The translational $\mu$-$\tau$ reflection symmetry of Majorana neutrinos

Zhi-zhong Xing1,2,3 ***E-mail<EMAIL_ADDRESS>
1Institute of High Energy Physics, Chinese Academy of Sciences, Beijing
100049, China
2School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China
3Center of High Energy Physics, Peking University, Beijing 100871, China

###### Abstract

The present neutrino oscillation data allow $m_{1}=0$ (or $m_{3}=0$) for the
neutrino mass spectrum and support $\theta_{23}\simeq\pi/4$ and
$\delta\simeq-\pi/2$ as two good approximations for the PMNS lepton flavor
mixing matrix $U$. We show that these intriguing possibilities can be a very
natural consequence of the translational $\mu$-$\tau$ reflection symmetry —
the effective Majorana neutrino mass term keeps invariant under the
transformations $\nu_{e\rm L}\to(\nu_{e\rm L})^{c}+U^{*}_{ei}z^{c}_{\nu}$,
$\nu_{\mu\rm L}\to(\nu_{\tau\rm L})^{c}+U^{*}_{\tau i}z^{c}_{\nu}$ and
$\nu_{\tau\rm L}\to(\nu_{\mu\rm L})^{c}+U^{*}_{\mu i}z^{c}_{\nu}$ (for $i=1$
or $3$), where $z^{c}_{\nu}$ is the charge conjugation of a constant spinor
field $z_{\nu}$. Extending such a working flavor symmetry to the canonical
seesaw mechanism at a superhigh-energy scale, we calculate its soft breaking
effects at the electroweak scale by using the one-loop renormalization-group
equations.

Keywords: Majorana neutrinos, flavor mixing and CP violation, translational
$\mu$-$\tau$ reflection symmetry, minimal seesaw mechanism, renormalization-
group equations

##  1 Introduction

The facts that the three known (active) neutrinos have nonzero but tiny masses
and lepton flavors are significantly mixed [1] tell us that the standard model
(SM) of particle physics is incomplete at least in its flavor aspects,
although this great theory of electromagnetic and weak interactions has proved
to be a huge success in describing how the Universe works. A particular
characteristic of massive neutrinos is the flavor oscillation — a pure quantum
phenomenon, as their flavor eigenstates $\nu_{\alpha}$ (for
$\alpha=e,\mu,\tau$) are the quantum superpositions of their mass eigenstates
$\nu_{i}$ (for $i=1,2,3$):

$\displaystyle\left(\begin{matrix}\nu_{e}\cr\nu_{\mu}\cr\nu_{\tau}\cr\end{matrix}\right)_{\rm
L}=\left(\begin{matrix}U_{e1}&U_{e2}&U_{e3}\cr U_{\mu 1}&U_{\mu 2}&U_{\mu
3}\cr U_{\tau 1}&U_{\tau 2}&U_{\tau
3}\cr\end{matrix}\right)\left(\begin{matrix}\nu_{1}\cr\nu_{2}\cr\nu_{3}\cr\end{matrix}\right)_{\rm
L}\;,$ (1)

where $U_{\alpha i}$ stand for the Pontecorvo-Maki-Nakagawa-Sakata (PMNS)
lepton flavor mixing matrix elements [2, 3, 4]. A global analysis of current
neutrino oscillation data in the three-flavor scheme leaves us with the
following information on neutrino masses, lepton flavor mixing and CP
violation [5, 6]:

  * •

The mass ordering of three neutrinos can be either normal (namely,
$m_{1}<m_{2}<m_{3}$) or inverted (namely, $m_{3}<m_{1}<m_{2}$), but the latter
is slightly disfavored as compared with the former. Now that the absolute
neutrino mass scale has been undetermined, the possibility of $m_{1}=0$ or
$m_{3}=0$ is actually allowed as an extreme but intriguing case of the
neutrino mass spectrum.

  * •

The $3\times 3$ PMNS matrix $U$ exhibits an approximate but suggestive
$\mu$-$\tau$ interchange symmetry $|U_{\mu i}|\simeq|U_{\tau i}|$ (for
$i=1,2,3$), which is rather different from the observed pattern of the
Cabibbo-Kobayashi-Maskawa (CKM) quark flavor mixing matrix [7, 8]. This
observation, together with the preliminary experimental evidence for
significant CP violation in neutrino oscillations [9, 10], implies that
$\theta_{23}=\pi/4$ and $\delta=-\pi/2$ may naturally hold in the standard
parametrization of $U$ at a given energy scale.

It is well known that making use of some proper discrete symmetry groups (such
as $\rm A_{4}$ and $\rm S_{4}$) to describe the flavors of charged leptons and
massive neutrinos constitutes the most popular way to achieve
$\theta_{23}=\pi/4$ and $\delta=\pm\pi/2$ [11, 12, 13, 14, 15, 16, 17]. In
particular, the minimal or residual flavor symmetry of this kind is expected
to be the $\mu$-$\tau$ reflection symmetry in the neutrino sector [18]. A very
natural way of obtaining $m_{1}=0$ or $m_{3}=0$ is to invoke the minimal
type-I seesaw model in which two right-handed neutrino fields are introduced
and lepton number violation is required [19, 20, 21]. The so-called Friedberg-
Lee symmetry [22, 23, 24, 25, 26, 27], which demands the effective neutrino
mass term to be invariant under the translational transformations
$\nu_{\alpha}\to\nu_{\alpha}+\eta_{\alpha}z_{\nu}$ with $z_{\nu}$ being a
constant spinor field and $\eta_{\alpha}$ denoting a flavor-dependent complex
number (for $\alpha=e,\mu,\tau$), is also an interesting option because it can
not only predict one of the neutrinos to be massless but also help constrain
the flavor texture of massive neutrinos if the values of $\eta_{\alpha}$ are
explicitly assumed.

So far some attempts have been made to build a simple but viable neutrino mass
model by combining the $\mu$-$\tau$ reflection symmetry with either the
minimal seesaw scenario [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]
or a generalized version of the Friedberg-Lee symmetry [41, 42, 43, 44, 45,
46, 47], in order to simultaneously obtain $m_{1}=0$ (or $m_{3}=0$),
$\theta_{23}=\pi/4$ and $\delta=\pm\pi/2$. Along the latter line of thought,
the present paper is intended to go further and show that such three
intriguing predictions can naturally result from a translational $\mu$-$\tau$
reflection symmetry in the neutrino sector; namely, the effective Majorana
neutrino mass term keeps invariant under the following transformations of
three left-handed neutrino fields:

$\displaystyle\nu_{e\rm L}$ $\displaystyle\to$ $\displaystyle(\nu_{e\rm
L})^{c}+U^{*}_{ei}z^{c}_{\nu}\;,$ $\displaystyle\nu_{\mu\rm L}$
$\displaystyle\to$ $\displaystyle(\nu_{\tau\rm L})^{c}+U^{*}_{\tau
i}z^{c}_{\nu}\;,$ $\displaystyle\nu_{\tau\rm L}$ $\displaystyle\to$
$\displaystyle(\nu_{\mu\rm L})^{c}+U^{*}_{\mu i}z^{c}_{\nu}\;,\hskip
14.22636pt$ (2)

where $U_{\alpha i}$ denote the elements of the $3\times 3$ PMNS flavor mixing
matrix $U$ in its first or third column (i.e., $i=1$ or $3$), and
$z^{c}_{\nu}$ is the charge conjugation of a constant spinor field $z_{\nu}$
which anticommutates with the neutrino fields [47, 48, 49].

As first pointed out in Ref. [47], a massless Majorana neutrino field
$\nu_{i}$ and its charge-conjugated counterpart $\nu^{c}_{i}=\nu_{i}$ satisfy
the same Dirac equation ${\rm i}\gamma^{\mu}\partial_{\mu}\nu_{i}=0$ which is
invariant under the translational transformation $\nu_{i}\to\nu_{i}+z_{\nu}$
or $\nu_{i}\to\nu^{c}_{i}+z^{c}_{\nu}$ in free space. Such a transformation of
$\nu_{i}$ is equivalent to $\nu_{\alpha\rm L}\to\nu_{\alpha\rm L}+U_{\alpha
i}z_{\nu}$ or $\nu_{\alpha\rm L}\to(\nu_{\alpha\rm L})^{c}+U^{*}_{\alpha
i}z^{c}_{\nu}$ in the flavor space as a consequence of lepton flavor mixing.
This novel observation clearly explains why the flavor-dependent coefficients
$\eta_{\alpha}$ of $z_{\nu}$ in the original Friedberg-Lee transformation or
its generalized form are not really arbitrary but can be uniquely identified
as the PMNS matrix elements $U_{\alpha i}$, and thus it naturally motivates us
to conjecture the translational $\mu$-$\tau$ reflection transformations of
$\nu_{\alpha\rm L}$ in Eq. (2). In this respect our conjecture is certainly
new as compared with those made before (see, e.g., Refs. [50, 51, 52]) and may
provide a new angle of view to understand the flavor issues of Majorana
neutrinos. Moreover, we shall extend such a simple working flavor symmetry to
the canonical seesaw framework [53, 54, 55, 56, 57, 58] so as to naturally
arrive at a constrained version of the minimal seesaw model.

The remaining parts of this paper are organized as follows. In section 2, we
are going to show that $m_{i}=0$ (for $i=1$ or $3$) will definitely hold if
the effective Majorana neutrino mass term is invariant under the
transformations made in Eq. (2). Section 3 is devoted to extending such a
translational $\mu$-$\tau$ reflection symmetry to the right-handed neutrino
sector in the well-known canonical seesaw mechanism at a superhigh-energy
scale, and section 4 is intended to calculate its soft breaking effects on
three neutrino masses, three flavor mixing angles and two CP-violating phases
at the electroweak scale. A brief summary and some concluding remarks will be
made in section 5.

##  2 The symmetry and its consequences

Regardless of how the tiny masses of three Majorana neutrinos are generated,
here let us simply focus on their effective mass term ${\cal L}_{\rm mass}$ at
a given energy scale:

$\displaystyle-{\cal L}_{\rm mass}=\frac{1}{2}\hskip
1.42271pt\overline{\nu_{\rm L}}\hskip 2.84544ptM_{\nu}\hskip
1.42271pt(\nu_{\rm L})^{c}+{\rm h.c.}\;,$ (3)

where $\nu_{\rm L}=(\nu_{e\rm L},\nu_{\mu\rm L},\nu_{\tau\rm L})^{T}$ denotes
the column vector of three left-handed neutrino fields, $(\nu_{\alpha\rm
L})^{c}\equiv{\cal C}\overline{\nu_{\alpha\rm L}}^{T}$ stands for the charge
conjugate of $\nu_{\alpha\rm L}$ (for $\alpha=e,\mu,\tau$) with $\cal C$ being
the charge conjugation operator which satisfies ${\cal C}\gamma^{T}_{\mu}{\cal
C}^{-1}=-\gamma_{\mu}$, ${\cal C}\gamma^{T}_{5}{\cal C}^{-1}=\gamma_{5}$ and
${\cal C}^{-1}={\cal C}^{\dagger}={\cal C}^{T}=-{\cal C}$, and $M_{\nu}$ is
the $3\times 3$ symmetric Majorana neutrino mass matrix. If the traditional CP
transformations are made for the left-handed neutrino fields, namely
$\nu_{\alpha\rm L}\to(\nu_{\alpha L})^{c}$ (for $\alpha=e,\mu,\tau$) as shown
in Fig. 1, then the invariance of ${\cal L}_{\rm mass}$ requires
$M_{\nu}=M^{*}_{\nu}$, implying that this effective Majorana neutrino mass
term must be CP-conserving.

Now let us introduce a real orthogonal $(\mu,\tau)$-associated permutation
matrix, or equivalently one of the six elements of the non-Abelian $\rm S_{3}$
group, of the form

$\displaystyle{\cal P}={\cal P}^{T}={\cal
P}^{\dagger}=\left(\begin{matrix}1&0&0\cr 0&~{}0~{}&1\cr
0&1&0\cr\end{matrix}\right)\;.$ (4)

With the help of this notation, we may rewrite the translational $\mu$-$\tau$
reflection transformations proposed in Eq. (2) as follows:

$\displaystyle\nu_{\rm L}\to{\cal P}\left[(\nu_{\rm
L})^{c}+\xi^{*}_{i}z^{c}_{\nu}\right]\;,$ (5)

together with the charge-conjugated transformation $(\nu_{\rm L})^{c}\to{\cal
P}\left(\nu_{\rm L}+\xi_{i}z_{\nu}\right)$, where $\xi_{i}$ is a column vector
of the PMNS lepton flavor mixing matrix $U$ defined as
$\xi_{i}\equiv(U_{ei},U_{\mu i},U_{\tau i})^{T}$ corresponding to $\nu_{i}$
(for $i=1,2,3$). Figure 1 provides us with a clear comparison between the
traditional CP transformations and the $\mu$-$\tau$-interchanging CP
transformations for three left-handed neutrino fields. Under the latter
transformations the effective Majorana neutrino mass term in Eq. (3) changes
to

$\displaystyle-{\cal L}^{\prime}_{\rm mass}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Big{[}\overline{\nu_{\rm L}}\left({\cal
P}M^{*}_{\nu}{\cal P}\right)(\nu_{\rm L})^{c}+\overline{\nu_{\rm
L}}\left({\cal P}M^{*}_{\nu}{\cal
P}\right)\xi^{*}_{i}z^{c}_{\nu}+\overline{z_{\nu}}\hskip
1.42271pt\xi^{\dagger}_{i}\left({\cal P}M^{*}_{\nu}{\cal P}\right)(\nu_{\rm
L})^{c}$ (6) $\displaystyle+\hskip 1.9919pt\overline{z_{\nu}}\hskip
1.42271pt\xi^{\dagger}_{i}\left({\cal P}M^{*}_{\nu}{\cal
P}\right)\xi^{*}_{i}z^{c}_{\nu}\Big{]}+{\rm h.c.}\;.$

The necessary and sufficient conditions for ${\cal L}^{\prime}_{\rm
mass}={\cal L}_{\rm mass}$ turn out to be the relation

$\displaystyle M_{\nu}={\cal P}M^{*}_{\nu}{\cal P}\;,$ (7)

together with three dependent relations

$\displaystyle M_{\nu}\xi^{*}_{i}={\bf 0}\;,\quad\xi^{\dagger}_{i}M_{\nu}={\bf
0}^{T}\;,\quad\xi^{\dagger}_{i}M_{\nu}\xi^{*}_{i}=0\;,$ (8)

where $\bf 0$ denotes the zero column vector. The condition in Eq. (7) is a
natural consequence of the $\mu$-$\tau$ reflection symmetry and provides a
quite strong constraint on the flavor texture of $M_{\nu}$; and those in Eq.
(8) are apparently required by the translational symmetry of ${\cal L}_{\rm
mass}$ and should simply lead us to the result $m_{i}=0$.

Figure 1: A straightforward comparison between the traditional CP
transformation and the $\mu$-$\tau$-interchanging CP transformation for three
left-handed neutrino fields.

To prove the last point, let us denote the elements of $M_{\nu}$ as $\langle
m\rangle_{\alpha\beta}$ (for $\alpha,\beta=e,\mu,\tau$) and express them in
terms of the neutrino masses $m_{i}$ and the PMNS matrix elements $U_{\alpha
i}$ and $U_{\beta i}$ (for $i=1,2,3$) in the basis where the mass eigenstates
of three charged leptons are identified with their flavor eigenstates. Namely,
$M_{\nu}=UD_{\nu}U^{T}$ with $D_{\nu}\equiv{\rm
Diag}\left\\{m_{1},m_{2},m_{3}\right\\}$ holds in this flavor basis, and
therefore

$\displaystyle\left(M_{\nu}\right)_{\alpha\beta}\equiv\langle
m\rangle_{\alpha\beta}=\sum^{3}_{i=1}\left(m_{i}U_{\alpha i}U_{\beta
i}\right)\;.$ (9)

With the help of the unitarity of $U$, we combine Eq. (8) with Eq. (9) and
then obtain 111In Eq. (8) the second condition is actually a transpose of the
first condition, and thus they are equivalent.

$\displaystyle\sum_{\beta}\langle m\rangle_{\alpha\beta}U^{*}_{\beta
i}=m_{i}U_{\alpha i}=0\;,$
$\displaystyle\sum_{\alpha}\sum_{\beta}U^{*}_{\alpha i}\langle
m\rangle_{\alpha\beta}U^{*}_{\beta i}=m_{i}=0\;.\hskip 14.22636pt$ (10)

So $m_{i}=0$ is proved to be another consequence of the translational
$\mu$-$\tau$ reflection symmetry of ${\cal L}_{\rm mass}$. Now that
$m_{2}>m_{1}$ has been fixed [1], we are left with the possibility of either
$m_{1}=0$ (normal ordering) or $m_{3}=0$ (inverted ordering) for the neutrino
mass spectrum. Combining $m_{1}=0$ or $m_{3}=0$ with the available data on the
neutrino mass-squared differences $\Delta m^{2}_{ij}\equiv
m^{2}_{i}-m^{2}_{j}$, one may immediately determine the three neutrino masses.
To be more explicit, we have the special normal neutrino mass hierarchy

$\displaystyle m_{1}$ $\displaystyle=$ $\displaystyle 0\;,$ $\displaystyle
m_{2}$ $\displaystyle=$ $\displaystyle\sqrt{\Delta
m^{2}_{21}}=8.61^{+0.12}_{-0.11}\times 10^{-3}~{}{\rm eV}\;,$ $\displaystyle
m_{3}$ $\displaystyle=$ $\displaystyle\sqrt{\Delta
m^{2}_{31}}=5.01^{+0.03}_{-0.03}\times 10^{-2}~{}{\rm eV}\;,\hskip 14.22636pt$
(11)

where $\Delta m^{2}_{21}=7.42^{+0.21}_{-0.20}\times 10^{-5}~{}{\rm eV}^{2}$
and $\Delta m^{2}_{31}=2.510^{+0.027}_{-0.027}\times 10^{-3}~{}{\rm eV}^{2}$
as the best-fit values plus the $\pm 1\sigma$ ranges [5, 6] have been input;
or the special inverted neutrino mass hierarchy

$\displaystyle m_{1}$ $\displaystyle=$ $\displaystyle\sqrt{\left|\Delta
m^{2}_{31}\right|}=4.92^{+0.03}_{{}_{-}0.03}\times 10^{-2}~{}{\rm eV}\;,$
$\displaystyle m_{2}$ $\displaystyle=$ $\displaystyle\sqrt{\left|\Delta
m^{2}_{32}\right|}=4.99^{+0.03}_{-0.03}\times 10^{-2}~{}{\rm eV}\;,\hskip
14.22636pt$ $\displaystyle m_{3}$ $\displaystyle=$ $\displaystyle 0\;,$ (12)

where $\Delta m^{2}_{21}=7.42^{+0.21}_{-0.20}\times 10^{-5}~{}{\rm eV}^{2}$
and $\Delta m^{2}_{32}=-2.490^{+0.026}_{-0.028}\times 10^{-3}~{}{\rm eV}^{2}$
[5, 6] have been used for a numerical illustration. The next-generation
neutrino oscillation experiments are going to determine which neutrino mass
ordering is really true, and the next-generation precision measurements of the
cosmic microwave background anisotropies and large scale structures may
hopefully help to probe the absolute neutrino mass scale and examine whether
$m_{1}$ or $m_{3}$ is essentially vanishing or vanishingly small [15].

No matter whether $m_{1}=0$ or $m_{3}=0$ holds, we are left with only two
nontrivial CP-violating phases $\delta$ and $\sigma$ in the PMNS matrix $U$
[37]:

$\displaystyle
U=P_{l}\left(\begin{matrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-{\rm
i}\delta}\cr-s_{12}c_{23}-c_{12}s_{13}s_{23}e^{{\rm
i}\delta}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{{\rm i}\delta}&c_{13}s_{23}\cr
s_{12}s_{23}-c_{12}s_{13}c_{23}e^{{\rm
i}\delta}&~{}-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{{\rm
i}\delta}~{}&c_{13}c_{23}\cr\end{matrix}\right)P_{\nu}\;,$ (13)

where $P_{l}={\rm Diag}\left\\{e^{{\rm i}\phi_{e}},e^{{\rm
i}\phi_{\mu}},e^{{\rm i}\phi_{\tau}}\right\\}$ has no physical effects, but
$P_{\nu}={\rm Diag}\left\\{1,e^{{\rm i}\sigma},1\right\\}$ with $\sigma$ being
the Majorana CP phase is sensitive to lepton number violation. Substituting
Eq. (13) into Eq. (9) and then taking account of

$\displaystyle\langle m\rangle_{ee}=\langle m\rangle^{*}_{ee}\;,\quad\langle
m\rangle_{\mu\tau}=\langle m\rangle^{*}_{\mu\tau}\;,\quad\langle
m\rangle_{e\tau}=\langle m\rangle^{*}_{e\mu}\;,\quad\langle
m\rangle_{\tau\tau}=\langle m\rangle^{*}_{\mu\mu}\;$ (14)

derived from Eq. (7), we simply arrive at

$\displaystyle\theta_{23}=\frac{\pi}{4}\;,\quad\delta=\pm\frac{\pi}{2}\;,\quad\sigma=0~{}~{}{\rm
or}~{}~{}\frac{\pi}{2}\;$ (15)

as a natural consequence of the $\mu$-$\tau$ reflection symmetry no matter
whether $m_{1}=0$ or $m_{3}=0$ holds, together with the conditions
$\phi_{e}=0$ or $\pm\pi$ and $\phi_{\mu}+\phi_{\tau}=0$ or $\pi$. The
preliminary T2K measurement favors the possibility of $\delta=-\pi/2$ [9]. At
present the measurement of a neutrinoless double-beta ($0\nu 2\beta$) decay
mode, whose half-life is sensitive to the magnitude of $\langle
m\rangle_{ee}$, is the only experimentally feasible way to probe the Majorana
nature of massive neutrinos and constrain the corresponding phase parameters.
In the above translational $\mu$-$\tau$ reflection symmetry limit, we have

$\displaystyle\left|\langle m\rangle_{ee}\right|_{m_{1}=0}$ $\displaystyle=$
$\displaystyle m_{2}s^{2}_{12}c^{2}_{13}\mp
m_{3}s^{2}_{13}\simeq\left\\{\begin{array}[]{l}1.45\times 10^{-3}~{}{\rm
eV}\quad\quad(\sigma=0)\;,\\\ \vspace{-0.6cm}\\\ 3.67\times 10^{-3}~{}{\rm
eV}\quad\quad(\sigma=\pi/2)\;;\end{array}\right.$ (19)
$\displaystyle\left|\langle m\rangle_{ee}\right|_{m_{3}=0}$ $\displaystyle=$
$\displaystyle c^{2}_{13}\left(m_{1}c^{2}_{12}\pm
m_{2}s^{2}_{12}\right)\simeq\left\\{\begin{array}[]{l}4.83\times
10^{-2}~{}{\rm eV}\quad\quad(\sigma=0)\;,\\\ \vspace{-0.6cm}\\\ 1.91\times
10^{-2}~{}{\rm eV}\quad\quad(\sigma=\pi/2)\;,\end{array}\right.\hskip
14.22636pt$ (23)

where the best-fit values $s^{2}_{12}=0.304$ and $s^{2}_{13}=0.022$ [5, 6]
together with the numerical results of $m_{i}$ (for $i=1,2,3$) obtained in
Eqs. (11) and (12) have been used as the typical inputs. Eq. (23) may give one
a ball-park but promising feeling that the translational $\mu$-$\tau$
reflection symmetry is so powerful that it can really lead us to a definite
prediction for the effective electron-neutrino mass of the $0\nu 2\beta$
decays. In this special case an experimental measurement of $\left|\langle
m\rangle_{ee}\right|$ will allow us not only to determine the neutrino mass
ordering but also to pin down the Majorana CP-violating phase $\sigma$.

As a by-product, the other three independent matrix elements of $M_{\nu}$ in
the $\mu$-$\tau$ reflection symmetry limit can similarly be determined. We
have the results as follows:

$\displaystyle\left|\langle m\rangle_{\mu\mu}\right|_{m_{1}=0}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\displaystyle\left[m_{2}\left(c^{2}_{12}-s^{2}_{12}s^{2}_{13}\right)\pm
m_{3}c^{2}_{13}\right]^{2}+4m^{2}_{2}c^{2}_{12}s^{2}_{12}s^{2}_{13}}$ (27)
$\displaystyle\simeq$ $\displaystyle\left\\{\begin{array}[]{l}2.75\times
10^{-2}~{}{\rm eV}\quad\quad(\sigma=0)\;,\\\ \vspace{-0.6cm}\\\ 2.15\times
10^{-2}~{}{\rm eV}\quad\quad(\sigma=\pi/2)\;;\end{array}\right.$
$\displaystyle\left|\langle m\rangle_{\mu\mu}\right|_{m_{3}=0}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\displaystyle\left[m_{1}\left(s^{2}_{12}-c^{2}_{12}s^{2}_{13}\right)\pm
m_{2}\left(c^{2}_{12}-s^{2}_{12}s^{2}_{13}\right)\right]^{2}+4\left(m_{1}\mp
m_{2}\right)^{2}c^{2}_{12}s^{2}_{12}s^{2}_{13}}\hskip 14.22636pt$ (31)
$\displaystyle\simeq$ $\displaystyle\left\\{\begin{array}[]{l}2.43\times
10^{-2}~{}{\rm eV}\quad\quad(\sigma=0)\;,\\\ \vspace{-0.6cm}\\\ 1.22\times
10^{-2}~{}{\rm eV}\quad\quad(\sigma=\pi/2)\;;\end{array}\right.$

and

$\displaystyle\left|\langle m\rangle_{e\mu}\right|_{m_{1}=0}$ $\displaystyle=$
$\displaystyle\frac{c_{13}}{\sqrt{2}}\sqrt{\displaystyle
m^{2}_{2}c^{2}_{12}s^{2}_{12}+\left(m_{2}s^{2}_{12}\pm
m_{3}\right)^{2}s^{2}_{13}}$ (35) $\displaystyle\simeq$
$\displaystyle\left\\{\begin{array}[]{l}6.13\times 10^{-3}~{}{\rm
eV}\quad\quad(\sigma=0)\;,\\\ \vspace{-0.6cm}\\\ 5.65\times 10^{-3}~{}{\rm
eV}\quad\quad(\sigma=\pi/2)\;;\end{array}\right.$ $\displaystyle\left|\langle
m\rangle_{e\mu}\right|_{m_{3}=0}$ $\displaystyle=$
$\displaystyle\frac{c_{13}}{\sqrt{2}}\sqrt{\displaystyle\left(m_{1}\mp
m_{2}\right)^{2}c^{2}_{12}s^{2}_{12}+\left(m_{1}c^{2}_{12}\pm
m_{2}s^{2}_{12}\right)^{2}s^{2}_{13}}\hskip 14.22636pt$ (39)
$\displaystyle\simeq$ $\displaystyle\left\\{\begin{array}[]{l}5.13\times
10^{-3}~{}{\rm eV}\quad\quad(\sigma=0)\;,\\\ \vspace{-0.6cm}\\\ 3.19\times
10^{-2}~{}{\rm eV}\quad\quad(\sigma=\pi/2)\;;\end{array}\right.$

as well as

$\displaystyle\left|\langle m\rangle_{\mu\tau}\right|_{m_{1}=0}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\left[m_{3}c^{2}_{13}\mp
m_{2}\left(c^{2}_{12}+s^{2}_{12}s^{2}_{13}\right)\right]$ (43)
$\displaystyle\simeq$ $\displaystyle\left\\{\begin{array}[]{l}2.15\times
10^{-2}~{}{\rm eV}\quad\quad(\sigma=0)\;,\\\ \vspace{-0.6cm}\\\ 2.75\times
10^{-2}~{}{\rm eV}\quad\quad(\sigma=\pi/2)\;;\end{array}\right.$
$\displaystyle\left|\langle m\rangle_{\mu\tau}\right|_{m_{3}=0}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[m_{2}\left(c^{2}_{12}+s^{2}_{12}s^{2}_{13}\right)\pm
m_{1}\left(s^{2}_{12}+c^{2}_{12}s^{2}_{13}\right)\right]\hskip 14.22636pt$
(47) $\displaystyle\simeq$ $\displaystyle\left\\{\begin{array}[]{l}2.54\times
10^{-2}~{}{\rm eV}\quad\quad(\sigma=0)\;,\\\ \vspace{-0.6cm}\\\ 9.68\times
10^{-3}~{}{\rm eV}\quad\quad(\sigma=\pi/2)\;.\end{array}\right.$

Note that all the results in Eqs. (23)—(47) are insensitive to the two-fold
uncertainties of $\delta$ (i.e., $\delta=\pm\pi/2$) in the translational
$\mu$-$\tau$ reflection symmetry limit. The smallness of these matrix elements
makes it extremely difficult, if not impossible, to probe the relevant lepton-
number-violating processes mediated by $\nu_{i}$ (for $i=1,2,3$), as such
reactions are directly associated with the magnitudes of $\langle
m\rangle_{\alpha\beta}$ [59, 60, 61].

For the sake of illustration, here let us briefly comment on a typical example
of this kind: the lepton-number-violating decay modes
$B^{-}_{u}\to\pi^{+}\alpha^{-}\beta^{-}$ which involve all the six effective
Majorana neutrino masses $\langle m\rangle_{\alpha\beta}$ (for
$\alpha,\beta=e,\mu,\tau$). The two tree-level Feynman diagrams responsible
for $B^{-}_{u}\to\pi^{+}\alpha^{-}\beta^{-}$ are shown in Fig. 2. Quite
similar to the more familiar case of those $0\nu 2\beta$ processes which are
directly associated with $|\langle m\rangle_{ee}|^{2}$, the decay rates
$\Gamma(B^{-}_{u}\to\pi^{+}\alpha^{-}\beta^{-})$ are simply proportional to
$|\langle m\rangle_{\alpha\beta}|^{2}$. So the $\mu$-$\tau$ reflection
symmetry under discussion allows us to obtain
$\Gamma(B^{-}_{u}\to\pi^{+}e^{-}\tau^{-})=\Gamma(B^{-}_{u}\to\pi^{+}e^{-}\mu^{-})$
and
$\Gamma(B^{-}_{u}\to\pi^{+}\tau^{-}\tau^{-})=\Gamma(B^{-}_{u}\to\pi^{+}\mu^{-}\mu^{-})$.
But there are only very preliminary upper bounds on the branching ratios of
$B^{-}_{u}\to\pi^{+}e^{-}e^{-}$, $B^{-}_{u}\to\pi^{+}e^{-}\mu^{-}$ and
$B^{-}_{u}\to\pi^{+}\mu^{-}\mu^{-}$ [1] which cannot provide any meaningful
constraints on $|\langle m\rangle_{ee}|$, $|\langle m\rangle_{e\mu}|$ and
$|\langle m\rangle_{\mu\mu}|$ as compared with the more realistic and feasible
$0\nu 2\beta$-decay experiments. This situation is expected to be improved to
some extent in the upcoming precision measurement era characterized by the
LHCb experiment at the high-luminosity LHC [62] and the Belle-II experiments
at the KEK super-$B$ factory [63].

Figure 2: The Feynman diagrams for a lepton-number-violating decay mode
$B^{-}_{u}\to\pi^{+}\alpha^{-}\beta^{-}$ (for $\alpha,\beta=e,\mu,\tau$),
where $\nu_{i}$ (for $i=1,2,3$) denote the neutrino mass eigenstates,
$U_{\alpha i}$ and $U_{\beta i}$ are the PMNS matrix elements, $V_{ud}$ and
$V_{ub}$ are the CKM matrix elements.

So far we have simply assumed the translational $\mu$-$\tau$ reflection
symmetry of three active Majorana neutrinos at low energies. It is certainly
appropriate to conjecture such a working flavor symmetry at a superhigh-energy
scale where there may exist a well-defined mechanism responsible for the
origin of tiny neutrino masses and a suitable flavor symmetry to constrain the
neutrino mass texture. In this regard the popular canonical seesaw mechanism
[53, 54, 55, 56, 57, 58], which attributes the smallness of three active
neutrino masses to the existence of three heavy Majorana neutrinos and the
violation of lepton number conservation, is no doubt the most natural
theoretical framework at hand.

##  3 An extension in the seesaw framework

The canonical seesaw mechanism is a straightforward and natural extension of
the SM with three right-handed neutrino fields $N_{\alpha\rm R}$ (for
$\alpha=e,\mu,\tau$) and lepton number violation [53, 54, 55, 56, 57, 58]. In
this theoretical framework the gauge- and Lorentz-invariant neutrino mass
terms can be written as

$\displaystyle-{\cal L}_{\nu}=\overline{\ell_{\rm L}}\hskip
1.42271ptY_{\nu}\widetilde{H}N_{\rm R}+\frac{1}{2}\hskip
1.42271pt\overline{(N_{\rm R})^{c}}\hskip 1.42271ptM_{\rm R}N_{\rm R}+{\rm
h.c.}\;,$ (48)

where $\ell_{\rm L}$ denotes the $\rm SU(2)_{\rm L}$ doublet of the left-
handed lepton fields, $\widetilde{H}\equiv{\rm i}\sigma_{2}H^{*}$ with $H$
being the Higgs doublet and $\sigma_{2}$ being the second Pauli matrix,
$N_{\rm R}=(N_{e\rm R},N_{\mu\rm R},N_{\tau\rm R})^{T}$ is the column vector
of three right-handed neutrino fields which are the $\rm SU(2)_{\rm L}$
singlets, $(N_{\rm R})^{c}\equiv{\cal C}\overline{N_{\rm R}}^{T}$ is the
charge conjugate of $N_{\alpha\rm R}$, $Y_{\nu}$ represents an arbitrary
$3\times 3$ Yukawa coupling matrix, and $M_{\rm R}$ stands for a symmetric
$3\times 3$ Majorana mass matrix. After spontaneous electroweak gauge symmetry
breaking, Eq. (48) becomes

$\displaystyle-\widetilde{\cal L}_{\rm mass}=\overline{\nu_{\rm L}}\hskip
1.42271ptM_{\rm D}N_{\rm R}+\frac{1}{2}\hskip 1.42271pt\overline{(N_{\rm
R})^{c}}\hskip 1.42271ptM_{\rm R}N_{\rm R}+{\rm h.c.}\;,$ (49)

where $\nu_{\rm L}$ is the column vector of $\nu_{e\rm L}$, $\nu_{\mu\rm L}$
and $\nu_{\tau\rm L}$ as already given in Eq. (3), and $M_{\rm D}\equiv
Y_{\nu}\langle H\rangle$ with $\langle H\rangle\simeq 174~{}{\rm GeV}$ being
the vacuum expectation value of the Higgs field. Note that $M_{\rm D}$ is in
general neither Hermitian nor symmetric. In a way similar to Eq. (5), we may
make the following transformations for the left- and right-handed neutrino
fields 222Here we only make the $\mu$-$\tau$ reflection transformation instead
of the translational $\mu$-$\tau$ reflection transformation for the left-
handed neutrino fields. Otherwise, the constraint on $M_{\rm D}$ would be too
strong to result in a viable texture for the effective Majorana neutrino mass
matrix $M_{\nu}$. If the $\mu$-$\tau$ reflection transformation $\nu_{\rm
L}\to{\cal P}(\nu_{\rm L})^{c}$ were not made, on the other hand, the
constraint on $M_{\nu}$ would be too weak to be desirable for our purpose of
enhancing the predictability of this model.:

$\displaystyle\nu_{\rm L}\to{\cal P}(\nu_{\rm L})^{c}\;,\quad N_{\rm
R}\to{\cal P}\left[(N_{\rm R})^{c}+\zeta^{*}_{i}z^{c}_{N}\right]\;,$ (50)

together with $(\nu_{\rm L})^{c}\to{\cal P}\nu_{\rm L}$ and $(N_{\rm
R})^{c}\to{\cal P}\left(N_{\rm R}+\zeta_{i}z_{N}\right)$, where the
$(\mu,\tau)$-associated permutation matrix $\cal P$ has been given in Eq. (4),
$\zeta_{i}$ is a column vector of the $3\times 3$ unitary flavor mixing matrix
$U^{\prime}$ used to diagonalize $M_{\rm R}$ and defined as
$\zeta_{i}\equiv(U^{\prime}_{ei},U^{\prime}_{\mu i},U^{\prime}_{\tau i})^{T}$
corresponding to the heavy neutrino mass eigenstate $N_{i}$ with the mass
$M_{i}$ (for $i=1,2,3$), and $z_{N}$ is another constant spinor field like
$z_{\nu}$ in Eq. (2). Then Eq. (49) becomes

$\displaystyle-\widetilde{\cal L}^{\prime}_{\rm mass}$ $\displaystyle=$
$\displaystyle\overline{\nu_{\rm L}}\hskip 1.42271pt\left({\cal P}M^{*}_{\rm
D}{\cal P}\right)N_{\rm R}+\frac{1}{2}\Big{[}\overline{(N_{\rm
R})^{c}}\left({\cal P}M^{*}_{\rm R}{\cal P}\right)N_{\rm R}+\overline{(N_{\rm
R})^{c}}\left({\cal P}M^{*}_{\rm R}{\cal P}\right)\zeta_{i}z_{N}$ (51)
$\displaystyle+\hskip 2.84544pt\overline{z^{c}_{N}}\hskip
2.84544pt\zeta^{T}_{i}\left({\cal P}M^{*}_{\rm R}{\cal P}\right)N_{\rm
R}+\overline{z^{c}_{N}}\hskip 2.84544pt\zeta^{T}_{i}\left({\cal P}M^{*}_{\rm
R}{\cal P}\right)\zeta_{i}z_{N}\Big{]}+{\rm h.c.}\;.$

In this case the necessary and sufficient conditions for $\widetilde{\cal
L}^{\prime}_{\rm mass}=\widetilde{\cal L}_{\rm mass}$ turn out to be

$\displaystyle M_{\rm D}={\cal P}M^{*}_{\rm D}{\cal P}\;,\quad M_{\rm R}={\cal
P}M^{*}_{\rm R}{\cal P}\;,$ (52)

together with the dependent relations

$\displaystyle M_{\rm D}\hskip 1.42271pt\zeta_{i}={\bf 0}\;,\quad M_{\rm
R}\hskip 1.42271pt\zeta_{i}={\bf 0}\;,\quad\zeta^{T}_{i}M_{\rm R}={\bf
0}^{T}\;,\quad\zeta^{T}_{i}M_{\rm R}\hskip 1.42271pt\zeta_{i}=0\;,$ (53)

where $\bf 0$ stands for the zero column vector. The flavor textures of
$M_{\rm D}$ and $M_{\rm R}$ constrained by Eq. (52) can therefore be expressed
as follows [64, 65]:

$\displaystyle M_{\rm D}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}a&b&b^{*}\cr e&c&d\cr
e^{*}&~{}~{}d^{*}~{}~{}&c^{*}\end{matrix}\right)\;,$ $\displaystyle M_{\rm R}$
$\displaystyle=$ $\displaystyle\left(\begin{matrix}A&B&B^{*}\cr B&C&D\cr
B^{*}&D&C^{*}\end{matrix}\right)\;,\hskip 14.22636pt$ (54)

where the matrix elements $a$, $A$ and $D$ are real. On the other hand, Eq.
(53) implies that $M_{i}=0$ must hold. To assure the seesaw mechanism to work,
however, only one of the three mass eigenstates $N_{i}$ (for $i=1,2,3$) is
allowed to vanish.

Without loss of any generality, let us take $M_{1}=0$. Now that there is no
way to measure $U^{\prime}$ in any realistic experiments, we simply choose the
flavor basis defined by $\zeta_{1}\equiv(1,0,0)^{T}$. Then Eq. (53)
immediately leads us to

$\displaystyle M_{\rm D}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}0&b&b^{*}\cr 0&c&d\cr
0&~{}~{}d^{*}~{}~{}&c^{*}\end{matrix}\right)\;,$ $\displaystyle M_{\rm R}$
$\displaystyle=$ $\displaystyle\left(\begin{matrix}0&0&0\cr 0&C&D\cr 0&\hskip
2.84544ptD\hskip 2.84544pt&C^{*}\end{matrix}\right)\;.\hskip 22.76228pt$ (55)

As argued in Ref. [43], the flavor eigenstate of the massless neutrino $N_{1}$
simply decouples and has only gravitational interactions due to its kinetic
energy term. We actually have no idea about whether such a massless sterile
neutrino state could really exist in nature. Here our focus is on the point
that the translational $\mu$-$\tau$ reflection symmetry of $\widetilde{\cal
L}_{\rm mass}$ can be regarded as a reasonable way to reduce the degrees of
freedom associated with the canonical seesaw mechanism and constrain its
flavor textures. Through this purely phenomenological operation, we only need
to consider the remaining two right-handed neutrino fields and may simply
rewrite $\widetilde{\cal L}_{\rm mass}$ in Eq. (49) as

$\displaystyle-\widehat{\cal L}_{\rm mass}=\overline{\nu_{\rm L}}\hskip
1.42271pt\widehat{M}_{\rm D}N^{\prime}_{\rm R}+\frac{1}{2}\hskip
1.42271pt\overline{(N^{\prime}_{\rm R})^{c}}\hskip 1.42271pt\widehat{M}_{\rm
R}N^{\prime}_{\rm R}+{\rm h.c.}\;,$ (56)

where $N^{\prime}_{\rm R}\equiv(N_{\mu\rm R},N_{\tau\rm R})^{T}$ is defined,
and the effective neutrino mass matrices $\widehat{M}_{\rm D}$ and
$\widehat{M}_{\rm R}$ are of the following textures:

$\displaystyle\widehat{M}_{\rm D}=\left(\begin{matrix}b&~{}b^{*}\cr c&~{}d\cr
d^{*}&~{}c^{*}\end{matrix}\right)\;,\quad\widehat{M}_{\rm
R}=\left(\begin{matrix}C&~{}D\cr D&~{}C^{*}\end{matrix}\right)\;,$ (57)

which assure $\widehat{\cal L}_{\rm mass}$ itself to satisfy the $\mu$-$\tau$
reflection symmetry. Such a scenario was first proposed in the minimal seesaw
framework [32], and here it is reproduced from the canonical seesaw mechanism
with the help of the translational $\mu$-$\tau$ reflection symmetry.

Given Eqs. (56) and (57), in which the heavy degrees of freedom can be
integrated out to obtain the unique dimension-five Weinberg operator for three
active Majorana neutrino fields [66], the resulting effective Majorana
neutrino mass matrix

$\displaystyle\widehat{M}_{\nu}\simeq-\widehat{M}_{\rm D}\widehat{M}^{-1}_{\rm
R}\widehat{M}^{T}_{\rm D}\;$ (58)

must be of rank two and thus have a zero eigenvalue (i.e., either $m_{1}=0$ or
$m_{3}=0$) 333Note that this observation is a natural consequence of the
“seesaw fair play rule” [67] which is independent of the approximation made in
obtaining the seesaw formula in Eq. (58).. Substituting Eq. (57) into Eq.
(58), one may easily show that $\widehat{M}_{\nu}$ satisfies

$\displaystyle\widehat{M}_{\nu}={\cal P}\widehat{M}^{*}_{\nu}{\cal P}\;,$ (59)

just as $M_{\nu}$ in Eq. (7). So the relevant matrix elements and flavor
mixing parameters of $\widehat{M}_{\nu}$ are constrained as in Eqs. (14) and
(15).

Now that $\widehat{M}_{\nu}$ is obtained at a superhigh-energy scale
$\Lambda_{\mu\tau}$ where both the minimal seesaw mechanism and the
translational $\mu$-$\tau$ reflection symmetry take effect, it will
unavoidably receive some quantum corrections at the electroweak scale
$\Lambda_{\rm EW}\sim 10^{2}~{}{\rm GeV}$ through the renormalization-group-
equation (RGE) running effects. At the one-loop level such RGE-induced quantum
corrections provide a very simple and natural way of softly breaking the exact
$\mu$-$\tau$ reflection symmetry but keeping one of the three active neutrinos
massless. Let us proceed to elaborate on this point.

##  4 RGE-induced soft symmetry breaking

The RGE approach, which was originally invented and developed in the 1950s
[68, 69, 70], is a very powerful theoretical tool. It has proved to be
staggeringly successful in the study of critical phenomena in condensed matter
physics [71, 72] and in the discovery of asymptotic freedom of strong
interactions in quantum field theories and particle physics [73, 74]. The key
point of the RGE tool is that the physical parameters at one renormalization
point or energy scale can be related to their counterparts at another
renormalization point or energy scale with the help of the relevant
differential equations, such that the theory keeps its form invariance or self
similarity between the two points or scales. Regarding the flavor issues of
charged fermions and massive neutrinos, one is especially interested in the
RGE-induced connection between a superhigh energy scale characterized by a
fundamental mechanism of neutrino mass generation and the electroweak scale
where the physical flavor parameters can be experimentally measured [15, 75].

The one-loop differential RGE for the effective Majorana neutrino mass matrix
is well known to us [76, 77, 78, 79], and its integral form can also be found
in the literature (see, e.g., Refs. [80, 81, 82, 83]). Given
$M_{\nu}(\Lambda_{\mu\tau})$ at the $\mu$-$\tau$ reflection symmetry scale and
in the basis where the mass and flavor eigenstates of three charged leptons
are chosen to be identical, its counterpart $M_{\nu}(\Lambda_{\rm EW})$ at the
electroweak scale is explicitly expressed as

$\displaystyle M_{\nu}(\Lambda_{\rm EW})=I_{\nu}(\Lambda_{\rm
EW})\left[T_{l}(\Lambda_{\rm EW})\cdot M_{\nu}(\Lambda_{\mu\tau})\cdot
T_{l}(\Lambda_{\rm EW})\right]\;$ (60)

in the safe $\tau$-flavor dominance approximation, where $T_{l}\simeq{\rm
Diag}\\{1,1,1+\Delta_{\tau}\\}$ and the scale-dependent one-loop RGE evolution
functions $I_{\nu}$ and $\Delta_{\tau}$ are defined as [83]

$\displaystyle I_{\nu}(\mu)$ $\displaystyle\equiv$
$\displaystyle\exp\left[-\frac{1}{16\pi^{2}}\int^{\ln(\Lambda_{\mu\tau}/\Lambda_{\rm
EW})}_{\ln(\mu/\Lambda_{\rm EW})}\alpha_{\nu}(t)~{}{\rm d}t\right]\;,$
$\displaystyle\Delta_{\tau}(\mu)$ $\displaystyle\equiv$
$\displaystyle-\frac{C_{\nu}}{16\pi^{2}}\int^{\ln(\Lambda_{\mu\tau}/\Lambda_{\rm
EW})}_{\ln(\mu/\Lambda_{\rm EW})}y^{2}_{\tau}(t)~{}{\rm d}t\;.$ (61)

In Eq. (61) $t\equiv\ln(\mu/\Lambda_{\rm EW})$ with $\Lambda_{\rm
EW}\leq\mu\leq\Lambda_{\mu\tau}$, $C_{\nu}=-3/2$ and
$\alpha_{\nu}\simeq-3g^{2}_{2}+6y^{2}_{t}+\lambda$ in the SM framework with
$g_{2}$ being the gauge coupling constant of weak interactions, $y_{t}$ being
the top-quark Yukawa coupling eigenvalue and $\lambda$ being the Higgs self-
coupling parameter; or $C_{\nu}=1$ and
$\alpha_{\nu}\simeq-6g^{2}_{1}/5-6g^{2}_{2}+6y^{2}_{t}$ in the minimal
supersymmetric SM (MSSM) with $g_{1}$ being the gauge coupling constant of
electromagnetic interactions. Moreover, $y_{\tau}=m_{\tau}/\langle H\rangle$
in the SM or $y_{\tau}=m_{\tau}/\left(\langle H\rangle\cos\beta\right)$ in the
MSSM, where $\cos\beta$ originates from $\tan\beta$ which is defined as the
ratio of the two Higgs doublets’ vacuum expectation values. The smallness of
$y_{\tau}$ means that $\Delta_{\tau}$ is expected to be small and become zero
at $\mu=\Lambda_{\mu\tau}$, as numerically illustrated in Ref. [83].
Therefore, we have

$\displaystyle M_{\nu}(\Lambda_{\rm EW})\simeq
M_{\nu}(\Lambda_{\mu\tau})+\left(\begin{matrix}0&0&\langle
m\rangle^{*}_{e\mu}\cr 0&0&\langle m\rangle_{\mu\tau}\cr\langle
m\rangle^{*}_{e\mu}&~{}\langle m\rangle_{\mu\tau}~{}&2\langle
m\rangle_{\tau\tau}\end{matrix}\right)\Delta_{\tau}(\Lambda_{\rm EW})\;,$ (62)

where the second term with $\langle m\rangle_{\alpha\beta}$ or its complex
conjugates at $\Lambda_{\mu\tau}$ characterizes some small effects of the
$\mu$-$\tau$ reflection symmetry breaking. Note that $M_{\nu}(\Lambda_{\rm
EW})$ is of rank two as $M_{\nu}(\Lambda_{\mu\tau})$, and thus it contains a
vanishing eigenvalue.

Adopting the standard parametrization of the $3\times 3$ PMNS matrix $U$ in
Eq. (13), we have the form invariance of $M_{\nu}=UD_{\nu}U^{T}$ with
$D_{\nu}\equiv{\rm Diag}\left\\{m_{1},m_{2},m_{3}\right\\}$ at any energy
scale between $\Lambda_{\rm EW}$ and $\Lambda_{\mu\tau}$. A combination of
this decomposition and Eq. (62) allows us to establish some intriguing
relations between the physical parameters of three active Majorana neutrinos
at these two scales. As for the neutrino masses, we explicitly have

$\displaystyle m_{1}(\Lambda_{\rm EW})$ $\displaystyle\simeq$ $\displaystyle
0\;,$ $\displaystyle m_{2}(\Lambda_{\rm EW})$ $\displaystyle\simeq$
$\displaystyle I_{\nu}(\Lambda_{\rm EW})\Big{[}1+\Delta_{\tau}(\Lambda_{\rm
EW})\left(1-s^{2}_{12}c^{2}_{13}\right)\Big{]}m_{2}(\Lambda_{\mu\tau})\;,$
$\displaystyle m_{3}(\Lambda_{\rm EW})$ $\displaystyle\simeq$ $\displaystyle
I_{\nu}(\Lambda_{\rm EW})\Big{[}1+\Delta_{\tau}(\Lambda_{\rm EW})\hskip
1.42271ptc^{2}_{13}\Big{]}m_{3}(\Lambda_{\mu\tau})\;$ (63)

in the $m_{1}(\Lambda_{\mu\tau})=0$ case; or

$\displaystyle m_{1}(\Lambda_{\rm EW})$ $\displaystyle\simeq$ $\displaystyle
I_{\nu}(\Lambda_{\rm EW})\Big{[}1+\Delta_{\tau}(\Lambda_{\rm
EW})\left(1-c^{2}_{12}c^{2}_{13}\right)\Big{]}m_{1}(\Lambda_{\mu\tau})\;,$
$\displaystyle m_{2}(\Lambda_{\rm EW})$ $\displaystyle\simeq$ $\displaystyle
I_{\nu}(\Lambda_{\rm EW})\Big{[}1+\Delta_{\tau}(\Lambda_{\rm
EW})\left(1-s^{2}_{12}c^{2}_{13}\right)\Big{]}m_{2}(\Lambda_{\mu\tau})\;,$
$\displaystyle m_{3}(\Lambda_{\rm EW})$ $\displaystyle\simeq$ $\displaystyle
0\;$ (64)

in the $m_{3}(\Lambda_{\mu\tau})=0$ case, where
$\theta_{23}(\Lambda_{\mu\tau})=\pi/4$, $\delta(\Lambda_{\mu\tau})=\pm\pi/2$
and $\sigma(\Lambda_{\mu\tau})=0$ or $\pi/2$ have been input, and
$\theta_{12}$ and $\theta_{13}$ take their values measured at $\Lambda_{\rm
EW}$. As for the flavor mixing angles and CP-violating phases, let us define

$\displaystyle\Delta\theta_{ij}$ $\displaystyle\equiv$
$\displaystyle\theta_{ij}(\Lambda_{\rm
EW})-\theta_{ij}(\Lambda_{\mu\tau})\;,\hskip 14.22636pt$
$\displaystyle\Delta\delta$ $\displaystyle\equiv$
$\displaystyle\delta(\Lambda_{\rm EW})-\delta(\Lambda_{\mu\tau})\;,$
$\displaystyle\Delta\sigma$ $\displaystyle\equiv$
$\displaystyle\sigma(\Lambda_{\rm EW})-\sigma(\Lambda_{\mu\tau})\;,$ (65)

to describe small deviations of their values at $\Lambda_{\rm EW}$ from those
at $\Lambda_{\mu\tau}$ where the translational $\mu$-$\tau$ reflection flavor
symmetry is exact. Following the generic analytical results obtained in Ref.
[83], we simply arrive at

$\displaystyle\Delta\theta_{12}$ $\displaystyle\simeq$
$\displaystyle-\frac{1}{2}c_{12}s_{12}\left(1-s^{2}_{13}\hskip
1.42271pt\zeta^{\eta_{\sigma}}_{32}\right)\Delta_{\tau}(\Lambda_{\rm EW})\;,$
$\displaystyle\Delta\theta_{13}$ $\displaystyle\simeq$
$\displaystyle-\frac{1}{2}c_{13}s_{13}\left(c^{2}_{12}+s^{2}_{12}\hskip
1.42271pt\zeta^{\eta_{\sigma}}_{32}\right)\Delta_{\tau}(\Lambda_{\rm
EW})\;,\hskip 14.22636pt$ $\displaystyle\Delta\theta_{23}$
$\displaystyle\simeq$
$\displaystyle-\frac{1}{2}\left(s^{2}_{12}+c^{2}_{12}\hskip
1.42271pt\zeta^{-\eta_{\sigma}}_{32}\right)\Delta_{\tau}(\Lambda_{\rm EW})\;$
(66)

in the $m_{1}(\Lambda_{\mu\tau})=0$ case; or

$\displaystyle\Delta\theta_{12}$ $\displaystyle\simeq$
$\displaystyle-\frac{1}{2}c_{12}s_{12}c^{2}_{13}\hskip
1.42271pt\zeta^{-\eta_{\sigma}}_{21}\Delta_{\tau}(\Lambda_{\rm EW})\;,\hskip
14.22636pt$ $\displaystyle\Delta\theta_{13}$ $\displaystyle\simeq$
$\displaystyle+\frac{1}{2}c_{13}s_{13}\Delta_{\tau}(\Lambda_{\rm EW})\;,$
$\displaystyle\Delta\theta_{23}$ $\displaystyle\simeq$
$\displaystyle+\frac{1}{2}\Delta_{\tau}(\Lambda_{\rm EW})\;$ (67)

in the $m_{3}(\Lambda_{\mu\tau})=0$ case, where
$\zeta_{ij}\equiv\left(m_{i}-m_{j}\right)/\left(m_{i}+m_{j}\right)$ is defined
at $\Lambda_{\rm EW}$ and $\eta_{\sigma}\equiv\cos 2\sigma=\pm 1$ denotes one
of the two possible options of $\sigma(\Lambda_{\mu\tau})$ in the $\mu$-$\tau$
reflection symmetry limit [84]. Furthermore, we have

$\displaystyle\Delta\delta$ $\displaystyle\simeq$
$\displaystyle-\frac{c_{12}s_{12}\eta_{\delta}}{2s_{13}}\left[1-\zeta^{-\eta_{\sigma}}_{32}+\frac{s^{2}_{13}}{c^{2}_{12}s^{2}_{12}}\left(1-s^{4}_{12}+c^{4}_{12}\hskip
1.42271pt\zeta^{-\eta_{\sigma}}_{32}\right)\right]\Delta_{\tau}(\Lambda_{\rm
EW})\;,\hskip 14.22636pt$ $\displaystyle\Delta\sigma$ $\displaystyle\simeq$
$\displaystyle
c_{12}s_{12}s_{13}\eta_{\delta}\left(1-\zeta^{-\eta_{\sigma}}_{32}\right)\Delta_{\tau}(\Lambda_{\rm
EW})\;$ (68)

in the $m_{1}(\Lambda_{\mu\tau})=0$ case; or

$\displaystyle\Delta\delta$ $\displaystyle\simeq$
$\displaystyle\frac{s_{13}\eta_{\delta}}{2c_{12}s_{12}}\hskip
1.42271pt\big{(}1-2s^{2}_{12}-\zeta^{\eta_{\sigma}}_{21}\big{)}\hskip
1.42271pt\Delta_{\tau}(\Lambda_{\rm EW})\;,\hskip 14.22636pt$
$\displaystyle\Delta\sigma$ $\displaystyle\simeq$
$\displaystyle\frac{s_{12}s_{13}\eta_{\delta}}{2c_{12}}\hskip
1.42271pt\big{(}1+\zeta^{\eta_{\sigma}}_{21}\big{)}\hskip
1.42271pt\Delta_{\tau}(\Lambda_{\rm EW})\;$ (69)

in the $m_{3}(\Lambda_{\mu\tau})=0$ case, where
$\eta_{\delta}\equiv\sin\delta=\pm 1$ stands for one of the two possible
options of $\delta(\Lambda_{\mu\tau})$ in the $\mu$-$\tau$ reflection symmetry
limit. Some immediate comments are in order.

  * •

$m_{i}(\Lambda_{\mu\tau})=0$ will automatically lead to $m_{i}(\Lambda_{\rm
EW})=0$ (for $i=1$ or $3$) as a natural consequence of the one-loop RGE
running. Note that a finite value of $m_{i}(\Lambda_{\rm EW})$ may result from
$m_{i}(\Lambda_{\mu\tau})=0$ at the two-loop level [85, 86, 87], but it is
extremely tiny (typically of ${\cal O}(10^{-13})~{}{\rm eV}$ in the SM or
${\cal O}(10^{-10})~{}{\rm eV}$ to ${\cal O}(10^{-8})~{}{\rm eV}$ in the MSSM)
and can thus be treated as zero in practice. Due to the smallness of
$\Delta_{\tau}$, the two nonzero neutrino masses essentially have the same
running effects characterized by $I_{\nu}$.

  * •

Among the three flavor mixing angles, $\theta_{13}$ is most insensitive to the
RGE-induced quantum correction because of its smallness. That is to say,
$\Delta\theta_{13}$ is proportional to $s_{13}$ no matter whether
$m_{1}(\Lambda_{\mu\tau})=0$ or $m_{3}(\Lambda_{\mu\tau})=0$. The smallness of
$\Delta m^{2}_{21}$ implies that only $\Delta\theta_{12}$ in the
$m_{3}(\Lambda_{\mu\tau})=0$ case with $\sigma(\Lambda_{\mu\tau})=0$ (i.e.,
$\eta_{\sigma}=1$) may be significantly enhanced. The deviation of
$\theta_{23}(\Lambda_{\rm EW})$ from $\pi/4$ is expected to be quite mild
unless $\tan\beta$ assumes a large value in the MSSM framework.

  * •

One can see that the magnitude of $\Delta\sigma$ is suppressed by the
smallness of $\theta_{13}$, no matter whether $m_{1}(\Lambda_{\mu\tau})=0$ or
$m_{3}(\Lambda_{\mu\tau})=0$. In the latter case the magnitude of
$\Delta\delta$ is also suppressed by $s_{13}$ but it can simultaneously be
enhanced by $\zeta^{\eta_{\sigma}}_{21}$ with $\eta_{\sigma}=-1$ (i.e.,
$\sigma(\Lambda_{\mu\tau})=\pi/2$), so can the magnitude of $\Delta\sigma$. In
comparison, the magnitude of $\Delta\delta$ is somewhat enhanced by $1/s_{13}$
in the $m_{1}(\Lambda_{\mu\tau})=0$ case.

It is worth mentioning that the signs of $\Delta\theta_{ij}$, $\Delta\delta$
and $\Delta\sigma$ in the SM and those in the MSSM are always opposite,
because of $\Delta_{\tau}\propto C_{\nu}$ having the opposite signs in these
two frameworks.

To numerically illustrate our results obtained above, let us only consider the
possibility of $\delta=-\pi/2$ (i.e., $\eta_{\delta}=-1$) at
$\Lambda_{\mu\tau}\simeq 10^{14}~{}{\rm GeV}$ and input the best-fit values
$s_{12}\simeq 0.551$ and $s_{13}\simeq 0.148$ at $\Lambda_{\rm EW}\simeq
10^{2}~{}{\rm GeV}$. Using Eqs. (11) and (12), we obtain $\zeta_{32}\simeq
0.707$ in the $m_{1}(\Lambda_{\mu\tau})=0$ case and $\zeta_{21}\simeq
7.564\times 10^{-3}$ in the $m_{3}(\Lambda_{\mu\tau})=0$ case at $\Lambda_{\rm
EW}$. As for the RGE-induced quantum corrections, we quote
$I_{\nu}(\Lambda_{\rm EW})\simeq 0.748$ and $\Delta_{\tau}(\Lambda_{\rm
EW})\simeq 2.822\times 10^{-5}$ in the SM; $I_{\nu}(\Lambda_{\rm EW})\simeq
0.879$ and $\Delta_{\tau}(\Lambda_{\rm EW})\simeq-1.354\times 10^{-3}$ in the
MSSM with $\tan\beta=10$; or $I_{\nu}(\Lambda_{\rm EW})\simeq 0.871$ and
$\Delta_{\tau}(\Lambda_{\rm EW})\simeq-1.346\times 10^{-2}$ in the MSSM with
$\tan\beta=30$ [83]. Then we arrive at the numerical result

$\displaystyle\frac{m_{2}(\Lambda_{\rm
EW})}{m_{2}(\Lambda_{\mu\tau})}\simeq\frac{m_{3}(\Lambda_{\rm
EW})}{m_{3}(\Lambda_{\mu\tau})}\simeq I_{\nu}(\Lambda_{\rm
EW})\simeq\left\\{\begin{array}[]{l}0.75\quad\quad({\rm SM})\;,\\\
\vspace{-0.6cm}\\\ 0.88\quad\quad({\rm MSSM}~{}{\rm
with}~{}\tan\beta=10)\;,\\\ \vspace{-0.6cm}\\\ 0.87\quad\quad({\rm
MSSM}~{}{\rm with}~{}\tan\beta=30)\;,\end{array}\right.$ (75)

in the $m_{1}(\Lambda_{\mu\tau})=0$ case from Eq. (63); and the same
approximate result for the ratios $m_{1}(\Lambda_{\rm
EW})/m_{1}(\Lambda_{\mu\tau})$ and $m_{2}(\Lambda_{\rm
EW})/m_{2}(\Lambda_{\mu\tau})$ in the $m_{3}(\Lambda_{\mu\tau})=0$ case can be
obtained from Eq. (64). The numerical values of $\Delta\theta_{ij}$ (for
$ij=12,13,23$), $\Delta\delta$ and $\Delta\sigma$ are calculated by using Eqs.
(66) — (69) for either $\sigma(\Lambda_{\mu\tau})=0$ or
$\sigma(\Lambda_{\mu\tau})=\pi/2$, and they are concisely listed in Table 1.
These results are certainly compatible with our above observations based on
the analytical approximations. Two particular remarks are in order.

Table 1: A numerical illustration of the RGE-induced quantum corrections to three flavor mixing angles and two CP-violating phases in the canonical seesaw model constrained by the translational $\mu$-$\tau$ reflection symmetry, where $\Lambda_{\mu\tau}\simeq 10^{14}~{}{\rm GeV}$ and $\delta(\Lambda_{\mu\tau})=-\pi/2$ have been taken, and the results of ${\cal O}(10^{-5})$ degrees or smaller have been denoted as $\sim 0^{\circ}$. $m_{1}(\Lambda_{\mu\tau})=0$ | SM | MSSM ($\tan\beta=10$) | MSSM ($\tan\beta=30$)
---|---|---|---
$\sigma(\Lambda_{\mu\tau})$ | $0$ | $\pi/2$ | $0$ | $\pi/2$ | $0$ | $\pi/2$
$\Delta\theta_{12}$ | $\sim 0^{\circ}$ | $\sim 0^{\circ}$ | $0.018^{\circ}$ | $0.017^{\circ}$ | $0.175^{\circ}$ | $0.172^{\circ}$
$\Delta\theta_{13}$ | $\sim 0^{\circ}$ | $\sim 0^{\circ}$ | $0.005^{\circ}$ | $0.006^{\circ}$ | $0.051^{\circ}$ | $0.064^{\circ}$
$\Delta\theta_{23}$ | $-0.001^{\circ}$ | $-0.0006^{\circ}$ | $0.050^{\circ}$ | $0.031^{\circ}$ | $0.497^{\circ}$ | $0.307^{\circ}$
$\Delta\delta$ | $-0.0004^{\circ}$ | $0.001^{\circ}$ | $0.021^{\circ}$ | $-0.056^{\circ}$ | $0.207^{\circ}$ | $-0.552^{\circ}$
$\Delta\sigma$ | $\sim 0^{\circ}$ | $\sim 0^{\circ}$ | $-0.002^{\circ}$ | $0.0015^{\circ}$ | $-0.022^{\circ}$ | $0.015^{\circ}$
$m_{3}(\Lambda_{\mu\tau})=0$ | SM | MSSM ($\tan\beta=10$) | MSSM ($\tan\beta=30$)
$\sigma(\Lambda_{\mu\tau})$ | $0$ | $\pi/2$ | $0$ | $\pi/2$ | $0$ | $\pi/2$
$\Delta\theta_{12}$ | $-0.048^{\circ}$ | $\sim 0^{\circ}$ | $2.306^{\circ}$ | $0.0001^{\circ}$ | $22.925^{\circ}$ | $0.001^{\circ}$
$\Delta\theta_{13}$ | $0.0001^{\circ}$ | $0.0001^{\circ}$ | $-0.006^{\circ}$ | $-0.006^{\circ}$ | $-0.056^{\circ}$ | $-0.056^{\circ}$
$\Delta\theta_{23}$ | $0.0008^{\circ}$ | $0.0008^{\circ}$ | $-0.039^{\circ}$ | $-0.039^{\circ}$ | $-0.386^{\circ}$ | $-0.386^{\circ}$
$\Delta\delta$ | $-0.0001^{\circ}$ | $0.034^{\circ}$ | $0.005^{\circ}$ | $-1.646^{\circ}$ | $0.048^{\circ}$ | $-16.360^{\circ}$
$\Delta\sigma$ | $-0.00008^{\circ}$ | $-0.011^{\circ}$ | $0.004^{\circ}$ | $0.505^{\circ}$ | $0.038^{\circ}$ | $5.019^{\circ}$

  * •

Now that the neutrino mass spectrum has been fixed by the translational flavor
symmetry (i.e., either $m_{1}=0$ or $m_{3}=0$ at $\Lambda_{\mu\tau}$) and the
observed neutrino mass-squared differences, there is no freedom to adjust the
strength of the RGE-induced quantum corrections in the SM case. That is why
the numerical results of $\Delta\theta_{ij}$, $\Delta\delta$ and
$\Delta\sigma$ are all too small to be experimentally distinguishable. In the
MSSM case taking relatively large values of $\tan\beta$ may help enhance the
significance of the RGE-induced effects, simply because
$\Delta{}_{\tau}(\Lambda_{\rm EW})\propto\left(1+\tan^{2}\beta\right)$ holds.

  * •

Table 1 shows that $\Delta\theta_{12}$ can be significantly enhanced for
$m_{3}(\Lambda_{\mu\tau})=0$ and $\sigma(\Lambda_{\mu\tau})=0$ in the MSSM
framework with a sufficiently large input of $\tan\beta$. The reason is simply
that $\Delta\theta_{12}$ is proportional to both
$\zeta^{-1}_{21}=\left(m_{1}+m_{2}\right)^{2}/\Delta m^{2}_{21}\simeq 132.2$
and $\left(1+\tan^{2}\beta\right)\gtrsim 10^{2}$ for $\tan\beta\gtrsim 10$ and
thus gets enhanced in this case. In comparison, $\Delta\delta$ and
$\Delta\sigma$ can be remarkably enhanced for the same reasons if
$m_{3}(\Lambda_{\mu\tau})=0$ and $\sigma(\Lambda_{\mu\tau})=\pi/2$ are taken
in the MSSM, although these two quantities are apparently suppressed by the
smallness of $\theta_{13}$ as shown in Eq. (69).

In addition, the small magnitude of $\Delta\theta_{23}$ in the MSSM with
$\tan\beta=30$ implies that the RGE-induced quantum correction to
$\theta_{23}(\Lambda_{\mu\tau})=\pi/4$ at $\Lambda_{\rm EW}$ is in general too
mild to resolve the octant issue of $\theta_{23}$ in the minimal seesaw
scenario.

If the values of $\theta_{23}$ and $\delta$ to be measured in the future long-
baseline neutrino oscillation experiments (e.g., Hyper-Kamiokande [88] and
DUNE [89]) turn out to be remarkably different from their values constrained
by the translational $\mu$-$\tau$ reflection symmetry under discussion, it
will be difficult to naturally account for such a discrepancy with the help of
the above RGE-induced soft symmetry breaking effects. In this case one may
either invoke an explicit $\mu$-$\tau$ reflection symmetry breaking scenario
to fit the experimental data (see, e.g., a recent scenario proposed in Ref.
[90]), which certainly seems contrived, or go beyond the $\mu$-$\tau$
reflection symmetry by invoking a larger flavor symmetry group for model
building. In either case it will be useful to explore the direct correlation
between thermal leptogenesis and CP violation in neutrino oscillations (see,
e.g., the pioneering attempts in Refs. [21, 91] and a recent comprehensive
review in Ref.[37]).

##  5 Summary

The strong hierarchies associated with the mass spectra of charged leptons,
up-type quarks and down-type quarks, together with a clear hierarchy of the
off-diagonal CKM matrix elements, strongly indicate that the flavors of
charged fermions should have specific structures instead of a random nature.
This expectation has been further strengthened in recent years with the very
observation that the PMNS lepton flavor mixing matrix $U$ exhibits an
approximate $\mu$-$\tau$ interchange symmetry, although whether the three
active neutrinos have a normal, inverted or nearly degenerate mass spectrum is
still an open question. So far a lot of phenomenological efforts have been
made towards deeper understanding of the issues of tiny neutrino masses and
significant lepton flavor mixing and CP violation with the help of current
neutrino oscillation data, but new ideas are always called for because we
still have a long way to go before the true theory finally emerges.

Along this line of thought, we have coherently combined two simple but
suggestive flavor symmetries — the $\mu$-$\tau$ reflection symmetry and the
translational flavor symmetry by proposing a translational $\mu$-$\tau$
reflection symmetry for the effective Majorana neutrino mass term ${\cal
L}_{\rm mass}$ in this work. Namely, ${\cal L}_{\rm mass}$ is required to keep
invariant under the transformations $\nu_{e\rm L}\to(\nu_{e\rm
L})^{c}+U^{*}_{ei}z^{c}_{\nu}$, $\nu_{\mu\rm L}\to(\nu_{\tau\rm
L})^{c}+U^{*}_{\tau i}z^{c}_{\nu}$ and $\nu_{\tau\rm L}\to(\nu_{\mu\rm
L})^{c}+U^{*}_{\mu i}z^{c}_{\nu}$, where $z_{\nu}$ is a constant spinor field
and $U_{\alpha i}$ are the PMNS matrix elements corresponding to $m_{i}=0$
(for $\alpha=e,\mu,\tau$ and $i=1$ or $3$). Extending such a flavor symmetry
to the canonical seesaw mechanism by requiring the relevant neutrino mass
terms to be invariant under similar translational $\mu$-$\tau$ reflection
transformations for the right-handed neutrino fields, we show that a minimal
seesaw scenario can naturally be reproduced and it automatically respects the
$\mu$-$\tau$ reflection symmetry. As a by-product, the soft breaking effects
of this flavor symmetry have been studied with the help of the one-loop RGEs.

We admit that the translational $\mu$-$\tau$ reflection symmetry is not a real
flavor symmetry for the whole Lagrangian of a Majorana neutrino mass model.
Instead, it is just an effective or working flavor symmetry for the effective
neutrino mass term, but it can greatly help constrain the corresponding flavor
textures. Motivated by the principle of Occam’s razor, we find that such
simple but viable and predictive neutrino mass models really deserve
attention. First, their predictions can be more easily tested in the near
future. Second, it is very likely that the empirical flavor symmetries of this
kind are actually the residual symmetries of some larger flavor symmetry
groups in an underlying fundamental and UV-complete flavor theory. In this
sense, we may argue that the translational $\mu$-$\tau$ reflection symmetry
under discussion, among many other existing flavor symmetries, might be
located in the neutrino landscape instead of the neutrino swampland [92, 93,
94].

Acknowledgments

I am grateful to Dr. Di Zhang for very useful discussions and for his help in
correcting a fatal “full stop” error in my LaTeX file. This work is supported
in part by the National Natural Science Foundation of China under grants No.
12075254 and No. 11835013.

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