Superconvergent Dispersion Relations and Pion-Nucleon Sum Rules

Abstract

Both the Adler-Weisberger sum rule and the spin-flip sum rule for pion-nucleon scattering have been derived from superconvergent dispersion relations for weak amplitudes. Our basic assumption is that the weak axial-vector-nucleon scattering amplitude ${{T}_{\ensuremath{\mu}\ensuremath{\nu}}}^{A}$ approaches the weak vector-nucleon scattering amplitude ${{T}_{\ensuremath{\mu}\ensuremath{\nu}}}^{V}$ at high energies. This allows us to write down superconvergent dispersion relations for certain invariant amplitudes in the decomposition of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}={{T}_{\ensuremath{\mu}\ensuremath{\nu}}}^{A}\ensuremath{-}{{T}_{\ensuremath{\mu}\ensuremath{\nu}}}^{V}$. We then use the hypotheses of partially conserved axial-vector current and of conserved vector current to obtain pion-nucleon scattering sum rules while avoiding the ambiguities of the $q\ensuremath{\rightarrow}0$ limit which is usually used in the current-algebra approach. We also discuss sum rules for ${G}_{A}({q}^{2})$ away from ${q}^{2}=0$.