Radiative correction to the ${\overline{\nu }}_e({\nu }_e)$ spectrum in $\beta$ decay

We derive an analytic expression for the $\mathcal{O}(\alpha )$ radiative correction to the ${\overline{\nu }}_e({\nu }_e)$ spectrum in allowed $\beta$ decay. The $m\to 0$ limit is convergent and leads to a very simple result ($m$ is the electron mass). This is in sharp contrast to the correction to the $e^{-}(e^{+})$ spectrum, that diverges as $m\to 0$, an important difference that we explain on theoretical grounds. After discussing some of their general properties, we use the corrections to the ${\overline{\nu }}_e$ and $e^{-}$ spectra to derive the corresponding correction to the conversion from the $e^{-}$ spectrum to the ${\overline{\nu }}_e$ spectrum, a relation that is of considerable interest for reactor studies of neutrino oscillations.