Sum Rules for the Axial-Vector Coupling-Constant Renormalization inβDecay

Abstract

Starting from the axial-vector current algebra suggested by Gell-Mann and the hypothesis of a partially conserved axial-vector current, we derive a sum rule relating $1-g_A^{}{}_{}{}^{-2\mathrm{}}$ to off-mass-shell pion-proton total cross sections. Numerical evaluation gives the theoretical prediction $g_A=1.24\mathrm{}$, in good agreement with experiment. A similar sum rule for pion-pion scattering can only be satisfied if there is a large low-energy $I=0\mathrm{}$, $S$-wave pion-pion scattering cross section. We suggest tests, in high-energy neutrino reactions, of an algebra suggested by Gell-Mann for the vector and axial-vector current octets.