Why are the isoscalar neutral current axial-vector couplings and isoscalar nucleon anomalous moments small?

Volume 88B, number 3,4

PHYSICS LETTERS

17 December 1979

ISOSCALAR NEUTRAL CURRENT AXIAL-VECTOR COUPLINGS AND ISOSCALAR NUCLEON ANOMALOUS MOMENTS SMALL? ~

WHY ARE THE

S. ONEDA and T. TANUMA Center for Theoretical Physics, University of Maryland, College Park, MD 20742, USA and Milton D. SLAUGHTER Theoretical Division, Los Alamos Scientific Laboratory, University o f California, Los Alamos, N M 87545, USA

Received 21 August 1979

By using an argument based on the realization of chiral current algebras, we explain why the isoscalar neutral current axial-vectors,~,c°uplingands,~ , t hisoscalar e nucleon anomalous moments are small. We give an argument which predicts, (G~(O))p/(G~(O)) n = (H~(O)),/(H~(O)) n = kp/k n ~ 1.79/-1.91, where kp and k n are the anomalous moments of the proton and neutron, respectivelY. (G/~(0))p and (HN(0))p are the weak neutral current axial-vector coupling and the induced pseudoscalar coupling for the proton, respectively.

The standard phenomenological analysis of weak neutral currents for hadrons is remarkably consistent with the predictions of the Weinberg-Salam (W-S) model [1] in which only the u and d quarks are considered. In this model the effective neutral current is given by J ~ = J~ - 2sin20wJeUm,

(1)

where J~ is the isovector V - A current, Je~m is the electromagnetic current and 0 w the Weinberg angle. No isoscalar axial-vector current appears in (1). However, in a more realistic model we certainly expect the appearance (with full strength) of the isoscalar axial-vector current. For example, in the celebrated model of Glashow et al. [2] constructed using quarks of four flavours, the neutral hadronic current is given by 1 It

J ~ = J~ - 2sin20wJe~m + ~Jc - - ~ J s~"

(2)

Je~ and Js~ are the isoscalar V - A currents of the charmed and strange quarks and in the language of flavor symmetry they involve the currents V~, V~5, A~ and A~5. In addition, there may also be currents involving only heavy (t, b .... ) quarks. Schematically Research supported in part by the Department of Energy.

the currents may look like .(flu - dd) + (~c - ~s + tt - bb) + .... In the test of the W - S model, one discards the terms (~c - ~s + tt - t~b) + ... on the grounds [3] that their hadronic matrix elements are small, probably because the hadrons under consideration consist primarily of u and d quarks. Therefore, here we once again introduce a quark-line rule. From the standpoint of pure group theory, it is rather mysterious why the currents A ~, A ~5, V~, V~5, ... which appear in the primary lagrangian with coupling constants a priori comparable with those of A~ and V~ do not produce appreciable contributions.As recently stressed by Wolfenstein [4], while some estimates on the matrix elements o f A ~ were made by using SU(6) [5], SU(3) [6] or n-pole dominance [4], there is a completely uncertain contribution from A ~5" Collins et al. [7] applied QCD perturbation theory and estimated the contribution of the currents (~c - ~s + tt - bb) + .... They estimated the strength of the effective axial-vector baryon number neutral current coupling induced by strong interactions as A~ ~ X (t~Tt~75u - dT~75d) + 0.05 (~7u3,5 u + dT~'5d). However, as they mentioned [7], it is questionable whether the strange quark is heavy enough to justify their calculations. (For an estimate of the higher order 343

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induced isoscalar axial-vector neutral current in gauge theory but neglecting the contribution of heavy quarks, see ref. [8].) Therefore, it is worthwhile to add a different argument which can predict the magnitudes of the isoscalar neutral current axial-vector couplings. As experiments progress, we then hope to test the predictions. The result adds further evidence to the claim [9] that the quark-line rule is intimately connected with certain realization patterns of chiral algebras. Our argument is based on the hypothesis [9] of asymptotic level realization of SU(3) in the well-known chiral SU(2) ® SU(2) charge algebra

[An+,An- ] = 2V3,

(3)

and also of SU(2) in the related charge-charge and charge-current algebras such as [[A3,An + ] , A n - ] = 2A3,

(4)

17 December 1979

s~ state, we obtain the dynamical selection rules, g~p, = 0 and g~n'r = 0. In the present problem, we derive below the conclusion that the isoscalar neutral-current axial-vector coupling constants as well as the isoscalar parts of the nucleon anomalous moments vanish as long as we keep exact SU(2) symmetry and neglect the SU(3) (SU(2)) mixing between the ground state baryons and their excited states. A heuristic way to express this selection rule for the isoscalar neutral current axial-vector couplings in the W-S model is to argue [3] that the isoscalar axial-vector current consists of s, c, ... quarks, whereas nucleons consist primarily of u and d quarks, We start illustrating the hypothesis by using the algebra of eq. (3) sandwiched between the same SU(3) multiplet states, (B(a, p, k)l and IB(a,p', k) with momentum p (p ~ oo), physical SU(3) index a and helicity k:

[ [ V~ (0) (A ~ (0)), A n + ], A n-] = 2 V~ (0) (.4 ~ (0)), (5)

nL

((B(a, k)lAn+ InL )(n L IA n- IB(u, X))

(7)

[[V~(0) (A ~(0)), An+], Vn-] = 2A~(0) (V~(0)). (6) The quark-line rule may be a reflection of some dynamical principle which is valid among hadrons composed of confined quarks. The quark current algebras, such as eqs. (3)-(6), are certainly important constraints imposed by confined quarks upon the world of hadrons. It is then a legitimate question to ask how SU(3) (SU(2)) is realized asymptotically, when we insert these algebras between SU(3) (SU(2)) states and then vary their appropriate SU(3) (SU(2)) indices. In the course of this study we find some interesting dynamical patterns for which the notion of levels of hadrons plays an important role. Some of the characteristics (notably the existence of dynamical selection rules) of these patterns can be explained - very intuitively - in terms of the quark-line rule. However, the dynamical patterns themselves - called asymptotic level realization of SU(3) (SU(2)) in certain chiral algebras - may provide us with theoretical understanding. Indeed, they provide a simple rigorous way to compute the degree of violation of some of the quark-line rules. For example, by using the algebras, eq. (3) and (5), respectively, the hypothesis enables us to give a good prediction [10] goon/gtoon =g~n'r/gton7 = - t a n ( 0 - 00), where 0 is the physical 6o-4) mixing angle (0 = - 4 0 °) and 00 is the ideal angle ( ~ -35°). In the ideal limit 0 = 00' i.e., when the C-meson becomes a pure 344

- (B(c~, ~)IA n- InL )(n L [An+ IB(a, X))) = C(a, X), where C(c~, k) = 2(B(~, X)I V3IB(a, ;k)) is a pure number. Among the sum over the single particle hadron intermediate states in eq. (7), we distinguish (for given and ~) the fractional contribution f L (a, X) coming from a//the states n L belonging to a level L (L = 0, 1,2, ...), i.e., eq. (7) becomes C(~, X) (f0(cL k) + f l (~, ~.) + f2(c~, ~) + ...) = C(a, ~). In eq. (7) we may vary the SU(3) index a Which produces non-vanishing C' s. Correspondingly the intermediate states n L also undergo SU(3) rotations. The hypothesis states that fL(~, k) for a given level L will depend on ~. but not on a, i.e., the fractional contribution from each level L is universal or invariant under SU(3) rotation. In other words "block realization" of asymptotic SU(3) in the chiral algebra is postulated as a possible manifestation of quark dynamics in the hadronic world. We now take as the external states the ground state baryons, i.e., 1/2 + octet and 3/2 + decuplet with heticity k = 1/2, and apply the hypothesis to the fractional contribution coming from the same ground state baryons. We define the relevant axial-vector matrix elements as follows. (n, 1/2lAn-lp,1/2)=f, (n, - 1 / 2 lA n-lp, - 1 / 2 > = - f ,

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17 December 1979

Using asymptotic SU(3) we can express all the SU(3) counterparts of eq. (8) in terms o f f , g, and h and the ratio D/F of the D and F type asymptotic axial-vector matrix elements ( 1/2 + IAa 11/2+). When external states B(a, 1/2) are the 1/2 + octet we can vary t~, for example, as t~ = p, Z+ and -0. The hypothesis implies the sum rules f0(p) = f0 (Z+) = f0(X 0) = k where k is the universal fraction of the L = 0 intermediate states. By solving the above sum rules, Oneda and Matsuda obtained [11]

straints.) This helps us to obtain a secure result free from the problem of SU(3) breaking. In varying the indices t~ and/3 for the states (B(tx)[ and IB'(fl)) we need to consider the following four independent combinations related by SU(2): (A) (p, p) and (n, n), (B) (A ++, A++), (A +, A+), (A0, A0) and (A-, A - ), (C) (p, A+) and (n, A0), and (D) (A+, p) and (A0, n). In contrast to eq. (7) we now have two sets of intermediate states inserted among the three factors in the commutators. We again consider the fractional contribution coming solely from the ground state baryons. We first sandwich the gauge commutator eq. (4) between the ground state baryons with helicity ;k = 1/2 and we then study how SU(2) is realized among the contributions from the L = 0 intermediate states. For the combinations (A) and (B), SU(2) is automatically satisfied. However, from the combinations (C) and (D) we obtain one constraint

D/F = 3/2,

8h 2 -- ( v ~ / + g) 2.

( A+, 1/2IA,-I A++, 1/2) -- --X/-3/2g, ( A+, - 1 / 2 IATr-IA++, - 1 / 2 ) = VC~-2g, (p, 1/2[A,r-[ A++, 1 / 2 ) - - X , / 6 h , (p, - 1 / 2 IA~-[ A++, --1/2) = - x / ~ h .

f 2 = (25/9) k,

h 2 = (4/9) k.

(8)

(9)

If we take k = 1 (saturation of eq. (3) by the ground states!), eq. (9) implies D/F = 3/2 and Ill = IgA(0)l = 5/3 which is essentially the result of SU(6). However, eq. (9) gives an improvement. By using PCAC, h 2 can be related to the rate of A -* prr decay which yields k ~ 0.6. Therefore, we obtain IgA(0)[ ~. 1.2. We repeat the same procedure by now taking as external state B(o~, 1/2) the 3/2 + decuplet (i.e., a = A, y, E*) states. However, we do not obtain further constraints, since invariance of the fractional contribution under SU(3) rotation is automatically satisfied. However, we have one statement about the universal fraction k' of the ground state contribution 1 2 + 2h 2 = k ' . ~g

(10)

An interesting question is whether k = k' or not. We derive an answer to this question later. Although we have derived eqs. (9) and (10) using the particular algebra, eq. (3), the result is actually valid for any choice of the algebra [As, A~] = if~vV v. We now proceed to apply the hypothesis to the double commutators of the type eqs. (4), (5) and (6). We insert these commutators between the ground state baryons (B(cz, s, ~)l and IB'(/3, t, ~')) with s oo and t ~ o0. In this case we can immediately obtain many constraints by studying how SU(2) is realized by each level. (For the algebra, eq. (3), we have to study SU(3), since SU(2) does not produce non-trivial con-

(11)

From eqs. (9) and (11) we obtain

f = (5/3)x/~,

h ---+(2/3)x/-~,

g = -(v~/3)x/k, (12)

or

f=-(S/3)x/~,

h = --- (2/3)x/~,

g = (x/~/3)V~. (13)

At the present stage in which not all the constraints are considered we discard the solution eq. (13) by physical arguments, since it predicts gA(0) = (--5/3) × X/~. Actually there exists another set of solutions with the same values for f a n d h but different values for g. However, these solutions can be excluded later. Using the values given by eqs. (12) or (13), we now see from eq. (10), k = k'. This is a very natural result from the point of view of level realization and it may allow us to introduce the concept of super realization * x. ,1

So far, external particles B (a, h) are assumed to belong to the same SU(3) (SU(2)) multiplet. A bolder but probably natural hypothesis is to allow, for the external particles, all the particles belonging to the samelevelM, B (ceM, h) and assume that the fractions do not depend on the individual choice (aM) of the particles belonging to the levelM. We may call this hypothesis super realization. With super realization, the algebra eq. (3) implies k = k' at the SU(2) level and all the results of this paper can be obtained by using

only SU(2) arguments. 345

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In order to derive quickly a result on the isoscalar current, we consider from now on the choice of the helicity states X = - 1 / 2 and X' = 1/2 (spin flip sum rules). We insert the algebra eq. (6), [[V~(0),ATr÷ ], VTr-] = 2A~ (0), between the ground states, (B(a, s, X = -1/2)1 and [B'(/3, t, X = 1/2)) with s ~ oo and t ~ ~. We define (suppressing index #),

17 December 1979

and sandwich it again between the same states. The SU(2) parametrization of the matrix elements of JeUm(0) can be made by using the commutators [[Je~m(0), VTr+], V~r-] = [IVy(0), VTr+], VTr-] = 2v~(0).

Suppressing index #, we have (pfJfp) = i + 6, (nlJln) = - i + 6, (A ++ 1/I A++) = 3/" + 3t~ +/3, (A + I/I A+) = / + a +/3, (A0 IJlA 0) = - ] - a +/3, ( A - IJI A - ) - 3 j - 3o~ +/3,(nlJI A °)= (p IJI A + ) = k and (A 0 IJln) = (A + [J[p) = I. i,j, k and I denote the isovector matrix elements whereas a,/3, and 6 denote the isoscalar matrix elements. With the information about the matrix elements of axial-charge A,r, eq. (12), and of the isovector current V~, eq. (14), in hand, we now study the SU(2) realization by the ground state baryons in the algebra eq. (15), We then find a =/3 = 6 = 0, i.e.

(A +, - 1 / 2 , s l V31A +, 1/2, t) = a , (A +, - 1 / 2 , slA31A +, 1/2, t) = A ,

=

( p , - 1 / 2 , s [ V 3 1 P , 1 / 2 , t ) - b, ( p , - l / 2 , s l A 3 1 P , 1/2, t ) - B , ( p , - 1 / 2 , s 1 V3 IA +, 1/2, t>=e, (p, - 1 / 2 , slA 3 ]A +, 1/2, t) = C,

(p, -1/21V~(O)lp, 1/2)= (n, -1/21V~(O)In, 1/2) = O, (A +, - 1 / 2 , sl V3 Ip, 1/2, t) = d ,

so that

(A +, - 1 / 2 , slA 3 Ip, 1/2, t) = D .

( p , - l/2,slJ~em(O)lp, 1/2, t)

If we write the ground state contribution, for example, in the case of (A ++, A ++) and ( A - , A - ) explicitly, they are - 6 x / 2 g a + 3x/~hc + ... = 6A and -6vr2ga - 3V~hd + ... = - 6 A , respectively. With the values of f, g and h given by eq. (12), ground state realization of SU(2) demands that: (B) ~ 4a + x/2c - x/~d = 0; (C) -~ 2x/--2a - 2x/~b - 3c = 0; and (D) ~ 2 x / - ~ - 2x/2b + 3d = 0. (No constraint from (A).) We thus obtain

a:b:c:d = l : 5 / 2 : - v t 2 : V ~ .

(14)

Next we consider eq. (5), [ [V~(0),A~r+] ,A n- ] = 2V~(0), sandwiched between the same states. We then find that, as far as the ground state contribution to the algebra is concerned, SU(2) is automatically realized by the values off, g and h and a, b, c and d already obtained. In fact, the fraction is found to be universally zero. At this point the set of extra solutions o f f , g and h mentioned before is excluded. We now treat the electromagnetic current JUern which can be written as JeUm = V~ + V~, where V~ is SU(2) singlet. We consider the algebra [ [JeUm(0),A ~r+1, A ~r-] = 2 V~ (0), 346

(15)

= ~n, --1/2,sIJ¢m

(O)Jn, 1/2, t>,

(16)

and (A, --1/2,sl V~(0)I A, 1/2, t) = 0 ,

(17)

for all charge states of A. a =/3 = 6 = 0 is also consistent with the realization of SU(2) by the ground state in the algebra, [ [JeUm(0),A 7r+], VTr-] = 2A ~(0).

(18)

For both the algebras eqs. (15) and (18), fractional contributions from the ground states are universally zero. The implication of the prediction, eq. (16), is as follows. We take a limit Itl -.oo where sz = Xltl with X > 0 (t is taken along the z-axis and s lies in the zx plane, i.e., Sy = 0). Then (p, -1/2,slJ°m(O)lp, 1/2, t) oc (Xltl 2)1/2 [-kp(e/2mp) [(1 + X)/X] SxF2(q2)], where kp is the anomalous magnetic moment of the proton in the unit o f proton magneton, i.e., F~(0) = 1. 2 Then in the limit- X 1 q 2 = -(1 - X)2 X- 1mp2 - SxX.

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and sx ~ 0, q2 ~ 0 and eq. (16) predicts kp = - k n.

GS(0)=0 (19)

That is, to the approximation we have made (i.e., exact SU(2) symmetry) isoscalar contributions to the nucleon anomalous moments vanish. Experimentally eq. (19) reads 1.79 ~ 1.91. We now claim below that the same mechanism is also responsible for the smallness of the neutral current isoscalar axial-vector couplings. Instead of eq. (15), we now use the algebra

[ [A~q (0), A n + ], A ~-I = 2A ~(0),

(20)

where A~q is the weak neutral axial-vector current "1 S.-'~ S is the isoscalar part o f A ~ . Again using the commutator

V.+], V.-] =

V.÷], V.-] = 2A (O),

the matrix elements of A ~ can be parametrized exactly in the same way as in the case OfJeUm. We may write (p IA~q I p) = i' + 6 ', (A ++ IA~[ A ++) = 3/' + 3c~' + t3', .... We then obtain a' = 13' = 6' = 0, i.e., (p, - 1 / 2 , s]A~(O)l p, 1/2) = (n, -1]21A~(0)ln, 1]2) = 0 ,

17 December 1979 and

HAS(0)=0.

This result is consistent with the neglect of isoscalar axial-vector couplings in the standard analysis of neutral currents. We note that in our argument the contributions from the strange, charmed .... quarks (i.e., A~, A~5 ..... V~, V~5 .... ) are already included in the SU(2) isoscalar currents V~(x) and A~(x) and we have a great advantage, since we can operate only in the framework of SU(2). We have thus shown that a kind of selection rule holds for the isoscalar part of the anomalous nucleon moments as well as the neutral-current isoscalar axialvector couplings. A small correction will arise, if we break SU(2) symmetry and include the effect of SU(2) (SU(3)) mixings of the nucleons and A's with their excited states (perhaps the L P = 0 + 70 states). However, as seen from the argument presented above there is no difference in the algebraic structure for the computation of the matrix elements of JeUm(0) and A ~ (0), as far as the spin-flip case is concerned. All the intermediate states are the same and so are the (broken) SU(2) parametrization of the spin-flip matrix elements of the vector and axial-vector currents. We, therefore, expect ,2 (GN(0))p _ (HNA(0))p _ kp

so that

(23)

(GN(0))n

(HN(0))n

kn

1.79 - -1.91'

(24)

( p , - 1 / 2 , s[A~q (0)l p, 1]2, t) = - ( n , -1/2,slA~q(O)ln, 1/2, t),

(21)

and (A, -112, slA~(0)l A, 1/2, t) = 0,

(22)

for all charge states of A. We write the proton matrix elements of A ~ ( 0 ) as (p, s l A ~ (O) l p, t) ~ 5(s)[iT57uGN(q 2)

+ T s q , HN(q2)] u(t), where qu = (s - t)u , since we now know that the second class term is quite small. We decompose G N and H N into isoscalar and isovector parts GAff= C3A + GSA and H N = H 3 + H S. By taking the same limit (s ~ ~ , t -~ oo) as employed in deriving eq. (19) for t~ = 1,2 and 3, eq. (21) then predicts

where (GAN(0))p and (HN(0))p, for example, denote the weak neutral-current axial-vector coupling and the so-called induced pseudoscalar coupling for the proton, respectively. This prediction may be tested by future experiments. Finally we note that, although we developed the discussion in this paper with the W - S model in mind, our result, is, in fact, model independent of the structure of neutral current. In the W - S model our result serves to explain why the effect of the isoscalar (~c - ~s + tt - fob) current is small. Since from our point of view the smallness of the neutral-current isoscalar axial-vector couplings has a dynamical origin, this cannot be used as an important criterion in selecting out the various models.

,2 If we use super realization, we can only use SU(2) arguments. 347

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One of us (S.O.) thanks Dr. G. Senjanovi6 and Professor G. Snow for useful discussion.

References [1] S. Weinberg, Phys. Rev. Lett. 19 (1967) 264; A. Salam, in: Elementary particle theory, ed.N. Svartholm (Almquist, Stockholm, 1968) p. 367. [2] S.L. Glashow, J. lliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [3] S. Weinberg, Phys. Rev. D5 (1972) 1412. [41 L. Wolfenstein, Phys. Rev. D19 (1979) 3450. [5] S.L. Adler, Phys. Rev. D l l (1975) 3309; D12 (1975) 2644; R.M. Barnett, Phys. Rev. D14 (1976) 2990; P.Q. Hung and J.J. Sakurai, Phys. Lett. 69B (1977) 323; 72B (1977) 208;

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P.Q. Hung, Phys. Rev. D17 (1978) 1893. [6] C.H. Albright et al., Phys. Rev. D14 (1976) 1780; D.P. Sidhu and P.L. Langacker, Phys. Rev. Lett. 41 (1978) 732. [7] J. Collins, F. Wilczek and A. Zee, Phys. Rev. D18 (1978) 242. [8] R.N. Mohapatra and G. Senjanovi6, Phys. Rev. D19 (1979) 2165. [9] S. Oneda, in: Proc. INS Intern. Symp. on New particles and structure of hadrons, (Tokyo, 1977), (Institute for Nuclear Study, Univ. of Tokyo) p. 94. [10] S. Oneda, J.S. Rno and M.D. Slaughter, Phys. Rev. 17 (1978) 1389. [ l l l S. Oneda and S. Matsuda, Phys. Rev. D5 (1972) 2287.