Partially Conserved Currents and Weak Form Factors

Abstract

A modification of the Goldberger-Treiman relations (for $G_A$ and for the induced pseudoscalar $G_P$) is obtained by assuming, in addition to the hypothesis of partially conserved axial-vector currents, that the matrix element of the axial-vector current between two nucleon states is dominated by the $A_1\mathrm{}\left(1080\right)\mathrm{}$ and $\pi$. This yields a more detailed structure of the weak form factors. The result is discussed. Extending this line of thought to the case of strangeness-changing currents, which we assume to be "partially conserved," we obtain corresponding expressions for the strangeness-changing induced scalar and pseudoscalar form factors. A relation is obtained relating the induced scalar coupling to the $K_{l3\mathrm{}}$ form-factor ratio $\xi =\frac{f_{-}\mathrm{}\left(0\right)\mathrm{}}{f_{+}\mathrm{}\left(0\right)\mathrm{}}$. These results are useful in a detailed study of the Cabibbo theory. A value for the $K_{l3\mathrm{}}$ parameter $\xi$ is obtained by combining our results with the soft-pion current-algebra result of Callen and Treiman. It ranges from -0.15 to -0.10, depending on the experimental value of $\frac{f_K}{f_{\pi }}$.