Corrections to the Goldberger-Treiman relation in a unified gauge-field model

The corrections to the Goldberger-Treiman relation are examined in a unified gauge-field model. The strong interactions are governed by the local chiral $\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}\otimes \mathrm{SU}\mathrm{}\left(2\right)\mathrm{}$ gauge group, whereas the weak and electromagnetic interactions are based on $\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}\otimes \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$ gauge invariance. We find that the Goldberger-Treiman formula is a zeroth-order relation, so that its corrections are finite. We estimate the corrections in the case that the pion is a pseudo-Goldstone boson. The result, of which the gauge independence is explicitly verified, is proportional to the weak and electromagnetic coupling constants.