Consequences of $\mu \text{−}\tau$ reflection symmetry for $3+1$ neutrino mixing

We investigate the consequences of $\mu \text{−}\tau$ reflection symmetry in presence of a light sterile neutrino for the $3+1$ neutrino mixing scheme. We discuss the implications of total $\mu \text{−}\tau$ reflection symmetry as well as partial $\mu \text{−}\tau$ reflection symmetry. For the total $\mu \text{−}\tau$ reflection symmetry we find that values of ${\theta }_{23}$ and $\delta$ remain confined near $\pi /4$ and $\pm \pi /2$, respectively. The current allowed region for ${\theta }_{23}$ and $\delta$ in case of inverted hierarchy lies outside the area preferred by the total $\mu \text{−}\tau$ reflection symmetry. However, interesting predictions on the neutrino mixing angles and Dirac $CP$ violating phases are obtained considering partial $\mu \text{−}\tau$ reflection symmetry. We obtain predictive correlations between the neutrino mixing angle ${\theta }_{23}$ and Dirac $CP$ phase $\delta$ and study the testability of these correlations at the future long baseline experiment DUNE. We find that while the imposition of $\mu \text{−}\tau$ reflection symmetry in the first column admits both normal and inverted neutrino mass hierarchy, demanding $\mu \text{−}\tau$ reflection symmetry for the second column excludes the inverted hierarchy. Interestingly, the sterile mixing angle ${\theta }_{34}$ gets tightly constrained considering the $\mu \text{−}\tau$ reflection symmetry in the fourth column. We also study the implications of $\mu \text{−}\tau$ reflection symmetry for the Majorana phases and neutrinoless double beta decay in the $3+1$ scenario.

Figure 1

Allowed region for total $\mu \text{−}\tau$ symmetry for $3+1$ neutrino scenario. Here the other mixing parameters are varied within $3\sigma$ range [76, 77, 78]

Figure 2

Correlation between ${\theta }_{23}$ and Dirac $CP$ phase $\delta$ for $|U_{\mu 1}|=|U_{\tau 1}|$ with ${\delta }_{14}$, ${\delta }_{24}=0°$ (left panel) and ${\delta }_{14}$, ${\delta }_{24}\ne 0°$ (right panel) respectively. Here all other mixing parameters are varied within their $3\sigma$ range as given in Table 1. The continuous and dashed contours represent $3\sigma$ allowed range in the ${\theta }_{23}\text{−}\delta$ plane and the $\odot$ and $\star$ represent the best-fit values for normal and inverted neutrino mass hierarchy respectively.

Figure 3

Correlation between ${\theta }_{23}$ and Dirac $CP$ phase $\delta$ for $|U_{\mu 2}|=|U_{\tau 2}|$ with ${\delta }_{14}$,${\delta }_{24}=0°$ (left panel) and ${\delta }_{14}$, ${\delta }_{24}\ne 0°$ (right panel) respectively. Here all other mixing parameters are varied within their $3\sigma$ range as given in Table 1. The continuous and dashed contours represent $3\sigma$ allowed range in the ${\theta }_{23}\text{−}\delta$ plane and the $\odot$ and $\star$ represents the best-fit values for normal and inverted neutrino mass hierarchy respectively.

Figure 4

Correlation between ${\theta }_{23}$ and Dirac $CP$ phase $\delta$ for $|U_{\mu 3}|=|U_{\tau 3}|$ with ${\delta }_{14}$,${\delta }_{24}=0°$ (left panel) and ${\delta }_{14}$, ${\delta }_{24}\ne 0°$ (right panel) respectively. Here all other mixing parameters are varied within their $3\sigma$ range as given in Table 1. The continuous and dashed contours represent $3\sigma$ allowed range in the ${\theta }_{23}\text{−}\delta$ plane and the $\odot$ and $\star$ represents the best-fit values for normal and inverted neutrino mass hierarchy respectively.

Figure 5

Constraints on ${\theta }_{34}$ obtained from $|U_{\mu 4}|=|U_{\tau 4}|$. Here the red dashed line (including the black line) represents this correlation. Shaded regions are current allowed range [78]. Here $3\sigma$ allowed range of ${\theta }_{24}$ [78] restricts ${\theta }_{34}$ within the range $4.9°–9.8°$ (as given by the dark black line).

Figure 6

Experimental constraints on ${\delta }_{CP}$ from the correlations $|U_{\mu 1}|=|U_{\tau 1}|$ and $|U_{\mu 2}|=|U_{\tau 2}|$ for DUNE. The left(right) column indicate correlation $|U_{\mu 1}|=|U_{\tau 1}|$ ($|U_{\mu 2}|=|U_{\tau 2}|$). The blue, grey, and green shaded regions depict $1\sigma$, $2\sigma$, and $3\sigma$ confidence regions, respectively. While the red contours represent the currently allowed $3\sigma$ region considering true normal hierarchy.

Figure 7

The effective neutrino mass $m_{\beta \beta }$ for $0\nu \beta \beta$ as a function of the lightest neutrino mass. The left panel shows the effective neutrino mass for NH while the right panel is for IH. The red region represents $m_{\beta \beta }$ in presence of $\mu \text{−}\tau$ reflection symmetry in $3+1$ neutrino mixing where the Majorana phases are kept as zero whereas the blue region represents the scenario when all the Majorana phases are fixed at 90°. The areas inside the black solid lines are the 3 neutrino allowed regions. The pair of purple dashed lines at 0.071 eV and 0.161 eV represent the upper limit for $m_{\beta \beta }$ by combined analysis of GERDA and KamLAND-Zen experiments.

Figure 8

The effective neutrino mass $m_{\beta \beta }$ for $0\nu \beta \beta$ as a function of $\sin {\theta }_{14}$.