Inequalities for the pion-nucleon partial waves

Nuclear Physics B31 (1971) 570-574. North-Holland Publishing Company

INEQUALITIES FOR THE PION-NUCLEON PARTIAL WAVES* A. P. BALACHANDRAN P h y s i c s D e p a r t m e n t , S y r a c u s e Uni~,ersity, S y r a c u s e , N . Y . 13210** and I s t i t u t o N a z i o n a l e di F i s i c a N u c l e a t e - S e z i o n e di Napoli

W. CASE P h y s i c s Deparlrrtent, S y r a c u s e U n i v e r s i t y , S y r a c u s e , N . Y . 13210

A. D E L L A S E L V A and S. SAITO I s t i t u t o N a z i o n a l e di F i s i c a N u c l e a t e - S e z i o n e di Napoli Istilzdo di F i s i c a T e o r i c a d e l l ' U u i v e r s i t & di Napoli ***

Received

1 June

1971

Abstract: We derive two inequalities for the pion-nucleon partial waves with angular momenta l ~< 4.

In a r e c e n t p a p e r [1], C a s e h a s d e v e l o p e d a m e t h o d f o r d e r i v i n g an i n f i n i t e n u m b e r of i n e q u a l i t i e s f o r the p a r t i a l w a v e s of a c e r t a i n c l a s s of s c a t t e r i n g a m p l i t u d e s . In t h i s note, we s h a l l u s e h i s m e t h o d to d e r i v e the e x p l i c i t f o r m of two of t h e s e i n e q u a l i t i e s for the p i o n - n u c l e o n p a r t i a l w a v e s . W h i l e t h e r e a r e in fact an i n f i n i t e n u m b e r of s u c h i n e q u a l i t i e s , the o n e s we d e r i v e c o n t a i n p a r t i a l w a v e s with the l e a s t p o s s i b l e v a l u e s for the a n g u l a r m o m e n t a . T h e y a r e t h e r e f o r e b e l i e v e d to be of the g r e a t e s t p r a c t i c a l i n terest. T h e d e r i v a t i o n of the ~N p a r t i a l - w a v e i n e q u a l i t i e s f o l l o w s c l o s e l y a m e t h o d d e v e l o p e d p r e v i o u s l y for the d e r i v a t i o n of the nTr p a r t i a l - w a v e i n e q u a l i t i e s [2, 3]. We s h a l l t h e r e f o r e only s k e t c h the m e t h o d . A n a t t e m p t will h o w e v e r be m a d e to i d e n t i f y and e x p l a i n the p o i n t s w h e r e t h e r e a r e new c o m p l i c a t i o n s due to the n u c l e o n s p i n . L e t A + ( s , t) and B + ( s , l) d e n o t e the s t a n d a r d A and B a m p l i t u d e s which i n the s - c h a n n e l r e f e r to the p r o c e s s n+p ~ u+p and l e t A _ ( s , t) a n d B _ ( s , t) d e note the c o r r e s p o n d i n g a m p l i t u d e s f o r ~ - p ~ ~r-p. A c c o r d i n g to Mahoux and M a r t i n [4] ( s e e a l s o C a s e [1]), the f u n c t i o n s * Supported in part by the U. S. Atomic Energy Commission. ** Permanent address. *** Postal address: Istituto di Fisica Teorica, Mostra d ' O l t r e m a r e , pad. 19-80125, Napoli, Italy.

A. P. Balaehandran et al., Pion-nucleon partial waves

571

(1)

M(±)(s, t) = A+(s, t) + (s - u - t)/4m B±(s, t)

h a v e the p r o p e r t y t h a t t h e i r a b s o r p t i v e p a r t s in both the s - and the u - c h a n n e l s a r e n o n - n e g a t i v e f o r 0 < I < 4 p 2. (We d e n o t e by t~ and m the p i o n and n u c l e o n m a s s e s . ) F u r t h e r it is p o s s i b l e to w r i t e 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 with at m o s t two s u b t r a c t i o n s f o r M(+). L e t u s d e n o t e by M(±) the f u n c t i o n s M(±) with the n u c l e o n pole t e r m s s u b t r a c t e d out. It f o l l o w s f r o m the F r o i s s a r t - G r i b o v r e p r e s e n t a t i o n of the i n t e g r a l 1

II±)(')= f d z t ( 1 - z ~ ) P ~ ( z t ) M ( ± ) ( s , t ) , -1

l >~ 3 ,

(2)

and the p o s i t i v i t y p r o p e r t i e s of M(±), t h a t *

I~±)(l) >/ 0 ,

0 ~< t < 4t~ 2

(3)

H e r e z t i s the c o s i n e of the s c a t t e r i n g a n g l e i n the t - c h a n n e l and P~ is the associated Legendre polynomial. T o t u r n (3) into a yN i n e q u a l i t y , we p r o c e e d a s f o l l o w s * * . L e t ~ ( s , l ) b e the K i b b l e f u n c t i o n [8]. We h a v e 22

2

• (s,t) = 4 t P t q t ( 1 - zt) , = 4 s k ( 1 - Zs) ,

(4)

w h e r e z s is the c o s i n e of the s - c h a n n e l s c a t t e r i n g a n g l e and

pt2qt2 = ~ ( t _ 4 m 2 ) ( t _ 4 9 )

,

k 2 : [s - (m + t~)2l[s - (rn - p)2]/4s . s

(5)

T h e n (3) i m p l i e s t h a t 1

-2Ptq t _ fl d z t ~ ( s , t ) [ p t q2t 2P,3 ( z t ) ] M ~ ± ) ( s , t ) >/ 0 ,

0 < l<4~ 2 ,

(6)

* In deriving (3), use is also made of the fact that the functions of the second kind associated with P~ have positivity properties very s i m i l a r to the Legendre functions of the second kind. (Cf. Case [1] or the first appendix to the third paper of Balachandran and Blackmon [3]. P~ is proportional to the Jacobi polynomial p~_~il). ).1 Note that (3) is the analogue of the Martin inequalities for ?rg partial waves [2] and that in factI~±)(t) >10 f o r l = 3, 5,7 . . . . and 0 --
A .P. Balachandran el al., Pion-nucleon partial waves

572

w h e r e we have c h o s e n to define Ptqt to be n o n - p o s i t i v e f o r 0 -< t -< 4 p 2.

Thus 4~ 2

f

1 22 , dt(-2Ptqt) -fl dzt 4'(s' t)[ptqtP'3(zt )] Fl(±l(s, t) >~0 .

(7)

0

Since

dt(-2Ptqt)dz ! = -dsdt ,

(8)

= ds(-2k2s)dZs , and s i n c e the d o m a i n

{tlo

~z

-.<4p2}®{zt]-l
-< +1}

(9)

has also the decomposition* z~: {sl(m- p) 2 -< s <(m+t~)2} ® { z s l - 1 < z s ~< +1},

(10)

it follows that 1 2 22 , (+) ~ m + p ) 2 ds S k6s fl dzs(1 - zs) [ p t q t P 3 ( z ? ] M (s,t) --< 0 . (m-p)2

(11)

2 2 P3(zt) , Now we can w r i t e ptql in the f o r m

2 2 t P3(zt) , ptq = a(s) +~(s)z s + ~(s)Z2s ,

(12)

where

a(s) = 3 [2k4 +k2 s +8k 2 Eco+8E2CO2] S

S

S

fi(s) : {k 2[s +4k2 +8ECO] , 7(s)

=

]k4s~ ,

(13)

and E and co a r e the n u c l e o n and pion e n e r g i e s with

Eco = [s 2 - (m 2 - /z2)2]/4s .

(14)

F u r t h e r , 2Er(+) has the p a r t i a l - w a v e e x p a n s i o n

* A is the so-called Euclidean region. The existence of such decompositions for the Euclidean region is provided by Balachandran et al. [7].

A. P. Balachandran et al., Pion-nucleon partial waves

573

M(±)(s, t) = ~ bl±)(s)P~(Zs) .

(15)

l T h e p r e c i s e definition of b~±) is given in eqs. (16)-(21) b e l o w * . When we i n s e r t (12) and (15) in (11)"and use the o r t h o g o n a l i t y p r o p e r t i e s of P~, we find that the s u m in (15) can be r e s t r i c t e d to l ~< 3 in so f a r as the i n t e g r a l (11) is c o n c e r n e d . We finally obtain the r e q u i s i t e i n e q u a l i t i e s on doing the remaining integrals. T h e i n e q u a l i t i e s a r e in detail as follows: L e t gl.+) denote the n+p -* n+p p a r t i a l w a v e s f o r total a n g u l a r m o m e n t u m J = l ± ½ with the n o r m a l i z a t i o n 1

=s2 exp[iSl3±'(s,]sinSl3±'(s,,

(16,

in the elastic region where 513, are the phase shuts in t~e isospin-~ channel. L e t gl~ ) d e n o t e the c o r r e s p o n d i n g p a r t i a l waves f o r ~-p ---7r-p with the normalization = s2

exp{iSi3±)}sin{Si3)}

+ ~ exp {i51:)} sin{511)}] ,

(17)

w h e r e 6 l+ (1) a r e the p h a s e shifts in the isospin-½ channel. Define

bli)(s) = g(i)

"s" + (i)'s" ~(i) _ ~(i) gl- ( )-Ul+ (s) ~(/+l)_(s),

(l-1)+ ( )

bl+)p(s) g2 s_m2_p2 ~Ql4~ 2k 2 1 -

(1+m2+2p2"s

2k 2

(18)

i = +,(1+

m2+292-s~7

•)- Ql+l

, (19)

=~~-1) 5~1,

bl_)p(s ) g2 { ~2

(20)

bli)(s) = b~i)(s)-b~i)P(s) ,

(21)

w h e r e g2/4~ ~- 14.6, and Ql+l a r e the L e g e n d r e functions of the s e c o n d kind. T h e n (m+~) 2 f ds s h 6 [ {k2(45s - 28k 2 - 40Ew) +40m2~z2}b~i)(s) (m- ~)2 s s

+ 3k2{s+4k~+8Ew}b~2i'(s'+~k4bli'(s)]<~ 0 , * See for example ref. [9].

i= +,-.

(22,

574

A . P . B a l a c h a n d r a n et al., P i o n - n u c l e o n p a r t i a l w a v e s

We note that we c o u l d h a v e c a r r i e d out the a n a l y s i s u s i n g S o m m e r ' s a m p l i t u d e [ 10] i n s t e a d of M(+). It t u r n s out h o w e v e r that the r e s u l t a n t i n e q u a l i t i e s c a n be d e r i v e d f r o m (22) by a p p l y i n g the ~N p a r t i a l - w a v e c r o s s i n g r e l a t i o n s of r e f s . [6, 7] to the l e f t - h a n d s i d e of (22) and t a k i n g s u i t a b l e l i n e a r c o m b i n a t i o n s with n o n - n e g a t i v e c o e f f i c i e n t s of the r e s u l t a n t i n e q u a l i t i e s . F u r t h e r , as c o m p a r e d to (22), t h e s e i n e q u a l i t i e s a l s o c o n t a i n m o r e v a l u e s of I. We have t h e r e f o r e p r e f e r r e d to s t a t e o u r r e s u l t s in the f o r m (22). It m a y f i n a l l y be o b s e r v e d t h a t i n e q u a l i t i e s of the f o r m (22) which one m a y d e r i v e by c o n s i d e r i n g a m p l i t u d e s of p r o c e s s e s l i k e ~Op ~ ~Op a r e not new and m a y a l s o be o b t a i n e d by t a k i n g l i n e a r c o m b i n a t i o n s with n o n - n e g a t i v e c o e f f i c i e n t s of the i n e q u a l i t i e s (22).

RE F E R E N C E S [1] W. Case, Phys. Rev. D3 (1971) 2472. [2] A. Martin, Nuovo Cimento 47A (1967) 265; See also S. M. Roy, Phys. Rev. Letters 20 (1968) 1016. [3] O. Piquet and G. Wanders, Phys. Letters 30B (1969) 418; R. Roskies. J. Math. Phys. 11 (1970) 2913; A. P. Balachandran and M. L. Blackmon, Phys. Letters 31B (1970) 655; Phys. Rev. D3 (1971) 3133; Phys. Rev. D3 (1971) 3142; Syracuse University preprint SU-1206-252; M. R. Pennington, Nucl. Phys. B24 (1970) 317; B25 (1970) 621. [4] G. Mahoux and A. Martin, Phys. Rev. 174 (1968) 2140. [5] A.K. Common, Nuovo Cimento 63A (1969) 863; F. J. Yndurain, Nuovo Cimento 64A (1969) 225. [6] A. P. Balachandran, W. Case and M. Modjtehedzadeh, Phys. Rev. D1 (1970) 1773; See also A. P. Balaehandran, The crossing and positivity properties of partial waves, A review, Syracuse University preprint SU-1206-243 (1971). [7] A. P. Balachandran, W. J. Meggs, J. Nuyts and P. Ramond, Phys. Rev. 187 (1969) 2080: See also J. L. Basdevant, G. Cohen-Tannoudji and A. Morel, Nuovo Cimento 64A (1969) 585. [8] T. W. B. Kibble, Phys. Rev. 117 (1960) 1159. [9] J. Hamilton and W. S. Woolcock, Rev. Mod. Phys. 35 (1963) 741. [10] G. Sommer, Nuovo Cimento 52A (1967) 373.