A model for baryon structure and its application to ...

A model for baryon structure and its application to magnetic moments and semileptonic decays V. Gupta a , R. Huerta", and G. Sanchez-Col6n"~~ .Departamento de Fisica Aplicoda Centm de Inwestigacion y de Estudios Avanzados del IPN. Unidad Mirida A.P. 73, Cordemez. Mkrida. Yucatan 97310, MEXICO bDepartment of Physics, University of California Riverside, CA 92521-0413, U.S.A.

'Abstract. The spin 112 baryons are pictured as a composite system made out of a "core" of three valence quarks (as in the simple quark model) surrounded by a "sea" (of gluon and qQ pairs) which is specified by its total quantum numbers. We assume the sea is a SU(3) flavor octet with spin 0 or 1 but no color. This model. considered earlier. is used to obtain simultaneous fits for magnetic moments and G A / G V for semileptonic decays. These fits give predictions for nucleon spin distributions in reasonable agreement with experiment.

I

INTRODUCTION

The simple quark model (SQM) though qualitatively successful, fails to account for low energy properties of baryons quantitatively. Experimentaly [I], it is found that quarks cannot even account for the proton spin and thus it is necessary to go beyond SQM. Since quarks interact through strong color forces mediated by gluons, a physical hadron, in reality, consists of valence quarks surrounded by a "sea" of gluons and quark-antiquark (qq) pairs. The effect of the sea contribution to hadron structure has been considered by several authors ['Z-41. In this paper, we study the static properties of the spin 112 baryons (p, n, .I, ...) following Refs. [3] and [4] where the general sea is specified by its total flavor, spin and color quantum numbers. The baryons are pictured as a composite system made out of a baryon "core" of the three valence quarks (as in SQM) and a flavor octet sea with spin 0 and 1 but no color. The purpose of this paper is to use this wavefunction to obtain a simultaneous fit to magnetic moments and semileptonic decays. CP444.Panicle Physics and Cosmology: First Tropical Workshop / High Energy Physics: Second Latin Amencan Symposium edited by J. F. Nieves O 1998 The American Institute of Physics 1 -56396-775-8/98/$15.00

553

I1 BARYON WAVE FUNCTIONS WITH SEA The physical baryon octet states. denoted by B(1/2 t ) are obtained by combining the "core" wavefunction B(8,1/2) (the usual SQM spin 1/2 baryon octet wave function) with the sea wavefunction with specific properties given below. We assume the sea is a color singlet but has flavor and spin properties which when comb i n d with those of the core barvons B give the desired properties of the physical bayon B. Since both the physical and core baryon have J P = ;+, this implies that the sea has even parity and spin 0 or 1. The spin 0 and 1 wavefunctions for the sea a$e denoted by Ho and H I , respectively. We also refer to a spin 0 (1) sea as a scalar (vector) sea. For SU(3) flavor we assume the sea has a SU(3) singlet component and an octet component described by wavefunctions S(1)and S ( 8 ) , respectively. The color singlet sea in our model is thus described by the wavefunctions S(l)Hol S ( l ) H l . S(8)Ho1and S(8)Hl. The total flavor-spin wavefunction of a spin up ( t ) physical b a ~ o which n consists of 3 valence quarks and a sea component (as discussed above) can be written schematically as B(1/2 t ) = 8 ( 8 , 1 / 2 t)HoS(l)+ 60 [ ~ ( 8 , 1 / 28 ) ~i]'~(l)

+ x a ( N ) [@8;1/2 t)Ho 8 ~ ( 8 1 1

(1)

N

+

b(N) ([B(8,1/2) 8 H I ] 8 N

SO),.

The first term is the usual q3-wavefunction of the SQM (with a trivial sea) and the second term (coefficient bo) comes from spin-1 (vector) sea which combines with t the spin 1/2 core baryon B to a spin 1/2t state. So that, [8(8,1/2) 8 HI] = 8~(8,1/2 - & ~ ( 8 , 1 / 2 T)Hl,o. In both these terms the sea is a flavor singlet. The third (fourth) term in Eq. (1) contains a scalar (vector) sea which transforms as a flavor octet. The various SU(3) flavor representations obtained from ~ ( 88)S(8)are labelled by N = 1 , 8F,8D,10, f0,27. As it stands, Eq. (1) represents a spin 1/2t baryon which is not a pure flavor octet but has an admixture of other SU(3) representations weighted by the unspecified constants a ( N ) and b(N). It will be a flavor octet if a ( N ) = b(N) = 0 for N = 1,10,13,27. The color wavefunctions have not been indicated as the three valence quarks in the core B and the sea (by assumption) are in a color singlet state. The sea isospin multiplets contained in the SU(3) flavor octet S(8)are denoted as (ST+, STo,ST-), (SK+,SKo), (SRo1 SK- ), and S,. The familiar pseudoscalar mesons are used here as subscripts to label the isospin and hypercharge quantum numbers of the sea states. Details of the wavefunction have been given earlier [3,4]. However, for completeness the explicit physical baryon states in terms of the core and sea states are given in Table 1. The normalization of a given baryon state (not indicated in

554

*-

TABLE 1. Contribution to the physical barvon state B(1', I , 13) formed out of B(Y,I , 13)and flavor octet states S ( Y , I , 13)(see 3rd term in Eq. (1)). The core baryon states B denoted by 2, ti, etc. are the normal 3 valence quark states of SQM. The sea octet states are denoted by ST+ = S(O,l, I ) , etc. Further. ( N s , ) I , I , , ( % s ~ ) ~ (, ~ S, , ) I , I ,.,. ., stand for total I , 13 normalized combinations of N and S,, etc. Only the contribution from the 3rd term is given. 4th term has exactly the same flavor symmetries & r:,6:). See Ref. (31 for explicit with coefficients (b,, Ti, &) -+ expressions for the coefficients di, A, . . .. and $, p:, ....

a,

(4,

Eq. (1))depends on the parameters which enter in the wavefunction and is different for different isospin multiplets (see Ref. [3]). For applications. we need the quantities (4q)B, q = u , d,s; for each spin-up baryon B. These are defined as (Aq) = n B (q t)- n B (q 4) + nB(Q t) - n B (Q I), where n B (q t) (nB(q I))are the number of spin-up (spin-down) quarks of flavor q in the spin-up baryon B. Also. nB(q t) and nB(Q 1) have a similar meaning for antiquarks. However. these are zero as there are no explicit antiquarks in the wavefunctions given by Eq. (1). The expressions for (Aq)B in terms of b,, $, D,, etc. are given in Table 3 of Ref. [3].

111 MAGNETIC MOMENTS (MM) AND SEMILEPTONIC DECAYS (SLD)

in

For any operator 0 which depends only on quarks, the matrix elements are easily obtained using the orthogonality of the sea components. Clearly (B t (01 B' t ) will be a linear combination of the matrix elements (B t 101~' t ) (known from SQM)

with coefficients which depend on the coefficients in the wavefunction, Eq. (1). We assume the baryon magnetic moment operator, b, to act solely on the valence quarks in B, so that b = C , p,af where p, = eq/2m,and e, and m, are quark charge and mass for q = u: d , s. It is possible to show that the M M of the spin 1 / 2 baryons. p~ ( B = p, n, A,. . .), and the transition magnetic moment, p=on. can be written as

n

,

) ~defined above. Expressions for ( L I ~in) terms ~ of the paramwhere the ( L I ~are eters bo, Pi and are given in Ref. [3]. From Eqs. ( 2 ) we see that the M M depend on the quark masses (or quark M M ) and-on the parameters bo, a ( N ) ,b(N) which determine the sea. of the charge For SLD, the detailed expressions for Gv,A(B+ B') = (B'IJ\,v.~IB) changmg hadronic vector ( J v ) and axial vector ( J A )currents using our wavefunction (Eq. ( 1 ) ) are given in Ref. [4]. Here we briefly summarize how they were calculated. The. AS = 0 and AS = 1 vector currents are the total isospin raising (I+ = I!) + I!")) and \/-spin lowering (1,: = 17!')) operators [4]. The operators I$') and \'!Q) act on the quarks in the core baryons and I!) and 1'2') act on the sea states in the wavefunction. However, the axial vector current has a quark part J!) and a sea part which may? in general, have different relative strengths, so that

+

JY)

J?)

where the constants -40 and .Al specify the strength of J!) relative to for AS = 0 and AS = 1 transitions respectively. In SQh4, J?)(AS = 0) = I?)U! and J.!)(AS = 1) = CqI'!~)u,~ so that, in analogy, we took J ~ ) ( A= S 0) = and J ~ ) ( A= S 1 ) = 2\'!)~!') where Sp) is the spin operator acting only on the sea states in the wavefunction. For AS = 0 transitions. the quark part was sufficient so that A. = 0 for all the fits. For AS = 1 transitions. a direct sea contribution is needed when the theoretical error on the M M is very small. through

z,

21Y)s:')

~2)

N

COMBINED FITS AND RESULTS

The excellent fits for the M M given in Ref. [3] can straightaway be used to predict the GA/Gvfor the 4 SLD's n + p, A + p, C- + n, and 2- + A for which data are available. The numerical predictions are poor especially for n -,p and C- + n. The minimum y2-fits to M M alone do not give acceptable fits to the SLD data. This situation changes profoundly when a combined fit to both the 8 magnetic moment d a t a and 4 SLD GA/Gvdata is made with 1 or 2 more parameters to describe the sea, but reducing one of the pqls as a parameter, for example, by having mu = md.

IVA. Fits with experimental errors for all data. A combined fit to the SLD and MM data with 5 parameters to describe the sea and 2 parameters and p, (we put mu = md) give an excellent fit (given in Table 2) with A. = 0 and Al = -1, which determine the strength of the direct sea contribution to G A ( A S= 0) and G,.,(AS = 1) respectively. Treating and A1 as free parameters does not affect the X2 as their values come to be A. = -0.016 and A, = -1.01. Thus our combined fit gives & = 0.7, xiLD= 0.3, with total x'/DOF = 1/5. If one considers A. and A1 as parameters then x2/DOF = 1/3. The values of the sea parameters, A,and p, are given a t the end of Table 2. For a comparison of this 7 parameter fit with the earlier 6 parameter fits of Refl [3] see Ref. [4]. ;Jn summary, at the expense of an extra parameter overall one obtains a better fit to MM data than before as well as fit the known SLD data using ezperimental e m r s throughout. IVB. Fits with theoretical errors of 0 . 1 for ~ ~MM. We consider such fits because all the fits in the literature (unlike our fits above) add an a r b i t r a ~ theoretical error. The motivation for adding this error is that all MM are treated "democratically". Otherwise, the extremely accurately measured p, and p, act as ~ quadratures ~ inputs to a minimum x2-fit. We add a theoretical, error of 0 . 1 in to the experimental errors for all the MM data. This is a popular choice [6,7]. An error of 0 . 1 is~ fairly ~ large (compared to the actual experimental errors) and facilitates a good fit with a few parameters only. This is true in our model also! A 3 parameter fit with inputs mu = md = 0.6m, and .40 = Al = 0 is given in column 4 of Table 2. In this fit, the scalar and vector seas are described by one parameter each, namely a ( 1 0 ) and b ( 8 ~respectively. ) How does our fit compare to other fits with O.lpN theoretical error? We give a comparison with the most recent fits [7] refered to as CS below. Unlike us the model of CS does not fit p ~ aso it ~is not clear how to include it in their picture of 3-quark correlation within a baryon. For MM alone our fit e v e XaM/DOF= 3.8/5 compared to 4.4/4 for Models A11 and AIII of CS. their best fits. An important difference in their and our model is reflected in the phenomenological values of (Aq)B. In particular, CS obtain (their Model AIII) ( h ) P = 0.783, (Ad), = -0.477, and (As)P = -0.147. This is to be contrasted with the fact that our fits yield (Au)P = 0.964, (Ad), = -0.296, and (As), = 0.008. Physically, our fits require a very tiny strange-quark content in the nucleon compared to their and other similar fits [6,7]. Another physical difference is that in our case the valence quarks carry 6% of the proton spin compared to about 16% in Model AIII of CS. IVC. Spin Distributions. The spin distribution. Ile, for baryon B is defined as IIB J,' gls(z)dz, where the spin structure function gle occurs in polarized electron-ba~onscattering.

=

In SQM,IIB is given by the expectation value IIB I ( B ~ ~ , ( ~where ) I B ) the quark = (1/2) C, e i c i . This gives operator iiq)

TABLE 2. Combined fits to the SLD and MM data. All the MM values are given in nuclear rnagnetons, p ~ a). Fit with experimental errors for all data, see Sec. IV A (column 3). b) Fit with theoretical errors of 0 . 1 added ~ ~ in quadratures for MM, see Sec. nr B tcolumn 4). P(P) An) r(A) r@+) rGO) rP-) ~ ( 9 ) r(E-1

IdWl

Inputs

Fitted parameters

Data 151 2.79284739 6 x -1.91~1428f 5 x lo-' -0.613 f 0.004 2.458 f 0.010

*

-

-1.16of 0.025 -1.250 f 0.014 -0.6507 f 0.0025 1.61 f 0.08

a) Experimental errors

2.79284739 - 1.9130428 -0.613 2.458 0.6396 -1.179 -1.251 -0.6506 1.59

b) Theoretical errors 2.79239

- 1.96330 -0.608 2.538 0.7186 -1.101 -1.151 -0.5331 1.55

In our model in addition to the quarks there can be a direct sea contribution IIBE (BIP~($B) where by analogy we take = e f ~ g ) Thus . only the charged states in the vector sea will contribute to I,(:. For the nucleons, one obtains

12

Putting the two contributions together we have I ~=B + BII$, where B1 determines the strength of the direct sea contribution to the valence quark contribution. Since the value of B1 is not known a priori, so phenomenologically it may be treated as a parameter. Exflriment [1,8] gives 11,= O.126f 0.018 and I!, = -0.O8f 0.06 which are very different from the SQM predictions Ilp= 5/18 = 0.2778 and Iln= 0. One must -noteihat the EMC experiment gives IIpfor (Q 2) = 10.7 ( G ~ V / Cand ) ~ this could be very different for the very low Q2(z0) result predicted by SQM or other theoretical models. This could mean that a model which gives values for IIBdiffering by 2-3 standard deviations from experiment may be quite acceptable. Using the fit to MM and SLD data with experimental errors given in Table 2 we can predict I l ~One . obtains I$ = 0.205, = -0.005. while, = 0.044 and I!:) = 0.057. where we have used (Au)P = 0.989, (Ad)P = -0.271, and (As)P = 0.009. If one keeps only the quark part, that is B1 = 0, then our Ilpis much lower than the SQM value but still 40 higher than experiment. This may be due to large (Q2) in the experiment. Another possibility is to invoke the direct sea contribution. For example. with B1 = -1 one obtains Ilp= 0.161 and 11, = -0.062 in good agreement with experiment.

1t)

V

I!;)

SUMMARY

In summary, we have shown that our model of the sea component in spin-112 baryons can fit their MM, weak decay constants G , I / G ~for both A S = 0 and 1 SLD as well as nuclear spin distributions using ezperimental errors. To accomplish this, one has to invoke a direct sea contribution for A S = 1 decays and nucleon spin distribution. The sea was found to be both scalar (spin 0) and vector (spin 1). Two physical features of our fits are that about 70% of the proton spin resides with the valence quarks and they give a tiny strange-quark content to the nucleon. Acknowledgments. This work was partially supported by CONACyT (Mexico).

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&.

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