Resonances in the S11 pion nucleon amplitude

Volume 21, number 1

PHYSICS LETTERS

RESONANCES

IN

THE

Sll

PION

15 April 1966

NUCLEON

AMPLITUDE

C. MICHAE L

Theoretical Physics Department, Clarendon Laboratory, Oxford Received 24 March 1966

The Sll yN phase shift is interpreted as the sum of an inelastic resonance at 1570 MeV and an elastic one at 1700 MeV using a general formulation for combining such contributions.

A r e s o n a n c e with a n o n - r e s o n a n t amplitude having the s a m e quantum n u m b e r s and so being in the s a m e p a r t i a l wave, is d i s c u s s e d . This background c o n t r i b u t i o n could be due to p a r t i c l e exchange or in the c a s e c o n s i d e r e d h e r e it is due to a not h er r e s o n a n c e . The u n i t a r i t y equation f o r the whole p a r t i a l wave a m p l it u d e m u s t be s a t i s f i e d and Dalitz [1] has d i s c u s s e d this p r o b l e m using a K m a t r i x f o r m a l i s m which e n s u r e s this. We p r e s e n t an a l t e r n a t i v e d i s c u s s i o n which g i v e s s i m i l a r r e s u l t s . The s e p a r a t i o n into r e s o nant and n o n - r e s o n a n t p a r t s is a r b i t r a r y u n l e s s s o m e definite choice is m a d e , and the definition chosen is that the n o n - r e s o n a n t contribution s h a l l s a t i s f y u n i t a r i t y . This will allow f u r t h e r n o n - r e sonant c o n t r i b u t i o n s to be r e m o v e d by r e p e a t i n g the s e p a r a t i o n s i n c e the u n i t a r i t y p r o p e r t i e s of the whole p a r t i a l wave amplitude and the a m p l i tude with one r e s o n a n t contribution r e m o v e d a r e the s a m e . In the l i m i t of no background the r e s u l t s r e d u c e to the u s u a l r e q u i r e m e n t s of u n i t a r i t y on r e s o n a n c e s ; the r e l a t i o n of width to coupling constant and the r e a l i t y of coupling c o n s t a n t s . In the l i m i t of v an i s h i n g r e s o n a n t coupling the r e s u l t r e d u c e s to the identity of the whole a m p l it u d e and the n o n - r e s o n a n t b a c k g r o u n d which is p o s s i b l e sinc e they both s a t i s f y u n i t a r i t y . This f o r m a l i s m is w e l l i l l u s t r a t e d by c o n s i d e r ing many coupled two-body channels, l a b e l l e d 1 through r to n. Then the r e s o n a n t contribution, s e p a r a t i n g only one r e s o n a n c e in the p a r t i a l wave f r o m the amplitude, will have a c o m m o n c o n t r i bution to al l channels, n a m e l y the r e s o n a n t p r o p a g a t o r P(s). The coupling or decay v e r t e x Vr(S ) to e a c h channel will depend on e n e r g y also. F o r the p a r t i a l wave under c o n s i d e r a t i o n , s u p p r e s s i n g the i n d i c e s of t o t a l a n g u l a r m o m e n tum and p a r i t y , the coupled u n i t a r i t y equations are

Im

--

pr(s)

(i)

and for n o n - r e l a t i v i s t i c n o r m a l i z a t i o n the phase s p a c e f a c t o r Pr(S) = qrO(S - T 2) w h e r e qr is cent r e of m a s s m o m e n t u m of channel r and T r is t h r e s h o l d e n e r g y . The d e c o m p o s i t i o n d e s c r i b e d above f o r the total amplitude aij into background bij and r e s o n a n t p a r t is

aij(s) = bij(s) + vi(s)P(s)vj(s ) .

(2)

Now the r e q u i r e m e n t that the background s a t i s f i e s the u n i t a r i t y equation is

biffs) :

bir(S) Or(s) b*j(s) .

(3)

T h e s e t h r e e r e l a t i o n s i m p l y conditions on the v e r t e x t e r m s and r e s o n a n t p r o p a g a t o r f r o m substituting (2) in (1) and using (3)

(4) *

*

* *

* *

*

2 i ~ ( v i P v r P r b r j + b i r P r V r P vj + v i P V r P r V r P vj). r This equation is i d e n t i c a l l y s a t i s f i e d if P . i s defined by Im P - l ( s )

=

- E

Pr(S) IVr(S)l 2

=

-mF(s)

(5)

r

and the c o n s t r a i n t on the v e r t e x t e r m s b e c o m e s

Im vi(s ) = ~r vr(s) Pr(S) b~i(s) "

(6)

Eq. (5) iS the usual r e l a t i o n for the width F(s) of a r e s o n a n c e , v a r y i n g with phase space. The r e a l p a r t of P - l ( s ) is not s p e c i f i e d by the above equations but we define the r e a l p a r t of the r e s o nant m a s s m by the r e q u i r e m e n t that Re P - l ( m 2 ) = = 0. The f o r m of Re P - l ( s ) that we s h a l l use in the subsequent cal cu l at i o n is a s i m p l e pole ex p r e s s i o n , or e s s e n t i a l l y the B r e i t W i g n e r a p p r o x i -

93

Volume 21, number 1

PHYSICS

LETTERS

15 April 1966

A

6

ReA

Fig. 1. The Argand diagram for A = P l a l l shews the addition of a background B = £o exp(iSo) sin 5o in the s a m e partial wave to a resonant amplitude R = ~R exp(iSR) sin 5 R after the latter has been rotated by twice the phase 5B of v 1 in accord with (6). The circular path shewn corresponds to increasing energy through the resonance at 5R = ½~ with B constant in energy. m a r i o n , Re P - l ( s ) = m 2 - s. In g e n e r a l one s h o u l d c a l c u l a t e Re P - l ( s ) f r o m a d i s p e r s i o n r e l a t i o n in P - l ( s ) w h i c h w i l l be d i s c u s s e d e l s e w h e r e , and l e a d s to the s i m p l e p o l e e x p r e s s i o n m u l t i p l i e d by a s l o w l y v a r y i n g f u n c t i o n w h i c h we take c o n s t a n t in the c a l c u l a t i o n . Eq. (6) is s a t i s f a c t o r y in that bij d o e s not c o n tain the r e s o n a n c e p e a k so v w i l l not c o n t a i n it, w h e r e a s the a n a l o g o u s f o r m f a c t o r e q u a t i o n s w h i c h h a v e aij i n s t e a d of bij w i l l not h a v e t h i s p r o p e r t y . One c o u l d s o l v e (6) by d i s p e r s i o n r e l a t i o n s a l s o , but we s h a l l a s s u m e t h a t vi(s) is g i v e n by the B o r n v e r t e x t e r m m u l t i p l i e d by a s l o w l y v a r y i n g f u n c tion, so f o r an s w a v e c a s e Ivi(s)! w i l l be j u s t the c o u p l i n g c o n s t a n t . H o w e v e r , the p h a s e of vi(s ) i s i m p o r t a n t and m a y be d e d u c e d f r o m (6). F o r only one open c h a n n e l , that is e l a s t i c s c a t t e r i n g b e l o w the i n e l a s t i c t h r e s h o l d b l l = exp(iSo) s i n S o , so s i n c e Im Vl(S ) m u s t be r e a l , (6) g i v e s the p h a s e of Vl(S) to be 5o a l s o . T h e n (2) r e a d s a l l = exp(iS) s i n 5 / p 1 = = b l l + exp(2i5 o) l v l l 2 ( R e P - l ( s ) - i p

(7) 1 IVl]2) - 1 ,

so that d e f i n i n g the r e s o n a n t p h a s e s h i f t by cot 5 R = Re P-l(s)/mF(s) we find (7) r e d u c e s to 6 = 5 o + 5 R so that one adds the r e s o n a n t and u n i t a r y n o n resonant phase shifts. F o r m a n y open c h a n n e l s (2) b e c o m e s f o r e l a s -

.q4

Fig. 2. The $11 YN phase shift amplitude A from the analysis of Bareyre et al. [2] is shewn by the crosses at each laboratory pion kinetic energy in MeV. The smooth curve represents (9) with the values of the r e sonance p a r a m e t e r s given by (10). The c i r c l e s are the values of A calculated at each energy where a phase shift determination exists.

t i c s c a t t e r i n g in c h a n n e l 1 Plall

= e o exp(i5 o) s i n 5 o +

(8)

+ exp(2i5 B) e R exp(i6R) s i n 5 R , w h e r e the p h a s e of v 1 is 5 B and m a y be found by s o l v i n g (6). Eo and E R a r e the e l a s t i c i t i e s of b a c k g r o u n d and r e s o n a n c e r e s p e c t i v e l y . T h i s r e l a t i o n is i l l u s t r a t e d on the A r g a n d d i a g r a m in fig. 1. A s i m p l e f o r m f o r 5 B o b t a i n s in a two c h a n n e l c a s e when IV2bl21 << I V l b l l l s i n c e the a r g u m e n t of (7) a g a i n h o l d s and 5 B = 5 o. A p a r t i c u l a r l y s u i t a b l e c a s e in w h i c h to e m ploy t h i s f o r m a l i s m is the S l l ~N p h a s e s h i f t w h e r e we h a v e u s e d the a n a l y s i s of B a r e y r e et ai. [2] s h o w n in fig. 2. T h e loop at 700 M e V pion l a b o r a t o r y k i n e t i c e n e r g y is i n t e r p r e t e d a s an i n e l a s t i c r e s o n a n c e R ~ with a b a c k g r o u n d s m o o t h ly v a r y i n g with e n e r g y and e s s e n t i a l l y e l a s t i c . The i n e l a s t i c i t y i s known [3-5] to be a s s o c i a t e d with 77 p r o d u c t i o n and in a two c h a n n e l a p p r o x i m a tion the y p r o d u c t i o n c r o s s s e c t i o n m a y be d e t e r m i n e d f r o m the p h a s e s h i f t data. The p r o d u c t i o n c r o s s s e c t i o n s o d e t e r m i n e d is c o n s i s t e n t with the d a t a [3] f o r ~ - p -~ yOn up to 700 M e V w h e r e p r o d u c t i o n w i l l not be only s w a v e . H e n d r y and M o o r h o u s e [4] u s i n g an e n e r g y d e p e n d e n t K m a t r i x a p p r o a c h found a t h i r d c h a n n e l n e c e s s a r y in t h e i r f i t up to 700 M e V of 71 p r o d u c t i o n but the phase shift analyses they used were substantially d i f f e r e n t f r o m [2]. H e n d r y and M o o r h o u s e found a

Volume 21, number 1

PHYSICS

LETTERS

Then it remains to vary the two m a s s e s and three coupling constants in (9) to give the best agreement with the phase shift analysis. Since the exact functional form of the resonance contributions are open to debate and the different phase shift analyses do not agree a m o n g themselves it was decided that a visual fit was sufficient. The errors were thus not determined statistically but the resonance m a s s e s are better determined than the widths. The results are shown in fig. 2, with

r e s o n a n c e at a b o u t 1510 M e V , j u s t a b o v e the 7/ t h r e s h o l d at 1488 M e V . We t a k e r e s o n a n c e Rot to h a v e m a s s mot and w i d t h rot (s) g i v e n by (5) w h e r e the rot 1 and rot2, c o u p l i n g to ~N and 7/N r e s p e c t i v e l y , a r e t a k e n as the coupling constants gotl and got2 near t h r e s h o l d . H o w e v e r the f i t w i l l be i m p r o v e d by t a k i n g the w i d t h c o n s t a n t at rot (m2a) f o r S > m 2 , it b e i n g k n o w n f o r m a n y o t h e r r e s o n a n c e s that t h e y s a t u r a t e f o r m o m e n t u m h i g h e r than the resonant momentum. The b a c k g r o u n d to Rot s e e m s to h a v e a r e s o n a n t s h a p e a l s o s o we c a n r e p e a t the f o r m u l a t i o n and e x p r e s s i t a s a s u m of a n o t h e r r e s o n a n t t e r m R~ and i t s b a c k g r o u n d w h i c h we t a k e z e r o . F o r c a l c u l a t i o n a l s i m p l i c i t y R B is t a k e n a s e l a s t i c s i n c e the p h a s e s h i f t i s s t i l l f a i r l y e l a s t i c at 900 M e V and we w i s h to k e e p the n u m b e r of p a r a m e t e r s a s low a s p o s s i b l e . The w i d t h F B (s) of R~ is t a k e n f r o m (5) f o r a f i x e d v a l u e of b ~ l , n a m e l y g/~ m u l t i p l i e d by the p h a s e s p a c e . So f r o m (8) we w r i t e the ~rN S l l a m p l i t u d e a s

A = Pl(S)all(S ) =

2 rnB-

ql

exp(215o)gotlql

m a = 1570 M e V ; 2

rnt3 = 1700 M e V ;

= 4.3

(10)

FB(m~) = 240 M e V .

The a u t h o r w o u l d l i k e to thank D r . G. R. S c r e a t o n and P r o f e s s o r R. H. D a l i t z f o r h e l p f u l d i s c u s s i o n s . , (9)

Refe~'ences 1. R.H. Dalitz, Ann. Rev. Nuel. Sci. 13 (1963) 339. 2. P . B a r e y r e , C . B r i c m a n , A.V.Stirling and G. Villet, Physics Letters 18 (1965) 342. 3. R.H. Dalitz, Proc. Oxford Int. Conf. on Elementary P a r t i c l e s (Rutherford High Energy Laboratory, Jan. 1966~. 4. A.W. Hendry and R. G. Moorhouse, Physics Letters 18 (1965) 171. 5. F.Uchiyama-Campbell, Physics Letters 18 (1965) 189.

cot So = (m2#- S)/g2#qI and

= m a rot(m 2j

2

= 130 M e V ;

The a s s i g n m e n t of t h e s e r e s o n a n c e s h a s b e e n d i s c u s s e d by D a l i t z [3] u s i n g a q u a r k m o d e l .

s_i ql m2_S_imotrot(s)

g2ot2q20(S- ~2)for

ra(m ~)

got2/gotl

where

mot rot(S) = g2otlq I +

15 April 1966

S -< m2ot

2. f o r S > rncz *****

95