#  Neutrino mass matrices from localization in M-theory on $G_{2}$ orbifold

Eric Gonzalez<EMAIL_ADDRESS>Leinweber Center for Theoretical Physics,
Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
Gordon Kane<EMAIL_ADDRESS>Leinweber Center for Theoretical Physics,
Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
Khoa Dang Nguyen<EMAIL_ADDRESS>Department of Mathematics, University of
Michigan, Ann Arbor, MI 48109, USA

###### Abstract

M-theory compactified on a $G_{2}$ manifold with resolved $E_{8}$ singularity
is a promising candidate for a unified theory. The experimentally observed
masses of quarks and charged leptons put a restriction on the moduli of the
$G_{2}$ manifold. These moduli in turn uniquely determine the Dirac
interactions of the neutrinos. In the paper, we explicitly compute the Dirac
terms for neutrino mass matrix using the moduli from a localized model with
resolved $E_{8}$ singularities on a $G_{2}$ manifold. This is a novel approach
as the Dirac terms are not assumed but derived from the structure of quarks’
and charged leptons’ masses. Using known mass splittings and mixing angles of
neutrinos, we show the acceptable region for Majorana terms. We also analyse
the theoretical region for Majorana terms induced from the expectation values
of right handed neutrinos through the Kolda-Martin mechanism. The intersection
of the two regions indicates a restriction on neutrino masses. In particular,
the lightest neutrino must have small but non-zero mass. Moreover, this also
puts constraints on possible Majorana contributions from Kähler potential and
superpotential, which can be traced down to a restriction on the geometry.We
conclude that the masses of the two heavier light neutrinos are about
$0.05\text{ eV}$ and $0.009\text{ eV}$ ($0.05\text{ eV}$ and $0.05\text{
eV}$)) for normal (inverted) hierarchy. In both hierarchies, we predict the
light neutrinos are mostly Dirac type. Hence neutrino-less double-beta decay
will be small. This is a testable result in a near future. Some bounds on
heavy neutrinos are also derived.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 General setup
    2. 2.2 Yukawa couplings
  3. 3 Terms
    1. 3.1 Neutrino-neutrino mixing terms
    2. 3.2 Mixing Matter with Higgs Superfields
    3. 3.3 Mixing Matter with Gauginos
    4. 3.4 General Mass Matrix
  4. 4 VEVs of right handed neutrinos and their conjugates
    1. 4.1 Case 1: No Mixing
    2. 4.2 Case 2: Mixing with Two Families
    3. 4.3 Case 3: Mixing with Three Families
  5. 5 Mass Matrix from Neutrino Mixing
    1. 5.1 Mass Matrix Setup
  6. 6 Majorana Mass Matrix
    1. 6.1 Majorana Mass Matrix from See-Saw Mechanism
    2. 6.2 Majorana mass from VEVs of $\nu^{c}_{i}$
  7. 7 Limit for Neutrinos
    1. 7.1 Lower Bound for $\epsilon_{i}$
    2. 7.2 Normal Hierarchy Analysis
    3. 7.3 Inverted Hierarchy Analysis
  8. 8 Ratios of Dirac and Majorana contributions
  9. 9 Heavy neutrino mass
  10. 10 Conclusion

##  1 Introduction

The origin of the light left handed neutrinos in the Standard Model (SM) has
been a mystery. Cosmological probes have constrained the sum of the left
handed neutrino masses to be $\Sigma m_{\nu}<0.12$ $(0.15)\ \text{eV}$ for
normal (inverted) ordering de Salas  _et al._ (2021). Neutrino mass splittings
observed from neutrino oscillation are $\Delta m_{12}^{2}=7.6\times 10^{-5}$
$eV^{2}$, and $\Delta m_{13}^{2}=2.5\times 10^{-3}$ $eV^{2}$ de Salas  _et
al._ (2021). Moreover, the oscillation angles are about
$\theta_{12}=33.44^{\circ}$, $\theta_{23}=49.0^{\circ}$, and
$\theta_{13}=8.57^{\circ}$ de Salas  _et al._ (2021, 2020), which can be used
to explicitly compute the flavor components of mass eigenstates.

Due to none-zero mixing angles, neutrino flavor eigenstates (electron, muon,
and tau) are not the same as the neutrino mass eigenstates (simply labeled
“1”, “2”, and “3”). It is not known which of these three is the heaviest. In
analogy with the mass hierarchy of the charged leptons, the configuration with
mass 2 being lighter than mass 3 is conventionally called the “normal
hierarchy”, while in the “inverted hierarchy”, the opposite would hold.
Several major experimental efforts are underway to help establish which is
correct. Current data favors the normal hierarchy, although the confidence for
this hierarchy has been decreasing over the years de Salas  _et al._ (2021).
In this paper we will assume the normal hierarchy first, and then apply
similar framework to the inverted hierarchy.

We show that viable neutrino masses can arise within the framework of M theory
with resolved $E_{8}$ singularities, which is a highly non-trivial result,
given the constrained nature of M theory constructions. From our previous work
Gonzalez  _et al._ (2020), we numerically compute a local solution for moduli
of $G_{2}$ manifold from the experimental masses of quarks and charged
leptons. As these moduli locally control the geometry structure of the
manifold, they determine all other interactions in the model. Therefore, we
can use them to compute the Dirac terms of the neutrinos. This distinguishes
our approach from previous works with neutrino Dirac mass Acharya  _et al._
(2016); Mohapatra  _et al._ (2000); Chen  _et al._ (2013); Barr and Dorsner
(2000); Rodejohann (2004); Goh  _et al._ (2003); Damanik (2010a, b); Akhmedov
_et al._ (2000); Ma (2021) as we do not make an estimation, instead we compute
the Dirac terms explicitly.

The origin of Majorana mass terms has been complicated to realize from the
string theory perspective Acharya  _et al._ (2016). For instance, it is
possible to obtain large Majorana mass terms from instanton effects Acharya
_et al._ (2006); Blumenhagen  _et al._ (2007); Ellis  _et al._ (2015), large
volume compactification Conlon and Cremades (2007), or orbifold
compactifications of the heterotic string Buchmuller  _et al._ (2007). In this
work, we use the Kolda-Martin mechanism Kolda and Martin (1996); Costa  _et
al._ (1987) to generate vaccumn expectation values (VEVs) for the scalar
components of right handed neutrino supermultiplets and their conjugates. The
Kolda-Martin (K-M) mechanism includes effects of non-perturbative terms via
the Kähler potential. A similar approach has been done by Acharya et al for an
$SO(10)$ gauge group Acharya  _et al._ (2016). Our work expands the idea to an
explicit resolved $E_{8}$ singularities model, with three generations fitting
the experimental data for quarks and charged leptons, and computes neutrino
Dirac terms. The computed Dirac terms put constraints on the Majorana terms
through the see-saw mechanism, and the Majorana terms are generated from the
VEVs of the conjugates of right handed neutrinos.

Additionally, when the right handed neutrinos get VEVs, we inevitably generate
bilinear R-parity violating terms of the form $\epsilon_{ij}L_{i}H_{j}$. There
are many works dedicated to study these terms Barbier  _et al._ (2005); Cohen
_et al._ (2019); Diaz (1997). In general, due to the presence of large
Majorana terms, the bilinear mixing between Higgs and leptons may spoil the
Higgs physics. It is more favorable to have a small $\epsilon_{ij}$. This puts
a stringent constraints on the aforementioned VEVs. In this paper, we show
that there are solutions for the VEVs in which the mixing between leptons and
Higgses is minimal. As a result of the constraints, with a generic un-
suppressed Kähler potential coefficient, the lightest neutrino can be neither
massless nor heavy.

Futhermore, the nature of lightest neutrinos are expected to be determined in
a near future. The most important process for this effort is the neutrinoless
double-beta decay, in which the total lepton number is violated by two units.
The prediction for the Standard Model including non-perturbative effects via
K-M mechanism is that the neutrinos are Dirac. That implies neutrinoless
double beta decay will be small under those assumptions, and therefore a good
window for new physics, e.g.neutralinos (which are Majorana particles),
R-parity violating interactions, and new physics in general. If the light
neutrinos are significantly Majorana, the experiments should be able to detect
them. Otherwise, the particles must be mostly Dirac Oberauer  _et al._ (2020).
In this paper, we predict that the light particles are mostly Dirac, hence the
decay will be small.

This paper is organized as following: section 2 will briefly cover the local
model of M theory compactified on a $G_{2}$ manifold with resolved $E_{8}$
singularities Gonzalez  _et al._ (2020). Section 3 will list all of the
contributions to the neutrino mass matrix. Section 4 discusses the VEVs for
the right handed neutrinos and their conjugates through the K-M mechanism
while discussing the $\epsilon_{ij}$ problem. Section 5 contains the computed
Dirac matrix and sets up the framework for the neutrino mass matrix. In
section 6 we discuss the Majorana mass matrix from the experimental data and
from the right handed neutrino VEVs. In section 7 we deduce a limit on the
neutrino masses. We predict the masses of the mass eigenstate neutrinos,
though we cannot yet exclude one of the normal or inverted case. This will
lead to section 8 where the ratio of Dirac to Majorana components are
estimated. Finally, some insight about heavy neutrino masses are presented in
section 9.

##  2 Background

###  2.1 General setup

M-theory is compactified on a $G_{2}$ manifold which is a 7-d manifold. We are
interested in $G_{2}$ manifold as an ALE fiberation of $\mathbb{C}^{2}/\Gamma$
on a 3-d base $M_{3}$. Locally the manifold looks like

$\displaystyle M_{3}\times\widehat{\mathbb{C}^{2}/\Gamma}$ (2.1)

where $\widehat{\mathbb{C}^{2}/\Gamma}$ is some resolution of ADE singular
space $\mathbb{C}^{2}/\Gamma$ which is a quotient of complex space
$\mathbb{C}^{2}$ by a finite subgroup $\Gamma$ of $SU(2)$, resulting in a
singularity at the origin. In our case, the singularity is $E_{8}$.

There is a metric $g$, 3-form $C$, gravitino spinor $\Psi$, and $G_{2}$
structure 3-form $\omega$ on the 7-d $X$ . The 3-form $\omega$ completely
determines the metric $g$ of the $G_{2}$ manifold; therefore, $C$ ,$\Psi$ and
$\omega$ govern all of the physics in M-theory. Integrating $C$ and $\omega$
on the basis of three cycles of $X$ gives axions $a_{i}$ and moduli fields
$s_{i}$ respectively Acharya  _et al._ (2007, 2008); Acharya and Gukov (2004)
. The superfield $\Phi_{i}$ is then of the form

$\displaystyle\Phi_{i}=a_{i}+is_{i}+\text{fermionic terms}$ (2.2)

The moduli $s_{i}$ controls the size of the three cycles in $X$. When $s_{i}$
varies along the base $M_{3}$, the singularity $E_{8}$ can be deformed and
resolved to lower singularities or even completely smooth points. These
singularities have a one-to-one correspondence with the gauge group of M
theory Witten (2001). The moduli $s_{i}$ corresponding to an $E_{8}$
singularity are listed in Bourjaily (2007). Vanishing moduli correspond to the
simple roots of the gauge group. When a simple root modulus vanishes the gauge
group is enhanced to a larger one. Furthermore, chiral fermions localize in
places along these ADE singularites where non-simple-roots three cycles vanish
to form conical singularities Acharya and Witten (2001). Bourjaily et al
Bourjaily (2007) and Gonzalez et al Gonzalez  _et al._ (2020) give detailed
explanations and example computations for the presentations of the chiral
fermions. In our model, the resolved $E_{8}$ singularity results in matter as
in Table 9 and Table 10 in Bourjaily (2007). In the following, the charges are
listed in the same order as in Table 10 in Bourjaily (2007), namely in order
$a,b,c,d,Y$. Then we have

$\displaystyle E_{8}\rightarrow SU(3)\times SU(2)\times U(1)_{a}\times
U(1)_{b}\times U(1)_{c}\times U(1)_{d}\times U(1)_{Y}.$ (2.3)

The hypercharge Y has a factor of $6$ compared to the conventional hypercharge
normalization to make all the charges integer and does not effect the
calculation. The charge $a$ is identically set to zero, following Gonzalez
_et al._ (2020), to drop terms dependent on a. Then, the relevant particles
for this study are

$\displaystyle H^{d}_{1}=({\bf 1},{\bf 1})_{(1,1,2,2,-3)}$ $\displaystyle
H^{u}_{1}=({\bf 1},{\bf 1})_{(1,1,2,2,3)}$ $\displaystyle L_{1}=({\bf 1},{\bf
2})_{(1,1,-1,3,-3)}$ $\displaystyle\nu_{1}^{c}=({\bf 1},{\bf
1})_{(1,1,-1,-5,0)}$ $\displaystyle H^{d}_{2}=({\bf 1},{\bf
1})_{(1,-1,2,2,-3)}$ $\displaystyle H^{u}_{2}=({\bf 1},{\bf
1})_{(1,-1,2,2,3)}$ $\displaystyle L_{2}=({\bf 1},{\bf 2})_{(1,-1,-1,3,-3)}$
$\displaystyle\nu_{2}^{c}=({\bf 1},{\bf 1})_{(1,-1,-1,-5,0)}$ $\displaystyle
H^{d}_{3}=({\bf 1},{\bf 1})_{(0,-1,2,2,-3)}$ $\displaystyle H^{u}_{3}=({\bf
1},{\bf 1})_{(0,-1,2,2,3)}$ $\displaystyle L_{3}=({\bf 1},{\bf
2})_{(-2,0,-1,3,-3)}$ $\displaystyle\nu_{3}^{c}=({\bf 1},{\bf
1})_{(-2,0,-1,-5,0)}.$ (2.4)

The reason for $a=0$ is to allow large top quark mass Gonzalez  _et al._
(2020). Note that the simple root cycles do not shrink under this condition,
so there is no enhanced gauge group. This is similar to taking the diagonal
$U(1)_{a}\times U(1)_{b}$. Notice that a $\mu$ term $H^{u}_{i}H^{d}_{j}$ is
generally not allowed, but can be generated by the Giudice-Masiero mechanism
Chen  _et al._ (2013).

###  2.2 Yukawa couplings

The couplings for the interactions in the superpotential are given by the
instanton effect Braun  _et al._ (2019); Atiyah and Witten (2003); Acharya
_et al._ (2007); Hubner (2021); Pantev and Wijnholt (2011)

$\displaystyle Y=\frac{1}{\Lambda}e^{-V_{\text{3 cycles}}}$ (2.5)

where $\Lambda$ is a scaling factor proportional to the volume of the $G_{2}$
manifold Acharya  _et al._ (2007); Braun  _et al._ (2019). In our model, the
local moduli are not enough to determine the volume, so we treat $\Lambda$ as
a parameter. $V_{\text{3 cycles}}$ is the volume of the three cycles
stretching between the three singularities where the three particles in the
cubic terms are located. This volume is a function of the moduli

$\displaystyle
Vol(\Sigma_{ABC})=\frac{1}{2}(-v_{A}^{T}H_{A}^{-1}v_{A}-v_{B}^{T}H_{B}^{-1}v_{B}$
$\displaystyle+(v_{A}+v_{B})^{T}(H_{A}+H_{B})^{-1}(v_{A}+v_{B})).$ (2.6)

Here, $\Sigma_{ABC}$ is a three cycle covering three particle singularities A,
B, and C. Moreover, each singularity’s location on $M_{3}$ is determined by
the critical point of

$\displaystyle f=\frac{1}{2}t^{T}Ht+v^{T}t+c$ (2.7)

where $t$ is the local 3-d coordinate on $M_{3}$, $H$ is a $3\times 3$ matrix,
$v$ is a 3-vector, and c is a scalar. Using this setup, we can write down the
mass matrix for quarks and charged leptons. Then, by fitting to experimental
data, we can find the solutions for $f_{i}$’s in the local model. We will use
the fit result of b, c, d, and Y from Gonzalez  _et al._ (2020). In a full
theory on a determined $G_{2}$ manifold, the moduli should uniquely determine
every other quantity in the theory as they determine the geometry of the
manifold. In our local model, as there is some global structure we are
missing, the $f_{i}$’s will determine many quantities, such as Dirac neutrino
terms, but leave some other quantities, such as Majorana terms and the soft
breaking mechanism Acharya  _et al._ (2016), subject to tuning. Nonetheless,
most of our main results will not depend of the tuning.

##  3 Terms

###  3.1 Neutrino-neutrino mixing terms

At tree level, the contribution from the superpotential is

$\displaystyle W_{tree}\supset
y_{123}H^{u}_{1}L_{2}\nu^{c}_{3}+y_{132}H^{u}_{1}L_{3}\nu^{c}_{2}+y_{312}H^{u}_{3}L_{1}\nu^{c}_{2}+y_{321}H^{u}_{3}L_{2}\nu^{c}_{1}$
(3.1)
$\displaystyle+y_{213}H^{u}_{2}L_{1}\nu^{c}_{3}+y_{231}H^{u}_{2}L_{3}\nu^{c}_{1}+y_{333}H^{u}_{3}L_{3}\nu^{c}_{3}$
(3.2)

where $y_{ijk}$ are coupling constants computed from Eq. 2.5. There are also
contributions to the same terms from the Kähler potential with coefficients of
order $\frac{1}{m_{pl}}$ which is negligible Acharya  _et al._ (2016). Similar
to the work done by Acharya et al Acharya  _et al._ (2016) to generate a
Majorana mass term, we get contributions to the superpotential of the form

$\displaystyle W\supset\sum_{0\leq h,l,m\leq
n}\sum_{i,j,k=1,2,3}\frac{C_{h,l,m}}{m_{pl}^{2n-3}}(\nu_{i}^{c}\bar{\nu_{i}^{c}})^{h}(\nu_{j}^{c}\bar{\nu_{j}^{c}})^{l}(\nu_{k}^{c}\bar{\nu_{k}^{c}})^{m}$
(3.3)

where $m_{pl}=2.4\times 10^{18}$ GeV is the reduced Planck mass. Ideally, the
constants $C_{h,k,l}$ should be determined completely by the moduli of the
$G_{2}$ manifold. In the local model of Gonzalez  _et al._ (2020), we do not
consider $\bar{\nu}_{j}$ fields as they are controlled by global moduli beyond
the local patch. As a result, $C_{h,l,m}$ is considered a tunable parameter in
our local model.

Contributions from the Kähler potential to the same terms are expected. They
can be computed from the full Kähler potential Beasley and Witten (2002);
Lukas and Morris (2004)

$\displaystyle K=-3\log\big{(}\frac{V}{2\pi}\big{)}$ (3.4)

where $V$ is the volume of $G_{2}$ manifold. Unfortunately, the precise
dependence of the volume on the global moduli in resloved $E_{8}$ orbifold is
unknown. We assume it is not significant due to the generic suppression as in
Acharya  _et al._ (2016).

By solving D term and F term equations from the terms in equation (3.3), one
can find the VEVs for right handed neutrinos. Assuming the leading term is
quartic which we will justify later, the Majorana mass terms in the
superpotential would have the form

$\displaystyle\sum_{i,j}\frac{C_{2,1}}{m_{pl}}(\langle\bar{\nu_{i}^{c}}\rangle\langle\bar{\nu_{j}^{c}}\rangle)\nu_{i}^{c}\nu_{j}^{c}.$
(3.5)

Additionally, we also receive terms of the form $L_{i}H^{u}_{j}$ from
expression (3.1) when right handed neutrinos get VEVs. We will discuss this in
section 4. In the same manner, the Dirac mass terms emerge from Eq. 3.1 when
the Higgses get VEVs.

###  3.2 Mixing Matter with Higgs Superfields

When the scalar components of the right handed neutrino superfields
$\nu^{c}_{i}$ get VEVs, cubic terms of the form
$Y_{ijk}H^{u}_{i}L_{j}\nu^{c}_{k}$ will give rise to the mixing between
$L_{j}$ and $H^{u}_{i}$ superfields. They appear in superpotential as

$\displaystyle\mu_{ij}H^{u}_{i}L_{j}$ (3.6)

where

$\displaystyle\mu_{ij}=Y_{ijk}\langle\nu^{c}_{k}\rangle.$ (3.7)

This mixing can potentially spoil the Higgs physics, so it is generally more
favorable to consider small $\mu_{ij}$ relative to Dirac mass terms in the
neutrino mass matrix. This creates a stringent condition which requires
$\langle\nu^{c}_{k}\rangle<\langle H^{u}_{i}\rangle$ while
$\langle\bar{\nu^{c}_{k}}\rangle$ remains large due to Eq. 3.5 and the see-saw
mechanism. This will be realized in section 4.

Furthermore, the presence of R-parity violating bilinear terms (B-RPV) induces
a sub-electroweak scale (EWS) VEV on the scalar components of the $\nu$-type
fields. In our case, below the EWS, we expect all $\nu$-type scalars to
acquire a non-vanishing VEV, generating a mixing between right handed neutrino
and Higgsinos Acharya  _et al._ (2016)

$\displaystyle\epsilon_{ij}H^{u}_{i}\nu^{c}_{j}$ (3.8)

Although this can create some correction to our analysis, the contribution is
usually expected to be smaller then the Dirac mass terms Acharya  _et al._
(2016).

###  3.3 Mixing Matter with Gauginos

Finally, as in the Minimal Supersymmetric Standard Model (MSSM), the presence
of VEVs will mix some fermions with gauginos through kinetic terms, namely the
Higgsinos with $\tilde{B}_{1},\tilde{W}_{0}$ due to the Higgses VEVs Binétruy
(2006). In our case we also have $\nu^{c}$-type and $\nu$-type scalar VEVs,
which will mix gauginos with matter fermions through kinetic terms.
Explicitly, we have, for the $SU(2)$ states (left-handed neutrinos),

$\displaystyle L\supset
g^{\prime}\tilde{B}\langle\tilde{\nu_{i}}\rangle\nu_{i}+g\tilde{W}^{0}\langle\tilde{\nu_{i}}\rangle\nu_{i}+g_{b}\tilde{B}_{b}\langle\tilde{\nu_{i}}\rangle\nu_{i}+g_{c}\tilde{B}_{c}\langle\tilde{\nu_{i}}\rangle\nu_{i}+g_{d}\tilde{B}_{d}\langle\tilde{\nu_{i}}\rangle\nu_{i}$
(3.9)

where the coefficients are gauge couplings. There will be an extra (charge
$\times\sqrt{(}2)$) coefficient for each specific particle Binétruy (2006).
For the $\nu^{c}$-states, which are singlets under the SM gauge group, mixing
takes the form

$\displaystyle L\supset
g_{b}\tilde{B}_{b}\langle\tilde{\bar{\nu_{i}}}\rangle\bar{\nu_{i}}+g_{c}\tilde{B}_{c}\langle\tilde{\bar{\nu_{i}}}\rangle\bar{\nu_{i}}+g_{d}\tilde{B}_{d}\langle\tilde{\bar{\nu_{i}}}\rangle\bar{\nu_{i}}.$
(3.10)

###  3.4 General Mass Matrix

Combining all of the previous arguments, we can write down the general mass
matrix for neutrinos. Considering the basis

$\displaystyle(\tilde{B},\tilde{W}^{0},\tilde{B}_{b,c,d},H^{u0}_{1,2,3},\nu_{1,2,3},\nu_{1,2,3}^{c}),$
(3.11)

the mass matrix will be

$\displaystyle M=\begin{pmatrix}M_{\chi^{0}}^{8\times 8}&M^{8\times
6}_{\chi\nu}\\\ (M^{8\times 6}_{\chi\nu})^{T}&M^{6\times 6}_{\nu}\\\
\end{pmatrix}.$ (3.12)

where $M^{8\times 6}_{\chi\nu}$ is the mixing sub-matrix between gauginos,
Higginos, and neutrinos which is insignificant in our analysis of the
magnitude of neutrino masses. $M_{\chi^{0}}^{8\times 8}$ is the pure gauginos-
Higginos sub-matrix. Although this sub-matrix can be significant in size, the
small mixing with neutrinos makes $M_{\chi^{0}}^{8\times 8}$ irrelevant for
the magnitude of neutrino masses. It would be interesting to study their
effects in detail in future works. Thus, for the scope of this paper, we will
focus only on the neutrino sub-matrix $M^{6\times 6}_{\nu}$.

##  4 VEVs of right handed neutrinos and their conjugates

In order to explicitly write down the entries for $M^{6\times 6}_{\nu}$, in
this section we will consider a semi-general method to give VEVs to right
handed neutrinos and their conjugates.

###  4.1 Case 1: No Mixing

First, we consider a standard superpotential that gives rise to right-handed
neutrino VEVs without mixing of families

$\displaystyle\mu\nu_{i}^{c}\bar{\nu}_{i}^{c}+\frac{C_{n,0,0}}{m_{pl}^{2n-3}}(\nu_{i}^{c}\bar{\nu^{c}}_{i})^{n}$
(4.1)

where $\mu=m_{3/2}\frac{s}{m_{pl}}=\mathcal{O}(10^{3})$ GeV with
$m_{3/2}=\mathcal{O}(10^{4})$ GeV is the mass of gravitino,
$\frac{s}{m_{pl}}\equiv 0.1$ GeV is a generic moduli VEVs contribution in
Kähler potential Acharya  _et al._ (2016). The latter should be determined
completely from the value of the moduli if we have a complete description of
$G_{2}$ manifold. Unfortunately, we will use this estimated value due to our
lack of knowledge for a complete $G_{2}$ structure.

D-flat directions implies

$\displaystyle\sum_{i}q^{j}_{i}\big{(}|\langle\nu_{i}^{c}\rangle|^{2}-|\langle\bar{\nu}_{i}^{c}\rangle|^{2}\big{)}-\xi_{j}=0$
(4.2)

for $j=b,c,d,Y$ and $\xi$’s are from Fayet–Iliopoulos terms. F-flat directions
give

$\displaystyle\mu\nu_{i}^{c}+\frac{nC_{n,0,0}}{m_{pl}^{2n-3}}(\nu_{i}^{c})^{n}(\bar{\nu^{c}}_{i})^{n-1}=0$
(4.3)
$\displaystyle\mu\bar{\nu}_{i}^{c}+\frac{nC_{n,0,0}}{m_{pl}^{2n-3}}(\nu_{i}^{c})^{n-1}(\bar{\nu}^{c}_{i})^{n}=0$
(4.4)

The VEVs for $\nu_{i}^{c}$ can be problematic because they can create terms
such as $y\langle\nu^{c}\rangle H^{u}L$ which may spoil Higgs physics. On the
other hand, large VEVs for $\bar{\nu_{i}}^{c}$ are needed to generate large
Majorana terms for right handed neutrinos and hence see-saw mechanism. Thus,
we consider
$\langle\nu_{i}^{c}\rangle=\epsilon_{i}\langle\bar{\nu}_{i}^{c}\rangle$. From
F-terms, this will imply

$\displaystyle\langle\nu_{i}^{c}\rangle=\epsilon_{i}\langle\bar{\nu}_{i}^{c}\rangle=\sqrt{\epsilon_{i}}\big{(}-\frac{\mu
m_{pl}^{2n-3}}{nC_{n,0,0}}\big{)}^{\frac{1}{2(n-1)}}.$ (4.5)

Plugging this into the D-term, we get a restriction for Fayet–Iliopoulos
coefficients.

$\displaystyle\xi_{b}$
$\displaystyle=(\epsilon_{1}^{2}-1)\langle\bar{\nu}_{1}^{c}\rangle-(\epsilon_{2}^{2}-1)\langle\bar{\nu}_{2}^{c}\rangle$
(4.6) $\displaystyle\xi_{c}$
$\displaystyle=-\sum_{i=1}^{3}(\epsilon_{i}^{2}-1)\langle\bar{\nu}_{i}^{c}\rangle$
(4.7) $\displaystyle\xi_{d}$
$\displaystyle=-5\sum_{i=1}^{3}(\epsilon_{i}^{2}-1)\langle\bar{\nu}_{i}^{c}\rangle$
(4.8) $\displaystyle\xi_{Y}$ $\displaystyle=0$ (4.9)

This cannot give too much texture to Majorana terms without tuning
$C_{n,0,0}$. From the observed data, as we will see later, a rich texture is
needed. Therefore, it is inviting to consider the mixing case.

###  4.2 Case 2: Mixing with Two Families

Consider the simplest mixing Käler potential

$\displaystyle\mu\nu_{i}\bar{\nu}_{i}+\mu\nu_{j}\bar{\nu}_{j}+\frac{C_{n-k,k,0}}{m_{pl}^{2n-3}}(\nu_{i}\bar{\nu}_{i})^{n-k}(\nu_{j}\bar{\nu}_{j})^{k}.$
(4.10)

The D-flat equations are the same as in Eq. 4.2. Again we consider
$\langle\nu_{i}^{c}\rangle=\epsilon_{i}\langle\bar{\nu}_{i}^{c}\rangle$.
F-flat directions give

$\displaystyle\mu\nu_{i}+(n-k)\frac{C_{n-k,k,0}}{m_{pl}^{2n-3}}(\nu_{i})^{n-k}(\nu_{j}\bar{\nu}_{j})^{k}(\bar{\nu}_{i})^{n-k-1}=0,$
(4.11)
$\displaystyle\mu\nu_{j}+(k)\frac{C_{n-k,k,0}}{m_{pl}^{2n-3}}(\nu_{i}\bar{\nu}_{i})^{n-k}(\nu_{j})^{k}(\bar{\nu}_{j})^{k-1}=0$
(4.12) $\displaystyle\text{Interchange }\nu\leftrightarrow\bar{\nu}.$ (4.13)

which imply

$\displaystyle\langle\nu_{i}^{c}\rangle=\epsilon_{i}\langle\bar{\nu}_{i}^{c}\rangle=\sqrt{\epsilon_{i}}\Bigg{[}-\frac{\mu}{C_{n-k,k,0}}\frac{(n-k)^{k-1}}{k^{k}}m_{pl}^{2n-3}\Bigg{]}^{\frac{1}{2(n-1)}},$
(4.14)
$\displaystyle\langle\nu_{j}^{c}\rangle=\epsilon_{j}\langle\bar{\nu}_{j}^{c}\rangle=\sqrt{\epsilon_{j}}\Bigg{[}-\frac{\mu}{C_{n-k,k,0}}\frac{k^{n-k-1}}{(n-k)^{n-k}}m_{pl}^{2n-3}\Bigg{]}]^{\frac{1}{2(n-1)}}.$
(4.15)

A hierarchy for Majorana terms is possible here as right handed anti-neutrinos
from different families get different VEVs.

###  4.3 Case 3: Mixing with Three Families

We can consider the simplest mixing of three families in the Kähler potential

$\displaystyle\mu\nu_{1}^{c}\bar{\nu}_{1}^{c}+\mu\nu_{2}^{c}\bar{\nu}_{2}^{c}+\mu\nu_{3}\bar{\nu}_{3}+\frac{C_{h,k,l}}{m_{pl}^{2n-3}}(\nu_{1}\bar{\nu}_{1})^{h}(\nu_{2}\bar{\nu}_{2})^{k}(\nu_{3}\bar{\nu}_{3})^{l}.$
(4.16)

where $h+k+l=n$. The D-flat equations are the same as in Eq. 4.2. Again we
consider
$\langle\nu_{i}^{c}\rangle=\epsilon_{i}\langle\bar{\nu}_{i}^{c}\rangle$. Then,
F-term equations are

$\displaystyle\mu+\frac{hC_{h,k,l}}{m_{pl}^{2n-3}}(\nu_{1}^{c}\bar{\nu}_{1}^{c})^{h-1}(\nu_{2}^{c}\bar{\nu}_{2}^{c})^{k}(\nu_{3}\bar{\nu}_{3}^{c})^{l}=0,$
(4.17) $\displaystyle\text{Permute 3 pairs }(1,h),(2,k)\text{, and }(3,l),$
(4.18) $\displaystyle\text{permute }\nu\leftrightarrow\bar{\nu}.$ (4.19)

The solution is

$\displaystyle\langle\nu_{1}^{c}\rangle=\epsilon_{i}\langle\bar{\nu}_{i}^{c}\rangle=\sqrt{\epsilon_{i}}\Bigg{[}-\frac{\mu
h^{k+l+1}m_{pl}^{2n-3}}{C_{h,k,l}k^{k}l^{l}}\Bigg{]}^{\frac{1}{2(n-1)}},$
(4.20) $\displaystyle\text{Permute 3 pairs }(1,h),(2,k)\text{, and }(3,l).$
(4.21)

Note that in all of the above cases, in practice, we can drop the negative
signs inside the brackets as they can be absorbed as a phase in the
oscillation matrix of neutrinos. Another scenario is that one of the right
handed neutrinos completely decouples from the other two. The Kähler potential
will then be a sum of case 1 and case 2, and the solutions are the same as
case 1 and case 2.

##  5 Mass Matrix from Neutrino Mixing

###  5.1 Mass Matrix Setup

We investigate the matrix with only right handed neutrinos and left handed
neutrinos. Using the moduli values computed from quarks and charged lepton
mass in Gonzalez  _et al._ (2020) 111Note that although we can only find one
solution in Gonzalez  _et al._ (2020), it is likely not unique. Study about
the uniqueness of local solution is left for future study., we compute Dirac
mass terms from the cubic yukawa couplings at tree level

$\displaystyle W_{tree}\supset
y_{123}H^{u}_{1}L_{2}\nu^{c}_{3}+y_{132}H^{u}_{1}L_{3}\nu^{c}_{2}+y_{312}H^{u}_{3}L_{1}\nu^{c}_{2}+y_{321}H^{u}_{3}L_{2}\nu^{c}_{1}$
(5.1)
$\displaystyle+y_{213}H^{u}_{2}L_{1}\nu^{c}_{3}+y_{231}H^{u}_{2}L_{3}\nu^{c}_{1}+y_{333}H^{u}_{3}L_{3}\nu^{c}_{3}$
(5.2)

where $y_{ijk}$’s are computed from the moduli. The Yukawa couplings $y_{ijk}$
form a matrix

$\displaystyle Y=\begin{pmatrix}&0&6.93\times 10^{-7}&4.52\times 10^{-10}\\\
&7.25\times 10^{-1}&0&3.19\times 10^{-1}\\\ &2.53\times 10^{-5}&1.71\times
10^{-2}&3.22\times 10^{-2}\end{pmatrix}.$ (5.3)

When the Higgs get VEVs, the Dirac terms (in GeV) are approximately

$\displaystyle D=\begin{pmatrix}&0&2.32\times 10^{-5}&-3.28\times 10^{-8}\\\
&2.42\times 10^{1}&0&-4.93\times 10^{1}\\\ &-1.83\times 10^{-3}&-2.64\times
10^{0}&1.08\times 10^{0}\end{pmatrix}.$ (5.4)

The first two diagonal entries vanish because there are no charge invariant
terms for those. This comes down to the fact that when breaking from $E_{8}$,
particles from the same family have the same $b$ charge. If their charges are
non-zero, they cannot couple in cubic level, which is the case for the first
two families with $b$ charge $\pm 1$. The explanation for the size of the rest
is complicated as the Yukawa is related to the moduli by exponentiated inverse
matrices. However, the significant different in sizes of the entries can be
traced back to the hierarchy of the up-type quarks whose b and c charges are
the same as the neutrinos.

The Majorana contribution comes form the superpotential

$\displaystyle W\supset y\nu_{i}^{c}\bar{\nu_{i}^{c}}\nu_{j}\bar{\nu_{j}^{c}}$
(5.5)

which was discussed in Sec. 3. When neutrino conjugate terms
$\bar{\nu_{i}^{c}}$ get VEVs, terms of the form in equation 5.5 constitute the
Majorana mass matrix $RM$. The mass matrix is in the basis of
$\\{L_{1},L_{2},L_{3},\nu^{c}_{1},\nu^{c}_{2},\nu^{c}_{3}\\}$

$\begin{pmatrix}&0&D\\\ &D^{\intercal}&RM\end{pmatrix}$

where $RM$ is the right-handed Majorana matrix. Notice that $RM$ must be
symmetric. $RM$ gets large entries when right-handed neutrinos get VEVs.
Before computing the VEVs for right-handed neutrinos through a variety of
methods, we want to see if it is possible to get a sensible left-handed
neutrino hierarchy and flavor-ratio for the mass eigenstates. According to the
experimental data, orthonormal eigenvectors are approximately

$\displaystyle
V\equiv\begin{pmatrix}&v_{1}&v_{2}&v_{3}\end{pmatrix}=\begin{pmatrix}&c_{13}c_{12}&c_{13}s_{12}&s_{13}\\\
&-c_{23}s_{12}-s_{13}s_{23}c_{12}&c_{23}c_{12}-s_{13}s_{23}s_{12}&c_{13}s_{23}\\\
&s_{23}s_{12}-s_{13}c_{23}c_{12}&-s_{23}c_{12}-s_{13}c_{23}s_{12}&c_{13}c_{23}\end{pmatrix}$
(5.6)

where $c_{ij}=\cos(\theta_{ij})$, $s_{ij}=\sin(\theta_{ij})$, and we omitted
the possible phase for simplicity. We use the oscillation angles

$\displaystyle\theta_{12}=33.44^{\circ}$
$\displaystyle\theta_{13}=8.57^{\circ}$
$\displaystyle\theta_{23}=49.0^{\circ}.$ (5.7)

Assuming normal hierarchy, the eigenvalues are

$\displaystyle\Lambda\equiv\mathop{\rm
diag}\nolimits(m_{1},m_{2},m_{3})=\mathop{\rm diag}\nolimits(x,\sqrt{\Delta
m_{21}^{2}+x^{2}},\sqrt{\Delta m_{31}^{2}+x^{2}},)$ (5.8)

where $x$ is the mass of the lightest left-handed neutrino and the mass-square
differences are

$\displaystyle\Delta m_{31}^{2}=2.32\times 10^{-21}\text{ GeV}^{2}$
$\displaystyle\Delta m_{21}^{2}=7.6\times 10^{-23}\text{ GeV}^{2}$ (5.9)

Finally, we denote the remaining components of the left-handed neutrino
eigenvectors as

$\displaystyle
E\equiv\begin{pmatrix}&\epsilon_{1}&\epsilon_{2}&\epsilon_{3}\end{pmatrix}$
(5.10)

which we expect to be small but non-zero. The final eigenvector expression is

$\displaystyle\begin{pmatrix}&0&D\\\
&D^{\intercal}&RM\end{pmatrix}\begin{pmatrix}&V\\\
&E\end{pmatrix}=\begin{pmatrix}&V\\\ &E\end{pmatrix}\Lambda$ (5.11)

##  6 Majorana Mass Matrix

###  6.1 Majorana Mass Matrix from See-Saw Mechanism

Performing the explicit multiplication in Eq. 5.11, we get

$\displaystyle DE=V\Lambda\implies E=D^{-1}V\Lambda,$ (6.1) $\displaystyle
D^{\intercal}V+RME=E\Lambda\implies RMD^{-1}V\Lambda=E\Lambda-D^{\intercal}.$
(6.2)

The lightest neutrino cannot be massless, otherwise $(RMD^{-1}V-E)\Lambda$
would have a vanishing third column while $D^{\intercal}$ does not. Thus,
$\Lambda$ is invertible. Combining the two equations we get an expression for
$RM$

$\displaystyle RM=D^{-1}V\Lambda V^{-1}D-D^{\intercal}V\Lambda^{-1}V^{-1}D.$
(6.3)

Notice that as $\Lambda$ has very small diagonal entries, the second term is
dominant

$\displaystyle RM\approx-D^{\intercal}V\Lambda^{-1}V^{-1}D.$ (6.4)

For convenience, we absorb negative signs by a phase in $V$. We can
investigate the small $x$ regime by writing

$\displaystyle
RM_{ij}\approx(D^{\intercal}V)_{i1}\frac{1}{x}(V^{-1}D)_{1j}=\frac{(V^{-1}D)_{1i}(V^{-1}D)_{1j}}{x}.$
(6.5)

Thus, at small $x$, the Majorana terms will behave as a hyperbolic curve with
respect to the lightest neutrino mass $x$, and the texture of $RM$, modulo the
magnitude of $x$, is given by the first column of $V^{-1}D$ which is fixed.

When $m_{1}$ is close to the largest mass splitting, all $m_{i}$ have the same
magnitude and the approximation becomes

$\displaystyle
RM_{ij}\approx\sum_{k}(D^{\intercal}V)_{ik}\frac{1}{m_{k}}(V^{-1}D)_{kj}=\frac{\sum_{k}(V^{-1}D)_{ki}(V^{-1}D)_{kj}}{x}$
(6.6)

which is also a hyperbola with respect to $x$, although the texture of $RM$
relies on all of $V^{-1}D$ here.

To build an intuition on the magnitude of $RM$, we plug in $x=10^{-11.5}GeV$
which is about the size of the second mass splitting. The diagonalized left
handed neutrino mass matrix is $\mathop{\rm diag}\nolimits(4.9\times
10^{-5},8.6\times 10^{-6},3.2\times 10^{-12})$, absorbing negative signs by a
phase in $V$, we get

$\displaystyle RM=\begin{pmatrix}&6.6\times 10^{13}&4.6\times
10^{12}&1.4\times 10^{14}\\\ &4.6\times 10^{12}&5.8\times 10^{11}&9.5\times
10^{12}\\\ &1.4\times 10^{14}&9.5\times 10^{12}&2.8\times
10^{14}\end{pmatrix}$ (6.7)

which is a symmetric matrix as we wanted. We will see that this matrix can be
constructed with appropriate right-handed neutrino VEVs. For readability, the
above entries of this Majorana matrix are being rounded from the actually
precise values needed for the hierarchy. In fact, the hierarchy and
oscillation of left-handed neutrinos can only be achieved with a high level of
precision in the entries of $RM$. We cannot round the entries up because that
would destroy the final hierarchy and oscillation. This is a consequence of
Eq. 6.2, where the entries of $RM$ are in general much larger than those of
$\Lambda$, independent of $E$. So for the equality in Eq.6.2 to happen,
entries of $RM$ need to cancel out in $RME$ precisely to very small non-zero
numbers.

###  6.2 Majorana mass from VEVs of $\nu^{c}_{i}$

We will argue that contributions beyond the order of equation 3.5 will be
insignificant. In fact, the contribution from order $2N$ in the superpotential
is

$\displaystyle\sum_{i,j}\frac{C_{N,N,0}}{m_{pl}^{4N-3}}\langle\bar{\nu_{i}^{c}}\rangle^{N}\langle\bar{\nu_{j}^{c}}\rangle^{N}\langle\nu_{i}^{c}\rangle^{N-1}\langle\nu_{j}^{c}\rangle^{N-1}\nu_{i}^{c}\nu_{j}^{c}$
(6.8)

Plugging in the VEVs from equation 4.20, the coefficients are of the form

$\displaystyle\frac{C_{N,N,0}m_{pl}^{\frac{n-2N}{n-1}}}{\epsilon}\Bigg{[}(hk)^{l+1}h^{k-h}k^{h-k}\frac{\mu^{2}}{C_{n}^{2}}\Bigg{]}^{\frac{2N-1}{2(n-1)}}$
(6.9)

where $h,k,l$ are permuted to get other terms. Instead of separate $n_{i}$ and
$n_{j}$ for $\langle\nu_{i}^{c}\rangle$ and $\langle\nu_{j}\rangle$, we can
consider $n_{i}=n_{j}=n$ for some fractional $n$. Assume
$\frac{C_{N,1}}{C_{N+1,1}}\approx\mathcal{O}(1)$. The $h,k,l$ dependent part
is also approximately $\mathcal{O}(1)$.and the coefficient is decreasing with
respect to $N$ if

$\displaystyle\frac{\mu}{C_{N,N,0}m_{pl}}<1$ (6.10)

which implies

$\displaystyle
C_{N,N,0}>\frac{\mu}{m_{pl}}\approx\frac{10^{3}}{10^{18}}=10^{-15}\text{
Gev}.$ (6.11)

Thus, as long as the suppression coefficient is not too small, the main
contribution is always at quadric order. Henceforth, we assume
$C\in[10^{-15},1]$ which is consistent with Acharya et al Acharya  _et al._
(2016).

##  7 Limit for Neutrinos

###  7.1 Lower Bound for $\epsilon_{i}$

When the right handed neutrinos get VEVs, along with familiar Dirac terms of
the form

$\displaystyle y\langle H^{u}_{j}\rangle L_{i}\nu^{c}_{k},$ (7.1)

there are terms of the from

$\displaystyle y\langle\nu^{c}_{k}\rangle L_{i}H^{u}_{j}.$ (7.2)

which may potentially spoil the Higgs’ physics. Therefore, it is desirable for
the couplings to be smaller than those of the $\mu$ terms $\mu
H^{u}_{i}H^{d}_{j}$ (generated at electroweak scale) and the Dirac terms. As
our computed Dirac coupling $y$ is $\mathcal{O}(10)$, it is sufficient to have
the right handed neutrino VEVs smaller than those of the Higgses

$\displaystyle\langle\nu^{c}_{i}\rangle\mathop{}_{\textstyle\sim}^{\textstyle<}10^{2}.$
(7.3)

Plugging the result from 4.20 in, we get

$\displaystyle\sqrt{\epsilon_{i}}\mathop{}_{\textstyle\sim}^{\textstyle<}10^{2}\Bigg{[}\frac{\mu
h^{k+l+1}m_{pl}^{2n-3}}{C_{h,k,l}k^{k}l^{l}}\Bigg{]}^{\frac{-1}{2(n-1)}}$
(7.4)

which implies

$\displaystyle\sqrt{\epsilon_{i}}\mathop{}_{\textstyle\sim}^{\textstyle<}10^{2}\Bigg{[}\frac{\mu
m_{pl}^{2n-3}}{C_{h,k,l}}\Bigg{]}^{\frac{-1}{2(n-1)}}$ (7.5)

where we have again assumed the $k,h,l$ dependent factor to be approximately
$\mathcal{O}(1)$.

###  7.2 Normal Hierarchy Analysis

Using the upper bound for $\epsilon$ we can find a lower bound for the
Majorana mass term

$\displaystyle
RM_{ij}=\frac{C_{1,1}}{m_{pl}}\langle\bar{\nu}^{c}_{i}\rangle\langle\bar{\nu}^{c}_{j}\rangle=\frac{C_{1,1}m_{pl}^{\frac{n_{ij}-2}{n_{ij}-1}}\mu^{\frac{1}{n_{ij}-1}}}{\epsilon
C_{h,k,l}^{\frac{1}{n_{ij}-1}}}\geq 10^{4}\Bigg{[}\frac{\mu
m_{pl}^{2n_{ij}-3}}{C_{h,k,l}}\Bigg{]}^{\frac{1}{(n_{ij}-1)}}\frac{C_{1,1}m_{pl}^{\frac{n_{ij}-2}{n_{ij}-1}}\mu^{\frac{1}{n_{ij}-1}}}{C_{h,k,l}^{\frac{1}{n_{ij}-1}}}$
(7.6) $\displaystyle=\frac{10^{4}\times C_{1,1}\times
m_{pl}^{\frac{3n_{ij}-5}{n_{ij}-1}}\times\mu^{\frac{2}{n_{ij}-1}}}{C_{h,k,l}^{\frac{2}{n_{ij}-1}}}$
(7.7)

Instead of considering separate $n_{i}$ and $n_{j}$ for
$\langle\nu_{i}^{c}\rangle$ and $\langle\nu_{j}^{c}\rangle$, we again consider
$n_{i}=n_{j}=n_{ij}$ for some fractional $n_{ij}$. Following the analysis of
the previous section, we find

$\displaystyle(D^{\intercal}V\Lambda^{-1}V^{-1}D)_{ij}=RM_{ij}=\frac{C_{2,1}}{m_{pl}}\langle\bar{\nu}^{c}_{i}\rangle\langle\bar{\nu}^{c}_{j}\rangle.$
(7.8)

We will analysis the upper bound for $m_{3}$ in many scenarios and deduce
those the rest of the neutrinos accordingly. For convenience, we let
$m_{1}=\frac{1}{k}m_{3}$ and $m_{2}=\frac{1}{h}m_{3}$. Then we get

$\displaystyle\frac{1}{m_{3}}\Big{[}(D^{\intercal}V)_{i3}(V^{-1}D)_{3j}+h(D^{\intercal}V)_{i2}(V^{-1}D)_{2j}+k(D^{\intercal}V)_{i1}(V^{-1}D)_{1j}\Big{]}=RM_{ij}$
(7.10)

which implies

$\displaystyle
m_{3}=\frac{(D^{\intercal}V)_{i3}(V^{-1}D)_{3j}+h(D^{\intercal}V)_{i2}(V^{-1}D)_{2j}+k(D^{\intercal}V)_{i1}(V^{-1}D)_{1j}}{RM_{ij}}.$
(7.11)

Now, before we use inequality in Eq. 7.6 to estimate the bound, we should
consider specific limiting cases and get the best bound.

First, we consider all masses are of the same order, i.e,
$k=h=\mathcal{O}(1)$. Consider $i=j=2$ the numerator is $\mathcal{O}(10)$, and
the upper bound is

$\displaystyle
m_{3}\approx\frac{10}{RM_{11}}\leq\frac{10C_{h,k,l}^{\frac{2}{n_{2}-1}}}{10^{4}\times
C_{1,1}\times
m_{pl}^{\frac{3n_{2}-5}{n_{2}-1}}\times\mu^{\frac{2}{n_{2}-1}}}<10^{-12}$
(7.12)

for all $n_{2}\geq 2$ where we use $C\in[10^{-15},1]$. As the largest mass
splitting is $10^{-10.5}$ GeV, it rules out the possibility of equal magnitude
for neutrino masses.

A second case is when $m_{1}$ and $m_{2}$ are of the same magnitude but much
smaller then $m_{3}$. Then $m_{3}$ will be approximately the mass splitting
which is $10^{-10.5}$ GeV and $h\approx k\gg 1$. However, due to the smaller
mass splitting $10^{-11.5}$ GeV, we need $m_{1}\approx m_{2}\gg 10^{-11.5}$
which implies $h\approx k\ll 10$. If we consider $(i,j)=(1,2)$, we find

$\displaystyle
m_{3}\leq\frac{10}{RM_{12}}\leq\frac{10C_{h,k,l}^{\frac{2}{n_{12}-1}}}{10^{4}\times
C_{1,1}\times
m_{pl}^{\frac{3n_{2}-5}{n_{12}-1}}\times\mu^{\frac{2}{n_{12}-1}}}<10^{-12}.$
(7.13)

Thus, $m_{3}$ fails to satisfy the mass splitting constraint in this case.

Finally, when $m_{1}\ll m_{2},m_{3}$, the magnitude of each entry in $RM_{ij}$
is determined by the magnitude of $m_{1}$. The estimate in Eq. 7.11 will be
dominated by $k$ and provide an upper bound larger than the mass splitting.
Hence this is a viable case that agrees with experimental observation.
Nonetheless, as mentioned in Sec. 6, $m_{3}$ cannot be massless in this model.
Thus, in general, we predict the lightest neutrino to be massive but light
comparing the other two. This implies

$\displaystyle m_{3}\approx 0.05\text{ eV}$ $\displaystyle m_{2}\approx
0.009\text{ eV}$ (7.14)

###  7.3 Inverted Hierarchy Analysis

We can carry out a similar analysis for the inverted hierarchy of left handed
neutrino masses. Notice that the oscillations for each label $i$ for $m_{i}$
do not change. The only thing we need to modify is the diagonal mass matrix

$\displaystyle\Lambda\equiv\mathop{\rm
diag}\nolimits(m_{1},m_{2},m_{3})=\mathop{\rm
diag}\nolimits(x,\sqrt{x^{2}+\Delta m_{21}^{2}},\sqrt{x^{2}-\Delta
m_{31}^{2}},).$ (7.15)

As $m_{2}$ is the largest, we will mimic the previous analysis as
$m_{1}=\frac{1}{h}m_{2}$ and $m_{3}=\frac{1}{k}m_{2}$ and end up with

$\displaystyle
m_{2}=\frac{k(D^{\intercal}V)_{i3}(V^{-1}D)_{3j}+(D^{\intercal}V)_{i2}(V^{-1}D)_{2j}+h(D^{\intercal}V)_{i1}(V^{-1}D)_{1j}}{RM_{ij}}.$
(7.16)

First, we consider all masses are of the same order, i.e,$k=h=O(1)$. Then,
consider $i=j=2$. We arrive at the same conclusion of $m_{2}<10^{-12}$ which
fails to satisfy the mass splitting constraint. Unlike the normal hierarchy,
the second case where $m_{1}\approx m_{3}\ll m_{2}$ is not possible with
inverted hierarchy. As the large mass splitting $\Delta m_{32}^{2}$ requires
$m_{1}\approx m_{3}>10^{10.5}$, the small mass splitting $\Delta
m_{12}^{2}\ll\Delta m_{32}^{2}$ will imply $m_{2}\approx m_{1}$ . Again, we
arrive at the conclusion the lightest left handed neutrino, in this case
$m_{3}$, is light compared to the other two. This implies

$\displaystyle m_{1}\approx m_{2}\approx 0.05\text{ eV}$ (7.17)

The results from both hierarchies are consistent with the current knowledge of
light neutrinos, for instance, the work of Gonzalo et al Gonzalo  _et al._
(2021).

##  8 Ratios of Dirac and Majorana contributions

It is also important to study the percentage of Dirac and Majorana components
in the three light neutrinos. From Eq. 6.1, we have

$\displaystyle E=D^{-1}V\Lambda$ (8.1)

Following the previous discussion, for the normal hierarchy, we use

$\displaystyle\Lambda=\mathop{\rm diag}\nolimits(x,0.009\times
10^{-9},0.05\times 10^{-9})$ (8.2)

where $x$ is nonzero and smaller then $10^{-12}$ GeV. Then, varying the value
of $x$, element of $E$ is of order at most $\mathcal{O}(10^{-6})$. Recall that
$V$ is chosen to be orthonormal and hence of order $\mathcal{O}(1)$. Thus, the
ratio of Majorana components to Dirac components is less than $10^{-6}$.

Similarly, we consider inverted hierarchy with

$\displaystyle\Lambda=\mathop{\rm diag}\nolimits(0.05\times
10^{-9},0.005\times 10^{-9},x)$ (8.3)

where $x$ is nonzero and smaller then $10^{-12}$ GeV. Although the Majorana
components look slightly different, we arrive at the same conclusion that the
ratio of Majorana components to Dirac components is less than $10^{-6}$.

This would predict light neutrinos are mostly Dirac type. Theoretically, this
is consistent with the current works, such as that of Gonzalo et al Gonzalo
_et al._ (2021). Experimentally, that implies they behave as four component
Dirac spinors, and the double beta decay would be small Oberauer  _et al._
(2020). There are several experiments in progress of testing this Bilenky and
Giunti (2015).

##  9 Heavy neutrino mass

We can also extract some information about heavy neutrinos by considering the
eigenvector equations similar to Eq. 5.11

$\displaystyle\begin{pmatrix}&0&D\\\
&D^{\intercal}&RM\end{pmatrix}\begin{pmatrix}&V^{\prime}\\\
&E^{\prime}\end{pmatrix}=\begin{pmatrix}&V^{\prime}\\\
&E^{\prime}\end{pmatrix}\Lambda^{\prime}$ (9.1)

where $\Lambda^{\prime}$ is the diagonal mass matrix of the heavy neutrinos.
In contrast with light neutrinos, We expect $V^{\prime}$ to be small compared
to $E^{\prime}$. Similarly to light neutrino case, we can pick $E$ to be
orthonormal.This would imply

$\displaystyle DE^{\prime}=V^{\prime}\Lambda^{\prime}$ (9.2) $\displaystyle
D^{\intercal}V^{\prime}+RME^{\prime}=E^{\prime}\Lambda^{\prime}$ (9.3)

As both $D$ and $V^{\prime}$ are small compared to $RM$ and $E^{\prime}$
respectively, we have the estimation

$\displaystyle RME^{\prime}\approx E^{\prime}\Lambda^{\prime}$ (9.4)

or

$\displaystyle E^{\prime-1}RME^{\prime}\approx\Lambda^{\prime}.$ (9.5)

As $E^{\prime}$ is orthonormal, we conclude that $\Lambda^{\prime}$ is
approximately the diagonalized matrix of $RM$. This means the lower bound for
the heaviest eigenvalue is

$\displaystyle\lambda_{max}\geq\frac{\mathop{\rm tr}(RM)}{3}\gtrsim
10^{14}\hskip 28.45274ptGeV$ (9.6)

Using this, we can estimate the upper bound for the lightest of the heavy
neutrinos.

$\displaystyle\prod_{i=1,2,3}\lambda^{heavy}_{i}=\det{RM}.$ (9.7)

Hence,

$\displaystyle\lambda^{heavy}_{min}\leq\big{(}\frac{\det(RM)}{\lambda_{max}}\big{)}^{\frac{1}{2}}.$
(9.8)

$\det(RM)$ is inversely proportional to the mass of the lightest neutrino, so
in general $\det(RM)$ is not bounded above when the lightest neutrino becomes
lighter and lighter. On the other hand, in the heaviest case, the lightest
neutrino is about $10^{-11.5}$ GeV, $\det(RM)$ is about
$\mathcal{O}(10^{39})$. Then, the upper bound for the lightest heavy neutrino
is

$\displaystyle\lambda^{heavy}_{min}\leq 10^{12.5}\hskip 28.45274ptGeV$ (9.9)

##  10 Conclusion

In this paper, our primary goal is to analyze the mass matrix of neutrinos
using the result from a localized model of M theory compactified on $G_{2}$
manifold with resolved $E_{8}$ singularity Gonzalez  _et al._ (2020). We learn
in this work that the neutrinos originate in the need for the full content of
the representations of the resolved $E_{8}$ singularity. Similar to the work
of Acharya et al Acharya  _et al._ (2016), there are two main contributions:
pure neutrino mixing, and neutralinos and higginos mixing with neutrinos. We
argue that the former is more significant and therefore the focus of the
paper.

Dirac terms of the neutrino mass matrix are explicitly computed from the
moduli of the localized model on $G_{2}$ manifold. We computed the
contribution on the cubic level. The texture of the neutrino masses is highly
hierarchical as a result of the correlation to hierarchy from the up-type
quark. From experimental data of the mixing angles and mass splittings,
assuming the normal ordering, we can use the Dirac terms to compute the
Majorana mass matrix as a function of the lightest neutrino mass.

The Kolda-Martin mechanism is the main theoretical tool to generate Majorana
terms in this paper. In this picture, the right handed neutrinos ( and their
anti particles) get VEVs and generate Majorana masses through quadric terms.
The VEVs along with the Dirac terms and experimental data oscillation angles
create an upper bound for the masses of left handed neutrinos. Considering
this upper bound in both scenarios of normal and inverted hierarchies, we
conclude that the last neutrino should always be light comparing the the other
two families regardless of the choice of hierarchy. However, the model and the
computed Dirac terms generally forbid the lightest neutrino to be massless.
The very light mass of the one of the neutrinos implies that the other two
left handed neutrinos have masses about $0.05\text{ eV}$ and $0.009\text{ eV}$
($0.05\text{ eV}$ and $0.05\text{ eV}$)) for normal (inverted) hierarchy.
Moreover, the ratio of Majorana components to Dirac components is less than
$10^{-6}$ for the three light neutrino in both hierarchy scenarios. This leads
to the prediction that in both hierarchies, the light neutrinos are mostly
Dirac type. Hence neutrinoless double-beta decay will be small. This is a
testable result in a near future. On the other hands, we achieve some
restriction on heavy neutrinos. The bounds are not stringent enough to make a
testable prediction.

For future work, we expect more predictive results when we understand better
about the contributions from the global structures which determine all the
coefficients, including those being tunable in our local theory. Locally, it
is also intriguing to explore the uniqueness of the solution. If other
solutions exist, it is interesting to see the implication on the physics,
especially the neutrinos. As our work can be repeated for other solutions in a
relatively straightforward way, it is inviting to examine a large class of
solutions using bigger computational power.

## References

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