Leptonic $CP$ violation search and the ambiguity of $\delta m_{31}^2$

We consider a search for the $CP$-violating angle ${\delta }_{CP}$ in long baseline neutrino oscillation experiments. We show that the subleading ${\delta }_{CP}$-dependent terms in the ${\nu }_{\mu }\to {\nu }_{\mathrm{e}}$ oscillation probability can be easily obscured by the ambiguity of the leading term which depends on $|\delta m_{31}^2|$. It is thus necessary to determine the value of $\delta m_{31}^2$ with a sufficient accuracy. The ${\nu }_{\mu }$ survival events, which can be accumulated simultaneously with the ${\nu }_{\mathrm{e}}$ appearance events, can serve for this purpose owing to its large statistics. Therefore, the combined analysis of ${\nu }_{\mathrm{e}}$ appearance and ${\nu }_{\mu }$ survival events is crucial to provide a restrictive constraint on ${\delta }_{CP}$. Taking a test experimental setup, we demonstrate in the ${\delta }_{CP}\mathrm{\text{−}}\delta m_{31}^2$ plane that the analysis of ${\nu }_{\mathrm{e}}$ appearance events leads to less restrictive constraints on the value of ${\delta }_{CP}$ due to the ambiguity of $\delta m_{31}^2$ and that the combined analysis efficiently improves the constraints.

Figure 1

The neutrino fluxes used in the example analysis. The flux of muon neutrinos (solid line) is obtained by fitting the flux presented in Fig. 3 of Ref. 18. The flux of electron neutrinos (dotted line) is assumed to be 0.7% of that of muon neutrinos with the same energy dependence.

Figure 2

Calculated energy spectra of ${\nu }_{\mathrm{e}}$ appearance events [(a), (b)] and ${\nu }_{\mu }$ survival events (c) for the setup described in text with $\delta m_{31}^2=\pm 2.5\times {10}^{-3}{\mathrm{eV}}^2$ [$+$ in (a), $-$ in (b), and both in (c)]. Other parameters are taken as $\delta m_{21}^2=8.2\times {10}^{-5}{\mathrm{eV}}^2$, ${sin}^{2}2{\theta }_{12}=0.84$, ${sin}^{2}2{\theta }_{23}=1.00$, and ${sin}^{2}2{\theta }_{13}=0.06$. The ${\nu }_{\mu }$ survival event spectra in (c) are presented only for ${\delta }_{CP}=\pi /2$ since they depend little on ${\delta }_{CP}$. The error bars include only statistical errors.

Figure 3

Allowed regions for 68.3% (gray solid line) and 95% (black solid line) confidence levels obtained from ${\nu }_{\mathrm{e}}$ appearance events between 0.7 and 3.1 GeV. The width of energy bins is taken as 0.2 GeV. Incident ${\nu }_{\mathrm{e}}$-contamination and statistical errors are taken into account. The true parameters (cross) are taken to be $\delta m_{31}^{2(\mathrm{true})}=2.5\times {10}^{-3}{\mathrm{eV}}^2$ and ${\delta }_{CP}^{(\mathrm{true})}=-\pi /2$, 0, $\pi /2$, and $\pi$ in (a), (b), (c), and (d), respectively, while $\delta m_{21}^2$ and ${sin}^{2}2{\theta }_{ij}$’s are fixed to those given in Fig. 2. The dotted line and the broken line show the solutions to Eqs. (11, 12) in text, respectively.

Figure 4

The same as Fig. 3, but $\delta m_{31}^{2(\mathrm{true})}=-2.5\times {10}^{-3}{\mathrm{eV}}^2$.

Figure 5

Allowed regions for 68.3% (gray solid line) and 95% (black solid line) confidence level obtained from ${\nu }_{\mu }$ survival events between 0.7 and 4.1 GeV. Incident ${\nu }_{\mathrm{e}}$-contamination as well as statistical errors are taken into account. The true parameters (cross) are ${\delta }_{CP}^{(\mathrm{true})}=\pi /2$ and (a) $\delta m_{31}^{2(\mathrm{true})}=2.5\times {10}^{-3}{\mathrm{eV}}^2$, (b) $\delta m_{31}^{2(\mathrm{true})}=-2.5\times {10}^{-3}{\mathrm{eV}}^2$, while $\delta m_{21}^2$ and ${sin}^{2}2{\theta }_{ij}$’s are fixed to those given in Fig. 2. The dotted line is explained in the text.

Figure 6

Allowed regions for 68.3% (gray solid line) and 95% (black solid line) confidence level. The values of the parameters are the same as in Fig. 3.

Figure 7

The same as Fig. 6, but $\delta m_{31}^{2(\mathrm{true})}=-2.5\times {10}^{-3}{\mathrm{eV}}^2$.