# The $\mu$-$\tau$ reflection symmetry of Majorana neutrinos111Dedicated to
the memory of Prof. Harald Fritzsch.

Zhi-zhong Xing1,2 1Institute of High Energy Physics and School of Physical
Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
2Center of High Energy Physics, Peking University, Beijing 100871, China

###### Abstract

The observed pattern of lepton flavor mixing and CP violation strongly
indicates the possible existence of a simple flavor symmetry in the neutrino
sector — the effective Majorana neutrino mass term keeps invariant when the
three left-handed neutrino fields transform as $\nu_{e\rm L}\to(\nu_{e\rm
L})^{c}$, $\nu_{\mu\rm L}\to(\nu_{\tau\rm L})^{c}$ and $\nu_{\tau\rm
L}\to(\nu_{\mu\rm L})^{c}$. A direct application of such a $\mu$-$\tau$
reflection symmetry to the canonical seesaw mechanism can help a lot to
constrain the flavor textures of active and sterile Majorana neutrinos. The
present article is intended to summarize the latest progress made in exploring
the properties of this minimal flavor symmetry, its translational and
rotational extensions, its soft breaking effects via radiative corrections
from a superhigh energy scale to the electroweak scale, and its various
phenomenological implications.

  * E-mail<EMAIL_ADDRESS>
Keywords: Majorana neutrino mass, flavor mixing, CP violation, flavor
symmetry, $\mu$-$\tau$ reflection transformation, seesaw mechanism,
leptogenesis

###### Contents

  1. 1 Introduction
    1. 1.1 Some key flavor issues of leptons and quarks
    2. 1.2 A minimal symmetry of the neutrino flavors
  2. 2 On the flavor-changing CP symmetry
    1. 2.1 On the CP transformation and CP violation
    2. 2.2 How to identify the $\mu$-$\tau$ reflection symmetry
    3. 2.3 The Jarlskog-like invariants of CP violation
    4. 2.4 A generalized case in the seesaw mechanism
    5. 2.5 An $\rm S_{3}$ extension of the $\mu$-$\tau$ flavor symmetry
  3. 3 Translational $\mu$-$\tau$ reflection symmetry
    1. 3.1 The translation of a massless neutrino field
    2. 3.2 A $\rm TM_{1}$ flavor mixing scenario with $m_{1}=0$
    3. 3.3 The translational $\mu$-$\tau$ reflection symmetry
  4. 4 Rotational $\mu$-$\tau$ reflection symmetry
    1. 4.1 A rotational $\mu$-$\tau$ reflection transformation
    2. 4.2 A constrained $\rm TM_{1}$ flavor mixing scenario
    3. 4.3 An extension to active-sterile flavor mixing
  5. 5 Soft $\mu$-$\tau$ reflection symmetry breaking
    1. 5.1 General remarks on $\mu$-$\tau$ symmetry breaking
    2. 5.2 RGE-induced effects on Majorana neutrinos
    3. 5.3 RGE-induced effects on Dirac neutrinos
    4. 5.4 Simplified RGEs when $m_{1}=0$ or $m_{3}=0$
  6. 6 Applications of $\mu$-$\tau$ reflection symmetry
    1. 6.1 Bounds on charged lepton flavor violation
    2. 6.2 Implications for lepton number violation
    3. 6.3 Constraints on cosmic UHE neutrino flavors
    4. 6.4 Constraints on the minimal seesaw scenario
    5. 6.5 Constraints on the leptogenesis mechanism
  7. 7 More specific examples and extensions
    1. 7.1 The littlest seesaw and partial $\mu$-$\tau$ symmetry
    2. 7.2 Zero textures with a $\rm Z_{2}$ reflection symmetry
    3. 7.3 The $\mu$-$\tau$ reflection symmetry of (3+1) flavors
    4. 7.4 Matter effects and nonstandard interactions
  8. 8 Summary and outlook
  9. A

##  1 Introduction

###  1.1 Some key flavor issues of leptons and quarks

The standard model (SM) of modern particle physics is composed of a unified
theory of electromagnetic and weak interactions established in the 1960s [1,
2, 3] and quantum chromodynamics (QCD) for strong interactions formulated in
the 1970s [4, 5, 6]. This model has proved to be a huge success in the past
five decades, as its consistency and predictive power have been tested
extensively and accurately [7]. In particular, recent ATLAS [8] and CMS [9]
measurements of the Higgs boson’s couplings to the vector bosons (i.e.,
$W^{\pm}$ and $Z^{0}$) and the heavier charged fermions (e.g., $t$, $b$,
$\tau$ and $\mu$) have convincingly verified the mass generation mechanism of
these fundamental particles in the SM. Figure 1 illustrates the ATLAS result
as an example, and it is fully compatible with the CMS measurement. In this
situation one may naturally anticipate that the masses of those lighter
charged fermions should also originate from their corresponding Yukawa
interactions with the Higgs field.

Figure 1: The reduced Higgs boson coupling strength modifiers defined as
$\kappa_{F}m_{F}/{\rm vev}$ for a charged fermion with mass $m_{F}$ and
$\sqrt{\kappa_{V}}m_{V}/{\rm vev}$ for a vector boson with mass $m_{V}$ [8],
where the scale factors $\kappa_{F}$ and $\kappa_{V}$ satisfy
$\kappa_{F}=\kappa_{V}=1$ in the SM and “vev” denotes the vacuum expectation
value of the Higgs field. Two fit scenarios with $\kappa_{c}=\kappa_{t}$
(coloured circle markers) or $\kappa_{c}$ left free-floating in the fit (grey
cross markers) are shown. The loop-induced processes are assumed to have the
SM structure, and the Higgs boson decays to the non-SM particles are not
allowed. The vertical bar on each point denotes the $68\%$ confidence
interval. The lower panel shows the values of the coupling strength modifiers.
The grey arrow points in the direction of the best-fit value and the
corresponding grey uncertainty bar extends beyond the lower panel range. The
relevant data are from the ATLAS Run II experiment [8].

But it is well known that the SM is actually incomplete and unsatisfactory in
several aspects, especially in its flavor sector [10]. Let us highlight the
flavor puzzles from the following two perspectives. On the one hand, the SM
does not lay a foundation for nonzero neutrino masses and significant lepton
flavor mixing effects; on the other hand, all of the flavor parameters in the
SM are theoretically undetermined and their values can only be extracted from
a variety of high-energy and low-energy experiments [7]. Nevertheless, the SM
has provided a canonical pathway for qualitatively unravelling the mysteries
of flavor mixing and CP violation in the quark sector — the Cabibbo-Kobayashi-
Maskawa (CKM) mechanism [11, 12]. A quite similar picture, known as the
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mechanism [13, 14], is expected to work
in the lepton sector once the neutrinos acquire their finite but tiny masses
in a natural and straightforward extension of the SM framework.

In this regard the key point is that the known fermions of the same electric
charge all have three families, and they interact not only with the gauge
fields but also with the scalar fields. As a consequence, there must be an
irremovable mismatch between the mass and flavor eigenstates of those fermion
fields. It should be noted that the mass eigenstates of quarks and leptons are
defined respectively by their participation in the strong interactions and by
their propagation in free space, and the flavor eigenstates of either leptons
or quarks are defined by their participation in the electroweak interactions.
Such a dynamical mismatch is highly nontrivial, and it is described by the CKM
quark flavor mixing matrix $V$ or the PMNS lepton flavor mixing matrix $U$ in
the standard weak charged-current interactions:

$\displaystyle-{\cal L}^{(q)}_{\rm cc}=\frac{g_{\rm w}}{\sqrt{2}}\hskip
1.42271pt\overline{\left(u\hskip 8.5359ptc\hskip 8.5359ptt\right)_{\rm
L}}\hskip 1.42271pt\gamma^{\mu}\hskip 1.42271ptV\pmatrix{\hskip
2.27626ptd\hskip 2.27626pt\cr s\cr b\cr}_{\rm L}W^{+}_{\mu}+{\rm h.c.}\;,$
$\displaystyle-{\cal L}^{(l)}_{\rm cc}=\frac{g_{\rm w}}{\sqrt{2}}\hskip
1.42271pt\overline{\left(e~{}~{}\mu~{}~{}\tau\right)_{\rm L}}\hskip
1.42271pt\gamma^{\mu}\hskip
1.42271ptU\pmatrix{\nu_{1}\cr\nu_{2}\cr\nu_{3}\cr}_{\rm L}W^{-}_{\mu}+{\rm
h.c.}\;,$ (1.1)

where $g_{\rm w}$ denotes the coupling constant of weak interactions, and all
the relevant fermion fields are in their mass eigenstates and have the left-
handed chirality. The SM tells us that the $3\times 3$ matrix $V$ must be
unitary and hence can be fully described in terms of four independent
parameters, usually chosen as three Euler-like flavor mixing angles (denoted
by $\vartheta_{12}$, $\vartheta_{13}$ and $\vartheta_{23}$) and one CP-
violating phase (denoted by $\delta_{q}$) as advocated by the Particle Data
Group [7]:

$\displaystyle V=P_{\rm
u}\pmatrix{c_{12}c_{13}&s_{12}c_{13}&\hat{s}^{*}_{13}\cr-
s_{12}c_{23}-c_{12}\hat{s}_{13}s_{23}&c_{12}c_{23}-s_{12}\hat{s}_{13}s_{23}&c_{13}s_{23}\cr
s_{12}s_{23}-c_{12}\hat{s}_{13}c_{23}&-c_{12}s_{23}-s_{12}\hat{s}_{13}c_{23}&c_{13}c_{23}\cr}P_{\rm
d}\;,$ (1.2)

where $c_{ij}\equiv\cos\vartheta_{ij}$, $s_{ij}\equiv\sin\vartheta_{ij}$ and
$\hat{s}_{ij}\equiv s_{ij}e^{{\rm i}\delta_{q}}$ (for $ij=12,13,23$) with
$\vartheta_{ij}$ lying in the first quadrant, $\delta_{q}$ is the irreducible
CP-violating phase, $P_{\rm u}={\rm Diag}\\{e^{{\rm i}\phi_{u}},e^{{\rm
i}\phi_{c}},e^{{\rm i}\phi_{t}}\\}$ and $P_{\rm d}={\rm Diag}\\{e^{{\rm
i}\phi_{d}},e^{{\rm i}\phi_{s}},e^{{\rm i}\phi_{b}}\\}$ are two basis-
dependent phase matrices which have no physical significance. In comparison,
whether the $3\times 3$ PMNS matrix $U$ is unitary or not depends on how the
tiny masses of three active neutrinos originate 111In a fermionic seesaw
mechanism of either type-I [15, 16, 17, 18, 19, 20] or type-III [21], the PMNS
matrix $U$ may slightly deviate from its exact unitarity limit because of
slight mixing between the active neutrino sector and the sterile neutrino
sector or the fermion triplet sector via the corresponding Yukawa
interactions. But $U$ is exactly unitary in the type-II seesaw mechanism with
a Higgs triplet being introduced into the SM [22, 23, 24, 25, 26, 27]..
Fortunately, $U$ is expected to be essentially unitary at least at the ${\cal
O}(1\%)$ accuracy level as constrained by both today’s neutrino oscillation
data and precision measurements of various electroweak and lepton-flavor-
violating processes [28, 29, 30, 31, 32]. That is why we are going to treat
$U$ as a unitary matrix unless otherwise specified. In this case the $3\times
3$ unitary matrix $U$ can simply be parametrized in terms of three lepton
flavor mixing angles (denoted as $\theta_{12}$, $\theta_{13}$ and
$\theta_{23}$) and one or three nontrivial CP-violating phases (denoted as
$\delta_{\nu}$ or as $\delta_{\nu}$, $\rho$ and $\sigma$), corresponding to
the Dirac or Majorana nature of massive neutrinos. Similar to the CKM matrix
$V$, the standard parametrization of the PMNS matrix $U$ is

$\displaystyle U=P_{l}\pmatrix{c_{12}c_{13}&s_{12}c_{13}&\hat{s}^{*}_{13}\cr-
s_{12}c_{23}-c_{12}\hat{s}_{13}s_{23}&c_{12}c_{23}-s_{12}\hat{s}_{13}s_{23}&c_{13}s_{23}\cr
s_{12}s_{23}-c_{12}\hat{s}_{13}c_{23}&-c_{12}s_{23}-s_{12}\hat{s}_{13}c_{23}&c_{13}c_{23}\cr}P_{\nu}\;,$
(1.3)

where $c_{ij}\equiv\cos\theta_{ij}$, $s_{ij}\equiv\sin\theta_{ij}$ and
$\hat{s}_{ij}\equiv s_{ij}e^{{\rm i}\delta_{\nu}}$ (for $ij=12,13,23$) with
all $\theta_{ij}$ lying in the first quadrant, $\delta_{\nu}$ is usually
referred to as the Dirac CP-violating phase, the diagonal Majorana phase
matrix $P_{\nu}$ is defined as $P_{\nu}\equiv{\rm Diag}\big{\\{}e^{{\rm
i}\rho},e^{{\rm i}\sigma},1\big{\\}}$, and the arbitrary phase matrix
$P_{l}={\rm Diag}\\{e^{{\rm i}\phi_{e}},e^{{\rm i}\phi_{\mu}},e^{{\rm
i}\phi_{\tau}}\\}$ has no physical significance. Although $P_{\rm u}$, $P_{\rm
d}$ and $P_{l}$ are irrelevant to any physical quantities, they should not be
ignored when a specific flavor symmetry is applied to constraining either the
pattern of $V$ or that of $U$. This point will become transparent later (see,
e.g., section 2.1).

A global analysis of current experimental data on quark flavor mixing and CP
violation leads us to the following results for nine elements of the CKM
matrix [7]:

$\displaystyle\big{|}V_{ud}\big{|}=0.97435\pm 0.00016\;,\quad$
$\displaystyle\big{|}V_{us}\big{|}=0.22500\pm 0.00067\;,$
$\displaystyle\big{|}V_{ub}\big{|}=0.00369\pm 0.00011\;,\quad$
$\displaystyle\big{|}V_{cd}\big{|}=0.22486\pm 0.00067\;,$
$\displaystyle\big{|}V_{cs}\big{|}=0.97349\pm 0.00016\;,\quad$
$\displaystyle\big{|}V_{cb}\big{|}=0.04182^{+0.00085}_{-0.00074}\;,$
$\displaystyle\big{|}V_{td}\big{|}=0.00857^{+0.00020}_{-0.00018}\;,\quad$
$\displaystyle\big{|}V_{ts}\big{|}=0.04110^{+0.00083}_{-0.00072}\;,$
$\displaystyle\big{|}V_{tb}\big{|}=0.999118^{+0.000031}_{-0.000036}\;.$ (1.4)

In comparison, the $3\sigma$ intervals of nine elements of the PMNS matrix
obtained from a global analysis of the latest atmospheric, solar, reactor and
accelerator neutrino (or antineutrino) oscillation data are [33]

$\displaystyle\big{|}U_{e1}\big{|}=0.801\to 0.845\;,\quad$
$\displaystyle\big{|}U_{e2}\big{|}=0.513\to 0.579\;,$
$\displaystyle\big{|}U_{e3}\big{|}=0.143\to 0.155\;,\quad$
$\displaystyle\big{|}U_{\mu 1}\big{|}=0.234\to 0.500\;,$
$\displaystyle\big{|}U_{\mu 2}\big{|}=0.471\to 0.689\;,\quad$
$\displaystyle\big{|}U_{\mu 3}\big{|}=0.637\to 0.776\;,$
$\displaystyle\big{|}U_{\tau 1}\big{|}=0.271\to 0.525\;,\quad$
$\displaystyle\big{|}U_{\tau 2}\big{|}=0.477\to 0.694\;,$
$\displaystyle\big{|}U_{\tau 3}\big{|}=0.613\to 0.756\;.$ (1.5)

To show the most salient features of $V$ and $U$ in a more intuitive way, let
us illustrate the relative magnitudes of their eighteen elements in Figure 2,
where each circle represents a given matrix element and its area is
proportional to $|V_{\alpha i}|$ (for $\alpha=u,c,t$ and $i=d,s,b$) for the
CKM quark flavor mixing or to $|U_{\alpha i}|$ (for $\alpha=e,\mu,\tau$ and
$i=1,2,3$) for the PMNS lepton flavor mixing on the same scaling. Two
observations are in order.

Figure 2: An illustration of the relative magnitudes of the CKM and PMNS
matrix elements, where the colored area of each circle is proportional to
$|V_{\alpha i}|$ (for $\alpha=u,c,t$ and $i=d,s,b$) or to $|U_{\alpha i}|$
(for $\alpha=e,\mu,\tau$ and $i=1,2,3$) on the same scaling.

  * •

The CKM matrix $V$ can be expressed as $V={\cal I}+\Delta V$, where $\Delta V$
denotes a kind of perturbation to the identity matrix $\cal I$ and its
magnitude is at most of ${\cal O}(\lambda)$ with $\lambda\equiv|V_{us}|\simeq
0.2$ being the Wolfenstein expansion parameter [34]. The strong structural
hierarchy of $V$ is characterized by three layers of its six off-diagonal
elements, ranging from ${\cal O}(\lambda)$ to ${\cal O}(\lambda^{2})$ and from
${\cal O}(\lambda^{2})$ to ${\cal O}(\lambda^{3})$.

  * •

The PMNS matrix $U$ does not have a clear structural hierarchy. Instead, it
exhibits an intriguing $\mu$-$\tau$ interchange symmetry [35]: $|U_{\mu
i}|=|U_{\tau i}|$ (for $i=1,2,3$). Some entries of $U$ are essentially equal
or very close to a few simple but typical Clebsh-Gordan coefficients of group
theory (i.e., the square roots of fractions formed from very small integers,
such as $1/\sqrt{2}$, $1/\sqrt{3}$ and $1/\sqrt{6}$), implying a kind of
underlying discrete flavor symmetry. It is therefore reasonable to conjecture
$U=U_{0}+\Delta U$, where $U_{0}$ denotes a constant flavor mixing pattern
originating from an underlying flavor symmetry group, and $\Delta U$ stands
for small corrections or perturbations to $U_{0}$ as a result of some soft
symmetry breaking effects.

Up to now many efforts have been made to try out various non-Abelian discrete
flavor symmetry groups which can help a lot to determine the explicit form of
$U_{0}$ in a chosen flavor basis [10, 35, 36, 37, 38]. Most of those early
attempts focused on $U=U_{0}$, simply because the available neutrino
oscillation data were quite preliminary at the time. To fit today’s more
precise experimental results, however, a viable neutrino mass model usually
needs the $\Delta U$ term which depends on either spontaneous or explicit
breaking of the flavor symmetry related to $U_{0}$.

Figure 3: An illustration of the mass spectra of three charged leptons and six
quarks at the electroweak scale, where the vertical coordinate is logarithmic.
The currently allowed region of neutrino masses is also included.

Another flavor puzzle is associated with the peculiar mass spectra of three
charged leptons and six quarks as shown in Figure 3, where the allowed region
of three active neutrino masses is also included. The quark mass values are
extracted from a number of different measurements [39] and can be renormalized
to the electroweak scale (i.e., $M_{Z}=91.1876~{}{\rm GeV}$) in the SM
framework as follows [40, 41, 42]:

$\displaystyle m_{u}=1.23\pm 0.21~{}{\rm MeV}\;,\quad$ $\displaystyle
m_{d}=2.67\pm 0.19~{}{\rm MeV}\;,$ $\displaystyle m_{c}=0.620\pm 0.017~{}{\rm
GeV}\;,\quad$ $\displaystyle m_{s}=53.16\pm 4.61~{}{\rm MeV}\;,$
$\displaystyle m_{t}=168.26\pm 0.75~{}{\rm GeV}\;,\quad$ $\displaystyle
m_{b}=2.839\pm 0.026~{}{\rm GeV}\;.$ (1.6)

The pole masses of three charged leptons have been precisely measured [39],
and the values of their running masses at the electroweak scale are given by
[40, 41, 42]

$\displaystyle m_{e}=0.48307\pm 0.00045~{}{\rm MeV}\;,$ $\displaystyle
m_{\mu}=101.766\pm 0.023~{}{\rm MeV}\;,$ $\displaystyle m_{\tau}=1728.56\pm
0.28~{}{\rm MeV}\;.$ (1.7)

Today’s neutrino oscillation experiments have determined the two independent
neutrino mass-squared differences to a quite good degree of accuracy (i.e.,
$\Delta m^{2}_{21}\equiv m^{2}_{2}-m^{2}_{1}=7.42^{+0.21}_{-0.20}\times
10^{-5}~{}{\rm eV}^{2}$ and $\Delta m^{2}_{31}\equiv
m^{2}_{3}-m^{2}_{1}=+2.517^{+0.026}_{-0.028}\times 10^{-3}~{}{\rm eV}^{2}$ for
the normal mass ordering $m_{1}<m_{2}<m_{3}$ or $\Delta m^{2}_{32}\equiv
m^{2}_{3}-m^{2}_{2}=-2.498^{+0.028}_{-0.028}\times 10^{-3}~{}{\rm eV}^{2}$ for
the inverted mass ordering $m_{3}<m_{1}<m_{2}$ [33]), but the absolute
neutrino mass scale remains unknown. The latest KATRIN experiment, which aims
to probe the electron energy spectrum’s endpoint for the tritium beta decay
${}^{3}{\rm H}\to\hskip 0.85355pt^{3}{\rm He}+e^{-}+\overline{\nu}_{e}$, has
set an upper bound for the effective electron-neutrino mass

$\displaystyle\langle
m\rangle_{e}=\sqrt{m^{2}_{1}\left|U_{e1}\right|^{2}+m^{2}_{2}\left|U_{e2}\right|^{2}+m^{2}_{3}\left|U_{e3}\right|^{2}}\;;$
(1.8)

namely, $\langle m\rangle_{e}<0.8~{}{\rm eV}$ at the $90\%$ confidence level
[43]. The future KATRIN experiment may ultimately reach a sensitivity down to
$0.2~{}{\rm eV}$ [44]. Precision measurements of both the cosmic microwave
background anisotropies and the large-scale structure formation of the
Universe, especially the one done by the Planck Collaboration, have put the
most stringent bound on the sum of three neutrino masses [45]:
$m_{1}+m_{2}+m_{3}<0.12~{}{\rm eV}$ at the $95\%$ confidence level. But one
should keep in mind that such a stringent cosmological bound is dependent more
or less on some assumptions associated with the neutrino mass spectrum and the
standard $\Lambda$CDM model [10, 46], where $\Lambda$ denotes the cosmological
constant (or dark energy) and CDM stands for cold dark matter. That is why we
have simply set $m_{i}<1~{}{\rm eV}$ (for $i=1,2,3$) in Figure 3 as the most
conservative upper limit for the absolute neutrino mass scale.

If the three massive neutrinos are their own antiparticles (i.e.,
$\nu_{i}=\nu^{c}_{i}$, where the superscript “$c$” denotes the charge
conjugation of $\nu_{i}$, and $i=1,2,3$), as first conjectured by Ettore
Majorana in 1937 [47], they can mediate the neutrinoless double-beta ($0\nu
2\beta$) decays of some even-even nuclei (such as ${}^{76}_{32}{\rm
Ge}\to\hskip 0.85355pt^{76}_{34}{\rm Se}+2e^{-}$, ${}^{130}_{~{}52}{\rm
Te}\to\hskip 0.85355pt^{130}_{~{}54}{\rm Xe}+2e^{-}$ and ${}^{136}_{~{}54}{\rm
Xe}\to\hskip 0.85355pt^{136}_{~{}56}{\rm Ba}+2e^{-}$) [48]. The rates of such
rare lepton-number-violating processes are governed by the magnitude of the
effective Majorana electron-neutrino mass

$\displaystyle\langle
m\rangle_{ee}=m_{1}U^{2}_{e1}+m_{2}U^{2}_{e2}+m_{3}U^{2}_{e3}\;.$ (1.9)

Today’s stringent upper bound on $|\langle m\rangle_{ee}|$ is $0.06$ eV to
$0.2$ eV at the $95\%$ confidence level [49, 50, 51], where the large
uncertainty originates mainly from the calculations of relevant nuclear matrix
elements. The next-generation $0\nu 2\beta$ experiments are expected to bring
the sensitivity down to the ${\cal O}(10)$ meV level, and the ultimate
sensitivity of such experiments is likely to reach the ${\cal O}(1)$ meV limit
[52].

It is commonly believed that the smallness of all the three quark flavor
mixing angles should be closely related to the strong quark mass hierarchies
[53]. The reason is simply that in an arbitrary flavor basis the CKM matrix
$V$ is actually a product of two unitary matrices used to diagonalize the up-
and down-type quark mass matrices $M_{\rm u,d}$ from their left-hand sides;
i.e., $V\equiv O^{\dagger}_{\rm u}O_{\rm d}$ with $O^{\dagger}_{\rm u}M_{\rm
u}O^{\prime}_{\rm u}={\rm Diag}\big{\\{}m_{u},m_{c},m_{t}\big{\\}}$ and
$O^{\dagger}_{\rm d}M_{\rm d}O^{\prime}_{\rm d}={\rm
Diag}\big{\\{}m_{d},m_{s},m_{b}\big{\\}}$, where $O_{\rm u,d}$ and
$O^{\prime}_{\rm u,d}$ are all unitary. So the elements of $V$ are naturally
expected to rely on $m_{u}/m_{c}\sim m_{c}/m_{t}\sim\lambda^{4}$ and
$m_{d}/m_{s}\sim m_{s}/m_{b}\sim\lambda^{2}$, although their specific
relations depend on specific textures of quark mass matrices. For example, the
simple but phenomenologically successful relations
$\big{|}V_{us}\big{|}\simeq\big{|}V_{cd}\big{|}\simeq\sqrt{m_{d}/m_{s}}$ and
$\big{|}V_{td}\big{|}/\big{|}V_{ts}\big{|}\simeq\sqrt{m_{d}/m_{s}}$ can be
easily obtained from the well-known Fritzsch texture of Hermitian $M_{\rm u}$
and $M_{\rm d}$ [54, 55].

In the lepton sector it seems less convincing to ascribe the largeness of two
of the three lepton flavor mixing angles to a very weak mass hierarchy of
three neutrinos, given the fact that the charged leptons have a rather strong
mass hierarchy and hence should not contribute much to the PMNS matrix $U$.
Especially in the $m_{1}\to 0$ (or $m_{3}\to 0$) limit, in which the neutrino
mass spectrum is expected to be most hierarchical (or to have a near
degeneracy between $m_{1}$ and $m_{2}$), one will be left with $m_{1}/m_{2}\to
0$ (or $m_{1}/m_{2}\to 1$) and $m_{2}/m_{3}\sim\lambda$ (or $m_{3}/m_{2}\to
0$). But it has been found that some viable flavor textures of up- and down-
type quarks can also be used to successfully describe the flavor textures of
charged leptons and massive neutrinos as constrained by current neutrino
oscillation data [10]. Note that small $\Delta m^{2}_{21}\simeq 7.4\times
10^{-5}~{}{\rm eV}^{2}$ and $|\Delta m^{2}_{31}|\simeq 2.5\times
10^{-3}~{}{\rm eV}^{2}$ do allow for a nearly degenerate neutrino mass
spectrum, but this special possibility does not mean that the large flavor
mixing angles $\theta_{12}$ and $\theta_{23}$ should be directly associated
with the mass ratios $m_{1}/m_{2}$ and $m_{2}/m_{3}$. Instead, a kind of non-
Abelian discrete flavor symmetry is more likely to lie behind such a neutrino
mass spectrum and two large flavor mixing angles [10, 35, 36, 37, 38].

All in all, the observed fermion mass spectra and flavor mixing patterns
strongly suggest that charged leptons, massive neutrinos and quarks should
have their respective flavor structures instead of a random nature. A real
challenge is therefore to explore how the flavors are structured, especially
how the active and sterile neutrino flavors are structured. In the absence of
a fundamental guiding principle regarding the origin of tiny neutrino masses,
it is realistic and greatly helpful to follow a phenomenological approach to
the flavor issues of massive neutrinos. In this regard a consensus is that no
matter how large or complicated the true flavor symmetry group can be, its
residual symmetry at low energies should serve as a minimal flavor symmetry in
the neutrino sector. The physical meaning of “minimal” is essentially two-
fold: (1) this flavor symmetry can be described by one of the simplest
discrete or continuous symmetry groups; (2) the pattern of lepton flavor
mixing and CP violation determined or constrained by such a simple flavor
symmetry should be as close as possible to the one extracted from current and
future experimental data on neutrino oscillations.

So the strategy of a successful study along this line of thought is also two-
fold. On the one hand, one may make every effort to identify the minimal
flavor symmetry behind the observed neutrino mass spectrum and flavor mixing
pattern. On the other hand, it makes a lot of sense to investigate possible
flavor structures of the active and sterile neutrinos as constrained by such
an instructive flavor symmetry. Up to now a $\rm Z_{2}$ flavor symmetry, the
$\mu$-$\tau$ interchange symmetry, has been commonly accepted to be the
desired minimal flavor symmetry associated with massive neutrinos. We are
going into details step by step in the following sections.

###  1.2 A minimal symmetry of the neutrino flavors

As shown in Figure 2, current neutrino oscillation data apparently favor
$|U_{\mu i}|=|U_{\tau i}|$ (for $i=1,2,3$), implying an intriguing
$\mu$-$\tau$ interchange symmetry for three pairs of the
$(\mu,\tau)$-associated PMNS matrix elements. This observation is a strong
reflection of the most salient feature of lepton flavor mixing, and it should
imply a kind of underlying flavor symmetry in the lepton sector. Note that a
lepton flavor symmetry of this kind must be basis-dependent, so let us choose
a convenient basis in which the flavor eigenstates of three charged leptons
$(l_{e},l_{\mu},l_{\tau})$ are identified with their mass eigenstates
$(e,\mu,\tau)$. Given the charged lepton mass term

$\displaystyle-{\cal L}_{l}=\overline{\pmatrix{l_{e}&l_{\mu}&l_{\tau}}_{\rm
L}}\hskip 1.9919ptM_{l}\pmatrix{l_{e}\cr l_{\mu}\cr l_{\tau}\cr}_{\rm R}+{\rm
h.c.}\;,$ (1.10)

where $M_{l}$ is in general neither Hermitian nor symmetric, a bi-unitary
transformation $O^{\dagger}_{l}M_{l}O^{\prime}_{l}=D_{l}\equiv{\rm
Diag}\big{\\{}m_{e},m_{\mu},m_{\tau}\big{\\}}$ allows us to change the flavor
basis $(l_{e},l_{\mu},l_{\tau})$ to the mass basis $(e,\mu,\tau)$. Given the
strong mass hierarchy of three charged leptons, the flavor structure of
$M_{l}$ itself is expected to have no way to respect a $\mu$-$\tau$
interchange (either permutation or reflection) symmetry. Without loss of
generality, we simply take $M_{l}=D_{l}$ or equivalently
$O_{l}=O^{\prime}_{l}={\cal I}$ with $\cal I$ being the identity matrix, such
that $l_{\alpha}=\alpha$ (for $\alpha=e,\mu,\tau$) holds. In this flavor basis
the PMNS matrix $U$ linearly links the mass eigenstates of three active
neutrinos to their flavor eigenstates as follows:

$\displaystyle\pmatrix{\nu_{e}\cr\nu_{\mu}\cr\nu_{\tau}\cr}_{\rm
L}=\pmatrix{U_{e1}&U_{e2}&U_{e3}\cr U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\cr U_{\tau
1}&U_{\tau 2}&U_{\tau 3}\cr}\pmatrix{\nu_{1}\cr\nu_{2}\cr\nu_{3}\cr}_{\rm
L}\;.$ (1.11)

Therefore, a special pattern of $U$ is strongly suggestive of an underlying
flavor symmetry associated with the flavor structure of massive neutrinos. Now
that the $\mu$-$\tau$ interchange symmetry is a sort of $\rm Z_{2}$
transformation symmetry with respect to the $\nu_{\mu}$ and $\nu_{\tau}$
flavors, it should be the simplest residual symmetry of any promising discrete
flavor symmetry groups that can be used to build viable neutrino mass models
[10, 35, 36, 37, 38]. That is why we are well motivated to explore what can be
maximally learnt from this minimal flavor symmetry in the neutrino sector.

For the sake of illustration, we consider the effective Majorana mass term of
three active neutrinos in the chosen flavor basis:

$\displaystyle-{\cal L}_{\nu}=\frac{1}{2}\hskip
1.42271pt\overline{\pmatrix{\nu_{e}&\nu_{\mu}&\nu_{\tau}}_{\rm L}}\hskip
1.42271ptM_{\nu}\pmatrix{\nu^{c}_{e}\cr\nu^{c}_{\mu}\cr\nu^{c}_{\tau}\cr}_{\rm
R}+{\rm h.c.}\;,$ (1.12)

where the Majorana neutrino mass matrix $M_{\nu}$ is symmetric,
$\nu^{c}_{\alpha}\equiv{\cal C}\overline{\nu_{\alpha}}^{T}$ and
$\big{(}\nu^{c}_{\alpha}\big{)}_{\rm R}=\big{(}\nu_{\alpha\rm L}\big{)}^{c}$
stand respectively for the charge conjugates of $\nu_{\alpha}$ and
$\nu_{\alpha\rm L}$ (for $\alpha=e,\mu,\tau$), and $\cal C$ is the charge
conjugation operator satisfying ${\cal C}\gamma^{T}_{\mu}{\cal
C}^{-1}=-\gamma_{\mu}$, ${\cal C}\gamma^{T}_{5}{\cal C}^{-1}=\gamma_{5}$ and
${\cal C}^{-1}={\cal C}^{\dagger}={\cal C}^{T}=-{\cal C}$ [56]. Then the
$\mu$-$\tau$ interchange symmetry for ${\cal L}_{\nu}$ can be classified into
the following two subsidiary sets [35].

(1) The $\mu$-$\tau$ permutation symmetry, which requires ${\cal L}_{\nu}$ to
keep unchanged under the partially flavor-changing transformations

$\displaystyle\nu_{e\rm L}\to\nu_{e\rm L}\;,\quad\nu_{\mu\rm L}\to\nu_{\tau\rm
L}\;,\quad\nu_{\tau\rm L}\to\nu_{\mu\rm L}\;.$ (1.13)

In this case the $3\times 3$ symmetric Majorana neutrino mass matrix $M_{\nu}$
is constrained to have the following texture [57]:

$\displaystyle M_{\nu}=\pmatrix{\langle m\rangle_{ee}&\langle
m\rangle_{e\mu}&\langle m\rangle_{e\mu}\cr\langle m\rangle_{e\mu}&\langle
m\rangle_{\mu\mu}&\langle m\rangle_{\mu\tau}\cr\langle m\rangle_{e\mu}&\langle
m\rangle_{\mu\tau}&\langle m\rangle_{\mu\mu}\cr}\;,$ (1.14)

where all the four independent matrix elements are in general complex. To
diagonalize $M_{\nu}$ with the Autonne-Takagi transformation
$U^{\dagger}M_{\nu}U^{*}=D_{\nu}\equiv{\rm
Diag}\big{\\{}m_{1},m_{2},m_{3}\big{\\}}$ [58], one immediately finds that the
unitary PMNS matrix $U$ contains two special flavor mixing angles in its
standard parametrization given by Eq. (1.3):

$\displaystyle\theta_{13}=0\;,\quad\theta_{23}=\frac{\pi}{4}\;,$ (1.15)

a result which assures the equalities $|U_{\mu i}|=|U_{\tau i}|$ (for
$i=1,2,3$) to hold. To generate nonzero $\theta_{13}$ and hence nontrivial
$\delta_{\nu}$, one may break the above $\mu$-$\tau$ permutation symmetry by
simply introducing a single phase parameter such that $\langle
m\rangle_{e\tau}=\langle m\rangle^{*}_{e\mu}$, $\langle
m\rangle_{\tau\tau}=\langle m\rangle^{*}_{\mu\mu}$ and $\arg\big{(}\langle
m\rangle_{e\mu}\big{)}=\arg\big{(}\langle m\rangle_{\mu\mu}\big{)}$ hold [59]
— a special texture of $M_{\nu}$ to be discussed below.

(2) The $\mu$-$\tau$ reflection symmetry, which requires ${\cal L}_{\nu}$ to
be invariant under the partially flavor-changing CP transformations

$\displaystyle\nu_{e\rm L}\to\big{(}\nu_{e\rm L}\big{)}^{c}\;,\quad\nu_{\mu\rm
L}\to\big{(}\nu_{\tau\rm L}\big{)}^{c}\;,\quad\nu_{\tau\rm
L}\to\big{(}\nu_{\mu\rm L}\big{)}^{c}\;.$ (1.16)

In this case the $3\times 3$ Majorana neutrino mass matrix $M_{\nu}$ takes the
special form

$\displaystyle M_{\nu}=\pmatrix{\langle m\rangle_{ee}&\langle
m\rangle_{e\mu}&\langle m\rangle^{*}_{e\mu}\cr\langle m\rangle_{e\mu}&\langle
m\rangle_{\mu\mu}&\langle m\rangle_{\mu\tau}\cr\langle
m\rangle^{*}_{e\mu}&\langle m\rangle_{\mu\tau}&\langle
m\rangle^{*}_{\mu\mu}\cr}\;,$ (1.17)

where the matrix elements $\langle m\rangle_{ee}$ and $\langle
m\rangle_{\mu\tau}$ are real. After $M_{\nu}$ is diagonalized by means of the
transformation $U^{\dagger}M_{\nu}U^{*}=D_{\nu}$, we are left with

$\displaystyle\theta_{23}=\frac{\pi}{4}\;,\quad\delta_{\nu}=\pm\frac{\pi}{2}\;,$
(1.18)

which give rise to the rephasing-independent equalities $|U_{\mu i}|=|U_{\tau
i}|$ (for $i=1,2,3$) too. Of course, the Majorana CP-violating phases of $U$
can also be determined or constrained by use of the $\mu$-$\tau$ reflection
symmetry [10, 35], as will be discussed in section 2.2.

At this point it is worth mentioning that the so-called trimaximal neutrino
mixing matrix proposed by Cabibbo [60] should be the first constant flavor
mixing pattern respecting the $\mu$-$\tau$ reflection symmetry:

$\displaystyle U_{\rm
C}=\pmatrix{\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}\cr\frac{1}{\sqrt{3}}&\frac{\omega}{\sqrt{3}}&\frac{\omega^{*}}{\sqrt{3}}\cr\frac{1}{\sqrt{3}}&\frac{\omega^{*}}{\sqrt{3}}&\frac{\omega}{\sqrt{3}}}\;,$
(1.19)

where $\omega=\exp\left({\rm i}2\pi/3\right)=-1/2+{\rm i}\sqrt{3}/2$. But
$U_{\rm C}$ leads us to a too large value of $\theta_{13}$ (i.e.,
$\theta_{13}=\arcsin\left(1/\sqrt{3}\right)\simeq 35.3^{\circ}$) and hence has
been ruled out by the experimental data on neutrino oscillations. A more
realistic constant neutrino mixing matrix, which exactly respects the
$\mu$-$\tau$ reflection symmetry, is the tetramaximal flavor mixing pattern
[61]

$\displaystyle U_{\rm
4M}=\pmatrix{\frac{1}{2}+\frac{1}{2\sqrt{2}}&\frac{1}{2}&\frac{1}{2}-\frac{1}{2\sqrt{2}}\cr-\frac{1}{2\sqrt{2}}+\frac{\rm
i}{4\left(\sqrt{2}+1\right)}&\frac{1}{2}-\frac{\rm
i}{2\sqrt{2}}&\frac{1}{2\sqrt{2}}+\frac{\rm
i}{4\left(\sqrt{2}-1\right)}\cr-\frac{1}{2\sqrt{2}}-\frac{\rm
i}{4\left(\sqrt{2}+1\right)}&\frac{1}{2}+\frac{\rm
i}{2\sqrt{2}}&\frac{1}{2\sqrt{2}}-\frac{\rm i}{4\left(\sqrt{2}-1\right)}}\;.$
(1.20)

It predicts $\theta_{12}=\arctan\big{(}2-\sqrt{2}\big{)}\simeq 30.4^{\circ}$
and
$\theta_{13}=\arcsin\big{[}\left(\sqrt{2}-1\right)/\left(2\sqrt{2}\right)\big{]}\simeq
8.4^{\circ}$ besides Eq. (1.18). We find that $U_{\rm 4M}$ can fit current
neutrino oscillation data after slight $\mu$-$\tau$ reflection symmetry
breaking effects are taken into account [62, 63]. A number of interesting
constant neutrino mixing patterns respecting the $\mu$-$\tau$ permutation
symmetry have also been proposed and discussed in the literature [10].

Here the emphasis is on the fact that the $\mu$-$\tau$ interchange symmetry
has proved to be powerful enough to constrain the texture of $M_{\nu}$ and
thus interpret the most salient feature of $U$ as revealed by the present
experimental data. It can be regarded as a very simple generalized CP symmetry
[64, 65, 66, 67]. Note, however, that in this regard one usually focuses on a
possible $\mu$-$\tau$ interchange symmetry of the effective neutrino mass term
after spontaneous gauge symmetry breaking. Otherwise, the transformations made
in Eq. (1.13) or Eq. (1.16) would affect some other parts of the Lagrangian of
electroweak interactions. To construct a consistent neutrino mass model which
respect both the electroweak gauge symmetry and the $\mu$-$\tau$ permutation
or reflection symmetry, it is in most cases unavoidable to introduce some
additional scalar fields and require them to couple to the charged-lepton and
neutrino sectors in quite different ways (see, e.g., Refs. [35, 57, 68, 69]
for details). In some viable models of this kind, such new hypothetical
degrees of freedom may lead to rich phenomenology in particle physics and thus
enhance their experimental testability.

That is why a less ambitious strategy has been commonly adopted: one
concentrates on how the flavor textures of three known active neutrinos and
possible sterile neutrinos can be constrained by the respective $\mu$-$\tau$
interchange symmetry and an analogous symmetry associated with the sterile
flavors, and how the constrained textures are further corrected by the
relevant spontaneous or explicit symmetry breaking effects. A very
comprehensive review of such phenomenological attempts was first made in 2016
[35], and since then a lot of new and in-depth research works have been done
along this line of thought. In particular, the $\mu$-$\tau$ reflection
symmetry has attracted much more attention after the smallest lepton flavor
mixing angle $\theta_{13}$ was measured by the Daya Bay Collaboration [70] and
a nearly $3\sigma$ evidence for the nonzero CP-violating phase $\delta_{\nu}$
was achieved by the T2K Collaboration [71].

The present paper is therefore intended to summarize both the latest progress
in studying possible flavor textures of the active neutrinos constrained by
the $\mu$-$\tau$ reflection symmetry and its $\rm S_{3}$ extension, which can
also be referred to as the flavor-changing CP symmetry as a whole, and the
recent developments in exploring possible flavor textures of the sterile
neutrinos with the help of a similar symmetry and the seesaw mechanism. Let us
stress that a flavor-changing CP transformation of the neutrino field
$\nu_{\alpha\rm L}$, namely $\nu_{\alpha\rm L}\to(\nu_{\beta\rm L})^{c}$ (for
$\alpha,\beta=e,\mu,\tau$), is actually a combined operation of the standard
CP transformation of $\nu_{\alpha\rm L}$ itself and the subsequent flavor-
changing $(\nu_{\alpha\rm L})^{c}\to(\nu_{\beta\rm L})^{c}$ transformation,
and thus it makes CP violation possible in the lepton sector.

The remaining parts of this article are organized as follows. In section 2 we
are going to describe the flavor-changing CP transformation for the three
active neutrino fields and identify the $\mu$-$\tau$ reflection symmetry as a
minimal flavor symmetry associated with either the effective Majorana neutrino
mass term or the canonical seesaw mechanism. An $\rm S_{3}$ extension of the
$\mu$-$\tau$ reflection symmetry is also discussed. Sections 3 and 4 are
devoted to the translational and rotational $\mu$-$\tau$ reflection
symmetries, respectively. Both of them can give rise to the phenomenologically
favored $\rm TM_{1}$ flavor mixing pattern, and they can also be combined with
the canonical seesaw mechanism. After a brief introduction to the
possibilities of explicit breaking of the $\mu$-$\tau$ reflection symmetry in
the beginning of section 5, we pay particular attention to its soft breaking
effects induced by the renormalization-group equations (RGEs) from a
superhigh-energy scale to the electroweak scale. The relevant RGEs are
simplified in the case of either $m_{1}=0$ or $m_{3}=0$. Some remarkable
consequences of the $\mu$-$\tau$ reflection symmetry in neutrino
phenomenology, including its implications for lepton flavor violation and
lepton number violation, its constraint on the flavor distribution of
ultrahigh energy cosmic neutrinos at a neutrino telescope, its application to
the minimal seesaw model and its impact on leptogenesis, are explored in
section 6. In section 7 we give a few more specific examples for how to extend
or utilize the $\mu$-$\tau$ reflection symmetry, such as the littlest seesaw
model, the zero textures of fermion mass matrices with a $\rm Z_{2}$
reflection symmetry, the $\mu$-$\tau$ reflection symmetry in a seesaw-induced
$(3+1)$ active-sterile flavor mixing scenario, and matter effects and
nonstandard interactions associated with neutrino oscillations. Section 8 is
devoted to a brief summary of what we have discussed, together with an outlook
for the prospects of applying Occam’s razor in neutrino physics.

##  2 On the flavor-changing CP symmetry

###  2.1 On the CP transformation and CP violation

Within the framework of the SM there is no leptonic CP violation in weak
interactions because a renormalizable gauge- and Lorentz-invariant neutrino
mass term is absent. To take a close look at how CP violation may naturally
occur in the lepton sector as in the quark sector, let us simply consider the
effective Majorana neutrino mass term ${\cal L}_{\nu}$ given in Eq. (1.12) and
rewrite it in the following way:

$\displaystyle-{\cal L}_{\nu}=\frac{1}{2}\left[\overline{\nu_{\rm L}}\hskip
2.84544ptM_{\nu}\hskip 1.42271pt(\nu_{\rm L})^{c}+\overline{(\nu_{\rm
L})^{c}}\hskip 2.84544ptM^{*}_{\nu}\hskip 1.42271pt\nu_{\rm L}\right]\;,$
(2.1)

in which $\nu_{\rm L}=(\nu_{e\rm L},\nu_{\mu\rm L},\nu_{\tau\rm L})^{T}$
denotes the column vector of three left-handed neutrino fields. By definition,
a left-handed neutrino field $\nu_{\alpha\rm L}(t,{\bf x})$ (for $\alpha=e$,
$\mu$ or $\tau$) transforms under charge conjugation (C) and parity (P) as
[56]

$\displaystyle\nu_{\alpha\rm L}(t,{\bf x})\to\big{[}\nu_{\alpha\rm L}(t,-{\bf
x})\big{]}^{c}={\cal C}\hskip 1.42271pt\overline{\nu_{\alpha\rm L}(t,-{\bf
x})}^{T},$ (2.2)

where the properties of $\cal C$ have been elucidated below Eq. (1.12).
Substituting Eq. (2.2) into Eq. (2.1), we find that ${\cal L}_{\nu}$ becomes

$\displaystyle-{\cal L}^{\prime}_{\nu}=\frac{1}{2}\left[\overline{\nu_{\rm
L}}\hskip 2.84544ptM^{*}_{\nu}\hskip 1.42271pt(\nu_{\rm
L})^{c}+\overline{(\nu_{\rm L})^{c}}\hskip 2.84544ptM_{\nu}\hskip
1.42271pt\nu_{\rm L}\right]\;,$ (2.3)

where ${\bf x}\to-{\bf x}$ is implied. One can immediate see that ${\cal
L}^{\prime}_{\nu}={\cal L}_{\nu}$ will hold (to be more exact, their
corresponding actions will be equal) if $M_{\nu}$ is real. That is to say, CP
will be a good symmetry if $M_{\nu}$ does not contain any nontrivial CP-
violating phase.

In general, $M_{\nu}$ is a symmetric complex matrix which certainly contains a
number of nontrivial phases. It can be diagonalized in an arbitrary basis by
the transformation

$\displaystyle
O^{\dagger}_{\nu}M_{\nu}O^{*}_{\nu}=D_{\nu}\equiv\pmatrix{m_{1}&0&0\cr
0&m_{2}&0\cr 0&0&m_{3}\cr}\;,$ (2.4)

where $O_{\nu}$ is a $3\times 3$ unitary matrix, and $m_{i}$ (for $i=1,2,3$)
denote the neutrino masses. Eq. (2.4) is actually equivalent to the basis
transformation $\nu_{\rm L}\to\tilde{\nu}_{\rm L}\equiv
O^{\dagger}_{\nu}\nu_{\rm L}$ for ${\cal L}_{\nu}$, where $\tilde{\nu}_{\rm
L}=(\nu_{1\rm L},\nu_{2\rm L},\nu_{3\rm L})^{T}$ stands for the column vector
of three left-handed neutrino mass eigenstates. So the PMNS matrix $U$
appearing in ${\cal L}_{\rm cc}$ is given by $U=O^{\dagger}_{l}O_{\nu}$, where
the unitary matrix $O_{l}$ has been used to diagonalize the charged lepton
mass matrix $M_{l}$ below Eq. (1.10). In the flavor basis of $M_{l}=D_{l}$, we
have $O_{l}={\cal I}$ and thus $U=O_{\nu}$. It is then possible to reconstruct
the elements $\langle m\rangle_{\alpha\beta}$ of $M_{\nu}$ in terms of $U$ and
$m_{i}$ as follows:

$\displaystyle\langle
m\rangle_{\alpha\beta}\equiv\left(UD_{\nu}U^{T}\right)_{\alpha\beta}=\sum^{3}_{i=1}m_{i}U_{\alpha
i}U_{\beta i}\;.$ (2.5)

It is obvious that a real $M_{\nu}$ will lead to a real $U$ in the chosen
flavor basis, or vice versa. The standard source of leptonic CP violation is
therefore the complex $M_{l}$ and (or) $M_{\nu}$, or equivalently the
nontrivial CP-violating phases of $U$ in the weak charged-current interactions
[72]. In other words, fermion mass generation, flavor mixing and CP violation
are closely related to one another and constitute the three key issues of
flavor physics. But in almost all the cases a discrete or continuous flavor
symmetry is not powerful enough to simultaneously predict the neutrino mass
spectrum, the flavor mixing pattern and the CP-violating phases.

Note that what can be really measured in the lepton-number-violating processes
are the rephasing invariants $|\langle m\rangle_{\alpha\beta}|$ instead of
$\langle m\rangle_{\alpha\beta}$ themselves (for $\alpha,\beta=e,\mu,\tau$).
So one may define the observable effective neutrino masses $\langle
m\rangle_{\alpha}$ (for $\alpha=e,\mu,\tau$) as follows:

$\displaystyle\langle m\rangle^{2}_{\alpha}\equiv\sum_{\beta}\left|\langle
m\rangle_{\alpha\beta}\right|^{2}=\sum_{i}m^{2}_{i}\left|U_{\alpha
i}\right|^{2}\;,$ (2.6)

where the unitarity of $U$ and Eq. (2.5) have been used. It is clear that
$\langle m\rangle_{e}$ is just the effective electron-neutrino mass in Eq.
(1.8), which can be probed in the beta decays. In comparison, the effective
neutrino masses $\langle m\rangle_{\mu}$ and $\langle m\rangle_{\tau}$ are
associated respectively with the decay modes $\pi^{+}\to\mu^{+}+\nu_{\mu}$ and
$\tau\to 5\pi+\nu_{\tau}$ (or $\tau\to 3\pi+\nu_{\tau}$) [73]. We easily see
that the three effective mass terms satisfy the sum rule

$\displaystyle\langle m\rangle^{2}_{e}+\langle m\rangle^{2}_{\mu}+\langle
m\rangle^{2}_{\tau}=m^{2}_{1}+m^{2}_{2}+m^{2}_{3}\;.$ (2.7)

It is worth pointing out that each $\big{|}\langle
m\rangle_{\alpha\beta}\big{|}$ contains at least two of the three independent
CP-violating phases of $U$, but each $\langle m\rangle_{\alpha}$ contains at
most a single phase parameter — the so-called Dirac phase $\delta_{\nu}$ which
can be determined from the lepton-flavor-violating but lepton-number-
conserving neutrino oscillations. If the massive neutrinos are the Majorana
particles, $\delta_{\nu}$ may also affect those lepton-number-violating
processes such as the $0\nu 2\beta$ decays. This point can be seen from Eq.
(1.9), where $\delta_{\nu}$ is hidden in $U_{e3}$; and it will become more
transparent in section 6.2.

###  2.2 How to identify the $\mu$-$\tau$ reflection symmetry

Now let us show that behind the observed equalities $\big{|}U_{\mu
i}\big{|}=\big{|}U_{\tau i}\big{|}$ (for $i=1,2,3$) and the T2K hint of CP
violation in neutrino oscillations [71] should lie a minimal flavor symmetry —
the $\mu$-$\tau$ reflection symmetry for the effective Majorana neutrino mass
term. The same proof is applicable to the effective Dirac neutrino mass term.

Given $\big{|}U_{\mu i}\big{|}=\big{|}U_{\tau i}\big{|}$ for the unitary PMNS
matrix $U$, the question is how to transform $U_{\mu i}$ to $U_{\tau i}$ or
$U^{*}_{\tau i}$ in a given flavor basis, in which $U_{ei}$ either stays
unchanged or becomes its complex conjugate such that $\big{|}U_{ei}\big{|}$
keeps invariant. We find that there are two typical possibilities of this kind
[74].

(1) One possibility is $U={\cal P}U\zeta$, where $\cal P$ is a real orthogonal
$(\mu,\tau)$-associated permutation matrix of the form

$\displaystyle{\cal P}={\cal P}^{T}={\cal P}^{\dagger}=\pmatrix{1&0&0\cr
0&0&1\cr 0&1&0\cr}\;,$ (2.8)

and $\zeta={\rm Diag}\\{\eta_{1},\eta_{2},\eta_{3}\\}$ with $\eta_{i}=\pm 1$.
In this case the $(\mu,\tau)$-associated orthogonality condition of $U$
becomes

$\displaystyle\sum^{3}_{i=1}U_{\mu i}U^{*}_{\tau
i}=\sum^{3}_{i=1}\eta_{i}\big{|}U_{\mu i}\big{|}^{2}=0\;,$ (2.9)

implying the corresponding unitarity triangle in the complex plane collapses
into lines. So the very possibility of $\eta_{1}=\eta_{2}=\eta_{3}$ is utterly
incompatible with the normalization condition of $U_{\mu i}$ (for $i=1,2,3$)
and hence has to be abandoned. Considering that $|U_{\mu 1}|$ is definitely
smaller than $|U_{\mu 3}|$, and $|U_{\mu 2}|$ is most likely smaller than
$|U_{\mu 3}|$ as one can clearly see in Eq. (1.5), we find that the
possibility of $\eta_{1}=\eta_{2}=-\eta_{3}$ is phenomenologically interesting
because it leads us to

$\displaystyle\big{|}U_{\mu 1}\big{|}^{2}+\big{|}U_{\mu
2}\big{|}^{2}=\big{|}U_{\mu 3}\big{|}^{2}=\frac{1}{2}\;,$ (2.10)

and a similar relation for $|U_{\tau i}|^{2}$ (for $i=1,2,3$). As a
consequence, $|U_{e3}|=0$ must hold. Combining these results with the
parametrization of $U$ in Eq. (1.3), we immediately arrive at $\theta_{13}=0$
and $\theta_{23}=\pi/4$ as given by Eq. (1.15).

This observation implies that $U={\cal P}U\zeta$ should give rise to the
$\mu$-$\tau$ permutation symmetry; namely, the effective neutrino mass term
${\cal L}_{\nu}$ in Eq. (1.12) or Eq. (2.1) should keep unchanged under the
transformations made in Eq. (1.13). To see this point more clearly, we
substitute $U={\cal P}U\zeta$ into Eq. (2.4) by taking $O_{\nu}=U$ in the
chosen $M_{l}=D_{l}$ basis. Then we obtain

$\displaystyle M_{\nu}=UD_{\nu}U^{T}={\cal P}U\zeta D_{\nu}\zeta U^{T}{\cal
P}={\cal P}UD_{\nu}U^{T}{\cal P}={\cal P}M_{\nu}{\cal P}\;.$ (2.11)

So the texture of $M_{\nu}$ can be constrained to the form given in Eq.
(1.14). If Eq. (2.11) is inserted into Eq. (2.1), one will have

$\displaystyle-{\cal L}_{\nu}=\frac{1}{2}\hskip 1.42271pt\overline{{\cal
P}\nu_{\rm L}}\hskip 2.84544ptM_{\nu}\hskip 1.42271pt{\cal P}(\nu_{\rm
L})^{c}+{\rm h.c.}\;,$ (2.12)

implying that ${\cal L}_{\nu}$ is invariant under the transformation $\nu_{\rm
L}\to{\cal P}\nu_{\rm L}$. The latter is simply equivalent to the $\mu$-$\tau$
permutation transformation given in Eq. (1.13). In this way we have identified
that behind a special pattern of $U$ as constrained by $U={\cal P}U\zeta$ is
the invariance of ${\cal L}_{\nu}$ under the $\mu$-$\tau$ permutation
transformation, or vice versa.

(2) The other typical and more intriguing possibility is $U={\cal
P}U^{*}\zeta$, where $\cal P$ and $\zeta$ have been given in and below Eq.
(2.8). In this case the $(\mu,\tau)$-associated orthogonality condition of $U$
turns out to be

$\displaystyle\sum^{3}_{i=1}U_{\mu i}U^{*}_{\tau
i}=\sum^{3}_{i=1}\eta_{i}U^{2}_{\mu i}=0\;,$ (2.13)

which defines a normal unitarity triangle in the complex plane. Taking
$\eta_{1}=\eta_{2}=\eta_{3}$ for example, one may combine Eq. (2.13) with the
$(\mu,\tau)$-associated normalization condition of $U$ and then obtain the
interesting results

$\displaystyle\sum^{3}_{i=1}\left({\rm Re}U_{\mu
i}\right)^{2}=\sum^{3}_{i=1}\left({\rm Im}U_{\mu
i}\right)^{2}=\frac{1}{2}\;,\hskip 34.14322pt$
$\displaystyle\sum^{3}_{i=1}\left({\rm Re}U_{\mu i}\hskip 1.42271pt{\rm
Im}U_{\mu i}\right)=0\;.$ (2.14)

A combination of Eqs. (1.3) and (2.14) will allow us to arrive at
$\theta_{23}=\pi/4$ and $\delta_{\nu}=\pm\pi/2$ in the $\mu$-$\tau$ reflection
symmetry, as already given in Eq. (1.18).

Table 1: A comparison between the $\mu$-$\tau$-interchanging CP transformation
and the traditional CP transformation for the left-handed neutrino fields
[75].

  * $\bigstar$ Traditional CP transformation |  $\bigstar$ $\mu$-$\tau$-interchanging CP transformation
---|---
$\left(t,{\bf x}\right)\to\left(t,-{\bf x}\right)$ | $\left(t,{\bf x}\right)\to\left(t,-{\bf x}\right)$
$\left\\{\begin{array}[]{rcl}\nu_{e\rm L}&\to&\big{(}\nu_{e\rm L}\big{)}^{c}\\\ \nu_{\mu\rm L}&\to&\big{(}\nu_{\mu\rm L}\big{)}^{c}\\\ \nu_{\tau\rm L}&\to&\big{(}\nu_{\tau\rm L}\big{)}^{c}\end{array}\right.$ | $\left\\{\begin{array}[]{rcl}\nu_{e\rm L}&\to&\big{(}\nu_{e\rm L}\big{)}^{c}\\\ \nu_{\mu\rm L}&\to&\big{(}\nu_{\tau\rm L}\big{)}^{c}\\\ \nu_{\tau\rm L}&\to&\big{(}\nu_{\mu\rm L}\big{)}^{c}\end{array}\right.$

Now let us prove that behind $U={\cal P}U^{*}\zeta$ is the $\mu$-$\tau$
reflection symmetry of ${\cal L}_{\nu}$ in Eq. (2.1). Thanks to $U={\cal
P}U^{*}\zeta$, we simply have

$\displaystyle M_{\nu}=UD_{\nu}U^{T}={\cal P}U^{*}\zeta D_{\nu}\zeta
U^{\dagger}{\cal P}={\cal P}\left(UD_{\nu}U^{T}\right)^{*}{\cal P}={\cal
P}M^{*}_{\nu}{\cal P}\;.$ (2.15)

That is why the texture of $M_{\nu}$ can be constrained to the form given in
Eq. (1.17). After Eq. (2.15) is substituted into Eq. (2.1), one will have

$\displaystyle-{\cal L}_{\nu}=\frac{1}{2}\hskip 1.42271pt\overline{{\cal
P}(\nu_{\rm L})^{c}}\hskip 2.84544ptM_{\nu}\hskip 1.42271pt{\cal P}\nu_{\rm
L}+{\rm h.c.}\;,$ (2.16)

implying that ${\cal L}_{\nu}$ keeps invariant under the transformation
$\nu_{\rm L}\to{\cal P}(\nu_{\rm L})^{c}$. Such a very simple
$\mu$-$\tau$-interchanging CP transformation is just equivalent to the
$\mu$-$\tau$ reflection transformation given in Eq. (1.16), and its similarity
to or difference from the traditional flavor-conserving CP transformation is
illustrated in Table 1. In this manner we have identified that behind a
special pattern of $U$ as constrained by $U={\cal P}U^{*}\zeta$ is actually
the invariance of ${\cal L}_{\nu}$ under the $\mu$-$\tau$ reflection
transformation, or vice versa. Since the $\mu$-$\tau$ reflection symmetry of
${\cal L}_{\nu}$ can naturally accommodate CP violation, it is expected to be
more helpful than the $\mu$-$\tau$ permutation symmetry of ${\cal L}_{\nu}$ in
revealing the salient features of current neutrino oscillation data.

Concentrating on the $\mu$-$\tau$ reflection symmetry of ${\cal L}_{\nu}$, we
take a close look at the constraints of $U={\cal P}U^{*}\zeta$ on the angles
and phases of the standard parametrization of $U$ in Eq. (1.3). It is easy to
find that the basis-dependent equality $U={\cal P}U^{*}\zeta$ leads us to the
following constraint conditions:

$\displaystyle e^{{\rm i}2\left(\phi_{e}+\rho\right)}=\eta_{1}\;,\quad e^{{\rm
i}2\left(\phi_{e}+\sigma\right)}=\eta_{2}\;,\quad e^{{\rm
i}2\left(\phi_{e}-\delta_{\nu}\right)}=\eta_{3}\;$ (2.17)

arising from $U_{ei}=\eta_{i}U^{*}_{ei}$ (for $i=1,2,3$), and

$\displaystyle\left(s_{12}c_{23}+c_{12}s_{13}s_{23}e^{{\rm
i}\delta_{\nu}}\right)e^{{\rm
i}\left(\phi_{\mu}+\phi_{\tau}+2\rho\right)}=-\eta_{1}\left(s_{12}s_{23}-c_{12}s_{13}c_{23}e^{-{\rm
i}\delta_{\nu}}\right)\;,$
$\displaystyle\left(c_{12}c_{23}-s_{12}s_{13}s_{23}e^{{\rm
i}\delta_{\nu}}\right)e^{{\rm
i}\left(\phi_{\mu}+\phi_{\tau}+2\sigma\right)}=-\eta_{2}\left(c_{12}s_{23}+s_{12}s_{13}c_{23}e^{-{\rm
i}\delta_{\nu}}\right)\;,\hskip 28.45274pt$ $\displaystyle s_{23}e^{{\rm
i}\left(\phi_{\mu}+\phi_{\tau}\right)}=\eta_{3}c_{23}\;$ (2.18)

coming from $U_{\mu i}=\eta_{i}U^{*}_{\tau i}$ (for $i=1,2,3$) with
$\eta_{i}=\pm 1$. We are therefore left with

$\displaystyle\theta_{23}=\frac{\pi}{4}\;,\quad\delta_{\nu}=\pm\frac{\pi}{2}\;,\quad\rho=0\hskip
5.69046pt{\rm or}\hskip 5.69046pt\frac{\pi}{2}\;,\quad\sigma=0\hskip
5.69046pt{\rm or}\hskip 5.69046pt\frac{\pi}{2}\;,$ (2.19)

together with $\phi_{e}=0$ or $\pi/2$ and
$\phi_{\mu}+\phi_{\tau}=\left(2n+1\right)\pi-2\phi_{e}$ with $n$ being an
integer, in which the ambiguity of $\eta_{i}$ has been taken into account.
Three immediate comments are in order. First, the same constraints on
$\theta_{23}$, $\delta_{\nu}$, $\rho$, $\sigma$ and $\phi_{\alpha}$ (for
$\alpha=e,\mu,\tau$) can actually be achieved from the texture of $M_{\nu}$
constrained by Eq. (2.15) without any ambiguity of $\eta_{i}$ [76, 77].
Second, it is very easy to check that Eq. (2.19) allows us to confirm the
parametrization-independent constraint conditions obtained in Eq. (2.14) with
$\eta_{1}=\eta_{2}=\eta_{3}$. Third, the recent T2K measurement of leptonic CP
violation in neutrino oscillations hints that the possibility of
$\delta_{\nu}=-\pi/2$ is somewhat preferable to that of $\delta_{\nu}=\pi/2$
[71]. Moreover, one may figure out that the area of the
$(\mu,\tau)$-associated unitarity triangle defined by Eq. (2.13) is given by

$\displaystyle
S_{\triangle}=\frac{1}{4}\big{|}U_{e1}\big{|}\big{|}U_{e2}\big{|}\big{|}U_{e3}\big{|}\sim
1.5\%\;$ (2.20)

in the $\mu$-$\tau$ reflection symmetry limit, where Eq. (1.5) has been
considered to make the numerical estimate. So a significant CP-violating
effect in normal neutrino-neutrino or antineutrino-antineutrino oscillations,
which is essentially measured by the magnitude of $S_{\triangle}$, is
naturally expected (see, e.g., Ref. [78] for an analysis of all the six
leptonic unitarity triangles which have the same area $S_{\triangle}$).

###  2.3 The Jarlskog-like invariants of CP violation

Note that the geometric quantity $S_{\triangle}$ is exactly equivalent to the
well-known Jarlskog invariant of CP violation defined by the rephasing-
independent relationship [79]

$\displaystyle{\rm Im}\left(U_{\alpha i}U_{\beta j}U^{*}_{\alpha
j}U^{*}_{\beta i}\right)={\cal
J}_{\nu}\sum_{\gamma}\sum_{k}\epsilon_{\alpha\beta\gamma}\epsilon_{ijk}\;,$
(2.21)

where $\epsilon_{\alpha\beta\gamma}$ and $\epsilon_{ijk}$ are the three-
dimensional Levi-Civita symbols with the Greek and Latin subscripts running
respectively over $(e,\mu,\tau)$ and $(1,2,3)$. It is very easy to prove
$S_{\triangle}=\big{|}{\cal J}_{\nu}\big{|}/2$ and obtain ${\cal
J}_{\nu}=c_{12}s_{12}c^{2}_{13}s_{13}c_{23}s_{23}\sin\delta_{\nu}$ in the
standard parametrization of $U$, and therefore $\big{|}{\cal
J}_{\nu}\big{|}=c_{12}s_{12}c^{2}_{13}s_{13}/2\sim 3\%$ holds in the
$\mu$-$\tau$ reflection symmetry limit. In comparison, the Jarlskog invariant
of quark flavor mixing and CP violation has been well determined by current
experimental data: ${\cal J}_{q}=\left(3.08^{+0.15}_{-0.13}\right)\times
10^{-5}$ [7], which is about three orders of magnitude smaller than its
leptonic analogue ${\cal J}_{\nu}$ given above. Whether ${\cal J}_{\nu}$ is
related to the observed matter-antimatter asymmetry of the Universe has been a
burning question in today’s neutrino cosmology [56].

If the three active neutrinos have the Majorana nature, then the lepton-
number-violating neutrino-antineutrino oscillations can in principle happen
[13, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91], although such rare
processes are strongly suppressed and almost unobservable in practice.
Nevertheless, we find that it is conceptually interesting to examine how the
Jarlskog-like invariants of CP violation, which govern the strengths of CP-
violating effects in possible neutrino-antineutrino oscillations, are
simplified by the $\mu$-$\tau$ reflection symmetry.

The probabilities of lepton-number-violating
$\nu_{\alpha}\to\overline{\nu}_{\beta}$ and
$\overline{\nu}_{\alpha}\to\nu_{\beta}$ oscillations (for
$\alpha,\beta=e,\mu,\tau$) can be expressed as [89, 90]

$\displaystyle P\left(\nu_{\alpha}\to\overline{\nu}_{\beta}\right)$
$\displaystyle=$ $\displaystyle\frac{\big{|}\langle
m\rangle_{\alpha\beta}\big{|}^{2}}{E^{2}}-4\sum_{i<j}\frac{m_{i}m_{j}}{E^{2}}{\rm
Re}\left(U_{\alpha i}U_{\beta i}U^{*}_{\alpha j}U^{*}_{\beta
j}\right)\sin^{2}\frac{\Delta m^{2}_{ji}L}{4E}$
$\displaystyle+2\sum_{i<j}\frac{m_{i}m_{j}}{E^{2}}{\rm Im}\left(U_{\alpha
i}U_{\beta i}U^{*}_{\alpha j}U^{*}_{\beta j}\right)\sin\frac{\Delta
m^{2}_{ji}L}{2E}\;,$ $\displaystyle
P\left(\overline{\nu}_{\alpha}\to\nu_{\beta}\right)$ $\displaystyle=$
$\displaystyle\frac{\big{|}\langle
m\rangle_{\alpha\beta}\big{|}^{2}}{E^{2}}-4\sum_{i<j}\frac{m_{i}m_{j}}{E^{2}}{\rm
Re}\left(U_{\alpha i}U_{\beta i}U^{*}_{\alpha j}U^{*}_{\beta
j}\right)\sin^{2}\frac{\Delta m^{2}_{ji}L}{4E}$ (2.22)
$\displaystyle-2\sum_{i<j}\frac{m_{i}m_{j}}{E^{2}}{\rm Im}\left(U_{\alpha
i}U_{\beta i}U^{*}_{\alpha j}U^{*}_{\beta j}\right)\sin\frac{\Delta
m^{2}_{ji}L}{2E}\;,$

where a universal kinematical factor has been neglected for these two
probabilities, $E$ is the average beam energy of neutrinos or antineutrinos,
$\Delta m^{2}_{ji}\equiv m^{2}_{j}-m^{2}_{i}$ denote the neutrino mass-squared
differences, $L$ stands for the baseline length between the neutrino (or
antineutrino) source and the detector, and the effective Majorana neutrino
mass term $\langle m\rangle_{\alpha\beta}$ has been defined in Eq. (2.5). We
see that CP violation in neutrino-antineutrino oscillations is measured by the
Jarlskog-like invariants

$\displaystyle{\cal J}^{ij}_{\alpha\beta}\equiv{\rm Im}\left(U_{\alpha
i}U_{\beta i}U^{*}_{\alpha j}U^{*}_{\beta j}\right)\;,$ (2.23)

where $i,j=1,2,3$ and $\alpha,\beta=e,\mu,\tau$. Different from the canonical
Jarlskog invariant ${\cal J}_{\nu}$ defined in Eq. (2.21), which is only
sensitive to the Dirac CP-violating phase $\delta_{\nu}$, ${\cal
J}^{ij}_{\alpha\beta}$ are in general sensitive to all the three CP-violating
phases of $U$ [90]. Taking account of Eq. (2.19) in the $\mu$-$\tau$
reflection symmetry limit, however, we can obtain the following simplified
results of ${\cal J}^{ij}_{\alpha\beta}$:

$\displaystyle{\cal J}_{ee}^{12}={\cal J}_{ee}^{13}={\cal J}_{ee}^{23}={\cal
J}_{\mu\tau}^{12}={\cal J}_{\mu\tau}^{13}={\cal J}_{\mu\tau}^{23}=0\;,$ (2.24)

and

$\displaystyle{\cal J}_{\mu\mu}^{12}$ $\displaystyle=$ $\displaystyle-{\cal
J}_{\tau\tau}^{12}=-{\cal J}_{e\mu}^{12}={\cal J}_{e\tau}^{12}={\cal
J}_{\nu}\eta_{\rho}\eta_{\sigma}\;,$ $\displaystyle{\cal J}_{\mu\mu}^{13}$
$\displaystyle=$ $\displaystyle-{\cal J}_{\tau\tau}^{13}=-{\cal
J}_{e\mu}^{13}={\cal J}_{e\tau}^{13}=+{\cal J}_{\nu}\eta_{\rho}\;,$
$\displaystyle{\cal J}_{\mu\mu}^{23}$ $\displaystyle=$ $\displaystyle-{\cal
J}_{\tau\tau}^{23}=-{\cal J}_{e\mu}^{23}={\cal J}_{e\tau}^{23}=-{\cal
J}_{\nu}\eta_{\sigma}\;,$ (2.25)

where ${\cal J}_{\nu}=\pm c_{12}s_{12}c^{2}_{13}s_{13}/2$ with “$\pm$”
corresponding to $\delta_{\nu}=\pm\pi/2$, and $\eta_{\rho}\equiv\cos 2\rho$
and $\eta_{\sigma}\equiv\cos 2\sigma$ are defined. As a result, the
$\mu$-$\tau$ reflection symmetry guarantees

$\displaystyle
P\big{(}\nu_{e}\to\overline{\nu}_{e}\big{)}=P\big{(}\overline{\nu}_{e}\to\nu_{e}\big{)}\;,\quad$
$\displaystyle
P\big{(}\nu_{\mu}\to\overline{\nu}_{\tau}\big{)}=P\big{(}\overline{\nu}_{\mu}\to\nu_{\tau}\big{)}\;,$
(2.26)

which are both T-conserving and CP-conserving; and

$\displaystyle
P\big{(}\nu_{e}\to\overline{\nu}_{\mu}\big{)}=P\big{(}\nu_{e}\to\overline{\nu}_{\tau}\big{)}\;,\quad$
$\displaystyle
P\big{(}\nu_{\mu}\to\overline{\nu}_{\mu}\big{)}=P\big{(}\nu_{\tau}\to\overline{\nu}_{\tau}\big{)}\;,$
(2.27)

which contain the CP-violating term. Of course, Eqs. (2.24)—(2.27) do not hold
when the effects of $\mu$-$\tau$ reflection symmetry breaking are taken into
account [89, 90]. If the unitarity of $U$ is slightly broken, the above
results will accordingly be modified [32].

It is worth pointing out that how to experimentally determine or constrain the
Majorana phases of $U$ will become a burning question once the Majorana nature
of massive neutrinos is verified in the CP-conserving $0\nu 2\beta$-decay
experiments someday. Although it is extremely difficult (if not impossible) to
observe neutrino-antineutrino oscillations even in the far future, the above
discussions should conceptually make sense as they can at least give one a
ball-park feeling of some interesting constraints on the Jarlskog-like
invariants of CP violation in such extremely rare processes as a natural
consequence of the $\mu$-$\tau$ reflection symmetry.

###  2.4 A generalized case in the seesaw mechanism

Given the fact that the canonical seesaw mechanism [15, 16, 17, 18, 19, 20] is
currently the most convincing and popular mechanism for interpreting the
origin of tiny neutrino masses, we are going to combine it with the
$\mu$-$\tau$ reflection symmetry in a rather generic approach that has
recently been developed in Ref. [74]. First of all, let us consider the SM
extended by adding three right-handed neutrino fields $N_{\alpha\rm R}$ (for
$\alpha=e,\mu,\tau$) and allowing for lepton number violation. In this case
the gauge- and Lorentz-invariant neutrino mass terms can be written as

$\displaystyle-{\cal L}_{\rm SS}=\overline{\ell_{\rm L}}\hskip
1.42271ptY_{\nu}\widetilde{H}N_{\rm R}+\frac{1}{2}\hskip
1.42271pt\overline{(N_{\rm R})^{c}}\hskip 1.42271ptM_{\rm R}N_{\rm R}+{\rm
h.c.}\;,$ (2.28)

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, $(N_{\rm R})^{c}\equiv{\cal
C}\overline{N_{\rm R}}^{T}$ with $\cal C$ having been described below Eq.
(1.12), $Y_{\nu}$ 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 symmetry breaking, Eq. (2.28) becomes

$\displaystyle-{\cal L}^{\prime}_{\rm SS}=\frac{1}{2}\hskip
1.42271pt\overline{\pmatrix{\nu_{\rm L}&(N_{\rm R})^{c}}}\pmatrix{{\bf
0}&M_{\rm D}\cr M^{T}_{\rm D}&M_{\rm R}\cr}\pmatrix{(\nu_{\rm L})^{c}\cr
N_{\rm R}\cr}+{\rm h.c.}\;,$ (2.29)

where $M_{\rm D}\equiv Y_{\nu}\langle H\rangle$ with $\langle H\rangle$ being
the vacuum expectation value of the Higgs field, and $\bf 0$ denotes the
$3\times 3$ zero matrix. Note that $M_{\rm D}$ is in general neither Hermitian
nor symmetric. The overall $6\times 6$ neutrino mass matrix in Eq. (2.29) can
be diagonalized by the following unitary transformation:

$\displaystyle\pmatrix{U&R\cr S&Q\cr}^{\dagger}\pmatrix{{\bf 0}&M_{\rm D}\cr
M^{T}_{\rm D}&M_{\rm R}\cr}\pmatrix{U&R\cr S&Q\cr}^{*}=\pmatrix{D_{\nu}&{\bf
0}\cr{\bf 0}&D_{N}\cr}\;,$ (2.30)

where $D_{\nu}$ has already been defined in Eq. (2.4), and $D_{N}\equiv{\rm
Diag}\\{M_{1},M_{2},M_{3}\\}$ with $M_{i}$ (for $i=1,2,3$) being the masses of
three heavy Majorana neutrinos. The four $3\times 3$ flavor mixing submatrices
in Eq. (2.30) satisfy the unitarity conditions

$\displaystyle UU^{\dagger}+RR^{\dagger}=SS^{\dagger}+QQ^{\dagger}={\cal
I}\;,$ $\displaystyle
U^{\dagger}U+S^{\dagger}S=R^{\dagger}R+Q^{\dagger}Q={\cal I}\;,$
$\displaystyle US^{\dagger}+RQ^{\dagger}=U^{\dagger}R+S^{\dagger}Q={\bf 0}\;;$
(2.31)

and among them, $U$ and $R$ also satisfy the exact seesaw relation [92, 93]

$\displaystyle UD_{\nu}U^{T}+RD_{N}R^{T}={\bf 0}\;.$ (2.32)

To be more specific, Eq. (2.32) leads us to the exact mass relations between
light (active) and heavy (sterile) neutrinos:

$\displaystyle\sum^{3}_{i=1}\left(m_{i}U_{\alpha i}U_{\beta
i}\right)=-\sum^{3}_{i=1}\left(M_{i}R_{\alpha i}R_{\beta i}\right)\;,$ (2.33)

where $\alpha$ and $\beta$ run over $e$, $\mu$ and $\tau$. In this case
$\nu_{\alpha}$ (for $\alpha=e,\mu,\tau$) can be expressed as a linear
combination of the mass eigenstates of three active neutrinos
$\nu_{i}=\nu^{c}_{i}$ and three sterile neutrinos $N_{i}=N^{c}_{i}$ (for
$i=1,2,3$). The standard weak charged-current interactions of these six
Majorana neutrinos are therefore expressed as

$\displaystyle-{\cal L}^{\prime}_{\rm cc}=\frac{g_{\rm w}}{\sqrt{2}}\hskip
2.84544pt\overline{\pmatrix{e&\mu&\tau}_{\rm L}}\hskip
2.84544pt\gamma^{\mu}\left[U\pmatrix{\nu_{1}\cr\nu_{2}\cr\nu_{3}\cr}_{\rm
L}+R\pmatrix{N_{1}\cr N_{2}\cr N_{3}\cr}_{\rm L}\right]W^{-}_{\mu}+{\rm
h.c.}\;,$ (2.34)

in which the PMNS matrix $U$ is responsible for describing the phenomenology
of active neutrino oscillations, and the matrix $R$ characterizes the
strengths of weak charged-current interactions of those heavy neutrinos. As a
consequence, the $(\mu,\tau)$-associated orthogonality condition of $U$ and
$R$ defines a unitarity hexagon in the complex plane as illustrated by Figure
4, but the sides $R_{\mu i}R^{*}_{\tau i}$ (for $i=1,2,3$) are expected to be
very short and thus can be treated as an effective vertex of the deformed
unitarity triangle formed by the sides $U_{\mu i}U^{*}_{\tau i}$ (for
$i=1,2,3$).

Figure 4: An illustration of the unitarity hexagon defined by the
$(\mu,\tau)$-associated orthogonality condition in the complex plane, where
the effective vertex in sky blue denotes small corrections of the active-
sterile flavor mixing effects to the three-flavor unitarity triangle in the
seesaw framework [74].

Although $U$ is not exactly unitary in the canonical seesaw framework, a
careful analysis of current electroweak precision measurements and neutrino
oscillation data has constrained $U$ to be essentially unitary at least at the
${\cal O}(10^{-2})$ level [28, 29, 30, 31, 32]. So possible non-unitarity
effects hiding in $U$ must be below ${\cal O}(10^{-2})$ and hence cannot be
identified up to the accuracy level of today’s neutrino oscillation
experiments. In other words, the possibility of $U={\cal P}U^{*}\zeta$ as
revealed by the numerical results listed in Eq. (1.5) is still expected even
though $U$ is slightly non-unitary. This observation is the starting point of
view for our subsequent discussions about the $\mu$-$\tau$ reflection symmetry
associated with the seesaw mechanism.

Now let us begin with $U={\cal P}U^{*}\zeta$. Inserting this $\mu$-$\tau$
reflection symmetry relation into the exact seesaw formula in Eq. (2.32) and
taking the complex conjugate for the whole equation, we simply arrive at

$\displaystyle UD_{\nu}U^{T}+{\cal P}R^{*}D_{N}({\cal P}R^{*})^{T}={\bf 0}\;.$
(2.35)

A comparison between Eqs. (2.32) and (2.35) immediately leads us to $R={\cal
P}R^{*}\zeta^{\prime}$ [74], where $\zeta^{\prime}={\rm
Diag}\\{\eta^{\prime}_{1},\eta^{\prime}_{2},\eta^{\prime}_{3}\\}$ with
$\eta^{\prime}_{i}=\pm 1$ (for $i=1,2,3$). This result implies that the novel
prediction $\big{|}R_{\mu i}\big{|}=\big{|}R_{\tau i}\big{|}$ is actually a
natural consequence of $\big{|}U_{\mu i}\big{|}=\big{|}U_{\tau i}\big{|}$ in
the canonical seesaw mechanism. Given a full Euler-like parametrization of the
$3\times 3$ matrices $U$, $R$, $S$ and $Q$ in Eq. (2.30) [92, 93], the
$\mu$-$\tau$ reflection symmetry allows us to obtain some quite strong
constraints on the relevant flavor mixing angles and CP-violating phases
appearing in $U$ and $R$ (see A for a detailed discussion [74]).

After substituting $U={\cal P}U^{*}\zeta$ and $R={\cal P}R^{*}\zeta^{\prime}$
into Eq. (2.31), we find $S={\cal T}S^{*}\zeta$ and $Q={\cal
T}Q^{*}\zeta^{\prime}$ with $\cal T$ being an arbitrary unitary matrix. We
proceed to insert these four relations into Eq. (2.30) and take the complex
conjugate for the whole equation. In this case we are left with

$\displaystyle\pmatrix{U&R\cr S&Q\cr}^{\dagger}\pmatrix{{\bf 0}&{\cal
P}M^{*}_{\rm D}{\cal T}\cr{\cal T}^{T}M^{\dagger}_{\rm D}{\cal P}&{\cal
T}^{T}M^{*}_{\rm R}{\cal T}\cr}\pmatrix{U&R\cr
S&Q\cr}^{*}=\pmatrix{D_{\nu}&{\bf 0}\cr{\bf 0}&D_{N}\cr}\;,$ (2.36)

which is apparently independent of the ambiguities of $\zeta$ and
$\zeta^{\prime}$. Then a direct comparison between Eqs. (2.30) and (2.36)
yields

$\displaystyle M_{\rm D}={\cal P}M^{*}_{\rm D}{\cal T}\;,\quad M_{\rm R}={\cal
T}^{T}M^{*}_{\rm R}{\cal T}\;.$ (2.37)

Note that the arbitrary unitary matrix $\cal T$ can be fixed in a given seesaw
model of massive neutrinos. In this case it will be possible to constrain the
textures of $M_{\rm D}$ and $M_{\rm R}$. Taking ${\cal T}={\cal P}$ for
example, just as considered in Refs. [68, 77], we have the flavor textures of
$M_{\rm D}$ and $M_{\rm R}$ as follows:

$\displaystyle M_{\rm D}=\pmatrix{X_{11}&X_{12}&X^{*}_{12}\cr
X_{21}&X_{22}&X_{23}\cr X^{*}_{21}&X^{*}_{23}&X^{*}_{22}\cr}\;,\quad M_{\rm
R}=\pmatrix{Z_{11}&\hskip 2.27626ptZ_{12}\hskip 2.27626pt&Z^{*}_{12}\cr
Z_{12}&Z_{22}&Z_{23}\cr Z^{*}_{12}&Z_{23}&Z^{*}_{22}\cr}\;,$ (2.38)

where $X_{11}$, $Z_{11}$ and $Z_{23}$ are real. In such a seesaw scenario the
number of real free parameters of $M_{\rm D}$ (or $M_{\rm R}$) is reduced by
half, from eighteen (or twelve) to nine (or six).

After Eq. (2.37) is substituted into Eq. (2.29), the overall neutrino mass
term ${\cal L}^{\prime}_{\rm SS}$ turns out to be

$\displaystyle-{\cal L}^{\prime}_{\rm SS}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\hskip 1.42271pt\overline{\pmatrix{{\cal P}\nu_{\rm
L}&{\cal T}^{*}(N_{\rm R})^{c}}}\pmatrix{{\bf 0}&M^{*}_{\rm D}\cr
M^{\dagger}_{\rm D}&M^{*}_{\rm R}\cr}\pmatrix{{\cal P}(\nu_{\rm
L})^{c}\cr{\cal T}N_{\rm R}\cr}+{\rm h.c.}$ (2.39) $\displaystyle=$
$\displaystyle\frac{1}{2}\hskip 1.42271pt\overline{\pmatrix{{\cal P}(\nu_{\rm
L})^{c}&{\cal T}N_{\rm R}}}\pmatrix{{\bf 0}&M_{\rm D}\cr M^{T}_{\rm D}&M_{\rm
R}\cr}\pmatrix{{\cal P}\nu_{\rm L}\cr{\cal T}^{*}(N_{\rm R})^{c}\cr}+{\rm
h.c.}\;,$

in which the apparent mass term of the second equation comes from the
Hermitian conjugation term of the first equation. Comparing between Eq. (2.29)
and Eq. (2.39), one may easily find that ${\cal L}^{\prime}_{\rm SS}$ keeps
unchanged under the transformations

$\displaystyle\nu_{\rm L}\to{\cal P}(\nu_{\rm L})^{c}\;,\quad N_{\rm
R}\to{\cal T}^{*}(N_{\rm R})^{c}\;.$ (2.40)

It is well known that CP would be a good symmetry in the canonical seesaw
mechanism if ${\cal L}^{\prime}_{\rm SS}$ were invariant under the charge
conjugation and parity transformations $\nu_{\alpha\rm L}(t,{\bf
x})\to\big{[}\nu_{\alpha\rm L}(t,-{\bf x})\big{]}^{c}$ and $N_{\alpha\rm
R}(t,{\bf x})\to\big{[}N_{\alpha\rm R}(t,-{\bf x})\big{]}^{c}$ (for
$\alpha=e,\mu,\tau$) [56]. But the $\mu$-$\tau$ reflection transformation of
the left-handed neutrino fields, together with an arbitrary unitary CP
transformation of the right-handed neutrino fields in Eq. (2.40), makes CP
violation possible even though ${\cal L}^{\prime}_{\rm SS}$ itself keeps
invariant in this case. The issue will certainly be simplified if a specific
form of ${\cal T}$ is assumed, such as ${\cal T}={\cal P}$ as taken above.

Integrating out those heavy degrees of freedom in the canonical seesaw
mechanism, one may get the effective Majorana mass term ${\cal L}_{\nu}$ for
three active neutrinos in Eq. (2.1) with the approximate but well-known seesaw
formula

$\displaystyle M_{\nu}\simeq-M_{\rm D}M^{-1}_{\rm R}M^{T}_{\rm D}\;.$ (2.41)

Substituting Eq. (2.37) into Eq. (2.41), we immediately arrive at
$M_{\nu}={\cal P}M^{*}_{\nu}{\cal P}$. This result is fully compatible with
Eq. (2.15), in which the PMNS matrix $U$ is unitary and satisfies $U={\cal
P}U^{*}\zeta$. As a direct consequence, ${\cal L}_{\nu}$ is invariant under
the $\mu$-$\tau$ reflection transformations $\nu_{e\rm L}\to(\nu_{e\rm
L})^{c}$, $\nu_{\mu\rm L}\to(\nu_{\tau\rm L})^{c}$ and $\nu_{\tau\rm
L}\to(\nu_{\mu\rm L})^{c}$.

###  2.5 An $\rm S_{3}$ extension of the $\mu$-$\tau$ flavor symmetry

It is well known that the real orthogonal $(\mu,\tau)$-associated permutation
matrix $\cal P$ in Eq. (2.8) is just one of the six elements of the non-
Abelian $\rm S_{3}$ group — a permutation group of three objects [94, 95]:

$\displaystyle{\cal S}^{(123)}=\pmatrix{1&0&0\cr 0&1&0\cr 0&0&1}\;,\quad{\cal
S}^{(231)}=\pmatrix{0&1&0\cr 0&0&1\cr 1&0&0}\;,$ $\displaystyle{\cal
S}^{(312)}=\pmatrix{0&0&1\cr 1&0&0\cr 0&1&0}\;,\quad{\cal
S}^{(213)}=\pmatrix{0&1&0\cr 1&0&0\cr 0&0&1}\;,$ $\displaystyle{\cal
S}^{(132)}=\pmatrix{1&0&0\cr 0&0&1\cr 0&1&0}\;,\quad{\cal
S}^{(321)}=\pmatrix{0&0&1\cr 0&1&0\cr 1&0&0}\;.$ (2.42)

Namely, ${\cal S}^{(123)}={\cal I}$ and ${\cal S}^{(132)}={\cal P}$ hold. Note
that ${\cal S}^{(231)}$ and ${\cal S}^{(312)}$ are non-symmetric but can be
expressed as the products of two symmetric $\rm S_{3}$ transformations:

$\displaystyle{\cal S}^{(231)}={\cal S}^{(213)}{\cal S}^{(321)}={\cal
S}^{(321)}{\cal S}^{(132)}={\cal S}^{(132)}{\cal S}^{(213)}\;,$
$\displaystyle{\cal S}^{(312)}={\cal S}^{(321)}{\cal S}^{(213)}={\cal
S}^{(213)}{\cal S}^{(132)}={\cal S}^{(132)}{\cal S}^{(321)}\;.$ (2.43)

Note also that the real orthogonal matrices ${\cal S}^{(213)}$ and ${\cal
S}^{(321)}$ describe the corresponding $e$-$\mu$ and $e$-$\tau$ flavor
permutations, but neither of them is phenomenologically favored to serve as a
simple working flavor symmetry in describing the observed pattern of lepton
flavor mixing. We have two immediate observations as follows.

(1) If the effective neutrino mass matrix $M_{\nu}$ satisfies

$\displaystyle{\cal S}^{(ijk)}M_{\nu}=M_{\nu}{\cal S}^{(ijk)}\;,$ (2.44)

where ${\cal S}^{(ijk)}$ stands for any of the six matrices in Eq. (2.42),
then it will be constrained to have a very particular texture:

$\displaystyle M_{\nu}=a{\cal I}+b{\cal D}\;,$ (2.45)

where the free parameters $a$ and $b$ are in general complex [96, 97], and
$\cal D$ is a rank-one constant matrix of the form

$\displaystyle{\cal D}\equiv{\cal S}^{(132)}+{\cal S}^{(213)}+{\cal
S}^{(321)}=\pmatrix{1&1&1\cr 1&1&1\cr 1&1&1}\;.$ (2.46)

The pattern of $\cal D$ is usually referred to as the flavor democracy texture
[98], which has often been used to define a special flavor basis of the
charged leptons, up-type quarks or down-type quarks by taking account of their
respective strong mass hierarchies [53, 99]. In this case $M_{\nu}$ in Eq.
(2.45) can be diagonalized by either the so-called “democratic” flavor mixing
pattern [98, 100]

$\displaystyle U_{\rm
DEM}=\pmatrix{\frac{1}{\sqrt{2}}&\frac{-1}{\sqrt{2}}&0\cr\vspace{-0.45cm}\cr\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{6}}&\frac{-2}{\sqrt{6}}\cr\vspace{-0.45cm}\cr\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}}\;$
(2.47)

via the real orthogonal transformation

$\displaystyle U_{\rm DEM}M_{\nu}U^{T}_{\rm DEM}=\pmatrix{a&0&0\cr 0&a&0\cr
0&0&a+3b}\;;$ (2.48)

or the so-called “tribimaximal” flavor mixing pattern [101, 102, 103]

$\displaystyle U_{\rm
TBM}=\pmatrix{\frac{2}{\sqrt{6}}&\frac{1}{\sqrt{3}}&0\cr\vspace{-0.45cm}\cr\frac{-1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{-1}{\sqrt{2}}\cr\vspace{-0.45cm}\cr\frac{-1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}}\;$
(2.49)

via the real orthogonal transformation

$\displaystyle U^{T}_{\rm TBM}M_{\nu}U_{\rm TBM}=\pmatrix{a&0&0\cr 0&a+3b&0\cr
0&0&a}\;.$ (2.50)

It is obvious that both $U_{\rm DEM}$ and $U_{\rm TBM}$ are far away from the
pattern of $U$ extracted from current neutrino oscillation data as shown in
Eq. (1.5), and hence proper perturbations to the $\rm S_{3}$-constrained
texture of $M_{\nu}$ in Eq. (2.45) have to be introduced 222Note that there
exists a mass eigenvalue degeneracy in either Eq. (2.48) or Eq. (2.50), and
hence the corresponding flavor mixing pattern cannot be uniquely fixed. In
other words, the $(1,2)$ mixing part of $U_{\rm DEM}$ in Eq. (2.47) or the
$(1,3)$ mixing part of $U_{\rm TBM}$ in Eq. (2.49) involves a degree of
arbitrariness. But such uncertainties can be removed by introducing
appropriate perturbations to $M_{\nu}$ in Eq. (2.45) from a point of view of
model building, as done in Refs. [99, 104, 105, 106] for example.

(2) If the effective neutrino mass matrix $M_{\nu}$ satisfies

$\displaystyle{\cal S}^{(ijk)}M_{\nu}=M^{*}_{\nu}{\cal S}^{(ijk)}\;,$ (2.51)

where ${\cal S}^{(ijk)}$ can be any of the six matrices listed in Eq. (2.42),
it will be constrained to the following specific form:

$\displaystyle M_{\nu}=a{\cal I}+b{\cal S}^{(231)}+b^{*}{\cal
S}^{(312)}=\pmatrix{a&b&b^{*}\cr b^{*}&a&b\cr b&b^{*}&a}\;,$ (2.52)

where $a$ is real and $b$ is in general complex. Now that $M_{\nu}$ is in
general Hermitian, it cannot coincide with the Majorana nature of massive
neutrinos unless $b$ is real. But in this case $M_{\nu}$ becomes real and thus
needs certain complex and symmetric perturbations so as to accommodate CP
violation for the Majorana neutrinos. Of course, a Hermitian texture of
$M_{\nu}$ is suitable for describing the Dirac neutrinos.

Note that the six $\rm S_{3}$ group elements can be categorized into three
conjugacy classes ${\cal C}_{0}=\left\\{{\cal S}^{(123)}\right\\}$, ${\cal
C}_{1}=\left\\{{\cal S}^{(231)},{\cal S}^{(312)}\right\\}$ and ${\cal
C}_{2}=\left\\{{\cal S}^{(213)},{\cal S}^{(132)},{\cal S}^{(321)}\right\\}$.
It is straightforward to show that the invariant subgroup of $\rm S_{3}$ is
the cyclic group of order three

$\displaystyle{\cal Z}_{3}=\left\\{{\cal S}^{(123)},{\cal S}^{(231)},{\cal
S}^{(312)}\right\\}=\left\\{{\cal I},{\cal Z},{\cal Z}^{2}\right\\}\;,$ (2.53)

where the generator of the $\rm Z_{3}$ group has been defined as ${\cal
Z}\equiv{\cal S}^{(231)}$. One may easily verify ${\cal Z}^{3}={\cal I}$ with
the help of Eq. (2.42). Moreover, it is worth pointing out that the $\rm
S_{3}$ group has three $\rm Z_{2}$ subgroups of order two,

$\displaystyle{\cal Z}^{(12)}_{2}=\left\\{{\cal S}^{(123)},{\cal
S}^{(213)}\right\\}\;,$ $\displaystyle{\cal Z}^{(23)}_{2}=\left\\{{\cal
S}^{(123)},{\cal S}^{(132)}\right\\}\;,$ $\displaystyle{\cal
Z}^{(31)}_{2}=\left\\{{\cal S}^{(123)},{\cal S}^{(321)}\right\\}\;,$ (2.54)

which can be identified as the $e$-$\mu$, $\mu$-$\tau$ and $e$-$\tau$
permutation symmetries, respectively. The symmetry breaking chains $\rm
S_{3}\to Z_{3}\to 0$ and $\rm S_{3}\to Z_{2}\to 0$ may help constrain the
flavor textures of charged leptons and massive neutrinos so as to explain
current neutrino oscillation data [107, 108].

One may also apply the $\rm S_{3}$ flavor symmetry to the canonical seesaw
framework in order to constrain the flavor textures of $M_{\rm D}$ and $M_{\rm
R}$ (see, e.g., Ref. [77] for a systematic analysis). In this case it is found
that the most favored $\rm S_{3}$ transformation for left- and right-handed
neutrino fields, which can naturally lead to a phenomenologically favored
pattern of the PMNS matrix $U$, is just the ${\cal S}^{(132)}$ transformation
— or equivalently the $(\mu,\tau)$-associated reflection transformation. As
shown in section 2.3, such an observation is also true even if the seesaw
relation between light and heavy neutrino masses is exact and the unitarity of
$U$ is slightly violated. That is why we are going to focus on the
$\mu$-$\tau$ reflection symmetry and its possible breaking effects for both
active and sterile neutrinos in the subsequent sections.

##  3 Translational $\mu$-$\tau$ reflection symmetry

###  3.1 The translation of a massless neutrino field

It is well known that current experimental data, either from the neutrino
oscillation experiments or from those non-oscillation experiments, do allow
the existence of a massless neutrino species — either $\nu_{1}$ with $m_{1}=0$
or $\nu_{3}$ with $m_{3}=0$. That is why the minimal seesaw model and some
other neutrino mass models, which can naturally predict $m_{1}=0$ or $m_{3}=0$
at least at the tree level, have aroused quite a lot of interest in the past
two decades (see, e.g., Ref. [109] for a recent review). One may try to see
the issue from a different perspective: is there any simple flavor symmetry
that can dictate one of the three active neutrinos to be massless?

The answer is actually affirmative. It was found that the Dirac equation of a
free neutrino would be invariant under a translation of the neutrino field in
the spinor space if the neutrino were a massless Goldstone-like fermion [110,
111]. Taking account of the very fact that there are three neutrino species
and their mass eigenstates (i.e., $\nu_{i}$ for $i=1,2,3$) do not match with
their flavor eigenstates (i.e., $\nu_{\alpha}$ for $\alpha=e,\mu,\tau$),
Friedberg and Lee put forward a novel idea that a Dirac neutrino mass term of
the form $\overline{\nu_{\rm L}}\hskip 1.42271ptM_{\rm D}N_{\rm R}+{\rm h.c.}$
should keep unchanged under the transformations $\nu_{\alpha\rm
L}\to\nu_{\alpha\rm L}+z_{\rm L}$ and $N_{\alpha\rm R}\to N_{\alpha\rm
R}+z_{\rm R}$ in the flavor space, where $z_{\rm L}$ and $z_{\rm R}$ are the
constant spinor fields which anticommute with the corresponding neutrino
fields $\nu_{\alpha\rm L}$ and $N_{\alpha\rm R}$ [112, 113, 114, 115, 116,
117]. Such a translational flavor symmetry, which is also referred to as the
Friedberg-Lee symmetry, can not only assure one of the neutrino masses $m_{i}$
to be vanishing as a consequence of $\det\big{(}M_{\rm D}\big{)}=0$ but also
predict an instructive neutrino mixing matrix which is equivalent to a
modified form of the tribimaximal flavor mixing pattern $U_{\rm TBM}$ in Eq.
(2.49). In particular, the original Friedberg-Lee symmetry can be regarded as
a simple extension of the $\mu$-$\tau$ reflection symmetry [118], and some
generalized scenarios of this kind have been proposed and discussed (see,
e.g., Refs. [119, 120, 121, 122, 123, 124, 125, 126, 127, 128]).

It is worth pointing out that ’t Hooft once considered an analogous
translational displacement $\phi(x)\to\phi(x)+\Phi$ for the Lagrangian ${\cal
L}_{\phi}$ in a renormalizable scalar field theory, where $\Phi$ is spacetime-
independent and commutes with $\phi$ [129]. In this case both the mass of
$\phi$ and its self-coupling parameter have to vanish so as to assure the
invariance of ${\cal L}_{\phi}$, which can be referred to as a Goldstone-type
symmetry [130, 131], under the above translational transformation. So it is
generally expected that the translational symmetry of an effective Lagrangian
may provide an easy way to understand why the mass of a fermion or boson in
this physical system is vanishing or vanishingly small 333If there were no
flavor mixing in the lepton or quark sector, the flavor and mass eigenstates
of a fermion would be identical with each other. In this case the kinetic
energy and mass terms of a fermion field, which involves no self-interaction,
would be rather analogous to those of a scalar field [127]., although its deep
meaning remains a big puzzle.

We find that there is also a close correlation between the zero mass of a
particle and the translational symmetry of its equation of motion [127]. For
example, the massless photon travels at the speed of light in free space and
its electromagnetic field obeys the equation of motion $\square{\bf A}=0$.
This equation is invariant under the translational transformation ${\bf
A}\to{\bf A}+{\bf A}_{0}$, where ${\bf A}_{0}$ stands for a constant vector
field commuting with ${\bf A}$. Analogously, a massless neutrino field
$\nu_{i}$ (for $i=1,2$ or $3$) also travels at the speed of light in free
space and it satisfies the Dirac equation ${\rm
i}\gamma^{\mu}\partial_{\mu}\nu_{i}=0$. The latter is invariant under the
translational transformation

$\displaystyle\nu_{i}\to\nu_{i}+z_{\nu}\;,$ (3.1)

where $z_{\nu}$ is a constant spinor field anticommuting with $\nu_{i}$. One
may argue that such a translation of the massless neutrino field is fully
compatible with Einstein’s principle of constancy of light velocity for all
the inertial reference systems in vacuum, just like the massless photon
itself. But different from the photon, the massless neutrino field $\nu_{1\rm
L}$ (or $\nu_{3\rm L}$) and its two massive counterparts $\nu_{2\rm L}$ and
$\nu_{3\rm L}$ (or $\nu_{1\rm L}$) form three quantum superposition states —
the neutrino flavor eigenstates $\nu_{e\rm L}$, $\nu_{\mu\rm L}$ and
$\nu_{\tau\rm L}$. The mismatch between flavor and mass eigenstates of the
three active neutrinos is described by the $3\times 3$ PMNS matrix $U$, as can
be seen from Eq. (1.11). In this case the translational transformation made
for the massless neutrino field $\nu_{i}$ (i.e., $m_{i}=0$) in Eq. (3.1)
demands that the neutrino flavor eigenstates transform in the following way
[127]:

$\displaystyle\nu_{\alpha\rm L}\to\nu_{\alpha\rm L}+U_{\alpha i}z_{\nu}\;,$
(3.2)

where $U_{\alpha i}$ denote the elements in the $i$th column of $U$. We must
emphasize that the coefficient $U_{\alpha i}$ of $z_{\nu}$ in Eq. (3.2) is a
natural consequence of Eq. (3.1) and lepton flavor mixing instead of a
phenomenological assumption. So the aforementioned Friedberg-Lee
transformation is equivalent to assuming $U_{e2}=U_{\mu 2}=U_{\tau
2}=1/\sqrt{3}$, corresponding to $m_{2}=0$. That is why the Friedberg-Lee
symmetry can give rise to a special neutrino mixing scenario which is very
close to the tribimaximal flavor mixing pattern $U_{\rm TBM}$, but a diagonal
and flavor-blind symmetry breaking term has to be introduced to acquire
$m_{2}\neq 0$ and keep the flavor mixing matrix unchanged [112].

Now let us start from Eq. (3.2) to prove that the invariance of the effective
Majorana neutrino mass term ${\cal L}_{\nu}$ in Eq. (1.12) under this
translational transformation must give rise to $m_{i}=0$. To do so, we rewrite
${\cal L}_{\nu}$ in the following form:

$\displaystyle-{\cal
L}_{\nu}=\frac{1}{2}\sum_{\alpha}\sum_{\beta}\overline{\nu_{\alpha\rm
L}}\hskip 2.84544pt\langle m\rangle_{\alpha\beta}\left(\nu_{\beta\rm
L}\right)^{c}+{\rm h.c.}\;,$ (3.3)

where $\alpha$ and $\beta$ run over the flavor indices $e$, $\mu$ and $\tau$,
and $\langle m\rangle_{\alpha\beta}$ has been given in Eq. (2.5). The
unitarity of $U$ leads us to

$\displaystyle\sum_{\alpha}U^{*}_{\alpha i}\langle
m\rangle_{\alpha\beta}=m_{i}U_{\beta i}\;,$ $\displaystyle\sum_{\beta}\langle
m\rangle_{\alpha\beta}U^{*}_{\beta i}=m_{i}U_{\alpha i}\;,$
$\displaystyle\sum_{\alpha}\sum_{\beta}U^{*}_{\alpha i}\langle
m\rangle_{\alpha\beta}U^{*}_{\beta i}=m_{i}\;,$ (3.4)

which are basically equivalent to one another and all proportional to the
neutrino mass $m_{i}$ (for $i=1$, $2$ or $3$). Under the translational
transformation made in Eq. (3.2), the effective neutrino mass term ${\cal
L}_{\nu}$ turns out to be

$\displaystyle-\widehat{\cal L}_{\nu}=-{\cal
L}_{\nu}+\frac{1}{2}m_{i}\left[\overline{z_{\nu}}\hskip
1.42271ptz^{c}_{\nu}+\sum_{\alpha}U_{\alpha i}\hskip
1.42271pt\overline{\nu_{\alpha\rm
L}}z^{c}_{\nu}+\overline{z_{\nu}}\sum_{\beta}U_{\beta i}\left(\nu_{\beta\rm
L}\right)^{c}\right]\;.$ (3.5)

It is therefore obvious that $\widehat{\cal L}_{\nu}={\cal L}_{\nu}$ will hold
if and only if $m_{i}=0$ holds. Namely, one of the three active neutrinos must
be massless if the effective Majorana neutrino mass term ${\cal L}_{\nu}$
keeps invariant under the discrete shifts of $\nu_{\alpha\rm L}$ made above.
Such a translational transformation helps provide a novel link between the two
sides of one coin — the mass and flavor mixing issues of the Majorana
neutrinos. The point is that the three flavor-dependent coefficients
$U_{\alpha i}$ of $z_{\nu}$ constitute the $i$th column of the PMNS matrix $U$
which corresponds to $m_{i}=0$ [127]. In comparison, most of the popular
global discrete flavor symmetries can predict very specific neutrino mixing
patterns but leave the neutrino mass spectrum unconstrained [36, 37, 132].
Given the fact of $m_{2}>m_{1}$, we are mainly interested in either $m_{1}=0$
(normal mass ordering) or $m_{3}=0$ (inverted mass ordering) for the neutrino
mass spectrum.

If the translational symmetry of a Majorana neutrino mass term is realized at
a superhigh energy scale $\Lambda$, it is in general expected to be broken at
the electroweak scale $\Lambda_{\rm EW}$ due to the RGE-induced quantum
corrections RGEs [133]. If the evolution of neutrino flavor parameters starts
from the heaviest Majorana neutrino mass scale $M_{\rm max}$ to the lightest
Majorana neutrino mass scale $M_{\rm min}$ and then to $\Lambda_{\rm EW}$, it
will in general be possible to generate a finite value of $m_{1}$ (or $m_{3}$)
at $\Lambda_{\rm EW}$ from $m_{1}=0$ (or $m_{3}=0$) at $M_{\rm max}$ via the
one-loop RGEs [134, 135, 136, 137]. In this case the seesaw threshold effects
play a critical role, but the numerical output of $m_{1}$ (or $m_{3}$)
strongly depends on the numerical inputs of those unknown Yukawa coupling
matrix elements. If the evolution of neutrino flavor parameters simply starts
from $M_{\rm min}$, the so-called seesaw scale where the heavy degrees of
freedom have been integrated out and the unique dimension-five Weinberg
operator is responsible for the origin of three active neutrino masses [138],
then a nonzero value of $m_{1}$ (or $m_{3}$) at $\Lambda_{\rm EW}$ cannot be
generated from $m_{1}=0$ (or $m_{3}=0$) at $M_{\rm min}$ unless the two-loop
RGE running behaviors are taken into account [139, 140, 141, 142]. In the
latter case the resulting mass of the lightest Majorana neutrino $\nu_{1}$ or
$\nu_{3}$ is typically of ${\cal O}(10^{-13})~{}{\rm eV}$ or smaller — such a
vanishingly small neutrino mass and the associated Majorana CP-violating phase
do not cause any seeable effects in neutrino physics.

###  3.2 A $\rm TM_{1}$ flavor mixing scenario with $m_{1}=0$

Now that the normal neutrino mass ordering is somewhat favored over the
inverted one by current neutrino oscillation data [7, 33, 143], let us focus
on the translational flavor symmetry of ${\cal L}_{\nu}$ in Eq. (3.3) by
assuming [144, 145]

$\displaystyle U_{e1}=-2U_{\mu 1}=-2U_{\tau 1}=\frac{2}{\sqrt{6}}\;,$ (3.6)

which corresponds to $m_{1}=0$. We find that the standard parametrization of
$U$ given in Eq. (1.3) is not very suitable for this case, because its first
column involves all the three flavor mixing angles. Instead, it is more
suitable to adopt the Kobayashi-Maskawa parametrization of $U$ [12] to
accommodate the condition in Eq. (3.6), simply because its first column is
much simpler:

$\displaystyle U=P_{l}\pmatrix{c_{1}&s_{1}c_{3}&s_{1}\hat{s}^{*}_{3}\cr-
s_{1}c_{2}&c_{1}c_{2}c_{3}+s_{2}\hat{s}_{3}&c_{1}c_{2}\hat{s}^{*}_{3}-s_{2}c_{3}\cr-
s_{1}s_{2}&c_{1}s_{2}c_{3}-c_{2}\hat{s}_{3}&c_{1}s_{2}\hat{s}^{*}_{3}+c_{2}c_{3}\cr}P_{\nu}\;,$
(3.7)

where $c_{i}\equiv\cos\theta_{i}$, $s_{i}\equiv\sin\theta_{i}$,
$\hat{s}_{3}\equiv s_{3}e^{{\rm i}\phi}$ and $P_{\nu}\equiv{\rm
diag}\\{1,e^{{\rm i}\sigma},1\\}$ with $\sigma$ being the single Majorana CP-
violating phase. A combination of Eqs. (3.6) and (3.7) leads us to
$\theta_{1}=\arcsin\left(1/\sqrt{3}\right)\simeq 35.3^{\circ}$ and
$\theta_{2}=45^{\circ}$. Then the PMNS matrix $U$ is simplified to

$\displaystyle U$ $\displaystyle=$ $\displaystyle P_{l}\hskip 1.42271ptU_{\rm
TBM}\pmatrix{1&0&0\cr 0&c_{3}&\hat{s}^{*}_{3}\cr
0&-\hat{s}_{3}&c_{3}\cr}P_{\nu}$ (3.8) $\displaystyle=$ $\displaystyle
P_{l}\pmatrix{\frac{2}{\sqrt{6}}&\frac{1}{\sqrt{3}}c_{3}&\frac{1}{\sqrt{3}}\hat{s}^{*}_{3}\cr\vspace{-0.45cm}\cr\frac{-1}{\sqrt{6}}&\frac{1}{\sqrt{3}}c_{3}+\frac{1}{\sqrt{2}}\hat{s}_{3}&\frac{-1}{\sqrt{2}}c_{3}+\frac{1}{\sqrt{3}}\hat{s}^{*}_{3}\cr\vspace{-0.45cm}\cr\frac{-1}{\sqrt{6}}&\frac{1}{\sqrt{3}}c_{3}-\frac{1}{\sqrt{2}}\hat{s}_{3}&\frac{1}{\sqrt{2}}c_{3}+\frac{1}{\sqrt{3}}\hat{s}^{*}_{3}}P_{\nu}\;,$

the so-called $\rm TM_{1}$ flavor mixing pattern as its “first” column is the
same as that of the tribimaximal flavor mixing pattern $U_{\rm TBM}$ [146,
147, 148]. The $3\times 3$ Majorana neutrino mass matrix $M_{\nu}$ can
therefore be reconstructed from $M_{\nu}=UD_{\nu}U^{T}$ with the help of the
$\rm TM_{1}$ form of $U$ in Eq. (3.8). To be explicit,

$\displaystyle M_{\nu}=P_{l}\left[a{\cal D}+\pmatrix{0&c&-c\cr
c&b+2c&-b\cr-c&-b&b-2c\cr}\right]P^{T}_{l}\;,$ (3.9)

where the flavor democracy texture $\cal D$ has been defined in Eq. (2.46),
and

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(\overline{m}_{2}c^{2}_{3}+m_{3}\hat{s}^{*2}_{3}\right)\;,$
$\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\overline{m}_{2}\hat{s}^{2}_{3}+m_{3}c^{2}_{3}\right)\;,$
$\displaystyle c$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\left(\overline{m}_{2}\hat{s}_{3}-m_{3}\hat{s}^{*}_{3}\right)c_{3}\;$
(3.10)

with $\overline{m}_{2}\equiv m_{2}e^{2{\rm i}\sigma}$. As a consequence, the
two nonzero neutrino masses and the Majorana CP-violating phase are given by

$\displaystyle m_{2}=\frac{1}{2}\left|3a+2b\hskip 0.85355pte^{-2{\rm
i}\phi}+\frac{\sqrt{6}\hskip 1.42271ptc}{c_{3}s_{3}}e^{-{\rm
i}\phi}\right|\;,$ $\displaystyle m_{3}=\frac{1}{2}\left|3a\hskip
0.85355pte^{+2{\rm i}\phi}+2b-\frac{\sqrt{6}\hskip
1.42271ptc}{c_{3}s_{3}}e^{+{\rm i}\phi}\right|\;,$ $\displaystyle\tan
2\sigma=\frac{c_{3}s_{3}{\rm Im}\left(3a+2b\hskip 0.85355pte^{-2{\rm
i}\phi}\right)+\sqrt{6}\hskip 1.42271pt{\rm Im}\left(c\hskip 0.85355pte^{-{\rm
i}\phi}\right)}{c_{3}s_{3}{\rm Re}\left(3a+2b\hskip 1.42271pte^{-2{\rm
i}\phi}\right)+\sqrt{6}\hskip 1.42271pt{\rm Re}\left(c\hskip 1.42271pte^{-{\rm
i}\phi}\right)}\;.$ (3.11)

One can see that the simple texture of $M_{\nu}$ depends on a very simple
choice of $U_{\alpha 1}$ and is strongly suggestive of certain simple discrete
flavor symmetries which can be used for a specific model building exercise
[147].

Note that the $\rm TM_{1}$ flavor mixing pattern obtained in Eq. (3.8) only
possesses a partial $\mu$-$\tau$ reflection symmetry characterized by
$\big{|}U_{\mu 1}\big{|}=\big{|}U_{\tau 1}\big{|}$ [144], if
$\phi\neq\pm\pi/2$ holds. But when $\phi=\pm\pi/2$ is taken, the $\rm TM_{1}$
scenario will assure $\big{|}U_{\mu i}\big{|}=\big{|}U_{\tau i}\big{|}$ (for
$i=1,2,3$) to hold and thus fully respect the $\mu$-$\tau$ reflection symmetry
[146]. Comparing the standard parametrization of $U$ in Eq. (1.3) with the
$\rm TM_{1}$ pattern in Eq. (3.8), we arrive at

$\displaystyle s^{2}_{13}=\frac{1}{3}s^{2}_{3}\;,\quad
s^{2}_{12}=\frac{1}{3}\left(1-t^{2}_{13}\right)\;,\quad
s^{2}_{23}=\frac{1}{2}-\sqrt{6}\hskip 1.42271pts_{12}t_{13}\cos\phi\;$ (3.12)

for the three flavor mixing angles, and

$\displaystyle\sin\delta_{\nu}=\frac{\sin\phi}{\sqrt{\left(1-4t^{2}_{13}\right)^{2}+8t^{2}_{13}\left(1-2t^{2}_{13}\right)\sin^{2}\phi}}\;$
(3.13)

for the Dirac CP-violating phase, where $t_{13}\equiv\tan\theta_{13}$ has been
defined [145]. It is obvious that $\phi=\pm\pi/2$ leads us to
$\theta_{23}=\pi/4$ and $\delta_{\nu}=\pm\pi/2$ as in the $\mu$-$\tau$
reflection symmetry limit. Furthermore, a combination of Eqs. (3.12) and
(3.13) predicts an instructive correlation between $\delta_{\nu}$ and
$\theta_{23}$:

$\displaystyle\sin^{2}\delta_{\nu}=\frac{8s^{2}_{13}\cos
2\theta_{13}-c^{4}_{13}\cos^{2}2\theta_{23}}{8s^{2}_{13}\left(\cos
2\theta_{13}\sin^{2}2\theta_{23}+s^{2}_{13}\cos^{2}2\theta_{23}\right)}\;.$
(3.14)

This interesting relation, together with
$s^{2}_{12}=\left(1-t^{2}_{13}\right)$ in Eq. (3.12), can be used to test the
$\rm TM_{1}$ flavor mixing scenario in the future precision neutrino
experiments. Inputting the best-fit and $3\sigma$ interval of $\theta_{13}$
[33, 143] for the normal neutrino mass ordering which is consistent with
$m_{1}=0$, we illustrate the numerical correlation between $\theta_{23}$ and
$\delta_{\nu}$ in Figure 5, where the grey bands of $\theta_{23}$ and
$\delta_{\nu}$ cover their respective $3\sigma$ intervals. One can see that
the best-fit point of $\theta_{23}$ and $\delta_{\nu}$ obtained today (the red
star) is not far away from the $\mu$-$\tau$ reflection symmetry point (the
blue star), and it is not far away from the correlation curve and band (the
red curve and orange band) either.

Figure 5: A numerical illustration of the correlation between $\theta_{23}$
and $\delta_{\nu}$ as predicted by the $\rm TM_{1}$ flavor mixing pattern with
$m_{1}=0$, where the best-fit (red curve) and $3\sigma$ interval (orange band)
of $\theta_{13}$ [33, 143] have been input. The grey bands of $\theta_{23}$
and $\delta_{\nu}$ cover their respective $3\sigma$ intervals, and the red
star “$\star$” denotes their best-fit values. The blue star “$\ast$” stands
for the $\mu$-$\tau$ reflection symmetry point.

It is certainly the assumption made in Eq. (3.6) for the invariance of ${\cal
L}_{\nu}$ under the translational transformation in Eq. (3.2) that allows us
to simultaneously obtain $m_{1}=0$ and the $\rm TM_{1}$ flavor mixing pattern.
To avoid such an empirical assumption, a possible way out is simply to combine
the translational flavor symmetry under discussion and the $\mu$-$\tau$
reflection symmetry for the effective Majorana neutrino mass term. Some
interesting attempts have been made along this line of thought [75, 123, 124,
126, 128, 149]. Let us proceed to go into details about this idea.

###  3.3 The translational $\mu$-$\tau$ reflection symmetry

Given the effective Majorana neutrino mass term ${\cal L}_{\nu}$ in Eq. (1.12)
or Eq. (2.1), one may consider the translational $\mu$-$\tau$ reflection
transformation

$\displaystyle\nu_{\rm L}\to{\cal P}\left[(\nu_{\rm
L})^{c}+n^{*}_{i}z^{c}_{\nu}\right]\;,$ (3.15)

and its charge-conjugated counterpart $(\nu_{\rm L})^{c}\to{\cal
P}\left(\nu_{\rm L}+n_{i}z_{\nu}\right)$, where $\cal P$ has been given in Eq.
(2.8), and $n_{i}\equiv(U_{ei},U_{\mu i},U_{\tau i})^{T}$ is a column vector
of the $3\times 3$ PMNS matrix $U$ corresponding to $\nu_{i}$ (for $i=1,2,3$).
Under these transformations ${\cal L}_{\nu}$ becomes

$\displaystyle-\widehat{\cal L}_{\nu}=\frac{1}{2}\Big{[}\overline{\nu_{\rm
L}}\left({\cal P}M^{*}_{\nu}{\cal P}\right)(\nu_{\rm
L})^{c}+\overline{\nu_{\rm L}}\left({\cal P}M^{*}_{\nu}{\cal
P}\right)n^{*}_{i}z^{c}_{\nu}+\overline{z_{\nu}}\hskip
1.42271ptn^{\dagger}_{i}\left({\cal P}M^{*}_{\nu}{\cal P}\right)(\nu_{\rm
L})^{c}$ $\displaystyle\hskip 40.40285pt+\hskip
1.42271pt\overline{z_{\nu}}\hskip 1.42271ptn^{\dagger}_{i}\left({\cal
P}M^{*}_{\nu}{\cal P}\right)n^{*}_{i}z^{c}_{\nu}\Big{]}+{\rm h.c.}\;.$ (3.16)

The conditions for $\widehat{\cal L}_{\nu}={\cal L}_{\nu}$ turn out to be
$M_{\nu}={\cal P}M^{*}_{\nu}{\cal P}$ obtained in Eq. (2.15) and

$\displaystyle M_{\nu}n^{*}_{i}={\bf 0}\;,\quad n^{\dagger}_{i}M_{\nu}={\bf
0}^{T}\;,\quad n^{\dagger}_{i}M_{\nu}n^{*}_{i}=0\;,$ (3.17)

where $\bf 0$ denotes the zero column vector. Note that the three dependent
conditions in Eq. (3.17) are exactly equivalent to setting those three
relations in Eq. (3.4) to be zero, leading to $m_{i}=0$. Moreover, the texture
of $M_{\nu}$ takes the $\mu$-$\tau$ reflection symmetry form shown in Eq.
(1.17). So the translational $\mu$-$\tau$ reflection symmetry of ${\cal
L}_{\nu}$ can predict not only $m_{1}=0$ or $m_{3}=0$ for the neutrino mass
spectrum but also $\theta_{23}=\pi/4$, $\delta_{\nu}=\pm\pi/2$ and $\sigma=0$
or $\pi/2$ for the PMNS flavor mixing parameters. The experimental testability
of such a simple flavor symmetry is therefore impressive.

The above observation reminds us to consider a combination of the
translational $\mu$-$\tau$ reflection symmetry with the canonical seesaw
mechanism at a superhigh energy scale, such that we can naturally arrive at a
minimal seesaw scenario constrained by the $\mu$-$\tau$ reflection symmetry.
To do so, let us rewrite ${\cal L}^{\prime}_{\rm SS}$ in Eq. (2.29) as

$\displaystyle-{\cal L}^{\prime}_{\rm SS}=\overline{\nu_{\rm L}}\hskip
1.42271ptM_{\rm D}N_{\rm R}+\frac{1}{2}\hskip 1.42271pt\overline{(N_{\rm
R})^{c}}\hskip 1.42271ptM_{\rm R}N_{\rm R}+{\rm h.c.}\;,$ (3.18)

and require it to be invariant under the transformations 444One may also
consider the possibility of allowing $\nu_{\rm L}$ to have an independent
translational $\mu$-$\tau$ reflection symmetry instead of just the
$\mu$-$\tau$ reflection symmetry. In this case we find that the resulting
constraint on the texture of $M_{\rm D}$ are so stringent that all the
elements of $M_{\rm D}$ have to vanish, making the seesaw mechanism completely
collapse.

$\displaystyle\nu_{\rm L}\to{\cal P}(\nu_{\rm L})^{c}\;,\quad N_{\rm
R}\to{\cal P}\big{[}(N_{\rm R})^{c}+n^{\prime*}_{i}z^{c}_{N}\big{]}\;,$ (3.19)

together with $(\nu_{\rm L})^{c}\to{\cal P}\nu_{\rm L}$ and $(N_{\rm
R})^{c}\to{\cal P}\left(N_{\rm R}+n^{\prime}_{i}z_{N}\right)$, where
$n^{\prime}_{i}\equiv(U^{\prime}_{ei},U^{\prime}_{\mu i},U^{\prime}_{\tau
i})^{T}$ is defined to be a column vector of the $3\times 3$ unitary flavor
mixing matrix $U^{\prime}$ used to diagonalize $M_{\rm R}$ in correspondence
to the heavy neutrino mass eigenstate $N_{i}$ with the mass $M_{i}$ (for
$i=1,2,3$) in the canonical seesaw mechanism, and $z_{N}$ is another constant
spinor field similar to $z_{\nu}$ in Eq. (3.2). Then the invariance of ${\cal
L}^{\prime}_{\rm SS}$ with respect to the above transformations leads us to
[75]

$\displaystyle M_{\rm D}={\cal P}M^{*}_{\rm D}{\cal P}\;,\quad M_{\rm R}={\cal
P}M^{*}_{\rm R}{\cal P}\;;$ $\displaystyle M_{\rm D}\hskip
1.42271ptn^{\prime}_{i}={\bf 0}\;,\quad M_{\rm R}\hskip
1.42271ptn^{\prime}_{i}={\bf 0}\;,\quad n^{\prime T}_{i}M_{\rm R}={\bf
0}^{T}\;,\quad n^{\prime T}_{i}M_{\rm R}\hskip 1.42271ptn^{\prime}_{i}=0\;,$
(3.20)

The textures of $M_{\rm D}$ and $M_{\rm R}$ turn out to have the familiar
$\mu$-$\tau$ reflection symmetry form

$\displaystyle M_{\rm D}=\pmatrix{a&b&b^{*}\cr e&c&d\cr
e^{*}&d^{*}&c^{*}}\;,\quad M_{\rm R}=\pmatrix{A&B&B^{*}\cr B&C&D\cr
B^{*}&D&C^{*}}\;,$ (3.21)

where the matrix elements $a$, $A$ and $D$ are real, and one of the mass
eigenvalues of $M_{\rm R}$ is vanishing (i.e., $M_{i}=0$ corresponding to
$n^{\prime}_{i}$). Without loss of generality, we simply take $M_{1}=0$ for
the sake of illustration [121]. Now that there is no way to measure
$U^{\prime}$ in any realistic experiments, let us just choose a flavor basis
defined by $n^{\prime}_{1}\equiv(1,0,0)^{T}$, which is consistent with
$M_{1}=0$. In this case Eq. (3.21) becomes

$\displaystyle M_{\rm D}=\pmatrix{0&b&b^{*}\cr 0&c&d\cr 0&\hskip
1.70709ptd^{*}\hskip 1.70709pt&c^{*}}\;,\quad M_{\rm R}=\pmatrix{0&0&0\cr
0&C&D\cr\hskip 1.13791pt0\hskip 1.13791pt&\hskip 2.84544ptD\hskip
2.84544pt&C^{*}}\;.$ (3.22)

As argued by Jarlskog [121], the flavor eigenstate of the massless sterile
neutrino $N_{1}$ should completely decouple from other particles (both the SM
particles and those massive sterile neutrinos) and have only gravitational
interactions due to its kinetic energy term [121]. In other words, such a
massless sterile neutrino state should have no impact on the effective number
of neutrino species, the Big Bang nucleosynthesis and some other physical
processes in the early Universe, but it might more or less affect the
formation of the large scale structure of the Universe. Here let us simply
focus on the point that the translational $\mu$-$\tau$ reflection symmetry of
${\cal L}^{\prime}_{\rm SS}$ can be treated as a phenomenologically viable way
to reduce the degrees of freedom associated with the canonical seesaw
mechanism and constrain its flavor textures. In this way ${\cal
L}^{\prime}_{\rm SS}$ in Eq. (3.18) is just simplified to

$\displaystyle-{\cal L}^{\prime}_{\rm MSS}=\overline{\nu_{\rm L}}\hskip
1.42271pt\widehat{M}_{\rm D}N^{\prime}_{\rm
R}+\frac{1}{2}\overline{(N^{\prime}_{\rm R})^{c}}\hskip
1.42271pt\widehat{M}_{\rm R}N^{\prime}_{\rm R}+{\rm h.c.}\;,$ (3.23)

where $N^{\prime}_{\rm R}=(N_{\mu\rm R},N_{\tau\rm R})^{T}$ is the reduced
column vector, $\widehat{M}_{\rm D}$ and $\widehat{M}_{\rm R}$ are the
$3\times 2$ and $2\times 2$ mass matrices of the forms

$\displaystyle\widehat{M}_{\rm D}=\pmatrix{b&b^{*}\cr c&d\cr
d^{*}&c^{*}}\;,\quad\widehat{M}_{\rm R}=\pmatrix{C&D\cr D&C^{*}}\;,$ (3.24)

with $D$ being real. It is obvious that $\widehat{M}_{\rm D}$ only contains
six real free parameters, and $\widehat{M}_{\rm R}$ consists of only three
real free parameters. In this case the approximate seesaw formula for the
effective Majorana mass matrix of three active neutrinos is

$\displaystyle\widehat{M}_{\nu}\simeq-\widehat{M}_{\rm D}\widehat{M}^{-1}_{\rm
R}\widehat{M}^{T}_{\rm D}=\pmatrix{\langle m\rangle_{ee}&\langle
m\rangle_{e\mu}&\langle m\rangle^{*}_{e\mu}\cr\langle m\rangle_{e\mu}&\langle
m\rangle_{\mu\mu}&\langle m\rangle_{\mu\tau}\cr\langle
m\rangle^{*}_{e\mu}&\langle m\rangle_{\mu\tau}&\langle
m\rangle^{*}_{\mu\mu}\cr}\;,$ (3.25)

where

$\displaystyle\langle m\rangle_{ee}$ $\displaystyle=$
$\displaystyle\frac{1}{\big{|}C\big{|}^{2}-D^{2}}\Big{[}2\big{|}b\big{|}^{2}D-2{\rm
Re}\left(b^{2}C^{*}\right)\Big{]}\;,$ $\displaystyle\langle m\rangle_{e\mu}$
$\displaystyle=$
$\displaystyle\frac{1}{\big{|}C\big{|}^{2}-D^{2}}\Big{[}b\left(dD-
cC^{*}\right)+b^{*}\left(cD-dC\right)\Big{]}\;,$ $\displaystyle\langle
m\rangle_{\mu\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{\big{|}C\big{|}^{2}-D^{2}}\Big{[}2cdD-c^{2}C^{*}-d^{2}C\Big{]}\;,$
$\displaystyle\langle m\rangle_{\mu\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{\big{|}C\big{|}^{2}-D^{2}}\Big{[}\left(\big{|}c\big{|}^{2}+\big{|}d\big{|}^{2}\right)D-2{\rm
Re}\left(cd^{*}C^{*}\right)\Big{]}\;.$ (3.26)

So the texture of $\widehat{M}_{\nu}$ also exhibits the $\mu$-$\tau$
reflection symmetry and contains only five real free parameters. In
particular, the rank of $\widehat{M}_{\nu}$ is two instead of three, implying
that one of its three mass eigenvalues must be vanishing (i.e., $m_{1}=0$ or
$m_{3}=0$). Note that the most salient feature of the minimal seesaw mechanism
is just the existence of a massless neutrino species, and this observation is
actually independent of the approximate seesaw formula [150]. A more generic
analysis of the minimal seesaw model constrained by the $\mu$-$\tau$
reflection symmetry will be made in section 6.4.

##  4 Rotational $\mu$-$\tau$ reflection symmetry

###  4.1 A rotational $\mu$-$\tau$ reflection transformation

Given the effective Majorana neutrino mass term ${\cal L}_{\nu}$ in Eq. (2.1),
let us now conjecture that it keeps invariant under a proper real and
orthogonal transformation $\nu_{\rm L}\to{\cal R}\nu_{\rm L}$ with ${\cal
R}={\cal R}^{T}={\cal R}^{\dagger}$. In this case it is easy to verify that
the condition $M_{\nu}={\cal R}M_{\nu}{\cal R}$ must hold for the symmetric
mass matrix $M_{\nu}$. We proceed to assume that ${\cal L}_{\nu}$ is also
invariant under the $\mu$-$\tau$ reflection transformation $\nu_{\rm
L}\to{\cal P}(\nu_{\rm L})^{c}$ as discussed in section 2.2, which requires
$M_{\nu}$ to satisfy $M_{\nu}={\cal P}M^{*}_{\nu}{\cal P}$. If ${\cal P}$ and
${\cal R}$ commute with each other such that

$\displaystyle M_{\nu}={\cal R}{\cal P}M^{*}_{\nu}{\cal P}{\cal R}={\cal
P}{\cal R}M^{*}_{\nu}{\cal R}{\cal P}$ (4.1)

holds together with $M_{\nu}={\cal P}M^{*}_{\nu}{\cal P}$ and $M_{\nu}={\cal
R}M_{\nu}{\cal R}$, then ${\cal L}_{\nu}$ will be invariant under the separate
but mutually exchangeable transformations

$\displaystyle\nu_{\rm L}\to{\cal R}\nu_{\rm L}\;,\quad\nu_{\rm L}\to{\cal
P}(\nu_{\rm L})^{c}\;.$ (4.2)

In this case ${\cal L}_{\nu}$ is of course invariant under the joint
transformation $\nu_{\rm L}\to{\cal R}{\cal P}(\nu_{\rm L})^{c}$ or $\nu_{\rm
L}\to{\cal P}{\cal R}(\nu_{\rm L})^{c}$. Such a simple flavor symmetry of
${\cal L}_{\nu}$ can therefore be referred to as the rotational $\mu$-$\tau$
reflection symmetry, in the sense that an extra $\rm Z_{2}$ symmetry is
imposed on the $\mu$-$\tau$ reflection symmetry. Note that the $\mu$-$\tau$
reflection symmetry itself can essentially be regarded as a kind of
generalized CP symmetry [64, 65], and hence the rotational $\mu$-$\tau$
reflection symmetry is actually a combination of the $\rm Z_{2}$ flavor
symmetry and the generalized CP symmetry which satisfy the consistency
conditions (see, e.g., Refs. [151, 152, 153, 154, 155]). The resulting texture
of $M_{\nu}$ should thus be more strongly constrained, unless ${\cal R}$ has a
rather trivial form (e.g., ${\cal R}={\cal I}$ or ${\cal P}$).

Up to an overall sign ambiguity, the real orthogonal matrix ${\cal R}$ that
represents another residual $\rm Z_{2}$ symmetry and satisfies the
commutativity with ${\cal P}$ (i.e., ${\cal R}{\cal P}={\cal P}{\cal R}$) can
in general be expressed as

$\displaystyle{\cal R}=\pmatrix{\cos 2\theta&\frac{1}{\sqrt{2}}\sin
2\theta&\frac{1}{\sqrt{2}}\sin 2\theta\cr\frac{1}{\sqrt{2}}\sin
2\theta&-\cos^{2}\theta&\sin^{2}\theta\cr\frac{1}{\sqrt{2}}\sin
2\theta&\sin^{2}\theta&-\cos^{2}\theta}\;,$ (4.3)

where $\theta$ is real. In Refs. [152, 153, 154] another parametrization of
${\cal R}$, which is equivalent to taking $\tan\theta=k/\sqrt{2}$ with $k$
being real, has been proposed. A special but extremely interesting case is
$\tan\theta=1/\sqrt{2}$ or $k=1$, as first pointed out in Ref. [151].
Substituting ${\cal P}$ in Eq. (2.8) and ${\cal R}$ in Eq. (4.3) into
$M_{\nu}={\cal P}M^{*}_{\nu}{\cal P}$ and $M_{\nu}={\cal R}M_{\nu}{\cal R}$,
we obtain the constraint conditions for the elements $\langle
m\rangle_{\alpha\beta}$ of $M_{\nu}$:

$\displaystyle\langle m\rangle_{ee}=\langle m\rangle^{*}_{ee}\;,\quad$
$\displaystyle\langle m\rangle_{\mu\mu}=\langle m\rangle^{*}_{\tau\tau}\;,$
$\displaystyle\langle m\rangle_{e\mu}=\langle m\rangle^{*}_{e\tau}\;,\quad$
$\displaystyle\langle m\rangle_{\mu\tau}=\langle m\rangle^{*}_{\mu\tau}\;,$
(4.4)

which are equivalent to the usual $\mu$-$\tau$ reflection symmetry conditions;
and

$\displaystyle{\rm Im}\langle m\rangle_{\mu\mu}=\sqrt{2}\hskip
0.85355pt\cot\theta\hskip 0.85355pt{\rm Im}\langle m\rangle_{e\tau}\;,$
$\displaystyle\langle m\rangle_{ee}={\rm Re}\langle m\rangle_{\mu\mu}+\langle
m\rangle_{\mu\tau}+2\sqrt{2}\hskip 0.85355pt\cot 2\theta\hskip 0.85355pt{\rm
Re}\langle m\rangle_{e\mu}\;,$ (4.5)

which arise from $M_{\nu}={\cal R}M_{\nu}{\cal R}$. It is clear that the
rotational $\mu$-$\tau$ reflection symmetry may provide two additional
constraints on $M_{\nu}$ as compared with the normal $\mu$-$\tau$ reflection
symmetry, if the free angle $\theta$ is fixed by an explicit flavor symmetry
group. Taking the notations $\langle m\rangle_{e\mu}=a+{\rm i}a^{\prime}$,
${\rm Re}\langle m\rangle_{\mu\mu}=b$ and $\langle m\rangle_{\mu\tau}=c$, let
us parametrize the flavor texture of $M_{\nu}$ as follows:

$\displaystyle M_{\nu}=\pmatrix{2\sqrt{2}\hskip 1.42271pta\cot
2\theta+b+c&a+{\rm i}a^{\prime}&a-{\rm i}a^{\prime}\cr a+{\rm
i}a^{\prime}&b-{\rm i}\sqrt{2}\hskip 1.42271pta^{\prime}\cot\theta&c\cr a-{\rm
i}a^{\prime}&c&b+{\rm i}\sqrt{2}\hskip 1.42271pta^{\prime}\cot\theta}\;$ (4.6)

with $a$, $a^{\prime}$, $b$ and $c$ being all real.

In the basis where the flavor eigenstates of three charged leptons are
identical with their mass eigenstates (i.e., $M_{l}=D_{l}$), the effective
Majorana neutrino mass matrix $M_{\nu}$ can be diagonalized by the PMNS matrix
$U$ via the transformation $U^{\dagger}M_{\nu}U^{*}=D_{\nu}$. After the
rotational $\mu$-$\tau$ reflection symmetry is taken into account, we have not
only

$\displaystyle UD_{\nu}U^{T}=\left({\cal P}U^{*}\right)D_{\nu}\left({\cal
P}U^{*}\right)^{T}\;,\quad UD_{\nu}U^{T}=\left({\cal
R}U\right)D_{\nu}\left({\cal R}U\right)^{T}\;,$ (4.7)

but also the joint constraint relationship

$\displaystyle UD_{\nu}U^{T}=\left({\cal P}{\cal
R}U^{*}\right)D_{\nu}\left({\cal P}{\cal R}U^{*}\right)^{T}\;.$ (4.8)

As a result, we obtain

$\displaystyle U={\cal P}U^{*}\zeta_{\cal P}\;,\quad U={\cal R}U\zeta_{\cal
R}\;,\quad U={\cal P}{\cal R}U^{*}\zeta\;,$ (4.9)

where $\zeta_{\cal P}={\rm
Diag}\\{\eta^{\prime}_{1},\eta^{\prime}_{2},\eta^{\prime}_{3}\\}$,
$\zeta_{\cal R}={\rm
Diag}\\{\eta^{\prime\prime}_{1},\eta^{\prime\prime}_{2},\eta^{\prime\prime}_{3}\\}$
and $\zeta={\rm Diag}\\{\eta_{1},\eta_{2},\eta_{3}\\}$ with the elements
$\eta^{\prime}_{i}=\pm 1$, $\eta^{\prime\prime}_{i}=\pm 1$ and $\eta_{i}=\pm
1$ (for $i=1,2,3$). So the pattern of $U$ is not only constrained by the
${\cal P}$ transformation but also restricted by the ${\cal R}$
transformation. Nevertheless, the concrete form of ${\cal R}$ should be
determined by the residual symmetry of an underlying flavor symmetry group in
a given neutrino mass model.

###  4.2 A constrained $\rm TM_{1}$ flavor mixing scenario

There have been some interesting model-building exercises based either on the
rotational $\mu$-$\tau$ reflection symmetry or on its analogs [35, 36, 37, 38,
67, 151, 152, 153, 154, 155], and among them a particularly successful example
is the derivation of a constrained $\rm TM_{1}$ lepton flavor mixing scenario
— a phenomenologically favored pattern of $U$ which can be expressed as a
product of the tribimaximal flavor mixing matrix $U_{\rm TBM}$ in Eq. (2.49)
and a special rotation matrix in the $(2,3)$ complex plane like that shown in
Eq. (3.8) [146, 147, 148]. Instead of repeating a specific flavor symmetry
model of this kind, here let us start from the flavor texture of $M_{\nu}$
obtained in section 4.1 and explore its consequence on flavor mixing and CP
violation. For this purpose we first make a transformation of $M_{\nu}$ in Eq.
(4.6) as follows:

$\displaystyle\left(P^{\prime}_{l}U_{\rm
TBM}\right)^{\dagger}M_{\nu}\left(P^{\prime}_{l}U_{\rm
TBM}\right)^{*}=M^{\prime}_{\nu}\;,$ (4.10)

where $P^{\prime}_{l}={\rm Diag}\\{1,-1,-1\\}$, and the matrix elements
$\langle m\rangle^{\prime}_{\alpha\beta}$ of $M^{\prime}_{\nu}$ are given by

$\displaystyle\langle m\rangle^{\prime}_{ee}$ $\displaystyle=$
$\displaystyle\frac{4}{3}\left(1+\sqrt{2}\hskip 0.85355pt\cot
2\theta\right)a+b+c\;,$ $\displaystyle\langle m\rangle^{\prime}_{e\mu}$
$\displaystyle=$ $\displaystyle\frac{\sqrt{2}}{3}\left(2\sqrt{2}\hskip
0.85355pt\cot 2\theta-1\right)a\;,$ $\displaystyle\langle
m\rangle^{\prime}_{e\tau}$ $\displaystyle=$ $\displaystyle{\rm
i}\frac{1}{\sqrt{3}}\left(2-\sqrt{2}\hskip
0.85355pt\cot\theta\right)a^{\prime}\;,$ $\displaystyle\langle
m\rangle^{\prime}_{\mu\mu}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\left(\sqrt{2}\hskip 0.85355pt\cot
2\theta-2\right)a+b+c\;,$ $\displaystyle\langle m\rangle^{\prime}_{\mu\tau}$
$\displaystyle=$ $\displaystyle{\rm i}\frac{2}{\sqrt{6}}\left(1+\sqrt{2}\hskip
0.85355pt\cot\theta\right)a^{\prime}\;,$ $\displaystyle\langle
m\rangle^{\prime}_{\tau\tau}$ $\displaystyle=$ $\displaystyle b-c\;.$ (4.11)

It is obvious that $\langle m\rangle^{\prime}_{e\mu}=\langle
m\rangle^{\prime}_{e\tau}=0$, or equivalently $\cot\theta=\sqrt{2}$ and $\cot
2\theta=1/\left(2\sqrt{2}\right)$, is the unique condition for the PMNS matrix
$U$ to have the $\rm TM_{1}$ flavor mixing pattern. In this case the real
orthogonal matrix ${\cal R}$ in Eq. (4.3) is fully determined as [151]

$\displaystyle{\cal R}=\frac{1}{3}\pmatrix{1&2&2\cr 2&-2&1\cr 2&1&-2}\;,$
(4.12)

and thus the effective Majorana neutrino mass matrix $M_{\nu}$ in Eq. (4.6)
and its reduced version $M^{\prime}_{\nu}$ in Eq. (4.10) are of the more
specific textures [67]

$\displaystyle M_{\nu}=\pmatrix{a+b+c&\hskip 5.40608pta+{\rm
i}a^{\prime}\hskip 5.40608pt&a-{\rm i}a^{\prime}\cr a+{\rm i}a^{\prime}&b-{\rm
i}2a^{\prime}&c\cr a-{\rm i}a^{\prime}&c&b+{\rm i}2a^{\prime}}\;,$
$\displaystyle M^{\prime}_{\nu}=\pmatrix{2a+b+c&0&0\cr 0&b+c-a&{\rm
i}\sqrt{6}\hskip 1.42271pta^{\prime}\cr 0&{\rm i}\sqrt{6}\hskip
1.42271pta^{\prime}&b-c}\;.$ (4.13)

The latter can be diagonalized via the transformation
$O^{\dagger}_{23}M^{\prime}_{\nu}O^{*}_{23}={\rm
Diag}\\{m^{\prime}_{1},m^{\prime}_{2},m^{\prime}_{3}\\}$, where the unitary
matrix $O_{23}$ is given by

$\displaystyle O_{23}=\pmatrix{1&0&0\cr 0&c_{*}&-{\rm i}s_{*}\cr 0&-{\rm
i}s_{*}&c_{*}}\;$ (4.14)

with $c_{*}\equiv\cos\theta_{*}$, $s_{*}\equiv\sin\theta_{*}$ and $\tan
2\theta_{*}=2\sqrt{6}\hskip 1.42271pta^{\prime}/\left(a-2b\right)$, and the
eigenvalues $m^{\prime}_{i}$ are

$\displaystyle m^{\prime}_{1}=2a+b+c\;,$ $\displaystyle
m^{\prime}_{2}=\pm\frac{1}{2}\sqrt{\left(a-2b\right)^{2}+24a^{\prime
2}}-\frac{1}{2}a+c\;,$ $\displaystyle
m^{\prime}_{3}=\pm\frac{1}{2}\sqrt{\left(a-2b\right)^{2}+24a^{\prime
2}}+\frac{1}{2}a-c\;.$ (4.15)

Note that the physical neutrino masses are $m_{i}=\left|m^{\prime}_{i}\right|$
(for $i=1,2,3$). Then one may easily figure out the PMNS matrix $U$ from
$U^{\dagger}M_{\nu}U^{*}=D_{\nu}={\rm Diag}\\{m_{1},m_{2},m_{3}\\}$ [146]:

$\displaystyle U=P^{\prime}_{l}U_{\rm
TBM}O_{23}=P^{\prime}_{l}\pmatrix{\frac{2}{\sqrt{6}}&\frac{1}{\sqrt{3}}c_{*}&\frac{\rm-i}{\sqrt{3}}s_{*}\cr\vspace{-0.45cm}\cr\frac{-1}{\sqrt{6}}&\frac{1}{\sqrt{3}}c_{*}+\frac{\rm
i}{\sqrt{2}}s_{*}&\frac{-1}{\sqrt{2}}c_{*}-\frac{\rm
i}{\sqrt{3}}s_{*}\cr\vspace{-0.45cm}\cr\frac{-1}{\sqrt{6}}&\frac{1}{\sqrt{3}}c_{*}-\frac{\rm
i}{\sqrt{2}}s_{*}&\frac{1}{\sqrt{2}}c_{*}-\frac{\rm
i}{\sqrt{3}}s_{*}}P^{\prime}_{\nu}\;,$ (4.16)

where $P^{\prime}_{\nu}$ contains the trivial Majorana CP phases arising from
possible negative signs of $m^{\prime}_{i}$. Comparing this constrained $\rm
TM_{1}$ flavor mixing pattern with the more general one in Eq. (3.8), we see
that the rotational $\mu$-$\tau$ reflection symmetry can further enhance the
predictive power of the $\mu$-$\tau$ reflection symmetry by fixing the phase
parameter of $U$. It is easy to verify $U={\cal P}{\cal R}U^{*}\zeta$ with
$\zeta={\rm Diag}\\{1,-1,1\\}$, as originally expected.

It is worth mentioning that the specific rotational $\mu$-$\tau$ reflection
symmetry discussed above has been referred to as the “trimaximal $\mu$-$\tau$
reflection symmetry” in Ref. [67], where a detailed analysis of its
phenomenological implications and a careful study of the soft symmetry
breaking effects induced by radiative corrections in the canonical seesaw
framework have been made. One may certainly consider some other possible forms
of ${\cal R}$ to obtain different but viable patterns of the PMNS matrix $U$
(see, e.g., Ref. [156]), especially when the future precision neutrino
oscillation data become available.

###  4.3 An extension to active-sterile flavor mixing

A combination of the rotational $\mu$-$\tau$ reflection symmetry with the
canonical seesaw mechanism is expected to help constrain the flavor structures
of both active and sterile neutrinos. Following the discussions in section
2.4, let us simply assume that the overall neutrino mass term ${\cal
L}^{\prime}_{\rm SS}$ in Eq. (2.29) is invariant not only under the
transformations

$\displaystyle\nu_{\rm L}\to{\cal P}(\nu_{\rm L})^{c}\;,\quad N_{\rm
R}\to{\cal T}^{*}(N_{\rm R})^{c}\;,$ (4.17)

but also under the transformations

$\displaystyle\nu_{\rm L}\to{\cal R}(\nu_{\rm L})^{c}\;,\quad N_{\rm
R}\to{\cal T}^{*}(N_{\rm R})^{c}\;,$ (4.18)

where ${\cal P}$ and ${\cal R}$ are mutually exchangeable and their
expressions are given in Eqs. (2.8) and (4.3) respectively, and $\cal T$ is an
arbitrary unitary matrix. Then we find that the invariance of ${\cal
L}^{\prime}_{\rm SS}$ under the transformations made in Eqs. (4.17) and (4.18)
requires

$\displaystyle\pmatrix{{\cal P}&{\bf 0}\cr{\bf 0}&{\cal
T}^{\dagger}\cr}\pmatrix{{\bf 0}&M_{\rm D}\cr M^{T}_{\rm D}&M_{\rm
R}\cr}\pmatrix{{\cal R}&0\cr{\bf 0}&{\cal T}^{*}\cr}=\pmatrix{{\bf
0}&M^{*}_{\rm D}\cr M^{\dagger}_{\rm D}&M^{*}_{\rm R}\cr}\;,$ (4.19)

where the symmetry of $M_{\rm R}$ has been used. As a result, we immediately
arrive at

$\displaystyle M_{\rm D}={\cal P}M^{*}_{\rm D}{\cal T}\;,\quad M_{\rm D}={\cal
R}M^{*}_{\rm D}{\cal T}\;,\quad M_{\rm R}={\cal T}^{T}M^{*}_{\rm R}{\cal
T}\;.$ (4.20)

Substituting these relations into the approximate seesaw formula for the
active Majorana neutrino mass matrix $M_{\nu}$ in Eq. (2.41), for example, we
have

$\displaystyle M_{\nu}$ $\displaystyle\simeq$ $\displaystyle-{\cal P}{\cal
R}M^{*}_{\rm D}{\cal T}\left({\cal T}^{T}M^{*}_{\rm R}{\cal
T}\right)^{-1}{\cal T}^{T}M^{\dagger}_{\rm D}{\cal R}{\cal P}$ (4.21)
$\displaystyle=$ $\displaystyle{\cal P}{\cal R}\left(-M_{\rm D}M^{-1}_{\rm
R}M^{T}_{\rm D}\right)^{*}{\cal P}{\cal R}$ $\displaystyle\simeq$
$\displaystyle{\cal P}{\cal R}M^{*}_{\nu}{\cal P}{\cal R}\;.$

So $M_{\nu}$ respects the rotational $\mu$-$\tau$ reflection symmetry. As
discussed in section 2.4, the textures of $M_{\rm D}$ and $M_{\rm R}$ can be
explicitly constrained by the rotational $\mu$-$\tau$ reflection symmetry if
both the patterns of $\cal R$ and $\cal T$ are fixed by an underlying flavor
symmetry group. As for the arbitrariness of $\cal T$, one may simply assume
${\cal T}={\cal I}$, ${\cal P}$, ${\cal R}$ or ${\cal P}{\cal R}$ as a
phenomenological example. Section 4.2 has given a typical pattern of $\cal R$
to produce a constrained $\rm TM_{1}$ flavor mixing pattern of $U$. An
extension of that interesting example to the seesaw mechanism is certainly
straightforward.

Now let us insert the rotational $\mu$-$\tau$ reflection symmetry constraints
on $U$ obtained in Eq. (4.9) into the exact seesaw formula in Eq. (2.32) and
take the complex conjugate for the whole equation. Then we are left with

$\displaystyle UD_{\nu}U^{T}+{\cal P}R^{*}D_{N}({\cal P}R^{*})^{T}={\bf 0}\;,$
$\displaystyle UD_{\nu}U^{T}+{\cal R}R^{*}D_{N}({\cal R}R^{*})^{T}={\bf 0}\;,$
$\displaystyle UD_{\nu}U^{T}+{\cal P}{\cal R}R^{*}D_{N}({\cal P}{\cal
R}R^{*})^{T}={\bf 0}\;.$ (4.22)

A comparison between Eqs. (2.32) and (4.22) can therefore lead us to

$\displaystyle R={\cal P}R^{*}\zeta_{\cal P}\;,\quad R={\cal R}R\zeta_{\cal
R}\;,\quad R={\cal P}{\cal R}R^{*}\zeta\;,$ (4.23)

where $\zeta_{\cal P}$, $\zeta_{\cal R}$ and $\zeta$ have been given below Eq.
(4.9). Such simple but instructive results mean that the active-sterile flavor
mixing matrix $R$ is constrained by the same rotational $\mu$-$\tau$
reflection symmetry as the PMNS matrix $U$. Note that both $U$ and $R$ appear
in the standard weak charged-current interactions of massive Majorana
neutrinos as described by Eq. (2.34), and they are correlated with each other
via the unitarity condition in Eq. (2.31) and the exact seesaw relation in Eq.
(2.32). So it is actually a natural result that both of them are constrained
by the same flavor symmetry.

In A, a full Euler-like parametrization of the active-sterile flavor mixing
matrix $R$ has been presented. It is therefore straightforward to constrain
the relevant flavor mixing angles and CP-violating phases of $U$ and $R$ in a
model-independent way after a specific pattern of ${\cal R}$ is assumed.

##  5 Soft $\mu$-$\tau$ reflection symmetry breaking

###  5.1 General remarks on $\mu$-$\tau$ symmetry breaking

In the $\mu$-$\tau$ reflection symmetry limit the unitary PMNS matrix $U$
satisfies $U={\cal P}U^{*}\zeta$, and therefore its flavor mixing angle
$\theta_{23}$ and three CP-violating phases $\delta_{\nu}$, $\rho$ and
$\sigma$ are well constrained, as can be seen from Eq. (2.19). To make sure of
how far away such constraints are from the corresponding experimental results,
let us quote the best-fit values of six neutrino oscillation parameters
extracted from a three-flavor global analysis of the latest experimental data
[33, 143], together with their 1$\sigma$ and 3$\sigma$ ranges, as shown in
Table 2. It is then trivial to see

$\displaystyle\Delta\theta_{23}\equiv\theta_{23}-45^{\circ}=\left\\{\begin{array}[]{l}-2.9^{+1.1^{\circ}}_{-0.9^{\circ}}~{}({\rm
NMO})\\\ \vspace{-0.4cm}\\\ +4.0^{+0.9^{\circ}}_{-1.3^{\circ}}~{}({\rm
IMO})\end{array}\right.\;$ (5.4)
$\displaystyle\Delta\delta_{\nu}\equiv\delta_{\nu}-270^{\circ}=\left\\{\begin{array}[]{l}-40^{+36^{\circ}}_{-25^{\circ}}~{}({\rm
NMO})\\\ \vspace{-0.4cm}\\\ +8^{+22^{\circ}}_{-30^{\circ}}~{}({\rm
IMO})\end{array}\right.\;$ (5.8)

at the $\pm 1\sigma$ level, where “NMO” (or “IMO”) denotes the normal (or
inverted) neutrino mass ordering, and the possibility of
$\delta_{\nu}=90^{\circ}$ in the $\mu$-$\tau$ reflection symmetry limit has
tentatively been discarded as it seems less compatible with current neutrino
oscillation data. We find that $|\Delta\theta_{23}|/\theta_{23}\lesssim 10\%$
holds up to the $\pm 1\sigma$ confidence level for the global-fit result of
$\theta_{23}$ as given in Table 2, and the ratio
$|\Delta\delta_{\nu}|/\delta_{\nu}$ involves much larger uncertainties which
are attributed to the large experimental uncertainties of $\delta_{\nu}$. This
observation implies that it is rather safe to treat $\Delta\theta_{23}$ as a
small perturbation to the $\mu$-$\tau$ reflection symmetry value
$\theta_{23}=45^{\circ}$, but whether $\Delta\delta_{\nu}$ can be treated as a
small perturbation to the $\mu$-$\tau$ reflection symmetry value
$\delta_{\nu}=270^{\circ}$ depends on which neutrino mass ordering is assumed
and what value of $\delta_{\nu}$ is adopted. For instance,
$\Delta\delta_{\nu}/\delta_{\nu}\lesssim 10\%$ is expected to hold in the IMO
case at the $\pm 1\sigma$ confidence level for the present global-fit result
of $\delta_{\nu}$, but the situation will be worse in the NMO case as
reflected by Eq. (5.8).

Table 2: The best-fit values and the 1$\sigma$ and 3$\sigma$ ranges of six
neutrino oscillation parameters in the standard parametrization of $U$,
obtained from a three-flavor global analysis of the latest experimental data
[33, 143].

  * Parameter | Best fit $\pm 1\sigma$ | 3$\sigma$ range
---|---|---
Normal mass ordering $(m_{1}<m_{2}<m_{3}$)
$\Delta m^{2}_{21}/10^{-5}~{}{\rm eV}^{2}$ | $7.42^{+0.21}_{-0.20}$ | $6.82\longrightarrow 8.04$
$\Delta m^{2}_{31}/10^{-3}~{}{\rm eV}^{2}$ | $+2.510^{+0.027}_{-0.027}$ | $+2.430\longrightarrow+2.593$
$\theta_{12}/^{\circ}$ | $33.45^{+0.77}_{-0.75}$ | $31.27\longrightarrow 35.87$
$\theta_{13}/^{\circ}$ | $8.62^{+0.12}_{-0.12}$ | $8.25\longrightarrow 8.98$
$\theta_{23}/^{\circ}$ | $42.1^{+1.1}_{-0.9}$ | $39.7\longrightarrow 50.9$
$\delta_{\nu}/^{\circ}$ | $230^{+36}_{-25}$ | $144\longrightarrow 350$
Inverted mass ordering $(m_{3}<m_{1}<m_{2}$)
$\Delta m^{2}_{21}/10^{-5}~{}{\rm eV}^{2}$ | $7.42^{+0.21}_{-0.20}$ | $6.82\longrightarrow 8.04$
$\Delta m^{2}_{32}/10^{-3}~{}{\rm eV}^{2}$ | $-2.490^{+0.026}_{-0.028}$ | $-2.574\longrightarrow-2.410$
$\theta_{12}/^{\circ}$ | $33.45^{+0.78}_{-0.75}$ | $31.27\longrightarrow 35.87$
$\theta_{13}/^{\circ}$ | $8.61^{+0.14}_{-0.12}$ | $8.24\longrightarrow 9.02$
$\theta_{23}/^{\circ}$ | $49.0^{+0.9}_{-1.3}$ | $39.8\longrightarrow 51.6$
$\delta_{\nu}/^{\circ}$ | $278^{+22}_{-30}$ | $194\longrightarrow 345$

On the other hand, there has been no convincing experimental information on
the Majorana nature of massive neutrinos, although a lot of efforts have been
made in searching for the $0\nu 2\beta$ decays and some other lepton-number-
violating processes [39]. So the two Majorana CP-violating phases $\rho$ and
$\sigma$ are completely unconstrained on the experimental side. Now that we
have no idea about what values of $\rho$ and $\sigma$ can be at low energies,
it certainly makes no sense for us to discuss to what extent they may deviate
from their corresponding $\mu$-$\tau$ reflection symmetry values
$\rho=0^{\circ}$ or $90^{\circ}$ and $\sigma=0^{\circ}$ or $90^{\circ}$ in an
explicit way. The key point is that an explicit symmetry breaking scenario is
usually guided by the available experimental data or a kind of
phenomenological judgement [53]. In view of the insurmountable uncertainties
associated with the three phase parameters of Majorana neutrinos (especially,
$\rho$ and $\sigma$), we conclude that it is very hard to describe how the
$\mu$-$\tau$ reflection symmetry can be explicitly broken for the whole PMNS
flavor mixing matrix $U$ at the present time.

But one may introduce the following three rephasing invariants to describe the
breaking of $\mu$-$\tau$ reflection symmetry for lepton flavor mixing and CP
violation no matter whether it is spontaneously or explicitly realized [157]: