We suppose that $\beta$ decay and $\mu$ capture are described by a universal vector and axial vector Lagrangian and we consider, via dispersion relation techniques, the properties of the corresponding $S$-matrix elements. Owing to the strong interactions of the nucleons, the structure of the $S$ matrix is expected to be more complicated than that of the Lagrangian. In the former, vector and axial vector terms appear, but with coefficients which in general depend on the invariant nucleon momentum transfer; they can be thought of as Fermi interaction form factors. Moreover, two additional kinds of terms can appear in the $S$-matrix elements: one which simulates a direct pseudoscalar coupling and one which simulates a direct coupling involving derivatives of the nucleon wave functions. The latter is probably too small to have any experimental significance. The former, though negligible in $\beta$ decay, may be appreciable in $\mu$ capture. We estimate the effective pseudoscalar coupling coefficient there to be about eight times as large as the axial vector coefficient. More generally, we investigate the structure of the various form factors; and we also reconsider, in further refinement, a recent quantitative discussion which we have given of $\pi \to \mu +\nu$ decay.

• Received 7 March 1958

DOI:https://doi.org/10.1103/PhysRev.111.354