Probing the leptonic Dirac $CP$-violating phase in neutrino oscillation experiments

The discovery of leptonic $CP$ violation is one of the primary goals of next-generation neutrino oscillation experiments, which is feasible due to the recent measurement of a relatively large leptonic mixing angle ${\theta }_{13}$. We suggest two new working observables $\Delta A_{\alpha \beta }^{\mathrm{m}}\equiv \max [A_{\alpha \beta }^{CP}(\delta )]-\min [A_{\alpha \beta }^{CP}(\delta )]$ and $\Delta A_{\alpha \beta }^{CP}(\delta )\equiv A_{\alpha \beta }^{CP}(\delta )-A_{\alpha \beta }^{CP}(0)$ to describe the $CP$-violating effects in long-baseline and atmospheric neutrino oscillation experiments. The former signifies the experimental sensitivity to the leptonic Dirac $CP$-violating phase $\delta$ and can be used to optimize the experimental setup, while the latter measures the intrinsic leptonic $CP$ violation and can be used to extract $\delta$ directly from the experimental observations. Both analytical and numerical analyses are carried out to illustrate their main features. It turns out that an intense neutrino beam with sub-GeV energies and a baseline of a few 100 km may serve as an optimal experimental setup for probing leptonic $CP$ violation.

Figure 1

Numerical results of $A_{\mu e}^{CP}(\delta )$ for $\delta =0$ and $\pi /2$ in the case of normal neutrino mass hierarchy (upper plots) and inverted neutrino mass hierarchy (lower plots), where the best-fit values ${\theta }_{12}=34°$, ${\theta }_{13}=9°$, ${\theta }_{23}=40°$, $\Delta m_{21}^2=7.5\times {10}^{-5}{\mathrm{eV}}^2$, and $|\Delta m_{31}^2|=2.5\times {10}^{-3}{\mathrm{eV}}^2$ have been used [14].

Figure 2

Numerical results of $\Delta P_{\mu e}^{\mathrm{m}}$ and $\Delta {\overline{P}}_{\mu e}^{\mathrm{m}}$ in the cases of normal neutrino mass hierarchy (upper plots) and inverted neutrino mass hierarchy (lower plots), where the input neutrino parameters are the same as in Fig. 1.

Figure 3

Numerical results of $\Delta A_{\mu e}^{CP}(\delta )$ for $\delta =\pi /2$ in the cases of normal neutrino mass hierarchy (left plot) and inverted neutrino mass hierarchy (right plot), where the input neutrino parameters are the same as in Fig. 1.

Figure 4

Numerical results of $\Delta A_{\mu e}^{\mathrm{m}}$ in the case of normal neutrino mass hierarchy (upper plots) and inverted neutrino mass hierarchy (lower plots), where the input neutrino parameters are the same as in Fig. 1. In the right plots, we have zoomed in the parameter region that is relevant for the long-baseline neutrino oscillation experiments T2K (green, at $h=88.7°$), $\mathrm{NO}\nu \mathrm{A}$ (cyan, at $h=86.4°$), LBNE (red, at $h=84.1°$), LAGUNA-LBNO (blue, $h=79.7°$), and ESS (black, at $h=87.8°$).