arXiv2004
Full one-loop radiative corrections are calculated for low energy neutrino-nucleon quasi-elastic scattering, $\bar ν_{e}+p \longrightarrow e^{+} + n$, which involves both Fermi and Gamow-Teller transitions, in the static limit of nucleons. The calculation is performed for both angular independent and dependent parts. We separate the corrections into the outer and inner parts {\` a} la Sirlin. The outer part is infrared and ultraviolet finite, and involves the positron velocity. The calculation of the outer part is straightforward, but that of the inner part requires a scrutiny concerning the continuation of the long-distance hadronic calculation to the short-distance quark treatment and the dependence on the model of hadron structure. We show that the logarithmically divergent parts do not depend on the structure of hadrons not only for the Fermi part, but also for the Gamow-Teller part. This observation enables us to deal with the inner part for the Gamow-Teller transition nearly parallel to that for the Fermi transition. The inner part is universal to weak charged-current processes and can be absorbed into the modification of the coupling constants up to the order of the inverse proton mass $O(1/m_{p})$. The resulting $O(α)$ corrections to the differential cross section take the form $ [1+δ_{\rm out}(E)] [ (1+δ_{\rm in}^{\rm F}) < 1 >^2 +g_A^2 (1+δ_{\rm in}^{\rm GT}) < {\mib σ} >^2]$, where $<1 >$ and $<{\mib σ}>$ stand for the Fermi and Gamow-Teller matrix elements; the outer correction $δ_{\rm out}$ is positron energy ($E$) dependent and takes different functional forms for the angular independent and dependent parts. All factors are explicitly evaluated.