Time-reversal violation in β decay in the standard model

We estimate the size of time-reversal violation in β decay due to the Kobayashi-Maskawa phase δ, and the θ term ${\mathcal{L}}_{\theta }.$ We find that the contribution of δ to the $D$ and $R$ correlations is not likely to be larger than of the order of ${10}^{-12}|a|\left(|s_2{s}_3{s}_{\delta }|{/10\mathrm{}}^{-3}\right)$ and ${10}^{-14}|a|\left(|s_2{s}_3{s}_{\delta }|{/10\mathrm{}}^{-3}\right),$ respectively, where $|a|\simeq 1$ for neutron and ${}^{19}\mathrm{Ne}$ decay. For the contribution of ${\mathcal{L}}_{\theta }$ to $D$ and $R$ we conclude that it is not likely to be larger than of the order of ${10}^{-14}|a|(|\theta |/3\mathrm{}\times {10}^{-10}{)Q/(m}_n-m_p)$ and ${10}^{-12}|a|(|\theta |/3\mathrm{}\times {10}^{-10}),$ respectively (the present experimental limits on $s_2{s}_3{s}_{\delta }$ and θ are $|s_2{s}_3{s}_{\delta }|\lesssim {10}^{-3}$ and $|\theta |\lesssim 3\times {10}^{-10}$), where $Q$ is the energy release in the decay. For both δ and ${\mathcal{L}}_{\theta }$ 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 (${10}^{-3}$ 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...