Hadronic Corrections to the Goldberger-Treiman Relation

Abstract

Using unsubtracted dispersion relations in momentum transfer for the matrix element of the divergence of the axial-vector current between nucleon states, we have examined the hadronic continuum corrections to the Goldberger-Treiman relation for ${\pi }^{+}$ decay. These are observed to be about + 10%. From a rigorous unitarity bound and the assumption that the pion propagator ${\Delta }_{\pi }\mathrm{}\left(0\right)\mathrm{}$ is dominated by the pion pole we show that the continuum states of energy greater than two nucleon masses contribute less than ½%. The $\pi \rho$ and $\pi \sigma$ states contributed negligibly. Using Weinberg's extrapolation for the $\pi \pi$ scattering amplitude and chiral dynamics, we find that the presumably dominant $3\pi$ state contributes with opposite sign and is more than an order of magnitude too small. In the absence of any simple explanation for the 10% correction, we conjecture that what is required is a $3\pi$ threshold enhancement or possible resonance, the tripion, with the quantum numbers of the pion and mass near threshold at 4.2 BeV/${\mathit{c}}^2$ or a possible subtraction in the dispersion relation.