Weinberg sum rules for one-particle matrix elements

8 7 A . 5 - N u c l e a r Physics B7 (1968) 30-36. North-Holland Publ. Comp., A m s t e r d a m

WEINBERG ONE-PARTICLE

SUM

RULES

MATRIX

FOR

ELEMENTS

G. P ( ) C S I K Ir~titute f o r Theoretical Physics, Roland EStvOs University, Budapest Received 7 March 1968

Abstract: Starting from the equality of the Schwinger t e r m s in the vector current commutator and the a x i a l - v e c t o r c u r r e n t commutator, new sum rules a r e derived. The sum rule containing only the isotriplet contributions is evaluated in an infinitemomentum limit. The r e s u l t imposes a r e s t r i c t i o n for the p a r a m e t e r s of the decays p---. 2~ and A 1 ---*py, from where reasonable values of the p , A 1 widths and the charge radius of the pion follow.

I. INTRODUCTION

It h a s b e e n s h o w n by W e i n b e r g [1] t h a t t h e e q u a l i t y of t h e v e c t o r a n d a x i a l - v e c t o r S c h w i n g e r t e r m s in the SU 2 ® SU 2 a l g e b r a ~ c o m b i n e d , w i t h c e r t a i n h i g h - e n e r g y a s s u m p t i o n s l e a d s to t h e s u c c e s s f u l rn~l__ = 2rn~ r e l a t i o n . T h e e q u a l i t y of t h e c - n u m b e r S c h w i n g e r t e r m s w a s p r o v e d a l s o f o r t h e m a s s i v e p i o n [2, 3]. (Ref. [3] m a k e s u s e of t h e J a c o b i i d e n t i t y . ) F o r o b t a i n ing t h e two W e i n b e r g s u m r u l e s a n o t h e r a p p r o a c h h a s b e e n u s e d b y D a s , M a t h u r a n d Okubo [4]. T h e b a s i s of t h e i r t r e a t m e n t i s the c l a i m t h a t t h e SU2 ® SU2 s y m m e t r y b e c o m e s e x a c t at h i g h e n e r g i e s . S u b s e q u e n t l y , t h e above-mentioned sum rules have been generalized also for larger groups b y m a k i n g u s e of t h e J a c o b i i d e n t i t y [5] a n d a s y m p t o t i c s y m m e t r y [6, 7], respectively. These considerations have led to several interesting results [ 8 - 1 7 ] . R e c e n t l y , an e x t e n s i o n of t h e a s y m p t o t i c s y m m e t r y h a s b e e n a t t e m p t e d f o r s c a t t e r i n g a m p l i t u d e s [18] in c o n s i s t e n c y w i t h e x p e r i m e n t . In t h e p r e s e n t p a p e r t h e a n a l o g u e of W e i n b e r g ' s f i r s t s u m r u l e i s d e r i v e d f o r o n e - p i o n m a t r i x e l e m e n t s of t h e c u r r e n t c o m m u t a t o r s u n d e r t h e a s s u m p t i o n t h a t S c h w i n g e r t e r m s g i v e no c o n t r i b u t i o n s in t h e J a c o b i i d e n t i t y . ( F o r i n s t a n c e , t h i s h a p p e n s f o r c - n u m b e r S c h w i n g e r t e r m s . In t h i s c o n n e c t i o n s e e r e f . [19].) T h e s u m r u l e i s s a t u r a t e d w i t h l o w - l y i n g s t a t e s . B e c a u s e of i s o t o p i c s p i n i n v a r i a n c e , one c a n s e l e c t a s u m r u l e c o n t a i n i n g o n l y t h e c o n t r i b u t i o n s of t h e s t a t e s ~, p a n d A 1 (A 2 i s n e g l e c t e d ) and t h i s i s e v a l u a t e d in t h e l i m i t PoTr~ ~. T h e r e l e v a n t m a t r i x e l e m e n t s a r e t a k e n f r o m r e f s . [20, 21]. T h e y a r e d e t e r m i n e d f r o m c u r r e n t a l g e b r a t a k e n b e t w e e n v a c u u m a n d ~, p a n d A1, r e s p e c t i v e l y , in the H a l l i d a y - L a n d s h o f f l i m i t [22]. In t h i s way, we a r e l e d to a s u m r u l e c o n c e r n i n g t h e p a r a m e t e r s ~2, )~ of

WEINBERG SUM RULES

31

B r o w n and West [23, 24], rn~/m 2 and the m o m e n t u m t r a n s f e r . T h i s is s a t i s fied with the e x p e r i m e n t a l data and it y i e l d s r e a s o n a b l e values of the p and A 1 widths and the c h a r g e r a d i u s of the pion.

2. DERIVATION OF THE SUM R U L E F o r the s a k e of s i m p l i c i t y we shall c o n s i d e r the a l g e b r a SU 2 ® SU2. L e t us denote the v e c t o r ( a x i a l - v e c t o r ) c u r r e n t by V~(x)a (Atz(X)a) and s t a r t with the J a c o b i identity [5]

[ Vo(X)a, [Ao(Y)b, AX(Z)c]] + [Ao(Y)b, [A).(Z)c, Yo(X)a]] + [A;~(Z)c , [ Vo(x)a , Ao(Y)b] ] = O, xo=Yo=Zo,

a,b,c=l,2,3,

~= 1,2,3.

(1)

T a k e eq. (1) between o n e - p i o n s t a t e s , I~r(p)t), and substitute the inner c o m m u t a t o r s f r o m the c u r r e n t a l g e b r a . As we mentioned, the total contribution of the Schwinger t e r m s is neglected in the m a t r i x e l e m e n t of eq. (1). Under this a s s u m p t i o n the equality of Schwinger t e r m s in the V-V and A-.4 c o m m u t a t o r s e a s i l y follows (between pion s t a t e s ) . F i r s t , f r o m g e n e r a l p r i n c i p l e s including a l s o C T P - i n v a r i a n c e we have (lr(P)t ][J/l (X)a, Ju(Y)b]

2i I~r(P)t> = (27r)3f

d4p' nabt (p;p,)j sin [(p-p')(x-y)]

ju = V ,Au

(2)

Then, the above c o n s i d e r a t i o n , eq. (2) and S c h u r ' s l e m m a lead to the f o l lowing s u m r u l e s

, abt (P',P+q, Po)V-Pox , abt (P,P'q, . , ) = 6ab ~) to~.(P,q) , f dPo(Pox Po)V ,f dPo(PoX , a b t (P;P+q, Po)A-PoX , a b t (P,P-q', Po)A) , = 5abed to

(3)

(t,,q )

w h e r e ~ t (p,q) is a function c o m i n g f r o m S c h u r ' s l e m m a , and q ¢ 0 , p2 = m2. U'~ Now, we u s e the i s o s p i n i n v a r i a n c e in eq. (2), then it is e a s y to see that eq. (3) d e c o m p o s e s into t h r e e s u m r u l e s . Eq. (3) is n o n - t r i v i a l only f o r a = b. In the p r e s e n t p a p e r we d i s c u s s the s u m r u l e t v a, which s t a t e s a r e l a t i o n between m a t r i x e l e m e n t s of v e c t o r and a x i a l - v e c t o r c u r r e n t s .

3. CONTRIBUTIONS OF L O W - L Y I N G STATES F o r s a t u r a t i n g the s u m r u l e t ¢ a we choose 7r, p, A 1 as i n t e r m e d i a t e s t a t e s , while A 2 is a s s u m e d to be negligible (only i s o t r i p l e t s give c o n t r i butions). As well a s , d i s c o n n e c t e d contributions with two pions running through a r e a l s o taken into account, C T P - i n v a r i a n c e is maintained. In this

32

G. P6CSIK

sense only 13~), 12~p) and 12~A1) contribute to the sum rule t ¢ a f r o m the possible two-, and t h r e e - p a r t i c l e states. Other kind of disconnected contributions would make the calculations very complicated. The spectral function of the axial-vector commutator in the approximation indicated is

paat, ...,, ~ = oX ~'~" JA

pa~p](p,p.)o(p,~6(p,,2_m2) + ½pa~xPl(_p,p.-2p)O(p'~-2Po)O((p"-2p)9"-m2p)

aat w h e r e [p] means the contribution of the p - m e s o n . PoX(p,p')v

(4)

contains the

contributions of A 1 and ~ in a f o r m s i m i l a r to eq. (4). Define the invariant functions which are p r e s e n t in PV,A as follows

(~(P)t l All (O)a [P(P', ~) c) - x/~(27r)3 ¢ tac[ K l (k2)gvt~ + K2(k2)pv(Pu+P ~) + K3(k2)pv(Pp-P'g) ] ,

(~(P)tl vu(O)a]AI(P',

W)c) - f2(2w)a

etac[Ll(k2)gv~ + L2(k2)pv(pu+P'u) + L3(k2)pv(P~-P'u)],

i Ctacf(k2)(Plz+P~) (~(P)t [ V~(O)a[~(P')c ) = (2~)3

(5)

Here k = p-p' and all the invariant functions are real. Considering the divergence equations, KI(L1) can be expressed [24] by K2,3(L2,3). The calculation of the one-particle contributions shows the following s t r u c t u r e of the spectral function for t ¢ a 2(2~)3 pa~P](p,p,)= p~p~ Tl(k2 ) + (pop~+p,oPx)T2(k 2) + popx T3(k2 )

(6)

and Tl(k2 ) = m~[KI+pp'(K2-K3)]2

- m2(K2-K3) 2 ,

la

p

T3(k2)=-2Kl(K2+K3)

C(PP')2

~(pp,)2

2pp'= m2+m2_k p ~ 2 • Similarly to eq. (6), the contribution of IA1) (ly)) defines the functions

(7)

WEINBERG SUM RULES

33

Si(k 2) (U~'(k2)); i = 1, 2, 3. The functions Si are r e p r e s e n t e d by eq. (7), when m~ ~ m~. and K i ~ Li, while Ui is s i m p l y 2 f 2. 1 In o r d e r to s i m p l i f y our s u m r u l e , the l i m i t Po - ~ will be c a r r i e d out, keeping q ¢ 0 p e r p e n d i c u l a r l y to p . Then k 2 = _q2. In this l i m i t the s u m r u l e t ¢ a h a s the f o r m T1 (_q2) + T2(_q2 ) = Sl(--q2 ) +S2(-q 2) + Ul(--q 2) + U2(--q 2) + D ,

q2 ¢ 0 .

(8)

H e r e D r e p r e s e n t s the d i s c o n n e c t e d contributions O =~t

2t J - T I ( ¢ ° ) + S I ( ~ ) - S 2 ( ° ° ) )

(9)

w h e r e the infinite a r g u m e n t is due to (p+p,)2 '4 o~.

4. EVALUATION OF THE po ~ ¢0 SUM RULE A p o s s i b l e way of evaluating the s u m r u l e (8) is to take the i n v a r i a n t functions Ki, Li, f f r o m ref. [21] w h e r e they a r e d e t e r m i n e d by the aid of Halliday and L a n d s h o f f ' s p r o c e d u r e f r o m c u r r e n t a l g e b r a t a k e n between v a c u u m and ~, p, and A1, r e s p e c t i v e l y . In what follows we shall apply the solutions f r o m ref. [21] except f o r the e x t r a p o l a t e d PDDAC equations (eqs. (13) and (15) of r e f . [20]). Then, all the m a t r i x e l e m e n t s needed can be exp r e s s e d by Brown and W e s t ' s p a r a m e t e r s 42 and ~ [23, 24] 42 = - _ m ~

2F~ G2p '

X = G p ~ =- 1-co .

2Gp

(10)

In our c a l c u l a t i o n s W e i n b e r g ' s [1] f i r s t s u m r u l e

is u s e d t o g e t h e r with m2A1 ; 2m02. The e x t r a p o l a t e d PDDAC equations would i m p o s e the r e l a t i o n [21] 242 + 4w-3 = 0 .

(12)

Now, a s i m p l e c a l c u l a t i o n shows that in the a p p r o x i m a t i o n applied h e r e the d i s c o n n e c t e d contributions a r e exactly c a n c e l l e d e a c h other, D = 0. F u r t h e r m o r e , if q2 = xm2

34

G. POCSIK

t 2(x+1)2 T l ( - q 2 ) + T2(-q2) = i ~- I - ( x - ~ - (242 +4w-3)(~2 + w - l )

+x(~2+w))][X42(6 2

+ 2i m2(w-l) + 1

+ m2+xm--2p-IX-!+ ~(242 +4w-3)((x

+ 1) 2

+ m-p2 (x - 1)) - ~ + ~ (642 +4o)-5) - (1 - x ) ( - 2 f 2 - 4w+5)

,

(13)

and

1 {,~ ( xx+2 2,^ + l - ) (z~2-1)

S l ( _ q 2 ) +$2(_q2 ) + Ul(_q2 ) + U2(_q2 ) _ 2 ~ . ~ × (2~2+4w-3)+

(6~2+4w-5)-2~2+4w-I

1 +~-x+li

(6~2+1200-9+4X(~2+¢0-1))]

[-~2+2w-½ l .^/I+o~XX 2 +lt)l'--l-+x--] "

(14)

F

2 1, 2 and E q s . (14) and (15) w e r e n u m e r i c a l l y e v a l u a t e d f o r s m a l l x, x=], x >2 642 +4w-5 at the v a r i o u s v a l u e s of 42 and X allowed by the p r e s e n t exp e r i m e n t a l p - d a t a , that is roughly 0.9 -< ~ -< 1.4, 0.75 -< )~ -< 1.35. ( F o r a c a r e f u l d i s c u s s i o n of the r e l e v a n t data, see r e f . [24].) T h e four p a i r s 42, ~ fitting m o s t l y the s u m r u l e in the l a r g e s t d o m a i n of q2 a r e indicated in table 1. F r o m (8-10)m 2 the a g r e e m e n t is, in g e n e r a l , f a i r l y good. At l o w e r q2, Table 1 Evaluation of the sum rule {8) for various values of q2

q2

20.

m'~/m2 .~/m2 ~2 =~

(13)

(14)

(13)

(14)

(13)

(14)

(13)

(14)

7.1

11.1

9.5

4.5

7.1

1.3

2

3.5

12.2

4.6

17.2

8.5

7.4

11.4

9.5

6.1

7.6

1.8

1.5

42=1 =0.75

4.4

16.3

7.7

8.5

9.3

10.7

4.7

8.9

2

2

~2 -_ 0.75 )~ = 0.75

4

14.5

12.5

10.2

6.6

12.3

6.4

10.2

1.7

2

k =1 =0.9

10

2.. 9

WEINBERG SUM RULES

35

Table 2 T h e o r e t i c a l p r e d i c t i o n s c o r r e s p o n d i n g the best fit of eq. (8) rp._~ 2~. MeV ~2 = 3~ ), = 1

97

42 = ~ ~, = 0 . 9 42 = 1

I'p__,2f/Fp__,2~ 8.8

f

FA1 ---~p~- MeV

g/mAlh

x 10 -5

0.63

110

h =0

75

10.36 × 10 - 5

0.59

132

-2.75

77

6.66 x 10-5

0.55

83

h =0

104

3.71 x 10 -5

0.55

16

0.25

4

42 = ~ exp. data

93 :~ 15 130 + 5 ref. [26]

( 5 . 5 ± 0 . 7 ) x 10 - 5 (5.8±1 )× 10 - 5 ref. [26]

0.7 + 0.2 ref. [27]

80 ~: 35 ref. [26]

h o w e v e r , t h e s u m r u l e p r e d i c t s t o o s m a l l r n ~ / m 2. T h e f i r s t a n d t h i r d s o l u t i o n s h a v e b e e n s t u d i e d in r e f . [21], w h i l e t h e o t h e r s do n o t s a t i s f y eq. (12), h e n c e t h e y l e a d to n o n - v a n i s h i n g d - w a v e c o u p l i n g c o n s t a n t s . T h e o r e t i c a l p r e d i c t i o n s c o r r e s p o n d i n g t o t a b l e 1 a r e w r i t t e n in t a b l e 2. W e h a v e u s e d t h e v a l u e s F ~ = 188 M e V , m p = 775 M e V , r e a l = /2mp. The first three possibilities are consistent with the rather uncertain exp e r i m e n t a l d a t a . In t h e c a s e 42 = 1.5; ), = 0.9 t h e r a t e of t h e s - a n d d - w a v e couplings became negative. I am grateful to Drs.

F. C s i k o r a n d I. M o n t v a y f o r v a l u a b l e i n f o r m a t i o n s .

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

S. W e i n b e r g , Phys. Rev. L e t t e r s 18 (1967) 507. E. Y. C. Lu, P h y s . Rev, 159 (1967) 1427. S. P. De Alwis, Nuovo C i m e n t o 51A (1967) 846. T. Das, V . S . Mathur and S. Okubo, Phys. Rev. L e t t e r s 18 (1967) 761. S. L. Glashow, H . J . S c h n i t z e r and S. Weinberg, Phys. Rev. L e t t e r s 19 (1967) 139. T. Das, V . S . Mathur and S.Okubo, Phys. Rev. L e t t e r s 19 (1967)470. S. Okubo, R o c h e s t e r p r e p r i n t (1967). J . J . S a k u r a i , Phys. L e t t e r s 24B(1967) 619. T. Das, G.S. Guralnik and V.S. Mathur, Phys. Rev. L e t t e r s 18 (1967) 759. T . D . L e e , S. W e i n b e r g and B. Zumino, Phys. Rev. L e t t e r s 18 (1967) 1029. I.S. Ge.rstein, B.W. Lee, H. T. Nieh and H. J. S c h n i t z e r , Phys. Rev. L e t t e r s 19 (1967) 1064. S. P . De Alwis, C a m b r i d g e p r e p r i n t (1967). T. Das, V . S . Mathur and S. Okubo, R o c h e s t e r p r e p r i n t (1967). C. H. Woo, Phys. Rev. L e t t e r s 19 (1967) 537. S.K. Bose and R. T o r g e r s o n , Phys. Rev. L e t t e r s 19 (1967) 1151.

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G. P~CSIK J. Rothleitner, Nucl. Phys. B3 {1967) 89. F. Capra, O r s a y p r e p r i n t {1968). D. S. Narayan, R . P . Saxena and P. P. S r i v a s t a v a , I C T P - p r e p r i n t s {1967) R. P e r r i n , Phys. Rev. L e t t e r s 20 {1968) 306. F. C s i k o r and I. Montvay, Nucl. P h y s . , to be published. F. C s i k o r and I. Montvay, Nucl. Phys., to be published. I.G. Halliday and P . V . Landshoff, Nuovo Cimento 51A ~1967) 980. S. G. Brown and G . B . West, Phys. Rev. L e t t e r s 19 (1967) 812. S. G. Brown and G. B. West, C o r n e l l p r e p r i n t {1967). T. Das, V . S . Mathur and S.Okubo, Phys. Rev. L e t t e r s 19 (1967) 1067. A . H . Rosenfeld et al., UCRL-8030 (1968). C.W. A k e r l o f et al., Phys. Rev. L e t t e r s 16 {1966) 147.