Deep-Inelastic Scattering and Static Properties of the Baryons in the Quark-Gluon Model

The longitudinal structure function [$W_L(q^2,\nu )$] of inelastic neutrino and electron scattering is studied in the deep-inelastic limit in the canonical quark-gluon model. Although $W_L(q^2,\nu )$ vanishes asymptotically in this model, $\nu W_L(q^2,\nu )$ should scale. Sum rules are derived which relate integrals over the scaling limit of $\nu W_L(q^2,\nu )$ and the well-known structure function $F_2\left(\frac{-q^2}{2\nu }\right)$ to octet baryon masses, the Gell-Mann—Oakes—Renner parameter (c), and the pion-nucleon sigma term (${\sigma }_{\pi }$). The sum rules are convergent, since leading Regge terms are to be subtracted off according to a well-known prescription and contain no arbitrary constants if the residues of $\alpha =0\mathrm{}$ singularities in forward current-hadron scattering are polynomials in $q^2$. The sum rules are derived using light-cone techniques. It is shown that the parton model and Bjorken-Johnson-Low commutators yield identical results. Similar sum rules are presented for other interactions and scalar "quarks." Estimates of $c$ and ${\sigma }_{\pi }$ allow numerical evaluation of the sum rules indicating that the integrals over $\nu W_L(q^2,\nu )$ are small. The pattern of chiral-symmetry breaking in the vector-gluon model is discussed. It is shown that the dictum that scaling laws may be abstracted from free-field theory leads to difficulties (in that it generates too trivial a theory) if applied to the chiral-symmetry-breaking structure functions of neutrino scattering. Abstraction from gluon models, however, remains adequate.