Static Coulomb Correction to Beta Decay Arising from the Nuclear Charge Change: A Nonperturbative Approach

Electron wave functions for nuclear $\beta$ decay are derived which include exactly the effects due to the nuclear charge change at the instant of decay. Although these wave functions are of general interest for $\beta$-decay theory, the main motivation for this work is that the first-order (in the fine-structure constant $\alpha$) static Coulomb charge-change correction to $\beta$ decay comprises a large part (two thirds) of the total first-order radiative correction. The present work does not require a perturbation expansion in powers of $\alpha$ for this part of the radiative correction. It is thus particularly suited for an investigation of some problems with the radiative corrections which are directly, or which may be indirectly, related to such an expansion: (1) the validity of previous assumptions concerning the $Z$ dependence of the radiative corrections to $\beta$ decay, and (2) the comparison of the exact point-nucleus limit of this part of the radiative correction with the logarithmically divergent character of the first-order result. The first problem is treated in the present paper and the second will be given later. We find that the two previous $Z$-dependence assumptions are incorrect for the static Coulomb charge-change part of the radiative correction.