Spontaneous μ-τ Reflection Symmetry Breaking in Neutrino Phenomenology

Abstract

The latest global analysis of neutrino oscillation data indicates that the normal neutrino mass ordering is favored over the inverted one at the $3\sigma$ level. The best-fit values of the largest neutrino mixing angle ${\theta }_{23}^{}$ and the Dirac CP-violating phase $\delta$ are located in the higher octant and the third quadrant, respectively. We show that these experimental trends can be naturally explained by the $\mu$-$\tau$ reflection symmetry breaking, triggered by the one-loop renormalization-group equations (RGEs) running from a superhigh energy scale down to the electroweak scale in the framework of the minimal supersymmetric standard model (MSSM). The complete parameter space is numerically explored for both the Majorana and Dirac cases, by allowing the smallest neutrino mass $m_1^{}$ and the MSSM parameter $tan\beta$ to vary within their reasonable ranges.

1. Introduction

In the last twenty years, we have witnessed compelling evidence of the neutrino oscillation phenomena [1], as recognized by both the 2015 Nobel Prize in Physics and the 2016 Breakthrough Prize in Fundamental Physics. The standard model (SM) of particle physics must be extended to explain the neutrino masses and the large lepton flavor mixing effects. One can describe the mixings between the three known neutrinos (${\nu }_e^{}$, ${\nu }_{\mu }^{}$, ${\nu }_{\tau }^{}$) and their mass eigenstates (${\nu }_1^{}$, ${\nu }_2^{}$, ${\nu }_3^{}$) in terms of a $3\times 3$ unitary matrix, the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix [2,3]:

$$\begin{array}{c}\hfill U=P_l^{}\left(\begin{array}{ccc}{c}_{12}^{}{c}_{13}^{}& s_{12}^{}{c}_{13}^{}& s_{13}^{}{e}^{-\mathrm{i}\delta }\\ -s_{12}^{}{c}_{23}^{}-c_{12}^{}{s}_{13}^{}{s}_{23}^{}{e}^{\mathrm{i}\delta }& c_{12}^{}{c}_{23}^{}-s_{12}^{}{s}_{13}^{}{s}_{23}^{}{e}^{\mathrm{i}\delta }& c_{13}^{}{s}_{23}^{}\\ s_{12}^{}{s}_{23}^{}-c_{12}^{}{s}_{13}^{}{c}_{23}^{}{e}^{\mathrm{i}\delta }& -c_{12}^{}{s}_{23}^{}-s_{12}^{}{s}_{13}^{}{c}_{23}^{}{e}^{\mathrm{i}\delta }& c_{13}^{}{c}_{23}^{}\end{array}\right)P_{\nu }^{},\end{array}$$

(1)

where $c_{ij}^{}\equiv cos{\theta }_{ij}^{}$ and $s_{ij}^{}\equiv sin{\theta }_{ij}^{}$ (for $ij=12,13,23$) with ${\theta }_{ij}^{}$ being the neutrino mixing angles, $P_l^{}=\mathrm{Diag}\{e^{\mathrm{i}{\varphi }_e^{}},e^{\mathrm{i}{\varphi }_{\mu }^{}},e^{\mathrm{i}{\varphi }_{\tau }^{}}\}$, and $P_{\nu }^{}=\mathrm{Diag}\{e^{\mathrm{i}\rho },e^{\mathrm{i}\sigma },1\}$ with $({\varphi }_e^{},{\varphi }_{\mu }^{},{\varphi }_{\tau }^{})$ and $(\rho ,\sigma )$ being three unphysical phases and two Majorana phases, respectively. Different from the quark sector, there are big mixing angles and a potentially large Dirac CP-violating phase in the lepton sector. On the other hand, the masses of the three known neutrinos are found to be finite, but tiny. It is popular to introduce some heavy degrees of freedom (i.e., the seesaw mechanism [4,5,6,7,8]) and certain flavor symmetries at a superhigh energy scale to explain the smallness of neutrino masses and the lepton flavor mixing patterns observed at low energies. In this case, we need to run the renormalization-group equations (RGEs) to bridge the gap between these two scales.

A credible global analysis of experimental data often points to the truth of particle physics [9,10]. The latest global analysis of available neutrino oscillation data indicates that the normal neutrino mass ordering ($m_1^{}<m_2^{}<m_3^{}$) is favored over the inverted one ($m_3^{}<m_1^{}<m_2^{}$) at the $3\sigma$ level [11], where $m_i^{}$ (for $i=1,2,3$) stand for the neutrino masses. Moreover, the best-fit value of the atmospheric neutrino mixing angle ${\theta }_{23}^{}$ is slightly larger than ${45}^{\circ }$ (i.e., located in the upper octant), and the best-fit value of the Dirac phase $\delta$ is somewhat smaller than ${270}^{\circ }$ (i.e., located in the third quadrant). The $\mu$-$\tau$ reflection symmetry [12,13,14], the minimal discrete flavor symmetry responsible for the nearly maximal atmospheric neutrino mixing and potentially maximal CP violation in neutrino oscillations, can naturally lead to ${\theta }_{23}^{}={45}^{\circ }$ and $\delta ={270}^{\circ }$. Assuming this symmetry is realized at a superhigh energy scale, such as the seesaw scale ${\Lambda }_{\mu \tau }^{}$, it can be spontaneously broken at the electroweak scale ${\Lambda }_{\mathrm{EW}}^{}\sim$10${}^2$ GeV because of the RGE running effect, leading to the deviations of $({\theta }_{23}^{},\delta )$ from $({45}^{\circ },{270}^{\circ })$. Whether such quantum corrections agree with the experimental results of ${\theta }_{23}^{}$ and $\delta$ at low energies depends on the neutrino mass ordering and the theoretical framework accommodating the RGEs [15,16,17,18,19,20,21,22]. We show that the normal neutrino mass ordering, the upper-octant of ${\theta }_{23}^{}$, and the third-quadrant of $\delta$ can be naturally correlated via the RGE-induced $\mu$-$\tau$ reflection symmetry-breaking effect in the minimal supersymmetric standard model (MSSM) framework. Based on more reliable experimental data, we numerically explore the almost complete parameter space in both the Dirac and Majorana cases by allowing the smallest neutrino mass $m_1^{}$ and the MSSM parameter $tan\beta$ to vary in their reasonable regions. From the final numerical results, one can tell which part of the parameter space is favored by current neutrino oscillation data and which part is ruled out. We also conclude that the best-fit values of ${\theta }_{23}^{}$ and $\delta$ [11] may be explained by the spontaneous $\mu$-$\tau$ reflection symmetry breaking. The scenario under consideration will be tested in future neutrino oscillation experiments and help to constrain the values of the smallest neutrino mass $m_1^{}$, the MSSM parameter $tan\beta$, and even the Majorana phases. Therefore, our in-depth analysis is timely, general, and suggestive.

2. Spontaneous $\mu$-$\tau$ Reflection Symmetry Breaking

2.1. The Majorana Case

If neutrinos are Majorana particles, the $\mu$-$\tau$ reflection symmetry requires the Majorana neutrino mass matrix $M_{\nu }^{}$ to be invariant under charge-conjugation transformations ${\nu }_{e\mathrm{L}}^{}\leftrightarrow {\nu }_{e\mathrm{R}}^{\mathrm{c}}$, ${\nu }_{\mu \mathrm{L}}^{}\leftrightarrow {\nu }_{\tau \mathrm{R}}^{\mathrm{c}}$, and ${\nu }_{\tau \mathrm{L}}^{}\leftrightarrow {\nu }_{\mu \mathrm{R}}^{\mathrm{c}}$. In other words, the corresponding Majorana mass matrix elements ${\langle m\rangle }_{\alpha \beta }^{}\equiv {\sum }_i^{}{m}_i^{}{U}_{\alpha i}^{}{U}_{\beta i}^{}$ (for $\alpha ,\beta =e,\mu ,\tau$ and i running over $1,2,3$) must be constrained by ${\langle m\rangle }_{ee}^{}={\langle m\rangle }_{ee}^{*}$, ${\langle m\rangle }_{e\mu }^{}={\langle m\rangle }_{e\tau }^{*}$, ${\langle m\rangle }_{\mu \mu }^{}={\langle m\rangle }_{\tau \tau }^{*}$ and ${\langle m\rangle }_{\mu \tau }^{}={\langle m\rangle }_{\mu \tau }^{*}$ [14]. Taking the parametrization form of U as in Equation (1), one immediately obtains the constraints on U as follows: ${\theta }_{23}^{}={45}^{\circ }$, $\delta ={90}^{\circ }$ or ${270}^{\circ }$, $\rho =0^{\circ }$ or ${90}^{\circ }$, and $\sigma =0^{\circ }$ or ${90}^{\circ }$, for the four physical parameters, as well as ${\varphi }_e^{}={90}^{\circ }$ and ${\varphi }_{\mu }^{}+{\varphi }_{\tau }^{}=0^{\circ }$ for the three unphysical phases. In the framework of the MSSM, the evolution of $M_{\nu }^{}$ from ${\Lambda }_{\mu \tau }^{}$ down to ${\Lambda }_{\mathrm{EW}}^{}$ through the one-loop RGE can be expressed as: [23,24]

$$\begin{array}{c}\hfill M_{\nu }^{}\left({\Lambda }_{\mathrm{EW}}^{}\right)=I_0^2\left[T_l^{}·M_{\nu }^{}\left({\Lambda }_{\mu \tau }^{}\right)·T_l^{}\right]\end{array}$$

(2)

with $T_l^{}=\mathrm{Diag}\{I_e^{},I_{\mu }^{},I_{\tau }^{}\}$, in which:

$$\begin{array}{ccc}\hfill I_0^{}& =& exp\left[+\frac{1}{16{\pi }^2}{\int }_0^{ln\left({\Lambda }_{\mu \tau }^{}/{\Lambda }_{\mathrm{EW}}^{}\right)}\left(\frac{3}{5}{g}_1^2\left(\chi \right)+3g_2^2\left(\chi \right)-3y_t^2\left(\chi \right)\right)\mathrm{d}\chi \right],\hfill \\ \hfill I_{\alpha }^{}& =& exp\left[-\frac{1}{16{\pi }^2}{\int }_0^{ln\left({\Lambda }_{\mu \tau }^{}/{\Lambda }_{\mathrm{EW}}^{}\right)}{y}_{\alpha }^2\left(\chi \right)\mathrm{d}\chi \right].\hfill \end{array}$$

(3)

Here, $\chi =ln\left(\mu /{\Lambda }_{\mu \tau }^{}\right)$, with $\mu$ being an arbitrary renormalization scale between ${\Lambda }_{\mathrm{EW}}^{}$ and ${\Lambda }_{\mu \tau }^{}$, $g_1^{}$ and $g_2^{}$ denoting the gauge couplings, and $y_t^{}$ and $y_{\alpha }^{}$ (for $\alpha =e,\mu ,\tau$) standing for the Yukawa coupling eigenvalues of the top quark and charged leptons, respectively. Because of the smallness of $y_e^{}$ and $y_{\mu }^{}$, one can safely take the approximation $T_l^{}$≃$\mathrm{Diag}\{1,1,(1-{\Delta }_{\tau }^{})\}$ with:

$$\begin{array}{c}\hfill {\Delta }_{\tau }^{}\equiv 1-I_{\tau }^{}\simeq \frac{1}{16{\pi }^2}{\int }_0^{ln\left({\Lambda }_{\mu \tau }^{}/{\Lambda }_{\mathrm{EW}}^{}\right)}{y}_{\tau }^2\left(\chi \right)\mathrm{d}\chi .\end{array}$$

(4)

In Figure 1, we show the two-dimensional maps of ${\Delta }_{\tau }^{}$ (left panel) and $I_o^{}$ (right panel) with respect to ${\Lambda }_{\mu \tau }^{}$ and $tan\beta$. One can see that the value of $I_0^{}$ does not change too much with different settings of ${\Lambda }_{\mu \tau }^{}$ and $tan\beta$. In contrast, ${\Delta }_{\tau }^{}$ can change from $0.001$–$0.05$. Shifting the energy scale ${\Lambda }_{\mu \tau }^{}$ is equivalent to altering $tan\beta$ to obtain the same value of ${\Delta }_{\tau }^{}$. In the numerical calculations, we have fixed the $\mu$-$\tau$ symmetry scale ${\Lambda }_{\mu \tau }^{}$ as ${10}^{14}\mathrm{GeV}$, with $tan\beta$ varying in a decent range, i.e, $tan\beta \in [10,50]$. We use $\Delta {\theta }_{23}^{}\equiv {\theta }_{23}^{}\left({\Lambda }_{\mathrm{EW}}^{}\right)-{\theta }_{23}^{}\left({\Lambda }_{\mu \tau }^{}\right)$ and $\Delta \delta \equiv \delta \left({\Lambda }_{\mathrm{EW}}^{}\right)-\delta \left({\Lambda }_{\mu \tau }^{}\right)$ to measure the strengths of the RGE-induced $\mu$-$\tau$ reflection symmetry-breaking effect that are relevant for oscillation experiments. They can be approximated as:

$$\begin{array}{ccc}\hfill \Delta {\theta }_{23}^{}& \simeq & \frac{{\Delta }_{\tau }^{}}{2}\left(s_{12}^2{\zeta }_{31}^{-{\eta }_{\rho }^{}}+c_{12}^2{\zeta }_{32}^{-{\eta }_{\sigma }^{}}\right),\hfill \\ \hfill \Delta \delta & \simeq & \frac{{\Delta }_{\tau }^{}}{2}\left[\frac{c_{12}^{}{s}_{12}^{}}{s_{13}^{}}\left({\zeta }_{32}^{-{\eta }_{\sigma }^{}}-{\zeta }_{31}^{-{\eta }_{\rho }^{}}\right)-\frac{s_{13}^{}}{c_{12}^{}{s}_{12}^{}}\left(c_{12}^4{\zeta }_{32}^{-{\eta }_{\sigma }}-s_{12}^4{\zeta }_{31}^{-{\eta }_{\rho }^{}}+{\zeta }_{21}^{{\eta }_{\rho }^{}{\eta }_{\sigma }}\right)\right],\hfill \end{array}$$

(5)

in which ${\theta }_{12}^{}$ and ${\theta }_{13}^{}$ take their values at ${\Lambda }_{\mathrm{EW}}^{}$, ${\eta }_{\rho }^{}\equiv cos2\rho =\pm 1$ and ${\eta }_{\sigma }^{}\equiv cos2\sigma =\pm 1$ denote the possible options of $\rho$ and $\sigma$ in their $\mu$-$\tau$ symmetry limit at ${\Lambda }_{\mu \tau }^{}$, and the ratios ${\zeta }_{ij}^{}\equiv (m_i^{}-m_j^{})/(m_i^{}+m_j^{})$ are defined with $m_i^{}$ and $m_j^{}$ at ${\Lambda }_{\mathrm{EW}}^{}$ (for $i,j=1,2,3$). In obtaining Equation (5), the $\mu$-$\tau$ reflection symmetry conditions ${\theta }_{23}^{}\left({\Lambda }_{\mu \tau }^{}\right)={45}^{\circ }$ and $\delta \left({\Lambda }_{\mu \tau }^{}\right)={270}^{\circ }$ have been applied. The corresponding RGE-induced corrections to the other six flavor parameters $(m_1^{}$, $m_2^{}$, $m_3^{}$, ${\theta }_{12}^{}$, ${\theta }_{13}^{}$, $\rho$, and $\sigma )$ have been listed in [22]. Unless otherwise specified, the parameters appearing in the subsequent text and equations are all the quantities at ${\Lambda }_{\mathrm{EW}}^{}$.

2.2. The Dirac Case

It is also theoretically interesting to combine a pure Dirac mass term $M_{\nu }^{}$ with the $\mu$-$\tau$ reflection symmetry for three known neutrinos [20]. In this case, $M_{\nu }^{}$ is invariant under the charge-conjugation transformations ${\nu }_{e\mathrm{L}}^{}\leftrightarrow {\left({\nu }_{e\mathrm{L}}^{}\right)}^{\mathrm{c}}$, ${\nu }_{\mu \mathrm{L}}^{}\leftrightarrow {\left({\nu }_{\tau \mathrm{L}}^{}\right)}^{\mathrm{c}}$, and ${\nu }_{\tau \mathrm{L}}^{}\leftrightarrow {\left({\nu }_{\mu \mathrm{L}}^{}\right)}^{\mathrm{c}}$ for the left-handed neutrino fields and $N_{e\mathrm{R}}^{}\leftrightarrow {\left(N_{e\mathrm{R}}^{}\right)}^{\mathrm{c}}$, $N_{\mu \mathrm{R}}^{}\leftrightarrow {\left(N_{\tau \mathrm{R}}^{}\right)}^{\mathrm{c}}$, and $N_{\tau \mathrm{R}}^{}\leftrightarrow {\left(N_{\mu \mathrm{R}}^{}\right)}^{\mathrm{c}}$ for the right-handed neutrino fields. Similar to the Majorana case, this leads to constraints on the mass matrix elements ${\langle m\rangle }_{\alpha \beta }^{}$ (for $\alpha ,\beta =e,\mu ,\tau$), i.e., ${\langle m\rangle }_{ee}^{}={\langle m\rangle }_{ee}^{*}$, ${\langle m\rangle }_{e\mu }^{}={\langle m\rangle }_{e\tau }^{*}$, ${\langle m\rangle }_{\mu e}^{}={\langle m\rangle }_{\tau e}^{*}$, ${\langle m\rangle }_{\mu \tau }^{}={\langle m\rangle }_{\tau \mu }^{*}$, and ${\langle m\rangle }_{\mu \mu }^{}={\langle m\rangle }_{\tau \tau }^{*}$. Diagonalizing this special mass matrix leads us to the following predictions: ${\theta }_{23}^{}={45}^{\circ }$, $\delta ={90}^{\circ }$ or ${270}^{\circ }$, and $2{\varphi }_e^{}-{\varphi }_{\mu }^{}-{\varphi }_{\tau }^{}={180}^{\circ }$ at ${\Lambda }_{\mu \tau }^{}$. Under the MSSM framework, the one-loop RGE evolution of $M_{\nu }^{}$ from ${\Lambda }_{\mu \tau }^{}$ to ${\Lambda }_{\mathrm{EW}}^{}$ can be described as [20]:

$$\begin{array}{c}\hfill M_{\nu }^{}\left({\Lambda }_{\mathrm{EW}}^{}\right)=I_0^{}\left[T_l^{}·M_{\nu }^{}\left({\Lambda }_{\mu \tau }^{}\right)\right],\end{array}$$

(6)

where the definitions of $I_0^{}$ and $T_l^{}$ are the same as those in Equations (2) and (3). Using the same conventions as in the Majorana case, the deviations of ${\theta }_{23}^{}$ and $\delta$ between ${\Lambda }_{\mathrm{EW}}^{}$ and ${\Lambda }_{\mu \tau }^{}$ can be expressed as:

$$\begin{array}{ccc}\hfill \Delta {\theta }_{23}^{}& \simeq \hfill & \frac{{\Delta }_{\tau }^{}}{4}\left[c_{12}^2\left({\zeta }_{32}^{}+{\zeta }_{32}^{-1}\right)+s_{12}^2\left({\zeta }_{31}^{}+{\zeta }_{31}^{-1}\right)\right],\hfill \\ \hfill \Delta {\delta }^{}& \simeq & \frac{{\Delta }_{\tau }^{}}{4}\left[\frac{c_{12}^{}\left(s_{12}^2-c_{12}^2{s}_{13}^2\right)}{s_{12}^{}{s}_{13}^{}}\left({\zeta }_{32}^{}+{\zeta }_{32}^{-1}\right)-\frac{s_{12}^{}\left(c_{12}^2-s_{12}^2{s}_{13}^2\right)}{c_{12}^{}{s}_{13}^{}}\left({\zeta }_{31}^{}+{\zeta }_{31}^{-1}\right)\right.\hfill \\ & & \left.-\frac{s_{13}^{}}{c_{12}^{}{s}_{12}^{}}\left({\zeta }_{21}^{}+{\zeta }_{21}^{-1}\right)\right].\hfill \end{array}$$

(7)

The expressions for deviations of $m_1^{}$, $m_2^{}$, $m_3^{}$, ${\theta }_{12}^{}$, ${\theta }_{13}^{}$, $\rho$, and $\sigma$ can be found in [22]. The analytical approximations made in Equations (5) and (7) are instructive and helpful for understanding the RGE-induced $\mu$-$\tau$ reflection symmetry-breaking effect. However, the accuracy will be quite poor if the neutrino masses are strongly degenerate. Therefore, we have to use the numerical approach to evaluate the spontaneous $\mu$-$\tau$ reflection symmetry-breaking effect and to explore the allowed parameter space by fitting the current experimental data.

3. Numerical Exploration

In the framework of MSSM, we input the constraints of the $\mu$-$\tau$ reflection symmetry as initial conditions at ${\Lambda }_{\mu \tau }^{}$∼${10}^{14}\mathrm{GeV}$ and numerically run the RGEs from ${\Lambda }_{\mu \tau }^{}$ down to ${\Lambda }_{\mathrm{EW}}^{}$. To be more concrete, the initial conditions include ${\theta }_{23}^{}={45}^{\circ }$ and $\delta ={270}^{\circ }$, as well as four different cases of $\rho$ and $\sigma$ for Majorana neutrinos: Case A: $\rho =\sigma =0^{\circ }$; Case B: $\rho =\sigma ={90}^{\circ }$; Case C: $\rho =0^{\circ }$ and $\sigma ={90}^{\circ }$; Case D: $\rho ={90}^{\circ }$ and $\sigma =0^{\circ }$. For any given values of the MSSM parameter $tan\beta$ and the smallest neutrino mass $m_1^{}$ at ${\Lambda }_{\mathrm{EW}}^{}$, we scan the other relevant neutrino oscillation parameters like $\{{sin}^2{\theta }_{12},{sin}^2{\theta }_{13},\Delta m_{\mathrm{sol}}^2,\Delta m_{\mathrm{atm}}^2\}$ at ${\Lambda }_{\mu \tau }^{}$ over properly wide ranges by utilizing the MultiNest program [25,26,27]. The conventions $\Delta m_{\mathrm{sol}}^2\equiv m_2^2-m_1^2$ and $\Delta m_{\mathrm{atm}}^2\equiv m_3^2-(m_1^2+m_2^2)/2$ have been adopted to keep consistent with the definitions in [11]. For each scan, the neutrino mixing parameters at ${\Lambda }_{\mathrm{EW}}^{}$ are yielded and then compared with their global-fit values by minimizing:

$$\begin{array}{c}\hfill {\chi }^2\equiv \sum _i\frac{{\left({\xi }_i-{\overline{\xi }}_i\right)}^2}{{\sigma }_i^2}.\end{array}$$

(8)

Here, ${\xi }_i^{}\in \{{sin}^2{\theta }_{12},{sin}^2{\theta }_{13},\Delta m_{\mathrm{sol}}^2,\Delta m_{\mathrm{atm}}^2\}$ stand for the parameters at ${\Lambda }_{\mathrm{EW}}^{}$, which are produced by the RGE running from ${\Lambda }_{\mu \tau }^{}$; ${\overline{\xi }}_i^{}$’s represent the best-fit values of ${\xi }_i^{}$ in the global analysis; and ${\sigma }_i^{}$’s are the corresponding symmetrized $1\sigma$ errors. We take the best-fit values and the $1\sigma$ errors of the six neutrino oscillation parameters in [11]: ${sin}^2{\theta }_{12}^{}=3.{04}_{-0.13}^{+0.14}\times {10}^{-1}$, ${sin}^2{\theta }_{13}^{}={2.14}_{-0.07}^{+0.09}\times {10}^{-2}$, ${sin}^2{\theta }_{23}^{}={5.51}_{-0.70}^{+0.19}\times {10}^{-1}$, $\delta ={1.32}_{-0.18}^{+0.23}\times \pi$, $\Delta m_{\mathrm{sol}}^2={7.34}_{-0.14}^{+0.17}\times {10}^{-5}{\mathrm{eV}}^2$, $\Delta m_{\mathrm{atm}}^2={2.455}_{-0.032}^{+0.035}\times {10}^{-3}{\mathrm{eV}}^2$. The smallest neutrino mass $m_1^{}$ is allowed to take values over the range $[0,0.1]\mathrm{eV}$, and the MSSM parameter $tan\beta$ may vary from 10–50. With this setup, we examine how significantly ${\theta }_{23}^{}$ and $\delta$ at ${\Lambda }_{\mathrm{EW}}^{}$ can deviate from their initial values at ${\Lambda }_{\mu \tau }^{}$ by incorporating the recent global-fit results [11].

3.1. The Majorana Case

In Figure 2 and Figure 3, we plot the allowed values of ${\theta }_{23}^{}$ and $\delta$ at ${\Lambda }_{\mathrm{EW}}^{}$ by taking different values of $m_1^{}$ and $tan\beta$ with ${\chi }^2=0$. Four different cases of the initial Majorana phases $(\rho ,\sigma )$ at ${\Lambda }_{\mu \tau }^{}$ have been considered. For each point in the $m_1^{}$-$tan\beta$ plane, ${\theta }_{23}^{}$ and $\delta$ are determined simultaneously. The boundary conditions for the RGEs include both the initial values of $\{{\theta }_{23}^{},\delta ,\rho ,\sigma \}$ at ${\Lambda }_{\mu \tau }^{}$ and the experimental constraints on $\{{sin}^2{\theta }_{12}^{},{sin}^2{\theta }_{13}^{},\Delta m_{\mathrm{sol}}^2,\Delta m_{\mathrm{atm}}^2\}$ at ${\Lambda }_{\mathrm{EW}}^{}$. In this case, the RGEs may not have a realistic solution for some combinations of $m_1^{}$ and $tan\beta$; the gray-gap regions in Figure 2 and Figure 3 (Cases C and D) are excluded. There is no such gap in Cases A and B with different initial values of $\rho$ and $\sigma$. The RGE running effect always pushes ${\theta }_{23}^{}$ to the higher octant and, in most cases, leads $\delta$ to the third quadrant, just the right direction as indicated by the best-fit values of these two quantities [11]. In Case A and Case D, ${\theta }_{23}^{}$ may be significantly corrected for $m_1^{}$≃$0.1\mathrm{eV}$ and $tan\beta$≃50. Of course, such an evolution will be strongly disfavored by the experimental information of ${\theta }_{23}^{}$, which has not been included for this analysis. Our main purpose is to show the general magnitudes of RGE corrections to ${\theta }_{23}^{}$ and $\delta$, without directly involving the experimental limits on them. In Case B, ${\theta }_{23}^{}$ is not sensitive to the RGE running. The RGE-induced corrections to $\delta$ in Cases A and B are very weak. We highlight Case D, in which the best-fit point $({\theta }_{23}^{},\delta )$≃$({48}^{\circ },{238}^{\circ })$ is reachable for the same settings of $m_1^{}$ and $tan\beta$.

The RGE corrections to ${\theta }_{23}^{}$ and $\delta$ in Figure 2 and Figure 3 can be understood with the help of the analytical expressions in Equation (5), if the neutrino masses are not so degenerate. The factor ${\Delta }_{\tau }^{}$ is essentially proportional to ${tan}^2\beta$ as a result of $y_{\tau }^2\propto (1+{tan}^2\beta )$≃${tan}^2\beta$ for $tan\beta \gtrsim 10$, and the magnitudes of $\Delta {\theta }_{23}^{}$ and $\Delta \delta$ always increase with ${tan}^2\beta$. Because the factors ${\Delta }_{\tau }^{}$ and ${\zeta }_{31}^{}$≃${\zeta }_{32}^{}$ are all positive for the normal mass ordering, the sign of $\Delta {\theta }_{23}^{}$ is always positive. The dependence of $\Delta {\theta }_{23}^{}$ and $\Delta \delta$ on the neutrino mass $m_1^{}$ is different for the four options of $\rho$ and $\sigma$ at ${\Lambda }_{\mu \tau }^{}$. In the region of small $m_1^{}$ and $tan\beta$, the radiative correction to ${\theta }_{23}^{}$ is proportional to $m_1^{}$ for Cases A, C, and D, but it is inversely proportional to $m_1^{}$ in Case B with ${\eta }_{\rho }^{}={\eta }_{\sigma }^{}=-1$. As for $\Delta \delta$, its value depends on two terms: one is enhanced by $1/sin{\theta }_{13}^{}$; and the other is suppressed by $sin{\theta }_{13}^{}$. However, the latter one can become dominant in some cases. For example, in Case A, the first term $\propto 1/sin{\theta }_{13}^{}$ is positive and dominant when the neutrino mass $m_1^{}$ is relatively small, while the second term $\propto sin{\theta }_{13}^{}$ is negative and will gradually dominate when the value of $m_1^{}$ increases.

To see the correlation between ${\theta }_{23}^{}$ and $\delta$ at ${\Lambda }_{\mathrm{EW}}^{}$, we marginalize $m_1^{}$ and $tan\beta$ over the reasonable ranges $m_1^{}\in [0,0.1]\mathrm{eV}$ and $tan\beta \in [10,50]$ and show the results in Figure 4.

The recent global-fit results [11] are given as the black contours for the $1\sigma$ (dashed), $2\sigma$ (solid), and $3\sigma$ (dotted) confidence levels. The best-fit point of the global analysis is marked as the black star. We notice that the $\mu$-$\tau$ reflection symmetry point $({\theta }_{23}^{},\delta )=({45}^{\circ },{270}^{\circ })$ at ${\Lambda }_{\mathrm{EW}}^{}$, which is marked as the red circled cross in the plot, is on the dashed contour. This means that ${\theta }_{23}^{}\left({\Lambda }_{\mathrm{EW}}^{}\right)={45}^{\circ }$ and $\delta \left({\Lambda }_{\mathrm{EW}}^{}\right)={270}^{\circ }$ are statistically disfavored at the $1\sigma$ level [11]. For the pink regions, the best-fit values of $\{{sin}^2{\theta }_{12}^{},{sin}^2{\theta }_{13}^{},\Delta m_{\mathrm{sol}}^2,\Delta m_{\mathrm{atm}}^2\}$ can be simultaneously reached (i.e., ${\chi }_{\min }^2=0$). If the value of ${\chi }_{\min }^2$ is relaxed to $11.83$ (i.e., the $3\sigma$ confidence level for two degrees of freedom), the wider green regions of ${\theta }_{23}^{}$ and $\delta$ will be allowed. In the two upper panels of Figure 4, which correspond to Cases A (left one) and B (right one), the allowed range of $\delta$ is very narrow; this feature is compatible with the two upper panels of Figure 3 where $\delta$ varies less than $1^{\circ }$. In these two cases, the green region almost overlaps with the pink region. In the two lower panels of Figure 4, corresponding to Cases C (left one) and D (right one), the RGE-induced corrections of ${\theta }_{23}^{}$ and $\delta$ are both significant. The separate shaded region around ${\theta }_{23}^{}\simeq {50}^{\circ }$ in Figure 4C is associated with the small upper-right corner of the parameter space in Figure 2C or Figure 3C. There is a similar separate shaded region in Case D, but it is outside the chosen ranges of ${\theta }_{23}^{}$ and $\delta$ in plotting Figure 4D, and its confidence level is much weaker, outside the $3\sigma$ region in the global analysis.

3.2. The Dirac Case

The corresponding numerical analysis for the Dirac neutrinos is much easier due to the absence of two Majorana phases. Similar to the Majorana case, the allowed regions of ${\theta }_{23}^{}$ and $\delta$ at ${\Lambda }_{\mathrm{EW}}$ and their intimate correlation are illustrated in Figure 5 and Figure 6. The analytical approximations of $\Delta {\theta }_{23}^{}$ and $\Delta \delta$ in Equation (7) can be further simplified to:

$$\begin{array}{c}\Delta {\theta }_{23}^{}\simeq \frac{{\Delta }_{\tau }^{}}{2}\frac{m_2^2+m_3^2}{\Delta m_{\mathrm{atm}}^2},\hfill \\ \Delta {\delta }^{}\simeq -\frac{{\Delta }_{\tau }^{}}{2}\frac{s_{13}^{}}{c_{12}^{}{s}_{12}^{}}\frac{m_1^2+m_2^2}{\Delta m_{\mathrm{sol}}^2}.\hfill \end{array}$$

(9)

It is easy to see that bigger values of $m_1^{}$ and $tan\beta$ lead to larger deviations of $\left({\theta }_{23}^{},\delta \right)$ from $\left({45}^{\circ },{270}^{\circ }\right)$. In particular, ${\theta }_{23}^{}$ and $\delta$ are always located in the upper octant and the third quadrant, respectively. In Figure 5, we have the gray regions for the same reason as that in the Majorana case. Similar to Case D in the Majorana case, the spontaneous $\mu$-$\tau$ reflection symmetry breaking can take $({\theta }_{23}^{},\delta )$ very close to their best-fit point in the global analysis.

4. Conclusions

The $\mu$-$\tau$ reflection symmetry is inclusive to explain the nearly maximal atmospheric neutrino mixing and the potentially maximal CP violation in neutrino oscillations. However, the latest global analysis suggests this symmetry should be broken at the low energy experimental scale. The RGE running effect can be responsible for the symmetry breaking. The normal neutrino mass ordering, the upper-octant of ${\theta }_{23}^{}$, and the third-quadrant of $\delta$ can be naturally correlated with the RGEs running effect in the MSSM framework. Some of our main observations are subject to the chosen theoretical framework, i.e., the MSSM. The reasons why we do not consider the standard model is: (a) in the SM, the RGE running effect is always very small; (b) in the SM, the running direction of ${\theta }_{23}^{}$ from ${\Lambda }_{\mu \tau }^{}$ down to ${\Lambda }_{\mathrm{EW}}^{}$ is opposite its current best-fit result in the normal mass ordering case; and (c) there may be the vacuum-stability problem as the energy scale is above ${10}^{10}$ GeV [28,29] in the SM. On the other hand, the best-fit values of ${\theta }_{23}^{}$ and $\delta$ will unavoidably fluctuate in the future when more experimental data are accumulated and included in the global analysis. Since we have generally explored the complete parameter space, our results still keep working. If the future precision measurements favor the inverted neutrino mass ordering, the lower octant of ${\theta }_{23}^{}$, and (or) another quadrant of $\delta$, one can perform the numerical analysis in the same way, either within or beyond the MSSM. In the same spirit, one may study other interesting flavor symmetries and their RGE-induced breaking effect, in order to link model building at high-energy scales effectively with the observed neutrino oscillation data at low energies.