Superconvergent Dispersion Relations and Pion-Nucleon Sum Rules

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 ${T_{\mu \nu }}^A$ approaches the weak vector-nucleon scattering amplitude ${T_{\mu \nu }}^V$ at high energies. This allows us to write down superconvergent dispersion relations for certain invariant amplitudes in the decomposition of $T_{\mu \nu }={T_{\mu \nu }}^A-{T_{\mu \nu }}^V$. 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 $q\to 0$ limit which is usually used in the current-algebra approach. We also discuss sum rules for $G_A\left(q^2\right)$ away from $q^2=0\mathrm{}$.