Sum Rules for the Axial-Vector Coupling-Constant Renormalization in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>β</mml:mi></mml:math>Decay

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 $1\ensuremath{-}g_{A}^{}{}_{}{}^{\ensuremath{-}2}$ to off-mass-shell pion-proton total cross sections. Numerical evaluation gives the theoretical prediction ${g}_{A}=1.24$, in good agreement with experiment. A similar sum rule for pion-pion scattering can only be satisfied if there is a large low-energy $I=0$, $S$-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.