Trimaximal $\mu$-$\tau$ reflection symmetry

TL;DR

This paper explores a highly predictive lepton mixing framework combining $$-$ au$ reflection symmetry and TM1 mixing, analyzing its stability under renormalization in different neutrino mass models.

Contribution

It introduces a combined $$-$ au$ reflection and TM1 mixing scheme and studies its renormalization stability in the Standard Model and supersymmetric contexts.

Findings

Predicts maximal CP violation and atmospheric mixing angle.

Shows stability of the framework under renormalization effects.

Provides constraints on Majorana phases and mixing angles.

Abstract

The $\mu$-$\tau$ reflection symmetry $(\nu_{e},\thinspace\nu_{\mu},\thinspace\nu_{\tau})\rightarrow(\overline{\nu}_{e},\thinspace\overline{\nu}_{\tau},\thinspace\overline{\nu}_{\mu})$ and the TM1 mixing (a PMNS matrix with the first column fixed to the TBM form) are both well compatible with experiments. If both approaches are simultaneously assumed, all lepton mixing parameters except for $\theta_{13}$ are predicted. In particular, one expects maximal CP violation ($|\delta|=90^{\circ}$), maximal atmospheric mixing ($\theta_{23}=45^{\circ}$), a slightly less-than-TBM solar mixing angle ($\theta_{12}\approx34^{\circ}$), as well as values of $0$ or $\pi$ for the two Majorana phases. We study the renormalization stability of this highly predictive framework when neutrino mass is described by an effective Weinberg operator and by the type I seesaw mechanism, both in the Standard Model and with supersymmetry.

Equations

$${\nu }_e\to {\overline{\nu }}_e,{\nu }_{\mu }\to {\overline{\nu }}_{\tau },{\nu }_{\tau }\to {\overline{\nu }}_{\mu }.$$

$${\theta }_{12}^{TM1}={\cos }^{-1}\left(\frac{\sqrt{2}}{\sqrt{3}\cos {\theta }_{13}}\right)\approx 34.2^{\circ },{\theta }_{12}^{TM2}={\sin }^{-1}\left(\frac{1}{\sqrt{3}\cos {\theta }_{13}}\right)\approx 35.8^{\circ }.$$

$${\mathbb{Z}}_2^{TM1}:\nu \to R^{TM1}\nu ,$$

$${\mathbb{Z}}_2^{\mathrm{CP}}:\nu \to R^{\mu \tau }\overline{\nu },$$

$$R^{{\rm TM1}}\equiv-\frac{1}{3}\left(\begin{array}[]{ccc}1&2&2\\ 2&-2&1\\ 2&1&-2\end{array}\right),$$

$$R^{\mu\tau}\equiv\left(\begin{array}[]{ccc}1&0&0\\ 0&0&1\\ 0&1&0\end{array}\right).$$

$$R^{TM1}{R}^{\mu \tau }=R^{\mu \tau }(R^{TM1}{)}^{\ast }.$$

$$\mathcal{L}\supset -{\nu }_{\alpha }{M}_{\alpha \beta }^{\nu }{\nu }_{\beta }+h.c.,$$

$$(R^{TM1}{)}^T{M}^{\nu }{R}^{TM1}=M^{\nu },$$

$$(R^{\mu \tau }{)}^T{M}^{\nu }{R}^{\mu \tau }=(M^{\nu }{)}^{\ast }.$$

$$M_{11}^{\nu },M_{23}^{\nu }$$

$$M_{12}^{\nu }$$

$$M_{22}^{\nu }$$

$$\mathrm{Im}(M_{22}^{\nu })$$

$$M_{11}^{\nu }$$

$$M^{\nu}=\left(\begin{array}[]{ccc}r+x_{1}+x_{2}&x_{1}&x_{1}\\ x_{1}&x_{2}&r\\ x_{1}&r&x_{2}\end{array}\right)+iy\left(\begin{array}[]{ccc}0&1&-1\\ 1&-2&0\\ -1&0&2\end{array}\right).$$

$$(U^{\nu }{)}^T{M}^{\nu }{U}^{\nu }=\mathrm{diag}(m_1^{\prime },m_2^{\prime },m_3^{\prime }).$$

$$U^{\nu}=\frac{1}{\sqrt{6}}\left(\begin{array}[]{ccc}2&\sqrt{2}c&\sqrt{2}s\\ 1&-\sqrt{2}c-i\sqrt{3}s&i\sqrt{3}c-\sqrt{2}s\\ 1&-\sqrt{2}c+i\sqrt{3}s&-i\sqrt{3}c-\sqrt{2}s\end{array}\right),$$

$$s=\sqrt{\frac{\Delta +x_1-2x_2}{2\Delta }},c=\mathrm{sign}(y)\sqrt{1-\frac{x_1-2x_2+\Delta }{2\Delta }},$$

$$(m_1^{\prime },m_2^{\prime },m_3^{\prime })=\left(r+2x_1+x_2,r+\frac{\Delta }{2}-\frac{x_1}{2},r-\frac{\Delta }{2}-\frac{x_1}{2}\right),$$

$$\Delta \equiv \sqrt{24y^2+\left(x_1-2x_2\right){}^2}.$$

$$U_{\mathrm{PMNS}}=U\mathrm{diag}(1,e^{i{\alpha }_{21}/2},e^{i{\alpha }_{31}/2}),$$

$$U=\left(\begin{array}[]{ccc}c_{12}c_{13}&c_{13}s_{12}&e^{-i\delta}s_{13}\\ -c_{23}s_{12}-e^{i\delta}c_{12}s_{13}s_{23}&c_{12}c_{23}-e^{i\delta}s_{12}s_{13}s_{23}&c_{13}s_{23}\\ -e^{i\delta}c_{12}c_{23}s_{13}+s_{12}s_{23}&-e^{i\delta}c_{23}s_{12}s_{13}-c_{12}s_{23}&c_{13}c_{23}\end{array}\right),$$

$${\rm diag}(1,\thinspace-e^{i\beta},\thinspace e^{-i\beta})U^{\nu}=\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}&\frac{\sqrt{1-s^{2}}}{\sqrt{3}}&\frac{is}{\sqrt{3}}\\ \frac{2i\sqrt{3}s-3\sqrt{2-2s^{2}}}{6\sqrt{3-s^{2}}}&\frac{6+is\sqrt{6-6s^{2}}}{6\sqrt{3-s^{2}}}&\frac{\sqrt{3-s^{2}}}{\sqrt{6}}\\ \frac{2i\sqrt{3}s+3\sqrt{2-2s^{2}}}{6\sqrt{3-s^{2}}}&\frac{-6+is\sqrt{6-6s^{2}}}{6\sqrt{3-s^{2}}}&\frac{\sqrt{3-s^{2}}}{\sqrt{6}}\end{array}\right){\rm diag}(1,\thinspace 1,\thinspace-i),$$

$$\beta =\arg (\sqrt{3}c-i\sqrt{2}s).$$

$${\theta }_{23}=45^{\circ },\delta =-90^{\circ },c_{12}=\sqrt{\frac{2}{3}}\frac{1}{c_{13}}.$$

$$(1,e^{i{\alpha }_{21}/2},e^{i{\alpha }_{31}/2})=\left(1,\sqrt{\mathrm{sign}(m_2^{\prime }/m_1^{\prime })},-i\sqrt{\mathrm{sign}(m_3^{\prime }/m_1^{\prime })}\right).$$

$${\rm diag}(1,\thinspace-e^{i\beta},\thinspace e^{-i\beta})U^{\nu}=\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}&\frac{\sqrt{1-s^{2}}}{\sqrt{3}}&-\frac{is}{\sqrt{3}}\\ \frac{-2i\sqrt{3}s-3\sqrt{2-2s^{2}}}{6\sqrt{3-s^{2}}}&\frac{6-is\sqrt{6-6s^{2}}}{6\sqrt{3-s^{2}}}&\frac{\sqrt{3-s^{2}}}{\sqrt{6}}\\ \frac{-2i\sqrt{3}s+3\sqrt{2-2s^{2}}}{6\sqrt{3-s^{2}}}&\frac{-6-is\sqrt{6-6s^{2}}}{6\sqrt{3-s^{2}}}&\frac{\sqrt{3-s^{2}}}{\sqrt{6}}\end{array}\right){\rm diag}(1,\thinspace-1,\thinspace i),$$

$$\beta =\arg (-\sqrt{3}c+i\sqrt{2}s).$$

$${\theta }_{23}=45^{\circ },\delta =90^{\circ },c_{12}=\sqrt{\frac{2}{3}}\frac{1}{c_{13}},$$

$$(1,e^{i{\alpha }_{21}/2},e^{i{\alpha }_{31}/2})=\left(1,\sqrt{\mathrm{sign}(m_2^{\prime }/m_1^{\prime })},i\sqrt{\mathrm{sign}(m_3^{\prime }/m_1^{\prime })}\right).$$

$${\theta }_{23}=45^{\circ },{\theta }_{12}={\cos }^{-1}(\sqrt{\frac{2}{3}}\frac{1}{c_{13}})\approx 34.2^{\circ },$$

Full text

I Introduction

The structure of neutrino mixing, the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix, is considered as an important clue for possible underlying symmetries of the three generations of fermions in the Standard Model (SM). Many discrete flavor symmetries have been proposed in trying to understand the observed mixing – see, e.g., the reviews Mohapatra and Smirnov (2006); Altarelli and Feruglio (2010); Ishimori et al. (2010); King and Luhn (2013); Feruglio (2015). In particular, it had long been speculated that the neutrino mixing could be tribimaximal (TBM) Harrison et al. (2002); Harrison and Scott (2002a); Xing (2002); Harrison and Scott (2002b, 2003), which could originate from non-Abelian discrete symmetries such as $A_{4}$ and $S_{4}$. However, the TBM mixing predicts zero $\theta_{13}$ which has been excluded by reactor neutrino experiments Abe et al. (2012); An et al. (2012); Ahn et al. (2012).

It is well understood that $\theta_{13}=0$ in TBM is attributed to $\mu$-$\tau$ symmetry Xing and Zhao (2016), which is defined as the invariance of the neutrino mass terms under the interchange of $\nu_{\mu}$ and $\nu_{\tau}$. Therefore in the light of non-zero $\theta_{13}$, breaking the $\mu$-$\tau$ symmetry has been considered and extensively studied in many references in the past. However, there is a variation of the $\mu$-$\tau$ symmetry which does not require any breaking and is still well compatible with experiments. It is called $\mu$-$\tau$ reflection symmetry Harrison and Scott (2002b); Ma (2002a); Babu et al. (2003); Ma (2002b); Grimus and Lavoura (2004), which attaches the CP transformation to the interchange of $\nu_{\mu}$ and $\nu_{\tau}$,

[TABLE]

The $\mu$-$\tau$ reflection symmetry allows non-zero $\theta_{13}$ and predicts $\theta_{23}=45^{\circ}$ and $\delta=\pm 90^{\circ}$. Consequently it has aroused a lot of interest recently Nishi (2013); Ma (2014); Fraser et al. (2014); Li and He (2015); Ma et al. (2015); Li and Ding (2015); Di Iura et al. (2015); Mohapatra and Nishi (2015); Zhou (2014); Joshipura and Patel (2015); He et al. (2015); Zhao (2017); Nishi and Sanchez-Vega (2017); Chen et al. (2016); Fukuyama (2017). To generate TBM mixing the $\mu$-$\tau$ symmetry determines the third column of this mixing matrix and there is another $\mathbb{Z}_{2}$ symmetry that is responsible for the first or second column Lam (2008a, b, 2011). Those $\mathbb{Z}_{2}$ symmetries are assumed to be “residual symmetries”, after the full flavor group is broken. They could be accidental or subgroups of the full flavor group. If the $\mu$-$\tau$ symmetry is replaced with $\mu$-$\tau$ reflection symmetry, then we get a variation of TBM with its first or second column fixed and at the same time we will have non-zero $\theta_{13}$, $\theta_{23}=45^{\circ}$ and $\delta=\pm 90^{\circ}$. We study the consequences of this assumption in this paper. General deviations of the TBM mixing with some part being fixed have been discussed in many references Bjorken et al. (2006); Xing and Zhou (2007); He and Zee (2007); Albright and Rodejohann (2009); Albright et al. (2010); Antusch et al. (2012); He and Zee (2011); de Medeiros Varzielas and Lavoura (2013); Luhn (2013); Li and Ding (2014) and the case that the first/second column is fixed is usually referred to as TM1/TM2 mixing, respectively Albright and Rodejohann (2009). In the TBM mixing, $\theta_{12}=\sin^{-1}\frac{1}{\sqrt{3}}\approx 35.3^{\circ}$ is a little higher than the global best-fit value $\theta_{12}^{{\rm exp}}=33.56_{-0.75}^{+0.77}$ Esteban et al. (2017), while in TM1 or TM2 it deviates from $35.3^{\circ}$ with a lower or a higher value, respectively Albright and Rodejohann (2009):

[TABLE]

Since $\theta_{12}^{{\rm TM1}}$ is well compatible with $\theta_{12}^{{\rm exp}}$ while $\theta_{12}^{{\rm TM2}}$ is disfavored at about $3\sigma$, in this paper we will consider TM1 only.

When the TM1 symmetry111For simplicity, we will refer to the symmetry responsible for the TM1 mixing as the TM1 symmetry in this paper. and $\mu$-$\tau$ reflection symmetry are imposed on the neutrino mass terms simultaneously, all the PMNS parameters except for $\theta_{13}$ are predicted (in addition to the predictions mentioned above, the two Majorana phases are [math] or $\pi$). In the near future, this framework can be tested not only by a precision measurement of $\theta_{12}$ and $\theta_{23}$, but also by the confirmation of a maximal Dirac CP phase $|\delta|=\pi/2$, for which hints have recently appeared in T2K Abe et al. (2014, 2017). Besides, its predictions on Majorana phases could be verified in neutrinoless double beta decay ($0\nu\beta\beta$) experiments Rodejohann (2011).

Note that both the TM1 symmetry and $\mu-\tau$ reflection symmetry may be residual symmetries of a larger flavor symmetry broken at a high energy scale. Since $\mu-\tau$ reflection symmetry is essentially a generalized CP symmetry, looking for a horizontal flavor symmetry that contains it as a subsymmetry is more interesting and also more complicated. This is an active subject of on-going research and some non-Abelian discrete groups in semidirect product form, such as $A_{4}\rtimes\mathbb{Z}_{2}^{CP}$, $S_{4}\rtimes\mathbb{Z}_{2}^{CP}$, $\Delta(6n^{2})\rtimes\mathbb{Z}_{2}^{CP}$ can be the origin of the mixing scheme that we study here Feruglio et al. (2013, 2014); Ding et al. (2014); Li et al. (2016).

It is most likely that the predictions of TM1 and $\mu-\tau$ reflection symmetries are exact only at the scale where the horizontal flavor symmetry breaks into these residual symmetries. When going to lower energy scales these predictions will unavoidably receive corrections from renormalization group (RG) running Ohlsson and Zhou (2014). Therefore in this paper, we will also study the RG corrections on the predictions from the joined TM1 and $\mu$-$\tau$ reflection symmetry. We consider the case in which neutrino mass is described by the effective Weinberg operator, as well as by the most popular realization of this operator, the type I seesaw Minkowski (1977); Yanagida (1979); Glashow (1979); Mohapatra and Senjanovic (1980). Both the SM and the (Minimal Supersymmetric Standard Model) MSSM are assumed.

The remainder of the paper is organized as follows. In Sec. II, we introduce the TM1 symmetry and the $\mu$-$\tau$ reflection symmetry, and study the phenomenology if both are simultaneously present. Then we study the RG running effects on the PMNS parameters in the cases we mentioned above, presented in Sec. III. Finally we summarize our result and conclude in Sec. IV.

II Trimaximal $\mu$-$\tau$ reflection symmetry

The TM1 mixing and its symmetry as well as model-building aspects have been studied in many references (see e.g. Albright and Rodejohann (2009); de Medeiros Varzielas and Lavoura (2013); Luhn (2013); Li and Ding (2014); Rodejohann and Zhang (2012); Shimizu et al. (2015); Ballett et al. (2017)). In the following we denote the TM1 symmetry as $\mathbb{Z}_{2}^{{\rm TM1}}$. The $\mu$-$\tau$ reflection symmetry was originally proposed in Refs. Harrison and Scott (2002b); Ma (2002a); Babu et al. (2003); Ma (2002b); Grimus and Lavoura (2004) and later extensively studied in, e.g., Nishi (2013); Ma (2014); Fraser et al. (2014); Li and He (2015); Ma et al. (2015); Di Iura et al. (2015); Mohapatra and Nishi (2015); Zhou (2014); Joshipura and Patel (2015); He et al. (2015); Zhao (2017); Nishi and Sanchez-Vega (2017); Chen et al. (2016); Fukuyama (2017). It can be regarded as a generalized CP symmetry Neufeld et al. (1988); Grimus and Rebelo (1997) so we use $\mathbb{Z}_{2}^{\rm CP}$ to denote it222We prefer the symbol $\mathbb{Z}_{2}^{\rm CP}$ to $\mathbb{Z}_{2}^{\mu\tau}$ for the $\mu$-$\tau$ reflection symmetry because the latter is widely used for the $\mu$-$\tau$ symmetry without CP transformation.. Although both symmetries as well as their phenomenology have been extensive studied in the literature, their combination which provides a very effective description of the neutrino mixing data with only one free parameter, has attracted much less attention. Therefore in this section, we will discuss the theoretical and phenomenological aspects of this combination.

The explicit transformations of $\mathbb{Z}_{2}^{{\rm TM1}}$ and $\mathbb{Z}_{2}^{\rm CP}$ in the flavor basis are given as

[TABLE]

[TABLE]

where $\nu\equiv(\nu_{e},\thinspace\nu_{\mu},\thinspace\nu_{\tau})^{T}$ and the two matrices $R^{{\rm TM1}}$ and $R^{\mu\tau}$ are

[TABLE]

[TABLE]

The matrix $R^{{\rm TM1}}$ has been derived in, e.g., Ref. Lam (2008b) while the form of $R^{\mu\tau}$ is obvious according to the meaning of interchanging the $\mu$ and $\tau$ flavor. Since the $\mu$-$\tau$ reflection symmetry is essentially a generalized CP symmetry, it is necessary to check the consistency condition of flavor symmetry and CP symmetry Holthausen et al. (2013):

[TABLE]

The neutrino mass terms

[TABLE]

should be invariant under the transformations in Eqs. (3) and (4). Therefore, the mass matrix $M^{\nu}$ should satisfy

[TABLE]

[TABLE]

The above two equations can be broken down into equations in terms of the entries of $M^{\nu}$, so one can obtain explicit constraints on those:

[TABLE]

The above equations are equivalent to Eqs. (9) and (10), which means they are sufficient and necessary conditions for Eq. (8) being invariant under the transformations. With the above constraints, $M^{\nu}$ can be parametrized by four real parameters $r$, $x_{1,2}$ and $y$:

[TABLE]

As one can check, Eq. (16) is the most general mass matrix that satisfied Eqs. (9) and (10). The mass matrix contains only four real parameters; those are the three neutrinos masses and one degree of freedom for the PMNS matrix. As we will show later, this degree of freedom is just $\theta_{13}$. Therefore, the mass matrix with the form in Eq. (16) is highly predictive. It predicts all the parameters except for $\theta_{13}$ in the PMNS matrix, including two mixing angles $(\theta_{12},\thinspace\theta_{23})$, one Dirac phase $\delta$ and two Majorana phases $(\alpha_{21},\thinspace\alpha_{31})$.

The mass matrix is diagonalized by

[TABLE]

For $M^{\nu}$ in Eq. (16), due to the residual symmetries, $U^{\nu}$ can be analytically solved:

[TABLE]

where $(s,\thinspace c)=(\sin\theta,\thinspace\cos\theta)$ are given by

[TABLE]

and

[TABLE]

[TABLE]

Here ${\rm sign}(y)$ implies that we have taken $c=\sqrt{1-s^{2}}$ for positive $y$ and $c=-\sqrt{1-s^{2}}$ for negative $y$. Note that $(m^{\prime}_{1},\thinspace m^{\prime}_{2},\thinspace m^{\prime}_{3})$ computed from Eq. (20) are not necessarily positive (but always real), so they may be different from the neutrino masses by some minus signs.

Comparing the above result to the standard parametrization of the PMNS matrix

[TABLE]

[TABLE]

we can extract the predictions on all the PMNS parameters. It turns out that the predictions differ for positive and negative $y$. Next we will discuss both cases:

• •

Positive $y$ ($y>0$)

If $y>0$, then $c=\sqrt{1-s^{2}}$ is positive. We extract some phases from $U^{\nu}$ so that

[TABLE]

has the same phase convention as the standard parametrization, which requires

[TABLE]

Comparing Eq. (24) to Eqs. (23) and (22), we get

[TABLE]

If $m^{\prime}_{1,2,3}\geq 0$, then the Majorana phases should be $(1,\thinspace e^{i\alpha_{21}/2},\thinspace e^{i\alpha_{31}/2})=(1,\thinspace 1,\thinspace-i)$. However, $m^{\prime}_{1,2,3}$ could be negative, which can be converted to positive by further adding some phases to the right-hand side of Eq. (24). Therefore the actual Majorana phases depend on the signs of $m^{\prime}_{1,2,3}$:

[TABLE]

• •

Negative $y$ ($y<0$)

If $y<0$, then $c=-\sqrt{1-s^{2}}$ is negative so we need to remove the minus sign of the 12-entry of (18). Therefore Eqs. (24) and (25) are modified to

[TABLE]

where now

[TABLE]

In this case, comparing with the standard parametrization of the PMNS matrix we have

[TABLE]

and the Majorana phases are

[TABLE]

As a summary, we have

[TABLE]

if the experimental value $\theta_{13}\approx 9^{\circ}$ is taken as an input, and

[TABLE]

where the positive/negative signs depending on the signs of $y$ and $(m^{\prime}_{1},\thinspace m^{\prime}_{2},\thinspace m^{\prime}_{3})$ computed from Eq. (20). Here the Majorana phases are predicted to be either [math] or $\pi$. There have been many studies Rodejohann (2001); Ge and Lindner (2017); Bilenky et al. (2001); Pascoli et al. (2002); Minakata et al. (2015); Simkovic et al. (2013); Joniec and Zralek (2006); Rodejohann (2002); Benato (2015) on the option to measure the Majorana phases with upcoming neutrinoless double beta decay ($0\nu\beta\beta$) experiments. It was demonstrated in particular that expected nuclear and experimental uncertainties allow in principle to measure the phases, or at least contrain them non-trivially. The actual physical observable for $0\nu\beta\beta$ is the effective mass $|M_{ee}|$, which has significant dependence on the Majorana phases. For the inverted mass ordering, $|M_{ee}|$ is always nonzero, which necessarily leads to $0\nu\beta\beta$ at some level. For the normal mass ordering, it is well known that $|M_{ee}|$ can be zero for very small neutrino mass; however, $|M_{ee}|=0$ does not mean that $0\nu\beta\beta$ experiments tell us nothing about the Majorana phases. As it has been noticed in Refs. Ge and Lindner (2017); Xing (2003), this case still gives some constraints on the Majorana phases. In the scenario of this work, the relation between $|M_{ee}|$ and the Majorana phases is more explicit because all the neutrino parameters except for the lightest neutrino mass $m_{L}$ have been determined by symmetries or by experiments, enabling us to compute $|M_{ee}|$ explicitly, as shown in Fig. 1. Note that in this scenario, $|M_{ee}|<10^{-3}$ eV is possible only if $\alpha_{21}=\pi$. So if the future experiments push the upper bound of $|M_{ee}|$ down to $10^{-3}$ eV and still do not observe $0\nu\beta\beta$ decay, then we can draw the conclusion that $\alpha_{21}=\pi$.

We can confront the predictions of the mixing scheme with current data Esteban et al. (2017). First we study the predictions of TM1 mixing, namely the first column of the PMNS matrix being $(\sqrt{\frac{2}{3}}$, $\sqrt{\frac{1}{6}},\sqrt{\frac{1}{6}})^{T}$. The $\chi^{2}$-function is defined as

[TABLE]

where $x_{i}^{0}$ represents the data of the $i$-th experimental observable, $\sigma_{i}$ the corresponding $1\sigma$ absolute error, and $x_{i}$ the prediction of the model. For the normal ordering, TM1 has a $\chi^{2}$-minimum of $1.14\,(=0.063+0.000+1.058+0.0223)$ at the values $\theta_{13}=8.5^{\circ}$ and $\theta_{23}=41.6^{\circ}$. The numbers in brackets denote the contributions of $\theta_{13},\theta_{23},\theta_{12}$ and $\delta$ to the total value. In case of an inverted ordering, the $\chi^{2}$-minimum is $1.20\,(=0.006+0.000+1.056+0.143)$ at the values $\theta_{13}=8.5^{\circ}$ and $\theta_{23}=50.0^{\circ}$. Note that TM1 has two free parameters. Combining TM1 with $\mu$-$\tau$ reflection symmetry, which in total has only one free parameter, gives for the normal ordering a $\chi^{2}$-minimum of $3.88\,(=0.063+2.730+1.058+0.0308)$ at the value $\theta_{13}=8.5^{\circ}$. In the inverted ordering, the $\chi^{2}$-minimum is $5.76\,(=0.006+4.672+1.056+0.0234)$ at the value $\theta_{13}=8.5^{\circ}$.

III RG corrections

The residual symmetries we discussed in the previous section may appear at a very high energy scale, which we refer to as the flavor symmetry scale. Due to radiative corrections, the predictions at the flavor symmetry scale may be modified at the low energy scale, at which they are confronted with experimental measurements. If there is no new physics between the two scales, the corrections can be computed without many unknown parameters involved. However, it is also possible that some new physics appear in the middle so that the RG corrections would depend on more unknown parameters. For example, in the type I seesaw mechanism, the masses of right-handed neutrinos could be below the flavor symmetry scale; in this case the RG corrections would also depend on the masses of right-handed neutrinos.

III.1 RG running based on the Weinberg operator

To avoid the dependence on too many parameters, we will first focus on the case that all other new physics scales are above the flavor symmetry scale. In this case, the calculation will be based on the RGE of the SM extended by the Weinberg operator,

[TABLE]

where $L$ is the lepton doublet and $H$ the Higgs doublet. After electroweak symmetry breaking $\langle\tilde{H}\rangle=(v/\sqrt{2},0)^{T}$, the neutrino mass matrix is given by

[TABLE]

Constrained by the residual symmetries, $M^{\nu}$ depends on four parameters $(r,\thinspace x_{1},\thinspace x_{2},\thinspace y)$ in Eq. (16). Those parameters are actually highly constrained by neutrino oscillation measurements on the two mass-squared differences

[TABLE]

and $\sin\theta_{13}$. In this section we will fix them at the best-fit values Esteban et al. (2017); Capozzi et al. (2016) as the result of our calculation varies very little within experimental uncertainties. If the lightest neutrino mass $m_{L}$ is also known, then $(r,\thinspace x_{1},\thinspace x_{2},\thinspace y)$ can be determined by $(\theta_{13},\thinspace\delta m^{2},\thinspace\Delta m^{2},\thinspace m_{L})$. In Sec. II we have demonstrated how to compute $(\theta_{13},\thinspace\delta m^{2},\thinspace\Delta m^{2},\thinspace m_{L})$ for given values of $(r,\thinspace x_{1},\thinspace x_{2},\thinspace y)$. Determining $(r,\thinspace x_{1},\thinspace x_{2},\thinspace y)$ from experimental values of $(\theta_{13},\thinspace\delta m^{2},\thinspace\Delta m^{2},\thinspace m_{L})$ is then of course also possible.

However there are some positive/negative signs one needs to choose in determining $(r,\thinspace x_{1},\thinspace x_{2},\thinspace y)$. The first one is the sign of $\Delta m^{2}$, known as the neutrino mass ordering. Both the normal (NO, $\Delta m^{2}>0$) and the inverted ordering (IO, $\Delta m^{2}<0$ ) should be taken into consideration. The next one is the sign of the Dirac phase $\delta$. The $\mu$-$\tau$ reflection symmetry only predicts $|\delta|=90^{\circ}$ but both $+90^{\circ}$ and $-90^{\circ}$ are possible. Besides, as summarized in Eq. (32), the two Majorana phases take values of $\frac{\pi}{2}\pm\frac{\pi}{2}$, where we have to choose between the positive/negative signs.

Therefore, there are four positive/negative signs (and thus $16$ physically inequivalent cases) relevant in determining $(r,\thinspace x_{1},\thinspace x_{2},\thinspace y)$. However, as it can be seen from the mass matrix, for $\delta=+90^{\circ}$ and $-90^{\circ}$, the mass matrix in one case is simply the complex conjugate of the other, so we only need to study one of the two cases. Actually, the result of RG running of both cases shows that the radiative corrections on both cases are the same except that for $\delta$ it differs by a minus sign. This reduces the 16 cases to 8 cases in our analysis. In addition, the case of positive $\delta=+90^{\circ}$ is disfavored by current global fits. For simplicity, we refer to the 8 cases as $N_{\pm\pm}$ and $I_{\pm\pm}$ where $N$/$I$ stands for the normal/inverted ordering and the two $\pm$ stand for the signs of $e^{i\alpha_{21}}$ and $e^{i\alpha_{31}}$, respectively.

We solve the RGEs using the code REAP Antusch et al. (2005) and compute the RG corrections. The results are presented in Fig. 2 for $N_{++}$ and Fig. 3 for all the 8 cases. We set the flavor symmetry scale at $\Lambda=10^{14}$ GeV. Actually as shown in Fig. 2 the RG corrections depend linearly on $\log\Lambda$, so if $\Lambda$ is changed to another value $\Lambda^{\prime}$, the RG corrections can be evaluated correspondingly by simply multiplying a factor of $\log\Lambda^{\prime}/\log\Lambda$. Another parameter that may have significant effect is the lightest neutrino mass $m_{L}$. In Fig. 3 we show the RG corrections for different values of $m_{L}$ by green, yellow, orange and red points, corresponding to $m_{L}=(1,\thinspace 20,\thinspace 40,\thinspace 60)$ meV respectively333We do not take $m_{L}=0$ here because for $m_{L}=0$, the RG corrections are almost the same as $m_{L}=1$ meV except for the Majorana phases which are not well defined when $m_{L}=0$. . We assume here that strong limits on the neutrino mass scale from cosmology are valid Archidiacono et al. (2017) and simply note that the effect of running roughly scales with $m_{L}$ for values larger than 60 meV.

As shown in Fig. 3, typically the corrections to $\theta_{12}$, $\theta_{13}$, $\theta_{23}$, $\delta$, $\alpha_{21}$, $\alpha_{31}$ are about $0.1$, $0.001$, $0.005$, $0.1$, $0.05$, $0.1$ degrees respectively, except for some cases where due to some cancellations the RG corrections are suppressed. To understand the cancellation, we take $\theta_{12}$ as example, for which the analytic expression reads Antusch et al. (2003)

[TABLE]

Here $y_{\tau}$ is the tau-lepton Yukawa coupling. The plot for $\theta_{12}$ in Fig. 3 shows that the corrections in the four cases $N_{-\pm}$ and $I_{-\pm}$ are suppressed, which can be understood from Eq. (37): the correction is proportional to $|m_{1}+m_{2}e^{i\alpha_{21}}|^{2}$, which can be small if $e^{i\alpha_{21}}=-1$ and $m_{1}\approx m_{2}$. The latter always happens in the inverted ordering and in the normal ordering when the smallest mass $m_{1}$ approaches $\sqrt{\delta m^{2}}$.

Except for some cases with cancellations, the RG corrections generally increases when $m_{L}$ increases. This behavior is very common regarding small perturbations to the mass matrix, which has been studied in Ref. Rodejohann and Xu (2015) from a more general point of view. The reason is because for larger $m_{L}$, the mass spectrum is closer to the quasi-degenerate situation, where the PMNS mixing becomes unstable when the mass matrix suffers perturbations. Besides, among the three mixing angles, $\theta_{12}$ generally receives the largest correction (except for cancellations); this is because the gap between $m_{1}$ and $m_{2}$ is much smaller than that of $m_{1}$ and $m_{3}$ or $m_{2}$ and $m_{3}$.

Since all the corrections are at the order of or even lower than $0.1^{\circ}$, we can draw the usual conclusion that in the context of the SM with the Weinberg operator only, the RG corrections are negligible when compared with current and near future experimental measurements. As well known, if we replace the SM with the MSSM, then according to Ref. Antusch et al. (2003) the RG corrections to the neutrino mixing would be amplified by a factor of $\tan^{2}\beta$. To illustrate this effect, we compute the RG corrections again in the context of the MSSM with $\tan\beta=20$, and the result is shown in Fig. 4. As one can see, the RG corrections in the MSSM with large $\tan\beta$ are significantly enhanced to measurable values compared to Fig. 3.

III.2 RG running based on type I seesaw

In this section, we consider new physics that appears below the flavor symmetry scale. The Weinberg operator itself is UV incomplete and is usually believed to be a low-energy effective operator. Here we consider the type I seesaw realization of this operator only. Heavy right-handed neutrinos $N_{i}$ ($i=1,\thinspace 2,\ldots$) are integrated out to generate the Weinberg operator. We consider the scenario that the right-handed neutrino masses (or the seesaw scale) are lower than the flavor symmetry scale. So at the flavor symmetry scale, we should consider the symmetry of the following Lagrangian instead of the Weinberg operator,

[TABLE]

Next we need to specify the transformation rules of $\mathbb{Z}_{2}^{{\rm TM1}}$ and $\mathbb{Z}_{2}^{\rm CP}$ for the right-handed neutrinos. This depends on how we assign the right-handed neutrinos to the representations of the flavor symmetry, which is rather model-dependent. For simplicity, we assume that the number of right-handed neutrinos is three and that they have the same transformation rule as the left-handed neutrinos. As a result, both the Dirac mass matrix $m_{D}$ and the heavy Majorana matrix $M$ will be in the form of Eq. (16). As one can check explicitly, if both $m_{D}$ and $M$ are in the form of Eq. (16), then the light-neutrino mass matrix

[TABLE]

is also of the form in Eq. (16). As we have discussed, each matrix of the form (16) contains four real parameters thus in the Lagrangian (38) we have 8 free parameters. The tree-level predictions in Eqs. (31) and (32) are independent of the values of these parameters. However, the RG corrections inevitably depend on these parameters. As we have argued, when some new physics such as the right-handed neutrinos appears below the flavor symmetry scale, the RG corrections would usually depend on many unknown parameters, which makes it difficult to evaluate the RG corrections exactly. To understand generally how large the RG corrections would be, we adopt random scattering in the allowed parameter space rather than focus on some specific parameter settings.

We randomly generate 1000 samples with right-handed neutrino masses $M_{1}$, $M_{2}$, $M_{3}$ distributed from $10^{6}$ GeV to $10^{13}$ GeV and the lightest neutrino mass $m_{L}$ from 1 meV to 60 meV. Rectangular distributions are used for $\log M_{1,2,3}$ and $m_{L}$. The positive/negative signs of $\Delta m^{2}$ and Dirac/Majorana phases are also chosen randomly. The Yukawa couplings can be computed once $(M_{1},\thinspace M_{2},\thinspace M_{3})$ and $(m_{1},\thinspace m_{2},\thinspace m_{3})$ have been set. We again use the code REAP Antusch et al. (2005), which automatically integrates out the heavy right-handed neutrinos when the energy scale goes below their masses.

The results are presented in Fig. 5, where we can see most RG corrections are distributed in small ranges, e.g. $\Delta\theta_{12}$, $\Delta\theta_{23}$ and $\Delta\theta_{13}$ are most likely less than $0.05^{\circ}$, $0.01^{\circ}$ and $0.005^{\circ}$ respectively. So generally, the deviations are similar to the results in Fig. 3 where right-handed neutrinos are not introduced. However, large corrections are also possible. We do not find any significant cut-off of the deviations when the number of samples are increased, though the distributions above remain almost the same. This implies the RG corrections could be very large, but would require fine-tuning in the parameter space. For example, when the number of samples is increased to $10^{4}$, we find only two samples with $|\Delta\theta_{23}|>3{}^{\circ}$. Therefore, we can draw the conclusion that generally the RG corrections in the type I seesaw scenario are of similar magnitude as with the Weinberg operator only.

Again, the RG corrections can be significantly amplified within supersymmetric scenarios. We compute the RG corrections in the MSSM extended by the type I seesaw with $\tan\beta=20$. The result is shown in Fig. 6 where we can see that compared to Fig. 5, the RG corrections in the MSSM with large $\tan\beta$ are significantly enhanced by up to two orders of magnitude to measurable values.

IV Conclusion

Combining $\mu$-$\tau$ reflection and TM1 symmetry leads to a very predictive framework. We have shown in Sec. II that it not only can accommodate non-zero $\theta_{13}$ but also predicts all other PMNS parameters, including all CP phases ($\delta=\pm\pi/2$ and the Majorana phases are [math] or $\pi$).

With these symmetries, the neutrino mass matrix can be constrained to the form (16) containing only four real parameters. Given the experimental values of $\theta_{13}$, $\delta m^{2}$ and $\Delta m^{2}$ as input, the mass matrix can be exactly reconstructed for a fixed value of the smallest mass $m_{L}$ and several choices of positive/negative signs. Therefore, for the SM extended by the Weinberg operator, the RG corrections can be exactly evaluated as the only free parameter is $m_{L}$.

We have computed the RG corrections to the scenario, which are in agreement with known results, namely that in the SM they are typically small, but can be enhanced to measurable values within supersymmetric scenarios and within explicit multi-scale scenarios such as the type I seesaw mechanism.

In summary, the mixing scheme we propose here is very well compatible with data and addresses the closeness of $\delta$ with $-\pi/2$, of $\theta_{23}$ with $\pi/4$ and that $\sin^{2}\theta_{12}$ is slightly less than $1/3$. If future data confirms those special values of the mixing parameters, the proposed scheme seems an attractive approach to the description of lepton mixing. On the other hand, some deviations could occur in the future, which could either be explained by RG corrections if the deviations are small, or exclude this mixing scheme if they are large. One particularly noteworthy example is the deviation of $\theta_{23}$ from $45^{\circ}$, which was recently hinted by the NOVA measurement Adamson et al. (2017) $\theta_{23}=39.5^{\circ}{}_{-1.3}^{+1.7}$ or $52.2^{\circ}{}_{-1.8}^{+1.3}$ . Such a large deviation $(\gtrsim 5^{\circ})$ if confirmed by future data, would exclude this mixing scheme embedded in the simple scenarios considered in this paper.

Acknowledgements.

We thank Stefan Bruenner and Ludwig Rauch for helpful discussions. WR is supported by the DFG with grant RO 2516/6-1 in the Heisenberg program.