Radiative Corrections to Fermi Interactions

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 $\delta$, $\xi$, 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 $\rho$ 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 $\rho$ value is now an increase of the order of 5.6% when the experimental and theoretical spectral distributions are compared in the region $0\le \frac{p}{p_{max}}\le 0.95$. The radiative corrections to the spectrum and lifetime of the nuclear $\beta$ 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 $G=(1.40\mathrm{}\pm 0.01)\times {10}^{-49\mathrm{}}$ erg/${\mathrm{cm}}^3$. In the universal $V-A$ theory of weak interactions, the calculated muon mean life becomes ${\tau }_{\mu }=(2.31\mathrm{}\pm 0.05)\times {10}^{-6\mathrm{}}$ sec. (These three values depend logarithmically on the ultraviolet cutoff $\lambda$ and the corrections to ${\tau }_{\mu }$ increase for increasing values of $\lambda$.) It is found that the corrections to the spectral shape of $\beta$ decay are rather large in the case in which the end-point energy $E_m\gg m_e{c}^2$. The radiative corrections to the lifetime and the total asymmetry for muon decay are found to be well defined and finite for $m_e\to 0$ 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 $\beta$ 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.