Pion-Mass Extrapolations and Partial Conservation of Axial-Vector Current

It is proposed, as suggested by a recent study, that the observed 10% corrections to the Goldberger-Treiman relation cannot be accounted for on the basis of unsubtracted dispersion relations. To incorporate the possibility of a small subtraction in the matrix elements of the divergence of the axial-vector current, a supersmooth pion field is defined from the weak interactions by $i{\partial }_{\mu }{A_{\mu }}^{\mathrm{}(+)\mathrm{}}\left(\mathrm{x}\right)=\left(\frac{2M{g}_A{\mu }^2}{\sqrt{2}g}\right){\pi }^{\mathrm{}(+)\mathrm{}}\mathrm{}\left(x\right)-\Delta f_{\pi }\square {\pi }^{\mathrm{}(+)\mathrm{}}\mathrm{}\left(x\right)\mathrm{}$, with $\Delta =1-\frac{2M{g}_A}{\sqrt{2}g f_{\pi }}\simeq +0.1\mathrm{}$ representing the 10% correction. With this definition and the supersmoothness hypothesis the extrapolation of exact current-algebra threshold theorems for weak amplitudes to relate hadron amplitudes is relatively simple and eliminates a major ambiguity in performing such extrapolations in the virtual pion mass.