# Running couplings and unitarity
in a 4-derivative scalar field theory

Bob Holdom<EMAIL_ADDRESS>Department of Physics, University of
Toronto, Toronto, Ontario, Canada M5S 1A7

###### Abstract

We obtain the $\beta$-functions for the two dimensionless couplings of a 4d
renormalizable scalar field theory with cubic and quartic 4-derivative
interactions. Both couplings can be asymptotically free in the UV, and in some
cases also in the IR. This theory illustrates the meaning of unitarity in the
presence of a negative norm state. A perturbative calculation that accounts
for the new minus signs shows that the optical theorem is identically
satisfied. These minus signs also enter a discussion of tree-level scattering.
For a certain setup involving colliding beams of particles we find even more
intricate cancellations and quite normal behaviour at high energies. The
$\beta$-functions for the Stuckelberg gauged version of the theory are also
obtained.

##  I Introduction

We consider the following real scalar field theory in four spacetime
dimensions, with four derivatives appearing in both the kinetic term and the
cubic and quartic interaction terms,

$\displaystyle{\cal
L}_{1}=\frac{1}{2}\partial_{\mu}\phi(\Box+m^{2})\partial^{\mu}\phi+\lambda_{3}(\partial_{\mu}\phi\partial^{\mu}\phi)\,\Box\phi+\lambda_{4}(\partial_{\mu}\phi\partial^{\mu}\phi)^{2}.$
(1)

The scalar field $\phi=\phi(x)$ is dimensionless, as are the couplings
$\lambda_{3}$ and $\lambda_{4}$. We shall allow these couplings to have either
sign. The mass term breaks a classical scale invariance. At low enough
energies only the mass term survives and the field can be re-scaled to absorb
the mass, leaving a free and massless theory. Our interest is then in the UV
behaviour of the theory. Many of our results will be obtained by calculating
with a finite mass and then taking the $m\to 0$ limit at the end. These mass
independent results will exhibit the manner in which the scale invariance is
broken quantum mechanically. We do not claim that this $m\to 0$ limit should
be taken as the definition of the theory where $m$ is identically zero to
begin with. That theory has problems with its definition and cannot be handled
directly by standard methods. The theory in (1) can be.

This theory also exhibits a simple shift symmetry $\phi(x)\to\phi(x)+c$. A
term has a shift symmetry if a derivative acts on every factor of $\phi(x)$,
or if not, the term is equivalent to one that is by integration by parts.
Since the field is dimensionless the shift symmetry excludes an infinite
number of terms with dimensionless couplings. If the $\phi$ field couples to
any other field then the shift symmetry must also be preserved by that
coupling. The shift symmetry, along with the four derivative kinetic term and
the lack of couplings with inverse mass dimensions, ensures that the theory is
renormalizable. The two simpler theories, with $\lambda_{3}=0$ or
$\lambda_{4}=0$ respectively, are also renormalizable, and sometimes our focus
will be on one of those theories.

Our first goal is to study the renormalization of (1) and to obtain the one-
loop $\beta$-functions for the couplings $\lambda_{3}$ and $\lambda_{4}$. We
will display the flow diagram of the running couplings that shows their
behaviour in both the UV and the IR, where by IR we mean scales that are still
above the arbitrarily small mass $m$. Perhaps most notable is that there is a
family of trajectories where both couplings are asymptotically free in both
the UV and the IR. Our other goals are to study the unitarity of the theory
and the behaviour of scattering at high energies. We comment on the
Stuckelberg extension of the theory at the end.

A higher derivative scalar field theory has been extensively studied in
another context [1, 2, 3]. This is the theory of the conformal factor of the
metric as extracted from an effective action related to the conformal anomaly,
which in turn is generated by loops of light particles. A simple form of such
a Lagrangian using conventional notation is

$\displaystyle{\cal L}_{\rm
conf}=-\theta^{2}(\Box\sigma)^{2}-\zeta[2(\partial\sigma)^{2}\Box\sigma+(\partial\sigma)^{4}]+\gamma
e^{2\sigma}(\partial\sigma)^{2}-\lambda e^{4\sigma}.$ (2)

The corresponding action has a global conformal symmetry, and this is what
relates the coefficients in the second term and brings in the exponential
factors in the last two terms, which are the mass and cosmological constant
terms. The interest in (2) is usually driven by its effective description of
possible infrared physics whereas our interest in (1) is in the UV.
Calculations of $\beta$-functions for (2) usually do not treat renormalization
of $(\Box\sigma)^{2}$ as a wave function renormalization, and so give
different results. $\beta$-functions of the more standard type were given in
[2], but there are various minus sign and factor of 2 deviations from our
results. Very recent results for the renormalization of (2) have been given in
[4], without giving $\beta$-functions, and our results agree where they
overlap.

Other recent studies of higher derivative scalar theories have appeared in
various contexts [5, 6, 7], but without considering both the $\lambda_{3}$ and
$\lambda_{4}$ couplings simultaneously. There is interest in these theories
even with an assumed lack of unitarity due to the existence of a negative norm
state. But there is more to the unitarity story than this. As far as we are
aware, the authors of [8] were the first to look more closely at the optical
theorem in higher derivative scalar theories and its relation to
renormalizability. (One of their criteria for renormalizability appears to be
equivalent to requiring the shift symmetry to apply only to terms with
dimensionless couplings.) These authors derive an inequality from the optical
theorem at weak coupling and show by example that it is satisfied only when
the theory is renormalizable, with their renormalizable example being the
$\lambda_{3}=0$ theory. The authors refer to this as S-matrix unitarity, the
fact that $SS^{\dagger}=1$ can still hold, even though it no longer implies
the standard bound on the size of amplitudes at high energies.

Another motivation for the study of (1) is its similarity to quantum quadratic
gravity. The latter is also a renormalizable quantum field theory [9] with
four derivatives in all terms except for a mass term with two derivatives. The
mass term is the Einstein term and the two four-derivative interaction terms
are curvature-squared terms with dimensionless couplings. The mass term again
breaks a classical scale invariance, while the coordinate invariance of
quadratic gravity is the analog of the shift symmetry of (1). Quadratic
gravity is a significantly more complex theory, and so the theory in (1) is
thus a useful proxy for quadratic gravity where we can much more easily obtain
loop amplitudes, and where we can in particular investigate the optical
theorem.

With the completeness relation

$\displaystyle\mathbb{1}=\sum_{X}\int d\Pi_{X}\frac{|X\rangle\langle
X|}{\langle X|X\rangle}$ (3)

inserted on the RHS of $-i(T-T^{\dagger})=TT^{\dagger}$, where $S=1+iT$, we
see that there are new minus signs on the RHS of the optical theorem when a
state $|X\rangle$ has a negative norm. These signs produce the cancellations
among the terms on the RHS as observed by [8]. By focusing on the inequality
satisfied by the RHS, those authors avoided calculating the one-loop amplitude
appearing on the LHS of the optical theorem. In Section III we shall use the
$\lambda_{3}=0$ theory to calculate both the LHS and RHS to show that the
optical theorem at order $\lambda_{4}^{2}$ is identically satisfied, both in
the $m\to 0$ limit and for finite $m$. We can refer to the theory as having
unitarity without positivity.

We then argue that the structure of the theory puts constraints on the
scattering experiments that can actually be physically realized. In Section IV
we calculate the lowest order event rates in such scattering experiments and
find that they can have high energy behaviour similar to theories with
standard unitarity bounds. This is due to even more intricate cancellations
brought about by the lack of positivity among the terms summed in an inclusive
process. We have also found these types of cancellations for the scattering of
gravitational degrees of freedom in [10] and for photon-photon scattering in
[11], both in the context of quantum quadratic gravity.

##  II Renormalization and running couplings

With the following transformations,

$\displaystyle\phi$ $\displaystyle=\psi_{1}-\psi_{2},$ (4)
$\displaystyle\psi_{1}$ $\displaystyle=\frac{1}{m^{2}}(\Box+m^{2})\phi,$ (5)
$\displaystyle\psi_{2}$ $\displaystyle=\frac{1}{m^{2}}\Box\phi,$ (6)

Lagrangian ${\cal L}_{1}$ can be converted to Lagrangian ${\cal L}_{2}$,

$\displaystyle{\cal L}_{2}$
$\displaystyle=-\frac{m^{2}}{2}\psi_{1}\Box\psi_{1}+\frac{m^{2}}{2}\psi_{2}(\Box+m^{2})\psi_{2}$
$\displaystyle+\lambda_{3}\left(\partial\psi_{1}-\partial\psi_{2}\right)^{2}(\Box\psi_{1}-\Box\psi_{2})$
(7)
$\displaystyle+\lambda_{4}\left(\partial\psi_{1}-\partial\psi_{2}\right)^{4}.$

${\cal L}_{2}$ is familiar for having the wrong sign kinetic term for
$\psi_{2}$. We have chosen to keep the $\psi_{1}$ and $\psi_{2}$ fields
dimensionless, since it is not difficult to account for the $m^{2}$
normalization of the kinetic terms in calculations. Note that the shift
symmetry $\phi\to\phi+c$ corresponds to $\psi_{1}\to\psi_{1}+c$ and
$\psi_{2}\to\psi_{2}$.111That these two Lagrangians are the same can be
deduced by substituting (4) into the quadratic term in (1), taking the
difference with the quadratic terms in (7), and then substituting (5) and (6)
into the result to find zero after an integration by parts.

Lagrangian ${\cal L}_{1}$ can be taken to define a propagator of a two pole
form,

$\displaystyle-\frac{i}{(k^{2}+i\epsilon)(k^{2}-m^{2}+i\epsilon)}.$ (8)

This gives Feynman diagrams where the number of propagators is effectively
doubled, where each pair of propagators with the same momentum can be
interpreted to bring in an additional zero-momentum vertex. The diagram can
thus be evaluated by standard methods. But when the effective number of
propagators grows to six or more then standard methods are not so well
developed and this approach quickly becomes cumbersome.

For the second Lagrangian ${\cal L}_{2}$, the number of fields is doubled, and
so each internal line can be one of the following two propagators,

$\displaystyle\frac{1}{m^{2}}\,\frac{i}{k^{2}+i\epsilon},\quad-\frac{1}{m^{2}}\,\frac{i}{k^{2}-m^{2}+i\epsilon}.$
(9)

Only the first has the normal sign. The number of diagrams quickly grows with
the number of internal lines, but standard methods can now more easily deal
with, say, four propagators in a loop, rather than eight that would occur from
${\cal L}_{1}$.

We can suppose that either of these Lagrangians are composed of renormalized
fields and renormalized couplings $m$, $\lambda_{3}$ and $\lambda_{4}$. These
renormalized quantities are related to bare quantities via the renormalization
constants $\phi_{0}=Z_{\phi}^{\frac{1}{2}}\phi$, $m_{0}^{2}=Z_{m}m^{2}$,
$\lambda_{30}=Z_{3}\lambda_{3}$ and $\lambda_{40}=Z_{4}\lambda_{4}$. A
counterterm Lagrangian is constructed using these renormalization constants
such that the total Lagrangian yields finite results at one-loop. The
respective counterterm Lagrangians are

$\displaystyle{\cal L}_{1}^{\rm c.t.}$
$\displaystyle=\frac{1}{2}\partial\phi\\!\left[(Z_{\phi}-1)\Box+(Z_{\phi}Z_{m}-1)m^{2}\right]\\!\partial\phi$
$\displaystyle+(Z_{\phi}^{\frac{3}{2}}Z_{3}-1)\lambda_{3}(\partial\phi)^{2}\Box\phi$
(10) $\displaystyle+(Z_{\phi}^{2}Z_{4}-1)\lambda_{4}(\partial\phi)^{4},$
$\displaystyle{\cal L}_{2}^{\rm c.t.}$
$\displaystyle=-\frac{m^{2}}{2}\psi_{1}(Z_{\phi}Z_{m}-1)\Box\psi_{1}+\frac{m^{2}}{2}\psi_{2}\\!\left[(Z_{\phi}Z_{m}-1)\Box+(Z_{\phi}Z_{m}^{2}-1)m^{2}\right]\\!\psi_{2}$
$\displaystyle+(Z_{\phi}^{\frac{3}{2}}Z_{3}-1)\lambda_{3}\left(\partial\psi_{1}-\partial\psi_{2}\right)^{2}(\Box\psi_{1}-\Box\psi_{2})$
(11)
$\displaystyle+(Z_{\phi}^{2}Z_{4}-1)\lambda_{4}\left(\partial\psi_{1}-\partial\psi_{2}\right)^{4}.$

The total Lagrangians ${\cal L}_{1}+{\cal L}_{1}^{\rm c.t.}$ and ${\cal
L}_{2}+{\cal L}_{2}^{\rm c.t.}$ when written in terms of bare quantities take
the same form as the original Lagrangians ${\cal L}_{1}$ and ${\cal L}_{2}$.
We have used $Z_{\psi_{1}}=Z_{\psi_{2}}=Z_{\phi}$.

Either ${\cal L}_{1}$ or ${\cal L}_{2}$ can be used to extract the various
renormalization constants. Using dimensional regularization with

$\displaystyle\frac{2}{\varepsilon}=\frac{2}{4-d}-\gamma_{E}+\ln(4\pi),$ (12)

we find the following results222Our calculations use a combination of Maple
and Package-X [12] for Mathematica.

$\displaystyle Z_{\phi}Z_{m}-1$
$\displaystyle=-\frac{3\lambda_{4}}{4\pi^{2}}\frac{1}{\varepsilon},$ (13)
$\displaystyle Z_{\phi}-1$
$\displaystyle=\frac{5\lambda_{3}^{2}}{8\pi^{2}}\frac{1}{\varepsilon},$ (14)
$\displaystyle Z_{\phi}^{\frac{3}{2}}Z_{3}-1$
$\displaystyle=-\frac{5\lambda_{4}}{4\pi^{2}}\frac{1}{\varepsilon},$ (15)
$\displaystyle Z_{\phi}^{2}Z_{4}-1$
$\displaystyle=-\frac{5\lambda_{4}}{4\pi^{2}}\frac{1}{\varepsilon}.$ (16)

The one-loop one-particle-irreducible diagrams that produce these
$\varepsilon$-poles are those that involve one or two propagators, and the
four diagrams corresponding to (13), (14), (15), (16) respectively are shown
in Fig. 1.333The use of ${\cal L}_{1}$ to calculate these diagrams is somewhat
simpler, and if ${\cal L}_{2}$ is used to extract $Z_{\phi}$ and $Z_{m}$, the
one-loop corrections that emerge are in the form of the kinetic terms in
${\cal L}_{1}$ after inserting $\phi=\psi_{1}-\psi_{2}$.

Figure 1: Divergent diagrams.

There are also 3 and 4-point one-particle-irreducible diagrams that have more
than two propagators, as shown in Fig. 2. Power counting arguments suggest
that these diagrams should also have log divergences, similar to the diagrams
in Fig. 1. But this is not the case; we find that the diagrams in Fig. 2 are
finite by explicit calculation using ${\cal L}_{2}$. (For the first diagram in
Fig. 2 this was noted in [5].) Why this is so is related to renormalizability.
The diagrams in Fig. 2 can be converted to 5, 6, 7 or 8-point functions by
converting cubic vertices to quartic vertices. Power counting suggests that
those diagrams also have log divergences. But there are no counter-terms with
these numbers of fields, due to the shift symmetry, and so the absence of
divergences for such diagrams must be due to the shift symmetry. It thus
appears that the diagrams in Fig. 2 are finite for this reason as well.

Figure 2: Finite diagrams.

The nature of the divergence in the first diagram in Fig. 1, which
renormalizes the $m^{2}\partial\phi\partial\phi$ term, is different from the
other three. Other than the factor of $p^{2}m^{2}$, there is no further
momentum dependence and thus no scale dependence from this tadpole-type
diagram. Effectively (13) should be replaced by $Z_{\phi}Z_{m}=1$ at order
$1/\varepsilon$. If we define the anomalous dimensions as
$\gamma=\frac{1}{2}dZ_{\phi}/d\ln\mu$ and
$\gamma_{m}=\frac{1}{2}dZ_{m}/d\ln\mu$ then we have

$\displaystyle\gamma=-\gamma_{m}=\frac{5\lambda_{3}^{2}}{16\pi^{2}}.$ (17)

The running of the mass is due entirely to the wave function renormalization,
due to the second diagram in Fig. 1, and the mass term as a whole has no scale
dependence.

The other three diagrams in Fig. 1 do introduce nontrivial scale dependence.
The $\beta$-functions in $d=4$ dimensions are given by the residues of the
$1/(4-d)$ poles in $Z_{3}\lambda_{3}$ and $Z_{4}\lambda_{4}$ respectively,

$\displaystyle\frac{d\lambda_{3}}{d\ln\mu}$
$\displaystyle=-\frac{5}{4\pi^{2}}(\lambda_{4}\lambda_{3}+\frac{3}{4}\lambda_{3}^{3}),$
(18) $\displaystyle\frac{d\lambda_{4}}{d\ln\mu}$
$\displaystyle=-\frac{5}{4\pi^{2}}(\lambda_{4}^{2}+\lambda_{4}\lambda_{3}^{2}).$
(19)

The second term in each $\beta$-function is due to the wave function
renormalization. Clearly both couplings are asymptotically free in the UV when
both are positive. For general signs, we show the flow diagram for the two
couplings in Fig. (3). Interesting behaviour occurs in the quadrants where
$\lambda_{4}<0$. Then there are trajectories where both couplings are
asymptotically free in both the UV and the IR, such that $|\lambda_{3}|$ and
$|\lambda_{4}|$ reach maximum values at intermediate scales. The decrease of
$|\lambda_{3}|$ and $|\lambda_{4}|$ in the IR will continue down to the scale
of the mass, below which the theory is free.

Our results for the $\beta$-functions depend on the normalizations and signs
in our definition of the theory in (1). For example it may be common to
reverse the signs of the interaction terms, in which case (18, 19) are
transformed with $\lambda_{3}\to-\lambda_{3}$ and
$\lambda_{4}\to-\lambda_{4}$. The resulting flow diagram is obtained from the
one in Fig. (3) by flipping it around the line $\lambda_{4}=0$. In [5] the
$\lambda_{3}=0$ and $\lambda_{4}=0$ theories are treated separately, and so
they obtain the $\lambda_{3}^{3}$ term in (18) and the $\lambda_{4}^{2}$ term
in (19) (the latter is also obtained in [7]). Our results agree.

Figure 3: Flow diagram for the two couplings where the arrows point towards
the UV. The extension to negative $\lambda_{3}$ is obtained by flipping the
above diagram around the line $\lambda_{3}=0$.

With dimensional regularization we can reduce the number of spacetime
dimensions $d$ and look for $1/(2-d)$ poles, which would correspond to
quadratic divergences in four dimensions. We find a $1/(2-d)$ pole only in one
case, in the tadpole diagram associated with the mass renormalization. The
other diagrams in Fig. 1 are finite in two dimensions.

##  III Unitarity and the optical theorem

First we give the full momentum dependent amplitudes corresponding to the last
three diagrams in Fig. 1, after taking the $m/E\to 0$ limit. No infrared
divergences are encountered in this limit. The Feynman amplitudes are a factor
of $i$ times the following two, three and four-point functions. The
$1/\varepsilon$ poles appearing here are cancelled by the Lagrangian
counterterms as determined by (14), (15), (16) respectively.444The overall
sign of these results depends on the number of $\psi_{2}$ fields, as is the
case for the tree-level vertices. The results presented assume an even number.
This is also the case of interest for the optical theorem, since elastic
scattering involves the same particles in the initial and final states.

$\displaystyle\Sigma(p)$
$\displaystyle=\frac{\lambda_{3}^{2}}{(4\pi)^{2}}(p^{2})^{2}\left(\frac{10}{\varepsilon}+3+5\log\left(-\frac{\mu^{2}}{p^{2}}\right)\right)$
(20) $\displaystyle\Gamma_{3}(p,q)$
$\displaystyle=\frac{\lambda_{3}\lambda_{4}}{(4\pi)^{2}}\left((\frac{80}{\varepsilon}+\frac{112}{3})((pq)^{2}-p^{2}q^{2})\right.$
(21)
$\displaystyle-\frac{8}{3}[(7p^{2}q^{2}+6(pq)p^{2}-(pq)^{2})\log(-\frac{\mu^{2}}{p^{2}})+(7p^{2}q^{2}+6(pq)q^{2}-(pq)^{2})\log(-\frac{\mu^{2}}{q^{2}})$
$\displaystyle\left.+(p^{2}q^{2}-6(pq)(p^{2}+q^{2})-13(pq)^{2})\log(-\frac{\mu^{2}}{(p+q)^{2}})]\right)$
$\displaystyle\Gamma_{4}(s,t)$
$\displaystyle=\frac{\lambda_{4}^{2}}{(4\pi)^{2}}\left((\frac{80}{\varepsilon}+\frac{112}{3})(s^{2}+st+t^{2})+i\frac{8}{3}\pi\,(7s^{2}+st+t^{2})\right.$
(22)
$\displaystyle\left.+\frac{8}{3}[(7s^{2}+st+t^{2})\log(\frac{\mu^{2}}{s})+(s^{2}+st+7t^{2})\log(-\frac{\mu^{2}}{t})+(7s^{2}+13st+7t^{2})\log(\frac{\mu^{2}}{s+t})]\right)$

Our focus shall be on the optical theorem for two-to-two scattering where we
restrict to the $\lambda_{3}=0$ theory, and for which the four-point function
$\Gamma_{4}(s,t)$ is of interest. We have assumed that the four external
particles are on-shell, and so the $m/E\to 0$ limit produces a four-point
function that is a function of the Mandelstam variables $s$ and $t$.555A
similar result is obtained in [7]. It is evaluated in the physical region
where $s>0,t<0,s+t>0$ such that the imaginary part is displayed explicitly.
For elastic scattering in the forward direction we are interested in
$\Gamma_{4}(s,0)$; twice its imaginary part is the $s$ cut discontinuity that
is related by the optical theorem to an inclusive cross section (with the
initial state normalization factor omitted). We see from (22) that this
discontinuity is $7s^{2}/3\pi$.

Before checking the optical theorem we first mention some factors appearing in
an exclusive cross section. Most importantly the $\psi_{2}$ state is a
negative norm state, and this results in a factor $(-1)^{n_{2}}$ where $n_{2}$
is the number of $\psi_{2}$’s in the initial and final states.666A cross
section is related to a probability and this involves dividing by the norms of
the states involved, similar to the appearance of the norm in the completeness
relation (3). The presence of the $m^{2}$ factors in the kinetic terms of
${\cal L}_{2}$ also implies an additional normalization factor of $1/m^{2}$
for each $\psi_{1}$ or $\psi_{2}$ state. These factors would seem to make the
$m\to 0$ limit problematic, but this is not the case for the optical theorem
and for our discussion in the next section. We can also comment on the LSZ
reduction formulas that extract scattering amplitudes from the residues of the
poles of Greens functions. For Lagrangian ${\cal L}_{1}$ each external line is
associated with two poles, and so the residues of the various products of
poles give the scattering amplitudes. For Lagrangian ${\cal L}_{2}$ the
reduction formula is more standard, and indeed, the definition of $\psi_{1}$
and $\psi_{2}$ in (5) and (6) can be seen to leave just one pole for each
external line.

The RHS of the optical theorem involves a sum of squares of tree-level, on-
shell, two-to-two amplitudes generated by the quartic interaction. These tree-
level amplitudes are

$\displaystyle
i(-1)^{n_{2}}8\lambda_{4}[(p_{1}p_{2})(p_{3}p_{4})+(p_{1}p_{3})(p_{2}p_{4})+(p_{1}p_{4})(p_{2}p_{3})].$
(23)

The $(-1)^{n_{2}}$ amplitude factor here is not to be confused with the same
factor in an exclusive cross section. The squares of these amplitudes are
summed over the possible final states for a fixed initial state, with the norm
and normalization factors included for the possible final states, but not for
the initial state. (The RHS of the optical theorem differs from a cross
section by this initial state factor.) We employ on-shell kinematics for all
particles and express each amplitude in terms of $s$ and $m$ and the
scattering angle in the centre-of-momentum frame. The angular integration of
the sum of squares is done with a factor of $\frac{1}{2}$ to avoid double
counting, after which we finally let $m/E\to 0$. We find that the result is
$7s^{2}/3\pi$ and so the optical theorem is satisfied. Any of the initial
states, $\psi_{1}\psi_{1}$, $\psi_{2}\psi_{2}$ or $\psi_{1}\psi_{2}$ gives the
same result.

We can also consider the finite mass version of the optical theorem, and for
this we restrict to the $\psi_{1}\psi_{1}$ initial state. For the LHS we
evaluate the fourth diagram of Fig. 1 without taking the $m/E\to 0$ limit and
find that the discontinuity across the $s$ cut is

$\displaystyle\frac{\lambda_{4}^{2}}{30m^{4}\pi}$
$\displaystyle\left(21s^{4}\,\theta(s)-\frac{2}{s}(s-m^{2})^{3}(21s^{2}+8m^{2}s+m^{4})\,\theta(s-m^{2})\right.$
(24)
$\displaystyle\left.+\;s\sqrt{s(s-4m^{2})}(21s^{2}-68m^{2}s+56m^{4})\,\theta(s-4m^{2})\right).$

For the RHS of the optical theorem we just avoid taking the $m/E\to 0$ limit
in the previous evaluation of the RHS. The result is again (24) and so this
confirms the optical theorem for any $m$. Due to the negative sign of the
$\theta(s-m^{2})$ term, at large $s$ the cancellations produce $s^{2}$
behaviour rather than the naive $s^{4}/m^{4}$ behaviour. This minus sign is
due to a negative propagator when the LHS is calculated, and it is due to a
negative norm when the RHS is calculated. So it is not a surprise that the
optical theorem is satisfied. The theory is displaying unitarity without
positivity. This more accurate description allows us focus on the actual
nonstandard property of the theory, which is the lack of positivity.

The RHS of the optical theorem becomes a cross section when multiplying by the
initial state factor $\propto 1/(sm^{4})$. It is this factor that introduces a
minus sign for the $\psi_{1}\psi_{2}$ initial state, meaning that this cross
section is negative. But $\psi_{2}$ does not describe an asymptotic state and
so this initial state cannot be treated in isolation. The massive $\psi_{2}$
particle is unstable, decaying into $2\psi_{1}$ or $3\psi_{1}$ or any other
light particles to which it couples. The proper time between the original
production of the $\psi_{2}$ and its scattering must be less than the
$\psi_{2}$ lifetime. There is another minus sign built into the $\psi_{2}$
production cross section, and so there are two minus signs, in the production
and scattering cross sections respectively, that cancel in the product. A
positive probability is associated with the complete $n_{2}$-even process.

We shall instead turn to high energies and short time scales so that both mass
and decay effects are small. Within this context we can consider a standard
scattering experiment involving the collision of two beams of particles. The
two beams should be composed of nearly equal numbers of $\psi_{1}$ and
$\psi_{2}$, since the $\psi_{2}$ or $\psi_{1}$ particles are produced through
couplings that always involves the combination $\psi_{1}-\psi_{2}$ and so the
two particles are equally likely to be produced. With this setup it is a sum
of probabilities that is positive, and it is this that is related to certain
physical observables in the high energy theory. We develop this picture in the
next section.

If it was possible to have pure $\psi_{1}$ beams, for example by utilizing
long enough times for the initial beams to be depleted in $\psi_{2}$’s, due to
their decay, then the cross section would behave like $s/m^{4}$ for large $s$.
This violation of an upper bound, that is normally attributed to unitarity, is
allowed when unitarity comes without positivity [8]. But there is a caveat
regarding this picture of on-shell scattering at large $s$. It may be that the
analog of parton showers occurs in both the initial and final states, which
would imply that the actual hard scattering occurs between highly virtual
excitations of the $\psi_{1}$ and $\psi_{2}$ fields. Then on-shell particles
and negative norm states are not actually involved in the hard scattering
process. So it is still not certain that the $s/m^{4}$ behaviour can be
realized. In any case, the on-shell picture with suitably inclusive
differential cross sections can still provide a useful dual description of the
off-shell scattering, as it does in perturbative QCD. It is the inclusive,
rather than the exclusive, on-shell cross sections that are meaningful in this
context. This view of the on-shell description was taken in [10] and the
discussion of the next section can be viewed in this light.

##  IV A scattering experiment

The sum or average over $\psi_{1}$ and $\psi_{2}$ in the initial state is
trickier to deal with than an averaging over, for example, polarizations. As
before we calculate for finite $m$ first before taking the $m/E\to 0$ limit.
Since intricate cancellations take place in this limit, we need to properly
account for all mass effects before the limit is taken. We shall introduce a
luminosity factor to better describe the initial state, since this factor is
another source of mass dependence. We assume short time scales so that the
effect of $\psi_{2}$ decay can be ignored.

Consider the collision of two finite beams of particles, each containing both
types of particles. A choice of initial state $i=(A,B)$ corresponds to
considering the collision of particles of type $A$ in one beam with particles
of type $B$ in the other beam. And for a choice $i$ we can consider the
integrated luminosity ${\cal L}^{\rm int}_{i}$ such that its product with the
cross section $\sigma_{i\to f}$ gives the number of scattering events from
initial state $i$ to final state $f$. The integrated luminosity is an integral
of the Lorentz-invariant luminosity density $S$, [13]

$\displaystyle{\cal L}^{\rm int}_{i}=\int dtd^{3}\textbf{x}\,S,\quad
S=\sqrt{(j_{A}j_{B})^{2}-j_{A}^{2}j_{B}^{2}}.$ (25)

The 4-vector $j^{\mu}$ is a particle current density. $j^{\mu}$ is
proportional to $p^{\mu}$, and in particular $j^{\mu}=(\rho/E)p^{\mu}$ where
$\rho$ is a particle density. Thus $\rho/E$ is a Lorentz invariant constant
and we can write

$\displaystyle S$
$\displaystyle=\frac{\rho_{A}\rho_{B}}{E_{A}E_{B}}\sqrt{(p_{A}p_{B})^{2}-m_{A}^{2}m_{B}^{2}}.$
(26)

The actual values of $\rho_{A}$ and $\rho_{B}$ are functions of the respective
momenta $p_{A}$ and $p_{B}$ of the particles $A$ and $B$ occurring in the
beams.

Now let us go to the centre-of-momentum frame where

$\displaystyle{\cal L}^{\rm int}_{i}\sigma_{i\to f}$
$\displaystyle=F_{AB}\frac{\sqrt{s}p_{i}}{E_{A}E_{B}}\sigma_{i\to f},$ (27)
$\displaystyle F_{AB}$ $\displaystyle=\int
dtd^{3}\textbf{x}\,\rho_{A}\rho_{B}.$

We have used
$\sqrt{(p^{\mu}_{A}p_{B\mu})^{2}-m_{A}^{2}m_{B}^{2}}=\sqrt{s}p_{i}$, where
$p_{i}$ is the magnitude of the equal and opposite 3-momenta describing the
initial state in this frame. $F_{AB}$ describes the beams and has dimensions
of inverse area. Also in this frame the exclusive differential cross section
is

$\displaystyle\frac{d\sigma_{i\to
j}}{d\Omega}=\frac{(-1)^{n_{2}}}{m^{8}}\frac{1}{64\pi^{2}s}\frac{p_{f}}{p_{i}}|{\cal
M}_{fi}(s,\theta)|^{2},$ (28)

where $\Omega=(\theta,\phi)$ characterizes the direction of a final state
3-momentum, of magnitude $p_{f}$, relative to the direction of an initial
state 3-momentum. Note that $p_{i}$, $p_{f}$, $E_{A}$, $E_{B}$ are all
determined from the value of $s$ and the masses and from the constraint that
all particles are on shell. We have mentioned the origin of the
$(-1)^{n_{2}}/m^{8}$ factor above.

It is the inclusive differential cross section, the sum of the exclusive ones,
that can have a physical meaning. In particular the combination of (27) and
(28) shows that the total number of scattering events in differential form is
given by

$\displaystyle\frac{dN}{d\Omega}=\sum_{i,f}\frac{(-1)^{n_{2}}}{64\pi^{2}}F_{AB}\frac{1}{E_{A}E_{B}}\frac{p_{f}}{\sqrt{s}}\frac{1}{m^{8}}|{\cal
M}_{fi}(s,\theta)|^{2}.$ (29)

The sum over $i$ and $f$ sums over all combinations of $\psi_{1}$ and
$\psi_{2}$ in the initial and final states that are kinematically allowed by
the value of $s$ being considered. Although we are only calculating this at
tree level, the imposition of the on-shell constraints produces a lengthy
expression. Cancellations occur in the $m/E\to 0$ limit as long as the
particle densities that determine $F_{AB}$ in the limit become independent of
whether particles $A$ and $B$ are massive or massless. For example $\rho_{A}$
and $\rho_{B}$ can be functions of the velocities $p_{i}/E_{A}$ and
$p_{i}/E_{B}$ respectively.

First we consider the $\lambda_{3}=0$ theory, where the tree level diagram is
just the quartic vertex. For our first example we simply set $F_{AB}$ to a
constant $F$. Then the leading term in the $m/E\to 0$ limit is

$\displaystyle\frac{dN}{d\Omega}=8\lambda_{4}^{2}\frac{F}{s}\left(5\cos\\!\left(\theta\right)^{4}+2\cos\\!\left(\theta\right)^{2}+1\right).$
(30)

For the second example we introduce an extra product of velocities,
$F_{AB}=(p_{i}^{2}/E_{A}E_{B})F$. Then the leading term in $m/E\to 0$ limit is

$\displaystyle\frac{dN}{d\Omega}=8\lambda_{4}^{2}\frac{F}{s}\left(35\cos\\!\left(\theta\right)^{4}+54\cos\\!\left(\theta\right)^{2}+15\right).$
(31)

As a third example we remove the luminosity factor altogether and simply sum
the cross sections. Then the result is

$\displaystyle\frac{dN}{d\Omega}=16\lambda_{4}^{2}\frac{1}{s}\left(5\cos\\!\left(\theta\right)^{2}+1\right).$
(32)

These results are all positive definite. Since the individual amplitudes grow
like $s^{2}$, any term in the sum in (29) has a factor that grows like
$(s/m^{2})^{4}/s$. We instead see that intricate cancellations produce $1/s$
behaviour at large $s$. This behaviour is the same as for theories that have
standard unitarity bounds. We see that the cancellations are quite robust,
leaving only the angular dependence sensitive to the precise beam description.

We can also consider the same scattering experiment for the $\lambda_{4}=0$
theory. In this case each of the tree diagrams is a one-particle exchange
diagram. Performing the sum over initial and final states as described above
again gives a result that falls like $1/s$ at large $s$. The $\theta$
dependence now features a $1/\sin(\theta)^{4}$ pole behaviour for forward and
backward scattering, due to the exchange of (near) massless particles. For the
same three examples the $\theta$ dependence is again positive definite.

Figure 4: The number of scattering events according to (29) after the angular
integration in the $\lambda_{3}=0$ theory for the choices of $F_{AB}=F$ (lower
curve) and $F_{AB}=(p_{i}^{2}/E_{A}E_{B})F$ (higher curve).

Going back to the $\lambda_{3}=0$ theory, we can consider the number of
scattering events in (29) without taking $m/E\to 0$. After doing the angular
integration we display the result in Fig. 4 for the two different choices of
$F_{AB}$, both of which give positive definite results. In the second case the
extra factors of velocity suppress the $\psi_{2}$ contribution close to the
thresholds where these massive particles are slow. The suppression of the
negative norm contributions means that the final positive result is larger.
Trouble finally arrives for the third example we considered, the naive sum
over cross sections, which gives a result that is not positive definite. Here
we notice the $1/p_{i}$ factor in (28), which is enhanced for a slow
$\psi_{2}$ in the initial state. We have arrived back with the problem of an
$n_{2}$-odd initial state that should not be treated in isolation.

##  V Comments

We have presented a number of standard calculations within a 4-derivative
quantum field theory with a negative norm state. The results show more
similarities to standard 2-derivative theories than might have been expected.
Rather paradoxically, it is the negative norm state itself that makes these
similarities possible. The optical theorem being satisfied and the good high
energy behaviour of inclusive scattering rates are both due to the
cancellations brought about by the negative norm state. In addition we have a
four dimensional scalar field theory that exhibits asymptotic freedom in the
UV with a possible concurrent asymptotic freedom in the IR.

One difference between 2 and 4 derivative theories involves the classical
limit. The negative norm is a consequence of the manner in which the theory is
quantized so as to ensure the propagation of positive energy only, and thus
stability. There is no analog of negative norms in the classical theory, which
instead directly exhibits negative energies and instabilities. We have
mentioned that the theory becomes a free 2-derivative theory for wavelengths
longer than $1/m$, and it is only here that the classical wave solutions make
sense. The classical limit of quantum quadratic gravity must similarly involve
long wavelengths where Einstein gravity applies. At wavelengths shorter than
$1/m$ (or $1/m_{\rm Pl}$) these theories are intrinsically quantum.

Our approach to the theory has been strictly perturbative where, besides some
unusual signs, standard Feynman methods apply. The perturbation theory
displays a standard analytic structure. An example of a deviation from this
strictly perturbative approach is the resummation involving the one-loop self-
energy diagram to obtain a dressed propagator. As soon as this is done the
causal structure of the theory changes due to the abnormal structure of the
dressed propagator. The dressed $\psi_{2}$ particle propagates backward-in-
time over time scales of order its lifetime [14, 15]. We have noted that a
positive probability is associated with an $n_{2}$-even process, that is a
process that involves both the appearance and disappearance of the $\psi_{2}$
particle.

At the higher energies and shorter times scales of interest to our study here,
it is appropriate to use the strictly perturbative approach. We have applied
this to the scattering within the $\psi_{1}$, $\psi_{2}$ sector, in particular
for $m/E\to 0$. A similar limit can be considered for the interactions between
the $\psi_{1}$, $\psi_{2}$ fields and a normal matter sector. The couplings
will involve the shift symmetric factor
$\partial_{\mu}\phi=\partial_{\mu}(\psi_{1}-\psi_{2})$ and so only this
combination of positive and negative norm states can be produced. The cross
section for some process that emits one $\phi$ can thus be calculated as a
difference of two contributions that only differ by the appearance or not of
$m$. The $1/m^{2}$ normalization factor is cancelled and the cross section
remains finite as $m/M\to 0$, where $M$ is some mass or energy scale
characterizing the process. In addition the virtual exchange of the $\phi$
mode produces scattering cross sections among normal particles that behave
sensibly as a function of energy. An example of this is photon scattering
$\gamma\gamma\to\gamma\gamma$ at arbitrarily high energies in the context of
quadratic gravity [11].

We end by considering a gauged extension of the theory. A $U(1)$ gauge field
$A_{\mu}$ can be introduced via the Stuckelberg construction by defining
$V_{\mu}=A_{\mu}+\partial_{\mu}\phi$. We also add the standard kinetic term
for the photon,

$\displaystyle{\cal L}_{1}$
$\displaystyle=-\frac{1}{4g^{2}}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}V_{\mu}(\Box+m^{2})V^{\mu}+\lambda_{3}V_{\mu}V^{\mu}\,\partial_{\mu}V^{\mu}+\lambda_{4}(V_{\mu}V^{\mu})^{2},$
(33) $\displaystyle
F_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$
(34)

The interest in this theory comes in the way it introduces cubic and quartic
interactions among photons in a gauge invariant and renormalizable manner. In
the appendix we give the $\beta$-functions of this theory.

*

##  Appendix A Running couplings in the gauged extension

We see from (33) that the photon kinetic terms come both from the standard
first term and from the $V^{2}$ term. We will continue to treat the
renormalization of the $\frac{1}{2}V_{\mu}\Box V^{\mu}$ term as the wave
function renormalization for the $\phi$ field, and thus also for the $A_{\mu}$
field. In turn this necessitates the independent renormalization of the
standard first term and thus the gauge coupling $g$ appears in this term.

From the two types of $A_{\mu}$ kinetic terms, the photon propagator in
Feynman gauge is

$\displaystyle\frac{-i\zeta\eta_{\mu\nu}}{k^{2}+i\epsilon},\quad\zeta=\left(1+\frac{1}{g^{2}}\right)^{-1}.$
(35)

Our results will then have factors of $\zeta$ corresponding to photon
propagators on internal lines.

The calculation of the renormalization constants will involve the same
diagrams as before, with the addition of diagrams that involve photon
propagators. We find

$\displaystyle Z_{\phi}Z_{m}-1$
$\displaystyle=-\left(\frac{3\lambda_{4}}{4\pi^{2}}+\frac{3\lambda_{3}^{2}}{8\pi^{2}}\zeta\right)\frac{1}{\varepsilon},$
(36) $\displaystyle Z_{\phi}-1$
$\displaystyle=\left(\frac{5}{8}(1+\zeta^{2})+\frac{1}{2}\zeta\right)\frac{\lambda_{3}^{2}}{\pi^{2}}\frac{1}{\varepsilon},$
(37) $\displaystyle Z_{\phi}^{\frac{3}{2}}Z_{3}-1$
$\displaystyle=-\left(\frac{5}{4}+2\zeta+3\zeta^{2}\right)\frac{\lambda_{4}}{\pi^{2}}\frac{1}{\varepsilon},$
(38) $\displaystyle Z_{\phi}^{2}Z_{4}-1$
$\displaystyle=-\left(\frac{5}{4}+3\zeta+6\zeta^{2}\right)\frac{\lambda_{4}}{\pi^{2}}\frac{1}{\varepsilon},$
(39) $\displaystyle Z_{\phi}Z_{g}^{-2}-1$
$\displaystyle=-\left(\frac{7}{12}(1+\zeta^{2})+\frac{5}{12}\zeta\right)\frac{\lambda_{3}^{2}}{\pi^{2}}\frac{1}{\varepsilon}.$
(40)

The new contribution in (36) still does not introduce scale dependence. Thus

$\displaystyle\gamma=-\gamma_{m}=\left(\frac{5}{16}(1+\zeta^{2})+\frac{1}{4}\zeta\right)\frac{\lambda_{3}^{2}}{\pi^{2}}.$
(41)

The $\beta$-functions are

$\displaystyle\frac{d\lambda_{3}}{d\ln\mu}$
$\displaystyle=-\left(\frac{5}{4}+2\zeta+3\zeta^{2}\right)\frac{\lambda_{4}\lambda_{3}}{\pi^{2}}-\frac{3}{4}\left(\frac{5}{4}(1+\zeta^{2})+\zeta\right)\frac{\lambda_{3}^{3}}{\pi^{2}},$
(42) $\displaystyle\frac{d\lambda_{4}}{d\ln\mu}$
$\displaystyle=-\left(\frac{5}{4}+3\zeta+6\zeta^{2}\right)\frac{\lambda_{4}^{2}}{\pi^{2}}-\left(\frac{5}{4}(1+\zeta^{2})+\zeta\right)\frac{\lambda_{4}\lambda_{3}^{2}}{\pi^{2}},$
(43) $\displaystyle\frac{dg}{d\ln\mu}$
$\displaystyle=\left(\frac{29}{48}(1+\zeta^{2})+\frac{11}{24}\zeta\right)\frac{\lambda_{3}^{2}}{\pi^{2}}.$
(44)

The runnings of $\lambda_{3}$ and $\lambda_{4}$ have all the same
characteristics as before. The running of the $U(1)$ gauge coupling is not
asymptotically free, as usual, and so this theory as a whole is not UV
complete.

## References

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###### Abstract

We develop an $A_{4}\times Z_{4}\times Z_{2}$ symmetry extension of Standard
Model under the minimal extended seesaw (MES) mechanism which successfully
predicts neutrino masses and mixings patterns. This model breaks $\mu-\tau$
symmetry of neutrino mass matrix and explains leptonic mixing with non-zero
$\theta_{13}$. We study the phenomenological results of the keV-scale sterile
neutrino as a dark matter candidate along with other phenomenologies such as
neutrino oscillation observables, neutrinoless double beta decay, baryogenesis
via leptogenesis, etc. Dirac CP-violating phase $\delta_{CP}$ and two Majorana
phases $\alpha$ and $\beta$ are also calculated from the leptonic mixing
matrix. Best-fit values of the model parameters and neutrino observables are
calculated from $\chi^{2}$ analysis. The model predicts best-fit values of
neutrino mixing angles to be $\sin^{2}\theta_{23}=0.555,\
\sin^{2}\theta_{12}=0.301$ and $\sin^{2}\theta_{13}=0.022$ for normal
hierarchy. Significant results consistent with experimental data are also
observed for effective neutrino mass $m_{\beta\beta}\sim(0.97-5.02)$ meV,
effective electron mass $m_{\beta}\sim(0.084-0.41)$eV and sum of active
neutrino masses $\sum m_{i}<0.12$ eV. The model does not favour Inverted
hierarchy at the 3$\sigma$ level with the given parameter space.

##  I Introduction

Inspite of enormous successes and outstanding discovery of Higg’s boson, the
Standard Model (SM) is incapable of explaining many problems and phenomena in
the Universe. Absence of neutrino mass is one of the main shortcomings of SM.
Observations of neutrino oscillation in SNO[1, 2], SK[3], etc. have verified
that neutrinos have non-zero mass and they can change from one flavour to
another. The global analysis of the latest data from various experiments such
as solar, atmospheric, reactor and accelerator experiments, gives the best fit
values of the neutrino oscillation parameters such as the two mass-squared
differences ($\Delta m_{atm}^{2}\sim 7.4\times 10^{-5}$eV2 and $\Delta
m_{sol}^{2}\sim 2.5\times 10^{-3}$eV2 ) and the three mixing angles
($\theta_{12}\sim 34^{o},\theta_{13}\sim 7.4^{o}$ and $\theta_{23}\sim
48^{o}$)[4]. Some of the burning questions in high energy physics are the
absolute masses and exact nature of the neutrinos (Dirac or Majorana), Dirac
CP-violating phase, baryon asymmetry of the Universe (BAU), dark matter (DM),
etc.

Anomalies in the observations of LSND[5] which detected an excess of electron
anti-neutrino $(\overline{\nu}_{e})$ in a muon anti-neutrino
$(\overline{\nu}_{\mu})$ beam produced at the Los Alamos laboratory could not
be explained by the three neutrino oscillation scheme. This result was further
supplemented by the MiniBooNE [6] experiment which observed an oscillation
from $(\overline{\nu}_{\mu})$ to $(\overline{\nu}_{e})$ in agreement with the
LSND data. These observations gave birth to the idea of the existence of
additional neutrino(s) called sterile neutrino. Sterile neutrinos are singlet
fermions under $SU(2)_{L}$ which do not have weak interactions but can mix
with the active neutrinos. LSND and MiniBooNE results could be explained by
adding at least one additional sterile neutrino having mass in the eV scale to
the SM in a $3+1$ framework. However, sterile neutrino with different mass
scales can play important role in many cosmological phenomena. Recent results
from MicroBooNE suggests the non-possibility for existence of an eV scale
sterile neutrino. But, keV to GeV scale sterile neutrinos are still well-
motivated theoretically and do not contradict any existing experiments[7].

parameter | best fit $\pm$ 1$\sigma$ | 3$\sigma$ range
---|---|---
$|\Delta m^{2}_{21}|:[10^{-5}eV^{2}]$ | $7.41^{+0.21}_{-0.20}$ | 6.82–8.03
$|\Delta m^{2}_{31}|:[10^{-3}eV^{2}](NO)$ | $2.511^{+0.028}_{-0.027}$ | 2.428–2.597
$|\Delta m^{2}_{32}|:[10^{-3}eV^{2}](IO)$ | $2.498^{+0.032}_{-0.025}$ | 2.408–2.581
$\sin^{2}\theta_{12}$ | $0.303^{+0.012}_{-0.011}$ | 0.270–0.341
$\sin^{2}\theta_{23}(NO)$ | $0.572^{+0.018}_{-0.023}$ | 0.406–0.620
$\sin^{2}\theta_{23}(IO)$ | $0.578^{+0.016}_{-0.021}$ | 0.412–0.623
$\sin^{2}\theta_{13}(NO)$ | $0.02203^{+0.00056}_{-0.00059}$ | 0.02029–0.02391
$\sin^{2}\theta_{13}(IO)$ | $0.02219^{+0.00060}_{-0.00057}$ | 0.02047–0.02396
$\delta_{\rm CP}/^{o}(NO)$ | $197^{+42}_{-0.25}$ | 108-404
$\delta_{\rm CP}/^{o}(IO)$ | $286^{+27}_{-32}$ | 192-360
Table 1: Updated global-fit data for three neutrino oscillation, $Nufit$
2022[8].

Cosmological and astrophysical measurements have indicated the presence of a
mysterious, non-luminous, non-baryonic matter called dark matter(DM) which
accounts for 26.8$\%$ of the total energy density of the Universe [9, 10].
Yet, the fundamental nature of DM i,e. their origin, its constituents and
interactions are still unknown. The requirements of a DM particle [11] rule
out all the SM particles. One of the most interesting beyond Standard
Model(BSM) particle which behave as a warm dark matter (WDM) is the sterile
neutrino. Particularly, sterile neutrinos with masses in the keV range having
very small mixing with the active neutrinos of the order of
$\sin^{2}2\theta\sim\mathcal{O}(10^{-10})$ can be a dark matter [12, 13, 14].
According to the latest Planck data [15], the relative abundance of DM in the
Universe is observed as

$\Omega_{DM}h^{2}=0.1187\pm 0.0017$ (1)

Dodelson-Widrow mechanism[16] provides the correct abundance of keV-scale
sterile neutrinos as DM. It requires only the mixing between active and
heavier sterile neutrino, and thus this mechanism is very well-motivated.
Further, the Shi-Fuller mechanism[17] which includes lepton asymmetries, can
result in resonantly enhanced production of sterile neutrinos. This allows it
to access a larger range of the $m_{s}-\theta_{s}$ parameter space. The
stability of keV sterile neutrino at the cosmological scale is one of the most
important criteria to be a DM candidate. Sterile neutrinos can radiatively
decay into an active neutrino and a monoenergetic photon through
$\nu_{s}\rightarrow\nu_{a}+\gamma$. The decay rate is given by [18, 19]

$\Gamma=1.38\times
10^{-32}\left(\frac{\sin^{2}2\theta}{10^{-10}}\right)\left(\frac{m_{s}}{keV}\right)^{5}s^{-1}$
(2)

where $\sin^{2}2\theta=4\sum_{i}^{e,\mu,\tau}|U_{i4}|^{2}$ is the effective
active-sterile mixing angle and $|U_{i4}|$ are the elements of active-sterile
mixing matrix. The sterile neutrino lifetime is thus sufficiently large and it
is essentially stable over timescales comparable to the age of the Universe
$t_{U}=4.4\times 10^{17}$sec [15], so that it is a good DM candidate.
Following Ref.[Das:2019kmn, 20], the sterile neutrino abundance is expressed
as a function of active-sterile mixing angle $\theta$ and the mass of sterile
neutrino $m_{s}$ as

$\Omega_{DM}h^{2}\simeq
0.3\left(\frac{\sin^{2}2\theta}{10^{-10}}\right)\left(\frac{m_{s}}{100keV}\right)^{2}$
(3)

There are constraints on the mass and mixing of the keV-scale sterile neutrino
with active neutrinos provided by various cosmological and laboratory
experiments such as observed DM relic density[21], X-ray searches for sterile
neutrino decay[22, 23, 24, 25, 26], Lyman-$\alpha$ forests[27], Tremaine-
Gunn[28] and phase-space analysis[29], etc. Combining these constraints, we
can infer that the mass of the sterile neutrino should be $m_{s}\geq 4$ keV,
and its mixing with SM neutrinos should be $\sin^{2}2\theta\leq 10^{-6}$[30].
Various works on neutrino models based on the study of keV-mass sterile
neutrinos as DM can be found in Refs.[31, 32, 33, 34, 20, 35]. Detailed
analysis on the production and decay of keV-scale sterile neutrino DM in the
early Universe are given in Refs.[36, 37, 38, 32, 39]. Searches for decaying
DM signal in the keV-MeV mass range have been conducted using a wide range of
X-ray telescope, XMM-Newton[40, 41, 42], Chandra[43, 44, 45], Suzaku[46,
47],Swift[48], NuStar[49, 50, 51] etc. There also has been an unknown
observation of a 3.5 keV line in the X-ray spectra of galaxy clusters[52] as
well as Andromeda[53] and Milky Way galaxies[54]. This could be interpreted as
a signal coming from the decay of DM particle having mass $\sim 7$ keV[55].
However, this result is widely disputed and future progress with our
understanding of the origin and nature of this signal may come with the next
generation of high-resolution X-ray missions, including XARM (Hitomi
replacement mission), LYNX and Athena+[56].

Another important problem that we shall address here is the origin of matter-
antimatter asymmetry of the Universe. The current data on this Baryon
Asymmetry of Universe (BAU) is given by

$Y_{B}=\frac{\eta_{B}-\eta_{\overline{B}}}{s}\simeq(8.7\pm 0.06)\times
10^{-11}$ (4)

Fukugida and Yanagida[57, 58] showed that if heavy right-handed Majorana
neutrinos exist, they can decay in a lepton number violating process to the SM
leptons and Higgs boson. If C and CP are violated in these decays, there can
be symmetry between leptons and anti-leptons. Finally, the lepton asymmetry is
converted into baryon asymmetry through sphaleron process and thus producing
matter-antimatter asymmetry. Many authors have discussed baryogenesis via
leptogenesis in different neutrino models[59, 34, 60, 61, 62].

Further, neutrinoless double beta decay $(0\nu\beta\beta)$ is also one of the
most important experiments to probe the Dirac or Majorana nature of
neutrinos[63, 64]. These experiments are also sensitive to the absolute
neutrino masses through the effective neutrino mass $m_{\beta\beta}$. KAMLand-
Zen[65] and GERDA provided an upper bound on $m_{\beta\beta}$ through a
combined analysis in the range $m_{\beta\beta}<(0.071-0.161$) eV [66]. There
are kinematic measurements of $\beta-$decay also and the absolute effective
neutrino mass is directly probed from the cut-off of the electron energy
spectrum emitted from $\beta-$decay. Recent result from KATRIN experiment [67]
constrains the effective electron neutrino mass $m_{\beta}$ to be less than
1.1 eV and the latest Planck data [21] provides the upper limit on the sum of
active neutrino masses $\sum m_{i}<0.12$eV. The parameters $m_{\beta\beta}$
and $m_{\beta}$ are expressed as the sum of mass eigenstates and elements of
lepton mixing matrix as [68].

$m_{\beta\beta}=|\sum_{j=1}^{4}|U_{ej}|^{2}m_{j}|,$ (5)

$m_{\beta}=\left(\sum_{i=1}^{4}|U_{ei}|^{2}m_{i}^{2}\right)^{1/2}.$ (6)

From these, we are motivated to study a neutrino mass model based on minimal
extended seesaw mechanism(MES) to explain some of the SM problems mentioned
above. Several other literatures are available based on similar studies [69,
70, 71, 72, 33, 60, 73, 34, 74] but the main shortcomings of such models are
the presence of many hypothetical flavon fields as well as additional discrete
symmetries. Particularly, in Ref.[34], the authors considered different sets
of five flavons extended with another two flavons as perturbations in the
Lagrangian seperately for normal heirarchy(NH) and inverted heirarchy(IH). In
Ref.[33], authors use $S_{4}$ symmetry as an extension to SM along with
$Z_{4}\times Z_{3}$ in Inverse Seesaw(ISS) scenario. They consider six flavons
: four $S_{4}$ triplets and two singlets, in order to generate the mass
matrices in the model. The present work is different and more efficient in the
fact that we use only four flavon fields $\phi,\psi,\zeta$ and $\chi$ to
construct the Dirac, Majorana and sterile neutrino mass matrices. Deviation
from $\mu-\tau$ symmetry is generated through another $A_{4}$ triplet flavon
$\eta$. The fields are given group charges different from other works and
hence a new structure of light neutrino mass matrix is obtained using the
minimal extended seesaw formula. We also consider the mass of sterile neutrino
in a broader range of $(4-50)$keV.

The detailed description of the model is given in the next section. The
structure of this paper is as follows. We present a detailed description of
the model along with the mechanism for generating neutrino masses in section
II followed by the numerical analysis of the model in section III. Results of
the analysis is presented in section IV. We conclude with a brief summary and
discussion in section V.

##  II Description of the model

In this model we have considered an extension of SM through $A_{4}\times
Z_{4}\times Z_{2}$ where an $A_{4}$ singlet sterile neutrino $S$ is added
along with three right handed neutrino singlets $\nu_{R1},\nu_{R2},\nu_{R3}$.
$A_{4}$ has four irreducible representations denoted by singlets
$1,1^{\prime\prime},1^{\prime}$ and a triplet $3$. The SM lepton doublet $l$
transforms as triplet under $A_{4}$ while the charged lepton singlets
$e_{R},\mu_{R},\tau_{R}$ transform as $1,1^{\prime\prime},1^{\prime}$
respectively. Two flavons $\phi$ and $\psi$ are used along with three Higgs
$H,H^{\prime},H^{\prime\prime}$ to give neutrino masses through electroweak
symmetry breaking. Another flavon $\eta$ is responsible for breaking the
$\mu-\tau$ symmetry of neutrino mass matrix and to generate non-zero
$\theta_{13}$. Two $A_{4}$ singlets $\chi$ and $\zeta$ are used to generate a
diagonal Majorana mass matrix $M_{R}$ and sterile neutrino mass matrix $M_{S}$
respectively. The particle contents and their corresponding group charges are
shown in Table 2. Additional discrete symmetry $Z_{2}$ is used in order to
remove some unwanted interactions in the Lagrangian which are otherwise
allowed by $A_{4}\times Z_{4},$ such as
$\frac{1}{\Lambda}(y_{m}\overline{l}\tilde{H^{\prime}}\psi)_{1}\nu_{R1}$,
$\frac{1}{\Lambda}(y_{n}\overline{l}\tilde{H^{\prime\prime}}\psi)_{1}S$, etc.
The charged lepton and Dirac neutrino mass matrices are constructed using the
Weinberg dim-5 operator[75].

The invariant Yukawa interaction terms of the model in the charged lepton
sector are given by

$-\mathcal{L}^{cl}_{y}\sim\
\frac{y_{e}}{\Lambda}(\overline{l}H\psi)_{1}e_{R}+\frac{y_{\mu}}{\Lambda}(\overline{l}H\psi)_{1^{\prime}}\mu_{R}+\frac{y_{\tau}}{\Lambda}(\overline{l}H\psi)_{1^{\prime\prime}}\tau_{R}$
(7)

Whereas, the Yukawa interaction for the neutrino sector which are invariant
under all the symmetry groups of the model are given by

$\displaystyle-\mathcal{L}^{mass}_{y}\sim$ $\displaystyle\
\frac{1}{\Lambda}(y_{1}\overline{l}\tilde{H}\phi+y_{3}\overline{l}\tilde{H}\eta)_{1}\nu_{R1}+\frac{1}{\Lambda}(y_{1}\overline{l}\tilde{H}^{\prime}\phi+y_{3}\overline{l}\tilde{H}^{\prime}\eta)_{1}\nu_{R2}$
(8)
$\displaystyle+\frac{1}{\Lambda}(y_{4}\overline{l}\tilde{H}\psi+y_{2}\overline{l}\tilde{H}^{\prime\prime}\phi+y_{3}\overline{l}\tilde{H}^{\prime\prime}\eta)_{1}\nu_{R3}$
$\displaystyle+\
\frac{1}{2}\lambda_{1}\chi\overline{\nu}_{R1}^{c}\nu_{R1}+\frac{1}{2}\lambda_{2}\chi\overline{\nu}_{R2}^{c}\nu_{R2}+\frac{1}{2}\lambda_{3}\chi\overline{\nu}_{R3}^{c}\nu_{R3}$
$\displaystyle+\ \frac{1}{2}\lambda_{s}\zeta\overline{S}^{c}\nu_{R1}$

where $\tilde{H}=i\sigma_{2}H^{*}$ is used to make the Lagrangian Gauge
invariant.

$\frac{Fields}{Charges}$ | $l$ | $e_{R},\mu_{R},\tau_{R}$ | H | H′ | H′′ | $\psi$ | $\phi$ | $\chi$ | $\zeta$ | $\nu_{R1}$ | $\ \nu_{R2}$ | $\ \nu_{R3}$ | S | $\eta$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ | 2 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$U(1)_{Y}$ | -1/2 | +1 | -1/2 | -1/2 | -1/2 | 1 | 1 | 0 | -1 | 0 | 0 | 0 | 1 | 1
$A_{4}$ | 3 | 1,1′′,1′ | 1 | 1′ | 1′′ | 3 | 3 | 1 | $1^{\prime\prime}$ | 1 | 1 | 1 | 1′ | 3
$Z_{4}$ | 1 | 1 | 1 | -i | i | 1 | -i | 1 | -1 | i | -1 | 1 | -i | -i
$Z_{2}$ | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1
Table 2: Particle content of the model and their group charges.

After electroweak symmetry breaking, the scalar flavon fields obtain their
vacuum expectation values (v.e.v.) along with alignments given as [76, 77, 62]

$\displaystyle\langle\psi\rangle=v(1,0,0);\
\langle\eta\rangle=v(0,1,0);\langle\phi\rangle=v(1,1,1);\
\langle\chi\rangle=v;\ \langle\zeta\rangle=u$ (9)

Using the multiplication rule of $A_{4}$[78], the charged lepton mass matrix
becomes diagonal,

$M_{L}=\frac{\langle H\rangle v}{\Lambda}diag(y_{e},y_{\mu},y_{\tau}).$ (10)

As a result, the charged-lepton diagonalization matrix becomes unity,
$U_{L}=1$. Thus, the lepton mixing matrix depends on the neutrino sector only.

From Eq.(8) using Eq.(9), the Dirac, Majorana and sterile neutrino mass
matrices will take the form

$\displaystyle M_{D}=\left(\begin{array}[]{ccc}a&a&c+h\\\ a&a&h\\\
a+t&a+t&h+t\\\ \end{array}\right),\hskip
2.84544ptM_{R}=\left(\begin{array}[]{ccc}d&0&0\\\ 0&e&0\\\ 0&0&f\\\
\end{array}\right),$ (17) $M_{s}=\left(\begin{array}[]{ccc}g&0&0\\\
\end{array}\right).$ (18)

where,

$a=\frac{\langle H\rangle v}{\Lambda}y_{1},\ h=\frac{\langle H\rangle
v}{\Lambda}y_{2},\ c=\frac{\langle H\rangle v}{\Lambda}y_{4},\ t=\frac{\langle
H\rangle v}{\Lambda}y_{3},\ d=\lambda_{1}v,$ $e=\lambda_{2}v,\ \ \ \
f=\lambda_{3}v,\ \ \ g=\lambda_{s}u.$

In order to achieve the sterile neutrino mass in the keV range, the v.e.v of
scalar $\chi$ is assumed to lie around TeV scale. An approximate estimate of
the mass scales of the parameters in the model are as follows,

$\Lambda\sim 10^{15}\ \mbox{GeV},\ \ v\sim 10^{14}\ \mbox{GeV},\ \ u\sim 10\
\mbox{TeV}\ \mbox{and}\ \langle H\rangle\sim 125\ \mbox{GeV}$

In our analysis, we have used minimal extended seesaw(MES) mechanism to
calculate the masses of active neutrino as well as sterile neutrino. In MES,
the $4\times 4$ active-sterile neutrino mass matrix is given by [79]

$M_{\nu}^{4\times
4}=-\left(\begin{matrix}M_{D}M_{R}^{-1}M_{D}^{T}&M_{D}M_{R}^{-1}M_{S}^{T}\\\
M_{S}(M_{R}^{-1})^{T}M_{D}^{T}&M_{S}M_{R}^{-1}M_{S}^{T}\end{matrix}\right).$
(19)

It is important to observe that $det(M_{\nu})=0$. Thus, at least one of the
neutrino mass eigenvalues is zero in MES mechanism. Applying the seesaw
condition, $M_{\nu}$ is further diagonalised and the $(3\times 3)$ active
neutrino mass matrix $m_{\nu}$ and the sterile neutrino mass $m_{s}$ are
expressed as

$\displaystyle m_{\nu}\simeq$ $\displaystyle\
M_{D}M_{R}^{-1}M_{S}^{T}\left(M_{S}M_{R}^{-1}M_{S}^{T}\right)^{-1}M_{S}\left(M_{R}^{-1}\right)^{T}M_{D}^{T}-M_{D}M_{R}^{-1}M_{D}^{T}\
;$ (20) $\displaystyle m_{s}\simeq$ $\displaystyle-M_{S}M_{R}^{-1}M_{S}^{T}.$
(21)

Using Eq.(20) and Eq.(21), the active neutrino mass matrix and the sterile
neutrino mass are given by

$m_{\nu}=-\left(\begin{array}[]{ccc}\frac{a^{2}}{e}+\frac{(c+h)^{2}}{f}&\frac{a^{2}}{e}+\frac{h(c+h)}{f}&\frac{a(a+t)}{e}+\frac{(c+h)(h+t)}{f}\\\
\frac{a^{2}}{e}+\frac{h(c+h)}{f}&\frac{a^{2}}{e}+\frac{h^{2}}{f}&\frac{a(a+t)}{e}+\frac{h(h+t)}{f}\\\
\frac{a(a+t)}{e}+\frac{(c+h)(h+t)}{f}&\frac{a(a+t)}{e}+\frac{h(h+t)}{f}&\frac{(a+t)^{2}}{e}+\frac{(h+t)^{2}}{f}\\\
\end{array}\right)$ (22)

and

$m_{s}=-\frac{g^{2}}{d}.$ (23)

The $\mu-\tau$ symmetry in $m_{\nu}$ can be realised if we put $t=0$ in
eq.(22), i,e.

$m_{\nu}=-\left(\begin{array}[]{ccc}\frac{a^{2}}{e}+\frac{(c+h)^{2}}{f}&\frac{a^{2}}{e}+\frac{h(c+h)}{f}&\frac{a^{2}}{e}+\frac{h(c+h)}{f}\\\
\frac{a^{2}}{e}+\frac{h(c+h)}{f}&\frac{a^{2}}{e}+\frac{h^{2}}{f}&\frac{a^{2}}{e}+\frac{h^{2}}{f}\\\
\frac{a^{2}}{e}+\frac{h(c+h)}{f}&\frac{a^{2}}{e}+\frac{h^{2}}{f}&\frac{a^{2}}{e}+\frac{h^{2}}{f}\\\
\end{array}\right)$ (24)

##  III Numerical analysis

For numerical analysis, we use the latest 3$\sigma$ bounds of neutrino
oscillation data shown in Table 3 for both normal hierarchy(NH) and inverted
hierarchy(IH). Diagonal elements of the heavy Majorana mass matrix $M_{R}$ are
given non-degenerate values $d=10^{13}$GeV , $e=10^{11}$GeV and $f=5\times
10^{11}$GeV. We numerically diagonalise the active neutrino mass matrix
$m_{\nu}$ using the relation
$U^{\dagger}\mathcal{M}U=diag(m_{1}^{2},m_{2}^{2},m_{3}^{2})$, where
$\mathcal{M}=m_{\nu}m_{\nu}^{\dagger}$ and $U$ is a unitary mixing matrix. For
NH: $m_{1}=0,\ m_{2}=\sqrt{\Delta m_{21}^{2}},\mbox{ and}\ m_{3}=\sqrt{\Delta
m_{21}^{2}+\Delta m_{31}^{2}}$ and for IH: $m_{1}=\sqrt{\Delta
m_{21}^{2}+\Delta m_{31}^{2}},\ m_{2}=\sqrt{\Delta m_{21}^{2}},\ \mbox{and
}m_{3}=0,$ where $\Delta m_{ij}^{2}=|m_{j}^{2}-m_{i}^{2}|.$ We can also define
a parameter $r$ which is given by the ratio between the mass squared
differences as

$r=\sqrt{\frac{\Delta m_{21}^{2}}{\Delta m_{31}^{2}}}=\frac{m_{2}}{m_{3}}\ \ \
\mbox{for\ NH}$ (25)

and

$r=\sqrt{\frac{\Delta m_{21}^{2}}{|\Delta
m_{32}^{2}|}}=\sqrt{1-\frac{m_{1}^{2}}{m_{2}^{2}}}\ \ \ \mbox{for\ IH}$ (26)

In PDG convention [4], $U$ is parameterised using three mixing angles
$\theta_{12},\theta_{13},\theta_{23}$, one Dirac phase $\delta_{CP}$ and two
Majorana phases $\alpha,\beta$. The values of $\alpha$ and $\beta$ are unknown
and they are randomly varied in the range (0, 2$\pi$). The light neutrino mass
matrix $m_{\nu}$ contains four unknown complex parameters $a,c,t$ and $h$. We
randomly choose the values of these parameters in the ranges given in Table 3.
The parameter space is constrained by upper bound on the sum of active
neutrino mass $\sum m_{i}<0.12$ eV and the 3$\sigma$ bounds of the neutrino
mixing angles. The ratio $r$ is independent of the mass scales and mixing
matrix $U$. From Table 1, the best-fit value of $r$ in NH is $r_{o}=0.172.$
The allowed parameter space is further constrained by the experimental values
of $r$.

The full $(4\times 4)$ active-sterile mass matrix is diagonalised by a unitary
$(4\times 4)$ mixing matrix given by [80]

$V\simeq\left(\begin{matrix}(1-\frac{1}{2}RR^{\dagger})U&R\\\
-R^{\dagger}U&1-\frac{1}{2}R^{\dagger}R\end{matrix}\right),$ (27)

where $R$ represents the strength of active-sterile mixing given by

$\displaystyle R=$ $\displaystyle
M_{D}M_{R}^{-1}M_{S}^{T}(M_{S}M_{R}^{-1}M_{S}^{T})^{-1}\ =\
\left(\begin{array}[]{c}\frac{a}{g}\\\ \frac{a}{g}\\\ \frac{a+t}{g}\\\
\end{array}\right)$ (31)

We can solve the neutrino mixing angles from the elements of active-sterile
mixing matrix $V$ [81, 82]. However, The deviation from unitarity of the
$3\times 3$ mixing matrix $U$ is given by $\frac{1}{2}|RR^{\dagger}|$ and it
is found to be very small $\leq\mathcal{O}(10^{-10})$. As a result, we can
ignore the effects of the keV-scale sterile neutrino in the mixing matrix $U$.
Neutrino mixing angles are obtained from $U$ using the general formula given
below

$\displaystyle\sin^{2}\theta_{13}=|U_{13}|^{2},\
\sin^{2}\theta_{12}=\frac{|U_{12}|^{2}}{1-|U_{13}|^{2}},\
\sin^{2}\theta_{23}=\frac{|U_{23}|^{2}}{1-|U_{13}|^{2}}$ (32)

In order to study the possibility of keV-scale sterile neutrino as a dark
matter, it is required that the active-sterile mixing angle is very small,
$\theta_{s}<10^{-6}$. Remaining parameter $g$ is solved by constraining the
sterile neutrino mass in the range $(4-50)$ keV.

Besides, one of the most important parameters in neutrino sector is the
Jarlskog invariant $J$. Dirac CP violating phase $\delta_{CP}$ is related to
$J$ and it is given by

$J=Im[U_{e1}U_{\mu 2}U^{*}_{e2}U^{*}_{\mu
1}]=s_{23}c_{23}s_{12}c_{12}s_{13}c_{13}^{2}\sin\delta_{CP}$ (33)

where $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$ are the neutrino
mixing angles calculated from the model. Similarly, the Majorana phases are
also evaluated from $U$ using the invariants $I_{1}$ and $I_{2}$ defined as
follows

$\displaystyle
I_{1}=Im[U_{e1}^{*}U_{e2}]=c_{12}s_{12}c^{2}_{13}\sin(\alpha/2),$ (34)
$\displaystyle
I_{2}=Im[U_{e1}^{*}U_{e3}]=c_{12}s_{12}c_{13}\sin\left(\frac{\beta}{2}-\delta_{CP}\right)$
(35)

To find the best-fit values of the free parameters in our model as well as the
neutrino observables, we use the $\chi^{2}$ function given by

$\chi^{2}(x_{i})=\sum_{j}\left(\frac{y_{j}(x_{i})-y_{j}^{bf}}{\sigma_{j}}\right)^{2}$
(36)

where $x_{i}$ are the free parameters in the model and $j$ is summed over the
observables
$\\{\sin^{2}\theta_{12},\sin^{2}\theta_{13},\sin^{2}\theta_{23},r\\}$.
$y_{j}(x_{i})$ denotes the model predictions for the observables and
$y_{j}^{bf}$ are their best-fit values obtained from the global analysis.
$\sigma_{j}$ denotes the corresponding uncertainties obtained by symmetrizing
$1\sigma$ range of the neutrino observables given in table 1. By minimizing
the overall $\chi^{2}$ function, we can calculate the best-fit values of our
model parameters.

Parameters | Allowed ranges (GeV) | Best-fit (GeV) | Parameter phases | Best-fit
---|---|---|---|---
$|a|$ | 0.46 - 0.60 | 0.574 | $\phi_{a}=(0,2\pi)$ | 0.59$\pi$
$|c|$ | 2.53 - 4.72 | 3.835 | $\phi_{c}=(0,2\pi)$ | 0.44$\pi$
$|t|$ | 0.80 - 1.59 | 0.949 | $\phi_{t}=(0,2\pi)$ | 2.19$\pi$
$|h|$ | 3.20 - 3.94 | 3.469 | $\phi_{h}=(0,2\pi)$ | -2.98$\pi$
Table 3: Allowed ranges of model parameters and their best-fit values
corresponding to $\chi^{2}_{min}$.

Figure 1: (a),(b). Variation among allowed model parameters. (c),(d).
Variation between the allowed phases of the parameters.

Figure 2: (a). Variations of effective mixing angle $\sin^{2}2\theta$ with
elements of active-sterile mixing strength $R$. (b) Variation of
$\sin^{2}2\theta$ with $|g|$.

Figure 3: (a). Plot between mixing angle $\sin^{2}\theta_{13}$ with
$\sin^{2}\theta_{12}$ and $\sin^{2}\theta_{23}$. (b) Variation of $m_{1}$ and
$m_{2}$ with $\sum m_{i}$.

Figure 4:  Plot between mixing angle $\sin^{2}\theta_{13}$ with model
parameters $|a|$ and $|h|.$

##  IV Results

For the given ranges of the parameters in Table 3, our model is compatible
with global fit results of the neutrino oscillation parameters at 3$\sigma$
level for NH only, while IH is disallowed at $3\sigma$ level. The variation
between model parameters allowed in the numerical analysis for NH are shown in
Fig.1 (a),(b). The phases of the allowed parameters are shown as scatter plots
in Fig.1 (c),(d). It is observed that the allowed parameter space is very
narrow for $a$ and $t$ compared to $c$ and $h$. Fig.3 (b) shows the plot
between active neutrino masses $m_{2}$ and $m_{3}$ with the sum of neutrino
masses $\sum m_{i}$. For NH, $m_{1}=0$ and upper bound on the sum of neutrino
mass is obtained at $\sum m_{i}\leq 0.07604$ eV while the maximum values of
$m_{2}$ and $m_{3}$ are found to be respectively $0.01103$ eV and $0.06541$
eV. From the $\chi^{2}$ analysis, we find that the best fit values of the
model parameters for $\chi^{2}_{min}=0.8575$ in NH are obtained at
$|a|=5.74\times 10^{8}\mbox{eV},|c|=3.83\times 10^{9}\mbox{eV},|t|=9.49\times
10^{8}\mbox{eV}\ \mbox{and}\ |h|=3.47\times 10^{9}\mbox{eV}$ while the
respective best-fit values of their phases are obtained at
$\phi_{a}=0.591\pi,\phi_{c}=0.443\pi,\phi_{t}=2.19\pi$ and
$\phi_{h}=-2.98\pi$. Corresponding best-fit values of neutrino observables are
found to be
$\sin^{2}\theta_{12}=0.30110,\sin^{2}\theta_{13}=0.02212,\sin^{2}\theta_{23}=0.55542$
and $r=0.17466.$ The complete parameter ranges and their best-fit values
determined from the numerical analysis of the model are shown in Table 3.
Effective active-sterile mixing angle $\sin^{2}2\theta$ as a function of
parameter $|g|$ and mixing matrix element $|V_{14}|,|V_{34}|$ are shown in
Fig.2.

For the charged lepton sector, by comparing Eq.(10) with the experimental
values for masses of the charged leptons given in Ref.[4], $m_{e}=0.51099$
MeV, $m_{\mu}=105.65837$ MeV, $m_{\tau}=1776.86$ MeV, we get the charged
lepton Yukawa constants as $y_{e}\sim 10^{-5},\ y_{\mu}\sim 10^{-3}$ and
$y_{\tau}\sim 10^{-2}$.

Figure 5: (a),(b). Plot between mixing angles $\sin^{2}\theta_{13}$ and
$\sin^{2}\theta_{23}$ with $J$. (c),(d). Variation of CP-violating Dirac phase
$\delta_{CP}$ with $\sin^{2}\theta_{23}$ and $\sum m_{i}$.

Figure 6: (a). Plot between the Majorana phases $\alpha$ and $\beta$. (b)
Variation of $\delta_{CP}$ with Majorana phase $\beta$.

Figure 7: (a). Plot between effective electron mass $m_{\beta}$ with $\sum
m_{i}$. The coloured strip shows the upper bound from latest KATRIN data
$m_{\beta}<1.1$eV. (b) Variation of effective neutrino mass $m_{\beta\beta}$
with $\sum m_{i}$.

###  IV.1 Mixing angles and phases $\delta_{CP}$, $\alpha$ and $\beta$

The variation plot among neutrino mixing angles $\sin^{2}\theta_{12}$,
$\sin^{2}\theta_{23}$ with $\sin^{2}\theta_{13}$ is shown in Fig.3(a). Values
of $\sin^{2}\theta_{23}$ are distributed within the 3$\sigma$ range with very
small crowding towards region above 0.5. Future generation neutrino
oscillation experiments such as JUNO [83],Hyper-Kamiokande [84], DUNE [85],
etc. will hopefully confirm the neutrino mass ordering and also determine the
octant degeneracy of $\theta_{23}$. However, data points for
$\sin^{2}\theta_{13}$ and $\sin^{2}\theta_{12}$ are equally distributed within
their $3\sigma$ ranges. In order to show the variation of neutrino mixing
angles as a function of the model parameters, we have plotted
$\sin^{2}\theta_{23}$ with the model parameters $|a|$ and $|h|$ in Fig.4. It
can be observed that larger value of $|h|$ results in larger values of
$\sin^{2}\theta_{23}$ whereas values of $\sin^{2}\theta_{23}$ does not show
any distinct correlation with parameter $a$.

Dirac CP-violating phase $\delta_{CP}$ is calculated through the Jarlskog
invariant $J$ as given in Eq. (33). We have plotted $J$ as a function of
$\sin^{2}\theta_{23}$ and $\sin^{2}\theta_{13}$ in Fig.5 (a),(b). Values of
$J$ determined from the model lie within the range $\pm 0.0352$. Fig.5 (c),(d)
show the plot between $\delta_{CP}$ with $\sin^{2}\theta_{23}$ and $\sum
m_{i}$ respectively. We obtain the Dirac CP-violating phase in the range
$180.01^{o}\leq\delta_{CP}\leq 359.96^{o}$. The best fit-value of
$\delta_{CP}$ is obtained at $326.09^{o}$. Fig.6 shows the predictions of
Majorana phases $\alpha$ and $\beta$.

###  IV.2 Effective masses $m_{\beta}$ and $m_{\beta\beta}$ from $\beta$-decay
and $0\nu\beta\beta$

The effective neutrino mass and effective electron mass are calculated from
$0\nu\beta\beta$ and $\beta$-decay experiments using Eq.(5) and Eq.(6)
respectively. Variations of $m_{\beta\beta}$ and $m_{\beta}$ with sum of
active neutrino masses $\sum m_{i}$ are shown in Fig.7. From these plots, we
observe that the effective mass parameter lies in the range
$m_{\beta\beta}\sim(0.9794-5.0291)$ meV which will be a great challenge for
future $0\nu\beta\beta$ experiments [86, 87]. On the other hand, Fig7.(a)
indicates that the effective electron mass is obtained in the range
$m_{\beta}\sim(0.0845-0.4103)$ eV which well below the upper bound provided by
the latest KATRIN experiment [67].

###  IV.3 Dark matter

Assuming the existence of keV-scale sterile neutrinos, we work out the
possibilities of it behaving as a dark matter candidate. We have used the
relation for relic density and decay width from Eq.(3) and Eq.(2) respectively
to check the validity of the present model in giving the observed DM density.
Fig.8 (a) shows the plot between decay width $\Gamma$ with the mass of sterile
neutrino $m_{s}\equiv M_{DM}$. In our analysis, we have considered the upper
limit of decay width to be $\mathcal{O}(10^{-25})sec^{-1}$ for the sterile
neutrino to behave as a dark matter. From Fig.8 (a), it is found that the
allowed mass range for sterile neutrino is $m_{s}\leq 20$ keV. This range of
$m_{s}$ corresponds to the effective mixing angle
$\sim\mathcal{O}(10^{-11}-10^{-9})$. Comparing with the results of the
analysis in Ref.[88] shown in Fig.9(a), we can observe that the narrow mass
range $m_{s}\leq 20\ \mbox{keV}$ corresponding to effective mixing angle
$\mathcal{O}\sim(10^{-10}-10^{-11})$ is still allowed by the various
experimental bounds. This observation may also supplement the unknown
observation of 3.5 keV signal in X-ray data from the decay of a 7 keV sterile
neutrino[25, 53]. Similarly, based on the plots of decay width
$\Omega_{DM}h^{2}$ in Fig.8(b), we find that the allowed mass of sterile
neutrino in a more constrained range of $(4-18)$ keV is observed to be giving
the correct abundance consistent with current experimental value given in
Eq.(1).

Figure 8: (a) Variation of decay width $\Gamma$ with mass of the sterile
neutrino $m_{s}\equiv M_{DM}$. (b) Variation of relic density
$\Omega_{DM}h^{2}$ with mass of sterile neutrino. Here, the horizontal line
represents the experimental data of relic abundance of DM in the Universe
which is $0.1187\pm 0.0017$

Figure 9: (a).Various bounds on the mass and effective mixing angle of the keV
sterile neutrino DM. This plot is taken as reference from Ref.[88]. (b)
Variation of effective mixing angle $\sin^{2}2\theta$ with mass of sterile
neutrino $m_{s}$.

###  IV.4 Baryogenesis via Leptogenesis

We have considered non-degenerate masses of heavy right-handed neutrinos such
that $e<f<<d$, i,e. $\nu_{R2}$ is the lightest in our case. The lightest
$\nu_{R2}$ can decay into a Higgs and lepton pair which will produce
sufficient lepton asymmetry giving rise to the experimentally observed baryon
asymmetry of the Universe. We have used the parameterization from Ref. [89]
and the baryon asymmetry is given by

$Y_{B}=ck\frac{\epsilon_{22}}{g*}$ (37)

where $c\sim 12/37$ is a constant that determines the fraction of lepton
asymmetry converted to baryon asymmetry. $k$ is the dilution factor which can
be parameterized in our model as

$k\simeq\ \frac{0.3}{K(lnK)^{0.6}}$ (38)

where $K$ is defined as

$K=\frac{\Gamma}{H(T=M_{2})}=\frac{(\lambda^{\dagger}\lambda)_{22}M_{2}}{8\pi}\frac{M_{Planck}}{1.66\sqrt{g*}M_{2}^{2}}$
(39)

Figure 10: Plot between Baryon asymmetry parameter $Y_{B}$ with the Yukawa
couplings present in the lepton sector of the model. The green strip
represents the experimental bound of $Y_{B}$ which is $(8.7\pm 0.06)\times
10^{-11}$.

Here, $\Gamma$ is the decay width of $\nu_{R2}$. Further, $g*$ is the massless
relativistic degree of freedom in the thermal bath and it is approximately
110. Finally, $\epsilon_{22}$ is the lepton asymmetry produced in the decay of
the lightest $\nu_{R2}$. It is given by

$\epsilon_{22}=\frac{\Gamma(\nu_{R2}\rightarrow
l_{L}+\bar{\phi})-\Gamma(\nu_{R2}\rightarrow\bar{l_{L}}+\phi)}{\Gamma(\nu_{R2}\rightarrow
l_{L}+\bar{\phi})+\Gamma(\nu_{R2}\rightarrow\bar{l_{L}}+\phi)}$ (40)

The lepton asymmetry is calculated as [89, 90, 91, 92]

$\epsilon_{22}=\frac{1}{8\pi}\frac{1}{(\lambda^{\dagger}\lambda)_{22}}\sum_{j}^{1,3}Im[(\lambda^{\dagger}\lambda)_{2j}]^{2}g(x_{j})$
(41)

where $x_{j}\equiv\frac{M_{j}^{2}}{M_{2}^{2}}$ and within the SM [90]
$g(x_{j})$ is defined as,

$g(x_{j})=\sqrt{x_{j}}\left(\frac{2-x_{j}-(1-x_{j}^{2})ln((1+x_{j})/x_{j})}{1-x_{j}}\right)$
(42)

In the above relations, $\lambda$ is the Yukawa matrix corresponding to the
Dirac mass matrix Eq.(17). We constructed the Yukawa matrix $\lambda$ from the
model parameters $a,c,h$ and $t$. We have solved $Y_{B}$ using these relations
and the variation plots of $Y_{B}$ with Yukawa constants are shown in Fig.10.
In our analysis, the Yukawa constants of the lepton sector are observed to be
$\mathcal{O}(10^{-2}-10^{-1})$ which are in the acceptable range of the
perturbativity limit $y\leq\sqrt{4\pi}$. The model parameters and their
respective Yukawa couplings are found to give observed experimental bound for
Baryon asymmetry $Y_{B}$.

##  V Summary and discussion

We have successfully developed a neutrino mass model based on the extension of
SM using $A_{4}\times Z_{4}\times Z_{2}$ symmetry. We use the Weinberg
dimension five operator to construct the neutrino mass matrices in the model.
Possibility of a keV scale sterile neutrino to behave as a dark matter is
studied using MES mechanism. The mass of the sterile neutrino is constrained
in the range $(4-50)$keV. We solved the model parameters using the 3$\sigma$
values of the neutrino observables. It is observed that the model parameters
can produce an effective active-sterile mixing $\sin^{2}\theta\leq 10^{-6}$
which is a requirement for sterile neutrino to be DM. We are able to reproduce
neutrino oscillation parameters within their 3$\sigma$ range including the
Planck bound on the sum of active neutrinos $\sum m_{i}<0.12$eV. Other
phenomenological studies such as effective neutrino mass and effective
electron mass from $0\nu\beta\beta$ and $\beta-$decay experiments,
Baryogenesis via Leptogenesis, etc. are carried out to check the validity of
our model. The effective mass parameter $m_{\beta\beta}$ is observed to be in
the range $\sim(0.9794-5.0291)$meV while effective neutrino mass is found to
be $m_{\beta}\sim(0.0845-0.4103)$ eV which is less than the upper bound of
latest KATRIN data. The keV sterile neutrino mass in range
$m_{s}\sim(4-18)$keV is found to give the correct abundance as well as decay
width for a DM particle. Baryogenesis via leptogenesis is studied from the
lepton asymmetry produced by the decay of the lightest Majorana right-handed
neutrino which is $\nu_{R2}$ in our case. The experimental value of Baryon
asymmetry $Y_{B}$ is found to be consistent for a very small range of Yukawa
couplings $y_{a},y_{c},y_{t}$ and $y_{t}$. We have also determined the best-
fit values of the model parameters as well as the neutrino observables using
the $\chi^{2}$ analysis. Further, it is important to note that our analysis is
found to be consistent with the global analysis of neutrino oscillation data
for NH only while IH is excluded at the 3$\sigma$ level. In fact, for the
given parameter space that we considered, the neutrino observables such as
$r,$ gives larger values outside the 3$\sigma$. Hence, IH case is disallowed
in this model structure with the given parameter space. However, it may be
checked by considering a different set of parameter spaces or by making a
small change in the structure of the model through new flavons [79, 74, 62],
which can be studied in future. A small part of this work with similar model
structure has been published in our previous work as a part of $NuDM-2022$
conference[93], where we studied the possibility of keV DM in a narrower mass
range of $(1-18.5)$keV only. However, the parameter $r$ was not imposed as a
constraint in the previous work and IH could be reproduced. In this work, we
have expanded our study with other phenomenological studies and stricter
constraints. In conclusion, we have presented a model which provides a well-
motivated DM candidate as well as consistent neutrino phenomenologies.

## Acknowledgments

One of the authors (MKS), would like to thank DST-INSPIRE, Govt. of India, for
providing financial support under DST-INSPIRE Fellowship (IF180349).

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