A new family of sum rules from current algebra

Volume 22, number 5

A NEW

PHYSICS LETTERS

FAMILY

OF

SUM

RULES

15 September 1966

FROM

CURRENT

ALGEBRA

D. AMATI

CERN, Geneva and Istituto di Fisica Teorica dell Universitt~, Trieste R. JENGO tstituto di Fisica Teorica dell'UniversitY, Trieste arid E. REMIDDI

International Centre for Theoretical Physics, Trieste Received 18 July 1966

On the basis of current commutation relations a new family of sum rules is obtained. They involve invariant functions and integration variables different from those appearing in the Dashen, Gall-Mann Fubini sum rules. The study of the singularities in the redundant variables allows to obtain conditions on the nucleon form factors.

In t h i s note we p r o p o s e a new f a m i l y of sum r u l e s b a s e d on the a l g e b r a of c u r r e n t s . We d i s c u s s a l s o s o m e of i t s p h y s i c a l i m p l i c a t i o n s , d e f e r r i n g to a n o t h e r p a p e r a thorough a n a l y s i s of the d e r i v a t i o n and the r e s u l t s that can be obtained on t h e i r b a s i s . We s t a r t f r o m the equal t i m e c o m m u t a t i o n r e l a t i o n of (vector o r a x i a l ) c u r r e n t s

w h e r e the i n d i c e s a,/3, ~ r e f e r to i s o s p i n (or SU3) and p a r i t y t r a n s f o r m a t i o n p r o p e r t i e s . The l a s t t e r m in the r i g h t hand side i s the Schwinger t e r m (i = 1, 2, 3) and 0 a[3 could be an o p e r a t o r . We will f r e q u e n t ly d r o p the a , ~ i n d i c e s in o r d e r to s i m p l i f y the notation. L e t us c o n s i d e r the m a t r i x e l e m e n t of the c o m m u t a t o r of two v e c t o r c u r r e n t s between two s t a t e s of m o m e n t a P l and P2 and i t s F o u r i e r t r a n s f o r m

' I 'J~ a ( '~x) , Jv"fl (-~x) ' 1 t P l ) d4x t ~fl = 3f e i K X ( p 2 L

(2)

and l e t u s i n t r o d u c e the following k i n e m a t i c a l notation: 1

2

2

1 P = ~(pl+P2 ),

k 1 =K _A,

S = (Pl+kl)2,

V = 2KA = ~(k 2 - k l ) ,

A = ~1 ( p l - P 2 ),

k 2 =K+A,

t = 4 A2,

W

=K2+A 2

l ' k 2 + k2). =~ 1

A s s u m i n g f o r s i m p l i c i t y that the s t a t e s ~ p l ) and I P2) r e p r e s e n t s p i n l e s s p a r t i c l e s with the s a m e m a s s , t~v can be d e v e l o p e d as:

tlzu = al P# Pv +a2 P# A v +asP# Kv + blA # Pv + b2 A~ Av+ b3 hDKv + cl K~ Pv + c2Kp ~ + c3K~Kv + d S~v, 2 2

(3)

w h e r e ai, ~ , c i, d a r e s c a l a r functions of s, t , k l , k 2. S i m i l a r l y , we w r i t e

(P21J~ (O)IPl)= F~(t)Pv + F~(t)Av, (P210aB lPl) = C a#(t). 674

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Volume 22, number 5

PHYSICS

LETTERS

15 September 1966

We m a k e no u s e of p a r t i c u l a r p r o p e r t i e s of c u r r e n t s s u c h a s c o n s e r v a t i o n , t i m e r e v e r s a l i n v a r i a n c e e t c . T h o s e p r o p e r t i e s can g i v e r i s e to r e s t r i c t i o n s on the i n v a r i a n t f u n c t i o n s in e q s . (3) and (4), i . e . , F 2 (t) = 0 and a r e l a t i o n a m o n g the ai, bi, and c i in eq. (3). T w o m e t h o d s h a v e b e e n s u b s t a n t i a l l y u s e d in o r d e r to d e r i v e s u m r u l e s f r o m c o m m u t a t o r s : the u s e of p ~ ~o f r a m e of r e f . 1 a n d the d i s p e r s i o n r e l a t i o n a p p r o a c h [2]. With both t h e s e m e t h o d s the g e n e r a l s u m r u l e t h a t h a s b e e n d e r i v e d (which we s h a l l c a l l D a s h e m G e l l = M a n n - F u b i n i s u m r u l e s ) and can be w r i t t e n as

~ f a l { 3 ( s ' t,

4~r'azlfal3(s,t

k 2l , k 22) d s

=fot~,FYl(t) '

,k1,2 k~)ds= Ga~(t).

T h i s s u m r u l e c o n t a i n s p a r t i c u l a r c a s e s that h a v e b e e n p r o p o s e d , f o r i n s t a n c e , the A d l e r - W e i s b e r g e r [3] c a s e w h e n the c u r r e n t s in (1)_are a x i a l _ a n d k12 = k 2 = t - 0 ; the C a b i b b o - R a d i c a t i [4] s u m r u l e when both c u r r e n t s a r e v e c t o r with, k~ = k 2 = k 2, t = 0 and d e r i v i n g e q s . (5) with r e s p e c t to k 2 at k 2 = 0 [5]. In o b t a i n i n g e q s . (5) one m a k e s the a s s u m p t i o n t h a t the i n t e g r a l s c o n v e r g e . T h i s can be p h r a s e d in d i f f e r e n t but e q u i v a l e n t w a y s : t h a t t h e r e i s a l i m i t e d (or c o n v e r g e n t ) s e t of i n t e r m e d i a t e s t a t e s that s a t u r a t e the s u m r u l e o r t h a t the a m p l i t u d e 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 . T h i s c o u l d h a p p e n of c o u r s e f o r only s o m e of the i n t e g r a l s of e q s . (5). U s i n g m e t h o d s v e r y s i m i l a r to t h o s e that a l l o w the d e r i v a t i o n of the D G - M F s u m r u l e s , one can o b t a i n s e v e r a l o t h e r s w h i c h i n v o l v e a d i f f e r e n t v a r i a b l e of i n t e g r a t i o n a s w e l l a s d i f f e r e n t c o m b i n a t i o n s of i n v a r i a n t f u n c t i o n s * . O f t h e s e n e w f a m i l i e s of s u m r u l e s we obtain, we want to d i s c u s s h e r e one that, in o u r p o i n t of v i e w , i s of p a r t i c u l a r i n t e r e s t , i . e . ,

a~'y 'y

f

t,,:,.) d,, --:

r 2 (t),

(6)

T o v i s u a l i z e b e t t e r the m e a n i n g of t h e s e s u m r u l e s we can say t h a t w h i l e t h e D a s h e n G e l l - M a n n F u b i n i s u m r u l e s can be o b t a i n e d by i n t r o d u c i n g a c o m p l e t e s e t of s t a t e s b e t w e e n the two c u r r e n t s in the m a t r i x e l e m e n t (P2[ ~u (½x), j, (-½x) i Pl) in a P ~ ~o s y s t e m , o u r s u m r u l e s c o m e by u s i n g an a n a l o g o u s • r v • 1 • 1 p r o c e d u r e m the m a t r x x e l e m e n t (PlP21 [3p (~x), J u (-~x) ] 10 )" By v a r y i n g

S

at f i x e d v a l u e s of k21, k 2 and t in e q s .

(5) we p i c k up a l l the c o n t r i b u t i o n s of i n t e r m e d i a t e

s t a t e s in the s and u c h a n n e l s in the c o m m u t a t o r of eq. (1), i . e . , a l l the c o n t r i b u t i o n s of the f o r m * We shall discuss in a subsequent paper the relation between the different methods of deriving sum rules. We shall show that an equivalent and simple way to obtain all the sum rules is to start from the F o u r i e r transform of the matrix elements of eq. (1). In this case one obtains [6]

!27r f t ov (K, Ko) d K

o

= Fl(t)

PU + F2(t)Av+ G¢)K v 5yi

This equation must hold for every f r a m e of r e f e r e n c e and for every K. All the sum rules can be obtained from it, in particular the ones of eqs. (5) using a frame of reference in which Pz--* ~, and those of eqs. (6) for a frame in w~ich AZh'-*Qo The a r b i t r a r i n e s s in the choice of K implies the independence of the right hand side of eqs. (5) on k~ and k~, and of eqs. (6) on w and s.

675

Volume 22, number 5

PHYSICS

LETTERS

15 September 1966

(7)

Pl

P2

Pl

P2

w h e r e , a s u s u a l , a s l a s h e d i n t e r m e d i a t e l i n e in a d i a g r a m i n d i c a t e s the s u m m a t i o n o v e r s p i n s of p o s i t i v e e n e r g y s t a t e s and not the whole F e y n m a n p r o p a g a t o r . On the. o t h e r . h a n d , w h e n we v a r y v , in e q s . (6) k e e p i n g f i x e d s , I and w, we s w e e p all the c o n t r i b u t i o n s in k~ and kS, so that we p i c k up a l l the s t a t e s t h a t can c o n n e c t e i t h e r of the c u r r e n t s with the v a c u u m . D i a g r a m m a t i c a l l y , we p i c k up a l l the c o n t r i b u t i o n s of the f o r m

~,

k,

L

k,

(8)

Of c o u r s e , a s in the D a s h e n - G e l l - M a n n - F u b i n i s u m r u l e s , we have i m p l i c i t l y a c c e p t e d that only a r a p i d l y c o n v e r g e n t s e t of s t a t e s would a c t u a l l y c o n t r i b u t e . T h i s a s s u m p t i o n i s p r o b a b l y m o r e j u s t i f i a b l e in our s u m r u l e s that t h o s e of D a s h e n G e l l - M a n n F u b i n i , the r e a s o n f o r t h i s b e i n g t h a t while in o u r s only the s t a t e s with a d e f i n i t e a n g u l a r m o m e n t u m can c o n t r i b u t e (J = 1 due to the f a c t that we a r e c o n s i d e r i n g v e c t o r and a x i a l c u r r e n t s ) , in the D a s h e n G e l l - M a n n Fubini s u m e r u l e s the i n t e r m e d i a t e s t a t e s can h a v e any a n g u l a r m o m e n t u m . I n c r e a s i n g s , m o r e and m o r e a n g u l a r m o m e n t a can in p r i n c i p l e c o n t r i b u t e . We s h a l l c o m e b a c k l a t e r to the p r o b l e m of the spin of i n t e r m e d i a t e s t a t e s . In e q s . (5) and (6) t h e r e a r e two v a r i a b l e s in the l e f t hand side which do not a p p e a r in the r i g h t hand s i d e . T h i s i m p l i e s that fbi(s, t, v, w)d. i s a c o n s t a n t with r e s p e c t to s and w o r , in o t h e r w o r d s , that it h a s no s i n g u l a r i t i e s in t h o s e v a r i a b l e s . If, f o r i n s t a n c e , the s i n g u l a r i t i e s in s of bi(s , t, v, w) a r e e x p r e s s e d by:

hi(s, t, v, w) = ~ then e q s .

f

bi, s(S', t,

w) ds' s'---~--'

(9)

(6) i m p l y

f

bi, s (s, t, v, w) dv = 0

(tOa)

f bi, w(S , t, v, w)dv : 0

(10b)

and e q u a l l y

f o r e v e r y v a l u e of s, t and w. The analogoues relations 2

2

2

( s , t , k 1, k 2 ) d s = f a i . k 2 ( s , t , k 1, k 2 ) d s = 0

(11)

o b t a i n a b l e f r o m (5) h a v e b e e n i n v e s t i g a t e d by F u b i n i e t a l . [7]. It i s c l e a r t h a t e q s . (10) and (11) a r e not r e a l l y r e l a t e d to the a l g e b r a of c o m m u t i n g c u r r e n t s but contain only the l o c a l c o m m u t a t i v i t y - m i c r o c a u s a l i t y c o n d i t i o n s - of the c u r r e n t o p e r a t o r s . 676

Volume 22, number 5

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15 September 1966

Eq. (lOa) can be r e p r e s e n t e d by the d i a g r a m m a t i c a l equation

k,

9 = 0

(195

for ever~y state Ira}. The b r a c k e t of eq. (125 contains a f o r m f a c t o r for the transition (PeiJ,, I rn) evaluated at k S = w - v t i m e s the absorptive part of a f o r m f a c t o r for the transition (mljvlpl} a'~ k22 = v + w, minus m e same e x p r e s s i o n in which the absorbitive p a r t is shifted f r o m the f i r s t to the second f o r m f a c t o r . The integral o v e r v of such an e x p r e s s i o n m u s t vanish by virtue of eq. (10a). F o r instance, let us fix the p a r t i c l e s 1 and 2 to be pions, and let us consider s = mz in eq. (10a), i.e., take the i n t e r m e d i a t e state I re)in the d i a g r a m (12) to be a pion. If we call F1r(k z) the pion f o r m factor, eq. (10a) r e a d s

f CIm

ld = o.

Eq. (135 is of c o u r s e automatically satisfied. Let us now c o n s i d e r s = m 2 , i.e. let !re)be an ~0-meson. Due to the spin 1 of the co, we shall have an e x t r a m o m e n t u m coming f r o m the sum over the w spin components, so that we obtain, b e s i d e s the analog of eq. (135, t r i v i a l l y satisfied, the condition

fv[Im Fw,(k~)Fwn(k22 5- Fc01r(k21)Im Fw~r(k~)]dv= O,

(1 4)

where F~0~r(k2) is the f o r m f a c t o r for the to ~ r ~ , transition. Let us write for Fw~r(k2) the r e p r e s e n t a t i o n

F~(~95 -- a + ~ f

Im Fto~(k'2) dk,2"

(15)

k,9_h9

It is easy to v e r i f y that satisfying eq. (14) for e v e r y w implies F (1) = F (0) F (2) = O, (D'/T

00~"

(16)

C0~T

where F(0) a ' c o=T r

F(1)=wTr-lfImFwlr

(k2)dk2'

F(2)=lfk2ImF~lr(k2)dk2"co~r

(17)

Of c o u r s e , we need that the i n t e g r a l s converge; we shall come back later to this problem. In a pole model, in which Fwlr(k 2) would be given by •

2--

z ~i Eq.

k2

(18)

'

-

(16) implies that the sum of the coupling constants vanishes, i.e., ~ •g i $

= 0. The quantities F COT (i)

of eqs. (17) have a simple meaning in the asymptotic expansion of Fw?r(k2) for k 2 -~ co. Indeed F (1) ~¢o?r +

F(25

k2 + - - ~ - +

"'"

(19) 677

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Choosing the i n t e r m e d i a t e s t a t e [ m > to have h i g h e r spin J, o t h e r e x t r a m o m e n t a p r e s s i o n s giving r e l a t i o n s a n a l o g o u s to eq. (14) with h i g h e r p o w e r s of v. If the sum c o n v e r g e , m o r e and m o r e c o e f f i c i e n t s of the a s y m p t o t i c e x p a n s i o n m u s t be z e r o . A (1) =. . . . . . F (j~ J ) =0, which i m p l i e s that Fj~(k 2) ~ 0 (k2)-(J ± 1). Again we Fj~(0) = F j~

e n t e r into the e x r u l e s obtained s t i l l s i m p l e solution i s m e e t the r e q u i r e -

ment that the involved integrals converge. This would be of course the case if a limited set of states saturates the sum rules. In this case, indeed, Im Fj~(k2) would be different from zero only in a finite range of k2. We note, however, that the smaller J, the weaker the requirements on the convergence of the sum rules of eq. (10a). Therefore it could happen that some of those are meaningful even if the whole sum rule (6) is not. This reflects the already stated fact that the basis for eqs. (10) is quite wider than that for eqs. (6). We have, up to now, investigated eq. (10a). It is easy to verify that eq. (10b) is automatically verified. The w dependence of the sum rule appears indeed always in polynominal form and has been exploited in obtaining eqs. (16). It is amusing to remark the complementary way in which the angular momentum J of intermediate states appear in our and in the Dashen-Gell-Mann-Fubini sum rules. Indeed while the k 2 and k 2 singularities involve states with a fixed J (1 for vector and axial currents) the s-singularitiesinvolve a~y J. Therefore in our sum rules the integral runs over states with a definite J and the study of s-singularities allows us to reach (one by one) any J. In the Dashen-Gell-Mann-Fubini sum rules,, the situation is just opposite: the integral runs over every J in the s-channel, while the study of k~ and k~2 singularities involve only states with a definite J. Let us now consider the case in which particles 1 and 2 are nucleons. In this case the expansion of eq. (3) is no longer complete, due to the spin of the external particles. Correspondingly, the sum rules assume a different and more complicated form, because of the Lorentz transformation properties of the Dirac spinors. The procedure described before* can be generalized by decomposing eqs. (3) and (4) into invariants that include y-matrices, and then considering the matrix elements of eq. (I) in the system Az --* ~obetween states of definite helicity. Again, the right hand side of the sum rules is free from singularities in s, so that the singularities in s of the left hand side must be equated to zero. This condition for s = m2N (i.e., nucleon i n t e r m e d i a t e state Ira> in the d i a g r a m (12)) i m p l i e s that s o m e i n t e g r a l s involving nucleon f o r m f a c t o r s m u s t vanish. Let us again c o n s i d e r v e c t o r c u r r e n t s , both i s o v e c t o r or i s o s c a l a r . Defining, a s u s u a l ,

{P21Jv(O)[Pl>=i ~2 IFl,u +2--~uF2~up(P2- 'l)plU l, we obtain

f[(ImFl(k21)+ ImF2(k~)) from the coefficient of

F2(k~)- (F1(k21) + F2(k~))ImF2(k~)Idv=O

ApPv in the expansion of t~v

f v I (ImFI(k~) +ImF2`k~))F2(}~) from the coeeficient

(20)

(FI(k~)+F2(k~))ImF2(k~) 1 dv =0

(21)

of AI~vKand f v I l m F2(k21)F2(k~) - F2(k~)Im F2(k22)I dv : 0

(22)

from ApAyK. Other invariants give linear combinations of eqs. (20), (21) and (22). Assuming for F I , F 2 expressions analogous to eq. (15), we obtain

F~I) = F~O) F~I)

_(0)

=/~2

F~2) = F~O) F ~ 2 ) +

F~2)

F~O) = o

w h e r e the F~i)2 a r g , ~ e f i n e d a s in eqs. (17). The condition F~. = 0 i m p l i e s that if we a n a l i z e F2(k2) in the p o l a r f o r m

* See footnote on page 675. 678

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Volume 22, n u m b e r 5

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F2(k2) = a +

LETTERS

15 S e p t e m b e r 1966

gi i

•2

- k2'

t h e s u m o v e r t h e r e s i d u a of t h e P a u l i m a g n e t i c f o r m f a c t o r m u s t v a n i s h , i . e . , ~ g z

= 0. W e k n o w t h a t

$

t h i s r e l a t i o n i s v e r y w e l l s a t i s f i e d i n a l l a n a l y s i s of t h e i s o v e c t o r o r i s o s c a l a r f r o m f a c t o r s [ e . g . 8], n e a r l y i n d e p e n d e n t l y of t h e m a s s of t h e s e c o n d v e c t o r r e s o n a n c e w h i c h i s a l w a y s n e e d e d i n t h e f i t s . T h e m o s t s t r a i g h t f o r w a r d s o l u t i o n of t h e w h o l e s y s t e m 1 s e e m s a l s o c o m p a t i b l e w i t h e x p e r i m e n t a l d a t a [ e . g . 8]. A s d i s c u s s e d p r e v i o u s l y , t h e f a c t t h a t t h e r i g h t h a n d s i d e of e q s . (6) i s a f u n c t i o n t h a t d e p e n d s on l e s s v a r i a b l e s t h a n t h e i n t e g r a n d in t h e l e f t h a n d s i d e m a k e s u s e o n l y i n a v e r y l i m i t e d w a y of e q . (1). In o r d e r t o s a y m o r e t h a n t h a t , we m u s t u s e e x p l i c i t p r o p e r t i e s of t h e c u r r e n t m a t r i x e l e m e n t (4) w h i c h a p p e a r s i n t h e r i g h t h a n d s i d e of e q s . (6). A p a r t of t h i s i n f o r m a t i o n c a n b e e x p r e s s e d i n t e r m s of t h e s i n g u l a r i t i e s i n t of t h e c u r r e n t m a t r i x e l e m e n t s *. F o r i n s t a n c e , we e x p e c t b o t h s i d e s of e q s . (6) t o s h o w t h e p - p o l e f o r The equality of t h e r e s i d u a a t t h e t - s i n g u l a r i t y c o r r e s p o n d i n g t o a n y s t a t e I r ) , on b o t h s i d e s of t h e m ~"a t r i x e l e m e n t of eq. (1), g i v e s r i s e to a n e q u a t i o n of t h e f o r m

t =rn2.

(PlP2[r)(rl IJo(½X),Jv(-½xI I0)5(xo) :(PlP21r)(rl Ifaf3"j~(O) + i Oa~ !ox~ 5~21 84(x)[°)" We s e e t h e r e f o r e t h a t t h e p r o b l e m i s r e d u c e d t o a s i m p l e r o n e i n w h i c h o n e c o n s i d e r s t h e m a t r i x e l e m e n t s of t h e s a m e c o m m u t a t o r b e t w e e n t h a t p a r t i c l e a n d t h e v a c u u m .

References 1. s. Fubini and G. F u r l a n , P h y s i c s 1 (1965) 229. R. F. Dashen and M. Gell-Mann, CALT-68-65 (1966, Coral Gables Conference). 2. S. Fubini, G. F u r l a n and C . R o s s e t t i , Nuovo Cimento 40A (1965) 1171. S. Fubini, Nuovo Cimento 43A (1966) 475. 3. W . I . W e i s b e r g e r , Phys. Rev. L e t t e r s 14 (1965) 1047. S . L . A d l e r , P h y s . Rev. L e t t e r s 14 (1965)1051. 4. N. Cabibbo and L . A . R a d i c a t i , P h y s i c s L e t t e r s 19 (1966) 697. 5. C.Bouchiat and P h . M e y e r , p r e p r i n t O r s a y T h / 1 4 3 (1966). F. Buccella, G. Veneziano and R . G a t t o , Nuovo Cimento 42A (1966) 1019. 6. C.G. Bollini and J. J. Giambiagi, U n i v e r s i d a d de Buenos A i r e s , p r e p r i n t . 7. S. Fubini and G. SegrS, Nuovo Cimento (to be published). V. De Alfaro, S. Fubini, G. F u r l a n and C . R o s s e t t i , P h y s i c s L e t t e r s 21 (1966) 576. 8. F . M . Pipkin, P r o c . Oxford Conf. on E l e m e n t a r y p a r t i c l e s (1965), p. 61. * Let us r e m a r k that in our c a s e the s i n g u l a r i t i e s lie in the region in which we find our sum r u l e s - t i m e l i k e A _ while for the D a s h e n , G e l l - M a n n , F u b i n i sum r u l e s a continuation f r o m the spacelike A region was needed. Indeed, the t - s i n g u l a r i t i e s appear in the Dashen, Gell-Mann, Fubini sum r u l e s in a l e s s natural way. * * * * *

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