and

In chiral models with SU(3) group structure, strange form factors of baryon octet are evaluated by constructing their sum rules to yield theoretical predictions comparable to the recent experimental data of SAMPLE Collaboration. We also study sum rules for the flavor singlet axial currents for the EMC experiment in a modified quark model.


Introduction
There have been many interesting developments concerning the strange flavor structures in the nucleon and the hyperons. Especially, the internal structure of the nucleon is still a subject of great interest to both experimentalists and theorists. In 1933, Frisch and Stern [1] performed the first measurement of the magnetic moment of the proton and obtained the earliest experimental evidence for the internal structure of the nucleon. However, it wasn't until 40 years later that the quark structure of the nucleon was directly observed in deep inelastic electron scattering experiments and we still lack a quantitative theoretical understanding of these properties including the magnetic moments.
Quite recently, the SAMPLE Collaboration [2] reported the experimental data of the proton strange form factor through parity violating electron scattering [3]. To be more precise, they measured the neutral weak form factors at a small momentum transfer Q 2 S = 0.1 (GeV/c) 2 to yield the proton strange magnetic form factor [2] G s M (Q 2 S ) = +0.14 ± 0.29 (stat) ± 0.31 (sys). This positive experimental value is contrary to the negative values of the proton strange form factor which result from most of the model calculations except the predictions [4,5] based on the SU(3) chiral bag model [6] and the recent predictions of the chiral quark soliton model [7] and the heavy baryon chiral perturbation theory [8]. (See Ref. [9] for more details.) On the other hand, the EMC experiment [10] also reported the highly nontrivial data that less than 30% of the proton spin may be carried by the quark spin, which is quite different from the well-known prediction from constituent quark model. To explain this discrepancy, it has been proposed [11] that the experimentally measured quantity is not merely the quark spin polarization ∆Σ but rather the flavor singlet axial current (FSAC) via the axial anomaly mechanism [12]. Recently, at the quark model renormalization scale, the gluon polarization contribution to the FSAC in the chiral bag model has been calculated [13] to yield a significant reduction in the relative fraction of the proton spin carried by the quark spin, consistent with the small FSAC measured in the EMC experiments.
In this paper, in the chiral models with SU(3) group structure, we will investigate the strange form factors of octet baryons in terms of the sum rules of the baryon octet magnetic moments to predict the proton strange form factor. We will also study the modified quark model with SU(3) group structure to obtain sum rules for the strange flavor singlet axial current of the nucleon in terms of the octet magnetic moments µ B and the nucleon axial vector coupling constant g A . In section 2, we construct the sum rules of the baryon octet magnetic moments in the SU(3) chiral models. In section 3 we construct the sum rules for the nucleon strange flavor singlet axial current in the modified quark model.

Strange form factors
Now we consider the baryon magnetic moments in the chiral models such as Skyrmion [14], MIT bag [15] and chiral bag [6] with the general chiral SU(3) group structure. In the higher dimensional irreducible representation of SU(3) group, the baryon wave function is described as [4,16] where the representation mixing coefficients are given by Here E λ is the eigenvalue of the eigen equation H 0 |B λ = E λ |B λ . (For explicit expressions for the Hamiltonian H = H 0 +H SB in the chiral models, see Ref. [5].) Using the above baryon wave function in the Hamiltonian H the spectrum of the magnetic moment has the hyperfine structure (For the other magnetic moments, see Ref. [5].) Here one notes that the coefficients are solely given by the SU(3) group structure of the chiral models and the physical informations such as decay constants and masses are included in the above inertia parameters, such as M, N and so on, calculable in the chiral models. Using the V-spin symmetry sum rules [5], one can obtain the relation which will be used later to obtain sum rules of the strange form factors of octet baryons. Now we consider the form factors of the octet baryons which, in the chiral models, are definitely extended objects with internal structure associated with the electromagnetic (EM) current, to which the photon couples, According to the Feynman rules the matrix element ofV µ γ for the baryon with transition from momentum state p to momentum state p + q is given by the following covariant decomposition where u(p) is the spinor for the baryon states and q is the momentum transfer and σ µν = i 2 (γ µ γ ν − γ ν γ µ ) and M B is the baryon mass and F γ 1 and F γ 2 are the Dirac and Pauli EM form factors, which are Lorentz scalars and p 2 = (p + q) 2 = M 2 B on shell so that they depend only on the Lorentz scalar variable q 2 (= −Q 2 ). We will also use the Sachs form factors, which are linear combinations of the Dirac and Pauli form factors which can be rewritten as On the other hand, the neutral weak current operator is given by an expression analogous to Eq. (2.4) but with different coefficients: Here the coefficients depend on the weak mixing angle, which has recently been determined [17] with high precision: sin 2 θ W = 0.2315 ± 0.0004 . In direct analogy to Eq. (2.7), we have expressions for the neutral weak form factors G Z E,M in terms of the different quark flavor components: Here one notes that the form factors G f E,M (f = u, d, s) appearing in this expression are exactly the same as those in the EM form factors, as in Eq. (2.7).
Utilizing isospin symmetry, one then can eliminate the up and down quark contributions to the neutral weak form factors by using the proton and neutron EM form factors and obtain the expressions It shows how the neutral weak form factors are related to the EM form factors plus a contribution from the strange (electric or magnetic) form factor. Measurement of the neutral weak form factor will thus allow (after combination with the EM form factors) determination of the strange form factor of interest. It should be mentioned that there are electroweak radiative corrections to the coefficients in Eq. (2.9), which are generally small corrections, of order 1-2%, and can be reliably calculated [18].
The EM form factors present in Eq. (2.10) are very accurately known (1-2 %) for the proton in the momentum transfer region Q 2 < 1 (GeV/c) 2 . The neutron form factors are not known as accurately as the proton form factors (the electric form factor G n E is at present rather poorly constrained by experiment), although considerable work to improve our knowledge of these quantities is in progress. Thus, the present lack of knowledge of the neutron form factors will significantly hinder the interpretation of the neutral weak form factors.
At zero momentum transfer, one can have the relations between the EM form factors and the static physical quantities of the baryon octet, namely G γ E (0) = Q B and G γ M (0) = µ B with the electric charge Q B and magnetic moment µ B of the baryon. In the strange flavor sector, the Sachs magnetic form factor in Eq. (2.6) yields the strange form factors of baryon octet degenerate in isomultiplets F s with the fractional strange EM charge Q s B . Here note that one can express the slope of G s E at Q 2 = 0 in the usual fashion in terms of a strangeness radius r s defined as r 2 s = −6 dG s E /dQ 2 Q 2 =0 . Now we construct model independent sum rules for the strange form factors of baryon octet in the chiral models with the SU(3) flavor group structure. Since the nucleon has no net strangeness the nucleon strange form factor is given by [5] which, at least within the SU (3)  On the other hand, the quantities G Z E,M in Eq. (2.10) for the proton can be determined via elastic parity-violating electron scattering to yield the experimental data G s M (Q 2 S ) = +0.14 ± 0.29 (stat) ± 0.31 (sys) [2] for the proton strange magnetic form factor. Here one notes that the prediction for the proton strange form factor (2.13) obtained from the sum rule (2.12) is comparable to the SAMPLE data. Moreover, from the relation (2.10) at zero momentum transfer, the neutral weak magnetic moment of the nucleon can be written in terms of the nucleon magnetic moments and the proton strange form factor [19] 4µ Z p = µ p − µ n − 4 sin 2 θ W µ p − F s 2N (0). (2.14) Next, we obtain the other octet baryon strange form factors [5] F s 2Λ (0) = which, similarly to the nucleon strange form factors, can be rewritten in terms of the octet magnetic moments to yield the sum rules for the other octet strange form factors  Table 1 are unreliably sensitive in the strange flavor channel.

Strange flavor singlet axial currents
In this section, we consider a modified quark model [20]. In the nonrelativistic quark model, the quarks possess the static properties such as mass, electromagnetic charge and magnetic moments, which are independent of their surroundings. However this assumption seems to be irrelevant to the realistic experimental situation. In the literature [20], the magnetic moments of the quarks were proposed to be different in the different isomultiplets, but to be the same within an isomultiplet. The magnetic moments are then given by where µ B f is an effective magnetic moment of the quark of flavor f for the baryon B degenerate in the corresponding baryon isomultiplet, and ∆f B is the spin polarization for the baryon.
Using the SU(3) charge symmetry one can obtain the magnetic moments of the octet baryons as follows [20] After some algebra we obtain the novel sum rules for spin polarizations ∆f with the flavor f in terms of the octet magnetic moments µ B and the nucleon axial vector coupling constant g A where we have assumed the isospin symmetry µ B u = −2µ B d . Here one notes that the above sum rules (3.3) are given only in terms of the physical quantities, the coupling constant g A and baryon octet magnetic moments µ B , which are independent of details involved in the modified quark model, as in the sum rules in Eqs. (2.12) and (2.16). Moreover these sum rules are governed only by the SU(3) flavor group structure of the models.
Using the experimental data for g A and µ B , we obtain the strange flavor spin polarization ∆s ∆s = −0.26 (3.5) which, together with the other flavor spin polarizations ∆u = 0.81 and ∆d = −0.44, one can arrive at the flavor singlet axial current of the nucleon as follows 2 ∆Σ = ∆u + ∆d + ∆s = 0.11 (3.6) which is comparable to the recent value ∆Σ = 0.28 obtained from the deep inelastic lepton-nucleon scattering experiments [22]. Here note that the strange flavor singlet axial current ∆s in Eq. (3.5) is significantly noticeable even though the flavor singlet axial current ∆Σ in Eq. (3.6) is not quite large. Now it seems appropriate to discuss the strange form factor in this modified quark model. Exploiting the relations (3.2), together with the isospin symmetry µ B u = −2µ B d , one can easily obtain We thus arrive at the sum rule for the nucleon strange form factor in the modified quark model Substituting the experimental values for µ p and µ n , and the above predictions ∆u = 0.81 and ∆d = −0.44, we obtain which reveals the discrepancy from the SAMPLE experimental values, differently from the prediction (2.13) of the SU(3) chiral model case. However, 2 In fact, in the literature [20], ∆Σ is evaluated using the sum rule for ∆Σ. However, here we have explicitly obtained the sum rules for its flavor components ∆f (f = u, d, s) and as expected, this result is quite comparable to the prediction in the literature [21] where, similar to Eq. (3.2), the SU(3) charge symmetry relations with the quark-loops are used. The difference between the preditions of F s(0) 2N in the SU(3) modified quark model and the SU(3) chiral model originates from the assumptions of these models, for instance, those in the SU(3) modified quark model that the magnetic moments of the quarks are different in the different isomultiplets, but do not change within an isomultiplet.

Conclusions
In summary, we have investigated the strange flavor structure of the octet baryon magnetic moments in the chiral models with SU(3) group structure. The strange form factors of octet baryons are explicitly constructed in terms of the sum rules of the baryon octet magnetic moments, which originate from the SU(3) flavor group structure, to yield the theoretical predictions. Especially in case of using the experimental data for the baryon magnetic moments as input data of the sum rules, the predicted proton strange form factor is comparable to the recent SAMPLE experimental data.
On the other hand, we have studied the modified quark model with SU(3) group structure, where the magnetic moments of the quarks are different in the different isomultiplets, but do not change within an isomultiplet. In this model, we have obtained the sum rules for the spin polarizations ∆f with the flavor f (f = u, d, s) in terms of the octet magnetic moments µ B and the nucleon axial vector coupling constant g A , to yield the flavor singlet axial current of the nucleon, comparable to the recent experimental data. Moreover, the strange flavor spin polarization has been shown to be quite noticeable. However, exploiting the sum rule for the nucleon strange form factor constructed in the modified quark model, we have obtained the value, which shows discrepancy from the SAMPLE experimental values but is comparable to the prediction in the previous literature. Through further investigation, it will be interesting to study deep structure of the sum rules of the models with SU(3) group structures, which could lead to unification of these models.
STH would like to thank Bob McKeown for helpful discussions and kind concerns at Kellogg Radiation Laboratory, Caltech where a part of this work has been done, and M. Rho for useful discussions and comments. He also