Unsubtracted Dispersion Relations in Weak Interactions and the Goldberger-Treiman Relation

Abstract

A theory of weak interactions is formulated in terms of unsubtracted dispersion relations and is applied to the calculation of the $\pi -\mu$ decay amplitude. First, derivation of the Goldberger-Treiman relation is reexamined and the relationship between two different approaches, one by Goldberger and Treiman and the other by Gell-Mann and others, is studied based on Ida's formulation of this problem. Then, it is shown that a consistent use of unsubtracted dispersion relations for weak interactions leads us to an eigenvalue restriction to be imposed on the choice of the parameters in strong interactions.