Perturbation theory and the goldberger-treiman relation

Abstract

The Goldberger-Treiman relation was originally derived (~) by postulating that the form factor c~(q), which is the decay constant for a pion of four-momentum qg, satisfies unsubtracted dispersion relations, and that these dispersion relations could be saturated by the contribution of the nueleon-antinueleon intermediate state. Under certain approximations, it was then possible to relate e~ to the pion-nucleon coupling constant g ~ , , the nucleon mass M, and the axial vector coupling constant in neutron beta decay, g~. The resulting relation, cr: =-g .~ M/gr~x.~c, holds to an accuracy of about 10%. The approximations in this derivation were quickly recognized as difficult to justify, but several other derivations were suggested (2), based on assumed properties of matrix elements of the derivative of the axial vector current, taken between single-nucleon states. Here we describe another derivation, related to but distinct from earlier derivations. Following GOLDBERGER and TREIMAN (~), we consider the decay of the pion as occurring through a nueleon-antinueleon loop; we later discuss the extent to which the result depends on this assumption. We assume isospin-invariant derivative coupling at the pion-nucleon vertex; the Lagrangian is (3) *~fi = i O~¢P~u7~5 ~ru. For the weak vertex, we assume that only first-class currents (4) contribute; there is a contribution to the axial vector current of the form g~(q2)~r~,~su, and for the induced pseudosealar term iH(q~)~ybuq~ we assume pion-pole dominance. However, instead of using dispersion relations as in ref. (~), we do simple perturbation theory. After taking a trace, doing some simple algebra and making a translation in momentum space, one obtains