Radiative Corrections to Fermi Interactions

Abstract

The radiative correction to the decay spectrum of polarized muons is recalculated taking into account a mistake in our previous work which was recently pointed out by Berman. The revised values for the radiative correction to $\ensuremath{\delta}$, $\ensuremath{\xi}$, and integrated asymmetries for the high- as well as low-energy decay electrons have turned out to be practically identical with the old values. The $\ensuremath{\rho}$ value determined from experiments, on the other hand, has to be increased by about 1% because of the new correction. Thus the over-all effect of the radiative correction to the $\ensuremath{\rho}$ value is now an increase of the order of 5.6% when the experimental and theoretical spectral distributions are compared in the region $0\ensuremath{\le}\frac{p}{{p}_{max}}\ensuremath{\le}0.95$. The radiative corrections to the spectrum and lifetime of the nuclear $\ensuremath{\beta}$ decay arising from the charge interactions of the electron and proton are also studied. Use of this expression gives a correction of -1.7% for the lifetime of ${\mathrm{O}}^{14}$. The corrected Feynman-Gell-Mann coupling constant is $G=(1.40\ifmmode\pm\else\textpm\fi{}0.01)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg/${\mathrm{cm}}^{3}$. In the universal $V\ensuremath{-}A$ theory of weak interactions, the calculated muon mean life becomes ${\ensuremath{\tau}}_{\ensuremath{\mu}}=(2.31\ifmmode\pm\else\textpm\fi{}0.05)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ sec. (These three values depend logarithmically on the ultraviolet cutoff $\ensuremath{\lambda}$ and the corrections to ${\ensuremath{\tau}}_{\ensuremath{\mu}}$ increase for increasing values of $\ensuremath{\lambda}$.) It is found that the corrections to the spectral shape of $\ensuremath{\beta}$ decay are rather large in the case in which the end-point energy ${E}_{m}\ensuremath{\gg}{m}_{e}{c}^{2}$. The radiative corrections to the lifetime and the total asymmetry for muon decay are found to be well defined and finite for ${m}_{e}\ensuremath{\rightarrow}0$ in spite of the fact that the differential spectrum itself diverges logarithmically in the same limit. The same situation is encountered in the case of radiative corrections to the nuclear $\ensuremath{\beta}$ decay. A physical explanation for such behavior of the radiative corrections is attempted. In Appendix A, a simplified expression is given for the determination of the Michel parameter.