# Prediction of non-SUSY AdS conjecture on the lightest neutrino mass
revisited

Cao H. Nam<EMAIL_ADDRESS>Phenikaa Institute for Advanced
Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha
Dong, Hanoi 12116, Vietnam

###### Abstract

We study the constraint of the non-SUSY AdS conjecture on the three-
dimensional vacua obtained from the compactification of the Standard Model
coupled to Einstein gravity on a circle where the three-dimensional components
of the four-dimensional metric are general functions of both non-compact and
compact coordinates. We find from studying the wavefunction profile of the
three-dimensional metric in the compactified dimension that the radius of the
compactified dimension must be quantized. Consequently, the three-dimensional
vacua are constrained by not only the non-SUSY AdS conjecture but also the
quantization rule of the circle radius, leading to both upper and lower bounds
for the mass of the lightest neutrino as $\sqrt{2}\leq
m_{\nu}/\sqrt{\Lambda_{4}}<\sqrt{3}$ where $\Lambda_{4}\simeq 5.06\times
10^{-84}$ GeV2 is the observed cosmological constant. This means that the
lightest neutrino should have a mass around $10^{-32}$ eV or it would be well-
approximately massless. With this prediction, we reconstruct the light
neutrino mass matrix that is fixed by the neutrino oscillation data and in
terms of three new mixing angles and six new phases for both the normal
ordering and inverted ordering. In the situation that the light neutrino mass
matrix is Hermitian, we calculate its numerical value in the $3\sigma$ range.

##  I Introduction

The observations of the neutrino oscillations have indicated the non-zero mass
of the neutrinos but they only provide information about the difference of the
squared neutrino masses given by Esteban2020

$\displaystyle\Delta m^{2}_{21}$ $\displaystyle=$ $\displaystyle
7.39^{+0.21}_{-0.20}\times 10^{-5}\text{eV}^{2},$ $\displaystyle\Delta
m^{2}_{31}$ $\displaystyle=$ $\displaystyle 2.525^{+0.033}_{-0.032}\times
10^{-3}\text{eV}^{2}\ \ \ \ \text{Normal ordering (ON)},$ (1)
$\displaystyle\Delta m^{2}_{32}$ $\displaystyle=$
$\displaystyle-2.512^{+0.034}_{-0.032}\times 10^{-3}\text{eV}^{2}\ \ \ \
\text{Inverted ordering (IN)}.$

In this way, the present neutrino experiments cannot tell us the mass of the
lightest neutrino by which the mass of three neutrino generations can be
determined. The determination of the neutrino masses is very important to
understand the role of neutrinos in the evolution of the universe and the
formation of large structures (like galaxies) because the mass density of the
universe would obtain significant contributions from the very light neutrinos
Lesgourgues2012 .

It is interesting in recent years that the consistency of the low-energy
effective theories with quantum gravity, related to the swampland program
Vafa2005 ; Ooguri2007 ; Palti2019 , can give constraints on the mass of the
lightest neutrino. The Standard Model (SM) coupled to Einstein gravity is
clearly a good effective theory in the low-energy regime and hence it must be
consistent with quantum gravity. This means that the effective field theories
that are derived from its dimensional reduction are too. On the other hand,
the lower-dimensional vacua in the landscape of the SM coupled to Einstein
gravity would be subject to the constraint of the swampland conjectures. In
Refs. Martin-Lozano2017 ; Hamada2017 , by considering the compactification of
the SM coupled to Einstein gravity on a circle Arkani-Hamed2007 , the
constraint of the non-SUSY AdS conjecture (stating that stable non-SUSY AdS
vacua are inconsistent with quantum gravity or belong to the swampland)
imposes an upper bound on the mass of the lightest neutrino as
$m_{\nu_{1(3)}}\lesssim(M^{2}_{\text{Pl}}\Lambda_{4})^{1/4}\sim 10^{-3}$ eV
where $M_{\text{Pl}}\sim 10^{19}$ GeV is the Planck energy scale and
$\Lambda_{4}\simeq 5.06\times 10^{-84}$ GeV2 is the observed cosmological
constant Tanabashi2018 .

In this work, we will revisit the circle compactification of the SM coupled to
Einstein gravity and the application of the non-SUSY AdS conjecture to the
corresponding three-dimensional vacua where unlike the previous investigations
in the literature we do not restrict the three-dimensional components of the
four-dimensional metric to be the functions of the non-compact coordinates
only. But, we consider that they are general functions of both non-compact and
compact coordinates. By using the most general setting of the circle
compactification described by a U(1) principal bundle, we expand the Einstein-
Hilbert (EH) action plus a positive cosmological constant in terms of the
three-dimensional metric, U(1) gauge vector, and radion fields, represented in
Sec. II. From studying the wavefunction profile of the three-dimensional
metric in the compactified dimension, we show in Sec. III that the radius of
the compactified dimension must be quantized due to its circle topology. In
Sec. IV, we indicate that the quantization rule of the size of the
compactified dimension and the constraint of the non-SUSY AdS conjecture on
the three-dimensional vacua predict a mass range for the lightest neutrino as
$\sqrt{2}\leq m_{\nu}/\sqrt{\Lambda_{4}}<\sqrt{3}$, which means that the mass
of the lightest neutrino should be around $10^{-32}$ eV or it would be well-
approximately massless. From this prediction, in Sec. V we reconstruct the
light neutrino mass matrix which can be given by the numerical values in the
$3\sigma$ range if the light neutrino mass matrix is Hermitian.

##  II  Circle compactification

We start introducing the four-dimensional action describing the SM coupled to
Einstein gravity as follows

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{\text{Pl}}}{2}\int
d^{4}x\sqrt{-g_{4}}\left[\mathcal{R}^{(4)}-\Lambda_{4}\right]+S_{\text{SM}},$
(2)

where $\mathcal{R}^{(4)}$ is the Ricci scalar and $S_{\text{SM}}$ denotes the
action of the SM. As we will see later, only the lightest field of the SM
which is the lightest neutrino plays a role under the constraint of the non-
SUSY AdS conjecture because it significantly contributes to the radion
potential whose minimum would determine a landscape of the three-dimensional
vacua. In this sense, the relevant action of the SM is given by

$\displaystyle S_{\text{SM}}=\int
d^{4}x\sqrt{-g_{4}}\bar{\nu}_{1(3)}\left(i\gamma_{\alpha}{e_{\alpha}}^{\mu}D_{\mu}-m_{\nu_{1(3)}}\right)\nu_{1(3)}+\cdots,$
(3)

where ${e_{\alpha}}^{\mu}$ and $D_{\mu}$ denote the vierbein and the covariant
derivative, respectively, and the ellipsis refers to the rest of the SM fields
which can be ignored under the constraint of the non-SUSY AdS conjecture.

Now we consider the compactification of the SM coupled to Einstein gravity on
a circle $S^{1}$ with $x^{3}$ and $x^{3}+2\pi$ identified where the setting of
this compactification is described in the most general way by a principal
bundle with the typical fiber U(1) Coquereaux1988 ; Bailin1987 ; Overduin1997
; Nam2019 ; Nam2021 . Using this setting, we can write the metric endowed on
the spacetime compactified on the circle $S^{1}$ as follows

$\displaystyle ds^{2}=g_{ij}dx^{i}dx^{j}+R^{2}\left[dx^{3}+\kappa
A_{i}dx^{i}\right]^{2}.$ (4)

Here $g_{ij}$ is the three-dimensional metric, $A_{i}$ is the connection or
the gauge vector field on the U(1) principal bundle and transforms under the
general coordinate transformations as $A_{i}\rightarrow
A_{i}-\partial_{i}\epsilon(x^{j})/\kappa$, $R$ is the radion (or dilaton)
field whose vacuum expectation value would fix the radius of the $S^{1}$
fibre, $i,j=0,1,2$ are the indices of the non-compact coordinates, and
$\kappa$ is the gauge coupling. Note that, the three-dimensional field
$g_{ij}$ and the radion field $R$ are in general dependent on both non-compact
and compact coordinates.

Replacing the ansatz (4) into the Ricci scalar $\mathcal{R}^{(4)}$, the EH
action plus the cosmological constant gives Nam2023-2

$\displaystyle S_{\text{EH}}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{\text{Pl}}}{2}\int
d^{4}x\sqrt{-g_{4}}\left[\hat{\mathcal{R}}+\frac{1}{4R^{2}}\left(\partial_{3}g^{ij}\partial_{3}g_{ij}+g^{ij}g^{kl}\partial_{3}g_{ij}\partial_{3}g_{kl}\right)-\frac{R^{2}\kappa^{2}}{4}F_{ij}F^{ij}-\Lambda_{4}\right],$
(5)

where $\hat{\mathcal{R}}$ and the field strength tensor $F_{ij}$ of the U(1)
gauge field are given by

$\displaystyle\hat{\mathcal{R}}$ $\displaystyle\equiv$ $\displaystyle
g^{ij}\left(\hat{\partial}_{k}\hat{\Gamma}^{k}_{ji}-\hat{\partial}_{j}\hat{\Gamma}^{k}_{ki}+\hat{\Gamma}^{k}_{ji}\hat{\Gamma}^{l}_{lk}-\hat{\Gamma}^{k}_{li}\hat{\Gamma}^{l}_{jk}\right),$
(6) $\displaystyle\hat{\Gamma}^{k}_{ij}$ $\displaystyle\equiv$
$\displaystyle\frac{g^{kl}}{2}\left(\hat{\partial}_{i}g_{jl}+\hat{\partial}_{j}g_{il}-\hat{\partial}_{l}g_{ij}\right),$
(7) $\displaystyle F_{ij}$ $\displaystyle=$
$\displaystyle\partial_{i}A_{j}-\partial_{j}A_{i}$ (8)

with $\hat{\partial}_{i}\equiv\partial_{i}-\kappa A_{i}\partial_{3}$. A novel
point that appears in the gravity action (5 ) upon the circle compactification
is the presence of the second term. This is due to the fact that in the
calculation we have not restricted the three-dimensional field $g_{ij}$ to be
the functions of $x^{i}$ only as considered in the previous investigations
about the circle compactification of the SM coupled to Einstein gravity
Martin-Lozano2017 ; Hamada2017 ; Arkani-Hamed2007 ; Arnold2010 ; Fornal2011 ;
Valenzuela2017 ; Gonzalo2018a ; Gonzalo2018b . As we will indicate late, this
novel term would lead to a significant change to the constraint of the
lightest neutrino mass. In addition, it should note that no kinetic term
related to the first-order derivatives of the radion filed appear in the
expansion of the Ricci scalar $\mathcal{R}^{(4)}$ because the $S^{1}$ fiber
has a zero curvature. However, the kinetic term of the radion field will
appear in Einstein frame obtained by rescaling the three-dimensional metric.

##  III  Quantization of the circle radius

We consider the theory in the vacuum $\langle R\rangle=\text{constant}$ which
corresponds to the minimum of the radion potential studied in the next
section. The radion stabilization $\langle R\rangle=\text{constant}$ would
lead to $F_{ij}F^{ij}=0$ or $\langle A_{i}\rangle=0$ which is derived from the
equations of motion for the U(1) gauge field $A_{i}$ and the radion field $R$
Overduin1997  (also, see Appendix B of Ref. Nam2023-2 ). In this vacuum, we
find the equations of motion for the three-dimensional metric $g_{ij}$ which
are obtained from the variation of the action (5) in $g_{ij}$ as follows

$\displaystyle\mathcal{R}-6\Lambda_{4}-\frac{1}{4R^{2}}\left[8g^{ij}\partial^{2}_{3}g_{ij}+5\partial_{3}g^{ij}\partial_{3}g_{ij}+9\left(g^{ij}\partial_{3}g_{ij}\right)^{2}\right]$
$\displaystyle=$ $\displaystyle 0,$ (9)

where

$\displaystyle\mathcal{R}$ $\displaystyle\equiv$ $\displaystyle
g^{ij}\left(\partial_{k}\Gamma^{k}_{ji}-\partial_{j}\Gamma^{k}_{ki}+\Gamma^{l}_{ji}\Gamma^{k}_{kl}-\Gamma^{l}_{ki}\Gamma^{k}_{jl}\right),$
(10) $\displaystyle\Gamma^{k}_{ij}$ $\displaystyle\equiv$
$\displaystyle\frac{g^{kl}}{2}\left(\partial_{i}g_{lj}+\partial_{j}g_{ki}-\partial_{l}g_{ij}\right).$
(11)

It is interesting that Eq. (9) can be solved by separating the variables as
follows

$\displaystyle g_{ij}(x^{i},x^{3})=\chi(x^{3})g^{(3)}_{ij}(x^{i}),$ (12)

where $g^{(3)}_{ij}(x^{i})$ is identified as the metric defining the line
element of the effective three-dimensional spacetime and $\chi(x^{3})$
describes its wavefunction profile in the compactified dimension. We
substitute Eq. (12) into Eq. (9), then we derive the following equations

$\displaystyle\mathcal{R}^{(3)}$ $\displaystyle=$ $\displaystyle
6\Lambda_{3},$ (13)
$\displaystyle\chi^{\prime\prime}+\frac{11}{4}\frac{\chi^{\prime
2}}{\chi}+\frac{\Lambda_{4}}{R^{-2}}\chi$ $\displaystyle=$
$\displaystyle\frac{\Lambda_{3}}{R^{-2}},$ (14)

where $\mathcal{R}^{(3)}$ is the Ricci scalar of the effective three-
dimensional spacetime whose geometry is determined by Eq. (13). The source of
the geometry of the effective three-dimensional spacetime is a cosmological
constant $\Lambda_{3}$ originating from the dynamics of the three-dimensional
metric in the compactified dimension. By solving Eq. (14), one can find the
wavefunction profile of the three-dimensional metric in the compactified
dimension. It is important to note here that the equation for the wavefunction
profile is nonlinear due to the nonlinear nature of the three-dimensional
metric. This means that the solution for the three-dimensional metric should
not be expressed as a linear combination of the partial solutions.

Due to the presence of the nonlinear term (which is the second term on the
left-hand side) in Eq. (14), it is not easy to find an analytical solution for
the wavefunction profile of the three-dimensional metric. But, in the case of
$\Lambda_{3}/R^{-2}\ll 1$ [which can be realized from Eq. (16) and Table. 1],
Eq. (14) can be perturbatively solved in the order of $\Lambda_{3}/R^{-2}$. At
the leading order, we can find an analytical solution for the wavefunction
profile of the three-dimensional metric as follows

$\displaystyle\chi(x^{3})=\left[\frac{1}{2}\left\\{1+\cos\left(R\sqrt{15\Lambda_{4}}x^{3}\right)\right\\}\right]^{2/15}.$
(15)

Importantly, the circle topology requires that the wavefunction profile of the
three-dimensional metric must be periodic with the period of $2\pi$ as
$\chi(x^{3})=\chi(x^{3}+2\pi)$. This lead to a quantization condition for the
circle radius and cosmological constant as follows

$\displaystyle R=\sqrt{\frac{1}{15\Lambda_{4}}}n.\ \ \text{with}\ \
n=1,2,3,\cdots.$ (16)

In this way, the nontrivial behavior of the wavefunction profile of the three-
dimensional metric in the compactified dimension requires both the radius of
the compactified dimension and the four-dimensional cosmological constant to
be quantized. In other words, their value is not arbitrary but only obtains
the discrete values satisfying the quantization rule (16). Such a quantization
has been used to interpret the radiative stability of tiny observed
cosmological constant under the quantum corrections Nam2023-2  as well as to
construct microscopic configurations for observed black holes Nam2023-4

##  IV  Constraint of non-SUSY AdS conjecture

The compactification of the SM coupled to GR on the circle $S^{1}$ would lead
to a landscape of vacua which corresponds to the extrema of the radion
potential. Here, the radion potential is generated by the cosmological
constant and the one-loop correction of the light particles. Depending on the
three-dimensional cosmological constant $\Lambda_{3}$ and the mass of the
light particles, this landscape of vacua includes both the dS and AdS
geometries which would be an object for the constraint of the non-SUSY AdS
conjecture.

The three-dimensional effective action of Einstein gravity derived upon the
circle compactification in the Einstein frame is given by

$\displaystyle S_{\text{3D}}$ $\displaystyle\supset$ $\displaystyle\int
d^{3}x\sqrt{-g_{3}}\left[\frac{M_{3}}{2}\mathcal{R}^{(3)}-2M_{3}\left(\frac{\partial_{\mu}R}{R}\right)^{2}-M_{3}\left(\frac{r}{R}\right)^{2}\Lambda_{3}\right],$
(17)

where we have rescaled the three-dimensional metric as
$g^{(3)}_{ij}\rightarrow\Omega^{-2}g^{(3)}_{ij}$ with $\Omega=R/r$ (the
parameter $r$ is introduced to keep the rescaled three-dimensional metric
dimensionless and it would be fixed equal to the vacuum expectation value of
the radion field), $g_{3}\equiv\text{det}\left[g^{(3)}_{ij}\right]$, and the
three-dimensional Planck energy scale is identified as

$\displaystyle M_{3}$ $\displaystyle\equiv$ $\displaystyle
rM^{2}_{\text{Pl}}\int^{\pi}_{-\pi}dx^{3}\chi^{1/2}$ (18) $\displaystyle=$
$\displaystyle\frac{2\sqrt{\pi}\Gamma(17/30)}{\Gamma(16/15)}rM^{2}_{\text{Pl}}.$

The last term in the action (17) is the tree-level potential of the radion
field generated by the dynamics of the three-dimensional metric along the
compactified dimension. In addition, the one-loop quantum corrections would
contribute to the radion potential as

$\displaystyle V_{\text{1L}}(R)$ $\displaystyle=$
$\displaystyle\sum_{i}(-1)^{s_{i}}n_{i}R\left(\frac{r}{R}\right)^{3}\rho_{i}(R)\int^{\pi}_{-\pi}dx^{3}\chi^{\frac{3}{2}}.\
\ $ (19)

Here $s_{i}$ is equal to $0(1)$ for the fermions(bosons), $n_{i}$ is the
number of degrees of freedom corresponding to the $i$-th particle, and the
Casimir energy density with respect to the $i$-th particle is given by Arkani-
Hamed2007

$\displaystyle\rho_{i}(R)=\sum^{\infty}_{n=1}\frac{2m^{4}_{i}}{(2\pi)^{2}}\frac{K_{2}(2\pi
nm_{i}R)}{(2\pi nm_{i}R)^{2}},$ (20)

where $m_{i}$ and $K_{2}(z)$ are the mass of the $i$-th particle and the
modified Bessel function, respectively. It should be noted here that due to
the function $K_{2}(z)$ suppressed for $z\ll 1$ the particles with their mass
which is much larger than $R^{-1}$ do not contribute significantly to the one-
loop term of the radion potential and hence we can ignore their contribution.
On the other hand, only the light degrees of freedom contribute significantly
to $V_{\text{1L}}(R)$.

The radion potential $V(R)$ thus is a sum of the tree and loop level
contributions, which is expanded in terms of $m_{i}R$ for $m_{i}R\ll 1$ as

$\displaystyle\frac{V(R)}{2\sqrt{\pi}r^{3}}$ $\displaystyle\simeq$
$\displaystyle\left[\frac{\Gamma(17/30)}{\Gamma(16/15)}\frac{M^{2}_{P}\Lambda_{3}}{R^{2}}+\frac{1}{16\pi^{2}}\frac{\Gamma(7/10)}{\Gamma(6/5)}\frac{1}{R^{6}}\right.$
(21)
$\displaystyle\left.\times\sum_{i}(-1)^{s_{i}}n_{i}\left\\{\frac{1}{90}-\frac{(m_{i}R)^{2}}{6}+\frac{(m_{i}R)^{4}}{48}\right\\}\right].$

A stable non-SUSY AdS vacuum would always be formed for the case of
$\Lambda_{3}\leq 0$ Gonzalo2021 . But, according to the non-SUSY AdS
instability conjecture the stable non-SUSY AdS vacua are inconsistent with the
UV embedding in quantum gravity or they belong to the swampland. Because the
SM coupled to Einstein gravity must be consistent with quantum gravity, the
radion potential must develop a non-AdS vacuum which can be guaranteed if the
three-dimensional cosmological constant $\Lambda_{3}$ is positive and there
are enough fermionic degrees of freedom that are sufficiently light.111In
order to apply the non-SUSY AdS conjecture here, we assume the absence of the
UV non-perturbative instabilities transferred to the effective three-
dimensional theory which can be avoided if the bubble radius is larger than
the $\text{AdS}_{3}$ curvature radius.The experimental value
$\Lambda_{4}\simeq 5.06\times 10^{-84}$ GeV2 Tanabashi2018  and Eq. (16) lead
to $R\gtrsim 1.15\times 10^{41}$ GeV-1. This means that only the lightest
neutrino can contribute to the radion potential if it is light enough.
However, even with the possible maximum degrees of freedom of the lightest
neutrino which are four if it is a Dirac particle, it is impossible to
generate a non-AdS vacuum. This is due to the fact that the contributions
coming from the massless bosonic degrees of freedom in the SM coupled to
Einstein gravity (which are two from graviton and two from photon) would make
the radion potential drop for small $R$. This thus implies the existence of
sufficiently light fermionic degrees of freedom beyond the SM with the upper
bound for the mass in the order of $\sqrt{\Lambda_{4}}$. With such a tiny
upper bound for the mass, we can consider an additional massless Dirac
fermion.

In Fig. 1, we depict the behavior of the radion potential in terms of the
radius of the compactified dimension for various values of the lightest
neutrino mass.

---
Figure 1: The scaled radion potential $\bar{V}(R)\equiv
V(R)/(2\sqrt{\pi}r^{3}\Lambda^{6}_{4})$ versus the scaled $S^{1}$ radius
$R\sqrt{\Lambda_{4}}$ for various values of $m_{\nu}/\sqrt{\Lambda_{4}}$ with
$M^{2}_{\text{Pl}}\Lambda_{3}/\Lambda^{2}_{4}=0.1$. The red and orange curves
correspond to the AdS vacuum and runaway dS vacuum, respectively, whereas the
blue and purple curves lead to the dS vacua.

In the region of the large radius, the contribution of the three-dimensional
cosmological constant is dominant, positive, and decreases with the growth of
$R$, hence the radion potential will go to zero from above when $R$ approaches
infinitive. On the contrary, in the region of the small radius, the Casimir
contribution of the bosons and fermions is dominant where the bosons and
fermions contribute negatively and positively, respectively. However, due to
the fermionic degrees of freedom larger than the bosonic one, the radion
potential will approach infinity for $R\rightarrow 0$. In particular, a
minimum or runaway behavior (runaway dS vacua) is formed, depending on the
mass of the lightest neutrino.

In the previous works Martin-Lozano2017 ; Hamada2017 , there is no constraint
on the runaway behavior of the radion potential. On the other hand, no lower
bound on the mass of the lightest neutrino is imposed, only the upper bound is
required to guarantee the absence of the stable non-SUSY AdS vacuum.
Interestingly, in the present work, the runaway behavior of the radion
potential must be excluded by the quantization of the circle compactification.
Indeed, in the large field region in which the gradients of the radion
potential are very small, or in other words the radion field exhibits ultra-
slow-rolling behavior, we can study the theory with the radius of the
compactified dimension kept fixed. In this situation, as found above the
quantization condition (16) requires that the radius of the compactified
dimension only obtains the discrete values. However, for the runaway behavior
the spectrum of the radius of the compactified dimension is continuous and
hence it would violate the quantization condition (16). This would allow us to
impose a lower on the mass of the lightest neutrino as given by Eq. (24).

A minimum will develop in the situation in which the mass of the lightest
neutrino is above the lower bound. From the minimum condition $\partial
V(R)/\partial R=0$, we derive an expression for the value of $R$ corresponding
to the minimum of the radion potential as follows

$\displaystyle
R_{\text{min}}=\frac{2}{\left[20m^{2}_{\nu}+\sqrt{30(13m^{4}_{\nu}-64\pi^{2}\alpha
M^{2}_{\text{Pl}}\Lambda_{3})}\right]^{1/2}},$ (22)

where $m_{\nu}$ refers to the mass of the lightest neutrino and

$\displaystyle\alpha\equiv\frac{\Gamma(17/30)}{\Gamma(16/15)}\frac{\Gamma(6/5)}{\Gamma(7/10)}.$
(23)

It is clear from Eq. (22) that a condition for the existence of the
(meta)stable minimum of the radion potential is

$\displaystyle m^{4}_{\nu}>\frac{64}{13}\pi^{2}\alpha
M^{2}_{\text{Pl}}\Lambda_{3}.$ (24)

As we indicated above, the radius of the compactified dimension must be
quantized according to Eq. (16). Hence, this equation and Eq. (22) lead
immediately to an expression for the lightest neutrino mass as

$\displaystyle
m_{\nu}=\frac{\sqrt{2}}{n}\left[60\Lambda_{4}-\sqrt{6\left(585\Lambda^{2}_{4}-8n^{4}\pi^{2}\alpha
M^{2}_{\text{Pl}}\Lambda_{3}\right)}\right]^{1/2},$ (25)

with the following condition

$\displaystyle\frac{n^{4}\pi^{2}\alpha
M^{2}_{\text{Pl}}\Lambda_{3}}{\Lambda^{2}_{4}}<\frac{117}{64}.$ (26)

In addition, in order for a stable non-SUSY AdS vacuum that can not develop in
the lower dimensions, the following condition must satisfy

$\displaystyle m^{4}_{\nu}\leq\frac{192}{29}\pi^{2}\alpha
M^{2}_{\text{Pl}}\Lambda_{3}.$ (27)

This constraint leads to a lower bound on the three-dimensional cosmological
constant $\Lambda_{3}$ as follows

$\displaystyle\frac{n^{4}\pi^{2}\alpha
M^{2}_{\text{Pl}}\Lambda_{3}}{\Lambda^{2}_{4}}\geq\frac{29}{48}.$ (28)

In summary, the quantization rule (16) and the non-SUSY AdS conjecture impose
both the upper and lower bounds on the three-dimensional cosmological constant
$\Lambda_{3}$ and the mass of the lightest neutrino as

$\displaystyle\frac{29}{48n^{4}\pi^{2}\alpha}$ $\displaystyle\leq$
$\displaystyle\frac{M^{2}_{\text{Pl}}\Lambda_{3}}{\Lambda^{2}_{4}}<\frac{117}{64n^{4}\pi^{2}\alpha},$
(29) $\displaystyle\frac{\sqrt{2}}{n}$ $\displaystyle\leq$
$\displaystyle\frac{m_{\nu}}{\sqrt{\Lambda_{4}}}<\frac{\sqrt{3}}{n}.$ (30)

For more explicitly, we show the allowed values of the three-dimensional
cosmological constant versus those of the lightest neutrino mass (which are
scaled by the 4D cosmological constant) for some values of the quantum number
$n$ in Fig. 2 and in Table. 1.

---
Figure 2: The lightest neutrino mass versus the three-dimensional cosmological
constant under the quantization rule (16) and the constraint of the non-SUSY
AdS conjecture.

Their allowed values correspond to the curves (which are blue, red, green, and
purple for $n=1$, $2$, $3$, and $4$, respectively, for example) belonging to
the yellow region.

$n$ |  $m_{\nu}/\sqrt{\Lambda_{4}}$ |  $M^{2}_{\text{Pl}}\Lambda_{3}/\Lambda^{2}_{4}$
---|---|---
1 |  $(1.414,1.732)\ \ $ | $(0.053,0.161)$
2 | $(0.707,0.866)$ | $(3.3\times 10^{-3},0.01)$
3 | $(0.471,0.577)$ | $\ \ (6.5\times 10^{-4},2\times 10^{-3})\ \ $
4 | $(0.354,0.433)$ | $(2\times 10^{-4},6\times 10^{-4})$
Table 1: The predicted value range for the lightest neutrino mass and the
three-dimensional cosmological constant for some values of the quantum number
$n$.

We observe that the allowed region of the lightest neutrino mass and the 3D
cosmological constant would narrow when the quantum number $n$ increases. The
largest allowed region of the lightest neutrino mass which corresponds to
$n=1$ is only $(1.414\sqrt{\Lambda_{4}},1.732\sqrt{\Lambda_{4}})$, as seen in
Table. 1, which is around $\sqrt{\Lambda_{4}}\sim 10^{-32}$ eV. With such a
tiny mass scale, the lightest neutrino can be well-approximately considered to
be massless. Hence, our present scenario provides an extremely predictive
picture of the lightest neutrino mass. This is an essential point to
distinguish our present scenario from the previous investigations Martin-
Lozano2017 ; Valenzuela2017  which predicted
$m_{\nu}\lesssim(M^{2}_{\text{Pl}}\Lambda_{4})^{1/4}\sim 10^{-3}$ eV.

##  V  Light neutrino mass matrix

With the mass of the lightest neutrino predicted and the data of the neutrino
oscillation Esteban2020 , we can determine the light neutrino mass matrix
$M_{\nu}$ as

$\displaystyle
M_{\nu}=U_{L}\text{diag}\left(m_{\nu_{1}},m_{\nu_{2}},m_{\nu_{3}}\right)U^{\dagger}_{R},$
(31)

where $m_{\nu_{1}}$, $m_{\nu_{2}}$, and $m_{\nu_{3}}$ are the masses of the
light neutrinos given for the case of NO

$\displaystyle m_{\nu_{1}}\approx 0,\ \
m_{\nu_{2}}=\sqrt{m^{2}_{\nu_{1}}+\Delta m^{2}_{21}},\ \
m_{\nu_{3}}=\sqrt{m^{2}_{\nu_{1}}+\Delta m^{2}_{31}}$ (32)

and for the case of IO

$\displaystyle m_{\nu_{1}}=\sqrt{m^{2}_{\nu_{3}}+\Delta m^{2}_{23}-\Delta
m^{2}_{21}},\ \ m_{\nu_{2}}=\sqrt{m^{2}_{\nu_{3}}+\Delta m^{2}_{23}},\ \
m_{\nu_{3}}\approx 0,$ (33)

and $U_{L,R}$ are two unitary matrices used to diagonalize the light neutrino
mass matrix which is a general $3\times 3$ complex matrix.

We are interested in the situation in which the charged lepton mass matrix is
diagonal which appears in the models beyond the SM with the lepton generations
carrying the different charges under new gauge symmetries, for instance, the
$U(1)_{L_{\mu}-L_{\tau}}$ model Foot1991 ; XHe1991 ; Foot1994 ;
Altmannshofer2016 . In this situation, the diagonalizing matrix $U_{L}$ is
determined by the lepton mixing matrix $U_{\text{PMNS}}$ as follows

$\displaystyle U_{L}=U_{\text{PMNS}}=\left(\begin{array}[]{ccc}1&0&0\\\
0&c_{23}&s_{23}\\\ 0&-s_{23}&c_{23}\\\
\end{array}\right)\left(\begin{array}[]{ccc}c_{13}&0&s_{13}e^{-i\delta}\\\
0&1&s_{23}\\\ -s_{13}e^{i\delta}&0&c_{13}\\\
\end{array}\right)\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\\\ \end{array}\right)P,$ (43)

where $c_{ij}$ and $s_{ij}$ denote the cosine and sine of the mixing angles,
respectively, $\delta$ refers to the CP phase of the lepton sector and $P$ is
a diagonal matrix related to the Majorana phases. The neutrinos of the Dirac
nature, the Majorana phases are zero and thus $P=\mathbb{1}$. The best-fit
values of the mixing angles and the CP violation phase in the $3\sigma$ range
are given in Table. 2

|  NO |  IO
---|---|---
$s^{2}_{12}$ |  $0.275\rightarrow 0.350$ |  $0.275\rightarrow 0.350$
$s^{2}_{23}$ | $0.418\rightarrow 0.627$ | $0.423\rightarrow 0.629$
$s^{2}_{13}$ | $\ \ 0.02045\rightarrow 0.02439\ \ $ |  $0.02068\rightarrow 0.02463$
$\delta$ | $125^{\circ}\rightarrow 392^{\circ}$ | $196^{\circ}\rightarrow 360^{\circ}$
Table 2: The $3\sigma$ range for the mixing angles and the CP phase of the
neutrino oscillation Esteban2020 .

The diagonalizing matrix $U_{R}$ can be parametrized by three angles (denoted
by $\bar{\theta}_{12}$, $\bar{\theta}_{23}$, and $\bar{\theta}_{13}$) and six
phases (denoted by $\omega_{1}$, $\omega_{2}$, $\omega_{3}$, $\omega_{4}$,
$\omega_{5}$, and $\delta_{R}$). Then, we can write $U_{R}$ as follows
Senjanovic2016

$\displaystyle U_{R}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}e^{i\omega_{1}}&0&0\\\
0&e^{i\omega_{2}}&0\\\ 0&0&e^{i\omega_{3}}\\\
\end{array}\right)\left(\begin{array}[]{ccc}1&0&0\\\
0&\bar{c}_{23}&\bar{s}_{23}\\\ 0&-\bar{s}_{23}&\bar{c}_{23}\\\
\end{array}\right)\left(\begin{array}[]{ccc}\bar{c}_{13}&0&\bar{s}_{13}e^{-i\delta_{R}}\\\
0&1&s_{23}\\\ -\bar{s}_{13}e^{i\delta_{R}}&0&\bar{c}_{13}\\\
\end{array}\right)$ (60)
$\displaystyle\times\left(\begin{array}[]{ccc}\bar{c}_{12}&\bar{s}_{12}&0\\\
-\bar{s}_{12}&\bar{c}_{12}&0\\\ 0&0&1\\\
\end{array}\right)\left(\begin{array}[]{ccc}e^{i\omega_{4}}&0&0\\\
0&e^{i\omega_{5}}&0\\\ 0&0&1\\\ \end{array}\right),$

where $\bar{c}_{ij}\equiv\cos\bar{\theta}_{ij}$ and
$\bar{s}_{ij}\equiv\sin\bar{\theta}_{ij}$.

Using the lepton mixing matrix and the parametrization (60), the elements of
the light neutrino mass matrix, denoted by $[M_{\nu}]_{ij}$, can be
reconstructed by the neutrino oscillation data and in terms of three mixing
angles and six phases in $U_{R}$. For the case of NO, these elements are given
by the following analytical expressions

$\displaystyle\left[M_{\nu}\right]_{11}$ $\displaystyle=$ $\displaystyle
e^{-i\omega_{1}}\left[c_{13}\bar{c}_{13}s_{12}\bar{s}_{12}\sqrt{\Delta
m^{2}_{21}}e^{-i\omega_{5}}+s_{13}\bar{s}_{13}\sqrt{\Delta
m^{2}_{31}}e^{i(\delta_{R}-\delta)}\right],$
$\displaystyle\left[M_{\nu}\right]_{12}$ $\displaystyle=$ $\displaystyle
e^{-i\omega_{2}}\left[c_{13}s_{12}\sqrt{\Delta
m^{2}_{21}}e^{-i\omega_{5}}\left(\bar{c}_{12}\bar{c}_{23}-\bar{s}_{12}\bar{s}_{13}\bar{s}_{23}e^{-i\delta_{R}}\right)+\bar{c}_{13}s_{13}\bar{s}_{23}\sqrt{\Delta
m^{2}_{31}}e^{-i\delta}\right],$ $\displaystyle\left[M_{\nu}\right]_{13}$
$\displaystyle=$ $\displaystyle e^{-i\omega_{3}}\left[c_{13}s_{12}\sqrt{\Delta
m^{2}_{21}}e^{-i\omega_{5}}\left(\bar{c}_{12}\bar{s}_{23}+\bar{c}_{23}\bar{s}_{12}\bar{s}_{13}e^{-i\delta_{R}}\right)+\bar{c}_{13}\bar{c}_{23}s_{13}\sqrt{\Delta
m^{2}_{31}}e^{-i\delta}\right],$ $\displaystyle\left[M_{\nu}\right]_{21}$
$\displaystyle=$ $\displaystyle
e^{-i\omega_{1}}\left[\bar{c}_{13}\bar{s}_{12}\sqrt{\Delta
m^{2}_{21}}e^{-i\omega_{5}}\left(c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}\right)+c_{13}s_{23}\bar{s}_{13}\sqrt{\Delta
m^{2}_{31}}e^{i\delta_{R}}\right],$ $\displaystyle\left[M_{\nu}\right]_{22}$
$\displaystyle=$ $\displaystyle e^{-i\omega_{2}}\left[\sqrt{\Delta
m^{2}_{21}}e^{-i\omega_{5}}\left(c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}\right)\left(\bar{c}_{12}\bar{c}_{23}-\bar{s}_{12}\bar{s}_{13}\bar{s}_{23}e^{-i\delta_{R}}\right)\right.$
(61) $\displaystyle\left.+c_{13}\bar{c}_{13}s_{23}\bar{s}_{23}\sqrt{\Delta
m^{2}_{31}}\right],$ $\displaystyle\left[M_{\nu}\right]_{23}$ $\displaystyle=$
$\displaystyle e^{-i\omega_{3}}\left[-\sqrt{\Delta
m^{2}_{21}}e^{-i\omega_{5}}\left(c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}\right)\left(\bar{c}_{12}\bar{s}_{23}+\bar{c}_{23}\bar{s}_{12}\bar{s}_{13}e^{-i\delta_{R}}\right)\right.$
$\displaystyle\left.+c_{13}\bar{c}_{13}\bar{c}_{23}s_{23}\sqrt{\Delta
m^{2}_{31}}\right],$ $\displaystyle\left[M_{\nu}\right]_{31}$ $\displaystyle=$
$\displaystyle e^{-i\omega_{1}}\left[c_{13}c_{23}\bar{s}_{13}\sqrt{\Delta
m^{2}_{31}}e^{i\delta_{R}}-\bar{c}_{13}\bar{s}_{12}\sqrt{\Delta
m^{2}_{21}}e^{-i\omega_{5}}\left(c_{12}s_{23}+c_{23}s_{12}s_{13}e^{i\delta}\right)\right],$
$\displaystyle\left[M_{\nu}\right]_{32}$ $\displaystyle=$ $\displaystyle
e^{-i\omega_{2}}\left[-\sqrt{\Delta
m^{2}_{21}}e^{-i\omega_{5}}\left(c_{12}s_{23}+c_{23}s_{12}s_{13}e^{i\delta}\right)\left(\bar{c}_{12}\bar{c}_{23}-\bar{s}_{12}\bar{s}_{13}\bar{s}_{23}e^{-i\delta_{R}}\right)\right.$
$\displaystyle\left.+c_{13}c_{23}\bar{c}_{13}\bar{s}_{23}\sqrt{\Delta
m^{2}_{31}}\right],$ $\displaystyle\left[M_{\nu}\right]_{33}$ $\displaystyle=$
$\displaystyle e^{-i\omega_{3}}\left[\sqrt{\Delta
m^{2}_{21}}e^{-i\omega_{5}}\left(c_{12}s_{23}+c_{23}s_{12}s_{13}e^{i\delta}\right)\left(\bar{c}_{12}\bar{s}_{23}+\bar{c}_{23}\bar{s}_{12}\bar{s}_{13}e^{-i\delta_{R}}\right)\right.$
$\displaystyle\left.+c_{13}c_{23}\bar{c}_{13}\bar{c}_{23}\sqrt{\Delta
m^{2}_{31}}\right].$

We observe that the elements of the light neutrino mass matrix for the case of
NO are independent of the phase $\omega_{4}$. While, for the case of IO, the
elements of the light neutrino mass matrix reads

$\displaystyle\left[M_{\nu}\right]_{11}$ $\displaystyle=$ $\displaystyle
c_{13}\bar{c}_{13}e^{-i\omega_{1}}\left(c_{12}\bar{c}_{12}\sqrt{\Delta
m^{2}_{23}-\Delta m^{2}_{21}}e^{-i\omega_{4}}+s_{12}\bar{s}_{12}\sqrt{\Delta
m^{2}_{23}}e^{-i\omega_{5}}\right),$ $\displaystyle\left[M_{\nu}\right]_{12}$
$\displaystyle=$ $\displaystyle c_{13}e^{-i\omega_{2}}\left[s_{12}\sqrt{\Delta
m^{2}_{23}}e^{-i\omega_{5}}\left(\bar{c}_{12}\bar{c}_{23}-\bar{s}_{12}\bar{s}_{13}\bar{s}_{23}e^{-i\delta_{R}}\right)-c_{12}\sqrt{\Delta
m^{2}_{23}-\Delta m^{2}_{21}}e^{-i\omega_{4}}\right.$
$\displaystyle\left.\times\left(\bar{c}_{23}\bar{s}_{12}+\bar{c}_{12}\bar{s}_{13}\bar{s}_{23}e^{-i\delta_{R}}\right)\right],$
$\displaystyle\left[M_{\nu}\right]_{13}$ $\displaystyle=$ $\displaystyle
c_{13}e^{-i\omega_{3}}\left[c_{12}\sqrt{\Delta m^{2}_{23}-\Delta
m^{2}_{21}}e^{-i\omega_{4}}\left(\bar{s}_{12}\bar{s}_{23}-\bar{c}_{12}\bar{c}_{23}\bar{s}_{13}e^{-i\delta_{R}}\right)+s_{12}\sqrt{\Delta
m^{2}_{23}}e^{-i\omega_{5}}\right.$
$\displaystyle\left.\times\left(\bar{c}_{12}\bar{s}_{23}+\bar{c}_{23}\bar{s}_{12}\bar{s}_{13}e^{-i\delta_{R}}\right)\right],$
$\displaystyle\left[M_{\nu}\right]_{21}$ $\displaystyle=$
$\displaystyle\bar{c}_{13}e^{-i\omega_{1}}\left[\bar{s}_{12}\sqrt{\Delta
m^{2}_{23}}e^{-i\omega_{5}}\left(c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}\right)-\bar{c}_{12}\sqrt{\Delta
m^{2}_{23}-\Delta m^{2}_{21}}e^{-i\omega_{4}}\right.$
$\displaystyle\left.\times\left(c_{23}s_{12}+c_{12}s_{13}s_{23}e^{i\delta}\right)\right],$
$\displaystyle\left[M_{\nu}\right]_{22}$ $\displaystyle=$ $\displaystyle
e^{-i\omega_{2}}\left[\sqrt{\Delta m^{2}_{23}-\Delta
m^{2}_{21}}e^{-i\omega_{4}}\left(c_{23}s_{12}+c_{12}s_{13}s_{23}e^{i\delta}\right)\left(\bar{c}_{23}\bar{s}_{12}+\bar{c}_{12}\bar{s}_{13}\bar{s}_{23}e^{-i\delta_{R}}\right)\right.$
(62) $\displaystyle\left.+\sqrt{\Delta
m^{2}_{23}}e^{-i\omega_{5}}\left(c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}\right)\left(\bar{c}_{12}\bar{c}_{23}-\bar{s}_{12}\bar{s}_{13}\bar{s}_{23}e^{-i\delta_{R}}\right)\right],$
$\displaystyle\left[M_{\nu}\right]_{23}$ $\displaystyle=$
$\displaystyle-e^{-i\omega_{3}}\left[\sqrt{\Delta m^{2}_{23}-\Delta
m^{2}_{21}}e^{-i\omega_{4}}\left(c_{23}s_{12}+c_{12}s_{13}s_{23}e^{i\delta}\right)\left(\bar{s}_{12}\bar{s}_{23}-\bar{c}_{12}\bar{c}_{23}\bar{s}_{13}e^{-i\delta_{R}}\right)\right.$
$\displaystyle\left.+\sqrt{\Delta
m^{2}_{23}}e^{-i\omega_{5}}\left(c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}\right)\left(\bar{c}_{12}\bar{s}_{23}+\bar{c}_{23}\bar{s}_{12}\bar{s}_{13}e^{-i\delta_{R}}\right)\right],$
$\displaystyle\left[M_{\nu}\right]_{31}$ $\displaystyle=$
$\displaystyle\bar{c}_{13}e^{-i\omega_{1}}\left[\bar{c}_{12}\sqrt{\Delta
m^{2}_{23}-\Delta
m^{2}_{21}}e^{-i\omega_{4}}\left(s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}\right)-\bar{s}_{12}\sqrt{\Delta
m^{2}_{23}}e^{-i\omega_{5}}\right.$
$\displaystyle\left.\times\left(c_{12}s_{23}+c_{23}s_{12}s_{13}e^{i\delta}\right)\right],$
$\displaystyle\left[M_{\nu}\right]_{32}$ $\displaystyle=$
$\displaystyle-e^{-i\omega_{2}}\left[\sqrt{\Delta m^{2}_{23}-\Delta
m^{2}_{21}}e^{-i\omega_{4}}\left(s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}\right)\left(\bar{c}_{23}\bar{s}_{12}+\bar{c}_{12}\bar{s}_{13}\bar{s}_{23}e^{-i\delta_{R}}\right)\right.$
$\displaystyle\left.+\sqrt{\Delta
m^{2}_{23}}e^{-i\omega_{5}}\left(c_{12}s_{23}+c_{23}s_{12}s_{13}e^{i\delta}\right)\left(\bar{c}_{12}\bar{c}_{23}-\bar{s}_{12}\bar{s}_{13}\bar{s}_{23}e^{-i\delta_{R}}\right)\right],$
$\displaystyle\left[M_{\nu}\right]_{33}$ $\displaystyle=$ $\displaystyle
e^{-i\omega_{3}}\left[\sqrt{\Delta m^{2}_{23}-\Delta
m^{2}_{21}}e^{-i\omega_{4}}\left(s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}\right)\left(\bar{s}_{12}\bar{s}_{23}-\bar{c}_{12}\bar{c}_{23}\bar{s}_{13}e^{-i\delta_{R}}\right)\right.$
$\displaystyle\left.+\sqrt{\Delta
m^{2}_{23}}e^{-i\omega_{5}}\left(c_{12}s_{23}+c_{23}s_{12}s_{13}e^{i\delta}\right)\left(\bar{c}_{12}\bar{s}_{23}+\bar{c}_{23}\bar{s}_{12}\bar{s}_{13}e^{-i\delta_{R}}\right)\right].$

It is interesting that it can take the mass matrix of the light neutrinos to
be Hermitian when the generations of right-handed neutrinos transform
universally. In Ref. LiuZhou2013 , the authors proved that a general complex
mass matrix $M_{\nu}$ of the Dirac neutrinos can be decomposed as
$M_{\nu}=S_{\nu}.V_{\nu}$ where $S_{\nu}$ and $V_{\nu}$ are a Hermitian matrix
and a unitary matrix, respectively. And, by redefining the right-handed
neutrino fields as $\nu^{\prime}_{R}=V_{\nu}\nu_{R}$, the neutrino mass matrix
would be Hermitian. In this situation, we have $U_{R}=U_{L}=U_{\text{PMNS}}$
which corresponds to $\bar{c}_{ij}=c_{ij}$ and $\bar{s}_{ij}=s_{ij}$,
$\omega_{1}=\omega_{2}=\omega_{3}=\omega_{4}=\omega_{5}=0$, and
$\delta_{R}=\delta$. As a result, the elements of the light neutrino mass
matrix are fixed only by the neutrino oscillation data. The numerical value of
the light neutrino mass matrix is given in the $3\sigma$ range for the case of
NO

$\displaystyle\frac{M_{\nu}}{10^{-3}\text{eV}^{2}}=\left(\begin{array}[]{ccc}3.22\rightarrow
4.32&\left(0.82\rightarrow 9.14\right)e^{i\left(-3.1\rightarrow
1.9\right)}&\left(1.46\rightarrow 9.2\right)e^{i\left(-3.13\rightarrow
2.66\right)}\\\ (12)^{*}&21.67\rightarrow 36.14&\left(16.41\rightarrow
27.15\right)e^{i\left(-0.03\rightarrow 0.03\right)}\\\
(13)^{*}&(23)^{*}&19.71\rightarrow 34.11\\\ \end{array}\right),$ (66)

and for the case of IO

$\displaystyle\frac{M_{\nu}}{10^{-3}\text{eV}^{2}}=\left(\begin{array}[]{ccc}43.79\rightarrow
53.32&\left(3.31\rightarrow
7.54\right)e^{i\left(-2.62\rightarrow\pi\right)}&\left(3.05\rightarrow
7.35\right)e^{i\left(-2.68\rightarrow\pi\right)}\\\ (12)^{*}&17.12\rightarrow
32.46&\left(17.47\rightarrow
32.28\right)e^{i\left(-\pi\rightarrow\pi\right)}\\\
(13)^{*}&(23)^{*}&19.43\rightarrow 35.24\\\ \end{array}\right),$ (70)

where $(ij)^{*}$ refers to the complex conjugate of the element
$[M_{\nu}]_{ij}\times 10^{3}$ eV-2. Note that, the diagonal elements of the
light neutrino mass matrix are real due to its Hermitian property.

##  VI Conclusion

The neutrino oscillation data only provides information about the difference
of the squared neutrino masses without telling us the value of the lightest
neutrino mass. Recently, it has been shown that the consistency of the
compactification of the SM coupled to Einstein gravity to lower dimensions
with quantum gravity can impose constraints on the mass of the lightest
neutrino. The circle compactification may yield stable non-SUSY AdS vacua
which are inconsistent with quantum gravity or belong to the swampland
according to the non-SUSY AdS conjecture, depending on the light neutrino
masses. Because the SM and Einstein gravity have described very well the
observed world and hence they cannot be lying in the swampland, these
dangerous AdS vacua must be absent if the mass of the lightest neutrino
satisfies the following bound
$m_{\nu}\lesssim\left(\rho_{4\text{D}}\right)^{1/4}$ where
$\rho_{4\text{D}}\approx 2.6\times 10^{-47}$ GeV4 is the observed vacuum
energy density.

In the present work, we revisit the constraint of the non-SUSY AdS conjecture
on the three-dimensional vacua which are obtained from the circle
compactification of the SM coupled to Einstein gravity with the radion
potential generated by the cosmological constant and the Casimir effect of the
light particles. Unlike the previous studies where the three-dimensional
components of the four-dimensional metric are restricted to be dependent on
the non-compact coordinates only, we consider them to be the general functions
of both non-compact and compact coordinates. From investigating the
wavefunction profile of the three-dimensional metric in the compactified
dimension, we find that the radius $R$ of the compactified dimension must be
quantized by the following rule $R=n/\sqrt{15\Lambda_{4}}$ (due to the circle
topology of the compactified dimension) where $n$ refers to the positive
integers and $\Lambda_{4}=\rho_{4\text{D}}/M^{2}_{\text{Pl}}$ with
$M_{\text{Pl}}$ being the observed Planck scale. Because of this quantization,
the existence or the absence of dangerous three-dimensional AdS vacua is very
sensitive to $\Lambda_{4}$ (instead of $\rho_{4\text{D}}$ as indicated in the
previous works) and the light particles with the mass in the order of
$\Lambda_{4}$. In addition, the quantization of the radius of the compactified
dimension forbids the runaway behavior of the radion potential. Therefore, it
is interesting that both the quantization rule and the non-SUSY AdS conjecture
impose constraints on the three-dimensional vacua leading to an upper bound
and a lower bound for the mass of the lightest neutrino as $\sqrt{2}\leq
m_{\nu}/\sqrt{\Lambda_{4}}<\sqrt{3}$. This constraint is very predictive
because the mass of the lightest neutrino is around $10^{-32}$ eV which
implies that the lightest neutrino would be nearly massless.

With the well-approximately vanishing mass of the lightest neutrino, we
reconstruct the light neutrino mass matrix in the situation that the charged
lepton mass matrix is diagonal. In general, the light neutrino mass matrix is
a $3\times 3$ complex matrix with nine independent complex elements that can
be diagonalized by the lepton mixing matrix (or the Pontecorvo-Maki-Nakagawa-
Sakata matrix) and another unitary matrix. In this way, we fix the elements of
the light neutrino mass matrix based on the neutrino oscillation data and in
terms of three new mixing angles and six new phases for the cases of the
normal ordering (NO) and inverted ordering (IO). Interestingly, the mass
matrix of the light neutrinos can be taken to be Hermitian when the
generations of right-handed neutrinos transform universally by redefining the
right-handed neutrino fields. In this case, we calculate the numerical value
of the light neutrino mass matrix in the $3\sigma$ range for both NO and IO.

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