# Phenomenology of Lepton Masses and Mixing with Discrete Flavor Symmetries

Garv Chauhan  P. S. Bhupal Dev  Ievgen Dubovyk  Bartosz Dziewit  Wojciech
Flieger  Krzysztof Grzanka  Janusz Gluza<EMAIL_ADDRESS>Biswajit
Karmakar  Szymon Zięba  Center for Neutrino Physics, Department of Physics,
Virginia Tech, Blacksburg, USA  Department of Physics and McDonnell Center for
the Space Sciences, Washington University, St. Louis, USA  Institute of
Physics, University of Silesia, Katowice, Poland  Max-Planck-Institut für
Physik, Werner-Heisenberg-Institut, München, Germany

###### Abstract

The observed pattern of fermion masses and mixing is an outstanding puzzle in
particle physics, generally known as the flavor problem. Over the years,
guided by precision neutrino oscillation data, discrete flavor symmetries have
often been used to explain the neutrino mixing parameters, which look very
different from the quark sector. In this review, we discuss the application of
non-Abelian finite groups to the theory of neutrino masses and mixing in the
light of current and future neutrino oscillation data. We start with an
overview of the neutrino mixing parameters, comparing different global fit
results and limits on normal and inverted neutrino mass ordering schemes.
Then, we discuss a general framework for implementing discrete family
symmetries to explain neutrino masses and mixing. We discuss CP violation
effects, giving an update of CP predictions for trimaximal models with nonzero
reactor mixing angle and models with partial $\mu-\tau$ reflection symmetry,
and constraining models with neutrino mass sum rules. The connection between
texture zeroes and discrete symmetries is also discussed. We summarize viable
higher-order groups, which can explain the observed pattern of lepton mixing
where the non-zero $\theta_{13}$ plays an important role. We also review the
prospects of embedding finite discrete symmetries in the Grand Unified
Theories and with extended Higgs fields. Models based on modular symmetry are
also briefly discussed. A major part of the review is dedicated to the
phenomenology of flavor symmetries and possible signatures in the current and
future experiments at the intensity, energy, and cosmic frontiers. In this
context, we discuss flavor symmetry implications for neutrinoless double beta
decay, collider signals, leptogenesis, dark matter, as well as gravitational
waves.

###### keywords:

Discrete Symmetries , Flavor mixing , CP Violation , Neutrino Oscillation ,
Phenomenology

††journal: Progress in Particle and Nuclear Physics

###### Contents

  1. 1 Introduction
  2. 2 Flavor Symmetry and Lepton Masses and Mixing: Theory
    1. 2.1 General Framework
    2. 2.2 Flavor Symmetry, Nonzero $\theta_{13}$ and Nonzero $\delta_{\rm CP}$
    3. 2.3 Flavor Symmetry and Neutrino Mass Models
    4. 2.4 Flavor and Generalized CP Symmetries
    5. 2.5 Higher Order Discrete Groups
    6. 2.6 Flavor Symmetry and Grand Unified Theory
    7. 2.7 Flavor Symmetry and the Higgs Sector
    8. 2.8 Modular Symmetry
  3. 3 Flavor Symmetry at Intensity Frontier
    1. 3.1 Neutrino Oscillation Experiments
    2. 3.2 Neutrinoless Double Beta Decay
    3. 3.3 Lepton Flavor and Universality Violation
  4. 4 Flavor Symmetry at Energy Frontier
    1. 4.1 Example Group : $\Delta(6n^{2})$
    2. 4.2 Decay Lengths and Branching Ratios of RHNs
    3. 4.3 Lepton Flavor Violation at Colliders
    4. 4.4 Correlation between Collider Signals and Leptogenesis
    5. 4.5 Collider Signals in Other Flavor Models
    6. 4.6 Higgs to Diphoton Decay
  5. 5 Flavor Symmetry and Cosmic Frontier
    1. 5.1 Flavor Symmetry and Dark Matter
    2. 5.2 Flavor Symmetry and Baryon Asymmetry of the Universe
    3. 5.3 Flavor Symmetry and Gravitational Waves
  6. 6 Summary and Outlook
  7. A $A_{4}$ symmetry

## 1 Introduction

Over the past few decades, we have seen spectacular progress in understanding
neutrinos. The neutrino quantum oscillation phenomenon established that at
least two neutrino quantum states are massive, although the masses are tiny,
at the sub-electronvolt level, $m_{\nu}\lesssim{\cal{O}}(0.1)$ eV [1, 2]. A
tremendous effort led to this result, as earlier experimental studies of
neutrino physics faced the challenge of low event statistics for a scarce set
of observables. It started around half a century ago with the pioneering
Homestake experiment [3] and the so-called solar neutrino problem [4] and
culminated with the 2015 Nobel Prize in Physics for the discovery of neutrino
oscillations by the Super-Kamiokande and SNO collaborations [5, 6, 7], which
showed that neutrinos have mass.

The simplest neutrino mass theory is based on the three-neutrino ($3\nu$)
paradigm with the assumption that flavor states
($\nu_{e},\nu_{\mu},\nu_{\tau}$) are mixed with massive states
($\nu_{1},\nu_{2},\nu_{3}$) with definite masses ($m_{1},m_{2},m_{3}$), at
least two of which are non-zero. The standard parametrization of the
Pontecorvo–Maki–Nakagawa–Sakata (PMNS) unitary mixing matrix111In an
equivalent parametrization [8, 9, 10], the lepton mixing matrix can be written
in a ‘symmetrical’ form where all three CP violating phases are ‘physical’.
reads [11, 12, 13]

$\displaystyle U$ $\displaystyle=U(\theta_{23})U(\theta_{13},\delta_{\rm
CP})U(\theta_{12})U_{M}(\alpha_{1},\alpha_{2})$
$\displaystyle=\begin{pmatrix}1&0&0\\\ 0&c_{23}&{s_{23}}\\\
0&-s_{23}&{c_{23}}\end{pmatrix}\begin{pmatrix}c_{13}&0&s_{13}e^{-i\delta_{\text{\rm
CP}}}\\\ 0&1&0\\\ -s_{13}e^{i\delta_{\text{\rm
CP}}}&0&c_{13}\end{pmatrix}\begin{pmatrix}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}e^{i\alpha_{1}}&0&0\\\
0&e^{i\alpha_{2}}&0\\\ 0&0&1\end{pmatrix},$ (1.1)

where $M$ stands for Majorana neutrinos, $c_{ij}\equiv\cos(\theta_{ij})$,
$s_{ij}\equiv\sin(\theta_{ij})$, and the Euler rotation angles $\theta_{ij}$
can be taken without loss of generality from the first quadrant,
$\theta_{ij}\in[0,\pi/2]$, and the Dirac CP phase $\delta_{\text{\rm CP}}$ and
Majorana phases $\alpha_{1},\alpha_{2}$ are in the range $[0,2\pi]$ [14]. This
choice of parameter regions is independent of matter effects [15].

In the $3\nu$ paradigm, there are two non-equivalent mass orderings: normal
mass ordering (NO) with $m_{1}<m_{2}<m_{3}$ and inverted mass ordering (IO)
with $m_{3}<m_{1}<m_{2}$. The neutrino masses can be further classified into
normal hierarchical mass spectrum (NH) with $m_{1}\ll m_{2}<m_{3}$, inverted
hierarchical mass spectrum (IH) with $m_{3}\ll m_{1}<m_{2}$ and quasi-
degenerate mass spectrum (QD) with $m_{1}\simeq m_{2}\simeq m_{3}$ [14].
Within QD, quasi-degenerate NH (QDNH) mass spectrum with $m_{1}\lesssim
m_{2}\lesssim m_{3}$ and quasi-degenerate IH (QDIH) mass spectrum with
$m_{3}\lesssim m_{1}\lesssim m_{2}$ [16] can be distinguished. NO/IO notation
is sometimes used interchangeably with NH/IH in the literature. However, NO/IO
notation is more general since both NH and QDNH are NO and IH and QDIH are IO.
For more discussion, see section 14.7 in the PDG review [14]. In what follows,
we will use the NO/IO notation. Neutrino masses can be expressed by the
smallest of the three neutrino masses ($m_{0}$) and experimentally determined
mass-squared differences ($\Delta m_{21}^{2},\Delta m_{31}^{2},\Delta
m_{32}^{2}$)

$\begin{array}[]{ccc}\mbox{\text@underline{Normal mass ordering
(NO)}}&\mbox{\text@underline{Inverted mass ordering (IO)}}&\\\
\begin{array}[]{l}m_{1}=m_{0},\\\ m_{2}=\sqrt{m_{0}^{2}+\Delta m_{21}^{2}},\\\
m_{3}=\sqrt{m_{0}^{2}+\Delta m_{31}^{2}},\\\
\end{array}&\begin{array}[]{l}m_{1}=\sqrt{m_{0}^{2}-\Delta m_{21}^{2}-\Delta
m_{32}^{2}},\\\ m_{2}=\sqrt{m_{0}^{2}-\Delta m_{32}^{2}},\\\
m_{3}=m_{0},\end{array}\end{array}$ (1.2)

where $\Delta m_{ij}^{2}\equiv m_{i}^{2}-m_{j}^{2}$. The observation of matter
effects in the Sun constrains the product $\Delta
m^{2}_{21}\cos{2\theta_{12}}$ to be positive [17]. By
definition222Considerations of the mass schemes with some negative $\Delta
m_{ij}^{2}$ are not necessary from the point of view of neutrino oscillation
parametrization (in vacuum and matter) and may cause double counting only, see
Ref. [15]., we choose $\Delta m^{2}_{21}>0$ and $\theta_{12}$ in the first
octant, although in presence of non-standard matter effects, a high-octant
$\theta_{12}$ is possible [18].

Depending on their origin, the neutrino oscillation parameters $\theta_{12}$,
$\theta_{23}$ and $\theta_{13}$ are often called solar, atmospheric and
reactor angles, respectively, while $\Delta m_{21}^{2}$ and $\Delta
m_{31}^{2}$ are called solar and atmospheric mass-squared differences,
respectively. This naming has a historical background and is related to the
neutrino source associated with measured oscillation parameters in the initial
neutrino experiments.

Tab. 1.1 summarizes recent global fits for the neutrino oscillation parameters
[19, 20, 21, 22, 23], which are used in the present analysis for NO/IO. The
data shows a hierarchy between the mass splittings, $\Delta
m_{21}^{2}\ll\Delta m_{31}^{2}\simeq\Delta m_{32}^{2}$ and preference of NO at
an overall level of $\sim 2.5\sigma$ corresponding to $\Delta\chi^{2}\sim
6.4-6.5$. Lower octant atmospheric angle $\theta_{23}$ best-fit is favored for
NO in NuFit 5.2 [19, 20] and Capozzi et al [23] results. These analysis
include the most recent Superkamiokande (SK) data (with
$\sin^{2}\theta_{23}<0.5$ preference) [24, 25]. The $\theta_{23}$ best-fit
octant preference discussed above is illustrated in Fig. 1.1 together with
other neutrino parameters as given in Tab. 1.1.

Parameter | Ordering |  NuFit 5.2 [19, 20] |  de Salas et al. (2021) [21, 22] |  Capozzi et al. (2021) [23]
---|---|---|---|---
|  |  bf$\pm 1\sigma$ |  $3\sigma$ range |  bf$\pm 1\sigma$ |  $3\sigma$ range |  bf$\pm 1\sigma$ |  $3\sigma$ range
$\sin^{2}\theta_{12}/10^{-1}$ | NO, IO | $3.03^{+0.12}_{-0.12}$ | $2.70-3.41$ | $3.18^{+0.16}_{-0.16}$ | $2.71-3.69$ | $3.03^{+0.13}_{-0.13}$ | $2.63-3.45$
$\sin^{2}\theta_{23}/10^{-1}$ | NO | $4.51^{+0.19}_{-0.16}$ | $4.08-6.03$ | $5.74^{+0.14}_{-0.14}$ | $4.34-6.10$ | $4.55^{+0.18}_{-0.15}$ | $4.16-5.99$
| IO | $5.69^{+0.16}_{-0.21}$ | $4.12-6.13$ | $5.78^{+0.10}_{-0.17}$ | $4.33-6.08$ | $5.69^{+0.12}_{-0.21}$ | $4.17-6.06$
$\sin^{2}\theta_{13}/10^{-2}$ | NO | $2.225^{+0.056}_{-0.059}$ | $2.052-2.398$ | $2.200^{+0.069}_{-0.062}$ | $2.000-2.405$ | $2.23^{+0.07}_{-0.06}$ | $2.04-2.44$
| IO | $2.223^{+0.058}_{-0.058}$ | $2.048-2.416$ | $2.225^{+0.064}_{-0.070}$ | $2.018-2.424$ | $2.23^{+0.06}_{-0.06}$ | $2.03-2.45$
$\delta_{\rm CP}/\pi$ | NO | $1.29^{+0.20}_{-0.14}$ | $0.80-1.94$ | $1.08^{+0.13}_{-0.12}$ | $0.71-1.99$ | $1.24^{+0.18}_{-0.13}$ | $0.77-1.97$
| IO | $1.53^{+0.12}_{-0.16}$ | $1.08-1.91$ | $1.58^{+0.15}_{-0.16}$ | $1.11-1.96$ | $1.52^{+0.15}_{-0.11}$ | $1.07-1.90$
$\Delta m_{21}^{2}/10^{-5}\mbox{eV}^{2}$ | NO, IO | $7.41^{+0.21}_{-0.20}$ | $6.82-8.03$ | $7.50^{+0.22}_{-0.20}$ | $6.94-8.14$ | $7.36^{+0.16}_{-0.15}$ | $6.93-7.93$
$\left|{\Delta m_{\rm atm}^{2}}\right|/10^{-3}\mbox{eV}^{2}$ | NO | $2.507^{+0.026}_{-0.027}$ | $2.427-2.590$ | $2.55^{+0.02}_{-0.03}$ | $2.47-2.63$ | $2.485^{+0.023}_{-0.031}$ | $2.401-2.565$
| IO | $2.486^{+0.028}_{-0.025}$ | $2.406-2.570$ | $2.45^{+0.02}_{-0.03}$ | $2.37-2.53$ | $2.455^{+0.030}_{-0.025}$ | $2.376-2.541$
$\Delta\chi^{2}$ | IO - NO | $6.4$ | $6.4$ | $6.5$
Table 1.1:  Selection of neutrino oscillation data from NuFit 5.2 with SK
atmospheric data [19, 20], de Salas et al. neutrino global fit [21, 22] and
Capozzi et al. global fit [23]. Notation used by different groups is unified;
namely, the atmospheric neutrino mass-squared difference parameter $\Delta
m_{\rm atm}^{2}$ is defined for NuFit as $\Delta m_{31}^{2}$ for NO and
$\Delta m_{32}^{2}$ for IO, for de Salas et al. as $\Delta m_{31}^{2}$ for
NO/IO and for Capozzi et al. as $\Delta m_{31}^{2}-\Delta m_{21}^{2}/2$ for
NO/IO. The lower octant atmospheric angle $\theta_{23}$ best-fit is favored
for NO in NuFit and Capozzi et al. results, which can be related to more
recent SK atmospheric data used in analyses. $\Delta\chi^{2}$ represents NO
preference over IO at an overall level of $\sim 2.5\sigma$.

Figure 1.1: Schematics of neutrino oscillation data from Tab. 1.1. On the
plots, green represents best fits, red 1$\sigma$ ranges, and blue 3$\sigma$
ranges. The golden vertical lines symbolize the historic tribimaximal (TBM)
pattern values, which will be explained in Chapter 2. The plot is an update of
plots presented in Refs. [26, 27].

The initial results by the T2K collaboration [28] indicated CP violation in
the lepton sector, a preference for NO at the 3$\sigma$ level and also for
$\theta_{23}$ in the second octant. These data are confirmed with an improved
analysis [29, 30]; they continue to prefer normal mass ordering and upper
octant of $\theta_{23}$ with a nearly maximal $\delta_{\rm CP}$. Also, the
NO$\nu$A collaboration [31] reports on nonvanishing $\delta_{\rm CP}$ effects,
though the best-fit values of CP phase differ between the two groups. Both T2K
and NO$\nu$A prefer NO over IO, but T2K prefers $\delta_{\rm CP}=-90^{o}$
whereas NO$\nu$A prefers a region $\delta_{\rm CP}\sim 180^{o}$.333The mild
tension between T2K and NO$\nu$A could in principle be resolved by invoking
new physics [32]. Joint fits between NO$\nu$A +T2K and Super-K+T2K are
ongoing, with the aim to obtain improved oscillation parameter constraints due
to resolved degeneracies, and to understand potentially non-trivial systematic
correlations [33]. The next-generation oscillation experiments, such as JUNO
[34], Hyper-K [35], DUNE [36] and IceCube upgrade [37], will significantly
improve the prospects of measuring $\delta_{\rm CP}$ and determining the mass
ordering and the octant of $\theta_{23}$ [33].

Oscillation experiments do not put a limit on the Majorana phases
$\alpha_{1}$, $\alpha_{2}$. However, predictions for the Majorana phases can
be obtained using the neutrinoless double beta decay in conjunction with
information on the neutrino masses [38, 39]. Also, the oscillation experiments
are only sensitivity to the squared mass differences, and not to the
individual masses of neutrinos. Therefore, the lightest neutrino mass $m_{0}$
is a free parameter and the other two masses are determined through Eq. (1.2).
However, there are limits on the absolute neutrino mass scale from other
experiments, namely, from tritium beta decay [40], neutrinoless double beta
decay [41], and precision measurements of the cosmic microwave background
(CMB) and large-scale structure (LSS) [42]. We discuss them now.

  * (i)

A direct and model-independent laboratory constraint on the neutrino mass can
be derived from the kinematics of beta decay or electron capture [40]. These
experiments measure an effective electron neutrino mass

$\displaystyle
m^{2}_{\beta}=\frac{\sum_{i}m^{2}_{i}|U_{ei}|^{2}}{\sum_{i}|U_{ei}|^{2}}=\sum_{i}m^{2}_{i}|U_{ei}|^{2}\,,$
(1.3)

assuming $U$ is unitary. This can be expressed through oscillation parameters
as [43]

$\displaystyle m^{2}_{\beta}$
$\displaystyle=c_{13}^{2}c_{12}^{2}m_{1}^{2}+c_{13}^{2}s_{12}^{2}m_{2}^{2}+s_{13}^{2}m_{3}^{2}=\begin{cases}\text{NO:
}&m^{2}_{0}+\Delta m^{2}_{21}c_{13}^{2}s_{12}^{2}+\Delta
m^{2}_{3\ell}s_{13}^{2}\,,\\\ \text{IO: }&m^{2}_{0}-\Delta
m^{2}_{21}c_{13}^{2}c_{12}^{2}-\Delta m^{2}_{3\ell}c_{13}^{2}.\end{cases}$
(1.4)

Here $\ell=1~{}(2)$ for NO (IO) in $\Delta m^{2}_{3\ell}$. The current
oscillation data impose an ultimate lower bound of $m_{\beta}>0.008~{}(0.047)$
eV for NO (IO). At present, the best direct limit on $m_{\beta}$ comes from
the tritium beta decay experiment KATRIN: $m_{\beta}<0.8$ eV at 90% CL [44],
with projected sensitivity down to $m_{\beta}<0.2$ eV at 90% CL [45]. The
future Project 8 experiment using the Cyclotron Radiation Emission
Spectroscopy (CRES) technique is expected to reach a sensitivity for
$m_{\beta}$ down to 0.04 eV [46]. Some recent advances in the CRES technique
were reported in Ref. [47]. Fig. 1.2 (bottom panel) summarizes present and
future experimental bounds with corresponding projections to $m_{0}$ axis in
NO scenario. As we can see, IO is completely within the future Project 8
sensitivity.

Figure 1.2: The effective electron neutrino mass $m_{\beta}$ (Eq. (1.4)) and
the effective Majorana neutrino mass $m_{\beta\beta}$ (Eq. (1.5)) plotted
against the lightest neutrino mass $m_{0}$. The gray, reddish and light
reddish shaded regions represent the KATRIN upper bound
($m_{\beta}<0.8\;\mbox{eV}$ at 90% CL) [44], KATRIN future bound
($m_{\beta}<0.2\;\mbox{eV}$ at 90% CL) [45] and Project 8 future bound
($m_{\beta}<0.04\;\mbox{eV}$) [46] respectively. The light gray and light
magenta shaded regions represent the current upper limit range from KamLAND-
Zen [48] ($36-156$ meV at 90% CL) and future sensitivity range from nEXO [49]
($4.7-20.3$ meV at 90% CL) respectively. All regions are presented with NO
projections to $m_{0}$ axis. The cosmology NO/IO upper limits for $m_{0}$
correspond to the Planck data [50] (see the discussion in item (iii)).

  * (ii)

If neutrinos are Majorana particles, neutrinoless double beta decay
($0\nu\beta\beta$) experiments [41] can also provide direct information on
neutrino masses via the effective Majorana mass [43]

$m_{\beta\beta}=\Big{|}\sum_{i}m_{i}U_{ei}^{2}\Big{|}=\Big{|}m_{1}c_{13}^{2}c_{12}^{2}e^{i2\alpha_{1}}+m_{2}c_{13}^{2}s_{12}^{2}e^{i2\alpha_{2}}+m_{3}s_{13}^{2}e^{-i2\delta_{\mathrm{CP}}}\Big{|}\\\
=\begin{cases}\text{NO:}&m_{0}\,\Big{|}c_{13}^{2}c_{12}^{2}e^{i2(\alpha_{1}-\delta_{\mathrm{CP}})}+\sqrt{1+\frac{\Delta
m^{2}_{21}}{m_{0}^{2}}}\,c_{13}^{2}s_{12}^{2}e^{i2(\alpha_{2}-\delta_{\mathrm{CP}})}+\sqrt{1+\frac{\Delta
m^{2}_{3\ell}}{m_{0}^{2}}}\,s_{13}^{2}\Big{|}\\\
\text{IO:}&m_{0}\,\Big{|}\sqrt{1-\frac{\Delta m^{2}_{3\ell}+\Delta
m^{2}_{21}}{m_{0}^{2}}}\,c_{13}^{2}c_{12}^{2}e^{i2(\alpha_{1}-\delta_{\mathrm{CP}})}+\sqrt{1-\frac{\Delta
m^{2}_{3\ell}}{m_{0}^{2}}}\,c_{13}^{2}s_{12}^{2}e^{i2(\alpha_{2}-\delta_{\mathrm{CP}})}+s_{13}^{2}\Big{|}.\end{cases}$
(1.5)

The present best upper limit on $m_{\beta\beta}$ comes from the KamLAND-Zen
experiment using ${}^{136}{\rm Xe}$: $m_{\beta\beta}<0.036-0.156$ eV at 90% CL
[48], where the range is due to the nuclear matrix element (NME)
uncertainties. Several next-generation experiments are planned with different
isotopes [51], with ultimate discovery sensitivities to $m_{\beta\beta}$ down
to 0.005 eV. Based on Tab. 1.1, the update for $m_{\beta\beta}$ predictions
from Eq. (1.5) as a function of the lightest neutrino mass is plotted in Fig.
1.2. The light gray shaded region shows the current upper limit range for
$m_{\beta\beta}$ ($36-156$ meV at 90% CL) from KamLAND-Zen [48] (comparable
limits were obtained from GERDA [52]), whereas the light magenta shaded region
gives the future upper limit range for $m_{\beta\beta}$ ($4.7-20.3$ meV at 90%
CL) from nEXO [49], with the shaded area in each case arising from NME
uncertainties and corresponding projections to the $m_{0}$ axis in the NO
scenario. Comparable future sensitivities are discussed for other experiments,
such as LEGEND-1000 [53] and THEIA [54], not shown in this plot. The dark gray
shaded region is disfavored by KATRIN [44]. The vertical dashed lines are the
cosmological upper limits (for NO and IO) on the sum of neutrino masses ($\sum
m_{i}<0.12\;\mbox{eV}$ at 95% CL) from Planck [50]; see the next item and Fig.
1.3.

  * (iii)

Massive neutrinos impact CMB and LSS. Thus, precision cosmological data
restrict neutrino masses [55]. Here we use the most stringent limit from
Planck [50]: $\sum m_{i}<0.12\;\mbox{eV}$ at 95% CL (Planck
TT,TE,EE+lowE+lensing+BAO). A slightly stronger limit of $\sum
m_{i}<0.09\;\mbox{eV}$ at 95% CL has been obtained in Refs. [56, 57], while
Ref. [58] has argued in favor of a weaker limit of $\sum
m_{i}<0.26\;\mbox{eV}$. Sum of light neutrino masses $\sum m_{i}$ is plotted
in Fig. 1.3 against the lightest neutrino mass $m_{0}$ (left), effective
electron neutrino mass $m_{\beta}$ (middle) and effective Majorana mass
$m_{\beta\beta}$ (right) with the NuFit $3\sigma$ oscillation parameters from
Tab. 1.1 for both NO and IO scenarios. The horizontal gray-shaded region
represents the current Planck upper limit [50]. Future cosmology sensitivity
forecast is represented by the brown-green shaded area, with an uncertainty of
the order of 15 meV [59] (see also [60]). The dashed lines represent the
lowest allowed values of $\sum m_{i}=58$ meV (NO) and 97 meV (IO) by current
oscillation data.

Figure 1.3: Sum of light neutrino masses $\sum m_{i}$ plotted against the
lightest neutrino mass $m_{0}$ (left), effective electron neutrino mass
$m_{\beta}$ (middle) and effective Majorana mass $m_{\beta\beta}$ (right) with
the NuFit $3\sigma$ oscillation parameters from Tab. 1.1. The horizontal gray-
shaded region represents the current Planck upper limit [50]. Future cosmology
sensitivity forecast is represented by a brown-green shaded area, with an
uncertainty of the order of 15 meV [59] (see also Ref. [60]). The lowest
allowed values of $\sum m_{i}$ for NO and IO from NuFit data are also shown by
the dashed lines. Other current and future exclusion regions shown here are
described in Fig. 1.2. The plot is an updated variation of the plot presented
in Ref. [57].

Figure 1.4: Flavor puzzle. Sizes of circles represent magnitudes of mixing
among quarks (on the left) and neutrinos (on the right). Figure taken from the
arXiv version of Ref. [61].

Understanding the pattern of neutrino mixing is crucial because it is part of
the long-standing flavor puzzle. As seen with the naked eye in Fig. 1.4, the
mixing patterns for quarks and neutrinos are intrinsically very different. The
neutrino case is more "democratic" except for the $U_{e3}$ parameter, which is
proportional to the small $s_{13}$ in Eq. (1.1). In fact, before the non-zero
$\theta_{13}$ discovery by Daya Bay [62] and RENO [63] in 2012, the reactor
mixing angle $\theta_{13}$ was thought to be vanishingly small. Guided by the
$\theta_{13}\sim 0^{\circ}$ assumption and to be consistent with the observed
solar and atmospheric mixing angles, several flavor mixing schemes were
postulated. By substituting $\theta_{13}=0^{\circ}$ and
$\theta_{23}=45^{\circ}$ in the general lepton mixing matrix given in Eq.
(1.1), up to the phase matrix $U_{M}$, most of the popular mixing schemes such
as bi-maximal (BM) [64, 65, 66, 67], tribimaximal (TBM) [68, 69], hexagonal
(HG) [70], and golden ratio (GR) [71, 72, 73, 74, 75, 76] mixing schemes can
be altogether written as

$\displaystyle U_{0}=\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-\frac{s_{12}}{\sqrt{2}}&\frac{c_{12}}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\\
-\frac{s_{12}}{\sqrt{2}}&\frac{c_{12}}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array}\right).$
(1.9)

Substituting $s_{12}=1/\sqrt{2},~{}1/\sqrt{3},~{}1/2$, and
$\tan\theta_{12}=1/\varphi$ (with $\varphi=(1+\sqrt{5})/2$ being the golden
ratio), one can explicitly obtain the fixed mixing schemes BM, TBM, HG and
GR444There exists an alternate version of GR mixing where
$\cos\theta_{12}=\varphi/2$ [74, 75]. respectively. In each case the Dirac CP
phase $\delta_{\rm CP}$ is undefined as $\theta_{13}=0$ and can be easily
extended to include the non-vanishing Majorana phases defined in Eq. (1.1) via
the definition $U_{0}\rightarrow U_{0}U_{M}$. Using the diagonalization
relation

$\displaystyle m_{\nu}=U_{0}^{\star}{\rm
diag}(m_{1},m_{2},m_{3})U_{0}^{\dagger},$ (1.10)

such a mixing matrix can easily diagonalize a $\mu-\tau$ symmetric
(transformations $\nu_{e}\rightarrow\nu_{e}$,
$\nu_{\mu}\rightarrow\nu_{\tau}$, $\nu_{\tau}\rightarrow\nu_{\mu}$ under which
the neutrino mass term remains unchanged) neutrino mass matrix of the form
[61]

$\displaystyle m_{\nu}=\left(\begin{array}[]{ccc}A&B&B\\\ B&C&D\\\
B&D&C\end{array}\right),$ (1.14)

where the elements $A,B,C$ and $D$ are in general complex. With $A+B=C+D$ this
matrix yields the TBM mixing pattern

$\displaystyle U_{\rm
TBM}=\left(\begin{array}[]{ccc}\sqrt{\frac{2}{3}}&\frac{1}{\sqrt{3}}&0\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\end{array}\right)\,.$
(1.18)

Such first-order approximations of the neutrino oscillation data motivated
theorists to find other symmetry-based aesthetic frameworks which can lead
towards these fixed mixing matrices.

In this regard, non-Abelian discrete groups turned out to be popular as
appropriate flavor symmetries for the lepton sector. Discrete groups have
always played a key role in physics starting from crystallographic groups in
solid-state physics, to discrete symmetries such as $C$, $P$, and $T$, which
have shaped our understanding of nature. In neutrino physics, for a long time,
various discrete groups such as $S_{3}$, $A_{4}$, $S_{4}$, $A_{5}$,
$T^{\prime}$, $\Delta(27)$, $D_{n}$, $T_{7}$, $\Delta(6n^{2})$ [77, 78, 26]
etc. have been extensively used to explain fermion mixing. Among the various
discrete groups used for this purpose, $A_{4}$ emerged as the most widely
adopted choice initially proposed as an underlying family symmetry for quark
sector [79, 80]. Interestingly, in the last decade, thanks to the reactor
neutrino experiments Double Chooz [81], Daya Bay [62], and RENO [63] (also T2K
[82], MINOS [83], and others [84]), the reactor neutrino mixing angle is
conclusively measured to be ‘large’ (see Tab. 1.1). In addition to this, as
mentioned earlier, a non-zero value of the Dirac CP phase $\delta_{\rm CP}$ is
favored by the oscillation experiments. Such an observation has ruled out the
possibility of simple neutrino mixing schemes like in Eq. (1.9). Therefore, it
is consequential to find modifications, corrections or the successors of the
above mixing schemes which are still viable; this will be discussed in the
next Chapter. Models based on non-Abelian discrete flavor symmetries often
yield interesting predictions and correlations among the neutrino masses,
mixing angles and CP phases. Involvement of such studies may have broader
applications in various aspects of cosmology (matter-antimatter asymmetry of
the Universe, DM, gravitational waves), collider physics and other aspects of
particle physics, which will be discussed in subsequent Chapters.

Moreover, the light neutrino sector and masses connected with three families
of neutrinos and charged leptons can be a reflection of a more general theory
where weakly interacting (sterile) neutrinos exist with much higher masses
(GeV, TeV, or higher up to the GUT scale). A related problem is the symmetry
of the full neutrino mass matrix, including the sterile sector. In this
framework, the known neutrino mass and flavor states can be denoted by
${|{\nu^{(m)}_{i}}\rangle}$ and ${|{\nu^{(f)}_{\alpha}}\rangle}$,
respectively, where $i=1,2,3$ and $\alpha=e,\mu,\tau$. Any extra, beyond SM
(BSM) sterile mass and flavor states (typically much heavier than the active
ones) can be denoted by $|{\widetilde{\nu}^{(m)}_{j}}\rangle$ and
$|{\widetilde{\nu}^{(f)}_{\beta}}\rangle$, respectively for
$j,\beta=1,\ldots,n_{R}$. In this general scenario mixing between an extended
set of neutrino mass states
$\\{|{\nu^{(m)}_{i}}\rangle,|{\widetilde{\nu}^{(m)}_{j}}\rangle\\}$ with
flavor states
$\\{|{\nu^{(f)}_{\alpha}}\rangle,|{\widetilde{\nu}^{(f)}_{\beta}}\rangle\\}$
is described by

$\displaystyle\begin{pmatrix}{|{\nu^{(f)}_{\alpha}}\rangle}\\\
|{\widetilde{\nu}^{(f)}_{\beta}}\rangle\end{pmatrix}$
$\displaystyle=\begin{pmatrix}{{U}}&V_{lh}\\\
V_{hl}&V_{hh}\end{pmatrix}\begin{pmatrix}{|{\nu^{(m)}_{i}}\rangle}\\\
|{\widetilde{\nu}^{(m)}_{j}}\rangle\end{pmatrix}\equiv{\cal
U}\begin{pmatrix}{{|{\nu^{(m)}_{i}}\rangle}}\\\
|{\widetilde{\nu}^{(m)}_{j}}\rangle\end{pmatrix}\;.$ (1.19)

The SM flavor states $|{\nu^{(f)}_{\alpha}}\rangle$ are then given by

$|{\nu^{(f)}_{\alpha}}\rangle=\sum_{i=1}^{3}\underbrace{\left(U\right)_{\alpha
i}{|{\nu^{(m)}_{i}}\rangle}}_{\rm
SM\;part}+\sum_{j=1}^{n_{R}}\underbrace{{\left(V_{lh}\right)}_{\alpha
j}|{\widetilde{\nu}^{(m)}_{j}}\rangle}_{\rm BSM\;part}\;.$ (1.20)

The mixing matrix $\cal{U}$ in (1.19) diagonalizes a general neutrino mass
matrix

$M_{\nu}=\left(\begin{array}[]{cc}M_{L}&M_{D}\\\
M_{D}^{T}&M_{R}\end{array}\right),$ (1.21)

using a congruence transformation

${\cal U}^{T}M_{\nu}{\cal U}\simeq{\rm diag}(m_{i},M_{j}).$ (1.22)

The structure and symmetry of the heavy neutrino sector $M_{R}$ in Eq. (1.21),
altogether with $M_{D}$ variants, influence the masses and mixing of the light
sector, beginning with the seesaw type of models [85]. The extended unitary
mixing matrix ${\cal U}$ in (1.19), with nonzero submatrices $V$, makes $U$
nonunitary. In fact, oscillation experiments do not exclude such cases, giving
the following ranges of elements [86] (present analysis and future
projections):

$\displaystyle\left\lvert U^{\rm Current}\right\rvert^{2}_{3\sigma}$
$\displaystyle=\left(\begin{array}[]{c c
c}\left[0.606,0.742\right]&\left[0.265,0.337\right]&\left[0.020,0.024\right]\\\
\left[0.051,0.270\right]&\left[0.198,0.484\right]&\left[0.392,0.620\right]\\\
\left[0.028,0.469\right]&\left[0.098,0.685\right]&\left[0.140,0.929\right]\end{array}\right),$
(1.26) $\displaystyle\left\lvert U^{\rm Future}\right\rvert^{2}_{3\sigma}$
$\displaystyle=\left(\begin{array}[]{c c
c}\left[0.653,0.699\right]&\left[0.291,0.311\right]&\left[0.020,0.024\right]\\\
\left[0.074,0.108\right]&\left[0.355,0.454\right]&\left[0.447,0.561\right]\\\
\left[0.129,0.359\right]&\left[0.212,0.423\right]&\left[0.349,0.595\right]\end{array}\right).$
(1.30)

As can be seen from these numbers, the current (and future) precision on the
nonunitarity of the neutrino mixing matrix is still far away from the ultra-
high precision achieved in the quark sector [87]. The interval matrices (1.26)
and (1.30) include nonunitary cases. This information can be used to derive
bounds between known three neutrino flavors and additional neutrino states
[88, 89]. Neutrino mixing constructions based on discrete flavor symmetries
discussed in the next chapters are based on unitary $3\times 3$ mixing
matrices and variants of non-unitary distortions.

The number of model-building options available with discrete flavor symmetries
is vast, and many have already been thoroughly reviewed in the literature. In
Ref. [77], the authors have presented a pedagogical review of various non-
Abelian discrete groups, including their characters, conjugacy classes,
representation, and tensor products, which are essential for particle physics
phenomenology. In Ref. [78], the authors discussed the application of non-
Abelian finite groups to the theory of neutrino masses and mixing with
$\theta_{13}\sim 0$ to reproduce fixed mixing schemes like TBM and BM, based
on finite groups like $A_{4},S_{4}$, etc. After measurement of the reactor
mixing angle, in Ref. [26], the authors reviewed various discrete family
symmetries and their (in)direct model-building approaches. They also discussed
combining grand unified theories with discrete family symmetry to describe all
quark and lepton masses and mixing. In Ref. [90], the author reviewed the
scenarios for flavor symmetry combined with generalized CP symmetry to
understand the observed pattern of neutrino mixing and the related predictions
for neutrino mixing angles and leptonic Dirac CP violation. Finally, along
with conventional flavor symmetric approaches, in Ref. [91], the authors also
reviewed the modular invariance approach to the lepton sector. Many other
excellent reviews also partially cover issues discussed here [92, 93, 94, 95,
96, 97, 98]; see also the Snowmass contributions [99, 100, 101, 102]. Apart
from the update to the mentioned reviews, our main focus in this review is a
discussion on the phenomenology and testability of discrete flavor symmetries
at the energy, intensity, and cosmic frontier experiments.

The rest of the review is organized as follows. In Chapter 2, we present a
general framework for understanding neutrino masses and mixing with non-
Abelain discrete flavor symmetries, discuss the compatibility of a few
surviving mixing schemes with present neutrino oscillation data, and elaborate
on explicit flavor models. We also mention various neutrino generation
mechanisms and possible consequences once we augment them with discrete flavor
symmetries. Then, we discuss the implications of combining flavor symmetries
with CP, higher order discrete groups, Grand Unified Theories, extended Higgs
sector, and finally, allude to the recently revived modular invariance
approach to address the flavor problem. In Chapter 3, we discuss the impact of
discrete flavor symmetries in intensity frontiers such as neutrino oscillation
experiments, neutrinoless double beta decay, lepton flavor and universality
violation. Then in Chapter 4, with some specific examples, we elaborate on the
role of flavor symmetry at colliders, which includes a discussion on prospects
of right-handed neutrino detection at colliders, lepton flavor violation and
constraints on the $h\to\gamma\gamma$ decay width. In Chapter 5, we elaborate
on the consequences of flavor symmetry at cosmic frontier, including studies
on DM, leptogenesis and gravitational waves. Finally, in Chapter 6, we
summarize and conclude.

## 2 Flavor Symmetry and Lepton Masses and Mixing: Theory

From Eq. (1.1) we find that the neutrino mixing matrix is expressed in terms
of mixing angles and CP violating phases, and we are yet to understand the
experimentally observed mixing pattern [20]. The masses and mixing of the
leptons (as well as of the quarks) are obtained from the Yukawa couplings
related to the families. Therefore, it is natural to ask whether any
fundamental principle governs such a mixing pattern.

### 2.1 General Framework

The primary approaches which try to address the issue of the neutrino mixing
pattern include (i) random analysis without imposing prior theories or
symmetries on the mass and mixing matrices [103, 104, 105]; (ii) more specific
studies with imposed mass or mixing textures for which models with underlying
symmetries can be sought [106, 107, 108, 109, 110], and finally, (iii)
theoretical studies where some explicit symmetries at the Yukawa Lagrangian
level are assumed and corresponding extended particle sector is defined. In
the anarchy hypothesis (i), the leptonic mixing matrix manifests as a random
draw from an unbiased distribution of unitary $3\times 3$ matrices and does
not point towards any principle or its origin. This hypothesis does not make
any correlation between the neutrino masses and mixing parameters. However, it
predicts probability distribution for the parameters which parameterize the
mixing matrix. Though random matrices cannot solve fundamental problems in
neutrino physics, they generate intriguing hints on the nature of neutrino
mass matrices. For instance, in Ref. [111], preference has been observed
towards random models of neutrino masses with sterile neutrinos. In the
intermediate approach (ii), some texture zeros of neutrino mass matrices can
be eliminated. For instance, in Ref. [109] 570 (298) inequivalent classes of
texture zeros in the Dirac (Majorana) case were found. For both cases, about
75% of the classes are compatible with the data. In the case of maximal
texture zeros in the neutrino and charged lepton mass matrices, there are only
about 30 classes of texture zeros for each of the four categories defined by
Dirac/Majorana nature and normal/inverted ordering of the neutrino mass
spectrum. Strict texture neutrino mass matrices can also be discussed in
phenomenological studies. For more, see section 2.3.

In what follows, we will discuss the symmetry-based approach (iii) to explain
the non-trivial mixing in the lepton sector known as family symmetry or
horizontal symmetry. Such fundamental symmetry in the lepton sector can easily
explain the origin of neutrino mixing, which is considerably different from
quark mixing. Incidentally, both Abelian and non-Abelian family symmetries
have the potential to shed light on the Yukawa couplings. The Abelian
symmetries (such as Froggatt-Nielsen symmetry [112]) only point towards a
hierarchical structure of the Yukawa couplings, whereas non-Abelian symmetries
are more equipped to explain the non-hierarchical structures of the observed
lepton mixing as observed by the oscillation experiments.

If we consider a family symmetry $G_{f}$, the three generations of leptons and
quarks can be assigned to irreducible representations or multiplets, hence
unifying the flavor of the generations. If $G_{f}$ contains a triplet
representation ($3$), all three fermion families can follow the same
transformation properties. For example, let us consider that non-zero neutrino
mass is generated through the Weinberg operator $HL^{T}cLH$ where the lepton
and Higgs doublets transform as a triplet ($\mathbf{\bar{3}}$) and singlet
under a family symmetry, say, $SU(3)$. To construct $SU(3)$ invariant
operator, an additional scalar field $\Phi$ (also known as flavon ) is
introduced, and the effective operator takes the form $HL^{T}\Phi^{T}\Phi LH$.
A suitable vacuum alignment
($\langle\Phi\rangle\propto(u_{1},u_{2},u_{3})^{T}$) for the flavon is
inserted in such a way that the obtained mass matrix is capable of appropriate
mixing pattern. As a result, $G_{f}$ is spontaneously broken once flavons
acquire non-zero vacuum expectation values (VEV). Continuous family symmetry
such as $U(3)$, $O(3)$ (and their subgroups $SU(3)$ and $SO(3)$) can in
principle be used for this purpose to understand the neutrino mixing. However,
the non-Abelian discrete flavor symmetric approach is much more convenient as
in such a framework, obtaining the desired vacuum alignment (which produces
correct mixing) of the flavon can be obtained easily [113, 114]. At this
point, it is worth mentioning that these non-Abelian discrete symmetries can
also originate from a continuous symmetry [115, 116, 117, 118, 119, 120, 121,
122, 123]. For example widely used discrete groups such as
$A_{4},S_{4},A_{5},\Delta(27),T_{7}$ can originate from the continuous group
$SU(3)$ [26]. In another example [120], the authors showed that continuous
$SO(3)$ can also give rise to $A_{4}$, further broken into smaller $Z_{3}$ and
$Z_{2}$ symmetries. A few years back, it was proposed that various non-Abelian
discrete symmetries can also originate from superstring theory through
compactification of extra dimensions and known as the modular invariance
approach [114, 94, 124].

In this report, we concentrate on all these aspects of discrete family
symmetries discussed above and their implications for understanding lepton
mixing and its extensions. The model building with flavor symmetries is not
trivial since the underlying flavor symmetry group $G_{f}$ must be broken.
Usually, this symmetry $G_{f}$ is considered to exist at some large scale
(sometimes with proximity to GUT scale [78]) and to be broken at lower
energies with residual symmetries of the charged lepton and neutrino sectors,
represented by the subgroups $G_{e}$ and $G_{\nu}$, respectively. Therefore,
to obtain definite predictions and correlations of the mixing, the choice of
the non-Abelian discrete group $G_{f}$ and its breaking pattern to yield
remnant subgroups $G_{e}$ and $G_{\nu}$ shapes the model building
significantly. Without any residual symmetry, the flavor $G_{f}$ loses its
predictivity markedly. For a detailed discussion on the choice of various
discrete symmetries and their generic predictions, see Refs. [26, 90, 95].

Group | Order | Irreducible Representations | Generators
---|---|---|---
$A_{4}$ | 12 | $1,1^{\prime},1^{\prime\prime},3$ | $S,T$
$S_{4}$ | 24 | $1,1^{\prime},2,3,3^{\prime}$ | $S,T(U)$
$T^{\prime}$ | 24 | $1,1^{\prime},1^{\prime\prime},2,2^{\prime},2^{\prime\prime},3$ | $S,T(R)$
$\Delta(27)$ | 27 | $1_{r,s}(r,s=0,1,2),3_{01,02}$ | $C,D$
$A_{5}$ | 60 | $1,3,3^{\prime},4,5$ |  $\tilde{S}$, $\tilde{T}$
Table 2.1: Basic characteristics of a few small groups with triplet
irreducible representations. For details, see Ref. [77]. For instance, a
possible representation for generators of the $S_{4}$ group is defined in the
text, see Eqs. (2.1) and (2.11).

In Tab. 2.1, we mention the basic details such as order or number of elements
(first and second columns), irreducible representations (third column) and
generators (fourth column) of small groups (which contain at least one
triplet) such as $A_{4},S_{4},T^{\prime}$ $\Delta(27)$ and $A_{5}$. A
pedagogical review, including catalogues of the generators and multiplication
rules of these widely used non-Abelian discrete groups, can be found in Ref.
[77].

Now, for model building purposes, there exist various approaches based on the
breaking pattern of $G_{f}$ into its residual symmetries, also known as
direct, semi-direct and indirect approaches [26, 95]. After breaking of
$G_{f}$, different residual symmetries exist for charged lepton (typically
$G_{e}=Z_{3}$) and neutrino sector (typically $G_{\nu}=Z_{2}\times Z_{2}$,
also known as the Kline symmetry). It is known as the direct approach. In a
semi-direct approach, one of the generators of the residual symmetry is
assumed to be broken. On the contrary, in the indirect approach, no residual
symmetry of flavor groups remains intact, and the flavons acquire special
vacuum alignments whose alignment is guided by the flavor symmetry. Usually,
different flavons take part in the charged lepton and neutrino sectors. To
show how the family symmetry shapes the flavor model building, let us consider
$G_{f}=S_{4}$ as a guiding symmetry. Geometrically, this group can be seen as
the symmetry group of a rigid cube, a group of permutation four objects.
Therefore, the order of the group is $4!=24$ and the elements can be
conveniently generated by the generators $S,T$ and $U$ satisfying the relation

$S^{2}=T^{3}=U^{2}=1~{}~{}{\rm and}~{}~{}ST^{3}=(SU)^{2}=(TU)^{2}=1.$ (2.1)

In their irreducible triplet representations, these three generators can be
written as [77, 78]

$\displaystyle S=\frac{1}{3}\left(\begin{array}[]{ccc}-1&2&2\\\ 2&-1&2\\\
2&2&-1\end{array}\right);T=\left(\begin{array}[]{ccc}1&0&0\\\
0&\omega^{2}&0\\\ 0&0&\omega\end{array}\right)~{}~{}{\rm
and}~{}~{}U=\mp\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&1\\\
0&1&0\end{array}\right).$ (2.11)

where $\omega=e^{2i\pi/3}$. These generators can also be expressed as
3-dimensional irreducible (faithful) real representation matrices

$\displaystyle S=\left(\begin{array}[]{ccc}-1&0&0\\\ 0&1&0\\\
0&0&-1\end{array}\right);T=\frac{1}{2}\left(\begin{array}[]{ccc}1&\sqrt{2}&1\\\
\sqrt{2}&0&-\sqrt{2}\\\ -1&\sqrt{2}&-1\end{array}\right)~{}~{}{\rm
and}~{}~{}U=\mp\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&-1\end{array}\right).$ (2.21)

In the direct approach the charged lepton mass matrix ($M_{\ell}$) respects
the generator $T$ whereas the neutrino mass matrix ($M_{\nu}$) respects the
generators $S,U$ satisfying the conditions

$T^{\dagger}M_{\ell}^{\dagger}M_{\ell}T=M_{\ell}^{\dagger}M_{\ell},~{}S^{T}M_{\nu}S=M_{\nu}~{}{\rm
and}~{}U^{T}M_{\nu}U=M_{\nu},$ (2.22)

which leads to [95]

$[T,M_{\ell}^{\dagger}M_{\ell}]=[S,M_{\nu}]=[U,M_{\nu}]=0.$ (2.23)

The non-diagonal matrices $S,U$ can be diagonalized by the TBM mixing matrix
given in Eq. (1.18). Therefore, the TBM mixing scheme can be elegantly derived
from the direct approach of the $S_{4}$ group. For generic features of semi-
direct and indirect approaches to the flavor model building, we refer the
readers to [26, 97, 95]. The TBM mixing pattern explained here can be
generated using various discrete groups. For detailed models and groups see
$A_{4}$ [125, 126, 113, 114], $S_{4}$ [127, 128], $\Delta(27)$ [129],
$T^{\prime}$ [130]. In addition, explicit models with discrete flavor symmetry
for BM [131, 132, 133, 134], GR [73, 135, 76], HG [136] mixing can easily be
constructed.

### 2.2 Flavor Symmetry, Nonzero $\theta_{13}$ and Nonzero $\delta_{\rm CP}$

After precise measurement of the non-zero value of the reactor mixing angle
$\theta_{13}$ [81, 62, 63, 83, 82] the era of fixed patterns (such as BM, TBM,
GR, HG mixing) of the lepton mixing matrix is over. Also, as mentioned
earlier, long baseline neutrino oscillation experiments such as T2K [137] and
NO$\nu$A [138] both hint at CP violation in the lepton sector. Therefore, each
of the fixed patterns needs some modification to be consistent with the global
fit of the neutrino oscillation data [19, 20, 21, 22, 23]. There are two
distinct ways of generating a mixing pattern that appropriately deviates from
fixed mixing schemes such as BM, TBM, GR, and HG. The first approach is based
on symmetry assertion, which demands considering larger symmetry groups that
contain a larger residual symmetry group compared to the fixed mixing schemes
such us TBM [94, 139, 140, 141, 142, 143]. On the other hand, in the second
approach, the setups for the BM, TBM, GR, and HG mixing schemes are
supplemented by an additional ingredient which breaks these structures in a
well-defined and controlled way [144]. This can be achieved in various ways.
An apparent source for such corrections can be introduced through the charged
lepton sector [145, 146, 147, 148, 149]. Thus, in models where in the neutrino
sector the mass matrix solely reproduces the mixing scheme, a non-diagonal
charged lepton sector will contribute to the PMNS matrix
$U=U_{\ell}^{\dagger}U_{\nu}$ where $U_{\ell}$ and $U_{\nu}$ respectively are
the diagonalizing matrices of the charged lepton and neutrino mass matries. In
addition, one can also consider small perturbations around the BM/TBM/GR/HG
vacuum-alignment conditions [150, 151, 152, 153], which can originate from
higher dimensional operators in the flavon potential yielding desired
deviation. The minimal flavon field content can also be extended to
incorporate additional contributions to the neutrino mass matrix to achieve
correct deviation from fixed mixing schemes [154, 155, 156, 157, 158, 159]. To
summarize, the fixed mixing schemes can still be regarded as a first
approximation, necessitating specific corrections to include non-zero
$\theta_{13}$ and $\delta_{\rm CP}$. For example, even if the TBM mixing is
obsolete, two successors are still compatible with data. These are called TM1
and TM2 mixing and are given by

$\displaystyle|U_{{\rm
TM}_{1}}|=\left(\begin{array}[]{ccc}\frac{2}{\sqrt{6}}&*&*\\\
\frac{1}{\sqrt{6}}&*&*\\\ \frac{1}{\sqrt{6}}&*&*\end{array}\right)~{}{\rm
and}~{}|U_{{\rm TM}_{2}}|=\left(\begin{array}[]{ccc}*&\frac{1}{\sqrt{3}}&*\\\
&\frac{1}{\sqrt{3}}&*\\\ &\frac{1}{\sqrt{3}}&*\end{array}\right),$ (2.30)

respectively. Clearly, Eq. (2.30) shows that TM1 and TM2 mixings preserve the
first and the second column of the TBM mixing matrix given in Eq. (1.18).
Here, the reactor mixing angle becomes a free parameter, and the solar mixing
angle can stick close to its TBM prediction.

To illustrate this, let us again consider the discrete flavor symmetry
$G_{f}=S_{4}$. In contrast to the breaking pattern mentioned in Eqs. (2.22),
(2.23), $S_{4}$ is considered to be broken spontaneously into
$Z_{3}=\\{1,T,T^{2}\\}$ (for the charged lepton sector) and $Z_{2}=\\{1,SU\\}$
(for the neutrino sector) such that it satisfies

$[T,M_{\ell}^{\dagger}M_{\ell}]=[SU,M_{\nu}]=0.$ (2.31)

Following the above prescription, the matrix that diagonalizes $SU$ (see Eq.
(2.11)) can be written as $U_{\rm TBM}U_{23}(\theta,\gamma)$ where the ‘23’
rotation matrix is given by ($c_{\theta}=\cos\theta$, $s_{\theta}=\sin\theta$
and $\gamma$ is the associated phase factor)

$\displaystyle U_{{23}}=\left(\begin{array}[]{ccc}1&0&0\\\
0&c_{\theta}&s_{\theta}e^{-i\gamma}\\\
0&-s_{\theta}e^{i\gamma}&c_{\theta}\end{array}\right).$ (2.35)

The obtained effective mixing matrix is called $U_{{\rm TM}_{1}}$ and can be
written as

$\displaystyle U_{{\rm
TM}_{1}}=\left(\begin{array}[]{ccc}\frac{2}{\sqrt{6}}&\frac{c_{\theta}}{\sqrt{3}}&\frac{s_{\theta}}{\sqrt{3}}e^{-i\gamma}\\\
-\frac{1}{\sqrt{6}}&\frac{c_{\theta}}{\sqrt{3}}-\frac{s_{\theta}}{\sqrt{2}}e^{i\gamma}&-\frac{s_{\theta}}{\sqrt{3}}e^{-i\gamma}-\frac{c_{\theta}}{\sqrt{2}}\\\
-\frac{1}{\sqrt{6}}&\frac{c_{\theta}}{\sqrt{3}}-\frac{s_{\theta}}{\sqrt{2}}e^{i\gamma}&-\frac{s_{\theta}}{\sqrt{3}}e^{-i\gamma}+\frac{c_{\theta}}{\sqrt{2}}\end{array}\right),$
(2.39)

The above matrix has the TM1 mixing structure mentioned in Eq. (2.30). This is
also an example of the method of a semi-direct approach to the flavor model
building. Similarly, the generic structure for the structure for TM2 mixing
matrix can be written as

$\displaystyle U_{{\rm
TM}_{2}}=\left(\begin{array}[]{ccc}\frac{2c_{\theta}}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{2s_{\theta}}{\sqrt{6}}e^{-i\gamma}\\\
-\frac{c_{\theta}}{\sqrt{6}}+\frac{s_{\theta}}{\sqrt{2}}e^{i\gamma}&\frac{1}{\sqrt{3}}&-\frac{s_{\theta}}{\sqrt{3}}e^{-i\gamma}-\frac{c_{\theta}}{\sqrt{2}}\\\
-\frac{c_{\theta}}{\sqrt{6}}+\frac{s_{\theta}}{\sqrt{2}}e^{i\gamma}&\frac{1}{\sqrt{3}}&-\frac{s_{\theta}}{\sqrt{3}}e^{-i\gamma}+\frac{c_{\theta}}{\sqrt{2}}\end{array}\right).$
(2.43)

Figure 2.1: $\delta_{\rm CP}$ plotted against $\sin^{2}\theta_{23}$ within TM1
and TM2 models using Eqs. (2.45) and (2.46) and with the NuFit 5.2 oscillation
data [19, 20] from Tab. 1.1. The $1\sigma$, $2\sigma$, $3\sigma$ regions (also
in next figures) were derived from $\chi^{2}$ tables (NuFit 5.2 data files
[160] with SK atmospheric data) for corresponding two-dimensional projections
of the global analysis, minimized for a given mass ordering. The blue (red)
shaded region represents TM1 (TM2) model predictions for $\sin^{2}\theta_{13}$
in the $3\sigma$ range.

Figure 2.2: $\delta_{\rm CP}$ plotted against $\sin^{2}\theta_{13}$ within TM1
and TM2 models using Eqs. (2.45) and (2.46) and with the NuFit 5.2 oscillation
data [19, 20] from Tab. 1.1. The blue (red) shaded region represents TM1 (TM2)
model predictions for $\sin^{2}\theta_{23}$ in the $3\sigma$ range.

Figure 2.3: $\sin^{2}\theta_{12}$ plotted against $\sin^{2}\theta_{13}$ for
TM1 and TM2 mixing [cf. Eqs. (2.45) and (2.46)] with the NuFit 5.2 oscillation
data [19, 20].

The above discussion shows special cases of the TBM mixing, which can still be
relevant for models with discrete flavor symmetries. Now, imposing sufficient
corrections to the other fixed mixing schemes like BM, GR, and HG we can make
them consistent with observed data [135, 161, 162, 163, 164, 165, 166]. The
modified mixing matrix can be obtained by lowering the residual symmetry
$G_{\nu}$ for the neutrino sector. This generates a correction matrix for
these fixed mixing patterns. The general form of these corrections can be
summarized as [133, 90, 167]

$U=U_{e}^{\dagger}U_{0}U_{p},$ (2.44)

where $U_{0}$ is the general form of the relevant fixed pattern mixing scheme
mentioned in Eq. (1.9), $U_{e}$ is the generic correction matrix and $U_{p}$
is additional phase matrix contributing in the Dirac and Majorana phases
mentioned in Eq. (1.1). _These corrections help us to obtain interesting
correlations among $\sin\theta_{12},\sin\theta_{23},\sin\theta_{13}$ and
$\delta_{\rm CP}$ of PMNS mixing matrix_ [168]. In Tab. 2.2, we mention the
typical predictions for TM1 and TM2 mixing matrices, including the Jarlskog
invariant $J_{CP}=c_{12}s_{12}c_{23}s_{23}c_{13}^{2}s_{13}\sin\delta_{\rm CP}$
[169].

| TM1 | TM2
---|---|---
$|U_{e2}|$ | $\left|\frac{\cos\theta}{\sqrt{3}}\right|$ | $\frac{1}{\sqrt{3}}$
$|U_{e3}|$ | $\left|\frac{\sin\theta}{\sqrt{3}}\right|$ | $\left|\frac{2\sin\theta}{\sqrt{6}}\right|$
$|U_{\mu 3}|$ | $\left|\frac{\cos\theta}{\sqrt{2}}+\frac{\sin\theta}{\sqrt{3}}e^{-i\gamma}\right|$ | $\left|-\frac{\cos\theta}{\sqrt{2}}-\frac{\sin\theta}{\sqrt{6}}e^{-i\gamma}\right|$
$\sin^{2}\theta_{12}$ | $1-\frac{2}{3-\sin^{2}\theta}$ | $\frac{1}{3-2\sin^{2}\theta}$
$\sin^{2}\theta_{13}$ | $\frac{1}{3}\sin^{2}\theta$ | $\frac{2}{3}\sin^{2}\theta$
$\sin^{2}\theta_{12}$ | $\frac{1}{2}\left(1-\frac{\sqrt{6}\sin 2\theta\cos\gamma}{3-\sin^{2}\theta}\right)$ | $\frac{1}{2}\left(1+\frac{\sqrt{3}\sin 2\theta\cos\gamma}{3-\sin^{2}\theta}\right)$
$J_{CP}$ | $-\frac{1}{6\sqrt{6}}\sin 2\theta\sin\gamma$ | $-\frac{1}{6\sqrt{3}}\sin 2\theta\sin\gamma$
$\sin\delta_{\rm CP}$ | -$\frac{(5+\cos 2\theta)\sin\gamma}{\sqrt{(5+\cos 2\theta)^{2}-24\sin^{2}2\theta\cos^{2}\gamma}}$ | -$\frac{(2+\cos 2\theta)\sin\gamma}{\sqrt{(2+\cos 2\theta)^{2}-3\sin^{2}2\theta\cos^{2}\gamma}}$
Table 2.2: Mixing parameters in the TM1 and TM2 scenarios. For more details,
see Ref. [168]. The elements $U_{e2},U_{e3},U_{\mu 3}$ are defined in Fig.
1.4.

The correlations among the neutrino mixing angles
($\theta_{23},\theta_{12},\theta_{13}$) and phase ($\delta_{\rm CP}$) for TM1
and TM2 respectively can be written as [170]

$\displaystyle\mbox{TM}_{1}$ $\displaystyle:$ $\displaystyle
s_{12}^{2}=\frac{1-3s_{13}^{2}}{3-3s_{13}^{2}},~{}~{}~{}\cos\delta_{\rm
CP}=\frac{(1-5s_{13}^{2})(2s_{23}^{2}-1)}{4s_{13}s_{23}\sqrt{2(1-3s_{13}^{2})(1-s_{23}^{2})}},$
(2.45) $\displaystyle\mbox{TM}_{2}$ $\displaystyle:$ $\displaystyle
s_{12}^{2}=\frac{1}{3-3s_{13}^{2}},~{}~{}~{}\cos\delta_{\rm
CP}=-\frac{(2-4s_{13}^{2})(2s_{23}^{2}-1)}{4s_{13}s_{23}\sqrt{(2-3s_{13}^{2})(1-s_{23}^{2})}}.$
(2.46)

This helps us illuminate the feasibility of these models in the context of
present neutrino oscillation data. In this regard, following Eqs. (2.45) and
(2.46), we have plotted correlations in $\sin^{2}\theta_{23}-\delta_{\rm CP}$,
$\sin^{2}\theta_{13}-\delta_{\rm CP}$ and
$\sin^{2}\theta_{12}-\sin^{2}\theta_{13}$ planes for both NO and IO in Figs.
2.1, 2.2 and 2.3, respectively. In these plots we have also shown the
1$\sigma$, 2$\sigma$, 3$\sigma$ allowed regions of neutrino oscillation data,
based on the two degrees of freedom (2 dof) tabularized data given in Ref.
[160]. The best-fit values are denoted by $\star$ ($\bullet$) for NO (IO). The
$\sin^{2}\theta_{23}-\delta_{\rm CP}$ correlation plotted in Fig. 2.1 is
important because of the existing ambiguities on octant of $\theta_{23}$
(i.e., whether it is $\theta_{23}>45^{\circ}$ or $\theta_{23}<45^{\circ}$ )
and precise value of $\delta_{\rm CP}$. Here the blue and red shaded regions
represent the predicted correlation of $\theta_{23}$ and $\delta_{\rm CP}$ for
TM1 and TM2 for 3$\sigma$ allowed range of $\sin^{2}\theta_{13}$. For
correlations $\sin^{2}\theta_{13}-\delta_{\rm CP}$ in Fig. 2.2, the spreading
of the shaded region depends on the 3$\sigma$ allowed range of $\theta_{23}$.
In Fig. 2.3, the $\sin^{2}\theta_{12}-\sin^{2}\theta_{13}$ correlations for
TM1 and TM2 mixing schemes yield tight constraints on the allowed ranges of
$\sin^{2}\theta_{12}$: NO (IO) $0.3168-0.3195~{}(0.3167-0.3195)$ for TM1 and
$0.3405-0.3413~{}(0.3405-0.3413)$ for TM2. Furthermore, the
$\sin^{2}\theta_{12}$ prediction for the TM2 mixing lies at the edge of the
3$\sigma$ allowed region. Thus, a precise measurement of $\sin^{2}\theta_{12}$
can potentially rule out the TM2 mixing scheme and corresponding flavor
symmetric models.

Explicit models to obtain TM1 and TM2 mixing can be found on various occasions
in the literature [171, 172, 144, 170]. We present examples of such models for
TM1 and TM2 mixing with $A_{4}$ non-Abelian discrete flavor symmetry. Here, we
will present an example of the TM1 mixing in the context of a hybrid flavor
symmetric scoto-seesaw scenario (FSS) [173, 174] where effective neutrino mass
is generated via both type-I seesaw and scotogenic contributions. In the next
subsection, we will discuss various ways to obtain light neutrino masses. The
particle content of our model and charge assignment under different symmetries
are shown in Tab. 2.3. The role of each discrete symmetry, particle content
and charge assignment in this table are described in detail in Refs. [173,
175].

Fields |  $e_{R}$, $\mu_{R}$, $\tau_{R}$ | $L_{\alpha}$ | $H$ | $N_{R}$ | $f$ | $\eta$ | $\phi_{S}$ | $\phi_{A}$ | $\phi_{T}$ | $\xi$
---|---|---|---|---|---|---|---|---|---|---
$A_{4}$ |  $1$ , $1^{\prime\prime}$ , $1^{\prime}$ | $3$ | $1$ | $1$ | $1$ | $1$ | $3$ | $3$ | 3 | $1^{\prime\prime}$
$Z_{4}$ | $i$ | $i$ | $1$ | $1$ | $1$ | $1$ | $i$ | $i$ | $1$ | $1$
$Z_{3}$ | $\omega^{2}$ | $\omega$ | $1$ | $\omega^{2}$ | 1 | $1$ | $\omega^{2}$ | $\omega$ | $\omega^{2}$ | $1$
$Z_{2}$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $1$ | $1$ | $1$ | $1$ | $-1$
Table 2.3:  Field contents and transformation under the symmetries of our
model. flavons field in second block of the table are introduced to implement
the $A_{4}$ symmetry.

In this setup, the charged lepton Lagrangian can be written up to the leading
order as

$\displaystyle\mathcal{L}_{l}$ $\displaystyle=$
$\displaystyle\frac{y_{e}}{\Lambda}(\bar{L}\phi_{T})_{1}He_{R}+\frac{y_{\mu}}{\Lambda}(\bar{L}\phi_{T})_{1^{\prime}}H\mu_{R}+\frac{y_{\tau}}{\Lambda}(\bar{L}\phi_{T})_{1^{\prime\prime}}H\tau_{R}+h.c.,$
(2.48) $\displaystyle=$
$\displaystyle{\frac{y_{e}}{\Lambda}(\bar{L}_{1}\phi_{T_{1}}+\bar{L}_{2}\phi_{T_{3}}+\bar{L}_{3}\phi_{T_{2}})He_{R}+\frac{y_{\mu}}{\Lambda}(\bar{L}_{3}\phi_{T_{3}}+\bar{L}_{1}\phi_{T_{2}}+\bar{L}_{2}\phi_{T_{1}})H\mu_{R}}$
$\displaystyle{+\frac{y_{\tau}}{\Lambda}(\bar{L}_{2}\phi_{T_{2}}+\bar{L}_{1}\phi_{T_{3}}+\bar{L}_{3}\phi_{T_{1}})H\tau_{R}}$

where $\Lambda$ is the cut-off scale of the FSS model. $y_{e}$, $y_{\mu}$ and
$y_{\tau}$ are the coupling constants. In Eq. (2.48), the terms in the first
parenthesis represent products of two $A_{4}$ triplets forming a one-
dimensional representation which further contract with 1, $1^{\prime\prime}$
and $1^{\prime}$ of $A_{4}$, corresponding to $e_{R}$, $\mu_{R}$ and
$\tau_{R}$, respectively. Following multiplication rules given in A, the
complete $A_{4}$ decomposition is written in Eq. (2.48). Now, when the flavon
$\phi_{T}$ gets VEV in the direction
$\langle\phi_{T}\rangle=(\langle\phi_{T_{1}}\rangle+\langle\phi_{T_{2}}\rangle+\langle\phi_{T_{3}}\rangle)^{T}=(v_{T},0,0)^{T}$,
the charged lepton Lagrangian can be written as

$\displaystyle\mathcal{L}_{l}$ $\displaystyle=$
$\displaystyle\frac{y_{e}}{\Lambda}\bar{L}_{1}v_{T}He_{R}+\frac{y_{\mu}}{\Lambda}\bar{L}_{2}v_{T}H\mu_{R}+\frac{y_{\tau}}{\Lambda}\bar{L}_{3}v_{T}H\tau_{R}.$
(2.49)

Finally, when the SM Higgs field $h$ also gets non-zero VEV as $\langle
h\rangle=v$, following Eq. (2.49), the diagonal charged lepton mass matrix can
be written as

$\displaystyle m_{l}$ $\displaystyle=$
$\displaystyle\frac{vv_{T}}{\Lambda}\begin{pmatrix}y_{e}&0&0\\\ 0&y_{\mu}&0\\\
0&0&y_{\tau}\end{pmatrix}.$ (2.50)

Now, the Lagrangian for neutrino mass contributions in FSS can be written as

$\displaystyle\mathcal{L}=\frac{y_{N}}{\Lambda}(\bar{L}\phi_{S})_{1}\tilde{H}N_{R}+\frac{1}{2}M_{N}\bar{N}_{R}^{c}N_{R}+\frac{y_{s}}{\Lambda^{2}}(\bar{L}\phi_{A})_{1^{\prime}}\xi
i\sigma_{2}\eta^{*}f+\frac{1}{2}M_{f}\bar{f}^{c}f+h.c.,$ (2.51)

where $y_{N}$ and $y_{s}$ are the coupling constant and $M_{N}$ is the
Majorana mass of the right-handed neutrino $N_{R}$ while $M_{f}$ is the mass
of the fermion $f$. Again, following A and Eq. (2.51), the $A_{4}$
decomposition for the contribution to the neutrino sector can be written as

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{y_{N}}{\Lambda}(\bar{L}_{1}\phi_{S_{1}}+\bar{L}_{2}\phi_{S_{3}}+\bar{L}_{3}\phi_{S_{2}})\tilde{H}N_{R}+\frac{1}{2}M_{N}\bar{N}_{R}^{c}N_{R}$
(2.52)
$\displaystyle+\frac{y_{s}}{\Lambda^{2}}(\bar{L}_{3}\phi_{A_{3}}+\bar{L}_{1}\phi_{A_{2}}++\bar{L}_{2}\phi_{A_{1}})\xi
i\sigma_{2}\eta^{*}f+\frac{1}{2}M_{f}\bar{f}^{c}f+h.c.,$ $\displaystyle=$
$\displaystyle\frac{y_{N}}{\Lambda}(\bar{L}_{2}v_{S}-\bar{L}_{3}v_{S})\tilde{H}N_{R}+\frac{1}{2}M_{N}\bar{N}_{R}^{c}N_{R}+\frac{y_{s}}{\Lambda^{2}}(\bar{L}_{1}v_{A}+2\bar{L}_{2}v_{A})\xi
i\sigma_{2}\eta^{*}f+\frac{1}{2}M_{f}\bar{f}^{c}f+h.c.$ (2.53)

In the above Lagrangian, we put VEVs of $\phi_{S}$, $\phi_{A}$ and $\xi$ in
directions
$\langle\phi_{S}\rangle=(\langle\phi_{S_{1}}\rangle,\langle\phi_{S_{2}}\rangle,\langle\phi_{S_{3}}\rangle)^{T}=(0,-v_{S},v_{S})^{T}$,
$\langle\phi_{A}\rangle=(\langle\phi_{A_{1}}\rangle,\langle\phi_{A_{2}}\rangle,\langle\phi_{A_{3}}\rangle)^{T}=(2v_{A},v_{A},0)^{T}$
and $v_{\xi}$, respectively, getting the appropriate flavor structure.
Following Eq. (2.51), the Yukawa coupling for Dirac neutrinos and scotogenic
contributions can be written as

$\displaystyle
Y_{N}=(Y_{N}^{e},Y_{N}^{\mu},Y_{N}^{\tau})^{T}=(0,y_{N}\frac{v_{S}}{\Lambda},-y_{N}\frac{v_{S}}{\Lambda})^{T},$
(2.54) $\displaystyle
Y_{F}=(Y_{F}^{e},Y_{F}^{\mu},Y_{F}^{\tau})^{T}=(y_{s}\frac{v_{\xi}}{\Lambda}\frac{v_{A}}{\Lambda},y_{s}\frac{v_{\xi}}{\Lambda}\frac{2v_{A}}{\Lambda},0)^{T}\equiv(\kappa,2\kappa,0)^{T}.$
(2.55)

Finally, with the above Yukawa couplings, the total effective light neutrino
mass matrix (with both type-I and scotogenic contributions) is given by

$\displaystyle m_{\nu}$ $\displaystyle=$
$\displaystyle-\frac{v^{2}}{M_{N}}Y_{N}^{i}Y_{N}^{j}+\mathcal{F}(m_{\eta_{R}},m_{\eta_{I}},M_{f})M_{f}Y_{f}^{i}Y_{f}^{j}$
(2.56) $\displaystyle=$ $\displaystyle\begin{pmatrix}b&2b&0\\\ 2b&-a+4b&a\\\
0&a&-a\end{pmatrix},$ (2.57)

where
$a=y_{N}^{2}\frac{v^{2}}{M_{N}}\frac{v_{S}^{2}}{\Lambda^{2}},b=y_{s}^{2}\frac{v_{\xi}^{2}}{\Lambda^{2}}\frac{v_{A}^{2}}{\Lambda^{2}}\mathcal{F}(m_{\eta_{R}},m_{\eta_{I}},M_{f})M_{f}=\kappa^{2}\mathcal{F}(m_{\eta_{R}},m_{\eta_{I}},M_{f})M_{f}$
and the loop function $\mathcal{F}$ is written as

$\displaystyle\mathcal{F}(m_{\eta_{R}},m_{\eta_{I}},M_{f})=\frac{1}{32\pi^{2}}\Big{[}\frac{m_{\eta_{R}}^{2}\log\Big{(}M_{f}^{2}/m_{\eta_{R}}^{2}\Big{)}}{M_{f}^{2}-m_{\eta_{R}}^{2}}-\frac{m_{\eta_{I}}^{2}\log\Big{(}M_{f}^{2}/m_{\eta_{I}}^{2}\Big{)}}{M_{f}^{2}-m_{\eta_{I}}^{2}}\Big{]},$
(2.58)

with $m_{\eta_{R}}$ and $m_{\eta_{I}}$ being the masses of the neutral
component of $\eta$. The total mass matrix $m_{\nu}$ therefore can be
diagonalized by a mixing matrix of TM1 mixing pattern given by

$\displaystyle
U=\begin{pmatrix}\sqrt{\frac{2}{3}}&\frac{\cos\theta}{\sqrt{3}}&\frac{e^{-i\psi}\sin\theta}{\sqrt{3}}\\\
-\frac{1}{\sqrt{6}}&\frac{\cos\theta}{\sqrt{3}}+\frac{e^{i\psi}\sin\theta}{\sqrt{2}}&-\frac{\cos\theta}{\sqrt{2}}+\frac{e^{-i\psi}\sin\theta}{\sqrt{3}}\\\
-\frac{1}{\sqrt{6}}&\frac{\cos\theta}{\sqrt{3}}-\frac{e^{i\psi}\sin\theta}{\sqrt{2}}&\frac{\cos\theta}{\sqrt{2}}+\frac{e^{-i\psi}\sin\theta}{\sqrt{3}}\end{pmatrix}U_{M}$
(2.59)

where $U_{M}$ is the Majorana phase matrix defined in Eq. (1.1). The
correlation among the oscillation parameters is given in Eq. (2.45). In
literature, the TM1 mixing has been reproduced using various discrete groups
such as $S_{4}$ in the context of type-I or type-II seesaw scenarios [144,
170, 176, 177].

To reproduce the TM2 mixing, we adopt a modified version of the Altarelli-
Feruglio (AF) model [114, 155]. In this scenario, light neutrino masses are
generated completely via type-I seesaw, and hence three copies of RHNs are
included which we consider to be a triplet under $A_{4}$. With the involvement
of the $A_{4}$ flavons $\phi_{s},\phi_{T}$ (both triplet) and $\xi$ (singlet)
one can obtain the TBM mixing. Now to accommodate nonzero $\theta_{13}$,
$\delta_{\rm CP}$, the desired TM2 mixing can be achieved in the involvement
of one additional flavon $\xi^{\prime}$ ($1^{\prime}$ under $A_{4}$) which
contributes to the right-handed neutrino mass. In addition to the $A_{4}$
discrete symmetry, we also consider a $Z_{3}$ symmetry which forbids the
exchange of $\phi_{s}$ and $\phi_{T}$ in the Lagrangian. The complete particle
content and their transformations under the symmetries are given in Tab. 2.4.

Fields |  $e_{R}$, $\mu_{R}$, $\tau_{R}$ | $L$ | $N_{R}$ | $H$ | $\phi_{S}$ | $\phi_{T}$ | $\xi$ | $\xi^{\prime}$
---|---|---|---|---|---|---|---|---
$SU(2)$ | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1
$A_{4}$ | 1,$1^{\prime\prime}$, $1^{\prime}$ | 3 | 3 | 1 | 3 | 3 | 1 | $1^{\prime}$
$Z_{3}$ | $\omega$ | $\omega$ | $\omega^{2}$ | 1 | $\omega^{2}$ | 1 | $\omega^{2}$ | $\omega^{2}$

Table 2.4: Transformation of the fields needed to realize the $TM_{2}$ mixing
. Here $\omega$ is the third root of unity. $\xi^{\prime}$ is essential to
generate non-zero $\theta_{13}$.

Here we have considered the VEVs for the scalar fields as
$\langle\phi_{S}\rangle=(v_{S},v_{S},v_{S})^{T}$,
$\langle\phi_{T}\rangle=(v_{T},0,0)^{T}$, $\langle\xi\rangle=v_{\xi}$,
$\langle\xi^{\prime}\rangle=v_{\xi^{\prime}}$ [114, 178, 155]. In Tab. 2.4,
$H$ is the $SU(2)$ Higgs doublet (with VEV $v$) and singlet under $A_{4}$.
Now, with the symmetries and particle content present in Tab. 2.4, the
Lagrangian for the charged leptons can be written as

$\displaystyle{\cal{L}}_{CL}$ $\displaystyle=$
$\displaystyle\big{(}{y}_{e}(L{\phi_{T}})_{1}e_{R}+{y}_{\mu}(L{\phi_{T}})_{1^{\prime}}\mu_{R}+{y}_{\tau}(L{\phi_{T}})_{1^{\prime\prime}}\tau_{R}\big{)}\frac{H}{\Lambda},$
(2.60)

where $\Lambda$ is the cutoff scale of the theory and $y_{e},y_{\mu},y_{\tau}$
are the corresponding coupling constants. Note that each term in the first
parentheses represents products of two $A_{4}$ triplets $L,\phi_{T}$ which
further contracts with $e_{R}$, $\mu_{R}$, $\tau_{R}$, which are charged under
$A_{4}$ as $1,1^{\prime\prime}$ and $1^{\prime}$, respectively. Following the
prescription given in Eq. (2.48), the charged lepton mass matrix can be
obtained as

$\displaystyle M_{\ell}$ $\displaystyle=$
$\displaystyle\frac{vv_{T}}{\Lambda}\begin{pmatrix}y_{e}&0&0\\\ 0&y_{\mu}&0\\\
0&0&y_{\tau}\end{pmatrix}.$ (2.61)

In the presence of the $A_{4}$ flavons $\phi_{s},\xi,\xi^{\prime}$ (with VEV
$\langle\phi_{S}\rangle=(v_{S},v_{S},v_{S})^{T}$, $\langle\xi\rangle=v_{\xi}$,
and $\langle\xi^{\prime}\rangle=v_{\xi^{\prime}}$), the Lagrangian for the
neutrino sector can be written as

$\displaystyle{\cal{L}}_{\nu}$ $\displaystyle=$ $\displaystyle
y(LN_{R})H+(x_{A}\xi+x_{B}\phi_{S}+x_{N}\xi^{\prime})\overline{N^{c}}_{R}N_{R},$
(2.63) $\displaystyle=$ $\displaystyle
y(L_{1}N_{R_{1}}+L_{2}N_{R_{3}}+L_{3}N_{R_{2}})H+x_{A}(\overline{N^{c}}_{R_{1}}N_{R_{1}}+\overline{N^{c}}_{R_{2}}N_{R_{3}}+\overline{N^{c}}_{R_{3}}N_{R_{2}})$
$\displaystyle+x_{B}\phi_{S_{1}}(2\overline{N^{c}}_{R_{1}}N_{R_{1}}-\overline{N^{c}}_{R_{2}}N_{R_{3}}-\overline{N^{c}}_{R_{3}}N_{R_{2}})/3+x_{B}\phi_{S_{1}}(2\overline{N^{c}}_{R_{1}}N_{R_{1}}-\overline{N^{c}}_{R_{2}}N_{R_{3}}-\overline{N^{c}}_{R_{3}}N_{R_{2}})/3$
$\displaystyle+x_{B}\phi_{S_{3}}(2\overline{N^{c}}_{R_{3}}N_{R_{3}}-\overline{N^{c}}_{R_{1}}N_{R_{2}}-\overline{N^{c}}_{R_{2}}N_{R_{1}})/3+x_{N}\xi^{\prime}(\overline{N^{c}}_{R_{2}}N_{R_{2}}+\overline{N^{c}}_{R_{1}}N_{R_{3}}+\overline{N^{c}}_{R_{3}}N_{R_{1}})$

where $y$, $x_{A}$, $x_{B}$ are the coupling constants. After spontaneous
breaking of electroweak and flavor symmetries, we obtain the Dirac and
Majorana mass matrices as

$\displaystyle m_{D}=yv\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix},M_{R}=\begin{pmatrix}a+2b/3&-b/3&-b/3\\\ -b/3&2b/3&a-b/3\\\
-b/3&a-b/3&2b/3\end{pmatrix}+\begin{pmatrix}0&0&d\\\ 0&d&0\\\
d&0&0\end{pmatrix},$ (2.64)

where $a=2x_{A}v_{\xi},b=2x_{B}v_{S}$ and $d=2x_{N}v_{\xi^{\prime}}$. The mass
matrices for Dirac, and Majorana neutrinos are obtained from the Lagrangian
written in Eqs.(2.60) and (2.63) following the $A_{4}$ multiplication rules
given in A. The light neutrino mass matrix can be obtained through the type-I
seesaw mechanism using the relation $M_{\nu}=-M_{D}^{T}M_{R}^{-1}M_{D}$. After
diagonalizing $M_{\nu}$ with the tribimaximal mixing matrix $U_{\rm TB}$ we
find,

$\displaystyle M^{\prime}_{\nu}$ $\displaystyle=$ $\displaystyle U^{T}_{\rm
TB}M_{\nu}U_{\rm TB}$ (2.65) $\displaystyle=$ $\displaystyle
y^{2}v^{2}\begin{pmatrix}\frac{2(a+b)-d}{2(a^{2}-b^{2}-ad+d^{2})}&0&\frac{-\sqrt{3}d}{2(a^{2}-b^{2}-ad+d^{2})}\\\
0&\frac{1}{a+d}&0\\\
\frac{-\sqrt{3}d}{2(a^{2}-b^{2}-ad+d^{2})}&0&\frac{2(b-a)+d}{2(a^{2}-b^{2}-ad+d^{2})}\end{pmatrix}.$
(2.66)

The above matrix is diagonal for $d=0$ (contribution corresponding to
$\xi^{\prime}$). Therefore, the light neutrino mass matrix will no longer be
diagonalized by $U_{TB}$ and a further rotation in the 13-plane can
diagonalize $m^{\prime}_{\nu}$ given in Eq. (2.66). So the final diagonalizing
matrix for the light neutrino mass matrix can be written as

$\displaystyle U$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\sqrt{\frac{2}{3}}&\frac{1}{\sqrt{3}}&0\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}\end{pmatrix}.\begin{pmatrix}\cos\theta&0&\sin\theta
e^{-i\psi}\\\ 0&1&0\\\ -\sin\theta e^{i\psi}&0&\cos\theta\end{pmatrix}U_{M},$
(2.67) $\displaystyle=$
$\displaystyle\begin{pmatrix}\sqrt{\frac{2}{3}}\cos\theta&\frac{1}{\sqrt{3}}&\sqrt{\frac{2}{3}}e^{i\psi}\sin\theta\\\
-\frac{\cos\theta}{\sqrt{6}}+\frac{e^{i\psi}\sin\theta}{\sqrt{2}}&\frac{1}{\sqrt{3}}&-\frac{\cos\theta}{\sqrt{2}}-\frac{e^{i\psi}\sin\theta}{\sqrt{6}}\\\
-\frac{\cos\theta}{\sqrt{6}}-\frac{e^{i\psi}\sin\theta}{\sqrt{2}}&\frac{1}{\sqrt{3}}&\frac{\cos\theta}{\sqrt{2}}-\frac{e^{i\psi}\sin\theta}{\sqrt{6}}\end{pmatrix}U_{M},$

where, as in the TM1 case, $U_{M}$ is the Majorana phase matrix defined in Eq.
(1.1). The structure of the mixing matrix coincides with the TM2 mixing
obtained in the context of $A_{4}$ non-Abelian discrete flavor symmetry.

As mentioned earlier, the mixing matrix in Eq. (1.18) and the corresponding
mass matrix Eq. (1.14) obeys the underlying $\mu-\tau$ symmetry (also known as
the $\mu-\tau$ permutation symmetry). As this feature is outdated now for
obvious reasons, there is another class of flavor CP model known as $\mu-\tau$
reflection symmetry [96]. This symmetry can be expressed as the
transformation:

$\nu_{e}\rightarrow\nu_{e}^{C},~{}\nu_{\mu}\rightarrow\nu_{\tau}^{C},~{}\nu_{\tau}\rightarrow\nu_{\mu}^{C}$
(2.68)

where ‘C’ stands for the charge conjugation of the corresponding neutrino
field under which the neutrino mass term remains unchanged. The scheme leads
to the predictions $\theta_{23}=45^{\circ}$, $\delta_{\rm
CP}=90^{\circ}~{}{\rm or}~{}270^{\circ}$. This mixing scheme is still
experimentally viable [20]. Under the discussed $\mu-\tau$ symmetry, the
elements of the lepton mixing matrix satisfy

$|U_{\mu i}|=|U_{\tau i}|~{}~{}~{}~{}~{}{\rm where}~{}~{}~{}{i=1,2,3}.$ (2.69)

Such a mixing scheme is also known as cobimaximal (CBM) mixing scheme [179].
Eq. (2.69) indicates that the moduli of $\mu$ and $\tau$ flavor elements of
the $3\times 3$ neutrino mixing matrix are equal. With these constraints, the
neutrino mixing matrix can be parametrized as [180, 181]

$\displaystyle U_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}u_{1}&u_{2}&u_{3}\\\
v_{1}&v_{2}&v_{3}\\\ v^{*}_{1}&v^{*}_{2}&v^{*}_{3}\end{array}\right),$ (2.73)

where the entries in the first row, $u_{i}$’s are real (and non-negative) with
trivial (vanishing) values of the Majorana phases. Here $v_{i}$ satisfies the
orthogonality condition ${\rm Re}(v_{j}v_{k}^{*})=\delta_{jk}-u_{k}u_{k}$. In
Ref. [181] it was argued that the mass matrix leading to the mixing matrix
given in Eq. (2.73) can be written as

$\displaystyle{M}_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}a&d&d^{*}\\\ d&c&b\\\
d^{*}&b&c^{*}\end{array}\right),$ (2.77)

where $a,b$ are real and $d,c$ are complex parameters. As a consequence of the
symmetry given in Eqs.(2.69)-(2.77), we obtain the predictions for maximal
$\theta_{23}=45^{\circ}$ and $\delta_{\rm CP}=90^{\circ}~{}{\rm
or}~{}270^{\circ}$ in the basis where the charged leptons are considered to be
diagonal. This scheme, however, still leaves room for nonzero $\theta_{13}$.
Realization of such a mixing pattern is possible with various discrete flavor
symmetries ($A_{4},\Delta(27)$, etc.), for example, see Refs. [182, 183, 184,
185, 186].

Earlier, we mentioned that fixed mixing schemes such as BM, TBM, GR, HG are
ruled out and require specific corrections to accommodate non-zero
$\theta_{13}$ and $\delta_{\rm CP}$. These corrections can also provide a
possible deviation from $\theta_{23}=45^{\circ}$. For example, to achieve
experimentally viable TM1 and TM2 mixing, we have shown instances where
additional contribution to the neutrino sector over TBM mixing generates
necessary corrections. However, this can be achieved in various ways. A
correction in the charged lepton sector is one such possibility. This can be
achieved by additional non-trivial contribution in the charged lepton sector
[150].

Figure 2.4: $\delta_{\rm CP}$ plotted against $\sin^{2}\theta_{23}$ within
partial $\mu-\tau$ reflection symmetry predictions (Eqs. (2.78) and (2.79))
and the NuFit 5.2 oscillation data [19, 20]. The blue (red) shaded region
represents $|U_{\mu 1}|=|U_{\tau 1}|$ ($|U_{\mu 2}|=|U_{\tau 2}|$) model
predictions with $\sin^{2}\theta_{12}$ and $\sin^{2}\theta_{13}$ in the
$3\sigma$ range.

Note that the most recent best-fit values for $\theta_{23}$ (Tab. 1.1) prefer
lower octant ($\theta_{23}<45^{\circ}$) for NO and upper octant
($\theta_{23}>45^{\circ}$) for IO. Given this, it seems well motivated to
introduce partial $\mu-\tau$ reflection symmetry, for which the $\delta_{\rm
CP}$ and $\theta_{23}$ are not fixed but correlated [187]

$\displaystyle|U_{\mu 1}|=|U_{\tau 1}|$ $\displaystyle:$
$\displaystyle~{}~{}~{}\cos\delta_{\rm
CP}=\frac{(c_{23}^{2}-s_{23}^{2})(c_{12}^{2}s_{13}^{2}-s_{12}^{2})}{4c_{12}s_{12}c_{23}s_{23}s_{13}},$
(2.78) $\displaystyle|U_{\mu 2}|=|U_{\tau 2}|$ $\displaystyle:$
$\displaystyle~{}~{}~{}\cos\delta_{\rm
CP}=\frac{(c_{23}^{2}-s_{23}^{2})(c_{12}^{2}-s_{12}^{2}s_{13}^{2})}{4c_{12}s_{12}c_{23}s_{23}s_{13}}.$
(2.79)

The $\delta_{\rm CP}-\theta_{23}$ correlations in Eqs. (2.78) and (2.79) for
the partial $\mu-\tau$ reflection symmetry ($|U_{\mu 1}|=|U_{\tau 1}|$ ,
$|U_{\mu 2}|=|U_{\tau 2}|$) are given in Fig. 2.4. These correlations
partially overlap the $1\sigma$, $2\sigma$, $3\sigma$ regions but do not
include the best-fit values.

A similar investigation of partial $\mu-\tau$ reflection symmetry can also be
performed for the 3+1 neutrino mixing scheme, denoted as (3+1)$\nu$, leading
to the $\delta_{\rm CP}/\theta_{23}$ correlations as given below [182]

$\displaystyle|U_{\mu 1}|=|U_{\tau 1}|$ $\displaystyle:$
$\displaystyle~{}~{}\cos\delta_{\rm
CP}=\frac{(a_{1}^{2}+b_{1}^{2})-(c_{1}^{2}+d_{1}^{2})}{2(c_{1}d_{1}-a_{1}b_{1})}$
(2.80) $\displaystyle|U_{\mu 2}|=|U_{\tau 2}|$ $\displaystyle:$
$\displaystyle~{}~{}\cos\delta_{\rm
CP}=\frac{(a_{2}^{2}+b_{2}^{2})-(c_{2}^{2}+d_{2}^{2})}{2(a_{2}b_{2}-c_{2}d_{2})}$
(2.81) $\displaystyle|U_{\mu 3}|=|U_{\tau 3}|$ $\displaystyle:$
$\displaystyle~{}~{}\cos\delta_{\rm
CP}=\frac{(a_{3}^{2}+b_{3}^{2})-(c_{3}^{2}+d_{3}^{2})}{2(a_{3}b_{3}-c_{3}d_{3})}$
(2.82) $\displaystyle{|U_{\mu 4}|=|U_{\tau 4}|}$ $\displaystyle:$
$\displaystyle~{}~{}{\tan^{2}\theta_{24}=\sin^{2}\theta_{34}}$ (2.83)

where

$\displaystyle a_{1}$
$\displaystyle=c_{12}s_{13}s_{23}s_{24}s_{34}-c_{12}c_{23}c_{34}s_{13},$
$\displaystyle b_{1}$
$\displaystyle=c_{34}s_{12}s_{23}-c_{12}c_{13}c_{24}s_{14}s_{34}+c_{23}s_{12}s_{24}s_{34},$
(2.84) $\displaystyle c_{1}$
$\displaystyle=c_{23}c_{24}s_{12}+c_{12}c_{13}s_{14}s_{24},$ $\displaystyle
d_{1}$ $\displaystyle=c_{12}c_{24}s_{13}s_{23},$ $\displaystyle a_{2}$
$\displaystyle=c_{12}(c_{34}s_{23}+c_{23}s_{24}s_{34})+s_{12}c_{13}c_{24}s_{14}s_{34},$
$\displaystyle b_{2}$
$\displaystyle=s_{12}(s_{13}s_{23}s_{24}s_{34}-c_{23}c_{34}s_{13}),$
$\displaystyle c_{2}$
$\displaystyle=c_{12}c_{23}c_{24}-s_{12}c_{13}s_{14}s_{24},$ $\displaystyle
d_{2}$ $\displaystyle=s_{12}c_{24}s_{13}s_{23},$ $\displaystyle a_{3}$
$\displaystyle=c_{13}(c_{23}c_{34}-s_{23}s_{24}s_{34}),$ $\displaystyle b_{3}$
$\displaystyle=c_{24}s_{13}s_{14}s_{34},$ $\displaystyle c_{3}$
$\displaystyle=s_{13}s_{14}s_{24},$ $\displaystyle d_{3}$
$\displaystyle=c_{13}c_{24}s_{23}.$

Figure 2.5: $\delta_{\rm CP}$ plotted against $\sin^{2}\theta_{23}$ in the
(3+1)$\nu$ case within partial $\mu-\tau$ reflection symmetry scenario [182]
and NuFit 5.2 (with SK atmospheric) oscillation data [19, 20]. The blue, red,
green regions represent $|U_{\mu 1}|=|U_{\tau 1}|$ , $|U_{\mu 2}|=|U_{\tau
2}|$, $|U_{\mu 3}|=|U_{\tau 3}|$ symmetry predictions respectively. The
$\sin^{2}\theta_{12}$ and $\sin^{2}\theta_{13}$ vary in $3\sigma$ range (Tab.
1.1) while the (3+1)$\nu$ oscillation parameters
($\theta_{14},\theta_{24},\theta_{34}$) vary in representative $3\sigma$
ranges given in Tab. I of Ref. [182] ($\theta_{14}:4^{\circ}-10^{\circ}$,
$\theta_{24}:5^{\circ}-10^{\circ}$, $\theta_{34}:0^{\circ}-11^{\circ}$,
$\delta_{14}=\delta_{24}=0^{\circ}$).

The results for the (3+1)$\nu$ scenario are gathered in Fig. 2.5. From the
overlapped regions in Fig. 2.5, it is clear that if we demand a total
$\mu-\tau$ reflection symmetry, i.e., $|U_{\mu i}|=|U_{\tau i}|$ for all four
columns then the atmospheric mixing angle $\theta_{23}$ is restricted within a
narrow region around $45^{\circ}$ and the Dirac CP phase is also restricted
around the maximal CP violating values. Since the present best fit values for
$\theta_{23}$ clearly favors a deviation from $45^{\circ}$ (see Fig. 1.1), a
partial $\mu-\tau$ reflection symmetry in the (3+1)$\nu$ scenario may
accommodate appropriate deviation. From Fig. 2.5, we find that $\delta_{\rm
CP}-\theta_{23}$ correlations are partly overlapping the ($1\sigma$, $2\sigma$
and $3\sigma$) $\delta_{\rm CP}-\theta_{23}$ regions for all three partial
$\mu-\tau$ reflection symmetries but only for the $|U_{\mu 1}|=|U_{\tau 1}|$
symmetry, the $\delta_{\rm CP}-\theta_{23}$ correlation region includes the
best-fit value for IO. In addition to this it is worth mentioning that the
partial $\mu-\tau$ reflection symmetry in the fourth column ($|U_{\mu
4}|=|U_{\tau 4}|$) restricts $\theta_{34}$ within the range
$4.9^{\circ}-9.8^{\circ}$ [182].

### 2.3 Flavor Symmetry and Neutrino Mass Models

Apart from the observed pattern of neutrino mixing, the origin of tiny
neutrino mass is still unknown to us. Over decades, the exclusive evidence for
non-zero neutrino masses stems solely from neutrino oscillation experiments,
which are sensitive to the mass-squared differences and not to the absolute
scale of neutrino masses (see Tab. 1.1). On the other hand, bounds on absolute
neutrino masses come from cosmological surveys [188, 50] and the end-point
spectrum of tritium beta decay. Combining all these results, we come to the
neutrino mass ranges discussed in the Introduction, which are at the mili-
electronvolt scale at the most. Based on the established neutrino mass
spectrum, different models try to explain it. For a detailed discussion of
many mechanisms to generate light neutrino masses, the readers are referred to
Refs. [189, 190, 191, 192, 193, 194] and references therein. Here, we briefly
mention a few of them that are frequently used in realistic flavor model
building.

Seesaw models

One of the most popular ways to generate tiny neutrino mass is to start with
the high scale suppressed lepton number violating Weinberg operator
$HL^{T}LH/\Lambda$ mentioned earlier, which is non-renormalizable. This can
give rise to various seesaw mechanisms such as type-I, type-II, type-III,
inverse and linear seesaw [195, 196, 197, 198, 199, 8, 193]. They can be
embedded in the form of Eq. (1.21), see Ref. [200]. However, to elucidate the
observed pattern of neutrino mixings, one must include additional ingredients
such as discrete flavor symmetries. Examples of discrete flavor symmetric
models for type-I, type-II, and inverse seesaw which are very efficient in
explaining tiny mass as well as correct mixing as observed by the neutrino
oscillation experiments can be found in Refs. [155, 156, 157, 201, 202, 203,
204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215]. Based on their
scale, these flavor symmetric seesaw mechanisms may include a wide range of
phenomenological implications in lepton flavor violation, collider
phenomenology or leptogenesis [216].

Radiative mass models

Another class of models that can be connected with flavor symmetries are
radiative neutrino mass models in which masses of neutrinos are absent at the
tree-level and are generated at 1- or higher-loop orders. These models explain
the lightness of neutrino masses with sizable Yukawa couplings and suppression
provided by the loop factor. A broad review of various radiative neutrino
models can be found in Ref. [194]. The key feature of these models is that
they can be verified experimentally because the masses of exotic particles
that take part in the neutrino mass generation are in the TeV range, which the
current colliders’ experiments can probe. Furthermore, these models may
contribute to electric dipole moments, anomalous magnetic moments and meson
decays, matter-antimatter asymmetry [194, 217, 218]. Most interestingly, some
radiative models naturally incorporate potential DM candidates [219, 220].
Additional symmetries that explain tiny neutrino masses also stabilize the DM.
On top of that, radiative neutrino mass models with discrete flavor symmetries
can also explain the observed lepton mixing for obvious reasons, for instance,
modular $S_{3}$ and $A_{4}$ [221, 222, 223, 224, 225].

Neutrino mass sum rules

Neutrino mass mechanisms augmented with discrete flavor symmetries predict a
range of neutrino masses and mixings and can yield interesting correlations
between several observables, such as leptonic mixing angles, phases, and
neutrino masses. Models that have these features enhance the testability at
neutrino experiments. Discrete flavor symmetric models can give rise to sum
rules for neutrino mixing angles and masses. The mixing sum rules relate the
leptonic mixing angles to the Dirac CP-violating phase $\delta_{\rm CP}$ [226,
227, 145, 228]. For example, considering appropriate deviations, the
approximate mixing sum rules can be written as [229, 230, 231]

$\displaystyle\sin^{2}\theta_{12}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}+\sin{\theta_{13}}\cos\delta,$ (2.85)
$\displaystyle\sin^{2}{\theta_{12}}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{3}+\frac{2\sqrt{2}}{3}\sin{\theta_{13}}\cos\delta,$
(2.86)

for BM and TBM mixing, respectively. For the implication of the mixing sum
rules at the neutrino oscillation experiments, see Ref. [228]. On the other
hand, the mass sum rules [232, 233, 234, 235, 236, 237], which describe the
interrelation between the three complex neutrino eigenvalues are particularly
important because of their substantial implication [238, 239, 240, 241] in the
prediction of the effective mass parameter ($m_{\beta\beta}$) appearing in the
neutrinoless double beta decay described in Eq. (1.5). Theoretically, it is
natural to wonder if these sum rules are related to residual or accidental
symmetry. The neutrino mass sum rules can originate from discrete flavor
symmetric models in which neutrino masses can arise from the aforementioned
mass generation mechanisms. However, the most general mass sum rule can be
written as [240]

$A_{1}\tilde{m}_{1}^{p}e^{i\chi_{1}}+A_{2}\tilde{m}_{2}^{p}e^{i\chi_{2}}+A_{3}\tilde{m}_{3}^{p}e^{i\chi_{3}}=0,$
(2.87)

where $\tilde{m_{i}}$ are three complex mass eigen values, $p\neq
0,\chi_{1}\in[0,2\pi],A_{i}>0$. Here $A_{i}$ and $\chi_{i}$ stand for
appropriate complex coefficients and phase factors (without the Majorana
phases). The power $q$ of the complex mass eigenvalues characterizes the sum
rule. For example, a simple sum rule can be obtained when a type-I seesaw mass
mechanism is augmented by $A_{5}$ non-Abelian discrete flavor symmetry leading
to [162]

$\displaystyle\frac{1}{\tilde{m}_{1}}+\frac{1}{\tilde{m}_{2}}=\frac{1}{\tilde{m}_{3}}.$
(2.88)

Comparing Eq. (2.87) and Eq. (2.88) we find that
$A_{i}=1,\chi_{1,2}=0,\chi_{3}=\pi$ and $p=-1$. Such mass sum rules (i.e.,
when $p=-1$) are called inverse sum rules, obtained from a diverse combination
of neutrino mass mechanisms and discrete flavor symmetries [114, 232, 242,
243, 155].

Sum Rule | Group | Seesaw Type
---|---|---
$\tilde{m}_{1}+\tilde{m}_{2}=\tilde{m}_{3}$ |  $A_{4}$[153, 244, 245, 246, 247, 248]; $S_{4}$[249]; $A_{5}$[73] | Weinberg
$\tilde{m}_{1}+\tilde{m}_{2}=\tilde{m}_{3}$ |  $\Delta(54)$[250]; $S_{4}$[251] | Type II
$\tilde{m}_{1}+2\tilde{m}_{2}=\tilde{m}_{3}$ |  $S_{4}$[252] | Type II
$2\tilde{m}_{2}+\tilde{m}_{3}=\tilde{m}_{1}$ |  $A_{4}$ [153, 113, 253, 254, 255, 256, 257, 258, 259, 245, 246, 247, 248, 114, 234] | Weinberg
|  $S_{4}$[128, 260]; $T^{\prime}$[261, 262, 263, 151, 264, 243]; $T_{7}$[265] |
$2\tilde{m}_{2}+\tilde{m}_{3}=\tilde{m}_{1}$ |  $A_{4}$[266] | Type II
$\tilde{m}_{1}+\tilde{m}_{2}=2\tilde{m}_{3}$ |  $S_{4}$[267] | Dirac
$\tilde{m}_{1}+\tilde{m}_{2}=2\tilde{m}_{3}$ |  $L_{e}-L_{\mu}-L_{\tau}$[268] | Type II
$\tilde{m}_{1}+\frac{\sqrt{3}+1}{2}\tilde{m}_{3}=\frac{\sqrt{3}-1}{2}\tilde{m}_{2}$ |  $A_{5}^{\prime}$[269] | Weinberg
$\tilde{m}_{1}^{-1}+\tilde{m}_{2}^{-1}=\tilde{m}_{3}^{-1}$ |  $A_{4}$[153]; $S_{4}$[251, 244]; $A_{5}$[76, 162] | Type I
$\tilde{m}_{1}^{-1}+\tilde{m}_{2}^{-1}=\tilde{m}_{3}^{-1}$ |  $S_{4}$[251] | Type III
$2\tilde{m}_{2}^{-1}+\tilde{m}_{3}^{-1}=\tilde{m}_{1}^{-1}$ |  $A_{4}$[153, 270, 232, 271, 272, 273, 235, 242, 274, 275, 276, 277, 278, 114, 234]; $T^{\prime}$[243] | Type I
$\tilde{m}_{1}^{-1}+\tilde{m}_{3}^{-1}=2\tilde{m}_{2}^{-1}$ |  $A_{4}$[279, 280, 281]; $T^{\prime}$[282] | Type I
$\tilde{m}_{3}^{-1}\pm 2i\tilde{m}_{2}^{-1}=\tilde{m}_{1}^{-1}$ |  $\Delta(96)$[283] | Type I
$\tilde{m}_{1}^{1/2}-\tilde{m}_{3}^{1/2}=2\tilde{m}_{2}^{1/2}$ |  $A_{4}$[233] | Type I
$\tilde{m}_{1}^{1/2}+\tilde{m}_{3}^{1/2}=2\tilde{m}_{2}^{1/2}$ |  $A_{4}$[284] | Scotogenic
$\tilde{m}_{1}^{-1/2}+\tilde{m}_{2}^{-1/2}=2\tilde{m}_{3}^{-1/2}$ |  $S_{4}$[237] | Inverse
Table 2.5:  Sum rules for the complex light neutrino mass eigenvalues
$\tilde{m}_{i}$ defined in Eq. (2.87) obtained with various combinations of
neutrino mass generation mechanisms and discrete flavor symmetries [236, 240,
241].

In Tab. 2.5, we have mentioned various simple sum rules for the complex light
neutrino mass eigenvalues ($\tilde{m}_{i}$) obtained with combinations of
neutrino mass generation mechanism and discrete flavor symmetries [236, 240,
241]. Similar but less simple mass sum rules can also be obtained for models
with modular symmetry. The authors reported in Ref. [285] four different mass
sum rules for models based on modular symmetries where a residual symmetry in
the lepton sector is preserved. The reported sum rules (SR) within these
modular invariance approaches (which also follow the most general form given
in Eq. (2.87)) are called SR 1 (Case I and Case II), SR 2, SR 3, and SR 4
[285]. Studies show that these sum rules can not be conclusively connected to
a particular mass generation mechanism, discrete symmetry, or any remnant
symmetry in the lepton sector [286]. Rather, they are more connected to
minimal breaking of the symmetries, which introduces a minimal number of
parameters related to nonzero neutrino mass eigenvalues. However, the Majorana
phases (connected with the complex mass eigenvalues $\tilde{m}_{i}$) also
appear in the mass sum rules, making them ideal observable to test them in the
neutrinoless double beta decay experiments. The occurrence of such sum rules
severely constrains the prediction for $m_{\beta\beta}$, see Fig. 3.3 in
Section 3.2. A detailed discussion of the role of individual sum rules can be
found in Refs. [238, 236, 239, 240, 241, 286].

Flavor Models: Majorana vs Dirac neutrinos

While neutrino oscillation experiments are insensitive to the nature of
neutrinos, experiments looking for lepton number violating signatures can
probe the Majorana nature of neutrinos. Neutrinoless double beta decay is one
such lepton number violating process which has been searched for at several
experiments without any positive result so far but giving stricter bounds on
the effective neutrino mass, as discussed in Chapter 1. Although negative
results at neutrinoless double beta decay experiments do not prove that the
light neutrinos are of Dirac nature, it is nevertheless suggestive enough to
come up with scenarios predicting Dirac neutrinos with correct mass and
mixing. There have been several proposals already that can generate tiny Dirac
neutrino masses [287, 288, 289, 290, 291, 292]. In Refs. [293, 294, 295, 296],
the authors showed that it is possible to propose various seesaw mechanisms
(type-I, inverse and linear seesaw) for Dirac neutrinos with $A_{4}$ discrete
flavor symmetry. Here the symmetry is chosen in such a way that it naturally
explains the hierarchy among different terms in the neutrino mass matrix,
contrary to the conventional seesaws where this hierarchy is ad-hoc.

Texture zeroes

When a flavor neutrino mass matrix contains zero elements, this is called a
texture zeros mass matrix. Texture zeroes make the neutrino mass and mixing
models simpler. Such constructions are interesting as they lead to a reduction
of independent mass parameters in theory555 The number of free parameters in
neutrino models can also be reduced with the requirement of zero mass
determinant [297, 298] or the zero trace (”zero-sum”
$m_{\nu_{1}}+m_{\nu_{2}}+m_{\nu_{3}}=0$) condition [299, 300].. There is a
vast literature on the subject, e.g. see the list of references in the recent
work [301].

In the three-generation scenario, the low energy Majorana neutrino mass matrix
$M_{\nu}$ is a $3\times 3$ complex symmetric matrix having six independent
elements given by

$\displaystyle M_{\nu}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}m_{ee}&m_{e\mu}&m_{e\tau}\\\
m_{e\mu}&m_{\mu\mu}&m_{\mu\tau}\\\ m_{e\tau}&m_{\mu\tau}&m_{\tau\tau}\\\
\end{pmatrix}.$ (2.89)

For three generations and one-zero textures, all six one-zero textures
$G_{1}-G_{6}$ in Tab. 2.6 can accommodate the experimental data [302], see
also Refs. [303, 304, 305].

$\displaystyle{\rm G}_{1}:\left(\begin{array}[]{*{20}{c}}0&\times&\times\\\
-&\times&\times\\\ -&-&\times\\\ \end{array}\right),\quad{\rm
G}_{2}:\left(\begin{array}[]{*{20}{c}}\times&0&\times\\\ -&\times&\times\\\
-&-&\times\\\ \end{array}\right),\quad{\rm
G}_{3}:\left(\begin{array}[]{*{20}{c}}\times&\times&0\\\ -&\times&\times\\\
-&-&\times\\\ \end{array}\right),\quad$ $\displaystyle{\rm
G}_{4}:\left(\begin{array}[]{*{20}{c}}\times&\times&\times\\\ -&0&\times\\\
-&-&\times\\\ \end{array}\right),\quad{\rm
G}_{5}:\left(\begin{array}[]{*{20}{c}}\times&\times&\times\\\ -&\times&0\\\
-&-&\times\\\ \end{array}\right),\quad{\rm
G}_{6}:\left(\begin{array}[]{*{20}{c}}\times&\times&\times\\\
-&\times&\times\\\ -&-&0\\\ \end{array}\right).$ Table 2.6: Possible one-zero
textures in the three-generation neutrino flavor scenario.

The crosses “$\times$” stand for the non-zero entries and "$-$" represent
symmetric elements (the matrices are assumed to be symmetric). There are 15
possible two-zero textures categorised in different classes666The constraints
can be released for non-Hermitian textures. For instance, in the case of light
Dirac neutrinos (less CP phases) and non-Hermitian textures, even textures
with four zeros are allowed. Most of the one-zero, two-zero and three-zero
textures in this case are allowed. Four-zero textures are tightly constrained,
with only six allowed out of 126 possibilities [306]., as shown in Tab. 2.7.

$A_{1}$ | $A_{2}$ |  |  | $D_{1}$ | $D_{2}$ |
---|---|---|---|---|---|---
$\left(\begin{array}[]{ccc}0&0&\times\\\ 0&\times&\times\\\ -&-&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&\times&0\\\ -&\times&\times\\\ 0&-&\times\end{array}\right)$ |  |  | $\left(\begin{array}[]{ccc}\times&\times&\times\\\ -&0&0\\\ -&0&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&\times&\times\\\ -&\times&0\\\ -&0&0\end{array}\right)$ |
$B_{1}$ | $B_{2}$ | $B_{3}$ | $B_{4}$ | $E_{1}$ | $E_{2}$ | $E_{3}$
$\left(\begin{array}[]{ccc}\times&\times&0\\\ -&0&\times\\\ 0&-&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&0&\times\\\ 0&\times&\times\\\ -&-&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&0&\times\\\ 0&0&\times\\\ -&-&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&\times&0\\\ -&\times&\times\\\ 0&-&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&\times&\times\\\ -&0&\times\\\ -&-&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&\times&\times\\\ -&\times&\times\\\ -&-&0\end{array}\right)$ | $\left(\begin{array}[]{ccc}0&\times&\times\\\ -&\times&0\\\ -&0&\times\end{array}\right)$
$C$ |  |  |  | $F_{1}$ | $F_{2}$ | $F_{3}$
$\left(\begin{array}[]{ccc}\times&\times&\times\\\ -&0&\times\\\ -&-&0\end{array}\right)$ |  |  |  | $\left(\begin{array}[]{ccc}\times&0&0\\\ 0&\times&\times\\\ 0&-&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&0&\times\\\ 0&\times&0\\\ -&0&\times\end{array}\right)$ | $\left(\begin{array}[]{ccc}\times&\times&0\\\ -&\times&0\\\ 0&0&\times\end{array}\right)$
Table 2.7: Possible two-zero textures in the three-generation neutrino flavor
scenario.

The sum of crosses in each matrix in Tabs. 2.6 and 2.7 gives the number of
independent parameters of 5(4) for textures with one (two) zeroes. Textures
with more than two independent zeroes appear to be excluded by the
experiments777Extending the flavor space to the 3+1 neutrino framework, 15 out
of 210 textures with four-zero textures are allowed [307]. (three independent
parameters are not enough to accommodate neutrino data).

In the $3\times 3$ scenario, among the 15 possible textures, only 7 are
phenomenologically allowed [308, 309, 106]. In Ref. [310], nine patterns were
compatible with data (two of them only marginally for considered experimental
data). The results, in general, depend strongly on available data. After
measurement of non-zero $\theta_{13}$, the updated analysis can be found in
Refs. [311, 312, 313] and the number of viable textures have been reduced
significantly, namely:

  1. 1.

Class $A$ is allowed only for NH.

  2. 2.

Class $B$ is allowed for both NH and IH. $B_{1}$ and $B_{4}$ predict negative
values of $\cos\delta_{\rm CP}$ whereas the classes $B2$ and $B3$ predict
positive values of $\cos\delta_{\rm CP}$. The textures $B1$ and $B3$ predict
$\theta_{23}$ in the lower octant and the textures $B2$ and $B4$ predict
$\theta_{23}$ in the upper octant for NH. The predictions are opposite for the
IH.

  3. 3.

Class $C$ class is allowed mainly in the IH. This class is marginally allowed
in the NH when $\theta_{23}$ is close to $45^{\circ}$. In this class when
$\theta_{23}<45^{\circ}$, one must have $-90^{\circ}<\delta_{\rm
CP}<90^{\circ}$ and when $\theta_{23}>45^{\circ}$, one must have
$90^{\circ}<\delta_{\rm CP}<270^{\circ}$.

  4. 4.

The textures in the classes $D$, $E$ and $F$ are forbidden by the data.

In Ref. [108] numerical scan over neutrino parameters using adaptive Monte
Carlo generator confirmed seven allowed patterns with two zeroes (the CP
conserving case) and identified additional cases for the non-degenerated
neutrino masses (mass ordered scenarios). In Ref. [314], the stability of
phenomenological consequences of texture zeros under radiative corrections in
the type-I see-saw scenario is discussed. It has been shown that additional
patterns are allowed under certain conditions due to these effects. Comparing
these results with the classification of two-zero textures, three of the six
forbidden textures turn out to agree with experimental data due to the
renormalization group evolution of the Yukawa couplings

$\begin{pmatrix}\times&0&0\\\ 0&\times&\times\\\
0&-&\times\end{pmatrix},\begin{pmatrix}\times&0&\times\\\ 0&\times&0\\\
-&0&\times\end{pmatrix},\begin{pmatrix}\times&\times&0\\\ -&\times&0\\\
0&0&\times\end{pmatrix}.$

The matrices of the form

$\begin{pmatrix}0&\times&\times\\\ -&0&\times\\\
-&-&\times\end{pmatrix},\begin{pmatrix}0&\times&\times\\\ -&\times&\times\\\
-&-&0\end{pmatrix},\begin{pmatrix}0&\times&\times\\\ -&\times&0\\\
-&0&\times\end{pmatrix},$

remain forbidden. Tab. 2.8, extracted from Ref. [314], summarizes the
situation. In this work the quasi-degenerate cases are also considered,
however, they are already excluded by cosmological and $(\beta\beta)_{0\nu}$
data, see Figs. 1.3 and 1.2 (so not repeated in Tab. 2.8).

Neutrino masses | Majorana phases |
---|---|---
Normal ordering, $m_{1}\approx 0$ | arbitrary | $\left(\begin{smallmatrix}\makebox[2.41112pt]{\rule[0.3014pt]{0.0pt}{1.80835pt}$\scriptstyle\cdot$}&\cdot&\cdot\\\ \cdot&\makebox[2.41112pt]{\rule[0.3014pt]{0.0pt}{1.80835pt}$\scriptstyle\cdot$}&\cdot\\\ \cdot&\cdot&\makebox[2.41112pt]{\rule[0.3014pt]{0.0pt}{1.80835pt}$\scriptstyle\cdot$}\end{smallmatrix}\right)$
Inverted ordering, $m_{3}\approx 0$ | $\varphi_{1}\approx\varphi_{2}$ | $\left(\begin{smallmatrix}\cdot&\circ&\circ\\\ \circ&\cdot&\cdot\\\ \circ&\cdot&\cdot\end{smallmatrix}\right)$
| $\varphi_{1}\not\approx\varphi_{2}$ | $\left(\begin{smallmatrix}\makebox[2.41112pt]{\rule[0.3014pt]{0.0pt}{1.80835pt}$\scriptstyle\cdot$}&\cdot&\cdot\\\ \cdot&\makebox[2.41112pt]{\rule[0.3014pt]{0.0pt}{1.80835pt}$\scriptstyle\cdot$}&\cdot\\\ \cdot&\cdot&\makebox[2.41112pt]{\rule[0.3014pt]{0.0pt}{1.80835pt}$\scriptstyle\cdot$}\end{smallmatrix}\right)$
Table 2.8: Possible positions of radiatively generated texture zeros in the
neutrino mass matrix, marked by a “$\scriptstyle\circ$”. For
$\varphi_{1}\approx\varphi_{2}\approx\pi$, at most, 3 of the four zeros can be
produced simultaneously. Table taken from Ref. [314] where a convention for
Majorana phases in the PMNS matrix is $\phi_{1,2}=-\alpha/2$, see Eq. (1.1).

The question is if models with zeroes in neutrino mass matrix constructions
(either in the effective mass matrix $M_{\nu}$ or in $M_{D},M_{R}$) have
anything to do with discrete symmetries. _The answer is positive._ There are
examples in the literature where texture zeros are related to discrete
symmetries. For instance, in Ref. [315], the $A_{4}$-based texture one-zero
neutrino mass model within the inverse seesaw mechanism for DM is discussed.
The obtained effective neutrino mass matrix is in the form

$M_{\nu}=\begin{pmatrix}X+X^{{}^{\prime}}&0&\Delta+\Delta^{{}^{\prime}}\\\
0&\Delta^{{}^{\prime}}&X^{{}^{\prime}}\\\
\Delta+\Delta^{{}^{\prime}}&X^{{}^{\prime}}&\Delta^{{}^{\prime\prime}}\
\end{pmatrix}.$ (2.92)

Not-primed and doubly-primed elements gather the light neutrino mass matrix
contributions while primed elements come from the inverse seesaw mass
mechanism construction. For more on a connection between mass matrix textures
and discrete symmetries and further references, see Ref. [316] (the inverse
neutrino mass matrix with one texture zero and TM mixing) or Ref. [317]
(textures of neutrino mass matrix with $S_{4}$ symmetry in which some pairs of
mass matrix elements are equal, up to the sign).

The textures of neutrino matrices can also be studied using the matrix theory,
in particular an inverse eigenvalue (singular value) problem (IEP). It is a
method that reconstructs a matrix from a given spectrum [318, 319]. This is an
important field on its own with many applications. One application which seems
natural from the neutrino physics point of view is the reconstruction of the
neutrino mass matrix from experimental constraints. Especially important would
be one class of IEP, namely an inverse eigenvalue problem with prescribed
entries whose goal can be stated as follows: given a set
$\mathcal{L}=\\{i_{\nu},j_{\nu}\\}_{\nu=1}^{l}$, $1\leq i_{\nu},j_{\nu}\leq
n$, a set of $l$ values $\\{a_{1},\ldots,a_{l}\\}$ and a set of n values
$\\{\lambda_{1},\ldots,\lambda_{n}\\}$ find a matrix $A\in\mathbb{M}^{n\times
n}$ such that

$\displaystyle\sigma(A)=\\{\lambda_{1},\ldots,\lambda_{n}\\}\,,$ (2.93)
$\displaystyle A_{i_{\nu}j_{\nu}}=a_{\nu}\text{ for }\nu=1\ldots,l\,,$ (2.94)

where $\sigma(A)$ is a spectrum of the matrix $A$. Some of the classical
results of IEP are Schur-Horn theorem [320, 321], Mirsky theorem [322], Sing-
Thompson theorem [323, 324]. For example, the Mirsky theorem says

###### Theorem 2.1

A square matrix with eigenvalues $\lambda_{1},\ldots,\lambda_{n}$ and main
diagonal elements $a_{1},\ldots,a_{n}$ exists if and only if

$\displaystyle\sum_{i=1}^{n}a_{i}=\sum_{i=1}^{n}\lambda_{i}\,.$ (2.95)

There are many extensions of these classical results, allowing arbitrary
location of the prescribed elements, see for example Ref. [325]. This method
can be applied to studies of discrete symmetries in the neutrino sector. As an
example one can systematically examine the structure of the mass matrix in the
texture-zeros approach, where one would like to reconstruct a matrix with a
prescribed spectrum agreeing with the experimental observations and with zero
matrix elements in specific locations. However, it is also well suited for
studies of more general structures of the neutrino mass and mixing matrices.
Some applications of the IEP in the neutrino sector have already been done,
e.g., in Ref. [85] the discussion of the mass spectrum in the seesaw scenario
is given. The application of the inverse singular value problem in the study
of neutrino mixing matrices was considered in Ref. [89].

A different approach, also using tools from the matrix theory, to the modeling
of the structure of the neutrino mass matrix has been proposed in Ref. [326],
where it has been applied to the Altarelli–Feruglio model [113, 114] and its
perturbation from the TBM regime. This approach is based on the observation
that one of the invariants of the $n\times n$ complex matrix $A$ is a square
of the Frobenius norm

$\displaystyle
R^{2}=\|A\|_{F}^{2}=Tr\left(AA^{{\dagger}}\right)=\sum_{i,j}^{n}|a_{ij}|^{2}\,,$
(2.96)

which can be interpreted as an equation of the $n^{2}$-dimensional hyper-
sphere. Thus, it makes it natural to express elements of the matrix $A$ in
terms of spherical coordinates. For the physically interesting case of $n=3$,
elements of the neutrino mass matrix $M_{\nu}$ (assumed to be real) can be
parametrized as

$\displaystyle
M_{11}=R\sin(\chi)\left(\prod_{i=1}^{6}\sin(\phi_{i})\sin(\phi_{7})\right),\;M_{12}=R\sin(\chi)\left(\prod_{i=1}^{6}\sin(\phi_{i})\cos(\phi_{7})\right)\,,$
(2.97) $\displaystyle
M_{13}=R\sin(\chi)\left(\prod_{i=1}^{5}\sin(\phi_{i})\cos(\phi_{6})\right),\;M_{21}=R\sin(\chi)\left(\prod_{i=1}^{4}\sin(\phi_{i})\cos(\phi_{5})\right)\,,$
(2.98) $\displaystyle
M_{22}=R\sin(\chi)\left(\prod_{i=1}^{3}\sin(\phi_{i})\cos(\phi_{4})\right),\;M_{23}=R\sin(\chi)\left(\prod_{i=1}^{2}\sin(\phi_{i})\cos(\phi_{3})\right)\,,$
(2.99) $\displaystyle
M_{31}=R\sin(\chi)\sin(\phi_{1})\cos(\phi_{2})\,,\;M_{32}=R\sin(\chi)\cos(\phi_{1})\,,\;M_{33}=R\cos(\chi)\,.$
(2.100)

As a consequence of this parametrization, we get interrelations between
different elements, and also, it is straightforward to produce texture-zeros
by a particular choice of angles. Moreover, the Frobenius norm can also be
expressed in terms of singular values

$\displaystyle\|A\|_{F}=\sqrt{\sum_{i=1}^{q}\sigma_{i}^{2}}\,,$ (2.101)

where $q$ is the rank of $A$ and similarly to (2.96) can be interpreted as a
$q$-dimensional sphere. For the normalized matrix in the case $n=3$,
$\bar{A}=\frac{A}{\|A\|_{F}}$ , the normalized singular values can then be
defined as $\bar{\sigma}_{1}=\sin\alpha\sin\beta$ ,
$\bar{\sigma}_{2}=\sin\alpha\cos\beta$, $\bar{\sigma}_{3}=\cos\alpha$, where
$\alpha,\beta\in[0,\frac{\pi}{2}]$. Thus, one can see that only two angles
$\alpha$ and $\beta$ are necessary to describe normalized singular values of
$\bar{A}$ reflecting the fact that only two independent mass ratios are
relevant. As we have seen the $9$-dimensional sphere (2.96) carries more
information than the one given in the singular value space, requiring in total
$8$ angles to describe it fully. The additional 6 angles are related to the
unitary matrices of the singular value decomposition of $\bar{A}$

$\displaystyle\bar{A}=L^{{\dagger}}\Sigma R\,.$ (2.102)

Finally, we can express angles $\alpha$ and $\beta$ in terms of singular
values as

$\displaystyle\sin\alpha=\sqrt{\frac{\bar{\sigma}_{1}^{2}+\bar{\sigma}_{2}^{2}}{\bar{\sigma}_{1}^{2}+\bar{\sigma}_{2}^{2}+\bar{\sigma}_{3}^{2}}}\,,\;\sin\beta=\sqrt{\frac{\bar{\sigma}_{1}^{2}}{\bar{\sigma}_{1}^{2}+\bar{\sigma}_{2}^{2}}}\,.$
(2.103)

Another interesting realization of discrete flavor symmetries with specific
mixing matrices can be realized in the framework of so-called magic matrices.
An $n\times n$ matrix $A$ is magic if the row sums and the column sums are all
equal to a common number $\alpha$:

$\sum_{i=1}^{n}A_{ij}=\sum_{j=1}^{n}A_{ij}=\alpha.$ (2.104)

Neutrino mass matrix [327] is magic, in the sense that the sum of each column
and the sum of each row are all identical, see also Ref. [328]. Zeros in the
magic neutrino mass matrix are discussed in Ref. [329]. In Ref. [330], the
magic neutrino mass model within the type-I and II seesaw mechanism paradigm
are investigated. It is based on the neutrino mass model triggered by the
$A_{4}$ discrete flavor symmetry where a minimal scenario with two right-
handed neutrinos and with broken $\mu-\tau$ symmetry leads to leptogenesis.

Neutrino and quark models

Neutrino and quark sectors are qualitatively different regarding their mixing
and mass patterns. However, there are ambitious trials to consider them
together. For instances, in Ref. [331] the quark-lepton complementarity (QLC)
relations [332, 333, 334, 335, 336] are considered with
$\theta_{12}+\theta_{12}^{q}\simeq 45^{o}$ and
$\theta_{23}+\theta_{23}^{q}\simeq 45^{o}$. The QLC relations indicate that
there could be a quark-lepton symmetry based on a flavor symmetry. In Ref.
[331] a discrete symmetry $A_{4}\times Z_{2}$ is considered in the context of
charged leptons and quarks, and tribimaximal neutrino mixing, see also Ref.
[151]. We should mention that there is also an intriguing, the so-called King-
Mohapatra-Smirnov (KMS) relation [333, 337, 338, 339]
$U=U^{\star}_{CKM}U^{\star}_{TBM}$, which gives
$|U^{13}|\simeq|\sin{\theta_{C}}/\sqrt{2}|\simeq 0.156$. For a recent
discussion on the possible origin of this relation, see Ref. [340]. The KMS
relation will also be discussed in section 2.6 on the discrete symmetries and
the GUT scale. For the application of dihedral groups to the lepton and quark
sectors, see also Ref. [341].

### 2.4 Flavor and Generalized CP Symmetries

As discussed earlier (see Eq. (1.1), the PMNS mixing matrix is characterized
by three mixing angles which are well measured as well as three CP phases,
namely the Dirac CP phase ($\delta_{\rm CP}$), and the two Majorana phases,
which are largely unconstrained. The flavor symmetry approach can predict the
three mixing angles, making exploring how CP phases can be predicted similarly
attractive. A hint of how this can be achieved can be seen in the
transformations given in Eq. (2.68) for the $\mu$-$\tau$ reflection symmetry.
This overall symmetry operation can be seen as a canonical CP transformation
augmented with the $\mu$-$\tau$ exchange symmetry, which will later be a case
of the “generalized" CP transformation. This example predicts both atmospheric
mixing angle and $\delta_{\rm CP}$ to be maximal. Therefore, such
transformations represent an interesting extension of the discrete symmetry
framework discussed earlier with an additional invariance under a CP symmetry.
We will first discuss the basic properties of CP transformations. We then
discuss combining CP and flavor symmetry group $G_{f}$ in a consistent
framework.

The action of a generalized CP transformation $X$ on a field operator is given
by

$\varphi^{\prime}(x_{0},\vec{x})\,=\,X\,\varphi^{*}(x_{0},-\vec{x})$ (2.105)

where $X$ is a constant unitary matrix i.e. $XX^{\dagger}=\mathbb{1}$. The
requirement of CP invariance on the neutrino mass matrix $M_{\nu}$ leads to
the condition

$X^{T}M_{\nu}X\,=\,M_{\nu}^{*}$ (2.106)

For example, in the case of the $\mu$-$\tau$ reflection symmetry, the mass
matrix given in Eq. (2.77) is invariant under

$\mathcal{S}^{T}{M}_{0}\mathcal{S}={M}^{*}_{0},$ (2.107)

where the transformation matrix is given by

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&1\\\
0&1&0\end{array}\right).$ (2.111)

On comparing Eq. (2.107) with Eq. (2.106), we can easily identify
$\mathcal{S}$ as the CP transformation $X$ in this case. Note that if
neutrinos are Dirac particles, condition in Eq. (2.106) is replaced by
$X^{\dagger}m_{\nu}^{\dagger}m_{\nu}X\,=\,(m_{\nu}^{\dagger}m_{\nu})^{*}$. It
can be shown explicitly that only a generalized transformation leads to
vanishing CP invariants, thus leading to vanishing CP phases [342, 343].

Now consider the case with both flavor and CP symmetry. The existence of both
discrete flavor and generalized CP symmetries determines the possible
structure of the generalized CP symmetry matrices. For example, if we consider
a discrete flavor symmetry $G_{f}$ in the lepton sector, the transformation
matrix given in Eq. (2.111) satisfy the consistency condition given by [343,
344]

$\mathcal{S}\rho(g)^{*}\mathcal{S}^{-1}=\rho(u(g)),$ (2.112)

(or equivalently in the general case $S\rightarrow X$) where $u$ is an
automorphism of a group of $G_{f}$ which maps an element $g\in G_{f}$ into
$g^{\prime}=u(g)\in G_{f}$ where the latter belong to the conjugacy class of
$g^{-1}$. Since this automorphism is class-inverting, it is an outer
automorphism888In an outer automorphism, the automorphism cannot be
represented by a conjugation with a group element i.e. $g\rightarrow
hgh^{-1},\,h\notin G_{f}$ of $G_{f}$. One can derive this condition by
applying a generalized CP symmetry transformation, subsequently, a flavor
transformation associated with the group element $g=G_{f}$ and an inverse
generalized CP symmetry transformation in order. As the Lagrangian remains
unchanged, the resulting transformation must correspond to an element of
$G_{f}$. This can lead to interesting predictions for the leptonic CP phases
(Dirac and Majorana) and the mixing angles.

Figure 2.6: General scheme considered in literature for ways of merging
discrete flavor groups with CP effects; figure taken from Ref. [345]. For a
concrete example, see Section 4.1 and associated phenomenological studies in
the subsequent Chapters.

Following Ref. [345], in Fig. 2.6, it is indicated that typically CP symmetry
is considered in the neutrino sector. As an example, we consider $G_{f}=S_{4}$
and CP [260, 346, 347, 348, 349]. Our discussion follows specifically the case
$G_{f}=S_{4}$, $G_{\nu}=Z_{2}\times\,CP$ and $G_{e}=Z_{3}$ (see Ref. [260] for
details). Given this choice of residual group for $G_{e}$, the only possible
choice for a generator of the group is $Q=T$. In this case, it is also found
that all choices of generator $Z$ and $X$ are related through similarity
transformations to the following three options: $Z=S$, $Z=SU$ and $Z=U$. Using
the consistency relation as given in Eq. (2.112) along with the unitary
symmetric nature of $X$, the following forms of CP transformations are (in the
3-dim real representation of $S_{4}$, see Eq. (2.21)) : $X_{1}\propto 1$,
$X_{2}\propto S$, $X_{3}\propto U$, $X_{4}\propto SU$, $X_{5}\propto
TST^{2}S$, $X_{6}\propto T^{2}STS$, where $S,T,U$ are the three generators of
the group $S_{4}$. Also note that $X_{i}$’s being proportional to the
generators of $G_{f}$ is just a coincidence in this case and does not hold
generally.

Few other such examples are listed here for generalized CP symmetry
transformation with various discrete groups such as $\Delta(27)$ [350, 351],
$A_{4}$ [352], $\Delta(48)$ [353, 354], $\Delta(6n^{2})$ [355, 356],
$\Delta(96)$ [357], $A_{5}$ [358, 359], $\Delta(3n^{3})$ [360].

### 2.5 Higher Order Discrete Groups

In the above, we have discussed fixed mixed schemes (BM, TBM, GR, HG), which
are already ruled out by data and mixing schemes (TM1,TM2,CBM) which are
consistent with observations. We have also seen that smaller discrete groups
(such as $A_{4},S_{4},\Delta(27),T_{7}$) still can explain the correct mixing
with ’appropriate adjustments’ in the old flavor symmetric models. We will
discuss a few aspects of explaining lepton mixing with larger groups. In this
technique, we look for new groups that predict a different leptonic mixing
pattern. Therefore, our new method is to start with much higher order groups
($G_{f}$), which essentially breaks down to two groups $G_{e}$ and $G_{\nu}$
for charged leptons and neutrino sector. For example, when we start with a
discrete group of the order less than 1536, and the residual symmetries are
fixed at $G_{e}=Z_{3}$ and $G_{\nu}=Z_{2}\times Z_{3}$, the only surviving
group which can correct neutrino mixing are $\Delta(6n^{2})$ with $n=10$,
$(Z_{18}\times Z_{6})\times S_{3}$ and $\Delta(6n^{2})$ with $n=16$ [361]. In
a more updated study, considering discrete subgroups of $U(3)$, it has been
shown that the smallest group which satisfies 3$\sigma$ allowed range of
neutrino oscillation data is $\Delta(6n^{2})$ with $n=18$, i.e., the order of
the group is 1944 [122]. This study also proposes an analytical formula to
predict full columns of the lepton mixing matrix. For a review of similar
studies, the reader is referred to Ref. [362] and references therein.

### 2.6 Flavor Symmetry and Grand Unified Theory

To have a complete understanding of the flavor problem in particle physics,
propositions are there to combine family symmetry and Grand Unified Theory
(GUT) [363, 364, 365]. These special versions of Grand Unified Theories of
Flavor include a discrete flavor symmetry giving GUT predictions through the
associated Clebsch factors [366, 367] explaining the observed lepton mixing.
This construction can provide a novel connection between the smallest lepton
mixing angle, e.g., as discussed in Section 2.3 in the form of the KMS
relation [333, 337, 338, 339] between the reactor mixing angle $\theta_{13}$
and the largest quark mixing angle, the Cabibbo angle $\theta_{C}$

$\theta_{13}=\theta_{C}/\sqrt{2}.$ (2.113)

See other works on the subject [368, 369, 229, 340]. The symmetry choice makes
several combinations of discrete flavor symmetry and GUT possible. For
classification of such models, the readers are referred to the review [370]
and references therein for explicit models. These models, however, greatly
depend on the fields’ symmetry-breaking pattern and vacuum alignment. In
addition, such unified constructions have consequences in leptogenesis [371,
372, 373, 374, 375] and at LHC [376]. Recently, GUT frameworks have also been
implemented in modular invariance approach [377, 378, 379, 380, 381].

### 2.7 Flavor Symmetry and the Higgs Sector

There are many possibilities to extend the SM Higgs sector, e.g. by
introducing singlet scalar fields, two- (2HDM) and multi-Higgs doublets
(NHDM), and triplet multiplets. These models include charged and neutral Higgs
bosons with rich phenomenology, including modifications of the SM-like Higgs
couplings, flavour-changing neutral currents (FCNC), CP violation effects, and
in cosmology, scalar DM candidates, and modification of the phase transitions
in the early Universe. The 2HDM and NHDM doublets as copies of the SM Higgs
doublet, in analogy to fermion generations, is a natural choice. Also, many
BSM models, including gauge unification models, supersymmetry, and even string
theory constructions, inherently lead to several Higgs doublets at the
electroweak scale [382]. However, SM scalar sector extensions lead to many
free parameters. For example, the free parameters exceed one hundred in the
three-Higgs-doublet model (3HDM). One can impose additional flavour symmetry
to reduce the number of free parameters. To discuss BSM Higgs sectors in the
context of flavor symmetries, the starting point is the Yukawa Lagrangian. The
imposition of a flavor symmetry on the leptonic part of the Yukawa Lagrangian
has been discussed in Refs. [383, 370, 78]. In the SM, the application of
family symmetry is limited due to Schur’s lemma [384], which implies that for
three-dimensional mass matrices of charged leptons and neutrinos, their
diagonalization matrices are proportional to identity. Thus, the PMNS matrix
becomes trivial. This drawback can be overcome in two ways. One approach is to
break the family symmetry group by a scalar singlet, flavon [26]. Despite many
attempts, it has failed to reconstruct the PMNS matrix. In the second
approach, a non-trivial mixing can be achieved by extending the Higgs sector
by additional multiplets [385, 386, 387]. The general classification of which
symmetry groups can be implemented in the scalar sector of the 2HDM is studied
in Refs. [388, 389, 390]. Analogous analysis of an overall possible set of
finite reparametrization symmetry groups in the 3HDM is presented in Ref.
[391]. For the four-Higgs-doublet model, a systematic study of finite non-
abelian symmetry groups that can be imposed on the scalar sector is given in
Ref. [392].

Recent activities in the field of flavour symmetry and the Higgs sector are
mostly related to either to new analytical and model building studies (see
e.g. Refs. [393, 394, 395]) or to numerical verification of existing ones. For
the second approach, multi-Higgs doublet models were examined, which include
two Higgs doublets (2HDM) [384] and three Higgs doublets (3HDM) [396]. Finite,
non-Abelian, discrete subgroups of the $U(3)$ group up to order 1035 were
investigated to search for specific groups that could explain the masses and
mixing matrix elements of leptons. For both 2HDM and 3HDM models, both Dirac
and Majorana neutrinos were examined. From the model building point of view,
it was also assumed that the total Lagrangian has a full flavor symmetry, but
the Higgs potential is only form invariant [397, 398], meaning that only the
scalar potential coefficients may change while the terms in the potential do
not vary. The scan was performed with the help of the computer algebra system
GAP [399]. A limitation to subgroups with irreducible three-dimensional
faithful representations has been performed to reduce an enormous number of
evaluated subgroups significantly. The results were utterly negative for 2HDM
[384] \- for such an extension of the SM, up to considered highest dimensional
discrete groups, there is no discrete symmetry that would fully match the
masses and parameters of the mixing matrix for leptons. However, 3HDM provides
nontrivial relations among the lepton masses and mixing angles, leading to
nontrivial results [396]. Namely, some of the scanned groups provide either
the correct neutrino masses and mixing angles or the correct masses of the
charged leptons. The group $\Delta(96)$ is the smallest group compatible with
the experimental data of neutrino masses and PMNS mixing whereas $S_{4}$ is an
approximate symmetry for Dirac neutrino mixings, with parameters staying
within 3$\sigma$ of the measured $\theta_{12},\theta_{23},\theta_{13}$, and
$\delta_{\rm CP}$. Thus, phenomenological investigations based on the 3HDM
results or further theoretical studies of discrete groups beyond 3HDM are
worth future studies.

### 2.8 Modular Symmetry

Recently, in an appealing proposal to understand the flavor pattern of
fermions, the idea of modular symmetry [113, 124] has been reintroduced. In
conventional approaches, a plethora of models exist based on non-Abelian
discrete flavor symmetries and finite groups. The spectrum of the models is so
large that it is difficult to obtain a clear clue of the underlying flavor
symmetry. Additionally, there are a few major disadvantages to using this
conventional approach. Firstly, the effective Lagrangian of a typical flavor
model includes a large set of flavons. Secondly, though a vacuum alignment of
flavons essentially determines the flavor structure of quarks and leptons,
often auxiliary symmetries are needed to forbid unwanted operators from
contributing to the mass matrix. The third and most crucial disadvantage of
conventional approaches is the breaking sector of flavor symmetry, bringing
many unknown parameters, hence compromising the minimality. On the contrary,
the primary advantage of models with modular symmetry [113, 124] is that the
flavon fields might not be needed, and the flavor symmetry can be uniquely
broken by the vacuum expectation value of the modulus $\tau$. Here, the Yukawa
couplings are written as modular forms, functions of only one complex
parameter i.e., the modulus $\tau$, which transforms non-trivially under the
modular symmetry. Furthermore, all the higher-dimensional operators in the
superpotential are completely determined by modular invariance if
supersymmetry is exact; hence auxiliary Abelian symmetries are not needed in
this case. Like in the conventional model-building approach, models with
modular symmetry can also be highly predictive. The fundamental advantage is
that the neutrino masses and mixing parameters can be expressed by a few input
parameters. Modular symmetry is a property of supersymmetric theories, in
which the action $\mathcal{S}$ remains unchanged under modular transformations
[124, 400]

$\displaystyle\gamma:\tau\rightarrow\frac{a\tau+b}{c\tau+d},~{}~{}\chi^{\mathcal{I}}\rightarrow(c\tau+d)^{-k}\rho^{\mathcal{I}}(\gamma)\chi^{\mathcal{I}},$
(2.114)

here $\gamma$ is the element of the modular group where $a$, $b$, $c$, and $d$
are integers satisfying $ad-bc=1$, $\tau$ is an arbitrary complex number in
the upper complex plane, $\rho^{(I)}(\gamma)$ denotes the representation
matrix of the modular transformation $\gamma$, and $k_{I}$ is the weight
associated with the supermultiplet $\chi^{(I)}$. With this setup, the
superpotential ${\cal W}(\tau,\chi)$ is also invariant under the modular
transformation and can be expanded in terms of the supermultiplets
$\chi^{(I_{i})}$ (for $i=1,\cdots,n$) as

$\displaystyle{\cal
W}(\tau,\chi)=\sum_{n}\sum_{\\{I_{1},\dots,I_{n}\\}}Y_{I_{1}\dots
I_{n}}(\tau)\chi^{(I_{1})}\cdots\chi^{(I_{n})}\;,$ (2.115)

where the modular forms transform as

$\displaystyle Y_{I_{1}\dots
I_{n}}(\tau)\rightarrow(c\tau+d)^{k_{Y}}\rho_{Y}(\gamma)Y_{I_{1}\dots
I_{n}}(\tau).\;$ (2.116)

Here the coefficients $Y_{I_{1}\dots I_{n}}(\tau)$ take the modular forms, and
are the key elements of the modular symmetry approach. In the theory, $k_{Y}$
and $\rho_{Y}$ must satisfy $k_{Y}=k_{I_{1}}+\cdots+k_{I_{N}}$ which can be
used to constrain the charge assignments of superfields and modular forms. For
example, for the modular group $\Gamma_{4}\simeq S_{4}$, the functions
$Y(\tau)$ are modular forms of the level $N=4$ and weight $2k_{I}$ with five
linearly independent modular forms ($Y_{i}(\tau)$ for $i=1,2,\cdots,5$). These
five linearly independent forms $Y_{i}(\tau)$ arrange themselves into two
irreducible representations of $S_{4}$, a doublet $2$ and a triplet
$3^{\prime}$ which can be written as [379]

$\displaystyle Y_{\bf 2}(\tau)\equiv\left(\begin{matrix}Y_{1}(\tau)\\\
Y_{2}(\tau)\end{matrix}\right)\;,\quad
Y_{3^{\prime}}(\tau)\equiv\left(\begin{matrix}Y_{3}(\tau)\\\ Y_{4}(\tau)\\\
Y_{5}(\tau)\end{matrix}\right).\;$ (2.117)

The modular forms $Y_{2}(\tau)$ and $Y_{3^{\prime}}(\tau)$ dictate the flavor
structure of the charged lepton and neutrino mass matrices, controlling the
neutrino masses and mixing pattern. There are already several activities
adopting this approach, for a few examples see Refs. [401, 402, 403, 400, 404,
405, 406, 407, 408, 409, 410, 411, 412]. As modular symmetric models make it
possible to understand fermionicmixing, there is a scope for phenomenological
studies of mixings and possible connections with matter-antimatter asymmetry
and DM [408, 413, 414, 222]. However, testing the modular symmetry in the
context of current and future neutrino experiments like T2HK JUNO, NO$\nu$A,
and DUNE, makes it very hard to obtain robust correlations among the neutrino
oscillation parameters. A discussion in this direction can be found in Section
3.1. Nonetheless, the sum rules obtained in models based on modular symmetries
with residual symmetries may shed some light in this regard [285].

## 3 Flavor Symmetry at Intensity Frontier

Flavor symmetry can potentially provide an important link between the
outstanding puzzle of neutrino mass-mixing with various other aspects of
particle physics, cosmology, and astroparticle physics, such as neutrinoless
double-beta decay, lepton flavor violating decays, the nature of DM, baryon
asymmetry of the Universe, nonstandard interactions, etc. If a flavor symmetry
connects these seemingly uncorrelated sectors, the constraints from various
cosmological, collider, and neutrino experiments may probe the existence of
such symmetry. Such connections and correlations allow us to probe discrete
flavor symmetries at many frontiers. In this Chapter we discuss the so-called
intensity frontier experiments which include low-energy rare processes like
neutrinoless double beta decay, LFV, and long-baseline neutrino experiments.
Since these rare LNV/LFV processes are predicted to be either absent or highly
suppressed in the SM, even a single unambiguous event detection would signal
new physics.

### 3.1 Neutrino Oscillation Experiments

As discussed earlier, a wide class of models with various discrete flavor
symmetry groups ($G_{f}$) exists. With high statistics and their ability to
measure the mixing parameters more precisely, the current and future neutrino
oscillation experiments provide an excellent testing ground for the flavor
symmetry models. Such studies crucially depend on the breaking pattern of
$G_{f}$ into its residual subgroups for charged lepton sector $G_{e}$ and
neutrino sector $G_{\nu}$. For example, in Ref. [415], the authors have
studied implication of breaking of $G_{f}\times{CP}$ (with
$G_{f}=A_{4},S_{4},A_{5}$) into $G_{e}>Z_{2}$, $G_{\nu}=Z_{2}\times{CP}$ in
the context of ESSnuSB experiment [416]. Such a breaking pattern is usually
observed in the semi-direct approach of flavor model building [417]. In this
approach the PMNS matrix depends on two free parameters for
$G_{f}=A_{4},S_{4}$, and $A_{5}$ [39]. In a similar approach [418, 419]
considered the breaking pattern into $G_{e}=Z_{k},k>2$ or $Z_{m}\times Z_{n}$,
$m,n\geq 2$ and $G_{\nu}=Z_{2}\times CP$ residual symmetry for charged lepton
and neutrino sectors, respectively, in the context of ESSnuSB [420, 421], T2HK
[422], DUNE [36], and JUNO [34] experiments. In this approach the PMNS matrix
is more constrained and depends on a single free angle [343, 423, 424, 425].
In each case, distinct constraints were obtained on the neutrino oscillation
parameters $\delta_{\rm CP}$ and $\theta_{23}$. In Ref. [415] it was
demonstrated that out of the 11 (7) one-(two-)parameter models, five (five)
are compatible with the present global data at $3\sigma$.

Figure 3.1: Compatibility of one and two-parameter models with any potentially
true values of $\sin^{2}\theta_{23}$ and $\delta_{\rm CP}$ in the context of
ESSnuSB,T2HK, DUNE, and their combination. The dark (light) green, blue, and
red shaded regions represent $3\sigma$ (5$\sigma$) allowed regions for
ESSnuSB, T2HK, DUNE respectively. Similarly, the continuous (doted) cyan and
yellow lines represent 3$\sigma$ (5$\sigma$) allowed regions for ESSnuSB
combined with atmospheric data and ESSnuSB long-baseline experiments. The
stars are the best fit values. Figure taken from the arXiv version of Ref.
[418].

In Fig. 3.1, we show the compatibility of one and two-parameter models with
any potentially true values of $\sin^{2}\theta_{23}$ and $\delta_{\rm CP}$ in
the context of ESSnuSB, T2HK, DUNE, and their combination [418]. The models
are based on $A_{5}\times CP$ (Model 1.1), $A_{5}\times CP$ (Model 1.2)
$S_{4}\times CP$ (Model 1.3), $S_{4}\times CP$ (Model 1.4), $A_{5}\times CP$
(Model 1.5), $A_{5}$ (Model 2.1), $S_{4}$ (Model 2.2) and $A_{5}$ (Model 2.3)
discrete groups. A detailed discussion on the specification of each model and
their individual compatibility with a combination of experiments can be found