r o c e s s Z - ~ ~-A. T h e c h a r a c t e r i s t i c s of the d e c a y a r e u s u a l l y s p e c i f i e d by the p a r a m e t e r s [20] a
=
2 Re(AoA1)
/3 = 2 I m ( A o n 1) ,
,
~ = [nol 2 - In1{ 2 , (28)
w h e r e A L is the d e c a y a m p l i t u d e f o r a final ~A s t a t e of o r b i t a l a n g u l a r m o m e n t u m L, n o r m a l i z e d so that l A o l 2 + {All = 1. Since a 2 + f i 2 + y ' 2 = 1, it is c u s t o m a r y to u s e the p a r a m e t r i z a t i o n 1
= ( 1 - a2)~ sin~b ,
y = ( 1 - ~2)½ c o s ~b.
(29)
A r e c e n t c o m p i l a t i o n of ~ - d e c a y d a t a g a v e the a v e r a g e v a l u e s [21] = -0.42 + 0.04 , It can be shown [22] that, if t i m e - r e v e r s a l the d e c a y p r o c e s s ,
q5 = 13 o + 7 o .
(30)
i n v a r i a n c e is a s s u m e d to hold in
t g ( 5 1 - 50) = f i / ~ ,
(31)
w h e r e 5 o and 51 a r e , r e s p e c t i v e l y , the S½ and P,_2 ~A s c a t t e r i n g p h a s e shifts at the d e c a y e n e r g y . F r o m a d y n a m i c a l m o d e l of ~A s c a t t e r i n g b a s e d on p a r t i a l - w a v e d i s p e r s i o n r e l a t i o n , M a r t i n [23] o b t a i n e d 51 ~ -1.2 °. U s i n g this e s t i m a t e , t o g e t h e r with eqs. (29) to (31), we c a n c a l c u l a t e 5 o. M a r t i n [23] d e t e r m i n e d the c o n t r i b u t i o n s to the r e a l p a r t of the S - w a v e a m p l i t u d e f r o m the n e a r b y s i n g u l a r i t i e s in the p a r t i a l - w a v e d i s p e r s i o n r e l a t i o n . If we a s s u m e that the e n e r g y d e p e n d e n c e of t h i s a m p l i t u d e a r i s e s m a i n l y f r o m t h e s e n e a r b y s i n g u l a r i t i e s , his r e s u l t s can be u s e d to e x t r a p o l a t e the a m plitude to t h r e s h o l d . In this way, we obtain the v a l u e f o r bhe s c a t t e r i n g l e n g t h given in t a b l e 1.
7TA MODEL
97
A f u r t h e r i n d e p e n d e n t p r e d i c t i o n i s o b t a i n e d f r o m an a n a l y s i s of l o w e n e r g y K--N s c a t t e r i n g d a t a by K i m [24], a c c o r d i n g to a m u l t i - c h a n n e l e f f e c t i v e - r a n g e e x p a n s i o n of t h e i n v e r s e K - m a t r i x . If K i m ' s p a r a m e t e r s a r e e m p l o y e d to d e s c r i b e ~A s c a t t e r i n g , we f i n d a s u r p r i s i n g l y l a r g e v a l u e f o r A ( s e e t a b l e 1). T h i s v a l u e s h o u l d p e r h a p s not b e t a k e n t o o s e r i o u s l y , s i n c e K i m ' s d e s c r i p t i o n of e l a s t i c ~A s c a t t e r i n g i s a r r i v e d at i n d i r e c t l y b y a n a l y z i n g t h e e n e r g y d e p e n d e n c e of v a r i o u s KN p r o c e s s e s . M o r e o v e r , t h e r e i s s t r o n g e v i d e n c e f r o m f o r w a r d d i s p e r s i o n s u m r u l e s [25] t h a t h i s p a r a m e t e r s g i v e an i n a d e q u a t e d e s c r i p t i o n of t h e l o w - e n e r g y ~[N i n t e r a c t i o n . It i s a m u s i n g to n o t e t h a t t h e f o u r a v a i l a b l e e s t i m a t e s of A g i v e n in t a b l e 1 a r e m u t u a l l y i n c o n s i s t e n t a n d c o v e r a v e r y w i d e r a n g e of p o s s i b l e v a l u e s .
5. CONCLUSIONS W e h a v e s h o w n in a s p e c i f i c m o d e l t h a t a n a l y t i c i t y i m p o s e s s t r o n g c o r r e l a t i o n s a m o n g t h e f e a t u r e s of nA s c a t t e r i n g at low, i n t e r m e d i a t e a n d high e n e r g i e s . T h e m o d e l p r e d i c t s an a s y m p t o t i c t o t a l c r o s s s e c t i o n ~ > 15 rob, which is consistent with the quark model. However, the resulting S-wave s c a t t e r i n g l e n g t h , A < - 0 . 2 0 f m , d i s a g r e e s w i t h e a c h of t h r e e o t h e r , m u t u ally incompatible predictions. F u r t h e r c a l c u l a t i o n s s u g g e s t e d t h a t t h e output v a l u e s of t h e s e q u a n t i t i e s a r e f a i r l y i n s e n s i t i v e to t h e p r e c i s e f o r m a s s u m e d f o r t h e f u n c t i o n p(z), p r o v i d e d t h a t t h i s f u n c t i o n s a t i s f i e s t h e t h r e e b a s i c h y p o t h e s e s of t h e m o d e l . S i n c e o u r c a l c u l a t i o n s a r e e x a c t , f o r a g i v e n input f u n c t i o n p ( z ) , we s u s p e c t t h a t v a l u e s of ~ o r A s u b s t a n t i a l l y d i f f e r e n t f r o m t h e e s t i m a t e s g i v e n a b o v e a r e p o s s i b l e only if t h e ~A s c a t t e r i n g a m p l i t u d e h a s c o n s i d e r a b l y m o r e s t r u c t u r e t h a n a s s u m e d in t h e m o d e l . A m o r e r e f i n e d a n a l y s i s of t h i s t y p e , f o r ~ a s w e l l a s ~A s c a t t e r i n g , m a y b e c o m e p o s s i b l e w h e n m o r e r e l i a b l e m u l t i - c h a n n e l p a r a m e t r i z a t i o n s of t h e KN i n t e r a c t i o n h a v e b e e n m a d e . I a m g r a t e f u l to D r . C. H. C h a n f o r p o i n t i n g out an e r r o r in t h e o r i g i n a l manuscript.
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