Resolving ${\theta }_{23}$ degeneracy by accelerator and reactor neutrino oscillation experiments

If the lepton mixing angle ${\theta }_{23}$ is not maximal, there arises a problem of ambiguity in determining ${\theta }_{23}$ due to the existence of two degenerate solutions, one in the first and the other in the second octant. We discuss an experimental strategy for resolving the ${\theta }_{23}$ octant degeneracy by combining reactor measurement of ${\theta }_{13}$ with accelerator ${\nu }_{\mu }$ disappearance and ${\nu }_e$ appearance experiments. The robustness of the ${\theta }_{23}$ degeneracy and the difficulty in lifting it only by accelerator experiments with conventional ${\nu }_{\mu }$ (and ${\overline{\nu }}_{\mu }$) beams are demonstrated by analytical and numerical treatments. Our method offers a way to overcome the difficulty and can resolve the degeneracy between solutions ${sin}^2{\theta }_{23}=0.4$ and ${sin}^2{\theta }_{23}=0.6$ if ${sin}^{2}2{\theta }_{13}\gtrsim 0.05$ at 95% CL by assuming the T2K phase II experiment and a reactor measurement with an exposure of 10 GW·kt·yr. The dependence of the resolving power of the octant degeneracy on the systematic errors of reactor experiments is also examined.

Figure 1

The upper (lower) four panels describe the process of how the ${\theta }_{23}$ octant degeneracy can be resolved for the case where the true value of ${sin}^2{\theta }_{23}=0.458(0.542)$, corresponding to ${sin}^{2}2{\theta }_{23}=0.993$. The other input mixing parameters are given as $\Delta m_{31}^2=2.5\times {10}^{-3}{\mathrm{eV}}^2$, ${sin}^{2}2{\theta }_{13}=0.1$ and $\delta =0$, $\Delta m_{21}^2=8.0\times {10}^{-5}{\mathrm{eV}}^2$, ${sin}^2{\theta }_{12}=0.31$ (the input values of ${sin}^{2}2{\theta }_{13}$ and ${sin}^2{\theta }_{23}$ are indicated by the symbol of star in the plot). (a) The regions enclosed by the solid and the dashed curves are allowed regions only by the results of appearance and disappearance accelerator measurement, respectively. (b) The regions that remain allowed when results of appearance and disappearance measurement are combined. (c) The regions allowed by reactor measurement. (d) The regions allowed after combining the results of appearance and disappearance accelerator experiments with the reactor measurement. The exposures for accelerator are assumed to be 2 (6) yr of neutrino (antineutrino) running with 4 MW beam power with Hyper-Kamionande whose fiducial volume is 0.54 Mt, whereas for the reactor we assume an exposure of 10 GW·kt·yr. The case of optimistic systematic error is taken. (See Sec. 4 and the appendix for details).

Figure 2

Two examples of the ${\theta }_{23}$ degenerate solutions are shown to have almost the same energy spectrum in Kamioka with baseline of 295 km from J-PARC. In the upper (lower) panels, we show the difference of the probabilities of two degenerate solutions for neutrino (antineutrino) channels. The solid and the dashed lines are for normal ($\Delta m_{31}^2>0$) and the inverted ($\Delta m_{31}^2<0$) mass hierarchies, respectively.

Figure 3

Examples of event number distributions are plotted as a function of the reconstructed neutrino energy for the appearance mode with and without oscillations. In the latter, the following values of the oscillation parameters are used: $\Delta m_{31}^2=2.5\times {10}^{-3}{\mathrm{eV}}^2$, ${sin}^2{\theta }_{23}=0.4$, ${sin}^{2}2{\theta }_{13}=0.1$, $\Delta m_{21}^2=8.0\times {10}^{-5}{\mathrm{eV}}^2$, ${sin}^2{\theta }_{12}=0.31$, $\delta =0$.

Figure 4

Examples of event number distributions are plotted as a function of the reconstructed neutrino energy for the disappearance mode without oscillation (left panel) and with oscillation (right panel) with the mixing parameters $\Delta m_{31}^2=2.5\times {10}^{-3}{\mathrm{eV}}^2$, ${sin}^2{\theta }_{23}=0.4$, ${sin}^{2}2{\theta }_{13}=0.1$, $\Delta m_{21}^2=8.0\times {10}^{-5}{\mathrm{eV}}^2$, ${sin}^2{\theta }_{12}=0.31$ and $\delta =0$. The histogram by the solid lines indicate the sum of the signal and background events whereas the ones by the dashed line indicate only the background events.

Figure 5

Expected sensitivity on ${sin}^{2}2{\theta }_{23}$ as a function of the true value of $\Delta m_{23}^2$ obtained with the ${\chi }^2$ defined in (15) assuming the pure 2 flavor oscillation.

Figure 6

Some examples of the allowed regions in the ${sin}^2{\theta }_{13}-{sin}^2{\theta }_{23}$ plane determined by the combined analysis for various different input parameters. For all plots, the input values of $\Delta m_{31}^2=2.5\times {10}^{-3}{\mathrm{eV}}^2$ and the $CP$ phase $\delta =0$. The input value of ${\theta }_{13}$ for the upper (lower) four panels is ${sin}^2{\theta }_{23}=0.08(0.04)$. The input values of ${\theta }_{23}$ are (a) ${sin}^2{\theta }_{23}=0.4$, (b) ${sin}^2{\theta }_{23}=0.6$, (c) ${sin}^2{\theta }_{23}=0.45$ and (d) ${sin}^2{\theta }_{23}=0.55$. The conservative values of the systematic errors are taken for reactor measurement; ${\sigma }_{\mathrm{DB}}={\sigma }_{\mathrm{Db}}=2.0\%$ and ${\sigma }_{\mathrm{dB}}={\sigma }_{\mathrm{db}}=0.5\%$.

Figure 7

Same as Fig. 6 but for smaller systematic errors for the reactor experiment, ${\sigma }_{\mathrm{DB}}={\sigma }_{\mathrm{Db}}=1\%$, ${\sigma }_{\mathrm{db}}=0.2\%$ and ${\sigma }_{\mathrm{dB}}=0.2\%$.

Figure 8

The region in ${sin}^{2}2{\theta }_{13}-{sin}^2{\theta }_{23}$ space where the ${\theta }_{23}$ octant degeneracy can be resolved at 90% (thin green) and 99% (thick red) CL. The solid (dashed) curve is for the case of taking the normal (inverted) hierarchy to perform the fit, assuming the normal hierarchy as input. Conservative systematic errors, as indicated in the figure, are considered here.

Figure 9

The same as in Fig. 8 but with optimistic systematic errors.