\( \mu \tau \) reflection symmetry embedded in minimal seesaw
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Abstract
We embed \(\mu \tau \) reflection symmetry into the minimal seesaw formalism, where two righthanded neutrinos are added to the Standard Model of particle physics. Assuming that both the left and righthanded neutrino fields transform under \(\mu \tau \) reflection symmetry, we obtain the required forms of the neutrino Dirac mass matrix and the Majorana mass matrix for the righthanded neutrinos. To investigate the neutrino phenomenology at low energies, we first consider the breaking of \(\mu \tau \) reflection symmetry due to the renormalization group running, and then systematically study various breaking schemes by introducing explicit breaking terms at high energies.
1 Introduction
Over past two decades, phenomenal neutrino oscillation experiments have established the formalism of three flavor neutrino oscillations and determined two mass squared differences and three mixing angles. At present, unknowns in neutrino oscillation physics are: the neutrino mass hierarchy, i.e., whether neutrinos obey normal hierarchy (NH, \(m_1< m_2 < m_3\) with \(m_i\)’s being neutrino masses) or inverted hierarchy (IH, \(m_3 < m_1 \sim m_2\)); the octant of atmospheric mixing angle \(\theta _{23}\), and the determination of Dirac CPviolating phase \(\delta \).^{1} Were neutrinos the Majorana particles, there would then exist two additional Majorana phases, which do not affect neutrino oscillation probabilities but can be probed by the neutrinoless double betadecay (\(0\nu \beta \beta \)) experiments [4].
In the Standard Model (SM) of particle physics neutrinos are massless. One economical way to incorporate nonzero neutrino masses is to add two righthanded neutrinos to the SM and allow lepton number violation,^{2} resulting in the socalled minimal seesaw scenario (see Ref. [5] for a review) within the context of the TypeI seesaw mechanism [6, 7, 8, 9, 10]. Integrating out the heavy righthanded neutrino fields results in the light neutrino mass matrix \(M_\nu \) as \( M_{\nu } \approx  M_D M^{1}_R M^{T}_D \). In this minimal seesaw setup, \(M_D\) is the \((3 \times 2)\) neutrino Dirac mass matrix whereas \(M_R\) is the (\(2 \times 2)\) Majorana mass matrix for the righthanded neutrinos. In the basis where the charged lepton Yukawa matrix \(Y_l\) is diagonal, diagonalizing the light neutrino mass matrix \(M_\nu \) then leads to the lepton mixing matrix, which is found to be sharply different from the quark mixing matrix. Namely, the former is highly nondiagonal while the latter almost diagonal. To explain the peculiar patterns in the lepton mixing matrix, various flavor symmetry models have been considered, e.g., in Refs. [11, 12, 13, 14, 15].
Other than assigning the \(\mu \tau \) reflection symmetry only to the lefthanded neutrino fields, in this work we apply the same symmetry to the righthanded neutrino fields as well. Consequently, both the neutrino Dirac mass matrix \(M_D\) and the Majorana mass matrix \(M_R\) need to satisfy certain relations among their entries. While the resultant light neutrino mass matrix \(M_\nu \) still obeys the relations given in Eq. (2), the \(\mu \tau \) reflection symmetry is now embedded in the minimal seesaw formalism, and both the left and righthanded neutrinos are treated on the same footing. Similar ideas have been studied for the \(\mu \tau \) permutation symmetry [43, 44], while for the \(\mu \tau \) reflection symmetry a detailed study on the scenario as ours is still missing.^{3} Some recent studies on the minimal seesaw model can be found in Refs. [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62].
In this paper we investigate the implications of the above embedding on neutrino phenomenology at low energies, while the possible discussions on the ultraviolet (UV) aspects, such as explaining the baryon asymmetry via leptogenesis [63] deferred to the future work. For the low energy neutrino phenomenology, since the resultant light neutrino mass matrix \(M_\nu \) in Eq. (10) still preserves the usual \(\mu \tau \) reflection symmetry, one may conclude that there is no new prediction in this seesaw embedded setup. However, we want to point out here at least two deserving issues which need careful scrutiny. The first one is to study the breaking of \(\mu \tau \) reflection symmetry due to the renormalization group (RG) running. As now we impose the \(\mu \tau \) reflection symmetry above the seesaw mass thresholds, an investigation on the RGrunning effects is then inevitable in order to confront with the current globalfit of neutrino oscillation data [1, 2, 3] at low energies. Secondly, although the latest experimental data of T2K [64] and NO\(\nu \)A [65] are in good agreement with the predictions of the exact \(\mu \tau \) reflection symmetry,^{4} there still exist large uncertainties in the measurement of \(\theta _{23}\). For instance, the bestfit values of \(\theta _{23}\) for the lower and higher octants are \(43.6^\circ \) and \(48.3^\circ \) in Ref. [65], respectively. Thus, it is tenacious to believe the exactness of \( \mu \tau \) reflection symmetry, especially that there may exist large discrepancies when upcoming experimental data will be included. Therefore, it is also worthwhile to study how we can perturb such an exact \(\mu \tau \) reflection symmetry, so that remarkable deviations can be observed in the lepton mixing parameters.
We organize our paper as follows. In Sect. 2, we introduce the \(\mu \tau \) reflection symmetry transformations to the left and righthanded neutrino fields, and discuss the required forms of \(M_D\) and \(M_R\). In Sect. 3, we proceed to discuss the breaking of \(\mu \tau \) reflection symmetry due to the RG running, followed by the systematic investigation on all the possible explicit breaking patterns of \(M_D\) and \(M_R\) in Sect. 4. Finally, we summarize our findings in Sect. 5. Details of derivations, explanations and numerical results are relegated to the Appendices.
2 \( \mu \tau \) reflection symmetry embedded in minimal seesaw
3 Breaking due to renormalization group running
We start with the investigation on the breaking of \({\mu }{\tau }\) reflection symmetry due to the RG running. As one possible ultraviolet extension of the SM, the minimal supersymmetric standard model (MSSM) is taken to be our theoretical framework at high energies.^{5} Within MSSM, the neutrino Yukawa coupling in Eq. (3) needs to be modified to \(\overline{\nu _L} Y_\nu N_R H_u\), where \(H_u\) is the Higgs field that also couples to the upquark sector. When \(H_u\) picking up the vacuum expectation value, i.e., \(\langle H_u \rangle = v_u = v \sin \beta \), the neutrino Dirac mass matrix \(M_D\) becomes as \(M_D = v \sin \beta Y_\nu \). Moreover, we take the scale of grand unified theories (GUTs) (\(\Lambda _\mathrm {GUT}\)) as the high energy boundary scale, at which the \(\mu \tau \) reflection symmetry is viewed to be exact in \(Y_\nu \) (or \(M_D\)) and \(M_R\).
The final stage of RG running starts from the mass threshold of \(N_1\) and stops at a chosen low energy scale. Here we take the low energy scale to be the electroweak scale \(\Lambda _{\mathrm {EW}}\). This stage of RG running is below the seesaw threshold, and its impact on the lepton mixing parameters has been extensively discussed in the literature, e.g., Refs. [68, 69, 70, 71]. In particular, the breaking of \(\mu \tau \) reflection symmetry due to this stage of RG running is investigated in Refs. [28, 72]. To save space, we then would not elaborate more on this stage of running.
In Fig. 1 we show the numerical result for the NH case. The black scatter points have \(\chi ^2 < 30\), and the bestfit (BF) point that has the minimal value of \(\chi ^2\), denoted as \(\chi ^2_\mathrm {min}\), is shown in red. In this NH case, we obtain \(\chi ^2_\mathrm {min} = 19.34\) for the BF point. In the \(M_2\) vs. \(M_1\) plot of Fig. 1, we display the spread of two mass thresholds \(M_{1,2}\) for the righthanded neutrinos. It can be seen that \(M_1\) and \(M_2\) are quite to close each other and both of the order of \(10^{14}~\mathrm {GeV}\). One possible explanation for such closeness of \(M_1\) and \(M_2\) is that the entries in two columns of \(Y_\nu \) are related by the \(\mu \tau \) reflection symmetry, particularly due to the symmetry transformation on \(N_R\). Therefore, no large hierarchy exists between the two columns of \(Y_\nu \), and then in order to yield mild hierarchy in the light neutrino mass matrix, the entries in \(M_R\) also tend to be close to each other, resulting in similar values of \(M_1\) and \(M_2\). Because of the closeness of \(M_1\) and \(M_2\), the second stage of RG running between two mass thresholds turns out to be insignificant, and thus we focus on the first and third stages of running in the following.

In this NH case all \(\Delta x^\mathrm {LH}\)’s (shown as the xaxis) are rather small, indicating the mixing angles and phases receive small deviations from the RG running. The small deviations at the third stage of RG running are expected, as it is known that in NH the RG running of mixing angles and phases are insignificant below the seesaw threshold [68, 69, 70]. For the first stage of running, the corrections to \(M_\nu ^{(2)}\) are at the order of \( \ln (\Lambda _\mathrm {GUT} / M_2) /(16\pi ^2) \sim 0.03\), assuming \(Y_\nu \) to be of \(\mathcal {O}(1)\). Therefore, in the NH case the contributions from the first stage of running are also small.

Regarding the relative contributions between the first and third stages of running, we notice that for the Dirac phase \(\delta \), the third stage of running tends to yield larger deviations than the first stage, as \(\Delta x^{\mathrm {LT}} / \Delta x^\mathrm {TH}  > 1\) for most of scatter points. However, for the three mixing angles and the Majorana phase \(\sigma \), the first stage of running turns out to be more important than, or as important as, the first stage. It then demonstrates that in this seesaw embedded setup, the RG running effects above the seesaw threshold can be comparable with that below the seesaw threshold.

As for the breaking of \(\mu \tau \) reflection symmetry, \(\theta _{23}\) at low energies tends to be always larger than \(45^\circ \), while \(\delta \) and \(\sigma \) can receive either positive or negative deviations from RG running. The correlation between the positive deviation of \(\theta _{23}\) and NH is in agreement with the previous RG running studies below the seesaw threshold [77, 78].
We next turn to the IH case, and the corresponding numerical results are shown in Fig. 3. We find that in IH it becomes harder to search for the scatter points that have small values of \(\chi ^2\). Thus, here we show the scatter points that satisfy \(\chi ^2 < 80\), and the BF point has \(\chi ^2_\mathrm {min} = 41.75\). The detailed RG running of mixing parameters for the BF point is shown in the middle three plots of Fig. 2. By inspecting plots in Fig. 3, we first observe that \(M_1\) and \(M_2\) are also close to each other, as in the NH case. The running of the three mixing angles is again quite mild, except that in \(\theta _{12}\) there exists a branch of scatter points that can have deviations as large as \(30^\circ \). To have better understanding of such large deviations in \(\theta _{12}\), in the right three plots of Fig. 2 we present the detailed RG running for the scatter point that has the least value of \(\chi ^2\) (\(\chi ^2_\mathrm {min} = 51.3\)) in that branch of scatter points. One then identifies a sharp decline in \(\theta _{12}\) during the first stage of running. Such a decline can be traced to the potential crossing of neutrino masses when \(\mu \sim 5 \times 10^{15}~\mathrm {GeV}\) (red arrow). As interchanging the order of eigenvalues would lead to a \(90^\circ \) rotation in the mixing angle, it also explains why the sum of the values of \(\theta _{12}\) before and after the decline is around \(90^\circ \). From the running plot of \(\delta \), such interchange of eigenvalues seems to induce a \(180^\circ \) change in \(\delta \) as well.
Corrections to neutrino masses and lepton mixing angles according to the three breaking patterns in \(M_D\). For simplicity, we assume that all entries in \(\Delta M_\nu \) are real, and only the leading order corrections in terms of \(\epsilon \), \(\theta _{13}\) and \(\zeta = m_2/m_3\) (\(\xi = \Delta m_{21}^2 / m_2^2\)) for NH (IH) are kept. Shorthand notations of \(\widehat{A}_{(23)} \equiv \widehat{A}_2 + \widehat{A}_3\), \(\widehat{A}_{[23]} \equiv \widehat{A}_2  \widehat{A}_3\), \(\widehat{A}_{(23\phi )} \equiv \widehat{A}_3 \cos 2\phi + \widehat{A}_2\) and \(\widehat{A}_{[23\phi ]} \equiv \widehat{A}_3 \cos 2\phi  \widehat{A}_2\) are adopted
NH  IH  

\(\mathbf {S1:}\) \(M_D = \begin{pmatrix}b &{}\quad b^* (1+\epsilon ) \\ c &{} \quad d \\ d^* &{} \quad c^* \end{pmatrix} \)  \(\Delta m_1\)  0  \(\epsilon \mathcal {B}_{12} \left[ 2\widehat{A}_1 c_{12}^{2}  \sqrt{2} \widehat{A}_{(23)} s_{12} c_{12} \sin \phi \right] \) 
\(\Delta m_2\)  \(\epsilon \mathcal {B}_{12} \left[ 2\widehat{A}_1 s_{12}^{2} + \sqrt{2} \widehat{A}_{(23)} s_{12} c_{12} \sin \phi \right] \)  \(\epsilon \mathcal {B}_{12} \left[ 2\widehat{A}_1 s_{12}^{2} + \sqrt{2} \widehat{A}_{(23)} s_{12} c_{12} \sin \phi \right] \)  
\(\Delta m_3\)  \(\sqrt{2} \epsilon \theta _{13} \mathcal {B}_{12} \widehat{A}_{(23)} \cos \phi \)  0  
\(\Delta M_\nu \simeq \epsilon \mathcal {B}_{12} \begin{pmatrix} 2 \widehat{A}_1 &{} \quad \widehat{A}_2 &{}\quad \widehat{A}_3 \\ \widehat{A}_2 &{}\quad 0 &{}\quad 0 \\ \widehat{A}_3 &{}\quad 0 &{}\quad 0 \\ \end{pmatrix} \)  \(\Delta \theta _{12}\)  \(\frac{\epsilon \mathcal {B}_{12}}{2 m_3 \zeta } \left[ \widehat{A}_1 \sin 2 \theta _{12} + \sqrt{2} \widehat{A}_{(23)} \cos 2\theta _{12} \sin \phi \right] \)  \(\frac{\epsilon \mathcal {B}_{12}}{ m_2 \xi } \left[ 2 \widehat{A}_1 \sin 2\theta _{12} + \sqrt{2} \widehat{A}_{(23)} \cos 2\theta _{12} \sin \phi \right] \) 
\(\Delta \theta _{13}\)  \( \frac{\epsilon \mathcal {B}_{12}}{\sqrt{2} m_3} \widehat{A}_{(23)} \cos \phi \)  \( \frac{\epsilon \mathcal {B}_{12}}{\sqrt{2} m_2} \widehat{A}_{(23)} \cos \phi \)  
\(\Delta \theta _{23}\)  \( \frac{\epsilon \mathcal {B}_{12}}{\sqrt{2} m_3} \theta _{13} \widehat{A}_{[23]} \cos \phi \)  \( \frac{\epsilon \mathcal {B}_{12} }{\sqrt{2} m_2} \theta _{13} \widehat{A}_{[23]} \cos \phi \)  
\(\mathbf {S2:}\) \(M_D = \begin{pmatrix}b &{}\quad b^* \\ c &{}\quad d (1+\epsilon ) \\ d^* &{}\quad c^* \end{pmatrix} \)  \(\Delta m_1\)  0  \(\epsilon \mathcal {B}_{22} \left[ \widehat{A}_{[23\phi ]} s^2_{12} + \sqrt{2} \widehat{A}_1 s_{12} c_{12} \sin \phi \right] \) 
\(\Delta m_2\)  \(\epsilon \mathcal {B}_{22} \left[ \widehat{A}_{[23\phi ]} c^2_{12}  \sqrt{2} \widehat{A}_1 s_{12} c_{12} \sin \phi \right] \)  \(\epsilon \mathcal {B}_{22} \left[ \widehat{A}_{[23\phi ]} s^2_{12}  \sqrt{2} \widehat{A}_1 s_{12} c_{12} \sin \phi \right] \)  
\(\Delta m_3\)  \(\epsilon \mathcal {B}_{22} \widehat{A}_{(23\phi )}\)  0  
\(\Delta M_\nu \simeq \epsilon \mathcal {B}_{22} \begin{pmatrix} 0 &{}\quad \widehat{A}_1 &{} 0 \\ \widehat{A}_1 &{}\quad 2 \widehat{A}_3 &{}\quad \widehat{A}_2 \\ 0 &{}\quad \widehat{A}_2 &{}\quad 0 \end{pmatrix} \)  \(\Delta \theta _{12}\)  \(\frac{\epsilon \mathcal {B}_{22}}{2 m_3 \zeta } \left[ \widehat{A}_{[23\phi ]} \sin 2\theta _{12} + \sqrt{2} \widehat{A}_1 \cos 2\theta _{12} \sin \phi \right] \)  \(\frac{\epsilon \mathcal {B}_{22}}{m_2 \xi } \left[ \widehat{A}_{[23\phi ]} \sin 2\theta _{12} + \sqrt{2} \widehat{A}_1 \sin \phi \cos 2\theta _{12} \right] \) 
\(\Delta \theta _{13}\)  \( \frac{\epsilon \mathcal {B}_{22}}{\sqrt{2} m_3} \widehat{A}_1 \cos \phi \)  \( \frac{\epsilon \mathcal {B}_{22}}{\sqrt{2} m_2} \widehat{A}_1 \cos \phi \)  
\(\Delta \theta _{23}\)  \( \frac{\epsilon \mathcal {B}_{22}}{ m_3}\widehat{A}_3\cos 2\phi \)  \( \frac{\epsilon \mathcal {B}_{22}}{m_2} \widehat{A}_3\cos 2\phi \)  
\(\mathbf {S3:}\) \(M_D = \begin{pmatrix}b &{}\quad b^* \\ c &{}\quad d \\ d^* &{} \quad c^* (1+\epsilon ) \end{pmatrix} \)  \(\Delta m_1\)  0  \(\epsilon \mathcal {B}_{32} \left[ \widehat{A}_{[23\phi ]} s^2_{12} + \sqrt{2} \widehat{A}_1 s_{12} c_{12} \sin \phi \right] \) 
\(\Delta m_2\)  \(\epsilon \mathcal {B}_{32} \left[ \widehat{A}_{[23\phi ]} c^2_{12}  \sqrt{2} \widehat{A}_1 s_{12} c_{12} \sin \phi \right] \)  \(\epsilon \mathcal {B}_{32} \left[ \widehat{A}_{[23\phi ]} c^2_{12}  \sqrt{2} \widehat{A}_1 s_{12} c_{12} \sin \phi \right] \)  
\(\Delta m_3\)  \(\epsilon \mathcal {B}_{32} \widehat{A}_{(23\phi )} \)  0  
\(\Delta M_\nu \simeq \epsilon \mathcal {B}_{32} \begin{pmatrix} 0 &{}\quad 0 &{}\quad \widehat{A}_1 \\ 0 &{}\quad 0 &{}\quad \widehat{A}_2 \\ \widehat{A}_1 &{}\quad \widehat{A}_2 &{}\quad 2 \widehat{A}_3 \end{pmatrix} \)  \(\Delta \theta _{12}\)  \( \frac{\epsilon \mathcal {B}_{32}}{2 m_3 \zeta } \left[ \widehat{A}_{[23\phi ]} \sin 2\theta _{12} + \sqrt{2} \widehat{A}_1 \sin \phi \cos 2\theta _{12} \right] \)  \(\frac{\epsilon \mathcal {B}_{32}}{m_2 \xi } \left[ \widehat{A}_{[23\phi ]} \sin 2\theta _{12} + \sqrt{2} \widehat{A}_1 \sin \phi \cos 2\theta _{12} \right] \) 
\(\Delta \theta _{13}\)  \( \frac{\epsilon \mathcal {B}_{32} }{\sqrt{2} m_3} \widehat{A}_1 \cos \phi \)  \( \frac{\epsilon \mathcal {B}_{32}}{\sqrt{2} m_2} \widehat{A}_1 \cos \phi \)  
\(\Delta \theta _{23}\)  \( \frac{\epsilon \mathcal {B}_{32}}{m_3} \widehat{A}_3 \cos 2 \phi \)  \( \frac{\epsilon \mathcal {B}_{32}}{m_2} \widehat{A}_3 \cos 2 \phi \) 
In Appendix D we present the numerical values of the neutrino Yukawa matrices and the Majorana mass matrices for the righthanded neutrinos at \(\Lambda _\mathrm {GUT}\) for the three scenarios shown in Fig. 2. The lepton mixing parameters at various energy scales are also shown.
4 Breaking \(\mu \tau \) reflection symmetry in \(M_D\) and \(M_R\)
From the previous RG running study we notice that in both NH and IH the breaking effects due to the RG running are quite mild. For instance, the deviations in \(\theta _{23}\) are only around one degree. Although such small deviations are in compatible with current experimental data, it may become necessary to consider large deviations when more accurate data will be included. In this section, we set out to discuss the breaking of \(\mu \tau \) reflection symmetry in the low energy neutrino mass matrix by introducing explicit breaking terms in the neutrino Dirac mass matrix \(M_D\) and the Majorana mass matrix \(M_R\) for the righthanded neutrinos. As the RG running effects are found to be mild, for simplicity we choose to ignore them in the following discussion.
4.1 Breaking \(\mu \tau \) reflection symmetry in \(M_D\)
Similarly, one can introduce breaking terms in the other entries of \(M_D\). Without loss of generality, in the middle and bottom blocks of Table 1 we show the results for the other two breaking patterns in \(M_D\), namely, assigning breaking terms in the (22) and (32) positions of \(M_D\) and resulting in the breaking scenarios of \(\mathbf {S2}\) and \(\mathbf {S3}\), respectively. We notice that the deviations of the neutrino mass matrix \(\Delta M_\nu \) can also be expressed in terms of the parameters \(\widehat{A}_i\)’s, except that the overall breaking parameters are modified to be \(\mathcal {B}_{22} = b/\mathrm {det}(M_R)\) and \(\mathcal {B}_{32} = c^*/\mathrm {det}(M_R)\) for \(\mathbf {S2}\) and \(\mathbf {S3}\), respectively. We also observe that the analytic expressions for \(\Delta \theta _{ij}\)’s and \(m_i\)’s in \(\mathbf {S2}\) and \(\mathbf {S3}\) are quite similar, and in both scenarios there is no suppression factor of \(\theta _{13}\) in \(\Delta \theta _{23}\). As a result, one may expect larger deviation in \(\theta _{23}\) in \(\mathbf {S2}\) and \(\mathbf {S3}\) than that in \(\mathbf {S1}\). Thus, the last two breaking patterns may be distinguishable from the first one via the future precision measurement of \(\theta _{23}\).

According to the top three plots, we find that \(\Delta \theta _{23}\) in \(\mathbf {S1}\) is less than one degree, much smaller than that in the other two breaking patterns. This numerical finding agrees with the analytical results in Table 1, i.e., \(\Delta \theta _{23}\) is suppressed by a factor of \(\theta _{13}\) in \(\mathbf {S1}\).

In all three breaking patterns we observe correlations between \(\theta _{23}^\prime \) and \(\delta ^\prime \). For \(\mathbf {S1}\), a “oscillatory” pattern is identified. In Refs. [80, 81, 82], a similar “oscillatory” correlation between \(\theta _{23}\) and \(\delta \) was also obtained under the assumption of partial \(\mu \tau \) symmetry in the lepton mixing matrix. However, the “oscillatory” pattern observed here differs from that in Refs. [80, 81, 82] in \(\Delta \theta _{23}\), i.e., here \(\Delta \theta _{23} \lesssim 1^\circ \) while \(\Delta \theta _{23} \gtrsim 5^\circ \) in Refs. [80, 81, 82]. In addition, in \(\mathbf {S1}\), \(\Delta \delta \) and \(\Delta \theta _{23}\) seem to have a negative correlation when \(\delta ^\prime \sim  90^\circ \), and the deviation in \(\delta ^\prime \) is much more dramatic than that in \(\theta _{23}^\prime \). \(\delta ^\prime \) can reach 0 or \(\pm 180^\circ \), while \(\theta _{23}\) only less than one degree away from \(45^\circ \). For \(\mathbf {S2}\) and \(\mathbf {S3}\), however, the differences between \(\Delta \delta \) and \(\Delta \theta _{23}\) are less dramatic. In both scenarios there exist two branches of predictions that \(\Delta \delta \) and \(\Delta \theta _{23}\) can have a positive or negative correlation. In the case with positive correlation \(\delta ^\prime \) and \(\theta _{23}^\prime \) deviate by almost the same amount, while \(\Delta \delta \) is about three times larger than \(\Delta \theta _{23}\) for the case with negative correlation.

From the plots in the second row of Fig. 4 we find that \(\Delta \theta _{12}\) is indeed larger than \(\Delta \theta _{13}\) and \(\Delta \theta _{23}\), and it can reach around \(15^\circ \) for all three breaking patterns. This is also in agreement with the analytical results given in Table 1. Moreover, the value of \(\Delta \delta _{12} \sim 15^\circ \) (\(5^\circ \)) indicates that \(\theta _{12}\) before breaking can be quite close to \(45^\circ \) (\(30^\circ \)), and this may have interesting implications in the flavor model building with the exact \(\mu \tau \) reflection symmetry at high energies.

Lastly, in the bottom row of Fig. 4 we show the results for the total neutrino mass \(\sum m_\nu \equiv m_1 + m_2 + m_3\) and \(m_{ee}\) after the breaking. Here \(m_{ee}\) is the (11) element of \(M_\nu ^\prime \), and it is responsible for the decay rates of neutrinoless double betadecay modes of various isotopes. As expected, in NH we have \(m_1 = 0\) and then satisfying the masssquared differences from neutrino oscillation experiments leads to \(\sum m_\nu \sim 0.06~\mathrm {eV}\). Also, because of NH and \(m_1 = 0\), the predicted \(m_{ee}\) is only a few meV’s. Such small values of \(\sum m_\nu \) and \(m_{ee}\) would be hard to probe by upcoming cosmological observations and \(0\nu \beta \beta \) experiments [83, 84, 85, 86, 87, 88], respectively.

In \(\mathbf {S1}\), \(\theta _{23}^\prime \) tends to be very close to \(45^\circ \), although the BF point has a large deviation in \(\theta _{23}\), which may originate from some special combination in the input parameters. The obtained \(\delta ^\prime \), however, has a large spread in \([180^\circ , 180^\circ )\). Regarding \(\theta _{12}\) and \(\theta _{13}\), the deviation in \(\theta _{13}\) is quite small in general, while for \(\theta _{12}\) large deviations of \(\mathcal {O}(10^\circ )\) can be easily achieved.

As for \(\mathbf {S2}\) and \(\mathbf {S3}\), the favored parameter space are almost the same. Interestingly, in both scenarios it seems that \(\delta ^\prime \) and \(\theta _{23}^\prime \) exhibit similar oscillatory patterns as in the case of \(\mathbf {S1}\) under NH. Unfortunately, it is analytically difficult to confirm if there indeed exist connections among these scenarios, especially two different mass orderings are involved. In contrast with \(\mathbf {S2}\) and \(\mathbf {S3}\) in NH, the favored \(\theta _{23}^\prime \)’s are now close to \(45^\circ \), while large spreads are observed in \(\delta ^\prime \). On the other hand, the deviations in \(\theta _{13}\) are less than one degree in both scenarios, while, as expected, \(\theta _{12}\) can easily achieve \(\mathcal {O}(10^\circ )\) deviations, due to the enhancement factor of \(1/\xi \).

Lastly, in the \(m_\mathrm {ee}\) vs. \(\sum m_\nu \) plots we observe that for all three breaking scenarios the obtained \(\sum m_\nu \)’s are close to \(0.1~\mathrm {eV}\). This finding agrees with our expectation that with \(m_3 =0 \) the other neutrino masses \(m_1\) and \(m_2\) need to be \(m_{1,2} \sim 0.05~\mathrm {eV}\) so as to satisfy the currently measured masssquared differences. For \(m_\mathrm {ee}\), although there exist some spread within \([10, 50]~\mathrm {meV}\), most of scatter points are located around \(15~\mathrm {meV}\). This is due to the fact that even with breaking the favored \(\sigma ^\prime \)’s after breaking are also quite close to \(90^\circ \). As a result, \(m_\mathrm {ee}\) can approximate to \(m_\mathrm {ee} \sim m_1 \cos ^2\theta _{12} + e^{2i\sigma } m_2 \sin ^2\theta _{12} \sim m_1 \cos ^2\theta _{12}  m_2 \sin ^2\theta _{12}\), then with \(m_1 \sim m_2\) we have a significant cancellation between the two terms in \(m_\mathrm {ee}\). Therefore, comparing with the NH case, although now we can have larger values of \(m_\mathrm {ee}\)’s, the value of \(\sigma ^\prime \sim 90^\circ \) still results in relatively small values of \(m_\mathrm {ee} \sim 15~\mathrm {meV}\). Such small values of \(m_\mathrm {ee} \) are close to the lower bound of \(m_\mathrm {ee}\) in IH, and thus future tonscale \(0\nu \beta \beta \) experiments are needed in order to fully cover the favored parameter space.
So far we have focused on the breaking parameters reside in the neutrino Dirac mass matrix \(M_D\). Next, we turn to the breaking patterns in the Majorana mass matrix \(M_R\) for the righthanded neutrinos.
4.2 Breaking \(\mu \tau \) reflection symmetry in \(M_R\)
Summary of various breaking scenarios in NH. Note that for \(\mathbf {S2}\) and \(\mathbf {S3}\) two rows correspond to black and gray patterns in Fig. 4, respectively
Breaking scenarios  \( \theta _{23}^{\prime } \) (deg)  \( \delta _{CP}^{\prime }\) (deg)  \( \Delta \theta _{12}^{\prime } \) (deg)  \( \Delta \theta _{13}^{\prime } \) (deg)  \( \sum m_\nu \) (eV)  \( m_{ee} \) (meV) 

\(\mathbf {S1}\)  \(44.3 \rightarrow 45.7\)  \( 180 \rightarrow 180\)  \( 15 \rightarrow 10\)  \( 1 \rightarrow 9\)  \(0.0575 \rightarrow 0.061\)  \(1 \rightarrow 4.2\) 
\(\mathbf {S2}\)  \(35 \rightarrow 46\)  \( 100 \rightarrow  88\)  \( 18 \rightarrow 1\)  \( 0.1 \rightarrow 1.3\)  \(0.057 \rightarrow 0.061\)  \(3 \rightarrow 4.5\) 
\(40 \rightarrow 45\)  \( 90 \rightarrow  70\)  \(0 \rightarrow 9\)  \(0 \rightarrow 1.2\)  –  –  
\(\mathbf {S3}\)  \(37.5 \rightarrow 47\)  \( 98 \rightarrow  88\)  \(2 \rightarrow 7\)  \( 1.4 \rightarrow 0.2\)  \(0.057 \rightarrow 0.0615\)  \(3 \rightarrow 4.5\) 
\(46 \rightarrow 47\)  \( 94 \rightarrow  56\)  \(20 \rightarrow 3\)  \(  1.7 \rightarrow 0.3\)  –  –  
\(\mathbf {S4}\)  \(43 \rightarrow 46\)  \( 100 \rightarrow  88\)  \( 0.2 \rightarrow 0.7\)  \( 3 \rightarrow 1\)  \(0.0575 \rightarrow 0.061\)  \(3.1 \rightarrow 4.4\) 
\(\mathbf {S5}\)  \(39 \rightarrow 46.5\)  \( 120 \rightarrow 84\)  \( 1 \rightarrow 2.6\)  \( 8 \rightarrow 8\)  \(0.057 \rightarrow 0.061\)  \(3 \rightarrow 4.5\) 
Summary of various breaking scenarios in IH
Breaking scenarios  \( \theta _{23}^{\prime } \) (deg)  \( \delta _{CP}^{\prime }\) (deg)  \( \Delta \theta _{12}^{\prime } \) (deg)  \( \Delta \theta _{13}^{\prime } \) (deg)  \( \sum m_\nu \) (eV)  \( m_{ee} \) (meV) 

\(\mathbf {S1}\)  \( \sim 45\)  \( 180 \rightarrow 180\)  \(0 \rightarrow 20\)  \( \sim 0\)  \(0.097 \rightarrow 0.104\)  \(10 \rightarrow 50\) 
\(\mathbf {S2}\)  \(44.4 \rightarrow 45.7\)  \( 180 \rightarrow 180\)  \( 60 \rightarrow 40\)  \( 0.5 \rightarrow 0.5\)  \(0.097 \rightarrow 0.104\)  \(10 \rightarrow 50\) 
\(\mathbf {S3}\)  \(44.4 \rightarrow 45.8\)  \( 180 \rightarrow 180\)  \( 60 \rightarrow 20\)  \( 0.5 \rightarrow 0.5\)  \(0.097 \rightarrow 0.104\)  \(10 \rightarrow 50\) 
\(\mathbf {S4}\)  \( \sim 45 \)  \( 180 \rightarrow 180\)  \(0 \rightarrow 40\)  \( \sim 0\)  \(0.097 \rightarrow 0.104\)  \(10 \rightarrow 50\) 
\(\mathbf {S5}\)  \( \sim 45\)  \( 180 \rightarrow 180\)  \( 30 \rightarrow 25\)  \( \sim 0\)  \(0.097 \rightarrow 0.104\)  \(10 \rightarrow 48\) 
We now turn to the numerical results for \(\mathbf {S4}\) and \(\mathbf {S5}\) in IH, see Fig. 7. Unexpectedly, we observe that in both \(\mathbf {S4}\) and \(\mathbf {S5}\) the deviations in \(\theta _{23}\) and \(\theta _{13}\) are exactly zero. A full understanding of such null deviations is hard to pursue by considering a generic form of \(M_D\) and \(M_R\), while in Appendix C we demonstrate such null deviations considering a special case. The deviations in \(\delta \) and \(\theta _{12}\), however, can be rather large. Similar to \(\mathbf {S2}/\mathbf {S3}\) in IH, we also have \(\sum m_\nu \sim 0.1~\mathrm {eV}\), and because the Majorana phase \(\sigma ^\prime \) still favors \(90^\circ \) after breaking, the preferred \(m_\mathrm {ee}\) is again around \(15~\mathrm {meV}\).
Above we have discussed various breaking scenarios of exact \(\mu \tau \) reflection symmetry, and finally we summarize these results in Tables 2 and 3. In the context of ongoing neutrino oscillation experiments some of the breaking scenarios can be ruled out. For example, as the latest results of both T2K [64] and NO\( \nu \)A [65] favor NH over IH, and if it remains true, all the breaking schemes corresponding to IH can be ruled out. Furthermore, if in the upcoming experiments \(\theta _{23}\) were found to be close to \(45^\circ \) within only one degree, the breaking scenarios S2, S3, S4 and S5 in NH would be disfavored. However, to fully exclude these scenarios precise measurement of \(\delta _{CP}\) in the future experiments, such as DUNE [89], T2HK [90] and MOMENT [91], may be needed.
5 Conclusion
In this work we explore the possibility of embedding the \(\mu \tau \) reflection symmetry in the minimal seesaw formalism, where two righthanded neutrinos are added to the SM. Different from the previous works, we apply the \(\mu \tau \) reflection symmetry transformations to both the left and righthanded neutrinos, resulting in some particular forms of neutrino Dirac mass matrix \(M_D\) and the Majorana mass matrix \(M_R\) for the righthanded neutrinos. The obtained light neutrino mass matrix \(M_\nu \) is found to still possess the usual \(\mu \tau \) reflection symmetry, which predicts maximal atmospheric mixing angle (\( \theta _{23} = 45^\circ \)) and Dirac CP phase (\( \delta = \pm 90 ^\circ \)) along with the trivial Majorana phases. We later extend our study by incorporating the breaking of such symmetry, keeping in mind that theoretical as well as experimental results may favor nonmaximal \( \theta _{23}\).

The RG running between the thresholds is insignificant, as the two seesaw mass thresholds are found to be quite close. Such closeness of two thresholds is due to the fact that the two columns of the neutrino Yukawa matrix are related by the \(\mu \tau \) reflection symmetry, particularly the symmetry on the righthanded neutrinos as proposed here.

For both NH and IH scenarios, we find that the RG running effects above the seesaw thresholds are comparable to those below the thresholds. This would raise the necessity of considering RG running above the seesaw thresholds, if some flavor symmetry were imposed on the righthanded neutrino fields.

For the three mixing angles, the deviations due to the RG running are all rather small, e.g., \(\Delta \theta _{23} \lesssim 1^\circ \), except that for \(\theta _{12}\) in IH can there exist a large deviation. The latter exception arises from the fact that the two light neutrino masses may cross each other, leading to an interchange of the order of two neutrino masses.

The RG running effects of the Dirac and Majorana phases are also quite mild in NH, while large deviations of \(\mathcal {O}(10^\circ )\) can be observed in the case of IH.

Lastly, we note that the known correlation between the positive/negative deviation of \(\theta _{23}\) and the neutrino mass hierarchies are again observed in this extended RG running above the seesaw thresholds.

In NH we find that \(\Delta \theta _{23} \lesssim 0.5^\circ \) in \(\mathbf {S1}\) while \(\Delta \theta _{23}\) of a few degrees can be easily observed for the other breaking patterns. On the other hand, in IH all breaking patterns tend to have \(\Delta \theta _{23} \lesssim 0.5^\circ \), especially \(\Delta \theta _{23} = 0\) seems to hold exactly for \(\mathbf {S4}\) and \(\mathbf {S5}\).

For the deviations in \(\theta _{13}\), we obtain \(\Delta \theta _{13} \lesssim 1^\circ \) for \(\mathbf {S2}\), \(\mathbf {S3}\) and \(\mathbf {S4}\) in NH, while for \(\mathbf {S1}\) and \(\mathbf {S5}\) deviations of a few degrees are possible. However, in the case of IH all breaking patterns tend to have small deviations in \(\theta _{13}\), and again \(\Delta \theta _{13} = 0\) seems to hold exactly in \(\mathbf {S4}\) and \(\mathbf {S5}\) as well.

The deviations in \(\theta _{12}\) are found to be around \(\mathcal {O}(10^\circ )\) in general, except that for \(\mathbf {S4}\) and \(\mathbf {S5}\) we observe \(\Delta \theta _{12} \lesssim 1^\circ \).

For the Dirac CPviolating phase \(\delta \), the resultant values after breaking are extended to the whole range of \([180^\circ , 180^\circ )\) for \(\mathbf {S1}\) in NH and all breaking patterns in IH. For \(\mathbf {S2}\), \(\mathbf {S3}\), \(\mathbf {S4}\) and \(\mathbf {S5}\) in NH we identify linear correlations between \(\delta \) and \(\theta _{23}\) when \(\delta \sim 90^\circ \). Such correlations may be tested in the upcoming neutrino experiments.

The Majorana phase \(\sigma \) after the breaking tends to favor \(90^\circ \), which causes the effective neutrino mass \(m_{ee}\) to be around \(15~\mathrm {meV}\) for IH while only about \(4~\mathrm {meV}\) for NH. Such small values of \(m_{ee}\) pose challenges for the upcoming \(0\nu \beta \beta \) experiments.
Footnotes
Notes
Acknowledgements
We like to thank Shun Zhou, Guoyuan Huang, Jingyu Zhu and Zhenhua Zhao for useful discussions. The research work of NN and ZZX were supported in part by the National Natural Science Foundation of China under Grant no. 11775231. JZ was supported in part by the China Postdoctoral Science Foundation under Grant no. 2017M610008.
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