On the determination of the leptonic CP phase

The combination of data from long-baseline and reactor oscillation experiments leads to a preference of the leptonic CP phase ${\delta }_{\mathrm{C}\mathrm{P}}$ in the range between $\pi$ and $2\pi$. We study the statistical significance of this hint by performing a Monte Carlo simulation of the relevant data. We find that the distribution of the standard test statistic used to derive confidence intervals for ${\delta }_{\mathrm{C}\mathrm{P}}$ is highly non-Gaussian and depends on the unknown true values of ${\theta }_{23}$ and the neutrino mass ordering. Values of ${\delta }_{\mathrm{C}\mathrm{P}}$ around $\pi \mathrm{/}2$ are disfavored at between $2\sigma$ and $3\sigma$, depending on the unknown true values of ${\theta }_{23}$ and the mass ordering. Typically the standard ${\chi }^2$ approximation leads to over-coverage of the confidence intervals for ${\delta }_{\mathrm{C}\mathrm{P}}$. For the 2-dimensional confidence region in the (${\delta }_{\mathrm{C}\mathrm{P}},{\theta }_{23}$) plane the usual ${\chi }^2$ approximation is better justified. The 2-dimensional region does not include the value ${\delta }_{\mathrm{C}\mathrm{P}}=\pi \mathrm{/}2$ up to the 86.3% (89.2%)~CL assuming a true normal (inverted) mass ordering. Furthermore, we study the sensitivity to ${\delta }_{\mathrm{C}\mathrm{P}}$ and ${\theta }_{23}$ of an increased exposure of the T2K experiment, roughly a factor 12 larger than the current exposure and including also anti-neutrino data. Also in this case deviations from Gaussianity may be significant, especially if the mass ordering is unknown.