General Properties of the Electromagnetic Corrections to the Beta Decay of a Physical Nucleon

Abstract

Some general properties of the electromagnetic corrections to the $\beta$ decay of a physical nucleon, that is, a nucleon in the presence of strong interactions, are discussed. The aim of the paper is to isolate and determine general properties which are independent of the details of the strong interactions and the assumptions about the existence of the intermediate boson. The method used consists essentially in separating out in a finite and gauge-invariant manner all the terms of order $\frac{1}{k}$ in the hadronic covariants, and then examining properties of the other contributions. Under some general and plausible mathematical assumptions, it is shown that all the terms of order $\alpha$ in the correction factor to the electron spectrum which explicitly depend on the electron energy $E$ and the electron mass $m$ can be rigorously computed, in spite of the complications of the strong interactions, provided that contributions of relative order $\frac{\alpha q}{M}$, $\alpha \left(\frac{E}{M}\right)\ln \left(\frac{M}{E}\right)$, and $\frac{\alpha v_N}{c}$ are neglected. In particular, the electromagnetic correction of order $\alpha$ to the shape of the allowed electron spectrum is given by a single universal function $g(E, E_m, m)$ of $E$, $m$, and the end-point energy $E_m$; this function is independent of the details of the strong interactions and the assumption that the weak interactions are mediated by an intermediate boson. It is furthermore independent of the ratio $\frac{M_{\mathrm{GT}}}{M_F}$, and is physically significant, particularly if applied to nuclear $\beta$ decays with $\frac{E_m}{m}\gg 1$. These propositions do not preclude the existence of model-dependent terms which give contributions to the lifetime. No attempt is made here to evaluate these quantities, but their role on the physical observables is discussed. It is pointed out that, aside from the electron spectrum, there are other observables for which the corrections can be evaluated in a model-independent manner. This assertion is illustrated by giving the expression for the virtual radiative corrections of order $\alpha$ to the longitudinal polarization of the electron in allowed transitions. The contribution of the universal function $g(E, E_m, m)$ to the lifetime is briefly discussed. This paper implicitly assumes the validity of the vector and axial-vector theory of weak interactions.