Sum rule for the isovector magnetic moment of the nucleon

Volume 19, number 8

SUM RULE

FOR

PHYSICS LETTERS

THE

ISOVECTOR

1 January 1966

MAGNETIC

MOMENT

OF

NUCLEON

THE

N. CABIBBO

CERN, Geneva and L. A. RADICATI

Scuola Normale Superiore, Pisa, Italy Received 4 December 1965

Dashen and G e l l - M a n n and Lee [1] have shown that by c o n s i d e r i n g the a l g e b r a g e n e r a t e d by the m o m e n t s of the v e c t o r and axial v e c t o r c u r r e n t s it is p o s s i b l e to deduce an a p p r o x i m a t e r e l a t i o n between the m a g n e t i c m o m e n t and the root m e a n s q u a r e r a d i u s of the proton. This r e l a t i o n is obt a i n e d by taking 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 m a g n e t i c m o m e n t o p e r a t o r s between proton s t a t e s at r e s t . Only s t a t e s of the octet and decuplet a r e kept in the s u m r u l e ; f u r t h e r m o r e the t r a n s i t i o n m a g n e t i c m o m e n t of N into N* is a s s u m e d to have the v a l u e given by SU(6) [2]. In this note we p r e s e n t an exact s u m r u l e obtained f r o m the c o m m u t a t i o n r e l a t i o n s of the e l e c t r i c dipole m o m e n t o p e r a t o r s

Di = f dSx ]L (x) x ,

(1)

where ]io(X) is the t i m e component of the i s o s p i n c u r r e n t (i = 1, 2, 3). F r o m the c o m m u t a t i o n r e l a t i o n s [3] * of the jio(X, O) it follows

[D'~, Di] = 2 f d3xj3(x)(xl)2 .

(2)

If we now take the m a t r i x e l e m e n t of eq. (2) between proton s t a t e s of m o m e n t u m p' and p along the t h i r d axis and s e p a r a t e the one n e u t r o n c o n t r i b u t i o n ** we obtain: E2\

2M ' J 5 ( p - p ' ) = A(p,p') , (3)

where r 2 is r e l a t e d to the i s o v e c t o r c h a r g e f o r m factor of the n u c l e o n by • We note that our results depend only on the commutation relations of the isospin densities, and not on the more controversial parts of the current algebra used in ref. 1. • * The technique used in deriving our sum rule is essentially that of Fubini and Furlan [4].

_~r2 =

dG~

1

(4)

- dq 2 + 8M 2 '

M i s the n u c l e o n m a s s , ~ and ~n the m a g n e t i c m o m e n t s of proton and neiltron, and A(p,p') the i n e l a s t i c c o n t r i b u t i o n to the s u m r u l e :

A(p,p') = ~ o~n

Q~'

[D~[~>+ -


(5) [

lp> •

o/

We s e p a r a t e the [ = ½ and I = ~ c o n t r i b u t i o n s i n the i n t e r m e d i a t e s t a t e s , and u s e i s o s p i n inv a r i a n c e to w r i t e

A(p,p') = 4 A l ( p , p ' ) - 2A3(p,p' ) ,

(6)

where

and A 1 is a s i m i l a r e x p r e s s i o n with the s u m extended to all [ = ½ s t a t e s different f r o m the p r o ton. The m a t r i x e l e m e n t s of eq. (7) can be c a s t i n the f o r m


Ep) "

(8)

Using c u r r e n t c o n s e r v a t i o n , we obtain:

<~[b311p> =
(9)

Ij3(o)Ip>.

It i s c o n v e n i e n t to c o n s i d e r the l i m i t of eq. (3) for # going to infinity. In this l i m i t the m a t r i x e l e m e n t of eq. (9) i s p r o p o r t i o n a l to the m a t r i x e l e m e n t for the p h y s i c a l p r o c e s s ~, + p ~ / 3 ; A 3 and A 1 can then be e x p r e s s e d a s i n t e g r a l s 697

Volume 19, number 8

PHYSICS

over the total cross sections a: and a’ for photoproduction on protons of I = $ and 1 =I$ states through the isovector part of the electromagnetic current. The proof of this statement is similar to Adler’s derivation of the sum rule [5] for GA/GV. We thus finally obtain n pp;Ln ’ - a:) = :r2 .(lO) sY(2c: > +&

(

We assume that the integral converges which implies that 2ay - a: should vanish as w - Q). To show that this is indeed a plausible assumption let us consider an I = 1 vector meson (e.g., the p-meson) whose source is the isospin current *. In this case the quantity 2ay - a: is proportional to the difference between p+-tp and p- - pvextr+polated to zero p mass. The vanishing of 2aI - a3 would thus be ensured by Pomeranchuk’ s theorem. An approximate evaluation of the integral is obtained by neglecting a’ and assuming that uy a$ a) resonance. We thus is dominated by the (2, obtain /_lp- j_ln= 5.5

LETTERS

1 January

We obtain a sum rule similar to that of eq. (10) namely: ~/~[~V(~+nO-I=O)+uV(y+a+~I=l)+ - &V(,+lro

(y

(b lb;la)lpe+de3 =$(b /@f141pe=o

.

(11)

Eq. (2) can be used to derive other sum rules. Let us take for example the matrix element of eq. (2) between ?I+ states. Since the pion has no magnetic moment we do not get in this case any contribution from one pion intermediate states. * The assumption of a link between the p meson and the isospin current is not new [e.g. 61. ** Since the matrix elements of moments of current and charge distribution involve derivatives of spinor expressions, their value depends on the spinorial basis which is chosen. The problem is that the relation among the spinors at momentum p + q and those at momentum p is to a large extent arbitrary. A relation like eq. (11) is only true in a well defined basis, but this is sufficient for our argument, since final results such as eq. (10) do not of course depend on the basis chosen. One of us (N’. C.) acknowledges an interesting exchange with Gourdin on this point. 698

- I = 2) = +; 1

. (12)

It is interesting to investigate the analogue of the Dashen - Gell-Mann - Lee relation, i.e., a relation obtained from (12) by keeping in the integral only terms coming from the resonant production of vector mesons belonging to the “35”. The main contribution comes then from y+lTow, and can be computed from the rate of the inverse process w -+ go + y. From the experimental limit [7] on q - no + y we can see that the cp contribution to eq. (12) will be at most one tenth of the w contribution; since cp - ITO+ y is forbidden in SU(6) the cp contribution will probably be even smaller, and we will neglect it. We also note that y + no - p” involves the isoscalar current, and does not contribute to eq. (12). Such a truncated sum rule gives rise to the relation:

+2!

to be compared with the experimental value 4.7. A more accurate evaluation is under way. The sum rule (10) reduces to the relation of ref. 1 if we limit the sum over the intermediate states in eq. (5) to the ($, $) resonance considered as a particle and we take the SU(6) result for the N- N transition. The relation between the two results depends upon the fact that if a and b are particles with the same mass, we have ** (by a Lorentz transformation):

1966

(-fK!!L3 > r(w-nOY) M”,_M;

*

(13)

Using the experimental number [7]: F(w” neutrals) = 1.3 f 0.3 MeV, and assuming wo -+ no + y to be the only neutral decay mode of w”, eq. (13) would predict a pion radius 7B = 0.5 fm, somewhat smaller than the known proton radius, yp = 0.8 fm. A p-dominance model for the pion form factor would give Y~ = ,J6/Mp = 0.63 fm a value reasonably close to our estimate. Unfortunately we do not have experimental information on the pion radius or on other contributions to eq. (12). As a final comment, we note that other relations similar to eq. (11) can be established between the matrix elements of odd parity operators between states of finite momentum, and those of even parity operators between states at rest. . An interesting example is the following: if g;(r) is one of the nine axial currents, a and b two particles of equal mass, we find that (b,pl g:(O)\ a,& = i (b, 0 I{P*&O)]l

a, 0) .

This means that, within a set of equal mass particles, there is a relation between the matrix elements of the algebra of chinzl SU(3) @ SU(3) and those of collinear SU(3)@ SU(3). Jf the matrix

Volume 19. number 8

P H Y S I C S L E T T ERS

e l e m e n t s of the c o m m u t a t i o n r e l a t i o n s of the f i r s t a l g e b r a b e t w e e n s t a t e s of l a r g e m o m e n t u m b e l o n g ing to a c e r t a i n s e t of equal m a s s p a r t i c l e s a r e s a t u r a t e d by i n t e r m e d i a t e s t a t e s belonging to the s a m e set, these s t a t e s would s a t u r a t e the s e c o n d a l g e b r a a s well. Since the collinear * SU(3) ® SU(3) g e n e r a t e d by ki and xi (a. 15) i s a subgroup of SU(6), it is then not s u r p r i s i n g tha~ s o m e of the SU(6) r e s u l t s can be obtained by r e q u i r i n g the c h i r a l SU(3) ® SU(3) a l g e b r a to be s a t u r a t e d , in the l i m i t of l a r g e m o m e n t u m , by the b a r y o n octet and decuplet [9]. One of u s (L. A. R. ) w i s h e s to thank the Theory D i v i s i o n of CERN for the h o s p i t a l i t y extended to • The collinear SU(3) ® SU(3) to which our argument applies is a subgroup of the U(6) x U(6) generated by axial and vector currents [8]. The collinear group has been studied in particular by H. Ruegg and D.V.Volkov, to be published in Nuovo Cimento.

1 January 1966

h i m d u r i n g the s u m m e r of 1965 when this work was begun.

References 1. B.W. Lee, Phys.Rev. Lettsrs 14 (1965) 676; R. F. Dashen and M. Gell-Mann, Physics Letters 17 (1965) 145. 2. M.A.B.B~g, B.W. Lee and A.Pais, Phys. Rev. Letters 13 (1964} 514. 3. M.Gell-Mann, Phys.Rev. 125 (1962) 1067. 4. S.Fubini and G. Furlan, Physics 1 (1965) 229. 5. S. Adler, Phys. Rev. Letters 14 (1965} 1047; W.Weisberger, Phys.Rev. Letters 14 (1965) 1051. 6. J.Sakurai, Ann. Phys. 11 (1960) 1. 7. A. H. Rosenfeld, A. Barbaro--Galtieri, W.H. Barkas, P. L. Bastien, J.Kirz and Marts Roos, Revs.Modern Phys. 36 (1964) 977. 8. R. P. Feynman, M.Gell-Mann and G. Zweig, Phys. Rev. Letters 13 (1964} 678. 9. Results of this kind have been obtained by I. Gerstein and by S. Bergia and F. Lannoy.

*****

REAL PART OF THE ZERO-ANGLE ELASTIC PION-NUCLEON SCATTERING AMPLITUDE AND THE CHARGE EXCHANGE CROSS SECTION AT HIGH ENERGIES V. S. B A R A S H E N K O V Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, Dubna

Received 2 December 1965

In the paper by Foley et al. [1] it was e s t a b l i s h e d that the m e a n e x p e r i m e n t a l v a l u e s of the r e a l p a r t of the ~ - - p s c a t t e r i n g amplitude D ( T ) at e n e r g i e s T ~ 10-20 GeV d e c r e a s e m u c h f a s t e r than the d i s p e r s i o n r e l a t i o n s p r e d i c t [2, 3]. However in r e f s . 2 and 3 it was pointed out that the v a l u e s of D± (T) at T > 10 GeV should be c o n s i d e r e d only a s tentative s i n c e the a s y m p t o t i c b e h a v i o u r of the e m p l o y e d total 7r~-p i n t e r a c t i o n c r o s s s e c t i o n s ~±(T) has been chosen v e r y app r o x i m a t e l y (while c a l c u l a t i n g it was a s s u m e d that ~±(T) = const, a l r e a d y at T >/ 25 GeV). The o b s e r v e d d i s c r e p a n c y with e x p e r i m e n t , and f a r m o r e a c c u r a t e data on the c r o s s s e c t i o n s ~ . ( T ) r e c e n t l y obtained, i m p e l l e d us to make a m o r e careful theoretical analysis. New d i s p e r s i o n c a l c u l a t i o n s w e r e made on the JINR c o m p u t e r s u n d e r the a s s u m p t i o n that the

a s y m p t o t i c s of the lr-N c r o s s s e c t i o n s at T > 19 GeV, on which t h e r e a r e no e x p e r i m e n t a l data, is of the f o r m

a±(T) = ~o

+

C*/TK ;

(1)

the c o n s t a n t s %, K, C+ w e r e v a r i e d in a wide r a n g e u n d e r the condition that at T = 19 MeV the v a l u e s c a l c u l a t e d by eq. (1) coincide with the m e a n e x p e r i m e n t a l c u r v e s ~+(T)exper" and (T)ex~ r The c ~ 6 u l a t i o n s have shown the following: 1. At T ~ 10-30 GeV a v a r i a t i o n of the p a r a m e t e r s ~o, ~, C~_ can change the v a l u e s of D~(T) by 1.5 - 2 t i m e s , however, in all c a s e s the theo r e t i c a l c u r v e s drop s l o w e r with i n c r e a s i n g e n e r g y than the m e a n e x p e r i m e n t a l v a l u e s of ref. 8. Best a g r e e m e n t with e x p e r i m e n t i s achieved if ao = 2 2 - 23 m b and K ~ 0.5 (see fig. 1). -

699