# Neutrino Physics in TeV Scale Gravity Theories

Manuel Ettengruber<EMAIL_ADDRESS>Max-Planck-Institut für Physik (Werner-
Heisenberg-Institut), Föhringer Ring 6, 80805 München, Germany

(August 27, 2024)

###### Abstract

In this paper, the general features of the neutrino sector in TeV scale
quantum gravity theories, such as ADD and Many Species Theory, is
investigated. This class of theories has an inherent way to generate small
neutrino masses. After reviewing this mechanism it is generalized to a
realistic three-flavour case. Furthermore, a procedure is presented how to
diagonalize a mass matrix which is generated by this class of theories and how
one can find the Standard Model flavour eigenstates. The developed general
approach is applied to two specific scenarios within ADD and Many Species
Theory and possible effects on neutrino oscillations and on unitarity of the
lepton mixing matrix are calculated. Finally, a short overview of
phenomenology which can be potentially testable by the nowadays neutrino
experiments is presented.

##  I Introduction

The Standard Model (SM) of particle physics is one of the most successful
theories. In particular, it fully accounts for all the processed data from
high energy particle physics accelerators.

Nevertheless, there are several hints that the SM is not complete. In
particular it produces two outstanding puzzles that will be the subject of the
present paper: 1) the origin and inexplicable smallness of the neutrino mass;
2) the Hierarchy Problem.

The hierarchy problem is perhaps the most prominent naturalness puzzle of the
SM. Gravity plays a defining role in its essence due to the following Dvali
(2017). The Higgs mass is quadratically sensitive towards the cutoff of the
theory. The ultimate cutoff is provided by gravity in form of the Planck mass.
This cutoff is fully non-perturbative, since the Planck mass is an absolute
upper boundary on the mass of elementary particles. Indeed, any elementary
object much heavier than the Planck scale is a classical black hole.

This raises the question what keeps the observed value of the Higgs mass-term
by some 34 orders of magnitude smaller than the expected upper limit. This
Hierarchy strongly hints towards some new stabilizing physics not far from the
weak scale.

One mechanism for stabilization is based on lowering the fundamental scale of
quantum gravity. In this framework, the Planck mass $M_{P}$ still sets the
coupling strength of graviton at large distances. However, the actual scale
$M_{*}$, at which the quantum gravitational effects are strong, is much lower.
Correspondingly, in such a scenario the cutoff-sensitive corrections to the
Higgs mass are regulated by the scale $M_{*}$ and not by $M_{P}$. This idea
was originally proposed in the ADD model Arkani-Hamed  _et al._ (1998, 1999).
(see, Antoniadis  _et al._ (1998), for string theoretic realization).

In this setup the fundamental scale of gravity is lowered due to a large
volume of extra dimensions. The reason is that, due to universal nature of
gravity, the graviton wavefunction spreads over the entire volume of extra
space and gets effectively “dilute”. As a result, the coupling scale of
graviton $M_{P}$ is hierarchically larger than the fundamental scale of
quantum gravity $M_{*}$.

Remarkably, more recently it was shown Dvali (2010a); Dvali and Redi (2008)
(for various aspects, see, Dvali (2010b); Dvali and Gomez (2009); Dvali and
Redi (2009); Dvali and Gomez (2010); Dvali (2021)) that the effect of lowering
the cutoff $M_{*}$ relative to Planck mass $M_{P}$ is an universal property of
any theory with large number of particle species. Correspondingly, a general
solution to the hierarchy problem based on this mechanism was proposed in
Dvali (2010a) and further studied is subsequent papers. We shall refer to this
as the “Many Species” framework.

As explained in Dvali (2010a), the ADD model of large extra dimensions
represents a particular manifestation of this very general phenomenon. There
the role of the species is assumed by the Kaluza-Klein (KK) excitations of
graviton. This connection enables to understand the dilution of graviton
wavefunction in extra space of ADD as a dilution in what one can call Dvali
(2010b) the “space of species”. In the latter paper, it was argued that, due
to unitarity and other general consistency properties, the “space of species”
in many respects behaves as ordinary geometric space.

In a particularly interesting realization of “many species” solution the role
of the species is played by the identical copies of the SM Dvali (2010a);
Dvali and Redi (2008). Various phenomenological aspects of this proposal were
studied in Dvali and Redi (2009).

Soon after the invention of low scale quantum gravity idea, it has been
realized, first in Arkani-Hamed  _et al._ (2001); Dvali and Smirnov (1999)
within ADD and later in Dvali and Redi (2009) within many species theory, that
this general framework, as a bonus, offers an universal solution to the
neutrino mass problem in the SM. Namely, the same mechanism that explains the
hierarchy between the weak and Planck scales, is responsible for the
suppression of the neutrino mass.

Naturally, this fact boosts the motivation for the above class of theories,
since the origin and the hierarchy of the neutrino mass is a fundamental open
question in SM. Due to the phenomenon of neutrino oscillations Fantini  _et
al._ (2018), we know that they have a mass and this is now a field of several
ongoing experiments. So far, neutrino mass has not been detected directly.
Just upper bounds of roughly $1$eV have been given Aker  _et al._ (2019). It
is also not known whether the neutrino mass is of Majorana or of Dirac nature
as this is the case with all other fermions of the SM. We also do not know why
neutrinos are so much lighter than the charged fermions.

If neutrino masses are of Dirac type and originate from an ordinary Higgs
mechanism, an unusually-small Yukawa coupling would be required for generating
the masses smaller than eV.

Lack of the explanation for this smallness, prompted thinking that perhaps the
neutrino mass is of Majorana type. In such a case, the mass term must be
generated from an effective high-dimensional operator Weinberg (1979) and the
smallness can be attributed to the high cutoff scale. In the traditional See-
Saw mechanism Minkowski (1977); Gell-Mann  _et al._ (1979); Yanagida (1980);
Mohapatra and Senjanovic (1980); Mohapatra (2004), such an effective operator
is generated by integrating out a neutrino’s hypothetical right-handed partner
with large Majorana mass.

However, after formulation of theories with low scale gravity, it became clear
that they offer an alternative possibility in form of naturally small Dirac
neutrino mass. Originally, this idea was realized in Arkani-Hamed  _et al._
(2001); Dvali and Smirnov (1999) within the ADD framework Arkani-Hamed  _et
al._ (1998). We shall refer to this as ADDM model. Later a complementary
mechanism of suppressed neutrino mass was introduced by Dvali-Redi (DR) in
Dvali and Redi (2009) within the “many species” framework with identical
copies of the SM 111Such scenarios have other potential bonuses. For example,
particles of other copies could just interact with each other gravitationally
and are good candidates for dark matter Dvali  _et al._ (2009); Dvali and Redi
(2009).

Although complementary, the above two scenarios are based on one and the same
fundamental mechanism of the suppression of the neutrino Yukawa coupling, very
similar to the suppression of the coupling of graviton. In both cases this can
be viewed as a consequence of the dilution of the wavefunction of the sterile
neutrino into the bulk of the extra space of species. This dilution is
identical to the dilution of the wavefunction of graviton in the same space.
In ADD this space is also organized as a real coordinate space but this does
not change the essence of the dilution. In summary, the theories with low
$M_{*}$ can solve both the hierarchy and the neutrino mass problems by one and
the same mechanism. Both hierarchies are controlled by the ratio
$M_{*}/M_{P}$.

The main focus of the present work is implications for neutrino physics. The
above class of theories predict certain universal features of phenomenological
interest. In particular as already discussed in Dvali and Smirnov (1999) for
ADDM and in Dvali and Redi (2009) for DR scenarios, the mixing with the tower
of sterile neutrino species results into oscillations of neutrinos into the
hidden sector. The implies non-conservation of neutrino number within SM and
correspondingly a seeming violation of unitarity. This is of obvious potential
experimental interest.

The aim of this paper is to focus on the neutrino sector of these theories and
work out a general framework for how neutrino physics can be treated in this
class of theories. Moreover, we will generalize this framework to a realistic
three flavour case and investigate their effects on low energy phenomena and
observables such as neutrino oscillations into the hidden modes and possible
deviations from the Standard Model PMNS matrix which can be tested by nowadays
experiments. This is a still ongoing project Caldwell  _et al._ .

This paper is organized as follows. In section 2 the structures of DR and ADDM
models are presented and reviewed. In section 3 we formulate a general
approach how neutrino masses are induced in this kind of theories and how we
can guarantee the smallness of their mass. In section 4 we present a
generalisation of the mass matrices to a realistic three flavour case and in
section 5 we investigate the special case of highly symmetric mass matrices
which are of interest for the DR model. In section 6 the phenomenology of TeV
scale gravity theories in neutrino physics are investigated. In section 7 we
wrap up our findings and give an outlook to possible experimental tests.

##  II ADD and ”Many Species Theory”

###  II.1 Many Species Theory

Originally the idea of many mirror copies of the SM has been proposed in Dvali
(2010a); Dvali and Redi (2008) as the framework for solving the hierarchy
problem. In this work, it has been shown that through introducing $N$ particle
species, the fundamental scale of gravity $M_{*}$ in lowered relative to
Planck scale $M_{P}$ in the following way

$M_{p}^{2}\geq NM_{*}^{2}.$ (1)

This result has been obtained using the well established properties of black
hole physics and is fully non-perturbative. In order to lower the scale of
gravity to TeV energies, $N$ must be of the order of $10^{32}$. Since the
bound (1) is independent of the nature of the particles, the species of
various types can be used for lowering the gravitational cutoff. A particular
version introduced in Dvali (2010a); Dvali and Redi (2008), assumes that the
species are identical copies of the SM. Phenomenological; aspects including
generation of neutrino mass were discussed in Dvali and Redi (2009).

In this work, it is assumed that the species obey the full permutation group
$P(N)$ at first. This means that all species are equidistant in what one can
call the “space of species”. This gives a certain predictive power to the
theory. Alternative choices such as the cyclic symmetry are also possible. It
was shown that many species framework can give various phenomenological
signatures, including micro black holes, in region of TeV energies. In the
present work we shall focus on the implications for neutrino masses.

We shall study generalizations of the mechanism, originally introduced in
Dvali and Redi (2009), which allows the generation of small neutrino masses in
many species framework. This mechanism represents an infrared alternative to
see-saw which cannot be used in frameworks with low cutoff.

Let us briefly review the results of Dvali and Redi (2009). As already said,
the framework represents $N$ identical copies of the SM. The copies are
permuted under $P(N)$. It is useful to visualize the copies as placed on
equidistant sites in the space of species. Fermions are each sector is charged
under their own gauge group. The exceptions are sterile neutrinos, which
represent the right-handed partners of corresponding active left-handed
neutrinos. We shall denote them by $\nu_{Rj}$, where $j=1,2,...,N$ is the
label of the SM copy. These particles do not carry any charges under the SM
gauge groups. Thus the notion of “belonging” is defined by their
transformation properties under the permutation group $P(N)$ as well as by
their couplings to particles of specific SM copies. In particular, the gauge
charges do not forbid sterile neutrinos to interact with neutrinos of the
other copies. One can say that sterile neutrinos are not confined to specific
sites in the space of species. The most generic renormalizable coupling has
the following structure,

$(HL)_{i}\lambda_{ij}\nu_{Rj}+h.c.,$ (2)

where $H$ and $L$ stand for the Higgs and lepton doublets of the $i$-th copy.
Here $\lambda_{ij}$ is a $N\times N$ Yukawa matrix interaction in the space of
species. This Yukawa coupling matrix is restricted by the permutation symmetry
group $P(N)$ and has therefore the following form:

$\lambda_{ij}=\begin{pmatrix}a&b&b&\dots\\\ b&a&b&\dots\\\ b&b&a&\dots\\\
\dots&\dots&\dots&\ddots\end{pmatrix}.$ (3)

For calculation of the mass matrix of neutrinos, one has to have a closer look
at the Higgs doublet $H_{i}$. The simplest case for calculation is when the
permutation symmetry is unbroken by the electroweak vacua. That is, VEV of the
Higgs doublet in every copy of the SM takes the same value $v$. In this
section we shall focus on this case. The generalization to the case of broken
permutation symmetry will be given later.

For now, let us therefore take $v$ as the VEV of the Higgses for all copies.
Then, the mass matrix takes the form $m_{ij}=\lambda_{ij}v$.

This mass matrix has the eigenvalues

$m_{1}^{\prime}=(a-b)v,$ (4) $m_{H}=[a+(N-1)b]v,$ (5)

corresponding to the eigenvectors

$\nu_{1}^{\prime}=\sqrt{\frac{N-1}{N}}\nu_{1}-\frac{1}{\sqrt{N}}\nu_{h},$ (6)

and

$\nu_{H}=\frac{1}{\sqrt{N}}\nu_{1}+\sqrt{\frac{N-1}{N}}\nu_{h}.$ (7)

It is worth noticing for later convenience that the light eigenvalue is $N-1$
times degenerated. Because

$b\leq\frac{1}{\sqrt{N}},$ (8)

and $a\approx 100b$ we see that the mass of the neutrino is suppressed by the
number of species. The mechanism presented here can explain the smallness of
the neutrino mass but has no phenomenological implications which can be tested
by experiments due to the huge mass of the heavy state which scales with the
number of species which is of order $N\approx 10^{32}$.

###  II.2 The ADDM model

The ADDM model Arkani-Hamed  _et al._ (1998); Antoniadis  _et al._ (1998);
Arkani-Hamed  _et al._ (1999) is based on the idea that in addition to
observed $3$ space dimensions, there exist $d$ additional compact space ones
with radii $R_{i},\leavevmode\nobreak\ i=1,2,...,d$ below the tenths of a
millimetre.

The role of the gravitational cutoff in this theory is played by the
fundamental Planck mass of the $4+d$-dimensional theory, $M_{f}$. The two
Planck scales are related via,

$M_{P}=M_{f}\sqrt{M_{f}^{d}V_{d}},$ (9)

where

$V_{d}=(2\pi)^{d}R_{1}...R_{d},$ (10)

is the volume of the extra dimensional space.

Like the DR, this theory provides a solution to the hierarchy problem by
lowering the cutoff $M_{f}$ relative to Planck mass due to large volume of the
extra space. $M_{f}\sim$TeV requires that the volume of the extra space,
measured in units of the fundamental Planck mass, be about $M_{f}^{d}V_{d}\sim
10^{32}$.

As noticed in Dvali (2010a) the lowering of the cutoff in ADD can be
understood as a particular case of many species effect. This is because the
quantity $M_{f}^{d}V_{d}$ measures the number of Kaluza-Klein species of
graviton. Thus, the relation (9) represents a particular manifestation of a
more general relation (1).

According to this theory, the Standard Model particles are localized on a
3-dimensional hyper-surface (brane) which is embedded in the bulk of $d$ large
extra dimensions. The graviton propagates into the entire high dimensional
space. Together with gravity, the bulk is a natural habitat for all possible
particles that carry no gauge quantum numbers under the Standard model group.

Notice that Arkani-Hamed  _et al._ (1998) the bulk particles cannot carry any
quantum numbers under the SM gauge group. This is a consistency requirement
that follows from the gauge invariance and is an intrinsic feature of
localization mechanism of the gauge field on the brane Dvali and Shifman
(1997). Correspondingly, the localization of SM gauge fields on the brane
automaticaly forbids existence of any bulk modes with such charges. Only the
particles carrying no SM gauge quantum numbers are permitted to represent bulk
modes. In particular, such are sterile neutrinos that play the role of the
right-handed partners of the ordinary left-handed neutrinos of the Standard
Model.

This setup generates a naturally small Dirac mass for neutrinos Arkani-Hamed
_et al._ (2001); Dvali and Smirnov (1999). This mass originates from the
mixing of the right-handed component of bulk sterile neutrino $\nu$ with the
Standard Model left-handed neutrino $\nu_{L}$ which is localized on the brane.
In the approximation of a zero-width brane, the part of the action responsible
for this mixing can be written as

$\int
d^{4}x\frac{h}{M_{f}^{d/2}}H(x_{\mu})\bar{\nu}_{L}(x_{\mu})\nu(x_{\mu},y_{i}=0)+h.c.,$
(11)

where $x_{\mu}$ stands for ordinary $4$-dimensional space-time coordinates and
$y_{i},\leavevmode\nobreak\ i=1,2,...,d$-s are the extra ones. The brane
location is taken at $y_{i}=0$ point. The canonically normalized
$4+d$-dimensional fermion field $\nu(x,y)$ has dimensionality $(3+d)/2$.
Correspondingly, the coupling constant has dimensionality $-d/2$. We have
parameterized this coupling constant in terms of the fundamental scale $M_{f}$
and an order-one dimensionless constant $h$.

From the point of view of $4$-dimensional theory, $\nu(x,y)$ represents a
tower of Kaluza-Klein modes with their masses quatized in units of the inverse
radii $m^{2}=\sum_{i}n_{i}^{2}/R_{i}^{2}$ where $n_{i}$ are integers.

Notice that, a high dimensional fermion field $\nu$, viewed from the point of
view of a $4$-dimensional theory, has no chirality. That is, at each Kaluza-
Klein level of mass $m$ it contains $4$-dimensional fermions of both
chiralities, $\nu_{R}^{(m)}$ and $\nu_{L}^{(m)}$. The $4$-dimensional
reduction of the coupling (11) gives

$\frac{hM_{f}}{M_{P}}H\bar{\nu}_{L}\sum_{m}\nu_{R}^{(m)}+h.c.\,,$ (12)

where the factor $1/M_{P}$ comes from the canonical normalization of the
Kaluza-Klein modes. Notice that only the right-handed components
$\nu_{R}^{(m)}$ of the Kaluza-Klein modes mix with SM neutrino.

After taking into account the VEV of the Higgs field, $\langle H\rangle\equiv
v$, the above couplings translate as the Dirac-type mass terms

$m_{D}\bar{\nu}_{L}\sum_{m}\nu_{R}^{(m)}+h.c.\,,$ (13)

with $m_{D}\equiv\frac{hvM_{f}}{M_{P}}$. This mixing generates a Dirac mass of
the SM neutrino. Below, for evaluating the mass matrix we shall restrict
ourselves to the case of a single extra dimension. In this case the masses of
Kaluza-Klein excitations are labeled by a single integer $m=n/R$.

Taking into account the Dirac mass terms of Kaluza-Klein modes coming from the
mixing between their left and right-handed components,

$\sum_{n=-\infty}^{\infty}\frac{n}{R}\bar{\nu}_{nR}\bar{\nu}_{nL}\,,$ (14)

the resulting mass matrix has the form:

$M=\begin{pmatrix}m_{D}&\sqrt{2}m_{D}&\sqrt{2}m_{D}&\dots&\sqrt{2}m_{D}\\\
0&\frac{1}{R}&0&\dots&0\\\ 0&0&\frac{2}{R}&\dots&0\\\
\dots&\dots&\dots&\dots&\dots\\\ 0&0&0&\dots&\frac{k}{R}\end{pmatrix}$ (15)

After the diagonalization of the mass matrix, one can express a neutrino of a
specific flavour with the following expression

$\nu=\frac{1}{\Omega}(\nu_{0}+\xi\sum_{n=1}\frac{1}{n}\tilde{\nu}_{n}),$ (16)

with

$\xi=\frac{\sqrt{2}vM_{f}Rh}{M_{P}}.$ (17)

The normalisation parameter is $\Omega^{2}=1+\frac{\pi^{2}}{6}\xi^{2}$.

The mass of the lightest eigenstate $\nu_{0}$ is

$m_{D}=\frac{hvM_{f}}{M_{P}}.$ (18)

The other eigenstates have

$m_{n}\approx\frac{n}{R}.$ (19)

One of the important phenomenological implications of this scenario is the
oscillation of active neutrino species into the KK neutrinos Dvali and Smirnov
(1999). The effect takes place already for a single flavor case. We shall
review this later and compare it with the case of three flavors of active
neutrino species.

##  III Generalisation of Neutrino Masses

We have seen that in ADDM and in DR one can generate small neutrino masses by
introducing a sterile neutrino which is uncharged under the SM gauge group and
can therefore propagate into an additional space which was introduced in this
class of theories. In the case of ADDM, this space is represented the bulk of
large extra dimensions and in the DR it is described as the ”space of
species”. In both cases, the neutrino mass is suppressed by the large
effective volume of this extra space.

This common structure we want to investigate further. We shall make a rather
general assumption of the existence of an extra space in which the sterile
neutrino can propagate. Also, we assume that this extra space lowers the scale
of gravity via

$M_{*}^{2}=\frac{M_{P}^{2}}{\Lambda},$ (20)

where $\Lambda$ is the volume of the extra space measured in fundamental
units. In ADDM the size of the extra space is a function of $R$ $\Lambda(R)$
and in DR of $N$ $\Lambda(N)$.

It is assumed that the particles that are not charged under the SM gauge
symmetries can propagate in this extra space. That is, the couplings of such
particles are more or less uniformly spread over this space. Correspondingly,
the coupling to individual copies is suppressed.

Within the known framework there are two candidates for such particles. First
one is of course graviton, since gravity interacts universally. The second
natural candidate is a sterile neutrino. Currently, it is not known whether
neutrino is a purely Majorana particle. If it is not, then there necessarily
exist a sterile partner $\nu_{R}$ that together with ordinary left-handed
neutrino forms a Dirac state. This sterile neutrino carries no gauge quantum
numbers under the standard model group. Correspondingly, it has no obligation
to be confined to the site where our Standard Model is located. Instead, just
like gravity, such particles can spread over the entire extra space,
irrespectively whether this space stands for extra space dimensions or the
space of species. This spread naturally suppressed the coupling of the sterile
fermion to SM neutrino, thereby resulting into small Dirac mass. The
suppression of the coupling with many mixing partners results from the
principle of unitarity and was shown in Dvali (2010b). This is the key
mechanism behind the small neutrino mass both in ADDM Arkani-Hamed  _et al._
(2001) as well as in DR Dvali and Redi (2009).

A possible operator for neutrino mass of the SM neutrino is the Dirac operator

$yH\bar{\nu}_{L}\nu_{R},$ (21)

where $y$ is a Yukawa coupling and $H$ is the SM Higgs doublet. In this
framework, the left-handed neutrinos of the SM can mix with different types of
right-handed neutrinos which are inhabitants of the extra space. So $\nu_{R}$
is a superposition of all possible mixing partners

$\nu_{R}=\frac{1}{\Lambda}\sum_{n}c_{n}\nu_{nR}\,.$ (22)

Of course, the superposition has to be normalized and this depends on the size
of the extra space the right-handed neutrinos live in. Therefore the different
contributions of all mixing partners have to be divided by the volume of space
in which they can propagate. The resulting form of (21) is then

$yH\bar{\nu}_{L}\nu_{R}=\frac{yv}{\Lambda}\bar{\nu}_{L}\sum_{n}c_{n}\nu_{nR}\,.$
(23)

With (20) one gets

$\frac{M_{f}}{M_{P}}yv\bar{\nu}_{L}\sum_{n}c_{n}\nu_{nR}.$ (24)

The factor in front of the operator represents the effective Dirac mass of
neutrino which we can denote by

$m_{D}=\frac{M_{f}}{M_{P}}yv\,.$ (25)

Here we want to point out that this prefactor is suppressed by the Planck
mass. We see that it induces a small Dirac mass for neutrinos. This captures
an universal essence of generating a small neutrino mass in ADDM Arkani-Hamed
_et al._ (2001) and in DR Dvali and Redi (2009) formulated in a theory-
independent way. It follows that a small mass for neutrinos is a natural
property for this class of theories. The new feature is that the suppression
of the mass of the neutrino comes from the size of the extra space to which
the sterile neutrino can propagate. This is very different from the
introduction of a heavy Majorana particle as this is the case in see-saw. In
other words, the spirit of the solution for the smallness of the neutrino mass
we presented here is an infrared solution and not an ultraviolet solution by
introducing a very heavy particle.

Of course, such a mixing can also occur between $\nu_{Rj}$ and the left-handed
inhabitants, $\nu_{Li}$, of the extra space. Therefore, we also include the
mass terms of the following form,

$m_{ij}\bar{\nu}_{Li}\nu_{Rj}.$ (26)

Let us label the neutrino of the SM with $i=1$ and redefine the Yukawa
coupling as $y=yc_{1}$. Moreover, let us assume that the interactions among
certain pairs of neutrinos are stronger than the mixing with other types. We
shall organize such mass terms as the diagonal entries $m_{ii}$.
Correspondingly the off-diagonal entries $\mu_{ij}$ will denote mixings with
other species. The resulting mass matrix is

$\begin{pmatrix}m_{D}&\mu_{12}&\dots&\dots\\\
\mu_{21}&m_{22}&\mu_{23}&\dots\\\ \vdots&\dots&\ddots&\dots\end{pmatrix},$
(27)

with $\mu_{1i}=c_{i}m_{D}$, and we ordered the diagonal entries according to
their hierarchy

$m_{D}<m_{22}<\dots<m_{kk}.$ (28)

Assuming that the mixing angles due to off-diagonal entries are small, we can
split this matrix into the diagonal and off-diagonal parts and treat the
latter one as a perturbation

$\begin{pmatrix}m_{D}&\mu_{12}&\dots&\dots\\\
\mu_{21}&m_{22}&\mu_{23}&\dots\\\
\vdots&\dots&\ddots&\dots\end{pmatrix}=\begin{pmatrix}m_{D}&0&\dots&\dots\\\
0&m_{22}&0&\dots\\\
\vdots&\dots&\ddots&\dots\end{pmatrix}+\begin{pmatrix}0&\mu_{12}&\dots&\dots\\\
\mu_{21}&0&\mu_{23}&\dots\\\ \vdots&\dots&\ddots&\dots\end{pmatrix},$ (29)

and we denote

$V\equiv\begin{pmatrix}0&\mu_{12}&\dots&\dots\\\ \mu_{21}&0&\mu_{23}&\dots\\\
\vdots&\dots&\ddots&\dots\end{pmatrix}.$ (30)

With this, we find that the eigenvalues do not become corrected in the first
order in mixing

$m_{i}=m_{ii}+\braket{n_{i}}{V}{n_{i}}=m_{ii}+\mathcal{O}^{2}.$ (31)

The correction to the mass eigenstates has the following form

$\ket{m_{1}}=\ket{1^{(0)}}+\sum_{k=2}\frac{\mu_{1k}}{m_{1}^{(0)}-m_{k}^{(0)}}\ket{k^{(0)}},$
(32)

where the $\ket{n}$ are the eigenstates of the unperturbed matrix. Of course
one has to normalise the expression with

$Norm^{2}=1+\sum_{k\neq n}(\frac{\mu_{nk}}{m_{n}^{(0)}-m_{k}^{(0)}})^{2}.$
(33)

This leads then to the following expression for the mass eigenstates

$\ket{\vec{m}}=\begin{pmatrix}1&\frac{\mu_{12}}{m_{1}-m_{2}}&\dots\\\
\frac{\mu_{21}}{m_{2}-m_{1}}&1&\dots\\\
\vdots&\dots&\ddots\end{pmatrix}\ket{\vec{n}},$ (34)

symbolically

$\ket{\vec{m}}=U\ket{\vec{n}}.$ (35)

Now one has to invert U in order to find the expression for the space states.
In order to invert the matrix U we use the equation

$(A+X)^{-1}=A^{-1}+Y,$ (36)

with

$Y=-A^{-1}XA^{-1},$ (37)

and X beeing in this case the perturbation matrix V. One therefore gets for
$U^{-1}$

$U^{-1}=\begin{pmatrix}1&-\frac{\mu_{12}}{m_{1}-m_{2}}&\dots\\\
-\frac{\mu_{21}}{m_{2}-m_{1}}&1&\dots\\\ \vdots&\dots&\ddots\end{pmatrix}.$
(38)

This is how the mixing with the states of extra space takes place in case of a
single flavour of SM neutrino. In particular, the above reproduces the results
of such mixings in ADDM Arkani-Hamed  _et al._ (2001) and in DR Dvali and Redi
(2009) for the case of a single flavor.

##  IV Generalisation to three flavour case

We now generalize the discussion for the case of three flavors of SM
neutrinos. The simplest (but unrealistic) case is if all the three flavor
neutrinos have their own mixing partners in the extra space. In such a case
the mass matrix has the following block-diagonal form

$\mathcal{M}=\begin{pmatrix}M_{e}&0&0\\\ 0&M_{\mu}&0\\\
0&0&M_{\tau}\end{pmatrix},$ (39)

where the $M_{\alpha}$ stand for the mass matrices of the different flavours.
Each of them has a form analogous to (27). Of course, we must take mixing
among the different flavours into account. This is necessary for
phenomenological consistency. In particular, to make the SM three flavour
neutrino oscillations possible. In order to incorporate this phenomenon we
have to depart from the above block-diagonal structure. We therefore write,

$\mathcal{M}=\begin{pmatrix}M_{e}&e\mu&e\tau\\\ e\mu&M_{\mu}&\mu\tau\\\
e\tau&\mu\tau&M_{\tau}\end{pmatrix},$ (40)

where we denote with the $\alpha\beta$ ($\alpha,\beta=e,\mu,\tau)$ entries the
mixing matrices among the different space state partners of different
flavours. In order to increase the precision of the perturbative calculation
we treat the mixing of the flavour ground states (i.e., the direct mixing
among SM neutrinos) as part of the perturbed matrix and not as a part of the
perturbation matrix V. This leads to the following expressions for the mass
eigenstates of the three active neutrinos (we denoted the entries of the SM
like mixing elements as $U_{ei}^{-1}$)

$\ket{m_{1}^{e}}=U_{e1}^{-1}\ket{e}+U_{e2}^{-1}\ket{\mu}+U_{e3}^{-1}\ket{\tau}+\sum_{k=2}\frac{U_{e1}^{-1}\mu_{1k}^{e}+U_{e2}^{-1}e\mu_{1k}+U_{e3}^{-1}e\tau_{1k}}{m_{1}^{e}-m_{k}^{e}}\ket{k_{1}^{e}}+\\\
\\\
\sum_{k=2}\frac{U_{e1}^{-1}e\mu_{1k}+U_{e2}^{-1}\mu_{1k}^{\mu}+U_{e3}^{-1}\mu\tau_{1k}}{m_{1}^{e}-m_{k}^{\mu}}\ket{k_{1}^{\mu}}+\sum_{k=2}\frac{U_{e1}^{-1}e\tau_{1k}+U_{e2}^{-1}\mu\tau_{1k}+U_{e3}^{-1}\mu_{1k}^{\tau}}{m_{1}^{e}-m_{k}^{\tau}}\ket{k_{1}^{\tau}}.$
(41)

This is the expression for the lightest mass eigenstate and we identify it
with the dominant mass eigenstate for the electron neutrino. We have to invert
this expression now in an analogue way as in the one flavour case and in order
to do so we assume that

$U_{e1}>>U_{e2},U_{e3}>>e\mu_{1i},e\tau_{1i}.$ (42)

Then we can write the interaction eigenstate approximately as

$\ket{\nu_{e}}=U_{e1}\ket{m_{1}^{e}}+U_{e2}\ket{m_{1}^{\mu}}+U_{e3}\ket{m_{1}^{\tau}}-U_{e1}(\sum_{k=2}\frac{U_{e1}^{-1}\mu_{1k}^{e}+U_{e2}^{-1}e\mu_{1k}+U_{e3}^{-1}e\tau_{1k}}{m_{1}^{e}-m_{k}^{e}}\ket{m_{k}^{e}}\\\
\\\
+\sum_{k=2}\frac{U_{e1}^{-1}e\mu_{1k}+U_{e2}^{-1}\mu_{1k}^{\mu}+U_{e3}^{-1}\mu\tau_{1k}}{m_{1}^{e}-m_{k}^{\mu}}\ket{m_{k}^{\mu}}+\sum_{k=2}\frac{U_{e1}^{-1}e\tau_{1k}+U_{e2}^{-1}\mu\tau_{1k}+U_{e3}^{-1}\mu_{1k}^{\tau}}{m_{1}^{e}-m_{k}^{\tau}}\ket{m_{k}^{\tau}}).$
(43)

If we assume that $U_{ei}$ are already normalized, the normalisation looks as
follows,

$N_{e}^{2}=1+U_{e1}(\sum_{k=2}\frac{U_{e1}^{-1}\mu_{1k}^{e}+U_{e2}^{-1}e\mu_{1k}+U_{e3}^{-1}e\tau_{1k}}{m_{1}^{e}-m_{k}^{e}})^{2}+(\sum_{k=2}\frac{U_{e1}^{-1}e\mu_{1k}+U_{e2}^{-1}\mu_{1k}^{\mu}+U_{e3}^{-1}\mu\tau_{1k}}{m_{1}^{e}-m_{k}^{\mu}})^{2}+\\\
\\\
(\sum_{k=2}\frac{U_{e1}^{-1}e\tau_{1k}+U_{e2}^{-1}\mu\tau_{1k}+U_{e3}^{-1}\mu_{1k}^{\tau}}{m_{1}^{e}-m_{k}^{\tau}})^{2}.$
(44)

We can simplify the expression for the flavour neutrino a little bit further
by assuming that the masses of the bulk states in the diagonal entries are the
same for all flavors. This means that

$m_{k}^{e}=m_{k}^{\mu}=m_{k}^{\tau}=m_{k}.$ (45)

We also want to assume that different cross mixing elements among different
flavors have the same structure as the mixing of bulk states with its own
flavor. This means that also the mixing parts $\alpha\beta_{1k}$ and
$\mu_{1k}^{\alpha}$ look like

$\mu_{1k}^{\alpha}=\mu f(m_{D}^{\alpha}),$ (46)

with the same overall constant $\mu$ and the same function $f$ depending on
the induced Dirac mass just differing by the argument. This leads then to the
following expression for the flavor eigenstate

$\ket{\nu_{e}}=U_{e1}\ket{m_{1}^{e}}+U_{e2}\ket{m_{1}^{\mu}}+U_{e3}\ket{m_{1}^{\tau}}-U_{e1}\sum_{\alpha=1}^{3}\sum_{k=1}\frac{\mu_{1k}^{e}U_{e1}^{-1}+\mu_{1k}^{\mu}U_{e2}^{-1}+\mu_{1k}^{\tau}U_{e3}^{-1}}{m^{e}-m_{k}}\ket{k^{\alpha}}.$
(47)

Now let us drop the assumption (42) and give for the simplified equation (47)
the expression for a larger cross mixing among the SM neutrinos which is a
more realistic scenario. Then the equation gets modified in the following way

$\ket{\nu_{e}}=U_{e1}\ket{m_{1}^{e}}+U_{e2}\ket{m_{1}^{\mu}}+U_{e3}\ket{m_{1}^{\tau}}-\sum_{\alpha=1}^{3}\sum_{k=1}\frac{\overrightarrow{U_{e}}\overrightarrow{C}}{m^{e}-m_{k}}\ket{k^{\alpha}},$
(48)

with

$\overrightarrow{U_{e}}=\begin{pmatrix}U_{e1}\\\ U_{e2}\\\
U_{e3}\end{pmatrix},$ (49)

and

$\overrightarrow{C}=\begin{pmatrix}\mu_{1k}^{e}U_{e1}^{-1}+\mu_{1k}^{\mu}U_{e2}^{-1}+\mu_{1k}^{\tau}U_{e3}^{-1}\\\
\mu_{1k}^{e}U_{\mu 1}^{-1}+\mu_{1k}^{\mu}U_{\mu 2}^{-1}+\mu_{1k}^{\tau}U_{\mu
3}^{-1}\\\ \mu_{1k}^{e}U_{\tau 1}^{-1}+\mu_{1k}^{\mu}U_{\tau
2}^{-1}+\mu_{1k}^{\tau}U_{\tau 3}^{-1}\end{pmatrix}.$ (50)

In an analogue way, the equation (43) can get modified.

With these developed tools we can now calculate a general expression for a
flavor eigenstate of a neutrino which has mixing with a large number of extra
states and also includes mixing with the other flavor states. The investigated
case of non-degenerated non-perturbed eigenstates can be used for the ADDM
scenario and via a cross check we can reproduce the one-flavor equation
obtained in Dvali and Smirnov (1999). In the following section we show how one
can calculate the flavor states for a highly degenerated mass matrix which are
important for the DR scenario.

##  V Highly Symmetric Mass Matrices

So far we investigated the case of a very general mass matrix which contains
mixing with all the states of the extra space. But also the cases where these
mass matrices have a specific structure and are highly symmetric are of
interest. One specific example for this is the ”Many Species Theory” with
exact copies of the SM.

We now want to present a way how we can deal with this kind of matrices when
they are block-wise grouped in their mass matrix. without a loss of generality
we illustrate this on example of the DR scenario. A grouping of the different
copies of the SM can occur according to the VEV of the Higgs doublets. Notice
that even if copies obey a strict permutation symmetry, this symmetry can be
spontaneously broken by the VEVs of the Higgs doublets. This is because, due
to low cutoff and the cross couplings among different doublets, the potential
can admit vacua in which Higgs doublets of different copies take different
VEVs, $\langle H_{j}\rangle=v_{j}$.

Also, because in principal a Majorana mass term for neutrinos is not forbidden
neither by gauge nor by permutation symmetry, we will investigate additionally
to the common Dirac operator

$(HL)_{i}\lambda_{ij}\nu_{Rj},$ (51)

and also a Weinberg operator of the form

$(\bar{L^{c}}i\sigma_{2}H)_{i}\lambda_{ij}(Hi\sigma_{2}L)_{j}\,,$ (52)

where the indices $i$ and $j$ label different copies and $L$ beeing the SU(2)
doublet and $\sigma_{2}$ acting in this space. As previously, we assume that
Yukawa couplings obey the $P(N)$-symmetry and therefore have the form of (3).

Notice that the operators (52) break the global lepton number symmetries
explicitly.

The key now is to assign different Higgs VEVs to different SM copies. We group
the copies with the same VEVs in diagonal blocks of the neutrino mass matrix.

Let us consider a minimal case of this sort in which the VEVs take two
possible values $v$ and $v^{\prime}$. We take a subgroup of size $N<N_{TOTAL}$
and assign the VEV $v$. To the rest of the species $M=N_{TOTAL}-N$ we assign
the VEV $v^{\prime}$. This assignment can be expressed as,

$v_{i}=\begin{cases}v&\text{for }i\leq N\\\ v^{\prime}&\text{for
}i>N\,.\end{cases}$ (53)

Taking this into account and plugging it into the operators (51) and (52) one
gets the following mass matrices respectively

$M^{\text{Majorana}}=\begin{pmatrix}av^{2}&bv^{2}&bv^{2}&...&bv^{2}&bv^{\prime}v&&...&&bv^{\prime}v\\\
bv^{2}&av^{2}&bv^{2}&...&bv^{2}&\vdots&&\ddots&&\vdots\\\
\vdots&&\ddots&&\vdots&\vdots&&\ddots&&\vdots\\\
bv^{2}&&...&&av^{2}&bv^{\prime}v&&...&&bv^{\prime}v\\\
bv^{\prime}v&&...&&bv^{\prime}v&av^{\prime^{2}}&bv^{\prime^{2}}&bv^{\prime^{2}}&...&bv^{\prime^{2}}\\\
\vdots&&\ddots&&\vdots&bv^{\prime^{2}}&av^{\prime^{2}}&bv^{\prime^{2}}&...&bv^{\prime^{2}}\\\
\vdots&&\ddots&&\vdots&\vdots&&\ddots&&\vdots\\\
bv^{\prime}v&&...&&bv^{\prime}v&bv^{\prime^{2}}&&...&&av^{\prime^{2}}\end{pmatrix},$
(54)

and

$M^{\text{Dirac}}=\begin{pmatrix}av&bv&bv&...&bv&bv&&...&&bv\\\
bv&av&bv&...&bv&\vdots&&\ddots&&\vdots\\\
\vdots&&\ddots&&\vdots&\vdots&&\ddots&&\vdots\\\ bv&&...&&av&bv&&...&&bv\\\
bv^{\prime}&&...&&bv^{\prime}&av^{\prime}&bv^{\prime}&bv^{\prime}&...&bv^{\prime}\\\
\vdots&&\ddots&&\vdots&bv^{\prime}&av^{\prime}&bv^{\prime}&...&bv^{\prime}\\\
\vdots&&\ddots&&\vdots&\vdots&&\ddots&&\vdots\\\
bv^{\prime}&&...&&bv^{\prime}&bv^{\prime}&&...&&av^{\prime}\end{pmatrix}.$
(55)

The diagonalization of the above mass matrices will be performed in the next
section.

###  V.1 Diagonalizing of the Majorana mass matrices

In this part, the Majorana mass matrices will be diagonalized. Because the
resulting expressions are rather complex the diagonalization procedure will be
done within certain limits. The two limits which will be discussed are
$v^{\prime}\gg v$ and vice versa.

####  V.1.1 The symmetric breaking limit of the mass matrix

Here the focus lies on the Majorana mass matrix (54) and we make the
assumption that the breaking of $P(N)$ is into two equally large sectors,
$M=N$. In order to simplify the resulting equations even further, we will also
assume that $v^{\prime}\gg v$. we put the value $v^{\prime}$ close to the
cutoff of the theory which is $\sim$TeV. This will lead later to very
interesting phenomenological implications.

We start diagonalizing (54) noticing that it is a $2\times 2$ block matrix. As
the first step, we multiply the matrix with the following transformation
matrix

$U^{\prime}=\begin{pmatrix}S&0\\\ 0&S\end{pmatrix},$ (56)

where S is the diagonalization matrix of a matrix of just ones (a matrix with
the same entry everywhere)

$S=\begin{pmatrix}1&-1&\dots&\dots\\\ \vdots&1&0&\dots\\\
\vdots&0&\ddots&0&\dots\end{pmatrix}.$ (57)

This leads then to the following expression

$U^{\prime-1}M^{Majorana}U^{\prime}=U^{\prime-1}\begin{pmatrix}A&B\\\
C&D\end{pmatrix}U^{\prime}=\begin{pmatrix}S^{-1}AS&S^{-1}BS\\\
S^{-1}CS&S^{-1}DS\end{pmatrix},$ (58)

where the matrices A, B, D, C denote the block entries of the mass matrix. One
can separate the diagonal entries of the matrices A and D from the rest of the
matrix and turn this one into a matrix with just the same entry

$A=v\lambda_{ij}=\begin{pmatrix}(a-b)v^{2}&0&\dots\\\ 0&\ddots&0\\\
\vdots&\dots&(a-b)v^{2}\end{pmatrix}+\begin{pmatrix}bv^{2}&\dots&\dots\\\
\vdots&\ddots&\vdots\\\ \vdots&\dots&bv^{2}\end{pmatrix}.$ (59)

The diagonal part commutes with $S$ and one is therefore left with the
following matrix

$\begin{pmatrix}(a-b)v^{2}+Nbv^{2}&0&\dots&Nbvv^{\prime}&0&\dots&0\\\
0&(a-b)v^{2}&0&\dots&0&\dots&0\\\ \vdots&0&\ddots&0&&\dots&0\\\
Nbvv^{\prime}&0&\dots&(a-b)v^{\prime^{2}}+Nbv^{\prime^{2}}&0&\dots&0\\\
0&\dots&&0&(a-b)v^{\prime^{2}}&\dots&0\\\
0&\dots&&\dots&0&\ddots&0\end{pmatrix}.$ (60)

Now one can take out the diagonal element and can bring it down to a $2\times
2$ matrix of the following form

$\begin{pmatrix}Nbv^{2}&Nbvv^{\prime}\\\
Nbvv^{\prime}&Nbv^{\prime^{2}}+(a-b)(v^{\prime^{2}}-v^{2})\end{pmatrix}.$ (61)

In order to find the mass eigenstates one has to manipulate (60) further with
the following rotation matrix,

$\begin{pmatrix}cos(\theta)&sin(\theta)\\\
-sin(\theta)&cos(\theta)\,\end{pmatrix},$ (62)

with the rotation angel

$\theta=\frac{1}{2}\arctan(2\frac{vv^{\prime}}{v^{\prime^{2}}-v^{2}})\,.$ (63)

The rotation matrix multiplied with the $U^{\prime}$ matrix gives the
transformation matrix of the mass matrix. The result is

$U=\begin{pmatrix}cos(\theta)&-1&\dots&-1&sin(\theta)&0&&\dots\\\
cos(\theta)&1&0&\dots&sin(\theta)&0&&\dots\\\
\vdots&0&\ddots&&\vdots&0&&\dots\\\
-sin(\theta)&0&\dots&0&cos(\theta)&-1&&\dots\\\
\vdots&0&\dots&0&\vdots&1&0&\dots\\\ \end{pmatrix}.$ (64)

From here we can see that just two states are affected by the symmetry
breaking and the rest stays degenerated with the eigenvalues $(a-b)v^{2}$ and
$(a-b)v^{\prime^{2}}$. Therefore we can rewrite the new heavy states in terms
of the heavy states of the unbroken permutation subset, which we already have
encountered in the equations (6) and (7). Again in order to simplify the
rotation angle (63) we use the limit $v^{\prime}\gg v$. The result is then the
following

$n_{H}^{b}=n_{H}-\frac{v}{v^{\prime}}\tilde{n}_{H},$ (65)
$\tilde{n}_{H}^{b}=\tilde{n}_{H}+\frac{v}{v^{\prime}}n_{H},$ (66)

where we used tilde for the $v^{\prime}$ sector. When one solves now for
species states of the two different sectors one gets the following two
expressions

$n_{1}=\sqrt{\frac{N-1}{N}}n_{1}^{\prime}+\frac{1}{\sqrt{N}}n_{H}^{b}+\frac{1}{\sqrt{N}}\frac{v}{v^{\prime}}\tilde{n}_{H}^{b},$
(67)

(notice that for the sake of simplicity the overall normalization factor is
suppressed)

$n_{N+1}=\sqrt{\dfrac{N-1}{N}}\tilde{n}_{1}^{\prime}+\dfrac{1}{\sqrt{N}}\tilde{n}_{H}^{b}-\dfrac{1}{\sqrt{N}}\dfrac{v}{v^{\prime}}n_{H}^{b}\,,$
(68)

with the Eigenvalues of the mass eigenstates:

$m_{1}^{\prime}=(a-b)v^{2},$ (69)
$\tilde{m}_{1}^{\prime}=(a-b)v^{\prime^{2}},$ (70) $m_{H}=2(a-b)v^{2},$ (71)
$\tilde{m}_{H}=\text{super massive}.$ (72)

This is a rather interesting result for phenomenology at which we will have a
closer look later. We want to point out that the common heavy eigenstate
$n_{H}$ has a mass independent of $N$, which was not the case in the original
mechanism. This means that the common heavy eigenstate is not super heavy and
neutrino oscillations into this state are therefore possible.

####  V.1.2 Asymmetric breaking pattern with a large heavy sector

One can also break the symmetry in a way that the sectors include different
amounts of copies, $N\neq M$, where $N$ stands for the sector with a VEV of
$v$ and $M$ for $v^{\prime}$. In order to keep the expressions for the final
results in a simple form, we take the limit $Mv^{\prime^{2}}\gg Nv^{2}$.
After repeating the same diagonalization procedure, the matrix (60) in this
case has the following form

$\begin{pmatrix}(a-b)v^{2}+Nbv^{2}&0&\dots&Mbvv^{\prime}&0&\dots&0\\\
0&(a-b)v^{2}&0&\dots&0&\dots&0\\\ \vdots&0&\ddots&0&&\dots&0\\\
Nbvv^{\prime}&0&\dots&(a-b)v^{\prime^{2}}+Mbv^{\prime^{2}}&0&\dots&0\\\
0&\dots&&0&(a-b)v^{\prime^{2}}&\dots&0\\\
0&\dots&&\dots&0&\ddots&0\end{pmatrix}.$ (73)

Before we can perform the rotation, we have to make an intermediate step which
brings the off-diagonal entries to the same value. Therefore one applies
another transformation matrix of the following form

$\begin{pmatrix}1&0&\dots&\dots&\dots&\dots\\\ 0&\ddots&0&\dots&\dots&\dots\\\
\vdots&\dots&\kappa&0&\dots&\dots\\\ \vdots&\dots&0&1&0&\dots\\\
\vdots&\dots&\dots&0&\ddots&0\end{pmatrix},$ (74)

with $\kappa$ being

$\kappa=\sqrt{\dfrac{N}{M}}.$ (75)

After this procedure the off-diagonal entries are equal and one can perform
the rotation like in the symmetric case. Correspondingly one gets a mixing
angle of the following form

$\theta=\dfrac{1}{2}\arctan(-2\frac{\sqrt{N}\sqrt{M}bvv^{\prime}}{Nbv^{2}-Mbv^{\prime^{2}}}).$
(76)

The resulting transformation matrix is

$U=\begin{pmatrix}cos(\theta)&-1&\dots&-1&sin(\theta)&0&&\dots\\\
cos(\theta)&1&0&\dots&sin(\theta)&0&&\dots\\\
\vdots&0&\ddots&&\vdots&0&&\dots\\\ -\kappa sin(\theta)&0&\dots&0&\kappa
cos(\theta)&-1&&\dots\\\ \vdots&0&\dots&0&\vdots&1&0&\dots\\\ \end{pmatrix}.$
(77)

and $\theta$ simplified to

$\theta=\sqrt{\frac{N}{M}}\frac{v}{v^{\prime}}.$ (78)

The resulting mass eigenstates are then

$n_{H}^{b}=n_{H}-\frac{N}{M}\dfrac{v}{v^{\prime}}\tilde{n}_{H},$ (79)
$\tilde{n}_{H}^{b}=\tilde{n}_{H}+\frac{v}{v^{\prime}}n_{H},$ (80)

with the eigenvalues

$m_{H}=(a-b)v^{2},$ (81) $\tilde{m}_{H}=\text{super massive}.$ (82)

The corresponding copy eigenstates are

$n_{1}=\sqrt{\frac{N-1}{N}}n_{1}^{\prime}+\frac{1}{\sqrt{N}}n_{H}^{b}+\frac{1}{\sqrt{N}}\frac{N}{M}\frac{v}{v^{\prime}}\tilde{n}_{H}^{b},$
(83)
$n_{N+1}=\sqrt{\frac{M-1}{M}}+\frac{1}{\sqrt{M}}\tilde{n}_{H}^{b}-\frac{1}{\sqrt{M}}\frac{v}{v^{\prime}}n_{H}^{b}.$
(84)

We see that the mass $m_{H}$ is the same as for the degenerated mass
eigenstates.

####  V.1.3 Asymmetric breaking pattern with a large light sector

One can also investigate the case with a large light sector $Nv^{2}\gg
Mv^{\prime^{2}}$. In this case the procedure is the same and (73) stays
untouched. The resulting mixing angle is

$\theta=-\sqrt{\frac{M}{N}}\frac{v^{\prime}}{v}.$ (85)

The eigenvalues are

$m_{H}=(a-b)v^{\prime^{2}},$ (86) $\tilde{m}_{H}=\text{super massive}.$ (87)

The corresponding eigenstates are given by

$\tilde{n}_{H}^{b}=\frac{v}{v^{\prime}}n_{H}+\tilde{n}_{H},$ (88)
$n_{H}^{b}=\tilde{n}_{H}-\frac{Mv^{\prime}}{Nv}n_{H}.$ (89)

The copy eigenstates are

$n_{1}=\sqrt{\dfrac{N-1}{N}}n^{\prime}_{1}-\frac{1}{\sqrt{N}}\frac{v^{\prime}}{v}n_{H}^{b}+\frac{1}{\sqrt{N}}\frac{v^{\prime}}{v}\tilde{n}_{H}^{b},$
(90)
$n_{N+1}=\sqrt{\frac{M-1}{M}}\tilde{n}_{1}^{\prime}+\frac{1}{\sqrt{M}}n_{H}^{b}+\frac{1}{\sqrt{M}}\frac{M}{N}(\frac{v^{\prime}}{v})^{2}\tilde{n}_{H}^{b}.$
(91)

Now the situation is or reversed. The $m_{H}$ goes to the eigenvalues of the
degenerated states of the heavy sector. Taking $v^{\prime}$ close to the
cutoff ($\sim$ TeV) the estimated values of $m_{H}$ could be up to $\sim$keV.

###  V.2 Diagonalizing of the Dirac mass matrix

Let us now turn to a diagonalization of the Dirac mass matrix which results
from the operator (51). Procedure is similar but some details differ from the
Majorana case.

####  V.2.1 The symmetric breaking limit of the Dirac mass matrix

After the first steps similar to the ones taken for the Majorana case, the
matrix has the form

$\begin{pmatrix}(a-b)v+Nbv&0&\dots&Nbv&0&\dots&0\\\
0&(a-b)v&0&\dots&0&\dots&0\\\ \vdots&0&\ddots&0&&\dots&0\\\
Nbv^{\prime}&0&\dots&(a-b)v^{\prime}+Nbv^{\prime}&0&\dots&0\\\
0&\dots&&0&(a-b)v^{\prime}&\dots&0\\\ 0&\dots&&\dots&0&\ddots&0\end{pmatrix}.$
(92)

Now the situation is different because the matrix (92) is not symmetric (60).
Because of this one has to introduce the auxiliary parameter $\kappa$ already
in the symmetric breaking limit

$\kappa=\sqrt{\frac{v^{\prime}}{v}}$ (93)

and the rotation angle is

$\theta=\sqrt{\frac{v}{v^{\prime}}}.$ (94)

The resulting heavy eigenstates are

$n_{H}^{b}=\frac{1}{\sqrt{2}}n_{H}-\frac{1}{\sqrt{2}}\tilde{n}_{H},$ (95)
$\tilde{n}_{H}^{b}=\tilde{n}_{H}+\frac{v}{v^{\prime}}n_{H},$ (96)

with the eigenvalues

$m_{H}=2(a-b)v,$ (97) $\tilde{m}_{H}=\text{super massive}.$ (98)

Solving for the the species states leads to

$n_{1}=\sqrt{\frac{N-1}{N}}n_{1}^{\prime}+\sqrt{\frac{2}{N}}n^{b}_{H}+\frac{1}{\sqrt{N}}\tilde{n}^{b}_{H},$
(99)

$n_{N+1}=\sqrt{\frac{N-1}{N}}n_{1}^{\prime}-\sqrt{\frac{2}{N}}\frac{v}{v^{\prime}}n^{b}_{H}+\frac{1}{\sqrt{N}}\tilde{n}^{b}_{H}.$
(100)

####  V.2.2 Asymmetric breaking pattern with a large heavy sector

Now we turn again to the cases of asymmetric breaking the permutation group.
We investigate the scenario with $M\gg N$. In order to do so, in the matrix
(92) we replace for one sector $N$ with $M$ like in the Majorana case. In this
scenario the auxiliary parameter becomes

$\kappa=\sqrt{\frac{Nv^{\prime}}{Mv}},$ (101)

and the resulting rotation angel is

$\theta=\sqrt{\frac{Nv}{Mv^{\prime}}}.$ (102)

The mass eigenstates are

$n_{H}^{b}=n_{H}-\frac{N}{M}\tilde{n}_{H},$ (103)
$\tilde{n}_{H}^{b}=\frac{v}{\sqrt{v^{2}+v^{\prime
2}}}n_{H}+\frac{v^{\prime}}{\sqrt{v^{\prime 2+v^{2}}}}\tilde{n}_{H},$ (104)

with the eigenvalues

$m_{H}=(a-b)v,$ (105) $\tilde{m}_{H}=\text{super massive}.$ (106)

The species states are

$n_{1}=\sqrt{\frac{N-1}{N}}n_{1}^{\prime}+\frac{1}{\sqrt{N}}n^{b}_{H}+\frac{1}{\sqrt{N}}\frac{N}{M}\tilde{n}^{b}_{H},$
(107)
$n_{N+1}=\sqrt{\frac{M-1}{M}}\tilde{n}_{1}^{\prime}+\frac{1}{\sqrt{M}}\tilde{n}^{b}_{H}-\frac{1}{\sqrt{M}}\frac{v}{v^{\prime}}n^{b}_{H}.$
(108)

Again the oscillation in our copy has an extremely small frequency because the
$\Delta m$ goes to $0$ but, on the other hand, it is suppressed as $1/N$ but
$N$ in the present case is not large.

####  V.2.3 Asymmetric breaking pattern with a large light sector

Finally, let us investigate the case with $N\gg M$ and $v^{\prime}\gg v$. The
auxiliary parameter $\kappa$ stays the same as in equation (101). The rotation
angel is

$\theta=-\sqrt{\frac{Mv^{\prime}}{Nv}},$ (109)

with the eigenstates

$n_{H}^{b}=\tilde{n}_{H}-\frac{M}{N}n_{H},$ (110)
$\tilde{n}_{H}^{b}=\tilde{n}_{H}+\frac{v}{v^{\prime}}n_{H}.$ (111)

The eigenvalues are

$m_{H}=(a-b)v^{\prime},$ (112) $\tilde{m}_{H}=\text{super massiv}.$ (113)

The corresponding species states are

$n_{1}=\sqrt{\frac{N-1}{N}}n_{1}^{\prime}-\frac{1}{\sqrt{N}}\frac{v^{\prime}}{v}n^{b}_{H},$
(114)
$n_{N+1}=\sqrt{\frac{M-1}{M}}\tilde{n}_{1}^{\prime}+\frac{1}{\sqrt{M}}n^{b}_{H}.$
(115)

##  VI Phenomenology

We now want to turn to the phenomenological implications of the theoretical
framework we built up in the previous sections. We will do this within a
specific theory. First, we want to point out that the first steps in this
topic were already done in Dvali and Smirnov (1999) for ADDM and Dvali and
Redi (2009) in DR. But in both cases, just one flavour case of the SM neutrino
were investigated. We now aim to generalize this analysis to the three flavour
case using the general framework which we presented before.

###  VI.1 Phenomenology of ADDM model

First, we want to discuss the Phenomenology of the ADDM scenario in a
realistic three flavour setting. In order to do so, we want to use the
framework of section 3 and apply our generally derived formulas to the ADDM
case. First we have to define the mass matrix we are investigating. For this
we take the Ansatz from Dvali and Smirnov (1999) and generalize it to the
three flavour case. To write down the resulting mass matrix we assume that the
flavor symmetry is preserved in the bulk. This leads to the effect that the
mixing among bulk states is diagonal. The resulting mass matrix is

$\begin{pmatrix}m_{ee}&\sqrt{2}m_{ee}&\dots&\dots&m_{e\mu}&\sqrt{2}m_{ee}&\dots&m_{e\tau}&\sqrt{2}m_{ee}&\dots\\\
0&\frac{1}{R}&0&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\\
0&0&\ddots&0&\dots&\dots&\dots&\dots&\dots&\dots\\\
0&0&0&\frac{k}{R}&0&\dots&\dots&\dots&\dots&\dots\\\ m_{\mu
e}&\sqrt{2}m_{\mu\mu}&\dots&\sqrt{2}m_{\mu\mu}&m_{\mu\mu}&\sqrt{2}m_{\mu\mu}&\dots&m_{\mu\tau}&\sqrt{2}m_{\mu\mu}&\dots\\\
0&\dots&\dots&\dots&0&\frac{1}{R}&0&\dots&\dots&\dots\\\
\vdots&\dots&\dots&\dots&\dots&0&\ddots&0&\dots&\dots\\\ m_{\tau
e}&\sqrt{2}m_{\tau\tau}&\dots&\dots&m_{e\tau}&\sqrt{2}m_{\tau\tau}&\dots&m_{e\tau}&\sqrt{2}m_{\tau\tau}&\dots\\\
0&\dots&\dots&\dots&\dots&\dots&\dots&0&\frac{1}{R}&\dots\\\ \end{pmatrix}.$
(116)

In order to perform the diagonalization of this mass matrix one has to define
the parametrization of the $U_{PMNS}$ matrix

$U_{PMNS}=\begin{pmatrix}c_{12}c_{13}&&c_{13}s_{12}&&s_{13}\\\
-c_{23}s_{12}e^{i\phi}-c_{12}s_{13}s_{23}&&c_{12}c_{23}e^{i\phi}-s_{12}s_{13}s_{23}&&c_{13}s_{23}\\\
s_{23}s_{12}e^{i\phi}-c_{12}c_{23}s_{13}&&-c_{12}s_{23}e^{i\phi}-c_{23}s_{12}s_{13}&&c_{13}c_{23}\end{pmatrix}.$
(117)

With this PMNS-matrix parametrization we can use the formula (48) to calculate
the the expression for e.g., the muon neutrino. The result is

$\ket{\nu_{\mu}}=U_{\mu 1}\ket{m_{1}^{e}}+U_{\mu 2}\ket{m_{1}^{\mu}}+U_{\mu
3}\ket{m_{1}^{\tau}}+\overrightarrow{U_{\mu}}\overrightarrow{C}\sum_{\alpha}\sum_{k}\frac{1}{k}\ket{k_{\alpha}}.$
(118)

with $\overrightarrow{C}_{ADD}$

$\overrightarrow{C}_{ADD}=\begin{pmatrix}\xi^{e}U_{e1}^{-1}+\xi^{\mu}U_{e2}^{-1}+\xi^{\tau}U_{e3}^{-1}\\\
\xi^{e}U_{\mu 1}^{-1}+\xi^{\mu}U_{\mu 2}^{-1}+\xi^{\tau}U_{\mu 3}^{-1}\\\
\xi^{e}U_{\tau 1}^{-1}+\xi^{\mu}U_{\tau 2}^{-1}+\xi^{\tau}U_{\tau
3}^{-1}\end{pmatrix}.$ (119)

and the normalisation

$N^{2}_{e}=1+\frac{\pi^{2}}{2}(\overrightarrow{U_{\mu}}\overrightarrow{C})^{2}.$
(120)

Notice that the parameters $\xi^{\alpha}$ are related with each other via

$\xi^{e}\propto
m_{e}\approx\mathcal{O}(1)m_{\mu}\approx\mathcal{O}^{\prime}(1)m_{\tau},$
(121)

and therefore the key parameter in this expression is just the size of the
dominant extra dimension $R$.

In order to get an impression on the dependence of this deviation of the
composition of an muon neutrino from the Standard Model composition, one can
calculate the survival probability. We assume that just the lowest modes of
the KK towers contribute to the oscillations, since the higher modes get
averaged out due to large mass-splittings. Then the survival probability reads
as

$P(\nu_{\mu}\rightarrow\nu_{\mu})=\frac{1}{|N_{\mu}|^{4}}\left[\sum_{i}\sum_{j}|U_{\mu
i}|^{2}|U_{\mu
j}|^{2}e^{\frac{i(m_{i}^{2}-m_{j}^{2})}{2E}}+3|\overrightarrow{U_{\mu}}\overrightarrow{C}|^{4}\left(\frac{\pi^{4}}{90}-1\right)\right].$
(122)

with $E$ beeing the energy of the investigated neutrino. This can be compared
to the original result in Dvali and Smirnov (1999) for the one flavor case

$P=\frac{1}{(1+(\pi^{2}/6)\xi^{2})^{2}}\left[(1+\xi^{2})^{2}+\left(\frac{\pi^{4}}{90}-1\right)\xi^{4}-\xi^{2}\sin^{2}\frac{(m_{n}^{2}-m_{D}^{2})t}{4E}\right].$
(123)

From these two equations one can see that some properties of the one flavor
case also appear in a modified way in the three flavor equation. Particularly
interesting is how in ADDM models the averaged out modes influence the
survival probability by a term proportional to
$\left(\frac{\pi^{4}}{90}-1\right)$ if just the lowest mode is not averaged
out. Of course the experimental setup and the specific mass splitting
determine how much modes can be resolved in the oscillations. As more modes
participate as less important the contribution of the averaged out modes are.

For comparizon of the three flavor scenario with SM prediction, we take the
latest results of the nu-fit collaboration Esteban  _et al._ (2020) which are:

$\theta_{12}=33,44^{\circ},\theta_{23}=49,0^{\circ},\theta_{13}=8,57^{\circ},\delta_{CP}=195^{\circ}\,.$
(124)

With this data one can calculate the survival probability of an muon neutrino
depending on the parameter $\xi$ of the ADDM model. Figure 1 shows the result
of this calculation and how it deviates from the SM case.

Figure 1: Survival probability of an electron neutrino in a three flavour ADD
mixing scenario

Thus, the precision measurements of neutrino oscillations can put bounds on
the critical value $\xi$ of the ADDM model.

Moreover, this deviation from the SM case also affects the unitarity of the
Lepton Mixing Matrix. If the familiar three flavors of the SM neutrinos
exhaust the spectrum of neutral leptons, the $3\times 3$ mixing matrix we
measure in neutrino experiments must be unitary. This does not hold if there
exist more than three neutrinos. In this case the Lepton Mixing Matrix is not
a $3\times 3$ matrix anymore but a $(3+n)\times(3+n)$ matrix where $n$ is the
number of additional neutrinos.

Nevertheless, in the experiments that are sensitive to active species, we
would still measure only the $3\times 3$ part of the full Lepton Mixing
Matrix. Since this restricted part in general will not be unitary, we will
effectively register a deviation from unitarity. This happens also in the ADDM
model where neutrinos can oscillate into the KK modes. Using the above
results, the $3\times 3$ part of the full Lepton Mixing Matrix would get
modified in the following way

$\begin{pmatrix}U_{ee}\frac{1}{N_{e}}&U_{e\mu}\frac{1}{N_{e}}&U_{e\tau}\frac{1}{N_{e}}\\\
U_{\mu
e}\frac{1}{N_{\mu}}&U_{\mu\mu}\frac{1}{N_{\mu}}&U_{\mu\tau}\frac{1}{N_{\mu}}\\\
U_{\tau
e}\frac{1}{N_{\tau}}&U_{\tau\mu}\frac{1}{N_{\tau}}&U_{\tau\tau}\frac{1}{N_{\tau}}\\\
\end{pmatrix}.$ (125)

Now the task is to measure the parameters of the well known PMNS matrix very
precisely and look for possible deviations from unitarity. Of course, this
feature is not unique to ADDM and something similar can be realized in other
models too. However, the above example provides us with concrete motivated
framework for setting bounds on the unitarity-violating parameters and using
them for discriminating between the different models. We are going to see this
explicitly in the next section when we discuss phenomenology of the DR model
and confront it with ADDM framework.

###  VI.2 Phenomenology of Dvali-Redi model

The generalisation of the mass matrix in the DR scenario goes as follows. Of
course, the general structure of the mass matrix is again similar to (40) but
this time the off-diagonal block matrices have the following form

$M_{\alpha\beta}=\begin{pmatrix}m_{\alpha\beta}&0&0&\dots\\\
0&m_{\alpha\beta}&0&\dots\\\ \vdots&0&\ddots&\vdots\end{pmatrix}.$ (126)

This specific structure comes from the fact that in this theory the mixing
among the different flavors can happen within a single copy since it is
determined by the physics of the SM. This leads to the following electron
neutrino eigenstate

$\ket{\nu_{e}}=\sqrt{\frac{N-1}{N}}(U_{e1}\ket{m_{1}}+U_{e2}\ket{m_{2}}+U_{e3}\ket{m_{3}})+\frac{1}{\sqrt{N}}(U_{e1}\ket{m_{1}^{H}}+U_{e2}\ket{m_{2}^{H}}+U_{e3}\ket{m_{3}^{H}}).$
(127)

The key parameter is the number of active species. Above we showed how we can
group the total number of species into light and heavy sectors. Now one can
investigate different scenarios with sectors which contain different numbers
of copies. Because we have access predominantly to our copy of the SM, for us
the scenarios with small number of active species in the sector our copy
belongs to are of special interest. Due to this reason we focus on the
scenarios with large heavy sectors that bring down the number of active
species in our sector.

Taking the two expressions we found earlier for the Weinberg- and Dirac
operator (83) and (107) and comparing them with each other, one sees that the
oscillation into the other sector is suppressed by the number of the active
species in the large sector. Therefore, one can safely ignore this
contribution, especially in the Weinberg case since there it is further
suppressed by the scale of the larger VEV $v^{\prime}$. Then the probability
of survival in the one-flavour case Dvali and Redi (2009) is given by

$P(t)=1-\frac{4}{N}\sin^{2}(\frac{\Delta m^{2}t}{2E}).$ (128)

In this scenario, the problem of observing the effect is shifted from the
large suppression by the amplitude into the extremely low frequency which
comes from very small splitting among the mass eigenstates. Nevertheless, this
case is still is of high potential interest for long-baseline experiments of
neutrino oscillations. Astrophysical sources of high neutrino fluxes could be
useful candidates for testing such scenarios. Of course, detection of
deviations from the expected neutrino flux in pure SM requires understanding
of the operation mechanisms of these sources to sufficiently high accuracy.

####  VI.2.1 Integrating out scenario

We observed that in small light sector scenarios the suppression of the
amplitude goes contrary to the frequency of the oscillations. A scenario that
can bring both parameters to the range of easier experimental accessibility is
the “integrating out” scenario which we will now consider. The goal is to
combine the advantages of the different above-studied scenarios into an unique
setup. Let us assume that the permutation symmetry is broken very heavily
among the two sectors: One sector containing a large number $M$ of copies and
another sector containing a smaller number $N$. This is the case which we have
already discussed above. However, let us now assume that due to additional
breaking of the permutation symmetry, the smaller sector is further split into
two sectors with the numbers $N^{\prime}$ and $M^{\prime}$. Obviously, the
primary breaking of perturbation symmetry into the $M$ and $N$ sectors is
still dominant and the secondary breaking does not affect physics up to
effects of order $\mathcal{O}(\frac{N}{M})$ which is already negligibly small.
Due to this reason, this sector can be considered as effectively decoupled
from the other sectors.

Let us now turn our attention to the leftover copies that are broken down into
two smaller sectors $N^{\prime}$ and $M^{\prime}$. Here we have a choice to
which sector our SM copy belongs. In particular, we can assume that the number
of copies in our sector $N^{\prime}$ is much larger than the other sector
$M^{\prime}$. This does not decrease the suppression of the amplitude very
much but allows us to liberate the value of the common heavy eigenstate
$m_{H}$ in which the neutrinos of both sectors oscillate and can make $\Delta
m$ large enough for bringing the frequency to a value comparable to the
ordinary oscillations of the SM. This scenario of splitting is analogous to
the large light sector scenario.

Overall this integrating out scenario enables us to free the both parameters
of the theory. It allows us to bring down the number of copies and
correspondingly oscillation frequencies to a scale that makes it observable
for experiments.

We can now calculate the oscillation in the three-flavor case with an equal
size splitting scenario. The equation for the survival probability can be
written down as

$P(\nu_{\mu}\rightarrow\nu_{\mu})=\sum_{i=1}^{6}\sum_{j=1}^{6}|U_{\mu
i}|^{2}|U_{\mu j}|^{2}e^{\frac{i(m_{i}^{2}-m_{j}^{2})}{2E}}.$ (129)

First we want to point out that in this expression no modes are averaged out
like in the ADDM scenario. The reason for this is because just three
additional mass eigenstates have to be included, meanwhile in ADDM scenarios
the KK tower can inhabit a very large number of additional mass eigenstates.
To analyze equation (129) further we split it up in the following way

$P(\nu_{\mu}\rightarrow\nu_{\mu})=\left(\frac{N-1}{N}\right)^{2}\sum_{i=1}^{3}\sum_{j=1}^{3}|U_{\mu
i}|^{2}|U_{\mu
j}|^{2}e^{\frac{i(m_{i}^{2}-m_{j}^{2})}{2E}}+\frac{N-1}{N^{2}}\sum_{i=1}^{3}\sum_{j=4}^{6}|U_{\mu
i}|^{2}|U_{\mu j}|^{2}e^{\frac{i(m_{i}^{2}-m_{j}^{2})}{2E}}\\\ \\\
+\frac{N-1}{N^{2}}\sum_{i=4}^{6}\sum_{j=1}^{3}|U_{\mu i}|^{2}|U_{\mu
j}|^{2}e^{\frac{i(m_{i}^{2}-m_{j}^{2})}{2E}}+\frac{1}{N^{2}}\sum_{i=4}^{6}\sum_{j=4}^{6}|U_{\mu
i}|^{2}|U_{\mu j}|^{2}e^{\frac{i(m_{i}^{2}-m_{j}^{2})}{2E}}.$ (130)

The first term in this expression represents the oscillations within the
flavors which are already known in the SM. For large $N$ these oscillations
are just slightly modified. One also sees that the dominant contributions are
coming from oscillations into the hidden species of order $\frac{1}{N}$ like
in the one flavor case in equation (129). The contributions of solely the BSM
terms are suppressed by $\frac{1}{N^{2}}$. Figure 2 shows the result of the
calculations for a muon neutrino.

Figure 2: Survival probability of an muon neutrino in a three flavor case in
DR model with an equal size splitting scenario

Let us also show the unitarity violation in the SM lepton mixing matrix which
is expected by the DR scenario. For this we have to look into the formula
(127). Picking out the $3\times 3$ block matrix in the upper left corner of
the resulting mixing matrix we can write

$\begin{pmatrix}\sqrt{\frac{N-1}{N}}U_{ee}&\sqrt{\frac{N-1}{N}}U_{e\mu}&\sqrt{\frac{N-1}{N}}U_{e\tau}\\\
\sqrt{\frac{N-1}{N}}U_{\mu
e}&\sqrt{\frac{N-1}{N}}U_{\mu\mu}&\sqrt{\frac{N-1}{N}}U_{\mu\tau}\\\
\sqrt{\frac{N-1}{N}}U_{\tau
e}&\sqrt{\frac{N-1}{N}}U_{\tau\mu}&\sqrt{\frac{N-1}{N}}U_{\tau\tau}\end{pmatrix}.$
(131)

This is the matrix which is measured by experiments. One can immediately see
that unitarity is violated by the overall factor of $\frac{N-1}{N}$. This is a
characteristic signature of the theory and it comes from the democratic
oscillation into the common heavy eigenstates of the neutrino matrix. The idea
of testing the violation of unitarity experimentally by using availible
experimental data is a still ongoing work Caldwell  _et al._ .

##  VII Conclusions

In this paper we have focused on neutrino masses in the class of theories in
which gravity cutoff is lowered down to $\sim$TeV scale. The two main
frameworks accomplishing this lowering of the cutoff are ADD Arkani-Hamed  _et
al._ (1998, 1999) and “many species” Dvali (2010a); Dvali and Redi (2008)
theories. In both cases, the decrease of the gravitational cutoff can be
understood as a result of “dilution” of the graviton wave function in certain
space labelled by a new coordinate. In both scenarios the volume of this space
can be measured by the number of particle species. Correspondingly the role of
the coordinate can be played by a species label. As shown in Dvali (2010a), in
case of ADD Arkani-Hamed  _et al._ (1998) the species represent the Kaluza-
Klein excitations. Correspondingly, the extra space has an actual geometric
meaning of large extra space dimensions. On the other hand, in “many species”
solution to the hierarchy problem Dvali (2010a); Dvali and Redi (2008), the
species can be arbitrary particles.

Previously it has been suggested that in both scenarios the small neutrino
masses emerge naturally due to the dilution of the wave-function of the
sterile (right-handed) neutrino in the extra space. Within ADDM this idea was
introduced in Arkani-Hamed  _et al._ (2001) and its phenomenological
implications were studied in Dvali and Smirnov (1999). In this case the wave-
function of the sterile neutrino is diluted in the actual geometric extra
space. This results into a highly suppressed Yukawa coupling between the
sterile and the active neutrino of the SM, thereby, generating a tiny neutrino
mass. As shown in Dvali and Smirnov (1999) due to the mixing of active left-
handed neutrino with the KK tower of the sterile partner, a non-trivial
oscillation pattern emerges.

More recently, it has been shown Dvali and Redi (2009) that a similar
suppression mechanism of the neutrino mass works in the DR scenario Dvali
(2010a); Dvali and Redi (2008) in which species represented the identical
copies of the SM and the role of the extra coordinate is played by their
label. Using this framework, it was shown in Dvali and Redi (2009) that the
dilution of the wave-function of the sterile neutrino in the space of species
results into a small neutrino mass. However, as discussed there the
phenomenological aspects of this scenario are very different from the case of
Arkani-Hamed  _et al._ (2001) which relies on ADDM framework.

In this paper, we have generalized the above original proposals in certain
directions. In particular, we included a more realistic case of three SM
neutrino flavors. We adopted the universal language of species which allows to
capture some general aspects of the neutrino mixing matrix and confront
different scenarios.

We calculated an approximate formula for the flavor eigenstates of a general
mass matrix using perturbation theory in the three flavor case. Next, we
showed how highly symmetric mass matrices can be calculated in an exact manner
and we investigated different symmetry breaking patterns of these highly
symmetric mass matrices. We gave the explicit expressions for flavor
eigenstates for each case.

Our further step was to apply this generally derived formulas to the explicit
theories of neutrino masses such as the proposal of ADDM Arkani-Hamed  _et
al._ (2001); Dvali and Smirnov (1999) within ADD and the one of DR Dvali and
Redi (2009) within Many Species frameworks respectively. Here we used the
derived formulas and gave a three flavour solution that depends on the
parameters of the specific theories.

As it was already pointed out in Dvali and Smirnov (1999) and Dvali and Redi
(2009) within ADDM and DR frameworks, the generic prediction of both scenarios
is the non-conservation of neutrino number within SM. This is due to the
mixing of SM active neutrinos with the tower of sterile partners. This mixing
results into the oscillations of neutrinos into hidden species as well as in
seeming violation of unitarity within the SM lepton sector.

Correspondingly, our calculations of these effects for three-flavor case have
important phenomenological implications in both directions. First is the
account of deviations of neutrino oscillations from the case of SM. Second, is
the parameterization of violation of unitarity of the PMNS-Matrix.

The structures unitarity-violation in the two different theories (ADDM Arkani-
Hamed  _et al._ (2001); Dvali and Smirnov (1999) of ADD versus Dvali-Redi
Dvali and Redi (2009) of Many Species) differ from each other. Our analyses
therefore has a discriminating power between these two theories.

In general, we can say that small neutrino mass generation via mixing with
large number of extra species is an exciting field with different
phenomenological effects on the low energy neutrino physics. These effects can
be searched for both in current neutrino experiments, such as IceCube Ahlers
_et al._ (2018), as well as in the planned ones like JUNO et al. (2016). Here,
the violation of unitarity can be tested and one can use their results to give
bounds on the parameters of the theories such as the size of the extra
dimensions in ADDM or the number of sterile neutrino species to which our
neutrino mixes within many species scenario.

## Acknowledgments

I am very thankful to Gia Dvali for his supervision and the advice and
comments he offered me during this work. I also would like to thank Goran
Senjanovic who taught me a lot about BSM models and neutrino masses in plenty
of discussions. This work was partly supported by the IMPRES-EPP and the
Sonderforschungsbereich SFB1258.

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