Time-reversal violation in β decay in the standard model

Abstract

We estimate the size of time-reversal violation in \ensuremath{\beta} decay due to the Kobayashi-Maskawa phase \ensuremath{\delta}, and the \ensuremath{\theta} term ${\mathcal{L}}_{\ensuremath{\theta}}.$ We find that the contribution of \ensuremath{\delta} to the $D$ and $R$ correlations is not likely to be larger than of the order of ${10}^{\ensuremath{-}12}|a|(|{s}_{2}{s}_{3}{s}_{\ensuremath{\delta}}|{/10}^{\ensuremath{-}3})$ and ${10}^{\ensuremath{-}14}|a|(|{s}_{2}{s}_{3}{s}_{\ensuremath{\delta}}|{/10}^{\ensuremath{-}3}),$ respectively, where $|a|\ensuremath{\simeq}1$ for neutron and ${}^{19}\mathrm{Ne}$ decay. For the contribution of ${\mathcal{L}}_{\ensuremath{\theta}}$ to $D$ and $R$ we conclude that it is not likely to be larger than of the order of ${10}^{\ensuremath{-}14}|a|(|\ensuremath{\theta}|/3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}{)Q/(m}_{n}\ensuremath{-}{m}_{p})$ and ${10}^{\ensuremath{-}12}|a|(|\ensuremath{\theta}|/3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}),$ respectively (the present experimental limits on ${s}_{2}{s}_{3}{s}_{\ensuremath{\delta}}$ and \ensuremath{\theta} are $|{s}_{2}{s}_{3}{s}_{\ensuremath{\delta}}|\ensuremath{\lesssim}{10}^{\ensuremath{-}3}$ and $|\ensuremath{\theta}|\ensuremath{\lesssim}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$), where $Q$ is the energy release in the decay. For both \ensuremath{\delta} and ${\mathcal{L}}_{\ensuremath{\theta}}$ the $D$ and $R$ coefficients are dominated by long-distance contributions. The small values of $D$ and $R$ in the standard model compared to the best experimental limit on $T$-odd correlations (${10}^{\ensuremath{-}3}$ for $D$ in ${}^{19}\mathrm{Ne}$ decay) give potentially a wide window in which $T$-violating interactions beyond the standard model can be searched for. However, the range in this window that one will be able to exploit depends on the level at which the contributions of the final-state interactions can be kept under control.