Radiative corrections to β-decay and nucleon form factors

Abstract

An attempt is made to estimate the radiative corrections to the vector part of the β-interaction. As previous calculations have given a result which is logarithmically divergent, we particularly emphasize the influence of nucleon structure on the radiative corrections. We find it convenient to work in a gauge where the field operator renormalization constant for a point particle is finite. An infinite number of such gauges exists. The result obtained is independent of the particular gauge chosen within this class. Further, the formulae are not written in terms of standard Feynman diagrams but in a way where on-shell-form-factors appear more directly than would otherwise be the case. This implies the use of an approach similar to dispersion theory. An approximation scheme is introduced where the contributions of states with large masses and involving mesons of various kinds are neglected in the sum over virtual states appearing inside the dispersion integral. However, the effect of all strongly interacting particles is taken into account through their influence on various nuclear form factors. This scheme leads to a finite result in first non-trivial order for the radiative β-decay. The result obtained contains integrals over several form factors some of which are experimentally known today from electron-nucleon scattering experiments while others have so far not been measured. However, in principle, they are all observable. Assuming that these unknown form factors have the same general behaviour as functions of momentum transfer as the known form factors, we arrive at an estimate of the effective value of the cut-off only slightly larger than the nucleon mass. Further, it turns out that rather wide variations in the unknown form factors can be allowed without changing this estimate appreciably. Finally, the result obtained here is applied to some pure Fermi decays using a single-particle model of the nucleus. Numerically, the new estimates differ very little from the older estimates. Therefore, our calculation can be considered as a justification of the usual procedure to put the cut-off in the standard formula for the radiative β-decay equal to the nucleon mass.