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

The longitudinal structure function [${W}_{L}({q}^{2},\ensuremath{\nu})$] of inelastic neutrino and electron scattering is studied in the deep-inelastic limit in the canonical quark-gluon model. Although ${W}_{L}({q}^{2},\ensuremath{\nu})$ vanishes asymptotically in this model, $\ensuremath{\nu}{W}_{L}({q}^{2},\ensuremath{\nu})$ should scale. Sum rules are derived which relate integrals over the scaling limit of $\ensuremath{\nu}{W}_{L}({q}^{2},\ensuremath{\nu})$ and the well-known structure function ${F}_{2}(\frac{\ensuremath{-}{q}^{2}}{2\ensuremath{\nu}})$ to octet baryon masses, the Gell-Mann---Oakes---Renner parameter (c), and the pion-nucleon sigma term (${\ensuremath{\sigma}}_{\ensuremath{\pi}}$). 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 $\ensuremath{\alpha}=0$ 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 ${\ensuremath{\sigma}}_{\ensuremath{\pi}}$ allow numerical evaluation of the sum rules indicating that the integrals over $\ensuremath{\nu}{W}_{L}({q}^{2},\ensuremath{\nu})$ 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.