Confronting tridirect $CP$-symmetry models with neutrino oscillation experiments

Tridirect $CP$ symmetry is an economical neutrino model building paradigm, and it allows for the description of neutrino masses, mixing angles, and $CP$ violation phases in terms of four free parameters. The viability of a class of tridirect $CP$ models is examined with a comprehensive simulation of current and future neutrino oscillation experiments. The full parameter space of four independent parameters is carefully scanned, and the problem of parameter degeneracy appears for the constraints from one group of neutrino oscillation experiments. Two benchmark models which are promising from a model building point of view are also examined. Complementary roles from accelerator neutrino experiments (e.g., T2HK and DUNE) and reactor neutrino experiments (e.g., JUNO) are crucial to break the degeneracy and nail down the fundamental neutrino mixing parameters of the underlying theory.

Figure 1

The $\mathrm{\Delta }{\chi }^2$ value of $x$, $\eta$, $r$, and $m_a$ in the framework of three-neutrino oscillations taking uncertainties of the NuFit4.0 results. True values for the model parameters are used $(x,\eta ,r,m_a)=(-3.65,1.1\pi ,0.5,3.7\mathrm{meV})$. The experimental configurations we considered are the combination of $\mathrm{NO}\nu \mathrm{A}$ and T2K (dashed brown curve); DUNE (dashed blue); T2HK (dashed green); and the combination of all LBLs (solid black), including all LBLs and JUNO (solid red).

Figure 2

Precision measurements of any two model parameters at the $3\sigma$ confidence level in the framework of three-neutrino oscillations taking uncertainties of the NuFit4.0 results. True values for the model parameters are used: $(x,\eta ,r,m_a)=(-3.65,1.1\pi ,0.5,3.7\mathrm{meV})$. We present the expected results from DUNE (light blue curve), T2HK (green), and the combination $\mathrm{NO}\nu \mathrm{A}$ and T2K (pink), as well as the synergy of all LBLs (light gray) and the interplay of LBLs and JUNO (brown). The black dot denotes the best-fit values, while the magenta triangle is for the local minimum, where $r\sim 0.1$, $\eta \sim 1.84\pi$, $x\sim -8$, and $m_a\sim 3.81\mathrm{meV}$.

Figure 3

The contours for ${\theta }_{12}\sim 35.3°$ (gray), ${\theta }_{13}\sim 8.6°$ (short-dashed green), ${\theta }_{23}\sim 47°$ (short-dashed blue), $\delta \sim 279°$ (short-dashed red), $\mathrm{\Delta }{m}_{21}^2\sim 7.4\times {10}^{-5}{\mathrm{eV}}^2$ (short-dashed orange), and $\mathrm{\Delta }{m}_{31}^2\sim 2.52\times {10}^{-3}{\mathrm{eV}}^2$ (black) on the $x-\eta$, $x-r$, and $\eta -r$ planes. In the upper panels, we let model parameters be the best-fit values, except for those which are varied. In the lower panels we focus on the degeneracy regions, where $r\sim 0.1$, $\eta \sim 1.84\pi$, $x\sim -8$, and $m_a\sim 3.81\mathrm{meV}$. The gray curve for ${\theta }_{12}$ is below $r=0.07$ so that it is hardly visible. We show the true values and the local minimum using black dots and magenta triangles, respectively.

Figure 4

The $\mathrm{\Delta }{\chi }^2$ value against ${\theta }_{12}$ (upper left), ${\theta }_{13}$ (upper right), ${\theta }_{23}$ (middle left), $\delta$ (middle right), $\mathrm{\Delta }{m}_{21}^2$ (lower left), and $\mathrm{\Delta }{m}_{31}^2$ (lower right), for DUNE (dashed blue curve), T2HK (dashed green), the combination of $\mathrm{NO}\nu \mathrm{A}$ and T2K (dashed brown), the synergy of these four LBLs (black), and including all LBLs and JUNO (red), assuming the tridirect $CP$ model.

Figure 5

The points on the $3\sigma$ sphere in the four-dimensional model-parameter space projected on ${\theta }_{23}\text{−}\mathrm{\Delta }{m}_{31}^2$ (upper left), ${\theta }_{13}\text{−}\mathrm{\Delta }{m}_{31}^2$ (upper right), ${\theta }_{12}\text{−}\mathrm{\Delta }{m}_{21}^2$ (lower left), and ${\theta }_{13}\text{−}\delta$ (lower right) for the synergy of these four LBLs (gray curve) and including all LBLs and JUNO (red). We also compare these results to those without the restrictions from tridirect models for LBL synergy (gray contour) and combing all experiments (red contour).

Figure 6

The $\mathrm{\Delta }{\chi }^2$ value against $r$ (left) and $m_a$ (right) assuming model A [solid curve; Eq. (13)] and model B [dotted curve; Eq. (19)], for DUNE (blue), T2HK (green), the combination of $\mathrm{NO}\nu \mathrm{A}$ and T2K (brown), the synergy of four LBLs (black), and the interplay of LBLs and JUNO (red). In model A (B), two conditions are assumed: $x=-7/2$ and $\eta =\pi$ ($x=-4$ and $\eta =5\pi /4$), while in the current global fit results the best fits for the other two parameters are located at $(r,m_a)=(0.557,3.716\mathrm{meV})$ [$(r,m_a)=(0.421,3.723\mathrm{meV})$].

Figure 7

The contours on ${\theta }_{23}\text{−}\delta$ (left) and ${\theta }_{23}\text{−}\mathrm{\Delta }{m}_{31}^2$ (right) for $\mathrm{NO}\nu \mathrm{A}$ (upper) planes, T2K (middle), and their combination (lower) at $1\sigma$ (red curve), $2\sigma$ (blue), and $3\sigma$ (green) precision. The true values are ${\theta }_{12}\sim 35.3°$, ${\theta }_{13}\sim 8.6°$, ${\theta }_{23}\sim 47°$, $\delta \sim 279°$, $\mathrm{\Delta }{m}_{21}^2\sim 7.4\times {10}^{-5}{\mathrm{eV}}^2$, and $\mathrm{\Delta }{m}_{31}^2\sim 2.52\times {10}^{-3}{\mathrm{eV}}^2$. These results include the NuFit4.0 results as priors.

Figure 8

The contours on ${\theta }_{23}\text{−}\delta$ (left) and ${\theta }_{23}\text{−}\mathrm{\Delta }{m}_{31}^2$ (right) for DUNE (upper) and T2HK (lower) at $1\sigma$ (red curve), $2\sigma$ (blue), and $3\sigma$ (green) precision. The true values are ${\theta }_{12}\sim 35.3°$, ${\theta }_{13}\sim 8.6°$, ${\theta }_{23}\sim 47°$, $\delta \sim 279°$, $\mathrm{\Delta }{m}_{21}^2\sim 7.4\times {10}^{-5}{\mathrm{eV}}^2$, and $\mathrm{\Delta }{m}_{31}^2\sim 2.52\times {10}^{-3}{\mathrm{eV}}^2$. These results include the NuFit4.0 results as priors.