Relativistic calculation of radiative muon capture in hydrogen andHe3

In view of the fundamental importance of radiative muon capture, particularly in light nuclei, and of the somewhat confused theoretical situation, a new calculation of radiative muon capture on the proton and on ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ has been made. This calculation is based on the standard set of diagrams but, unlike previous calculations, has been performed without making nonrelativistic approximations. Results are given for the rate and photon spectrum for both ${}_{}{}^1{}_{}{}^{}\mathrm{H}$ and ${}_{}{}^3{}_{}{}^{}\mathrm{He}$. A detailed comparison of this calculation is made with the approach developed by Hwang and Primakoff which is based on general constraints of conserved vector current, partially conserved axial vector current, and gauge invariance and on a special linearity hypothesis to try to understand why the Hwang-Primakoff results differ markedly from all previous results. It is shown that this calculation, as well as other standard ones, satisfy the general constraints of Hwang and Primakoff and that the differences arise because of their linearity hypothesis and other approximations. These differences are examined in detail, and it is shown that the numerically most important one arises because the linearity hypothesis has been used in such a way that it leads to a Hwang-Primakoff amplitude which violates the Low soft photon theorem.