# Dirac CP phases in a 3+1 neutrino scenario with $\mu-\tau$ symmetry

Eduardo Becerra-García<EMAIL_ADDRESS>Abdel Pérez-Lorenzana
<EMAIL_ADDRESS>Departamento de Física, Centro de Investigación y de
Estudios Avanzados del I.P.N.
Apdo. Post. 14-740, 07000, Ciudad de México, México

###### Abstract

A sterile neutrino in the $3+1$ scheme, where the sterile accounts for
neutrino anomalies not explained solely by the weak active neutrinos, arises
as a natural source for the breaking of the $\mu-\tau$ symmetry suggested by
oscillation neutrino data. We explore the predictions for the Dirac CP phases
in this scenario, with and without sterile neutrino decay, and show that
current limits on $\delta_{CP}$ suggest a normal hierarchy and a lightest
neutrino scale below 0.1 eV as the most plausible explanation for that, when
Majorana phases are null. Other Dirac phases turn out to be non zero as well.

Neutrino mixings, Flavor symmetries, Sterile neutrinos

##  I Introduction

Although the Standard Model (SM) has proved to be a very precise theory to
explain most phenomena regarding fundamental particles and their interactions,
it is by no means a complete theory, in the sense that there are still some
issues for which it has not a solution, at least not in its original form,
i.e., as a quantum field theory with the gauge group $SU(2)_{L}\times
U(1)_{Y}$. One of these issues is related to the flavor sector of the SM,
which introduces a numerous quantity of free parameters, an undesirable
quality for a fundamental theory. If we could give a theoretical explanation
for the value of those parameters, maybe we would be in the right direction in
the search of a more fundamental theory. From the theoretical point of view
that is one of the reason to study flavor symmetries, which relates the
various generations of quarks and leptons, and which seem to be respected by
all interactions described by the model, but for Yukawa interactions which are
responsible for providing masses to fundamental fermions.

Furthermore, in the SM, neutrinos are predicted to be massles particles,
nonetheless, neutrino oscillations experiments, measuring neutrino fluxes that
come from the sun, from cosmic ray interactions at the upper atmosphere and
from human made sources including nuclear reactors and particle accelerators,
have provided evidence beyond doubt that neutrinos are rather massive and
change their flavor as they travel from source to detector, a phenomena known
as flavor oscillation. This last can be comprehended from the fact that flavor
neutrinos are not the actual mass eigenstates, but rather an admixture of
them.

In the three standard neutrino scheme, both flavor and mass neutrino basis are
related by a unitary mixing matrix, that in the PMNS parametrization [1, 2] is
written in terms of three successive rotations, as
$U_{mix}=R(\theta_{23})U(\theta_{13},\delta)R(\theta_{12})K$, where
$R(\theta_{ij})$ is the real rotation in the $ij$ plane, with the angle
$\theta_{ij}$. $U(\theta_{13},\delta)$ is a complex rotation on the $13$
plane, with an angle $\theta_{13}$ and a Dirac phase $\delta$, and $K$ is a
diagonal matrix containing two Majorana phases. Oscillation probability along
travel distance then becomes a function of the squared mass differences
$\Delta m^{2}_{ij}=m_{i}^{2}-m_{j}^{2}$, for $i,j=1,2,3$, and mixing angles,
through mixing matrix elements. Global fit analysis of most oscillation data
[3] indicate that
$\sin^{2}\theta_{12}=\sin^{2}\theta_{\odot}=0.304^{+0.012(3)}_{-0.012}$,
$\sin^{2}\theta_{23}=\sin^{2}\theta_{ATM}=0.573(5)^{+0.016}_{-0.020(19)}$, and
$\sin^{2}\theta_{13}=\sin^{2}\theta_{reac}=0.02219(38)^{+0.00062(3)}_{-0.00063(2)}$,
corresponding to solar, atmospheric and reactor oscillation mixings,
respectively, for the normal (inverted) hierarchy of the mass spectrum. Here,
normal (inverted) refers to whether (or not) $m_{3}^{2}$ is larger than
$m_{1}^{2}$. Due to MSW matter effects within the sun, the sign of the mass
squared difference that governs solar neutrino oscillations is well defined,
and one gets $\Delta m_{21}^{2}=\Delta
m_{\odot}^{2}=7.42^{+0.21}_{-0.20}\times 10^{-5}~{}eV^{2}$. In contrast, as
the hierarchy is still unknown, $\Delta m^{2}_{ATM}$ corresponds to $\Delta
m^{2}_{31}$ for normal hierarchy (NH), or to $\Delta m^{2}_{23}$ for inverted
hierarchy (IH). Its measured value as obtained from the same fits is $\Delta
m_{ATM}^{2}=2.517^{+0.026}_{-0.028}(-2.498^{+0.028}_{-0.028})\times
10^{-3}~{}eV^{2}$. Dirac CP phase comes to $\delta_{CP}=197^{+27}_{-24}$
($282^{+26}_{-30}$) and within the interval $[120,369]$ ($[193,352])$ at three
sigma level.

The observation that the reactor mixing is very small compared to other ones,
and that $\sin^{2}\theta_{ATM}$ appears to be close to $1/2$, has suggested an
approximate $\mu-\tau$ exchange symmetry in the neutrino sector [4,
*Mohapatra1999, *Lam2001, *Ma2001] which implies that $|(U_{mix})_{\mu
i}|=|(U_{mix})_{\tau i}|$. As a matter of fact, such a symmetry emerges
explicitly when a null $\theta_{13}$ and a maximal $\theta_{23}$ are used to
reconstruct the neutrino mass matrix from mass eigenvalues. Because of this,
$\mu-\tau$ symmetry has been widely studied as an appealing flavor symmetry
for the sector (for some earlier works see [8, *Babu2002, *Ohlsson2002,
*Kitabayashi2003, *Grimus2003, *Koide2004, *Mohapatra2004, *Ma2004,
*Goshal2004, 17, *Joshipura2007, *Fuki2007, *Riazuddin2007, *Luhn2007,
*Koide2008, *Honda208, *Ishimori2008, *Merle2014, *Rivera2016, 27, 28]. An
extensive review can be found in [29]). $\mu-\tau$ symmetry, however, is at
most an approximated one, because neither $\theta_{13}$ is consistent with
zero, nor $\theta_{23}$ is maximal, not even at three sigma level. The source
of the breaking of the symmetry is unknown, but it can be encoded within mass
matrix elements as corrections to symmetry conditions that account for the
final values of neutrino oscillation parameters (for flavor models exploring
the realization of the mixing matrix see for instance [30, 31]).

Above mentioned results, though, do ignore data coming from LSND [32] and
MiniBooNe [33, *miniboone2018] experiments, which have observed some events
identified as the appearance above background of electron neutrinos
(antineutrinos) within an originally muon neutrino (antineutrino) flux. This
observations are troublesome for the three standard neutrino scheme, since to
explain such events within the picture of flavor oscillations, data implies a
larger mass scale, about $\Delta m^{2}\sim 1~{}eV^{2}$, with a rather small
mixing angle, $\sin^{2}\theta_{LSND}\approx 0.0023$. To account for these, a
fourth neutrino needs to be added, which does not participate from weak
interactions. Such a sterile neutrino, $\nu_{s}$, has also been motivated by
the discovery of the reactor antineutrino anomaly [35, *Mention2011,
*Huber2011], associated to a deficit of the reactor antineutrino detection
rate in several experiments, and by the gallium neutrino anomaly [38,
*Lavede2007, *Giunti2007b]. With the presence of a sterile neutrino, the most
favorable arrangement of the spectrum would be that of the $3+1$ hierarchy
[41, 42], where all other neutrino oscillation experiments are explained by
the lighter three states. Global fits with $3+1$ scheme, nonetheless, have
shown that a serious tension exist among LSND/MiniBooNE and other short base
line experiment appearance data when results from disappearance experiments
are included [42, 43, 44], that excludes the sterile solution up to 4.5 sigmas
when the analysis is done without the low energy MiniBooNE data. This tension
can be alleviated, although not solved, if the sterile is allowed to fast
decay into electron neutrinos [45] such that the appearance excess can be
explained with smaller mixings, inducing a smaller effect on disappearance
channels [46, 47]. As shown in the analysis of Ref. [46], that includes a one
year data from IceCube, the global fit with the $3+1$ scenario with sterile
neutrino decay provides a 2.8$\sigma$ improvement over the simple $3+1$ model
predictions, with the best fit values $\Delta m^{2}\sim 1.35~{}eV^{2}$,
$|U_{e4}|=0.238$, $|U_{\mu 4}|=0.105$, and a sterile lifetime
$t=4.5~{}eV^{-1}$. New data regarding searches for a sterile neutrino, and the
confirmation (or refutal) of mentioned anomalies, is expected along the
forthcoming years from ongoing and planned short base line neutrino
experiments. Meanwhile, the possible existence of an $eV$ scale sterile
neutrino seems to stand up (for a recent review of current sterile search
results see [48]).

If it were to exist, sterile neutrinos can play a special roll in the
understanding of the flavor neutrino problem. Since it does not participate of
the weak interactions, sterile carries no standard flavor, and thus, it comes
natural for it to violate flavor symmetries. This observation suggest in
particular that the sterile could be the source for the breaking of the
apparent $\mu-\tau$ symmetry we observe from oscillation data. This idea was
already explored in a previous work [28], but no CP phases were considered in
there. Our goal here is to revise the idea and explore possible implications
of it for CP violation in the sector. For that, we shall assume that all the
relevant violation of the $\mu-\tau$ symmetry comes from the mass terms that
involve the sterile neutrino, whereas the active sector does obey the
$\mu-\tau$ symmetry. High energy flavour model realizations of this idea were
explored in Refs. [49, 50] were some general hints on Dirac phase correlations
were given for some scenarios (see also [51, 52] for similar ideas). Besides
the preference for the normal hierarchy, which is also hinted to from
cosmology [53], we will also see a significant difference for the allowed mass
values reported in this work when compared with the results obtained in [28].
Such distinction is due to the complex nature of the mass matrix elements in
the sterile sector. Our results also improve those of [49, 50] on the search
for solutions that are consistent with the expected PMNS CP phase. The present
work is organized as follows. In section 2 we briefly describe our active-
sterile neutrino ansatz and the method of approximated diagonalization we use
to calculate the mixings. Some important results and the system of equations
that connect neutrino oscillation observables with mass matrix parameters is
also presented there. In section 3 we explore the solutions to the system. In
particular, we found the parameter space for the CP phases in terms of the
lightest neutrino mass, in both normal and inverted hierarchy, for null
Majorana phases, within the $3+1$ and $3+1+decay$ models. In section 4 we
provide a brief discussion on the rephasing invariants. In section 5 we
present a brief analysis for the case when the Majorana phases are not null in
order to explore the sensitivity of our results to these last. We conclude in
section 6 with some perspectives and outlook of this work. An appendix with a
more detailed description of the diagonalization of the mass matrix has been
added.

##  II Active-sterile neutrino mass matrix and oscillation parameters

Under the hypothesis that active neutrino mass terms,
$m_{\ell\ell^{\prime}}\bar{\nu}_{\ell}^{c}\nu_{\ell^{\prime}}$, for
$\ell,\ell^{\prime}=e,\mu,\tau$, are invariant upon the discrete
$\nu_{\mu}\leftrightarrow\nu_{\tau}$ exchange, the mass matrix elements are
forced to satisfy that $m_{e\mu}=m_{e\tau}$, and $m_{\mu\mu}=m_{\tau\tau}$,
reducing the independent matrix elements to four. Furthermore, it can be shown
that in such a case CP phases can all be factored out, and so, no CP violation
is implied by the symmetry in the mixing matrix ($\delta_{CP}=0$). The four
remaining real parameters just account for the solar mixing and the three mass
eingenvalues. This is easily seen when we reconstruct the active neutrino mass
matrix starting from the PMNS mixing matrix, setting $\theta_{13}=0$ and
$\theta_{23}=-\pi/4$, which gives the following relations among the parameters

$\displaystyle m_{ee}$ $\displaystyle=$ $\displaystyle
m_{1}\cos^{2}\theta_{12}+m_{2}\sin^{2}\theta_{12}~{},$ $\displaystyle
m_{e\mu}$ $\displaystyle=$ $\displaystyle\frac{\sin
2\theta_{12}}{\sqrt{8}}\left(m_{2}-m_{1}\right)~{},$ $\displaystyle
m_{\mu\tau}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(m_{1}\sin^{2}\theta_{12}+m_{2}\cos^{2}\theta_{12}-m_{3}\right)~{},$
$\displaystyle m_{\mu\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(m_{1}\sin^{2}\theta_{12}+m_{2}\cos^{2}\theta_{12}+m_{3}\right)~{}.$
(1)

The last are easily inverted in favor of masses, as

$\displaystyle m_{1}$ $\displaystyle=$ $\displaystyle
m_{ee}-\sqrt{2}{m}_{e\mu}\tan\theta_{12},$ $\displaystyle m_{2}$
$\displaystyle=$ $\displaystyle m_{ee}+\sqrt{2}{m}_{e\mu}\cot\theta_{12},$
$\displaystyle m_{3}$ $\displaystyle=$
$\displaystyle{m}_{\mu\mu}-m_{\mu\tau}~{},$ (2)

where the solar to be $\theta_{12}$ mixing is given by

$\tan
2\theta_{12}=\sqrt{8}\left[\frac{{m}_{e\mu}}{{m}_{\mu\mu}+\left(m_{\mu\tau}-m_{ee}\right)}\right]~{}.$
(3)

As already stated, the just described $\mu-\tau$ symmetry in the standard
neutrino sector is at best an approximated symmetry, because reactor mixing is
not null nor atmospheric one is maximal. The problem then becomes to identify
possible sources that produce the breaking. Of course, one possibility is that
the same high energy mechanism that generates neutrino masses be responsible
for it. But it is also possible that the symmetry would be naturally violated
in some other sector, and the effect communicated through interactions to weak
flavor neutrinos. An example of the last is the mass difference among charged
muon and tau leptons, which is far from zero. This violation to $\mu-\tau$
symmetry is, as a matter of fact, communicated to neutrino masses through weak
charged interactions at one loop. Nevertheless, such induced mass correction
turns out to be suppressed by the $W$ mass and too small to account for the
observed value of $\theta_{reac}$ [27].

Along this same line of thought, a natural candidate to look at is the sterile
neutrino. If present, it would carry no standard flavor, but it has to couple
to the weak flavours through mass terms. It is then perfectly possible that
the sterile sector should not comply with the symmetry. If sterile to active
mass terms explicitly violate $\mu-\tau$, the induced effective mass
corrections on the active sector would also break the symmetry. The effect
should be expected to be of the order of $m_{\nu}/m_{s}$, for $m_{s}$ the
sterile mass, which should be just about what is needed to understand
$\theta_{reac}$. The idea has been previously discussed in Ref. [28], where it
has been shown that a sterile with a mass about $eV$ scale do allows for a
successful reconstruction of mixings, within the known limitations of the
sterile hypothesis, although the analysis there was done without considering
CP violation. Here we will revise this idea, including CP phases, to explore
its implications, with particular interest on the possible predicted values
for $\delta_{CP}$, and additional CP phases.

With a sterile neutrino, in the $3+1$ scheme where low scale neutrino
oscillations are mainly explained by active mixings, whereas LSND/MiniBooNE is
due to oscillations dominated by the sterile scale, the most general Majorana
mass matrix that conserves $\mu-\tau$ symmetry in the weak flavor sector has
the form

$M_{\nu}=\begin{pmatrix}M_{S}&\bm{\alpha}m_{s}\hskip 2.84526pt\vspace{1mm}\\\
\bm{\alpha}^{T}m_{s}&m_{s}\hskip 2.84526pt\\\ \end{pmatrix},$ (4)

where $M_{S}$ is a $3\times 3$ $\mu-\tau$ symmetric mass matrix, which alone
would produce the active neutrino masses given in Eq. (2), howbeit, due to the
sterile, we are now forced to maintain the former Majorana phases of the two
off-diagonal terms, $m_{e\mu}$ and $m_{\mu\tau}$ along the analysis. In above,
$m_{s}$ stands for the sterile neutrino mass and the vector
$\bm{\alpha}^{T}=(\alpha_{e},\alpha_{\mu},\alpha_{\tau})$ represents the
dimensionless and complex sterile to active neutrino mixing parameters, which
by construction do not obey the symmetry and thus
$\alpha_{\mu}\neq\alpha_{\tau}$. As already stated, in this scenario we are
assuming that the couplings with the sterile neutrino would be the sole
responsible for both, the deviation on $\mu-\tau$ predicted mixings, as well
as for CP violation on the neutrino sector.

A small scale see-saw approximation, given by the decoupling of the sterile,
shows that, at lower level, effective active neutrino mass terms would become
$m^{\prime}_{\rho\delta}\simeq
m_{\rho\delta}-{\alpha}_{\rho}{\alpha}_{\delta}m_{s}$. Thus, the breaking of
the symmetry, as induced by the sterile to active mass mixings, can be encoded
in the effective (and complex) parameters defined as
$\delta=m^{\prime}_{e\tau}-m^{\prime}_{e\mu}\simeq\alpha_{e}\,\Delta\alpha\,m_{s}$
and $\epsilon=m^{\prime}_{\tau\tau}-m^{\prime}_{\mu\mu}\simeq
2\bar{\alpha}_{\mu}\,\Delta\alpha\,m_{s}$, where
$\bar{\alpha}_{\mu}=(\alpha_{\mu}+\alpha_{\tau})/2$, and
$\Delta\alpha=\alpha_{\tau}-\alpha_{\mu}$. Note that $\Delta\alpha$ is the
only parameter that actually measures the amount of breaking of the symmetry,
and the one that would be responsible for the nonzero value of reactor mixing.
$\alpha_{e}$, on the other hand, would have the role of fixing the initial
value of $\theta_{12}$ to produce the solar mixing. CP conserving analysis has
indicated that a consistent solution can be obtained from
$\alpha^{\prime}s\approx{\cal O}(10^{-1})$ [28]. The addition of CP phases
should slightly modify those results, but a positive solution should still be
expected. The further exploration of how this affects masses and mixings could
be pursued in this small seesaw approximation. Of course, the limitations of
such an approach resides in the decoupling of the sterile that deprive us from
explicitly accounting for the sterile neutrino evidences themselves. To
provide a more complete analysis, all masses and mixings derived from the
complete mass matrix in Eq. (4) should be taken into account. We shall follow
this path hereafter, with the main aim of studying the phase space that is
consistent with the expected oscillation neutrino parameters. The interest on
this arises when we realize that $M_{\nu}$ is parametrized by at most thirteen
parameters. Five real mass terms, four in $M_{S}$ plus $m_{s}$, two free
phases therein, and three $\alpha$ mixing parameters and their three phases.
On the other hand, these parameters should account for sixteen physical
observables in the sector, given by six mixing angles, three in the active
sector and three active to sterile mixings, six CP phases (three Majorana and
three Dirac like ones) and four mass eigenvalues. As it is clear, since there
are less parameters than observables, there would have to be some definite
predictions from the model. As most mixings and mass square mass differences
are known, it is natural to think that the possible predictions could better
be seen in a so far less constrained sector, the CP phase space, that we now
proceed to explore. To this aim, we shall next diagonalize the neutrino mass
matrix and force its parameters to reproduce the observed oscillation
parameters, and then, move into exploring the remaining phase space, in
particular the one associated to Dirac like CP phases.

By replacing the sterile parameters by
$\alpha_{\ell}\rightarrow\alpha_{\ell}\,e^{i\phi_{\ell}}$, to explicitly
express their CP phases, the mass matrix $M_{\nu}$ is written as

$M_{\nu}=\begin{pmatrix}m_{ee}&m_{e\mu}&m_{e\mu}&\alpha_{e}e^{i\phi_{e}}m_{s}\hskip
2.84526pt\\\
m_{e\mu}&m_{\mu\mu}&m_{\mu\tau}&\alpha_{\mu}e^{i\phi_{\mu}}m_{s}\hskip
2.84526pt\\\
m_{e\mu}&m_{\mu\tau}&m_{\mu\mu}&\alpha_{\tau}e^{i\phi_{\tau}}m_{s}\hskip
2.84526pt\\\
\alpha_{e}e^{i\phi_{e}}m_{s}&\alpha_{\mu}e^{i\phi_{\mu}}m_{s}&\alpha_{\tau}e^{i\phi_{\tau}}m_{s}&m_{s}\hskip
2.84526pt\\\ \end{pmatrix}.$ (5)

This matrix can be diagonalized by the unitary transformation
$U^{T}M_{\nu}U=M_{D}$, with $U$ the four by four mixing matrix defined through
the relation $\nu_{\alpha L}=\sum_{i}U_{\alpha i}\nu_{iL}$ where hereafter
$i\in\\{1,2,3,4\\}$ and $\alpha\in\\{e,\mu,\tau,s\\}$. $U$ can be parametrized
in terms of six mixing angles and six phases. These last would be in general
non trivial expressions involving the parameters of $M$ and the $\phi_{\ell}$
phases. Those are the result of the diagonalization process and the
parametrization used for $U$. Nevertheless, we should emphasize that, for the
purpose of our study, it is not necessary to display those expressions
explicitly, but rather knowing the relation of the physical phases with all
other observables, as we show below. In this work we use the ‘symmetrycal
parametrization’ of the mixing matrix proposed by J. Schechter and J.W.F.
Valle in Ref. [54],

$U=\omega_{34}(\theta_{34},\gamma)\omega_{24}(\theta_{24},\beta)\omega_{14}(\theta_{14},\alpha)\omega_{23}(\theta_{23},\delta_{3})\omega_{13}(\theta_{13},\delta_{2})\omega_{12}(\theta_{12},\delta_{1}),$
(6)

with the $\omega_{ij}$ matrices given by

$[\omega_{ij}(\theta_{ij},\phi)]_{pq}=\begin{cases}\cos{\theta_{ij}}&\quad
p=q=i,j,\\\ 1&\quad p=q\neq i,j,\\\ \sin{\theta_{ij}}\hskip
2.84526pte^{-i\phi}&\quad p=i,q=j,\\\ -\sin{\theta_{ij}}\hskip
2.84526pte^{i\phi}&\quad p=j,q=i,\\\ 0&\quad\text{otherwise.}\end{cases}$ (7)

In this parametrization, three phases can be factored out and recombined into
the Majorana phases, but there are three phase combinations that cannot be
extracted from $U$, and they would become the physical Dirac phases appearing
in the oscillation probabilities. Following Ref. [55], these are

$\displaystyle I_{1}$ $\displaystyle=\delta_{1}+\delta_{3}-\delta_{2},$ (8)
$\displaystyle I_{2}$ $\displaystyle=\delta_{1}+\beta-\alpha,$ (9)
$\displaystyle I_{3}$ $\displaystyle=\delta_{2}+\gamma-\alpha,$ (10)

where $I_{1}$ corresponding to the $\delta_{CP}$ appearing in the PMNS mixing
matrix. $I_{2,3}$, on the other hand, are additional phases for which we still
do not have experimental inputs.

We block diagonalize the mass matrix (see Appendix) following a procedure
analogous to the one presented in Ref. [56], assuming the angles
$\theta_{14},\theta_{24},\theta_{34}$ and $\theta_{13}$ to be small enough
such that a perturbative approach is justified. As stated in the Appendix,
along the process we use a systematic step by step redefinition of neutrino
phases that warrants the real value of the obtained mixings and amounts to
define the phases of $U$ as given in Eq. (6). Those results, formally, should
allow us to express the more useful mixing matrix parameters in terms of the
mass matrix elements. We will use the former in the rest of our discussion.

In order to make the connection of the proposed ansatz with the experimental
data, we invert the diagonalization and rewrite it as
$M=U^{*}M_{D}U^{\dagger}$. From this relation we obtain the following
approximate relations, valid for either mass ordering, between the mixing
angles of the sterile sector and the $\alpha$ parameters of the mass matrix,

$\displaystyle\theta_{14}$ $\displaystyle\approx\alpha_{e},$ (11)
$\displaystyle\theta_{24}$ $\displaystyle\approx\alpha_{\mu},$ (12)
$\displaystyle\theta_{34}$ $\displaystyle\approx\alpha_{\tau}.$ (13)

This expressions can also be obtained from the approximated diagonalization of
the mass matrix, as discussed in the Appendix, in the limit of sterile mass
dominance (upon phase redefinitions).

The general expression for the transition probability is given by

$\displaystyle
P_{\nu_{\alpha}\rightarrow\nu_{\beta}}(L,E)=\delta_{\alpha\beta}$
$\displaystyle-4\sum_{k>j}Re[U^{*}_{\alpha k}U_{\beta k}U_{\alpha
j}U^{*}_{\beta j}]\sin^{2}{\Bigg{(}\frac{\Delta m^{2}_{kj}L}{4E}\Bigg{)}}$
$\displaystyle+2\sum_{k>j}Im[U^{*}_{\alpha k}U_{\beta k}U_{\alpha
j}U^{*}_{\beta j}]\sin{\Bigg{(}\frac{\Delta m^{2}_{kj}L}{2E}\Bigg{)}},$ (14)

from where, the survival probability becomes

$P_{\nu_{\alpha}\rightarrow\nu_{\alpha}}(L,E)=1-4\sum_{k>j}|U_{\alpha
k}|^{2}|U_{\beta k}|^{2}\sin^{2}{\Bigg{(}\frac{\Delta
m^{2}_{kj}L}{4E}\Bigg{)}}.$ (15)

Due to the hierarchy of the neutrino masses in the $3+1$ scenario, the above
expressions can be simplified as follows,

$\displaystyle P_{\nu_{\alpha}\rightarrow\nu_{\beta}}\approx$ $\displaystyle
A^{\alpha\beta}_{LSND}\sin^{2}{\Delta_{LSND}}+A^{\alpha\beta}_{ATM}\sin^{2}{\Delta_{ATM}}+A^{\alpha\beta}_{\odot}\sin^{2}{\Delta_{\odot}}$
$\displaystyle+B^{\alpha\beta}_{ATM}\sin{2\Delta_{ATM}}+B^{\alpha\beta}_{\odot}\sin{2\Delta_{\odot}}$
(16)

for the transition probabilities (for $\alpha\neq\beta$), and

$\displaystyle P_{\nu_{\alpha}\rightarrow\nu_{\alpha}}\approx
1-A^{\alpha}_{LSND}\sin^{2}{\Delta_{LSND}}-A^{\alpha}_{ATM}\sin^{2}{\Delta_{ATM}}-A^{\alpha}_{\odot}\sin^{2}{\Delta_{\odot}}$
(17)

for the survival probabilities, where the amplitudes are given by

$\displaystyle A^{\alpha\beta}_{LSND}$ $\displaystyle=$ $\displaystyle
4|U_{\alpha 4}|^{2}|U_{\beta 4}|^{2},$ $\displaystyle A^{\alpha\beta}_{ATM}$
$\displaystyle=$ $\displaystyle 4|U_{\alpha 3}|^{2}|U_{\beta
3}|^{2}+4Re(U_{\alpha 3}^{*}U_{\beta 3}U_{\alpha 4}U_{\beta 4}^{*}),$
$\displaystyle A^{\alpha\beta}_{\odot}$ $\displaystyle=$
$\displaystyle-4Re(U_{\alpha 2}^{*}U_{\beta 2}U_{\alpha 1}U_{\beta 1}^{*}),$
$\displaystyle A^{\alpha}_{LSND}$ $\displaystyle=$ $\displaystyle 4|U_{\alpha
4}|^{2}(1-|U_{\alpha 4}|^{2}),$ $\displaystyle A^{\alpha}_{ATM}$
$\displaystyle=$ $\displaystyle 4|U_{\alpha 3}|^{2}(|U_{\alpha
2}|^{2}+|U_{\alpha 1}|^{2}),$ $\displaystyle A^{\alpha}_{\odot}$
$\displaystyle=$ $\displaystyle 4|U_{\alpha 1}|^{2}|U_{\alpha 2}|^{2},$
$\displaystyle B^{\alpha\beta}_{ATM}$ $\displaystyle=$
$\displaystyle-2Im(U_{\alpha 3}^{*}U_{\beta 3}U_{\alpha 4}U_{\beta 4}^{*}),$
$\displaystyle B^{\alpha\beta}_{\odot}$ $\displaystyle=$ $\displaystyle
2Im(U_{\alpha 2}^{*}U_{\beta 2}U_{\alpha 1}U_{\beta 1}^{*}),$

and the oscillation arguments given by the neutrino scales
$\Delta_{LSND}=\Delta m^{2}_{LSND}L/4E$, $\Delta_{ATM}=\Delta
m^{2}_{ATM}L/4E$, and $\Delta_{\odot}=\Delta m^{2}_{\odot}L/4E$. Note that in
general, all three CP violating Dirac phases, $I_{1,2,3}$, would be involved
in the amplitude coefficients, in particular those entering the appearance
probabilities. The effective mixing angles associated to different types of
neutrino oscillations (solar, reactor, atmospheric and LSND/MiniBooNE) can be
read out from above amplitudes, considering the two neutrino interpretation of
each experiment, and so, they can be written as

$\displaystyle\sin^{2}{2\theta_{\odot}}$ $\displaystyle\approx
4|U_{e1}|^{2}|U_{e2}|^{2},$ (18) $\displaystyle\sin^{2}{2\theta_{reac}}$
$\displaystyle\approx 4|U_{e3}|^{2}(|U_{e1}|^{2}+|U_{e2}|^{2}),$ (19)
$\displaystyle\sin^{2}{2\theta_{ATM}}$ $\displaystyle\approx 4|U_{\mu
3}|^{2}|U_{\tau 3}|^{2}+4Re(U_{\mu 3}^{*}U_{\tau 3}U_{\mu 4}U_{\tau 4}^{*}),$
(20) $\displaystyle\sin^{2}{2\theta_{LSND}}$ $\displaystyle\approx
4|U_{e4}|^{2}|U_{\mu 4}|^{2}.$ (21)

It is worth noticing that the effective form of solar and reactor mixings is
the same as in the standard three neutrino flavor, in terms of mixing matrix
elements, whereas only the atmospheric mixing gets and explicit correction
coming from the sterile sector.

With the use of the parametrization given in (6) for the mixing matrix, one
straightforwardly gets the following explicit expressions for the required
mixing matrix elements

$\displaystyle U_{e1}$ $\displaystyle=$
$\displaystyle\cos\theta_{14}\cos\theta_{13}\cos\theta_{12},$ $\displaystyle
U_{e2}$ $\displaystyle=$
$\displaystyle\cos\theta_{14}\cos\theta_{13}\sin\theta_{12}e^{-i\delta_{1}},$
$\displaystyle U_{e3}$ $\displaystyle=$
$\displaystyle\cos\theta_{14}\sin\theta_{13}e^{-i\delta_{2}},$ $\displaystyle
U_{e4}$ $\displaystyle=$ $\displaystyle\sin\theta_{14}e^{-i\alpha},$
$\displaystyle U_{\mu 3}$ $\displaystyle=$
$\displaystyle\cos\theta_{24}\cos\theta_{13}\sin\theta_{23}e^{-i\delta_{3}}-\sin\theta_{14}\sin\theta_{24}\sin\theta_{13}e^{i(\alpha-\beta-\delta_{2})},$
$\displaystyle U_{\mu 4}$ $\displaystyle=$
$\displaystyle\cos\theta_{14}\sin\theta_{24}e^{-i\beta},$ $\displaystyle
U_{\tau 4}$ $\displaystyle=$
$\displaystyle\cos\theta_{24}\cos\theta_{14}\sin\theta_{34}e^{-i\gamma}$

and

$\displaystyle U_{\tau 3}=$
$\displaystyle\cos\theta_{34}\cos\theta_{13}\cos\theta_{23}-\sin\theta_{34}\cos\theta_{24}\sin\theta_{14}\sin\theta_{13}e^{i(\alpha-\gamma-\delta_{2})}$
$\displaystyle-\sin\theta_{34}\sin\theta_{24}\sin\theta_{23}\cos\theta_{13}e^{i(\beta-\gamma-\delta_{3})}~{}.$

Thus, with the results in (11-13), the right-hand side of Eqs. (18-21) can be
written, up to second order in the small mixing angle $\theta_{13}$, and
$\alpha_{e,\mu,\tau}$, as

$\displaystyle\sin^{2}{2\theta_{\odot}}\approx$
$\displaystyle\,4(1-2\theta_{13}^{2}-2\alpha_{e}^{2})\sin^{2}{\theta_{12}}\cos^{2}{\theta_{12}},$
(22) $\displaystyle\sin^{2}{2\theta_{reac}}\approx$
$\displaystyle\,4\theta_{13}^{2},$ (23)
$\displaystyle\sin^{2}{2\theta_{ATM}}\approx$
$\displaystyle\,4[(1-2\theta_{13}^{2}-\alpha_{\mu}^{2}-\alpha_{\tau}^{2})\cos{\theta_{23}}$
$\displaystyle-2\alpha_{\mu}\alpha_{\tau}\cos(\beta-\delta_{3}-\gamma)\sin{\theta_{23}}]\sin^{2}{\theta_{23}}\cos{\theta_{23}}$
$\displaystyle+4\alpha_{\mu}\alpha_{\tau}\cos(\beta-\delta_{3}-\gamma)\sin\theta_{23}\cos\theta_{23},$
(24) $\displaystyle\sin^{2}{2\theta_{LSND}}\approx$
$\displaystyle\,4\alpha_{e}^{2}\alpha_{\mu}^{2}.$ (25)

As expected, the above expressions reduce to the standard ones in the limit
when there is no sterile, i.e., with $\alpha_{e,\mu,\tau}$=0.

Since we have imposed $\mu-\tau$ symmetry to the active sector, then we have
another pair of equations, $m_{e\mu}=m_{e\tau}$ and $m_{\mu\mu}=m_{\tau\tau}$,
which in terms of the parameters of the mixing matrix and the mass eigenvalues
are given by

$\displaystyle\begin{split}&(-m_{1}e^{-i\delta_{1}}+m_{2}e^{i\delta_{1}})(\cos{\theta_{23}}+e^{-i\delta_{3}}\sin{\theta_{23}})\sin{\theta_{12}}\cos{\theta_{12}}\\\
&+m_{3}e^{i\delta_{2}}\theta_{13}(e^{i\delta_{3}}\sin{\theta_{23}}-\cos{\theta_{23}})+m_{4}e^{i\alpha}\alpha_{e}(\alpha_{\mu}e^{i\beta}-\alpha_{\tau}e^{i\gamma})\approx
0,\end{split}$ (26)
$\displaystyle\begin{split}&(m_{1}e^{-2i\delta_{1}}\sin^{2}{\theta_{12}}+m_{2}\cos^{2}{\theta_{12}})(\cos^{2}{\theta_{12}}-e^{-2i\delta_{3}}\sin^{2}{\theta_{12}})\\\
&+m_{3}(e^{2i\delta_{3}}\sin^{2}{\theta_{12}}-\cos^{2}{\theta_{12}})+m_{4}(\alpha_{\mu}^{2}e^{2i\beta}-\alpha_{\tau}^{2}e^{2i\gamma})\approx
0.\end{split}$ (27)

Notice that above relations, which are simply derived from mass matrix
diagonalization, are valid regardless whether sterile neutrino decay is
considered as part of the explanation of LSND/Miniboone data, but for
$\sin\theta_{LSND}$, because the decay mainly affects the residual
contributions of the heavy state to lighter neutrino state oscillations. To be
clear, with sterile neutrino decay, the transition probability calculation has
to separate the simple mixing contributions to the amplitude coming from
standard (and stable) neutrino flavors from those of the heavier (unstable)
state. Thus, the transition probability has to be written as

$P_{\nu_{\alpha}\rightarrow\nu_{\beta}}=P^{STD}_{\nu_{\alpha}\rightarrow\nu_{\beta}}+P^{s}_{\nu_{\alpha}\rightarrow\nu_{\beta}}+P^{decay}_{\nu_{\alpha}\rightarrow\nu_{\beta}}~{},$
(28)

where $P^{STD}_{\nu_{\alpha}\rightarrow\nu_{\beta}}$ is given by the standard
oscillation formula (14) including only the lighter states. On the other hand,
sterile contributions come now with an exponential suppression over distance
in the amplitude probability, due to the decay, that gives

$P^{s}_{\nu_{\alpha}\rightarrow\nu_{\beta}}=|U_{\alpha 4}|^{2}|U_{\beta
4}|^{2}\left(e^{-L/2t}-1\right)+2Re\sum_{j}^{3}U^{*}_{\alpha 4}U_{\beta
4}U_{\alpha j}U^{*}_{\beta j}\left(e^{-2i\Delta_{LSND}}e^{-L/2t}-1\right)~{},$
(29)

where $t$ stands for the life time of the sterile.
$P^{decay}_{\nu_{\alpha}\rightarrow\nu_{\beta}}$, on the other hand, provides
the contribution that comes from the probability that the heavy component of a
neutrino of flavor $\alpha$ decays at a shorter distance than $L$ and its
product interacts as a $\beta$ neutrino. The explicit form of this
contribution has no relevance for the identification of the effective
oscillation mixings, though. A detailed discussion of this contribution can be
found in Ref. [45].

Considering above arguments, even for the sterile decay scenario, Eqs. (22-24)
remain the same due to unitarity of the mixing matrix. Although the formula
(21) can still be used for our proposes as a reference, given that the global
fits with sterile decay give values for $|U_{e4}|$ and $|U_{\mu 4}|$, we will
use these last instead for this case.

The left hand side in Eqs. (22-25) is known from experiment, and the mass
eigenvalues that appear in Eqs. (26, 27) can be expressed in terms of the
squared-mass differences and a neutrino absolute scale parameter, $m_{0}$,
which by convention is given as the lightest neutrino mass, which means to
write

$m_{1}=m_{0},\quad|m_{2}|=\sqrt{m_{0}^{2}+\Delta
m_{\odot}^{2}},\quad|m_{3}|=\sqrt{m_{0}^{2}+|\Delta
m_{ATM}^{2}|},\quad\text{for NH, and}$ (30) $|m_{1}|=\sqrt{m_{0}^{2}+|\Delta
m_{ATM}^{2}|},\quad|m_{2}|=\sqrt{m_{0}^{2}++|\Delta m_{ATM}^{2}+\Delta
m_{\odot}^{2}},\quad m_{3}=m_{0},\quad\text{for IH,}\\\ $ (31)

and $\Delta m^{2}_{LSND/MiniBooNE}\approx m_{4}^{2}$. Therefore, by using this
data as inputs for our analysis, the set of eight real Eqs. (22-27) gets
defined by twelve unknowns, given by the three active mixing angles and six
phases used in the parametrization (7) aside to the three $\alpha_{\ell}$
parameters. Without any further knowledge of other observable parameters, we
are forced to make some reasonable assumptions on the unknowns in order to
explore the parameter space in a more comprehensive way. First we focus on
exploring the Dirac like phase space, and thus restrict our study to the
particular case of null Majorana phases, i.e., we set
$\alpha=\delta_{1}=\delta_{2}=0$. The case for other non-zero values of these
phases shall be considered later on. We consider these parameters as the
Majorana phases since they are the ones appearing in the effective mass
$|m_{ee}|^{2}$ of neutrinoless double beta decay, whose maximum value
corresponds to the null values of such phases. Then, the relevant set of
phases in (8-10) is restricted to $\delta_{3}$, $\beta$ and $\gamma$. In the
parametrization we are using and with the previous consideration, the phase
$\delta_{3}$ corresponds to the usual Dirac phase $\delta_{CP}$ that appears
in the PMNS parametrization. Furthermore, following Eq. (25), we notice that
$\alpha_{e}$ and $\alpha_{\mu}$ are tightly correlated, so we can use some
given values that comply with the perturbative condition $\alpha_{e,\mu}\ll 1$
to be able to depict the phase parameter space on two dimensional graphs, as
we explain next.

##  III Dirac CP phases

As stated above, we seek for solutions of the system of equations (22-27)
using the known neutrino oscillation parameter data, as reported in [3], as
constraints on the parameter space of our model. For the analysis we use for
the sterile sector two previously known results, corresponding to the $3+1$
and $3+1+decay$ scenarios. For the first one we use
$\sin^{2}{2\theta_{LSND}}\approx 0.0023$, $m_{4}=1~{}eV$ according to [41],
whereas for the second one we take the best fit values $|U_{e4}|=0.238$ and
$|U_{\mu 4}|=0.105$, meaning $\sin^{2}{2\theta_{LSND}}\approx 0.0025$, and
$m_{4}=1.35~{}eV$ according to [46]. We also use the best fit values for
$\Delta m_{\odot}^{2}$ ($7.42\times 10^{-5}eV^{2}$ for both hierarchies),
$\Delta m_{ATM}^{2}$ ($2.517\times 10^{-3}eV^{2}$ for NH, $-2.498\times
10^{-3}eV^{2}$ for IH), $\theta_{reac}$ ($8.57\degree$ for NH, $8.6\degree$
for IH), and $\theta_{\odot}$ ($33.44\degree$ for NH, $33.45\degree$ for IH).
Also, we explore our remnant parameter space using a linear distribution in
the range $[0,0.2]~{}eV$ for the lightest neutrino scale $m_{0}$, for any
given hierarchy.

Figure 1: Allowed values for $\alpha_{e}$ and $\alpha_{\mu}$ for the best fit
value of $\sin^{2}\theta_{LSND}$, as discussed in the text.

Next, we observe that, according to Eq. (23),
$\theta_{reac}\approx\theta_{13}$, which fixes the value of this parameter.
The values for $\alpha_{e,\mu}$ that solve Eq. (25), in the 3+1 scenario, and
which are of interest in this work are shown in figure 1. Choosing a pair of
values from this plot allows us to solve Eq. (22) for $\theta_{12}$. In the
sterile decay scenario, $\alpha_{e}$ and $\alpha_{\mu}$ are fixed by the best
fit values from Ref. [46] that give $\alpha_{e}\approx|U_{e4}|=0.238$ and
$\alpha_{\mu}\approx|U_{\mu 4}|=0.105$, same that we take for such a case
along the analysis. With such values for $\alpha_{e,\mu}$, Eq. (22) has no
real solution for $\theta_{12}$ with the best fit value for $\theta_{\odot}$,
but we can certainly find a set of solutions for sligthly smaller values of
$\theta_{\odot}$ lying within the $1\sigma$ interval. We are left to determine
which values of the following five unknown parameters, $\theta_{23}$,
$\alpha_{\tau}$, $\beta$, $\gamma$, and $\delta_{CP}$ (which is given by
$\delta_{3}$ as already mentioned) provide the right value for $\theta_{ATM}$
satisfying the four real conditions derived from Eqs. (26) and (27). For this
purpose, since Eq. (24) already suggests that $\theta_{23}$ should be close to
the actual value of $\theta_{ATM}$, we proceed to numerically explore the five
parameter space by allowing $\theta_{23}$ angle to vary within the ($3\sigma$)
interval [$40.1\degree,51.7\degree$] for NH and [$40.3\degree,51.8\degree$]
for IH, and solve all five constraining conditions to determine the remaining
parameter values that are consistent with current neutrino mixing observables
(up to 3$\sigma$ level in $\theta_{ATM}$), with the main goal of determining
the potential predictions for $\delta_{CP}$.

In figure 2 we show the results for $\delta_{CP}$ as a function of $m_{0}$ for
appropriate values of $\alpha_{e}$ and $\alpha_{\mu}$ as required in Ref.
[28]. The horizontal dashed and dashdotted lines indicate the $1\sigma$
($3.0194\rightarrow 3.9095$ for NH, $4.3982\rightarrow 5.3756$ for IH) and
$3\sigma$ ($2.0944\rightarrow 6.4402$ for NH, $3.3685\rightarrow 6.1436$ for
IH) current intervals for the Dirac CP phase $\delta_{CP}$, respectively. The
plots presented in this section correspond to $\alpha_{e}\approx 0.1910$,
$\alpha_{\mu}\approx 0.1176$, which imply that $\theta_{12}\approx
39.03\degree$ for the $3+1$ scenario, which is one solution to Eqs. (22) and
(25). We have explored solutions for different values of the parameters
$\alpha_{e},\alpha_{\mu}$, and found only small changes in the results for the
whole parameter space. Therefore we show here only the given set of solutions
to visually represent the main conclusions of the whole analysis and comment
on the effect of changing the values of $\alpha_{e,\mu}$ on a parameter case
basis. In fact, with a larger value for $\alpha_{e}$ (hence smaller for
$\alpha_{\mu}$), it is possible to reproduce very similar results to those
obtained in the $3+1+decay$ scenario, as expected, since these values would be
near the best fits already mentioned. We found a notorious difference between
the normal and inverted hierarchies for the phase $\delta_{CP}$. In the first
case we see there are solutions lying in the $1\sigma$ as well as in the
$3\sigma$ range, these are presented in figure 2(a) for the $3+1$
($3+1+decay$) scenario with light blue (light green) and dark blue (dark
green) dots, respectively, while gray (black) dots correspond to solutions of
the system of equations that are outside the $3\sigma$ interval. The same
color map is applied to all the plots presented in this section and the next
one.

(a) Normal hierarchy.

(b) Inverted hierarchy.

Figure 2: Predictions for the Dirac phase $\delta_{CP}$ as a function of the
lightest neutrino mass $m_{0}$ for normal (a) and inverted (b) hierarchy. For
the $3+1$ scenario light (dark) blue dots lie within the current $1\sigma$
($3\sigma$) region represented by the horizontal lines, analogously for the
$3+1+decay$ scenario. Gray and black dots lie outside the $3\sigma$ range.

Note that the vast majority of predicted points within the $1\sigma$ and
$3\sigma$ intervals correspond to solutions for which the mass $m_{0}$ is
below $0.075eV$, favoring a hierarchical neutrino spectrum, and we see from
figure 3(a) that the corresponding $\alpha_{\tau}$ values lie in the
approximated interval $(0.07,0.15)$ for the NH, but there is not any clear
differentiation among $1\sigma$ and $3\sigma$ sets. For larger (smaller)
values of $\alpha_{e}$, the gap between the two regions in figure 2(a)
decreases (increases) which implies there are more (fewer) points in the
$1\sigma$ region, but in any case the mass $m_{0}$ remains below $0.1eV$. The
region for $\alpha_{\tau}$ changes slightly remaining within the allowed
perturbative values. The results contrast with those found in Ref. [28], where
the lightest neutrino mass was required to be greater than $0.1eV$ (with no CP
violation and without decay). Hence the allowed parameter space for the
neutrino mass is drastically changed by the presence of the phases. For the
inverted hierarchy we only obtained a few solutions within the $3\sigma$
interval as can be seen in figure 2(b), indicating that the analysis favours
the normal hierarchy according to the present experimental data. The few dark
blue points in this case are also below $0.1eV$, as in the NH case. For other
values for $\alpha_{e,\mu}$ we obtained similar results, with even fewer
points lying in the $3\sigma$ region.

(a) Normal hierarchy.

(b) Inverted hierarchy.

Figure 3: Parameter space for $\alpha_{\tau}$, consistent with neutrino
oscillation data, as a function of the lightest neutrino mass $m_{0}$,
corresponding to $\alpha_{e}\approx 0.1910$, $\alpha_{\mu}\approx 0.1176$ and
$\theta_{12}\approx 39.03\degree$ for the $3+1$ scenario, and for the
$3+1+decay$ scenario as discussed in the text.

(a) Normal hierarchy.

(b) Inverted hierarchy.

Figure 4: $\sin{\beta}$ as a function of the lightest neutrino mass $m_{0}$.
The majority of the positive values for NH correspond to the $3\sigma$
interval for $\delta_{CP}$ while the negative ones are within $1\sigma$.

In figures 4 and 5 we present the results for the other two CP phases, $\beta$
and $\gamma$. Since $\alpha=\delta_{1}=\delta_{2}=0$, the Dirac phases $I_{2}$
and $I_{3}$ in Eqs. (9,10) reduce to $\beta$ and $\gamma$, respectively.
Therefore, these are the corresponding physical CP phases appearing in
neutrino oscillations. We plot the sine of these parameters as a function of
the lightest neutrino mass $m_{0}$. For the NH case we see a pronounced
difference for positive and negative values of $\sin{\beta}$ in the sense that
the negative values correspond to the solutions lying within the $1\sigma$
interval, while the positive values correspond to the $3\sigma$ interval. For
the IH case, the majority of the solutions lying within the $3\sigma$ interval
correspond to positive values of $\sin{\beta}$. This character of the plots is
also true for the $\sin{\gamma}-m_{0}$ plots, as we can see in figure 5. It is
also remarkable that null values of these phases do not appear to be
consistent with the current bounds on $\delta_{CP}$. Larger values of
$\alpha_{e}$, generate more positive solutions for $\sin{\beta}$ and
$\sin{\gamma}$ within the $1\sigma$ region in NH, while smaller values
distinguish even more the two regions. For the IH case, we found a similar
behaviour of the plots for different values of $\alpha_{e,\mu}$ but always
outside the $3\sigma$ region. Furthermore, note that a future improvement on
the measurement of $\delta_{CP}$ would better constrain the parameter space
for these extra phases, since each dot in these plots correspond to one single
value of $\delta_{CP}$.

(a) Normal hierarchy.

(b) Inverted hierarchy.

Figure 5: $\sin{\gamma}$ as a function of the lightest neutrino mass $m_{0}$.
The majority of the positive values for NH correspond to the $3\sigma$
interval for $\delta_{CP}$ while the negative ones are within $1\sigma$.

##  IV Rephasing invariants

One very convenient approach to the study of CP violation in the flavor sector
of the standard model is by means of the rephasing invariants, since these are
independent of the parametrization of the mixing matrix. In general, they are
given by [57]

$\mathcal{J}_{\alpha\beta}^{kj}=Im[U_{\alpha k}U_{\beta j}U_{\alpha
j}^{*}U_{\beta k}^{*}].$ (32)

Due to the unitarity of $U$ they are not all independent, for our case, with
four neutrino mixing, there are nine of such invariants, which we can identify
as $\mathcal{J}_{\tau s}^{13}$, $\mathcal{J}_{\tau s}^{14}$,
$\mathcal{J}_{\tau s}^{34}$, $\mathcal{J}_{se}^{13}$, $\mathcal{J}_{se}^{24}$,
$\mathcal{J}_{se}^{34}$, $\mathcal{J}_{e\mu}^{23}$, $\mathcal{J}_{e\mu}^{24}$,
$\mathcal{J}_{e\mu}^{34}$ according to Ref. [58]. Here we are interested in
the invariant defined by the elements of the active sector,
$\mathcal{J}_{e\mu}^{23}$, because it is the one that quantifies the CP
violation in weak neutrino oscillations, besides, it reduces to the Jarlskog
invariant when we switch off the sterile sector, i.e., with $\theta_{i4}=0$.
This invariant can be written, with null Majorana phases, as follows,

$\mathcal{J}_{e\mu}^{23}=\left[s_{13}s_{14}s_{24}c_{12}c_{23}\sin{\beta}-s_{23}c_{12}c_{13}c_{23}c_{24}\sin{\delta_{3}}-s_{12}s_{14}s_{23}s_{24}\sin{(\beta-\delta_{3})}\right]~{}s_{12}s_{13}c_{13}c_{14}^{2}c_{24}~{},$

where, to simplify matters, $c_{ij}$ ($s_{ij}$) stands for $\cos\theta_{ij}$
($\sin\theta_{ij}$).

By writing this same expression as a function of the $\alpha_{e}$,
$\alpha_{\mu}$, $\alpha_{\tau}$ parameters up to second order we have

$\mathcal{J}_{e\mu}^{23}=\theta_{13}\left(-\alpha_{e}\alpha_{\mu}s_{12}\sin{\left(\beta-\delta_{3}\right)}+\left(0.5\alpha_{e}^{2}+\alpha_{\mu}^{2}-1\right)c_{12}c_{23}\sin{\delta_{3}}\right)s_{12}s_{23}~{}.$
(33)

(a) Normal hierarchy.

(b) Inverted hierarchy.

Figure 6: Jarslkog rephasing invariant $J_{e\mu}^{23}$ as a function of the
smallest neutrino mass for both hierarchies.

In figure 6 we plot the latter quantity as a function of the lightest neutrino
mass $m_{0}$ for both normal and inverted hierarchies where the color map
indicates the same attributes as before. As we would expect, there is a
significant difference for the NH an IH scenarios. We see the same tendency
for the solutions within $1\sigma$ and $3\sigma$ to be clearly distinguished
for the NH case, in the $3+1$ scheme, the former are the mirror of the latter
with respect to $J_{e\mu}^{23}=0$, while in the case with sterile decay, the
majority of the solutions lie within the $1\sigma$ interval, hence such
scenario is preferred for CP violation. For the IH case the majority of the
points lie outside the $3\sigma$ interval of $\delta_{CP}$, as expected, since
the invariant is a function of the phases.

Regarding the other invariants, up to second order in the $\alpha$ parameters
and the small mixing angle $\theta_{13}$, we obtained the following
expressions,

$\mathcal{J}_{e\mu}^{24}\approx\alpha_{e}\alpha_{\mu}\left[\theta_{13}s_{12}s_{23}\sin{\left(\beta-\delta_{3}\right)}+\left(\frac{1}{2}~{}\theta_{13}^{2}-1\right)c_{12}c_{23}\sin{\beta}\right]s_{12}~{},$
(34)
$\mathcal{J}_{e\mu}^{34}\approx-\alpha_{e}\alpha_{\mu}\theta_{13}s_{23}\sin{\left(\beta-\delta_{3}\right)}~{},$
(35) $\displaystyle\mathcal{J}_{\tau s}^{13}/\cos{\theta_{23}}$
$\displaystyle\approx$
$\displaystyle\alpha_{e}\alpha_{\mu}\Bigg{(}\theta_{13}^{2}\left(\left(\frac{3}{2}s^{2}_{23}-1\right)s_{12}c_{12}\sin{\beta}-2s_{12}s^{2}_{23}c_{12}\cos{\left(\beta-\delta_{3}\right)}\sin{\delta_{3}}\right)$
(36) $\displaystyle+$
$\displaystyle\theta_{13}\left(2s^{2}_{12}-1\right)s_{23}c_{23}\sin{\left(\beta-\delta_{3}\right)}+s_{12}s^{2}_{23}c_{12}\sin{\beta}\Bigg{)}$
$\displaystyle+$
$\displaystyle\alpha_{e}\alpha_{\tau}\Bigg{(}\theta_{13}\left(1-2s^{2}_{23})s^{2}_{12}-c^{2}_{23}\right)\sin{\gamma}$
$\displaystyle+$
$\displaystyle\left(\theta_{13}^{2}\left(-\frac{1}{2}\sin{\delta_{3}}\cos{\gamma}+\frac{3}{2}\sin{\gamma}\cos{\delta_{3}}\right)+\sin{\left(\delta_{3}+\gamma\right)}\right)s_{12}c_{12}s_{23}c_{23}\Bigg{)}$
$\displaystyle+$
$\displaystyle\alpha_{\mu}\alpha_{\tau}\Bigg{(}\frac{1}{2}\theta_{13}^{2}s^{2}_{12}s_{23}\sin{\left(\delta_{3}+\gamma-\beta\right)}-\theta_{13}\left(s^{2}_{23}+c^{2}_{23}\right)s_{12}c_{12}c_{23}\sin{\left(\beta-\gamma\right)}$
$\displaystyle-$ $\displaystyle
s^{2}_{12}s_{23}\sin{\left(\delta_{3}+\gamma-\beta\right)}\Bigg{)}+\theta_{13}\left(\alpha_{e}^{2}-\alpha_{\mu}^{2}\right)s_{12}s_{23}c_{12}\sin{\delta_{3}}~{},$
$\displaystyle\mathcal{J}_{\tau s}^{14}$ $\displaystyle\approx$
$\displaystyle\alpha_{e}\alpha_{\tau}c_{12}\left(\theta_{13}c_{12}c_{23}\sin{\gamma}-\sin{\theta_{12}}\sin{\theta_{23}}\sin{\left(\delta_{3}+\gamma\right)}\right)$
(37) $\displaystyle+$
$\displaystyle\alpha_{\mu}\alpha_{\tau}c_{23}\Big{(}\theta_{13}\left(\cos{(2\theta_{23})}\sin{\left(\beta-\gamma\right)}+2s^{2}_{23}s_{12}c_{12}\sin{\delta_{3}}\cos{\left(\delta_{3}+\gamma-\beta\right)}\right)$
$\displaystyle+$
$\displaystyle\left(s^{2}_{12}-\theta_{13}^{2}c^{2}_{12}\right)s_{23}\sin{\left(-\beta+\delta_{3}+\gamma\right)}\Big{)}~{},$
$\mathcal{J}_{\tau
s}^{34}\approx\alpha_{\tau}\left(\alpha_{\mu}\left(\frac{1}{2}\theta_{13}^{2}-1\right)s_{23}\sin{\left(\delta_{3}+\gamma-\beta\right)}-\alpha_{e}\theta_{13}\sin{\gamma}\right)c_{23}~{},$
(38) $\displaystyle\mathcal{J}_{se}^{13}$ $\displaystyle\approx$
$\displaystyle\theta_{13}\Big{(}\alpha_{e}\alpha_{\tau}\left(\theta_{13}s_{12}s_{23}\sin{\left(\delta_{3}+\gamma\right)}-c_{12}c_{23}\sin{\gamma}\right)$
(39) $\displaystyle-$
$\displaystyle\alpha_{e}\alpha_{\mu}\left(\theta_{13}s_{12}c_{23}\sin{\beta}+c_{12}s_{23}\sin{\left(\beta-\delta_{3}\right)}\right)$
$\displaystyle+$
$\displaystyle\alpha_{\mu}\alpha_{\tau}\left(2s^{2}_{23}\sin{\delta_{3}}\cos{\left(\delta_{3}+\gamma-\beta\right)}-\sin{\left(\beta-\gamma\right)}\right)s_{12}$
$\displaystyle+$
$\displaystyle\left(\alpha_{\tau}^{2}-\alpha_{\mu}^{2}\right)s_{12}s_{23}c_{23}\sin{\delta_{3}}\Big{)}c_{12}~{},$
$\displaystyle\mathcal{J}_{se}^{24}$ $\displaystyle\approx$
$\displaystyle\alpha_{e}s_{12}\Bigg{(}\alpha_{\mu}\left(\theta_{13}s_{12}s_{23}\sin{\left(\beta-\delta_{3}\right)}+\left(\frac{1}{2}\theta_{13}^{2}-1\right)c_{12}c_{23}\sin{\beta}\right)$
(40) $\displaystyle+$
$\displaystyle\alpha_{\tau}\left(\theta_{13}s_{12}c_{23}\sin{\gamma}+\left(1-\frac{1}{2}\theta_{13}^{2}\right)s_{23}c_{12}\sin{\left(\delta_{3}+\gamma\right)}\right)\Bigg{)}~{},$

and

$\mathcal{J}_{se}^{34}\approx-\alpha_{e}\theta_{13}\big{(}\alpha_{\mu}s_{23}\sin{\left(\beta-\delta_{3}\right)}+\alpha_{\tau}c_{23}\sin{\gamma}\big{)}~{}.$
(41)

##  V Switching on Majorana phases

In this section we briefly discuss the case when the Majorana phases are not
null, within the $3+1+decay$ model. First, because it is more consistent with
data than the simple $3+1$ scenario, and second, since there is actually
little visual difference among parameter spaces, as the results of section III
had shown. Since we do not have any experimental data regarding these phases,
we fix them to certain arbitrary values in order to explore how they affect
the solutions previously reported. We study three cases, in the first one all
of them are equal, (i) $\alpha=\delta_{1}=\delta_{2}=\pi/4$, in the second
case two phases are equal, (ii) $\alpha=\pi/2$, $\delta_{1}=\delta_{2}=\pi/4$,
and in the third all three phases are fixed to different values, (iii)
$\alpha=3\pi/4$, $\delta_{1}=\pi/2$, $\delta_{2}=\pi/4$. In the following
plots, green, blue and purple dots represent the solutions for each of these
cases, respectively. Since now we have non-zero Majorana phases, we have to
take into account the relations (8-10), with $I_{1}$ corresponding to the
usual $\delta_{CP}$ phase. In figure 7 we show the parameter space for
$\delta_{CP}$, and we see there are still solutions within the $1\sigma$ and
$3\sigma$ range for the normal hierarchy case. For cases (i) and (ii) in the
IH scenario, almost all the solutions lie outside the $3\sigma$ interval,
while for case (iii) we now have the majority of the points within the same
interval. Very few poits in the last case, though, do drop within the
$1\sigma$ region, which show the robustness of the preference for NH of
current neutrino oscillation data. The most noticeable effect of Majorana
phases is the appearance of acceptable solutions in NH for the larger values
of the neutrino scale.

(a) Normal hierarchy.

(b) Inverted hierarchy.

Figure 7: Predictions for the Dirac phase $\delta_{CP}$, for the normal (a)
and inverted (b) hierarchies, in three scenarios of non-zero Majorana phases.
As before, the horizontal dashed and dashdotted lines indicate the $1\sigma$
and $3\sigma$ current intervals for the Dirac CP phase $\delta_{CP}$,
respectively.

In figure 8 we show the allowed values for $\alpha_{\tau}$ as a function of
the mass $m_{0}$. For NH there is an approximately defined region for each of
the cases (i-iii), while in the case of IH, the regions overlap between each
other, nevertheless, this case would be of little phenomenological interest
because the NH is preferred according to the present experimental data.

(a) Normal hierarchy.

(b) Inverted hierarchy.

Figure 8: Parameter space for $\alpha_{\tau}$, with non-zero Majorana phases,
as a function of the lightest neutrino mass $m_{0}$.

In the last two figures, the results for $\sin{I_{2}}$ and $\sin{I_{3}}$ are
shown. Although these results seem quite arbitrary, they illustrate that the
model has consistent solutions even when the Majorana phases are not null.
Although these extra Dirac phases seem quite sensitive to the actual values of
the Majorana phases, the presence of clear voids around zero values is
noticeable. This indicates that for some ranges of the absolute neutrino
scale, Dirac phases must definitively have non zero values.

(a) Normal hierarchy.

(b) Inverted hierarchy.

Figure 9: $\sin{I_{2}}$ as a function of the lightest neutrino mass $m_{0}$
for non-zero Majorana phases.

(a) Normal hierarchy.

(b) Inverted hierarchy.

Figure 10: $\sin{I_{3}}$ as a function of the lightest neutrino mass $m_{0}$
for non-zero Majorana phases.

##  VI Concluding remarks

To summarize, we have explored the parameter space of a $3+1$ scheme for
neutrino oscillations, where exact $\mu-\tau$ symmetry is assumed to be valid
in the active neutrino sector, but which is explicitly broken in the active-
sterile neutrino mass matrix terms, searching for its predictions about Dirac
like phases. By using the available neutrino data to constraint potential
predictions, in the scenario where double beta decay amplitude is maximized,
by assuming null Majorana like phases, we found that suitable solutions to the
system exist that may accommodate a $\delta_{CP}$ (the Dirac phase appearing
in the PMNS mixing matrix) well within current expected values within
$1\sigma$ and $3\sigma$ deviations. Considering the decay of the sterile
neutrino in the $3+1+decay$ scheme we found there is no a significant
distinction for the parameters we explored as compared to the simpler $3+1$
model, hence, the results are robust enough to also fit the less constrained
model in which the sterile neutrino is allowed to decay. It is noticeable that
if actual $\delta_{CP}$ were found within the $1\sigma$ current interval, the
normal hierarchy on active neutrino masses would be favored in the present
model, since our explorations did not find any consistent region for such
$\delta_{CP}$ for the inverted hierarchy. Moreover, our analysis also
indicates that for such values of the active CP phase, a lightest neutrino
mass below $0.1~{}eV$, and thus a non-degenerate neutrino spectrum, is
preferred. It is also worth stressing the fact that the presented solutions on
any hierarchy do predict that other Dirac like phases within the complete
mixing matrix to be non zero. Although we have centered our present discussion
on the results obtained from an specific set of values for $\alpha_{e}$ and
$\alpha_{\mu}$ parameters, varying those does not actually add any new
valuable information, as consistent solution are actually found for other
value sets, in basically the same parameter regions. Switching on the Majorana
phases shows the robustness of the NH scenario, incorporating consistent
solutions that reproduce the experimental data even for the larger values of
$m_{0}$, although the predicted parameter regions for the other Dirac phases
may substantially change. Majorana phases may also bring back the IH scenario
for the model at hand, provided future measurements of $\delta_{CP}$ keep
values within the current $3\sigma$ region.

Although the scenario at hand may still be seen as quite marginal, we believe
it is still of interest as far it is not yet completely ruled out by data, as
the addition of sterile neutrino decay has some potential of lessen the
constraints on the sterile hypothesis. Given the potential to reproduce the
observed neutrino oscillation parameters, it would be interesting to further
explore complete flavor model constructions that may reproduce all these
features. As already mentioned, earlier models were addressed in [49, 50, 51,
52] based on $A_{4}$ symmetry, but other discrete symmetries, as $S_{3}$ or
$S_{4}$, could be also explored. A consistent introduction of heavy neutrino
decay in the same flavor model basis is also a feature that remains to be
fully explored.

###### Acknowledgements.

This work has been partially supported by Conacyt, México, under FORDECYT-
PRONACES Grant No. 490769. The work of E. Becerra-García was supported by a
CONACyT graduate fellowship.

##  Appendix A Diagonalization of the neutrino mass matrix

Here we outline the method for the diagonalization of the mass matrix, which
is an extension of the one presented in [56] for the four neutrino scenario.
The diagonalization is achieved by the operation $U^{T}M_{\nu}U=M_{D}$ with
$U$ given in (6) and considering the mixing angles $\theta_{i4}$ and
$\theta_{13}$ to be sufficiently small for a perturbative analysis. In the
following, $s_{ij}$ ($c_{ij}$) stands for $\sin{\theta_{ij}}$
($\cos{\theta_{ij}}$).

  1. 1.

Perform the rotation $34$ as $\omega_{34}^{T}M_{\nu}\omega_{34}$, and set to
zero the $34,43$ elements of the matrix, which allows us to obtain an
expression for the mixing angle $\theta_{34}$, then we redefine the phases by
demanding the angle to be real.

$\omega_{34}^{T}M_{\nu}\omega_{34}=\begin{pmatrix}\times&\times&\times&\times\hskip
2.84526pt\\\ \times&\times&\times&\times\hskip 2.84526pt\\\
\times&\times&\times&0\hskip 2.84526pt\\\ \times&\times&0&\times\hskip
2.84526pt\\\ \end{pmatrix},$ (42)

from the entry $34$ we obtain

$\theta_{34}=\frac{\alpha_{\tau}m_{s}e^{i\gamma}}{m_{s}e^{2i\gamma}-m_{\tau\tau}}$
(43)

  2. 2.

Now we apply the rotation $24$ to the resultant matrix of the previous step,
and set to zero the entries $24,42$ after the operation, from which we obtain
the mixing angle $\theta_{24}$, and again, in order for this angle to be real
we need to redefine the phases. This second transformation produce a new
quantity in the $34,43$ entries, but we neglect it because another rotation in
the block $34$ to get rid of this term would produce contributions to second
order in the small mixing angles and we are restricting the analysis to lower
order. This argument also applies for the following transformation.

$\omega_{24}^{T}\omega_{34}^{T}M_{\nu}\omega_{34}\omega_{24}=\begin{pmatrix}\times&\times&\times&\times\hskip
2.84526pt\\\ \times&\times&\times&0\hskip 2.84526pt\\\
\times&\times&\times&0\hskip 2.84526pt\\\ \times&0&0&\times\hskip 2.84526pt\\\
\end{pmatrix},$ (44)

that set zeros in the $24$ and $42$ elements, so that we obtain the following
expression, after proper phase redefinition,

$\theta_{24}=\frac{(m_{\mu\tau}\theta_{34}+\alpha_{\mu}m_{s}e^{i\gamma})e^{i(\beta-\gamma)}}{m_{s}e^{2i\beta}-m_{\mu\mu}}$
(45)

  3. 3.

Next, we make the third rotation with $14$, and set to zero the elements
$14,41$ to obtain the angle $\theta_{14}$ after redefining the corresponding
phases.

$\omega_{14}^{T}\omega_{24}^{T}\omega_{34}^{T}M_{\nu}\omega_{34}\omega_{24}\omega_{14}=\begin{pmatrix}\times&\times&\times&0\hskip
2.84526pt\\\ \times&\times&\times&0\hskip 2.84526pt\\\
\times&\times&\times&0\hskip 2.84526pt\\\ 0&0&0&\times\hskip 2.84526pt\\\
\end{pmatrix}.$ (46)

The corresponding mixing angle is found to be

$\theta_{14}=\frac{(m_{e\mu}\theta_{24}e^{i(\alpha+\gamma)}+m_{e\tau}\theta_{34}e^{i(\alpha+\beta)}+\alpha_{e}m_{s}e^{i(\alpha+\beta+\gamma)})e^{-i(\beta+\gamma)}}{m_{s}e^{2i\alpha}-m_{ee}}.$
(47)

  4. 4.

The resultant matrix of the last step has about null elements on the fourth
column and fourth row, except of course on the entry $44$. Now, we should
diagonalize the upper $3\times 3$ block corresponding to the active neutrino
sector, starting with the $23$ rotation and proceeding as before with

$\omega_{23}^{T}\omega_{14}^{T}\omega_{24}^{T}\omega_{34}^{T}M_{\nu}\omega_{34}\omega_{24}\omega_{14}\omega_{23}=\begin{pmatrix}\times&\times&\times&0\hskip
2.84526pt\\\ \times&\times&0&0\hskip 2.84526pt\\\ \times&0&\times&0\hskip
2.84526pt\\\ 0&0&0&\times\hskip 2.84526pt\\\ \end{pmatrix}~{}.$ (48)

The equation for the $\theta_{23}$ angle then results into

$\tan{2\theta_{23}}=\frac{2(m_{\mu\tau}-\alpha_{\mu}m_{s}\theta_{34}e^{i\gamma})}{(m_{\tau\tau}-2\alpha_{\tau}m_{s}\theta_{34}e^{i\gamma})e^{i\delta_{3}}-(m_{\mu\mu}-2\alpha_{\mu}m_{s}\theta_{24}e^{i\beta})e^{-i\delta_{3}}}$
(49)

  5. 5.

The next rotation in the block $13$ sets to zero the $13,31$ entries to obtain
the mixing angle $\theta_{13}$, after redefining the phases,

$\omega_{13}^{T}\omega_{23}^{T}\omega_{14}^{T}\omega_{24}^{T}\omega_{34}^{T}M_{\nu}\omega_{34}\omega_{24}\omega_{14}\omega_{23}\omega_{13}=\begin{pmatrix}\times&\times&0&0\hskip
2.84526pt\\\ \times&\times&0&0\hskip 2.84526pt\\\ 0&0&\times&0\hskip
2.84526pt\\\ 0&0&0&\times\hskip 2.84526pt\\\ \end{pmatrix}~{}.$ (50)

For the $\theta_{13}$ angle we obtain the following expression

$\theta_{13}=\frac{(m_{e\mu}s_{23}+m_{e\tau}c_{23}e^{i\delta_{3}}-\alpha_{e}m_{s}(\theta_{24}s_{23}e^{i\beta}+\theta_{34}c_{23}e^{i(\delta_{3}+\gamma)}))e^{i(\delta_{2}+\delta_{3})}}{(m_{\mu\mu}s_{23}^{2}+m_{\tau\tau}c_{23}^{2}e^{2i\delta_{3}}+m_{\mu\tau}s2_{23}e^{i\delta_{3}})e^{2i\delta_{2}}+(2\alpha_{e}m_{s}\theta_{14}e^{i\alpha}-m_{ee})e^{2i\delta_{3}}}$
(51)

  6. 6.

Finally, we perform the rotation $12$ and follow the same procedure as before.
Thus, we get

$\tan{2\theta_{12}}=\frac{2(m_{12}-m_{14}\theta_{24}e^{i\beta})c_{23}-2(m_{13}-m_{14}\theta_{34}e^{i\gamma})e^{i\delta_{3}}s_{23}}{(k_{1}c_{23}^{2}+k_{2}e^{2i\delta_{3}}s_{23}^{2}-k_{3}e^{i\delta_{3}}s2_{23})e^{i\delta_{1}}-(m_{11}-2m_{14}\theta_{14}e^{i\alpha})e^{-i\delta_{1}}}$
(52)

where the quantities $k_{1},k_{2},k_{3}$ are given by

$\displaystyle k_{1}$ $\displaystyle=m_{22}-2m_{24}\theta_{24}e^{i\beta},$
(53) $\displaystyle k_{2}$
$\displaystyle=m_{33}-2m_{34}\theta_{34}e^{i\gamma},$ (54) $\displaystyle
k_{3}$ $\displaystyle=m_{23}-m_{24}\theta_{34}e^{i\gamma}.$ (55)

This completes the diagonalization process, with which we have obtained
approximate expressions for all the mixing angles in terms of the elements of
the mass matrix.

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# GANcrop: A Contrastive Defense Against Backdoor Attacks in Federated
Learning

Xiaoyun Gan  Guangxi Normal University15 Yucai RdGuilinGuangxiChina541004
<EMAIL_ADDRESS>0009-0002-6154-0855 ,  Shanyu Gan  Guangxi Normal
University15 Yucai RdGuilinGuangxiChina541004<EMAIL_ADDRESS>0009-0007-6014-0290 ,  Taizhi Su  Guangxi Normal University15 Yucai
RdGuilinGuangxiChina541004<EMAIL_ADDRESS>0009-0003-7646-5664 and  Peng
Liu  Guangxi Normal University15 Yucai RdGuilinGuangxiChina541004
<EMAIL_ADDRESS>0000-0003-2583-9112

(2018)

###### Abstract.

With heightened awareness of data privacy protection, Federated Learning (FL)
has attracted widespread attention as a privacy-preserving distributed machine
learning method. However, the distributed nature of federated learning also
provides opportunities for backdoor attacks, where attackers can guide the
model to produce incorrect predictions without affecting the global model
training process. This paper introduces a novel defense mechanism against
backdoor attacks in federated learning, named GANcrop. This approach leverages
contrastive learning to deeply explore the disparities between malicious and
benign models for attack identification, followed by the utilization of
Generative Adversarial Networks (GAN) to recover backdoor triggers and
implement targeted mitigation strategies. Experimental findings demonstrate
that GANcrop effectively safeguards against backdoor attacks, particularly in
non-IID scenarios, while maintaining satisfactory model accuracy, showcasing
its remarkable defensive efficacy and practical utility.

Federated Learning; Attack Defense; Backdoor Attack; Contrastive Learning; GAN

††copyright: acmlicensed††journalyear: 2018††doi: XXXXXXX.XXXXXXX††conference:
Make sure to enter the correct conference title from your rights confirmation
emai; June 03–05, 2018; Woodstock, NY††isbn: 978-1-4503-XXXX-X/18/06††ccs:
Security and privacy Intrusion detection systems

##  1\. Introduction

With the rise in awareness of data privacy protection, centralized data
collection faces significant challenges, making collecting training data a
pressing issue in machine learning. Federated learning, as a novel distributed
machine learning method that protects privacy (McMahan et al., 2017), cleverly
bypasses data collection challenges and has thus received widespread
attention.

In Federated Learning (FL), training data is kept locally on the user’s
device, and only the model initialization and trained model parameters are
transmitted between the server and participating users. This method allows for
secure training collaboration among multiple parties while protecting user
privacy. Nevertheless, owing to the data being distributed among various
participating entities, this distributed characteristic, while rendering the
model training more flexible and secure, concurrently offers attackers a
foothold(Xie et al., 2020; Fung et al., 2020). They can stealthily guide the
model to produce specific predictions for certain triggers without impacting
the global model training process (Bagdasaryan et al., 2020), thereby
diminishing the model’s accuracy and credibility. Thus, researching effective
defenses against backdoor attacks in federated learning is crucial for
ensuring the model’s security and reliability.

In the field of deep learning, there has been considerable research dedicated
to defending against backdoor attacks. Methods for recovering backdoor
triggers, such as NC (Wang et al., 2019) and GANsweep (Zhu et al., 2020), aim
to mitigate backdoor attacks by recovering the triggers of poisoned models,
thereby achieving defense objectives. However, due to the specific data
distribution in federated learning, applying methods from centralized learning
directly to FL may lead to poor defense outcomes or model performance, often
resulting in the global model failing to converge. Furthermore, the model
updating process in federated learning involves parameter exchange and model
aggregation among servers and multiple users, which adds to the defense’s
complexity. Therefore, developing new defense mechanisms tailored to the
peculiarities of federated learning is necessary to address potential backdoor
attack threats.

Existing federated learning backdoor defences schemes can be mainly divided
into two categories: methods based on anomaly detection and those utilizing
pruning or noise addition techniques. Methods based on anomaly detection
usually require extensive computations on the server side (Gao et al., 2021),
detecting backdoor attacks by monitoring the similarity between models or the
abnormal changes in update behaviors. However, due to the complexity of
federated learning models and the heterogeneity of data distribution, these
methods often incur excessive computational costs and have high prerequisites
for the scenarios (Ye et al., 2023; Radford et al., 2021). On the other hand,
methods based on pruning or adding noise work by diluting or reducing the
impact of malicious model updates on the global model through the insertion of
noise into the model updates (Liu et al., 2018; Sun et al., 2019). Although
this approach can enhance the robustness of the model, the added noise often
leads to a decrease in model performance and accuracy, thus affecting the
overall performance of the model. Therefore, it is imperative to seek a more
effective method for defending against federated learning backdoor attacks to
balance security and performance demands.

To overcome the limitations of existing methods, this paper proposes a
federated learning backdoor attack defense method based on Contrastive
Learning (Chen et al., 2020) and Generative Adversarial Networks (GAN)
(Goodfellow et al., 2020), named GANcrop. This method utilizes contrastive
learning to delve into the differences between malicious and benign models,
achieving effective attack identification. Then, on the predicted malicious
models, it uses GAN to recover the trigger of poisoned models and carries out
targeted backdoor mitigation to achieve the defense purpose. The contributions
of this paper are as follows:

  * •

We have implemented a new federated learning backdoor attack defense method,
GANcrop, based on contrastive learning and generative adversarial networks.

  * •

We introduced a model detection method based on contrastive learning, which
can effectively distinguish between malicious and benign models under the
multi-model scenario of federated learning, achieving effective attack
detection. This is one of the rare model-level contrastive learning methods in
existing research.

  * •

We applied the backdoor trigger recovery approach from deep learning to the
defense work against backdoor attacks in federated learning. In non-IID
scenarios, our method achieves sound defense effects while ensuring model
accuracy.

##  2\. Preliminary Knowledge

Backdoor attacks in neural networks aim to guide the neural network model to
produce incorrect output results by making subtle, humanly imperceptible
modifications to the input data. Attackers inject backdoors into the model
during the training or deployment phase, causing the model to produce
incorrect predictions under specific triggering conditions. In backdoor
attacks, the attacker’s task can be described as a multi-objective
optimization problem, where the attacker needs to maintain the accuracy of the
main task while achieving a high success rate of backdoor attacks on targeted
attack class samples (Nguyen et al., 2024). The optimization objective
function of a backdoor attack can be represented by Formula (1):

(1)
$\theta^{*}=\min_{\theta}\left(\sum_{i\in|D|}L(x_{{}_{i}},y_{{}_{i}})+\sum_{i\in\left|D_{p}\right|}L(\psi(x_{{}_{i}}),\tau(y_{{}_{i}}))\right)$

Where D is the test set for the main task; ${D_{p}}$ is the poisoned dataset
containing backdoor samples, these samples are manipulated by the
transformation function $\psi$, outputting a specific $y$ under the backdoor
task.

Figure 1. GANcrop architecture diagram

A woman and a girl in white dresses sit in an open car.

##  3\. Proposed Method

This section will introduce the federated learning backdoor attack defense
framework based on Generative Adversarial Networks (GAN). This framework
mainly consists of three modules: attack detection, backdoor mitigation, and
model aggregation. The framework of our method is shown in Fig. 1.

###  3.1. Attack Detection

In devising strategies to defend against backdoor attacks, directly comparing
the similarity between models is often insufficient to identify contaminated
models, as triggers are usually concealed. It is necessary to delve into the
analysis of model parameter differences and sensitivities to distinguish
between benign and poisoned models. In backdoor attacks, triggers are
generally placed around the periphery of images, not disturbing the main
task’s accuracy. It’s challenging to analyze the trigger location from model
parameters directly, so we employ contrastive learning to train an anomaly
detection model sensitive to edge position parameters, conducting direct
contrastive training on model data, which is particularly crucial in federated
learning.

We construct a poisoned dataset with non-central triggers to train the
poisoned models. Under the contrastive learning framework, we build positive
and negative sample pairs from the parameters of poisoned and clean models.
Unlike traditional methods (Chen et al., 2020), in our construction, different
poisoned model samples from different poisoned or benign models benign model
samples are considered positive samples, while poisoned and benign models are
considered negative samples. This method of constructing sample pairs allows
the model to extract as many common features among poisoned models as
possible. Our framework optimizes the contrastive learning model by conducting
direct contrastive training on model data and learning the patterns of trigger
locations to distinguish between model categories accurately. This is key to
our model-level contrastive detection in federated learning.

Our model contrast detection diagram is shown in Fig. 2. The specific
contrastive loss function is shown in Formula (2):

(2)
$L_{\text{InfoNCE}}=-\log\frac{\exp(d(u,v)^{-}/\tau)}{\sum_{k=1}^{K}\exp(d(u,v_{k})^{+}/\tau)}$

Where $d(u,v)^{-}$represents the distance between positive sample pairs,
$d(u,v_{k})^{+}$ represents the distance between negative sample pairs, and
$\tau$ is the temperature parameter of the contrastive loss, used to control
the degree of softening.

Figure 2. Model Contrastive Learning

Through the extraction of model representations using contrastive learning,
it’s possible to identify the common characteristics of potentially anomalous
models. Utilizing these characteristics for model classification training
allows for precisely excluding malicious influences and identifying
potentially poisoned models. This training process is completed on the server
side using a public dataset to distinguish potential poisoned models through
feature extraction and classifier training. The algorithm for GANcrop, as
shown in Algorithm 1.

###  3.2. Backdoor Mitigation

To effectively defend against backdoor attacks, we further mitigate backdoors
based on the results of parameter comparison detection, reducing the impact of
the attacks. We use GAN to recover the backdoor triggers of poisoned models,
reintroduce the recovered backdoors into the server’s clean dataset with
correct labels and retrain the poisoned models to mitigate the backdoors.

The GANGSWEEP framework inspires the backdoor recovery strategy, a defense
method in deep neural networks that uses GANs to recover backdoor triggers and
mitigate their effects. However, directly applying it to FL may cause
convergence difficulties for the global model. Therefore, this paper adopts a
method of attack detection followed by backdoor mitigation to defend against
backdoor attacks in FL.

In this paper, we attempt to recover the backdoor triggers in the contaminated
model F using the generator of a Generative Adversarial Network (GAN). The
backdoor triggers are usually designed as special vectors with the same
dimensions as the image. Such a design ensures the feasibility of generating
triggers from the image model. Unlike traditional GANs, we do not optimize the
discriminator during training; instead, we use the model from potentially
malicious users as a substitute to guide the generator in recovering the
original triggers more accurately. To verify the effectiveness of the
generated triggers, this paper injects the generated triggers one by one into
the images (marked as $x$) in the validation dataset, forming new images
$G(x)+x$, and observes the predictions of the malicious model F on these new
images to guide the generator in iteratively improving, gradually learning
trigger features. Our architecture can effectively recover backdoor triggers
through continuous iterative training, as illustrated in Fig. 3.

Input: Number of simulated clients $N$, learning rate $\eta$, ResNet network
model $R$, initial feature extraction $SimModel$, simulated user model
parameters $w_{i}^{t}$ , epoch $E$, trigger $TR$, public dataset $D_{c}$,
proportion of poisoned dataset $\gamma$, simulated attack user set $S_{p}$,
simulated benign user set $S_{b}$, positive and negative sample pair
construction function $g(\cdot)$

Output: Model feature extractor $SimModel$, discriminator

1

2 Function  _ModContract(_$D_{c},TR$_)_:

3 $D_{p}=\gamma D_{c}+TR$;

4 Send the initial model to each simulation client;

5 for  _$t=0$ to $T-1$_ do

6 for  _$j=0$ to $N-1$_ do

7 Send global model parameters to the customer;

8 if  _$j\in S_{p}$_ then

9 $w_{i}^{t}\leftarrow\text{Local Training}(j,w^{t},D_{p})$;

10

11 end if

12 else

13 $w_{i}^{t}\leftarrow\text{Local Training}(j,w^{t},(D_{c}-D_{p}))$;

14

15 end if

16 $(u,v)=g(w_{i}^{t}(j\in S_{p}),w_{i}^{t}(i\in S_{b}))$;

17
$L_{\text{InfoNCE}}=-\log\frac{\exp(d(u,v)^{-}/\tau)}{\sum_{k=1}^{K}\exp(d(u,v_{k})^{+}/\tau)}$;

18 $SimModel\leftarrow SimModel-\eta\nabla L$;

19

20 end for

21

22 end for

23 return _IdentifyMod_ ;

24

25

26

27Function  _ModClassify(_$SimModel,S_{p},S_{b},w_{i}^{t}$_)_:

28 $IdentifyMod\leftarrow$ LinearEvaluate($SimModel,\text{classes}$);

29 for  _$j=1$ to $E$_ do

30 $IdentifyMod_{j}\leftarrow IdentifyMod_{j-1}-\eta\nabla
F_{j}(S_{p},S_{b})$;

31

32 end for

33 return _IdentifyMod_ ;

34

35

Algorithm 1 Model Contrastive Detection Algorithm Figure 3. Trigger Generator

After obtaining the generated triggers highly similar to the original poisoned
model’s triggers, this paper cleverly injects these generated triggers into a
new, clean dataset while keeping the dataset’s real labels unchanged for model
retraining. The purpose is to allow the model to correctly identify and handle
the correspondence between image data and labels during retraining, thereby
effectively eliminating the impact of the backdoor triggers. The retraining
process is a process of knowledge updating and forgetting, where the model
gradually adjusts its internal parameters, forgetting the original backdoor
features to adapt to the new data distribution.

###  3.3. Model Aggregation

Based on the results of attack detection, this paper identifies and filters
out benign models and potentially malicious user models. Then, through the
backdoor recovery of the GAN model, pseudo triggers similar to the original
triggers of potentially malicious models are generated, and these pseudo
triggers are used for targeted backdoor mitigation. The models retrained
afterward will be restored to a state unaffected by any malicious influence.

To enhance the robustness and reliability of the model, fully utilizing the
training success of each user model, this repaired model is weighted and
aggregated with the identified benign models to obtain a global, more robust
federated learning model.

##  4\. Experiment

###  4.1. Experimental Setup

This experiment uses the RestNet18 neural network model as the base
architecture. The experiment employs a 3$\times$3 convolution kernel, with
stride and padding values set to 1. This experiment omits pooling layers to
avoid excessive compression of image information. The experiment involves 40
clients, selects CrossEntropyLoss as the loss function, and adopts classic
stochastic gradient descent (SGD) as the optimizer, with a learning rate set
to 0.1. In each iteration round, each client conducts training for four
epochs.

#### Dataset:

The experiment utilizes the widely used Cifar10 dataset in the same domain
applications. The dataset contains 60,000 data samples, 50,000 training
samples, and 10,000 test samples, with 10 categories in the dataset.

(a)

(b)

(c)

(d)

Figure 4. Attack Schematic

#### Data Division:

To achieve a non-IID data distribution, the experiment adopts the data skew
and label skew methods proposed in the literature (Li et al., 2022), setting
the Dirichlet coefficient to 0.7 to control the degree of data skew.

#### Attack Method:

The trigger injection (Li et al., 2021) backdoor attack method is adopted.
Fig. 4 shows the effect of trigger injection using the cifar10 dataset, where
the first two subfigures are original pictures, and the last two subfigures
are poisoned pictures after trigger injection.

###  4.2. Experimental Results and Analysis

#### Verifying the Effectiveness of GANcrop in Defending Against Backdoor
Attacks:

To compare the performance of GANcrop, control methods were selected including
FedAvg (McMahan et al., 2017) without defense measures as a baseline and four
representative defense methods: Krum(Blanchard et al., 2017), Trimmed-mean
(Melis et al., 2018), Fang(Li et al., 2019), and GANsweep(Zhu et al., 2020).
An attack ratio of 30$\%$ was set for the experiments, with these attacking
users launching attacks in each round of iteration. Fig. 5 (a) depicts the
rounds of successful defense against backdoor attacks for six methodologies on
the CIFAR-10 dataset, across a total of 50 experimental rounds. Here, we
define a successful defense against backdoor attacks as instances where
backdoor accuracy falls below 30%. Typically, the level of backdoor accuracy
is directly correlated with the success rate of the backdoor attack.

(a)

(b)

Figure 5. Successful rounds of defending against attacks and main task
accuracy of six methods

The experimental results indicate that the models trained using FedAvg, Krum,
Trimmed-mean, and Fang methods exhibit minimal effectiveness in defending
against backdoor attacks. In contrast, GANsweep demonstrates a significantly
higher defense capability when confronted with corresponding trigger attacks.
On the other hand, the GANcrop scheme presented in this paper achieves a more
balanced effect in defending against backdoor attacks. The reason is that
GANcrop is a federated scheme aiming to balance maintaining the accuracy of
the model’s main task with reducing the success rate of attacks.

#### Verifying the main task accuracy of GANcrop:

Fig. 5 (b) shows the change in the main task accuracy of the six experimental
methods on the cifar10 dataset. FedAvg, Trimmed-mean, and Fang do not
effectively defend against backdoor attacks despite having high main task
accuracy. However, combined with the experimental results of backdoor accuracy
above, it’s clear that FedAvg, Trimmed-mean, and Fang, despite having high
main task accuracy, do not effectively defend against backdoor attacks. By
discarding a portion of user models during the global aggregation phase, Krum
shows weakness in main task accuracy. Meanwhile, GANsweep, although it has
backdoor solid defense capabilities, also leads to lower main task accuracy.

Additionally, the GANcrop method presents a compromise in terms of the model’s
main task accuracy, achieving a level of backdoor defense effectiveness while
maintaining a main task accuracy that is higher than that of the GANsweep
method but lower than the other four methods.

#### Verify the main task accuracy and the backdoor task accuracy of the two
sub-models:

To validate the model’s attack detection and repair performance, this paper
analyzed the main task accuracy and the backdoor task accuracy of two sub-
models. Fig. 6 shows these results: Fig. 6 (a) displays the main task accuracy
of both the benign prediction model and the repair model, while Fig. 6 (b)
presents their backdoor task accuracy. Among them, a higher backdoor task
accuracy represents a higher backdoor attack success rate. Fig. 6 (a) reveals
that the benign model outperforms the repair model in main task accuracy,
potentially due to the repair model’s dataset. Fig. 6 (b) shows that the
backdoor task accuracy is significantly lower in the repair model. This
highlights the importance of integrating the benign and repair models to
lessen the main task accuracy loss from backdoor mitigation and diminish the
effects of undetected attacks.

(a)

(b)

Figure 6. The main task accuracy and backdoor attack success rate of two
submodels

#### Verifying Execution Time of Six Approaches:

The experiment further compared the execution time of the six different
methods, aiming to assess their total time consumption from local computation
to global aggregation in a single round of iteration. The execution times
obtained through experimental verification are shown in Table 1.

Table 1. Execution Time Time |  Fedavg |  Krum |  Trimmed _mean |  Fang |  GANsweep |  GANcrop
---|---|---|---|---|---|---
s |  258 |  303 |  302 |  533 |  451 |  477

##  5\. Conclusions

This paper addresses the problem of backdoor attacks in federated learning and
proposes a defense method based on contrastive learning and Generative
Adversarial Networks, GANcrop. Model comparison and sensitivity analysis of
parameters effectively distinguish malicious and benign models. Then,
utilizing GAN technology to recover and mitigate backdoor triggers in models,
significantly reduces the success rate of backdoor attacks. Experiments have
proven that compared to existing methods, GANcrop not only enhances the
defense against backdoor attacks but also mitigates the risk of backdoor
attacks in non-IID data scenarios while maintaining the accuracy of the main
task of federated learning models. In future work, efforts will be directed
towards further optimizing the mitigation strategy of GANcrop, aiming to
enhance the post-mitigation model accuracy and thereby augmenting the method’s
practical applicability.

##  6\. Acknowledgments

The research was supported in part by the Guangxi Science and Technology Major
Project (No.AA22068070), the National Natural Science Foundation of China
(Nos.62166004,U21A20474), the Key Lab of Education Blockchain and Intelligent
Technology, the Center for Applied Mathematics of Guangxi, the Guangxi ”Bagui
Scholar” Teams for Innovation and Research Project, the Guangxi Talent
Highland Project of Big Data Intelligence and Application, the Guangxi
Collaborative Center of Multisource Information Integration and Intelligent
Processing.

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