Chiral-symmetry-breaking corrections to the Goldberger-Treiman relation

The problem of accounting for the corrections to the Goldberger-Treiman relation (GTR) in neutron $\beta$ decay, ${\Delta }_{\pi }$, is reexamined. It is argued that the failure of the traditional approach, based on dispersion relations or Reggeized one-pion-exchange fits to high-energy hadronic reactions, is probably due to the fact that the radial excitations of the pion have not been explicitly incorporated into these calculations. Although there is now experimental confirmation of the existence of a three-pion resonance, more data is needed in order to update these analyses. A different approach to the calculation of ${\Delta }_{\pi }$ is proposed here. Starting from the chiral-SU(2)XSU(2)-symmetry limit, where the GTR is exact, one interprets all the GTR parameters as referring strictly to that limit. By calculating the renormalizations induced by symmetry breaking in all four parameters, one can then predict ${\Delta }_{\pi }$. As a result of this interpretation a new relation between ${\Delta }_{\pi }$ and the $\pi N$ $\sigma$ commutator ${\sigma }_{\pi N}$ is obtained. Using the most recent theoretical value of ${\sigma }_{\pi N}$ (35±10 MeV), one predicts ${\Delta }_{\pi }=0.06\mathrm{}\pm 0.02$, in good agreement with experiment (${\Delta }_{\pi }^{\exp }=0.06\mathrm{}\pm 0.01$). On the other hand, a large $\sigma$ term such as ${\sigma }_{\pi N}\simeq 60$ MeV, which in itself is very hard to understand in the framework of QCD and SU(3), would lead to ${\Delta }_{\pi }\simeq -0.30\mathrm{}$, in obvious conflict with experiment.