#  $(g-2)_{e,\,\mu}$ and strongly interacting dark matter with collider
implications

Anirban Biswas<EMAIL_ADDRESS>Department of Physics, Sogang
University, Seoul, Korea, 121-742  Sarif Khan<EMAIL_ADDRESS>Institut für Theoretische Physik, Georg-August-Universität Göttingen,
Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

###### Abstract

The quest for new physics beyond the Standard Model is boosted by the recently
observed deviation in the anomalous magnetic moments of muon and electron from
their respective theoretical prediction. In the present work, we have proposed
a suitable extension of the minimal $L_{\mu}-L_{\tau}$ model to address these
two experimental results as the minimal model is unable to provide any
realistic solution. In our model, a new Yukawa interaction involving first
generation of leptons, a singlet vector like fermion ($\chi^{\pm}$) and a
scalar (either an SU(2)L doublet $\Phi^{\prime}_{2}$ or a complex singlet
$\Phi^{\prime}_{4}$) provides the additional one loop contribution to $a_{e}$
only on top of the usual contribution coming from the $L_{\mu}-L_{\tau}$ gauge
boson ($Z_{\mu\tau}$) to both electron and muon. The judicious choice of
$L_{\mu}-L_{\tau}$ charges to these new fields results in a strongly
interacting scalar dark matter in $\mathcal{O}({\rm MeV})$ range after taking
into account the bounds from relic density, unitarity and self interaction.
The freeze-out dynamics of dark matter is greatly influenced by $3\rightarrow
2$ scatterings while the kinetic equilibrium with the SM bath is ensured by
$2\rightarrow 2$ scatterings with neutrinos where $Z_{\mu\tau}$ plays a
pivotal role. The detection of dark matter is possible directly through
scatterings with nuclei mediated by the SM $Z$ bosons. Moreover, our proposed
model can also be tested in the upcoming $e^{+}e^{-}$ colliders by searching
opposite sign di-electron and missing energy signal i.e.
$e^{+}e^{-}\rightarrow\chi^{+}\chi^{-}\rightarrow e^{+}e^{-}\cancel{E}_{T}$ at
the final state.

##  I Introduction

The Standard Model (SM) is a very well established theory of nature and is
confirmed fully with the discovery of the Higgs boson. But with time, we have
understood from various astrophysical phenomena Sofue:2000jx ;
Bartelmann:1999yn ; Clowe:2003tk  that the matter content of the Universe is
not only made up of with the elementary particles described by the SM but more
than 80$\%$ matter content of the Universe is unknown to us. This mysterious
part is the so called dark matter (DM), and its presence has been confirmed by
many pieces of evidence from large scale to small scale observations. The most
precise determination of dark matter abundance at the present epoch is
$\Omega_{\rm DM}h^{2}=0.120\pm 0.001$ by the Planck satellite Planck:2018vyg .
Therefore, in order to have a viable dark matter candidate(s), it is essential
to extend the particle spectrum of the SM. Moreover, a long standing
discrepancy exits over the last two decades between the theoretical prediction
of the SM and the experimental measurement of the anomalous magnetic moment of
muon Muong-2:2001kxu ; Muong-2:2006rrc ; Jegerlehner:2009ry . Recently,
Fermilab has announced a $4.2\sigma$ discrepancy between the experimental and
theoretical value Muong-2:2021ojo ,

$\displaystyle\Delta a_{\mu}=a^{\rm exp}_{\mu}-a^{\rm SM}_{\mu}=\left(2.51\pm
0.59\right)\times 10^{-9}\,.$ (1)

The uncertainty in $\Delta{a_{\mu}}$111The anomalous magnetic moment
$a_{\ell}$ of a lepton $\ell$ is defined as
$a_{\ell}\equiv\dfrac{(g-2)_{\ell}}{2}$, where $g$ is the Lande $g$ factor.
will go down in future when more data will be available from the ongoing
experiment at Fermilab Muong-2:2021vma  as well as the future experiment at
JPARC Abe:2019thb . Besides $(g-2)_{\mu}$, there is also an inconsistency in
$(g-2)$ for electron between theoretical and experimental values. However, the
magnitude of the deviation is four orders smaller than that of muon. Apart
from that, the sign of the discrepancy depends on the measurement of the fine
structure constant $\alpha_{em}$. The discrepancy in $(g-2)_{e}$ is negative
and around $2.4\sigma$ when we use the value of $\alpha_{em}$
($\alpha^{-1}_{em}=$137.035999046(27) with an accuracy of $2.0\times
10^{-10}$) obtained from Caesium atom 137Cs at Berkeley Parker:2018vye ,

$\displaystyle\Delta a_{e}=a^{\rm exp}_{e}-a^{\rm B}_{e}=\left(-8.8\pm
3.6\right)\times 10^{-13}\,.$ (2)

On the other hand, the experimental value becomes larger than the SM
prediction when we consider the fine structure constant
($\alpha^{-1}_{em}=137.035999206(11)$ with an accuracy of 81 parts per
trillion) obtained from Rubidium atom (87Rb) at LKB and it results $1.6\sigma$
deviation between experiment and theoretical value Morel:2020dww ,

$\displaystyle\Delta a_{e}=a^{\rm exp}_{e}-a^{\rm LKB}_{e}=\left(4.8\pm
3.0\right)\times 10^{-13}\,,$ (3)

where $a^{\rm exp}_{e}$ is the direct experimental measurement of the
electron’s anomalous magnetic moment Hanneke:2008tm . Although, both the
measurements of the fine structure constant from 137Cs and 87Rb atoms differ
by more than $5\sigma$, our model can address both positive and negative
discrepancies, and in particular, we have proceeded with our analysis by
considering $\Delta{a}_{e}=\left(-8.8\pm 3.6\right)\times 10^{-13}$ as this
result predicts the larger deviation ($2.4\sigma$) from the theoretical value.

In this work, we have considered an extension of the minimal
$L_{\mu}-L_{\tau}$ model He:1990pn ; He:1991qd  to address both the anomalies
in $(g-2)$ of muon and electron. The $L_{\mu}-L_{\tau}$ gauge extension of the
SM is not only an anomaly free theory but also is very well motivated from the
neutrino mass generation and producing correct mixing angles as measured by
several experiments over the last two decades Ma:2001md ; Grimus:2005jk ;
Rodejohann:2005ru ; Aizawa:2005yy ; Xing:2006xa ; Adhikary:2006rf ;
Fuki:2006ag ; Haba:2006hc ; Joshipura:2009tg ; Adhikary:2009kz ;
Joshipura:2009fu ; Xing:2010ez ; Araki:2010kq ; He:2011kn ; Heeck:2011wj ;
Grimus:2012hu ; Altmannshofer:2014pba ; Xing:2015fdg ; Asai:2017ryy ;
Dev:2017fdz ; Chen:2017gvf ; Nomura:2018vfz ; Nomura:2018cle ;
Banerjee:2018eaf . Moreover, both thermal as well as nonthermal dark matter
have also been earlier in $L_{\mu}-L_{\tau}$ model by several authors
Altmannshofer:2016jzy ; Patra:2016shz ; Biswas:2016yan ; Biswas:2016yjr ;
Biswas:2017ait ; Arcadi:2018tly ; Singirala:2018mio ; Foldenauer:2018zrz ;
Escudero:2019gzq ; Biswas:2019twf ; Okada:2019sbb ; Borah:2020jzi ;
Asai:2020qlp ; Borah:2021khc ; Drees:2021rsg ; Hapitas:2021ilr ;
Tapadar:2021kgw . We first have considered the minimal model and found that
within this framework it is not possible to address the deviation in anomalous
magnetic moment of electron after taking into account the $e-\nu$ scattering
result from Borexino Bellini:2011rx ; Borexino:2013zhu ; Borexino:2017rsf .
Therefore, we have introduced a single scalar $\Phi^{\prime}_{4}$, a scalar
doublet $\Phi^{\prime}_{2}$ and a “vector-like” SU(2)L and colour singlet
charged fermion $\chi^{\pm}$ in the minimal model. We have assigned suitable
$L_{\mu}-L_{\tau}$ charges to all these new fields so that two additional
Yukawa interaction terms can be generated. This results in an extra one loop
contribution to $\Delta{a}_{e}$ over the contribution due to $Z_{\mu\tau}$
through $Z-Z_{\mu\tau}$ mixing. On the other hand, $\Delta{a}_{\mu}$ gets only
the $Z_{\mu\tau}$ induced one loop contribution and it does not depend
significantly on the $Z-Z_{\mu\tau}$ mixing as unlike electron, muon has
nonzero $L_{\mu}-L_{\tau}$ charge. There are earlier works where authors have
explored both electron and muon anomalous magnetic moments which can be found
in Liu:2018xkx ; Crivellin:2018qmi ; Davoudiasl:2018fbb ; Bauer:2019gfk ;
Endo:2019bcj ; Badziak:2019gaf ; Bigaran:2020jil ; Dorsner:2020aaz ;
Haba:2020gkr ; Calibbi:2020emz ; Chen:2020jvl ; Dutta:2020scq ;
Abdallah:2020biq ; Chen:2020tfr ; Botella:2020xzf ; Jana:2020joi ;
Hati:2020fzp ; Chun:2020uzw ; Banerjee:2020zvi ; DelleRose:2020oaa ;
Escribano:2021css ; Frank:2021nkq ; Borah:2021khc ; Bharadwaj:2021tgp .

Apart from addressing the anomalous magnetic moments of muon and electron in
the light of recent experimental data, we have also studied the phenomenology
of a viable dark matter candidate which is an admixture of two neutral complex
scalars namely $\Phi_{4}^{\prime}$ and $\phi^{\prime}_{2}$ (neutral component
of the scalar doublet $\Phi_{2}^{\prime}$). The dynamics of the dark sector
especially our dark matter candidate $\phi_{4}$ is greatly influenced by the
choice of $L_{\mu}-L_{\tau}$ charges of the fields involving in the new Yukawa
interaction necessary for $(g-2)_{e}$. This results in the strongly
interacting dark matter scenario as we get a cubic self interaction term for
$\phi_{4}$ when $L_{\mu}-L_{\tau}$ symmetry is broken by the vacuum
expectation value (VEV) of $\Phi^{\prime}_{3}$. Therefore, the freeze-out era
of $\phi_{4}$ is predominantly determined by the competition between
$3\rightarrow 2$ interaction rates with the Hubble expansion rate. Moreover,
the kinetic equilibrium of the dark matter with the SM bath, as required for
the Strongly Interacting Massive Particle (SIMP) scenario Hochberg:2014dra ,
is achieved by the elastic scattering between $\phi_{4}$ and $\nu_{\alpha}$
($\alpha=\mu$, $\tau$) where $Z_{\mu\tau}$ plays the role of the dominant
mediator. Therefore, in this way parameters of the new gauge interaction such
as $g_{\mu\tau}$ and $M_{Z_{\mu\tau}}$ have large impact on the cosmic
evolution of dark matter and at the same time they are tightly constrained by
the precise experimental measurement of $(g-2)_{\mu}$. We have shown that the
$2\sigma$ parameter space for addressing $(g-2)_{\mu}$ has some overlap with
the region in $g_{\mu\tau}-M_{Z_{\mu\tau}}$ plane that keeps an MeV scale dark
matter in kinetic equilibrium till freeze-out. For earlier works focusing on
the SIMP scenario see Refs. Hochberg:2014kqa ; Hochberg:2015vrg ; Daci:2015hca
; Bernal:2015xba ; Bernal:2015bla ; Lee:2015gsa ; Choi:2015bya ; Choi:2016tkj
; Choi:2017zww ; Choi:2017mkk ; Ho:2017fte ; Davis:2017noy ; Hochberg:2018rjs
; Smirnov:2020zwf ; Katz:2020ywn ; Choi:2021yps . Finally, we have looked for
the prospect of collider signature of the charged fermion ($\chi^{\pm}$) at
the $e^{+}e^{-}$ linear collider for the centre of mass (c.o.m) energies
$\sqrt{s}=1000$ GeV and $3000$ GeV respectively. Here we have investigated the
opposite sign di-electron and missing energy signal at the final state i.e.
$e^{+}e^{-}\rightarrow\chi^{+}\chi^{-}\rightarrow e^{+}e^{-}\cancel{E}_{T}$.
We have shown that the signal strength of the charged fermion significantly
improved for the presence of the $t$-channel process mediated by SIMP dark
matter $\phi_{4}$ which remains absent at the hadron collider. This
enhancement in the cross section will ensure the detection of the present
model in the early run of $e^{+}e^{-}$ collider.

The rest of the paper is organised in the following way. In the Section II we
have described the minimal $L_{\mu}-L_{\tau}$ model and have shown that it is
not possible to address both the anomalies simultaneously in the minimal
model. The extended model has been described in detail in the Section III. The
new contribution in the anomalous magnetic moment of electron is given in the
Section IV. A brief discussion on neutrino masses via Type-I seesaw mechanism
in the context of the present model is presented in the Section V. The Section
VI is devoted to a comprehensive study on the SIMP dark matter and related
numerical analyses. The signature of the new charged fermion $\chi^{\pm}$ has
been studied in the Section VII. Finally, we summarise in the Section VIII.
The couplings necessary for calculating all the Feynman diagrams are listed in
the Appendix A.

##  II The minimal $L_{\mu}-L_{\tau}$ Model and anomalous magnetic moment

As discussed in the previous section, one of our prime motivations is to
address both the anomalies reported in the anomalous magnetic moments of $\mu$
and $e$ within a single framework. It is well known for quite a while that the
minimal U(1)${}_{L_{\mu}-L_{\tau}}$ model can resolve the enduring discrepancy
between the experimentally measured value of $(g-2)_{\mu}$ and the SM
prediction efficiently, where an MeV scale ($\mathcal{O}(10\,\,{\rm MeV}\sim
100\,\,{\rm MeV})$) new gauge boson ($Z_{\mu\tau}$) provides the require
deficit on top of the SM contribution to match the experimental prediction
Biswas:2016yan ; Biswas:2019twf . Keeping this in mind, we have investigated
the possibility of addressing $(g-2)$ of electron in the minimal
$L_{\mu}-L_{\tau}$ model alongside $(g-2)_{\mu}$. Since the electrons are not
charged under U(1)${}_{L_{\mu}-L_{\tau}}$ symmetry, the effect of
$L_{\mu}-L_{\tau}$ gauge boson on the anomalous magnetic moment comes only
through the kinetic mixing between U(1)${}_{L_{\mu}-L_{\tau}}$ and U(1)Y of
the SM. Before going into the details of anomalous magnetic moments of $e$ and
$\mu$ we would first like to describe the minimal $L_{\mu}-L_{\tau}$ model
briefly.

In the minimal $L_{\mu}-L_{\tau}$ model, in addition to the SM gauge symmetry,
we demand another local U(1)${}_{L_{\mu}-L_{\tau}}$ gauge invariance, where
$L_{\ell}$ represents the lepton number corresponding to the lepton $\ell$.
Therefore, the ${L_{\mu}-L_{\tau}}$ charges for three generations of the SM
leptons are $Q^{{}^{e,\,\nu_{e}}}_{\mu\tau}=0$,
$Q^{{}^{\mu,\,\nu_{\mu}}}_{\mu\tau}=+1$ and
$Q^{{}^{\tau,\,\nu_{\tau}}}_{\mu\tau}=-1$ respectively while all the quarks
possess zero ${L_{\mu}-L_{\tau}}$ charge. One of the biggest advantages of
${L_{\mu}-L_{\tau}}$ gauge extension is that it does not introduce any axial
vector anomaly Bilal:2008qx ; Adler:1969gk ; Bardeen:1969md  and gauge-
gravitational anomaly Delbourgo:1972xb ; Eguchi:1976db  since they cancel
automatically between second and third generations of the SM leptons. In
addition to the usual SM fields, we only need a scalar field having nonzero
${L_{\mu}-L_{\tau}}$ charge to break this local U(1) symmetry spontaneously
and thereby generating a massive neutral gauge boson $Z_{\mu\tau}$. The
${L_{\mu}-L_{\tau}}$ symmetric Lagrangian for the minimal model is given by

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\mathcal{L}_{\rm
SM}+\left(D_{\alpha}{\Phi^{\prime}_{3}}\right)^{\dagger}\left(D^{\alpha}{\Phi^{\prime}_{3}}\right)-\frac{1}{4}\hat{X}_{\alpha\beta}\hat{X}^{\alpha\beta}+\frac{\epsilon}{2}\hat{X}_{\alpha\beta}\hat{B}^{\alpha\beta}-g_{\mu\tau}\sum_{\begin{subarray}{c}\ell=\mu,\nu_{\mu},\\\
\,\,\,\tau,\nu_{\tau}\end{subarray}}Q^{\ell}_{\mu\tau}\bar{\ell}\gamma^{\alpha}\ell\hat{X}_{\alpha}$
(4)
$\displaystyle-\lambda_{13}\left({\Phi^{\prime}_{1}}^{\dagger}\Phi^{\prime}_{1}\right)\left({\Phi^{\prime}_{3}}^{\dagger}\Phi^{\prime}_{3}\right),$

where $\mathcal{L}_{\rm SM}$ denotes the SM Lagrangian. Here we have
considered a SM singlet scalar field ($\Phi^{\prime}_{3}$) having
${L_{\mu}-L_{\tau}}$ charge equal to unity which breaks the $L_{\mu}-L_{\tau}$
symmetry spontaneously. As mentioned above, the fourth term represents kinetic
mixing between the hypercharge gauge boson ($\hat{B}_{\mu}$) and the
${L_{\mu}-L_{\tau}}$ gauge boson ($\hat{X}_{\mu}$), where the respective
abelian field strength tensor is denoted by the same letter but with two
Lorentz indices. The last but one term corresponds to the interactions of
second and third generation leptons with $L_{\mu}-L_{\tau}$ gauge boson while
the last term is the only gauge invariant interaction between the SM Higgs
doublet $\Phi^{\prime}_{1}$ and the singlet scalar $\Phi^{\prime}_{3}$.

To obtain the physical gauge boson $Z_{\mu\tau}$, first we need a basis
transformation from the “hat” states ($\hat{B}_{\alpha}$, $\hat{X}_{\alpha}$)
to the “un-hat” states (${B}_{\alpha}$, ${X}_{\alpha}$) so that the off-
diagonal term proportional to $\epsilon$ vanishes. This requires a
transformation like

$\displaystyle\hat{B}_{\alpha}$ $\displaystyle=$ $\displaystyle
B_{\alpha}-\epsilon\,X_{\alpha}\,,$ $\displaystyle\hat{X}_{\alpha}$
$\displaystyle=$ $\displaystyle\,X_{\alpha}+\mathcal{O}(\epsilon^{2})\,.$ (5)

Where, we have considered terms up to linear order of $\epsilon$ as there
exist various experimental constraints on the kinetic mixing parameter
($\epsilon$), which forces us to consider $\epsilon\ll 1$ Bauer:2018onh . We
would like to mention here that the above transformation is neither an
orthogonal transformation nor it is a unique one. Although, the basis
transformation given in Eq. (5) removes the off-diagonal term (proportional to
$\epsilon$), it reintroduces $\epsilon$ again in the mass matrix of neutral
gauge bosons written in the “un-hat” basis
($W^{3}_{\alpha},\,\,B_{\alpha},\,\,X_{\alpha}$) after both electroweak
symmetry breaking and $L_{\mu}-L_{\tau}$ breaking as

$\displaystyle\mathcal{M}^{2}_{\rm
gauge}=\left(\begin{array}[]{ccc}\frac{1}{4}g^{2}_{2}v^{2}&-\frac{1}{4}g_{2}g_{1}v^{2}&-\frac{1}{4}g_{2}g_{1}v^{2}\epsilon\\\
-\frac{1}{4}g_{2}g_{1}v^{2}&\frac{1}{4}g^{2}_{1}v^{2}&\frac{1}{4}g^{2}_{1}v^{2}\epsilon\\\
-\frac{1}{4}g_{2}g_{1}v^{2}\epsilon&\frac{1}{4}g^{2}_{1}v^{2}\epsilon&g^{2}_{\mu\tau}v^{2}_{\mu\tau}\end{array}\right)\,\,,$
(9)

where $g_{1}$, $g_{2}$ and $g_{\mu\tau}$ are the gauge couplings of U(1)Y,
SU(2)L and U(1)${}_{L_{\mu}-L_{\tau}}$ respectively while the VEVs of
$\Phi^{\prime}_{1}$ and $\Phi^{\prime}_{3}$ are $v/\sqrt{2}$ and
$v_{\mu\tau}/\sqrt{2}$ respectively. The above mass matrix has a special
symmetry that if we rotate the upper $2\times 2$ block between
$W^{3}_{\alpha}$ and $B_{\alpha}$ by the Weinberg angle $\theta_{W}$
($\tan\theta_{W}=\frac{g_{1}}{g_{2}}$), not only the $2\times 2$ block becomes
diagonal but also the entire mass matrix reduces to a block diagonal form
where a new $2\times 2$ block is formed between $\mathcal{Z}_{\alpha}$
($\equiv\cos\theta_{W}W^{3}_{\alpha}-\sin\theta_{W}B_{\alpha}$) and
$X_{\alpha}$ while the state $A_{\alpha}$
($\equiv\sin\theta_{W}W^{3}_{\alpha}+\cos\theta_{W}B_{\alpha}$), orthogonal to
the state $\mathcal{Z}_{\alpha}$, decouples completely with a zero eigenvalue.
It is then natural to identify the state ${A_{\alpha}}$ as the photon, the
gauge boson corresponding to the unbroken U(1)em symmetry. This is the reason
behind our choice of Eq. (5) among many other possibilities.

Finally, to obtain the other two physical gauge bosons, we need to diagonalise
the $2\times 2$ block between $\mathcal{Z}_{\alpha}$ and $X_{\alpha}$, the
elements of which are given by

$\displaystyle\mathcal{M}^{2}_{\mathcal{Z}X}\,=\,\dfrac{1}{4}\left(\begin{array}[]{cc}(g_{1}^{2}+g_{2}^{2})v^{2}&-{\epsilon}g_{1}\sqrt{g_{1}^{2}+g_{2}^{2}}\,v^{2}\\\
-{\epsilon}g_{1}\sqrt{g_{1}^{2}+g_{2}^{2}}\,v^{2}&4\,g^{2}_{\mu\tau}v^{2}_{\mu\tau}\end{array}\right)\,.$
(12)

After diagonalisation, the physical gauge bosons are given by

$\displaystyle Z^{\alpha}$ $\displaystyle=$
$\displaystyle\cos\theta_{\mu\tau}\mathcal{Z}^{\alpha}-\sin\theta_{\mu\tau}X^{\alpha}=\cos\theta_{\mu\tau}\left(\cos\theta_{W}W^{3}_{\alpha}-\sin\theta_{W}B_{\alpha}\right)-\sin\theta_{\mu\tau}X^{\alpha}\,,$
(13) $\displaystyle Z_{\mu\tau}^{\alpha}$ $\displaystyle=$
$\displaystyle\sin\theta_{\mu\tau}\mathcal{Z}^{\alpha}+\cos\theta_{\mu\tau}X^{\alpha}=\sin\theta_{\mu\tau}\left(\cos\theta_{W}W^{3}_{\alpha}-\sin\theta_{W}B_{\alpha}\right)+\cos\theta_{\mu\tau}X^{\alpha}\,$
(14)

having masses as follows

$\displaystyle M_{Z}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(g^{2}_{1}+g^{2}_{2})v^{2}}{4}\cos^{2}\theta_{\mu\tau}+g^{2}_{\mu\tau}v^{2}_{\mu\tau}\sin^{2}\theta_{\mu\tau}+\epsilon\,\frac{g_{1}\sqrt{g^{2}_{1}+g^{2}_{2}}\,v^{2}}{4}\sin
2\theta_{\mu\tau}}\;,$ (15) $\displaystyle M_{Z_{\mu\tau}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(g^{2}_{1}+g^{2}_{2})v^{2}}{4}\sin^{2}\theta_{\mu\tau}+g^{2}_{\mu\tau}v^{2}_{\mu\tau}\cos^{2}\theta_{\mu\tau}-\epsilon\,\frac{g_{1}\sqrt{g^{2}_{1}+g^{2}_{2}}\,v^{2}}{4}\sin
2\theta_{\mu\tau}}\;,$ (16)

and the $Z-Z_{\mu\tau}$ mixing angle $\theta_{\mu\tau}$ has the following
expression

$\displaystyle\theta_{\mu\tau}=\frac{1}{2}\,\tan^{-1}\left(\dfrac{\dfrac{2\,g_{1}\,\epsilon}{\sqrt{g^{2}_{1}+g^{2}_{2}}}}{1-\dfrac{4g^{2}_{\mu\tau}}{g^{2}_{1}+g^{2}_{2}}\dfrac{v^{2}_{\mu\tau}}{v^{2}}}\right)\,\,.$
(17)

As expected the mixing angle is proportional to the kinetic mixing parameter
$\epsilon$. The Eq. (13) represents the neutral gauge boson of weak
interaction namely, the $Z$ boson. If we set $\epsilon=0$, Eq. (13) reduces to
the well known SM $Z$ boson with mass
$M_{Z}=\frac{1}{2}\sqrt{g_{1}^{2}+g_{2}^{2}}\,v$ (see Eq. (15)) as given in
the SM. Moreover, one can notice that in spite of the kinetic mixing between
U(1)Y and U(1)${}_{L_{\mu}-L_{\tau}}$, the state representing photon
($A_{\alpha}$) remains unaltered. The effect of $\epsilon$ enters only into
the expressions of $Z$ and $Z_{\mu\tau}$.

Due to this $Z-Z_{\mu\tau}$ mixing all the SM fermions, particularly the first
generation of leptons which do not possess any $L_{\mu}-L_{\tau}$ charge, will
now be able to interact with $Z_{\mu\tau}$. Consequently, we have an
additional contribution in the anomalous magnetic moment of electron,
analogous to muon, besides the usual SM contribution involving photon.
However, the only difference is that the magnitude of such BSM effect in the
context of electron will be much less compared to that of $\mu$ as the
$e^{+}e^{-}Z_{\mu\tau}$ vertex is suppressed by tiny $Z-Z_{\mu\tau}$ mixing
angle $\theta_{\mu\tau}$. The general structure of $\overline{\ell}\ell
Z_{\mu\tau}(Z)$ interaction is
$\overline{\ell}\gamma_{\alpha}(g^{Z_{\mu\tau}(Z)}_{{}_{V}}+g^{Z_{\mu\tau}(Z)}_{{}_{A}}\gamma_{{}_{5}})\ell\,Z^{\alpha}_{\mu\tau}(Z^{\alpha})$
for any lepton $\ell$. The expressions of $g_{{}_{V}}$ and $g_{{}_{A}}$ for
both $e$ and $\mu$ are given in Table 1. One can easily recover the familiar
$\overline{\ell}\ell Z$ vertex of the SM in the limit $\epsilon\rightarrow 0$.
Moreover, we can see that in the $\epsilon\rightarrow 0$ limit, both vector
and axial vector couplings of the $\overline{e}eZ_{\mu\tau}$ vertex disappear
while the $\overline{\mu}\mu Z_{\mu\tau}$ vertex becomes purely vectorial with
$g^{Z_{\mu\tau}}_{{}_{V}}=-g_{\mu\tau}$.

Field | $Z_{\mu\tau}$ | $Z$
---|---|---
| $g^{Z_{\mu\tau}}_{{}_{V}}$ | $g^{Z_{\mu\tau}}_{{}_{A}}$ | $g^{Z}_{{}_{V}}$ | $g^{Z}_{{}_{A}}$
$\ell$ | $-\dfrac{g_{2}\left(T^{\ell}_{3}-2\,Q^{\ell}_{em}\sin^{2}\theta_{W}\right)}{2\,\cos\theta_{W}}\sin\theta_{\mu\tau}$ | $\dfrac{g_{2}}{2\,\cos\theta_{W}}\Big{(}\sin\theta_{\mu\tau}$ | $-\dfrac{g_{2}\left(T^{\ell}_{3}-2\,Q^{\ell}_{em}\sin^{2}\theta_{W}\right)}{2\,\cos\theta_{W}}\cos\theta_{\mu\tau}$ | $\dfrac{g_{2}}{2\,\cos\theta_{W}}\Big{(}\cos\theta_{\mu\tau}$
| $-\left[g_{\mu\tau}\,Q^{\ell}_{\mu\tau}-\dfrac{g_{2}\,\epsilon\left(T^{\ell}_{3}-2\,Q^{\ell}_{em}\right)}{2\,\cot\theta_{W}}\right]\cos\theta_{\mu\tau}$ | $-\,\epsilon\sin\theta_{W}\cos\theta_{\mu\tau}\Big{)}\,T^{\ell}_{3}$ | $+\,\left[g_{\mu\tau}\,Q^{\ell}_{\mu\tau}-\dfrac{g_{2}\,\epsilon\left(T^{\ell}_{3}-2\,Q^{\ell}_{em}\right)}{2\,\cot\theta_{W}}\right]\sin\theta_{\mu\tau}$ | $+\,\epsilon\sin\theta_{W}\sin\theta_{\mu\tau}\Big{)}\,T^{\ell}_{3}$
$e$ | $-\dfrac{g_{2}\left(-1+4\,\sin^{2}\theta_{W}\right)}{4\,\cos\theta_{W}}\sin\theta_{\mu\tau}$ | $-\dfrac{g_{2}}{4\,\cos\theta_{W}}\Big{(}\sin\theta_{\mu\tau}$ | $-\dfrac{g_{2}\left(-1+4\,\sin^{2}\theta_{W}\right)}{4\,\cos\theta_{W}}\cos\theta_{\mu\tau}$ | $-\dfrac{g_{2}}{4\,\cos\theta_{W}}\Big{(}\cos\theta_{\mu\tau}$
| $+\left(\dfrac{3\,g_{2}\,\epsilon}{4\,\cot\theta_{W}}\right)\cos\theta_{\mu\tau}$ | $-\,\epsilon\sin\theta_{W}\cos\theta_{\mu\tau}\Big{)}$ | $-\left(\dfrac{3\,g_{2}\,\epsilon}{4\,\cot\theta_{W}}\right)\sin\theta_{\mu\tau}$ | $+\,\epsilon\sin\theta_{W}\sin\theta_{\mu\tau}\Big{)}$
$\mu$ | $-\dfrac{g_{2}\left(-1+4\,\sin^{2}\theta_{W}\right)}{4\,\cos\theta_{W}}\sin\theta_{\mu\tau}$ | $-\dfrac{g_{2}}{4\,\cos\theta_{W}}\Big{(}\sin\theta_{\mu\tau}$ | $-\dfrac{g_{2}\left(-1+4\,\sin^{2}\theta_{W}\right)}{4\,\cos\theta_{W}}\cos\theta_{\mu\tau}$ | $-\dfrac{g_{2}}{4\,\cos\theta_{W}}\Big{(}\cos\theta_{\mu\tau}$
| $-\left(g_{\mu\tau}-\dfrac{3\,g_{2}\,\epsilon}{4\,\cot\theta_{W}}\right)\cos\theta_{\mu\tau}$ | $-\,\epsilon\sin\theta_{W}\cos\theta_{\mu\tau}\Big{)}$ | $+\left(g_{\mu\tau}-\dfrac{3\,g_{2}\,\epsilon}{4\,\cot\theta_{W}}\right)\sin\theta_{\mu\tau}$ | $+\,\epsilon\sin\theta_{W}\sin\theta_{\mu\tau}\Big{)}$
Table 1: $\overline{\ell}\ell Z_{\mu\tau}(Z)$ vertex factors for a general
lepton $\ell$ with third component of weak isospin $T^{\ell}_{3}$, electric
charge $Q^{\ell}_{em}$ and $L_{\mu}-L_{\tau}$ charge $Q^{\ell}_{\mu\tau}$. The
vertex factors for electron
($T^{e}_{3}=-\frac{1}{2},\,Q^{e}_{em}=-1,\,Q^{e}_{\mu\tau}=0$) and muon
($T^{\mu}_{3}=-\frac{1}{2},\,Q^{\mu}_{em}=-1,\,Q^{\mu}_{\mu\tau}=+1$) are also
listed.

The Feynman diagrams contributing to the anomalous magnetic moments of both
$\mu$ and $e$ at one loop level are shown in Fig. 1. The expression
$\Delta{a}_{{}_{\ell}}$ ($\ell=e$, $\mu$) due to the $L_{\mu}-L_{\tau}$ gauge
boson $Z_{\mu\tau}$ is given by Biswas:2019twf

$\displaystyle\Delta
a_{{}_{\ell}}=\frac{1}{8\pi^{2}}~{}\left((g^{Z_{\mu\tau}}_{{}_{V}})^{2}\,F^{V}_{Z_{\mu\tau}}(R_{Z_{\mu\tau}})-(g^{Z_{\mu\tau}}_{{}_{A}})^{2}\,F^{A}_{Z_{\mu\tau}}(R_{Z_{\mu\tau}})\right)\,,$
(18)

where $R_{Z_{\mu\tau}}=\dfrac{M^{2}_{Z_{\mu\tau}}}{m^{2}_{\ell}}$ and

$\displaystyle g^{Z_{\mu\tau}}_{{}_{V}}$ $\displaystyle=$
$\displaystyle-\dfrac{g_{2}\left(T^{\ell}_{3}-2\,Q^{\ell}_{em}\sin^{2}\theta_{W}\right)}{2\,\cos\theta_{W}}\sin\theta_{\mu\tau}-\left[g_{\mu\tau}\,Q^{\ell}_{\mu\tau}-\dfrac{g_{2}\,\epsilon\left(T^{\ell}_{3}-2\,Q^{\ell}_{em}\right)}{2\,\cot\theta_{W}}\right]\cos\theta_{\mu\tau}\,,$
(19) $\displaystyle g^{Z_{\mu\tau}}_{{}_{A}}$ $\displaystyle=-$
$\displaystyle\dfrac{g_{2}}{2\,\cos\theta_{W}}\Big{(}\sin\theta_{\mu\tau}-\,\epsilon\sin\theta_{W}\cos\theta_{\mu\tau}\Big{)}\,T^{\ell}_{3}\;.$
(20)

The loop functions for the vectorial and axial vectorial interactions are
given by Biswas:2019twf

$\displaystyle F^{V}_{Z_{\mu\tau}}(R_{Z_{\mu\tau}})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\,\frac{2x(1-x)^{2}}{(1-x)^{2}+R_{Z_{\mu\tau}}x}\;,$
(21) $\displaystyle F^{A}_{Z_{\mu\tau}}(R_{Z_{\mu\tau}})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}dx\,\frac{2x(1-x)(3+x)}{(1-x)^{2}+R_{Z_{\mu\tau}}x}\;.$
(22)

Figure 1: Feynman diagrams for one loop contributions in the anomalous
magnetic moments of muon and electron.

From Eq. (18), it is clearly seen that the vectorial part and the axial
vectorial part of the interaction between $Z_{\mu\tau}$ and leptons act
oppositely (true for any gauge boson) in the anomalous magnetic moment and the
net effect due to a new gauge boson is the difference between the
contributions of both interactions. In the limit of small kinetic mixing
($\epsilon\rightarrow 0$), $\Delta{a}_{e}$ goes to zero while
$\Delta{a}_{\mu}\propto g^{2}_{\mu\tau}$, gets contribution from the vectorial
part only.

The recent measurement of $(g-2)_{\mu}$ shows a $4.2\sigma$ deviation from the
SM prediction with a magnitude of
$\Delta{a}_{\mu}=a^{exp}_{\mu}-a^{SM}_{\mu}=(2.51\pm 0.59)\times 10^{-9}$
Muong-2:2021ojo . On the other hand, there are two existing values of
$\Delta{a}_{e}$ due to two different measurements of the fine structure
constant ($\alpha_{em}$) at Berkeley with ${}^{137}{\rm Cs}$ atom
Parker:2018vye  and LKB with ${}^{87}{\rm Rb}$ atom Morel:2020dww
respectively. Using these measurements, the SM prediction for
$a_{e}\equiv\dfrac{(g-2)_{e}}{2}$ is either higher ($2.4\sigma$ deviation) or
lower ($1.6\sigma$ deviation) than the experiment value Hanneke:2008tm . More
importantly, the nature of new physics is determined by the measurement of
$\alpha_{em}$, where the former case requires a destructive BSM contribution
while an opposite situation is need for the latter. In the case of muon we
always need a positive contribution to $a_{\mu}$ from the BSM theory as the SM
prediction is lower than the experimental measurement. Therefore, depending
upon the value of $\alpha_{em}$, we either need positive values of
$\Delta{a_{\ell}}$ for both $e$ and $\mu$ or we require a positive
$\Delta{a_{\mu}}$ along with a negative $\Delta{a_{e}}$. Although, there is a
relative sign difference between the contributions coming from the vectorial
and the axial vectorial parts of the interaction to $\Delta{a}_{\ell}$ as seen
from Eq. (18), it is not possible to achieve the second possibility in the
minimal $L_{\mu}-L_{\tau}$ model. The reason behind this is that for muon we
need dominance of the vectorial part of interaction while for $e$ the axial
vectorial dominance, which mainly comes from tiny $Z-Z_{\mu\tau}$ mixing, is
required. On the other hand, the first possibility where we need same sign of
$\Delta{a_{\ell}}$ can be achieved easily in the minimal model. However, the
parameter space addressing both the experimental values for
$\Delta{a_{\mu}}=(2.51\pm 0.59)\times 10^{-9}$ and $\Delta{a_{e}}=(4.8\pm
3.0)\times 10^{-13}$ is already excluded by the measurement of $\nu_{e}\,e$
scattering at Borexino experiment. In Fig. 2 we have summarised all results in
the familiar $g_{\mu\tau}-M_{Z_{\mu\tau}}$ plane.

Figure 2: Parameter space for $(g-2)_{e}$ (circular points) and $(g-2)_{\mu}$
(plus shaped points) with relevant experimental constraints in
$g_{\mu\tau}-M_{Z_{\mu\tau}}$ plane

In this figure, the $2\sigma$ contour of $(g-2)_{\mu}$ has been shown by plus
shaped points while the corresponding contour for $(g-2)_{e}$ is denoted by
circular points. The colour-bar indicates variation of the kinetic mixing
parameter in the following range $10^{-6}\leq\epsilon\leq 10^{-3}.$ It is
clearly seen that to obtain a sufficient contribution in $\Delta{a_{e}}$ from
$Z_{\mu\tau}$ we need $\epsilon\gtrsim 10^{-4}$ while for $\mu$, the parameter
$\epsilon$ can be as low as $10^{-6}$ or even smaller. This is mainly due to
the fact that the interaction of $e^{+}\,e^{-}$ with $Z_{\mu\tau}$ is entirely
governed by the $Z-Z_{\mu\tau}$ mixing since $e^{\pm}$ do not have any
$L_{\mu}-L_{\tau}$ charge. However, the vectorial part of interaction for
$\mu^{\pm}$, responsible for getting a positive $\Delta{a_{\mu}}$, is
dominated by a factor proportional to the $L_{\mu}-L_{\tau}$ charge. The
overlapping region in Fig. 2 satisfies both the experimental values of
anomalous magnetic moments and the corresponding parameters are $3\times
10^{-4}\lesssim g_{\mu\tau}\lesssim 10^{-3}$, $M_{Z_{\mu\tau}}\lesssim 0.3$
GeV and $\epsilon\gtrsim 10^{-4}$. However, as one can see from Fig. 2 that
the above parameter space for $\epsilon\gtrsim 10^{-4}$ is already ruled-out
from the observation of $e-\nu$ scatterings at Borexino experiment. We have
depicted excluded regions in $g_{\mu\tau}-M_{Z_{\mu\tau}}$ plane for three
different values of the kinetic mixing parameters for which the $e-\nu$
scattering cross section in the minimal $L_{\mu}-L_{\tau}$ model lies outside
the $2\sigma$ range obtained from Borexino experiment i.e.
$0.88<\dfrac{\sigma^{L_{\mu}-L_{\tau}}_{e\nu}}{\sigma^{\rm SM}_{e\nu}}<1.24$
Altmannshofer:2019zhy . The other relevant existing/proposed experimental
bounds from CCFR, DUNE using neutrino tridents
($\nu_{\mu}N\rightarrow\nu_{\mu}N\mu^{+}\mu^{-}$) Altmannshofer:2014pba  and
the four $\mu$ production at LHC ($p\,p\rightarrow
Z_{\mu\tau}\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-}$) CMS:2018yxg
are also shown in Fig. 2. Therefore, looking at the current bounds in
$g_{\mu\tau}-M_{Z_{\mu\tau}}$ plane, it is clearly evident that the minimal
$L_{\mu}-L_{\tau}$ model is unable to address the experimental values of both
electron and muon anomalous magnetic moments simultaneously.

##  III Extended $L_{\mu}-L_{\tau}$ Model

In the previous section, we have tried to expound both the anomalies reported
in $g-2$ of muon and electron in a common framework. While doing so we have
noticed that the minimal model is not sufficient and we need some extension of
the minimal model. Moreover, we would also like to see that the extended model
is good enough to address issues related to neutrino mass and dark matter.
Therefore, in order to accomplish these unresolved issues, we extend particle
contents of the minimal $L_{\mu}-L_{\tau}$ model. In particular, we introduce
a vector like fermion $\chi$ with nonzero U(1)Y and
U(1)${}_{L_{\mu}-L_{\tau}}$ quantum numbers and has no colour and weak-isospin
charges. Additionally, in the fermionic sector we include three right handed
neutrinos, having transformation properties similar to the SM leptons under
U(1)${}_{L_{\mu}-L_{\tau}}$, for neutrino mass generation via Type-I seesaw
mechanism. In the scalar sector, besides the SM Higgs doublet
$\Phi^{\prime}_{1}$ and previously introduced $L_{\mu}-L_{\tau}$ breaking
scalar $\Phi^{\prime}_{3}$, we have one SU(2)L doublet $\Phi^{\prime}_{2}$ and
a SU(2)L singlet $\Phi^{\prime}_{4}$ with suitable $L_{\mu}-L_{\tau}$ charges
to construct new Yukawa interactions between the SM Leptons and $\chi$. As we
will see later, both these scalars along with the charged fermion $\chi$ have
played a pivotal role in generating $(g-2)_{e}$ in the ballpark of
experimental measurements. In Tables (2, 3), we have shown the complete
particle spectrum and associated charges under the complete gauge group
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}\times U(1)_{L_{\mu}-L_{\tau}}$ of
the present model. Here we would like to note that since all fermions are
vector like under the $L_{\mu}-L_{\tau}$ symmetry the extended model is also
anomaly free as it was for the minimal model.

| Gauge
---
Group
${\rm SU(2)}_{\rm L}$
${\rm U(1)}_{\rm Y}$
| Baryon Fields
---
$Q_{L}^{i}=(u_{L}^{i},d_{L}^{i})^{T}$ | $u_{R}^{i}$ | $d_{R}^{i}$
$2$ | $1$ | $1$
$1/6$ | $2/3$ | $-1/3$
| Lepton Fields
---
$L_{L}^{i}=(\nu_{L}^{i},e_{L}^{i})^{T}$ | $e_{R}^{i}$ | $N_{R}^{i}$ | $\chi_{{}_{L}}$ | $\chi_{{}_{R}}$
$2$ | $1$ | $1$ | $1$ | $1$
$-1/2$ | $-1$ | $0$ | $-1$ | $-1$
| Scalar Fields
---
$\Phi^{\prime}_{1}$ | $\Phi^{\prime}_{2}$ | $\Phi^{\prime}_{3}$ | $\Phi^{\prime}_{4}$
$2$ | $2$ | $1$ | $1$
$1/2$ | $1/2$ | $0$ | $0$
Table 2: Particle contents and their corresponding charges under the SM gauge
group.

| Gauge
---
Group
${\rm U(1)}_{L_{\mu}-L_{\tau}}$
| Baryonic Fields
---
$(Q^{i}_{L},u^{i}_{R},d^{i}_{R})$
$0$
| Lepton Fields
---
$(L_{L}^{e},e_{R},N_{R}^{e})$ | $(L_{L}^{\mu},\mu_{R},N_{R}^{\mu})$ | $(L_{L}^{\tau},\tau_{R},N_{R}^{\tau})$ | $\chi_{{}_{L}}$ | $\chi_{{}_{R}}$
$0$ | $1$ | $-1$ | $1/3$ | $1/3$
| Scalar Fields
---
$\Phi^{\prime}_{1}$ | $\Phi^{\prime}_{2}$ | $\Phi^{\prime}_{3}$ | $\Phi^{\prime}_{4}$
$0$ | $-1/3$ | $1$ | $-1/3$
Table 3: $L_{\mu}-L_{\tau}$ charges for fermions and scalars of the present
model.

The full Lagrangian of the present model is given by

$\displaystyle\mathcal{L}=\mathcal{L}_{\rm
SM}-\frac{1}{4}\hat{X}_{\alpha\beta}\hat{X}^{\alpha\beta}+\frac{\epsilon}{2}\hat{X}_{\alpha\beta}\hat{B}^{\alpha\beta}-g_{\mu\tau}\sum_{\begin{subarray}{c}\ell=\mu,\nu_{\mu},\\\
\,\,\,\tau,\nu_{\tau}\end{subarray}}Q^{\ell}_{\mu\tau}\bar{\ell}\gamma^{\alpha}\ell\hat{X}_{\alpha}+\bigg{[}i\,\bar{\chi}\not{\partial}\chi-\bar{\chi}\gamma^{\alpha}\chi\left(\dfrac{g_{\mu\tau}}{3}\hat{X}_{\alpha}-{g_{1}}\hat{B}_{\alpha}\right)$
$\displaystyle-
M_{\chi}\overline{\chi}\chi-\biggl{(}\beta^{R}_{e\chi}\overline{L_{e}}\Phi^{\prime}_{2}\chi_{{}_{R}}+\beta^{L}_{e\chi}\overline{e_{R}}\,\chi_{{}_{L}}\Phi^{\prime}_{4}+{\it
h.c.}\biggr{)}\bigg{]}+\mathcal{L}_{N}+\sum_{i=2}^{4}\left(D_{\alpha}{\Phi^{\prime}_{i}}\right)^{\dagger}\left(D^{\alpha}{\Phi^{\prime}_{i}}\right)-\mathcal{V}(\Phi^{\prime}_{1},\Phi^{\prime}_{2},\Phi^{\prime}_{3},\Phi^{\prime}_{4})\,.$
(23)

Apart from the terms which have already defined in the previous section, the
terms within the square brackets represent the Lagrangian of vector like
fermion $\chi$ including the new Yukawa interactions with couplings
$\beta^{R}_{e\chi}$ and $\beta^{L}_{e\chi}$ respectively. The Lagrangian for
three right handed neutrinos are denoted by $\mathcal{L}_{N}$. We have written
the exact form of $\mathcal{L}_{N}$ below.

$\displaystyle\mathcal{L}_{N}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\sum_{\begin{subarray}{c}i=e,\mu,\\\
\tau\end{subarray}}\overline{N_{i}}\not{D}N_{i}+\sum_{\begin{subarray}{c}i=e,\mu,\\\
\tau\end{subarray}}\big{(}y_{ii}\,\overline{L_{i}}\Phi^{\prime}_{1}N_{i}+{\it
h.c.}\big{)}+\big{(}M_{ee}N_{e}N_{e}+h_{e\mu}N_{e}N_{\mu}{\Phi^{\prime}_{3}}^{\dagger}$
(24)
$\displaystyle+h_{e\tau}N_{e}N_{\tau}\Phi^{\prime}_{3}+M_{\mu\tau}e^{i\theta}N_{\mu}N_{\tau}+{\it
h.c.}\big{)}\,,$

where, the first term is the kinetic term of $N_{i}$ while the rest are
interaction terms responsible for the light neutrino mass generation via
Type-I seesaw mechanism. We refrain further discussion on this topic and will
discuss again when we talk about neutrino mass in Section V. Finally, the last
two terms in Eq. (23) are the kinetic and interaction terms for the BSM
scalars ($\Phi^{\prime}_{i},\,i=2,\,3,\,4$). In the kinetic term, $D_{\alpha}$
being the usual covariant derivative involving gauge boson(s) and generator(s)
of each group under which $\Phi^{\prime}_{i}$ transforms non-trivially. The
explicit form of $\mathcal{V}$, invariant under the full gauge group, is given
below

$\displaystyle\mathcal{V}(\Phi^{\prime}_{1},\Phi^{\prime}_{2},\Phi^{\prime}_{3},\Phi^{\prime}_{4})$
$\displaystyle=$
$\displaystyle\sum^{4}_{i=2}\left[\mu^{2}_{i}\left({\Phi^{\prime}}^{\dagger}_{i}\Phi^{\prime}_{i}\right)+\lambda_{i}\left({\Phi^{\prime}}^{\dagger}_{i}\Phi^{\prime}_{i}\right)^{2}\right]+\sum_{i,j,j>i}\lambda_{ij}\left({\Phi^{\prime}}^{\dagger}_{i}\Phi^{\prime}_{i}\right)\left({\Phi^{\prime}}^{\dagger}_{j}\Phi^{\prime}_{j}\right)$
(25)
$\displaystyle+\lambda^{\prime}_{12}\left({\Phi^{\prime}}^{\dagger}_{1}\Phi^{\prime}_{2}\right)\left({\Phi^{\prime}}^{\dagger}_{2}\Phi^{\prime}_{1}\right)+\left[\mu\left({\Phi^{\prime}_{1}}^{\dagger}\Phi^{\prime}_{2}\right){\Phi^{\prime}_{4}}^{\dagger}+\xi\left(\Phi^{\prime}_{3}{\Phi^{\prime}_{4}}^{3}\right)+{\it
h.c.}\right].$

Here for necessary vacuum alignment we need $\mu^{2}_{1,3}<0$,
$\mu^{2}_{2,4}>0$ and $\lambda_{i}>0$. The second condition,
$\mu^{2}_{2,4}>0$, ensures that two new $L_{\mu}-L_{\tau}$ charged scalars
$\Phi^{\prime}_{2}$ and $\Phi^{\prime}_{4}$ do not have any VEV. Apart from
the usual self-conjugate terms, we have two non-self-conjugate terms also in
the Lagrangian which have utmost importance in the present context. The
trilinear term with coefficient $\mu$ introduces mixing between the neutral
component of $\Phi^{\prime}_{2}$ and $\Phi^{\prime}_{4}$. Since the VEVs of
both these scalars are zero, the lightest one is automatically stable and can
be a viable dark matter candidate. Therefore, for a singlet like dark matter
candidate (dominated by $\Phi^{\prime}_{4}$), which is precisely the case we
are considering, this mixing opens up a direct detection prospect through
exchange of the SM $Z$ boson. On the other hand, the quartic term proportional
to $\xi$ is responsible for cubic interaction among the dark matter particles,
which results in some higher order number changing processes ($3\rightarrow
2$). After both EWSB and $L_{\mu}-L_{\tau}$ breaking, the $2\times 2$ mass
matrix for the neutral component of $\Phi^{\prime}_{2}$ ($\phi^{\prime}_{2}$)
and $\Phi^{\prime}_{4}$ is given by

$\displaystyle
M^{2}_{\phi^{\prime}_{2}-\Phi^{\prime}_{4}}=\begin{pmatrix}a&b\\\
b&c\end{pmatrix}\,.$ (26)

where
$a=\mu^{2}_{2}+(\lambda_{12}+\lambda^{\prime}_{12})\frac{v^{2}}{2}+\lambda_{23}\frac{v_{\mu\tau}}{2}$,
$c=\mu^{2}_{4}+\lambda_{14}\frac{v^{2}}{2}+\lambda_{34}\frac{v_{\mu\tau}}{2}$
and $b=\frac{\mu v}{\sqrt{2}}$. One can easily diagonalise the above mass
matrix using an orthogonal transformation by an angle $\theta_{D}$ and the
resultant eigenstates are related to the old basis sates
$(\phi^{\prime}_{2},\,\,\Phi^{\prime}_{4})$ in the following way

$\displaystyle\begin{pmatrix}\phi_{2}\\\
\phi_{4}\end{pmatrix}=\begin{pmatrix}\cos\theta_{D}&-\sin\theta_{D}\\\
\sin\theta_{D}&\cos\theta_{D}\end{pmatrix}\,\,\begin{pmatrix}\phi^{\prime}_{2}\\\
\Phi^{\prime}_{4}\end{pmatrix}\,,$ (27)

where, the mixing angle $\theta_{D}$ can be expressed in terms of the
parameters of the Lagrangian as,

$\displaystyle\theta_{D}$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\tan^{-1}\left(\dfrac{2b}{c-a}\right)\,,$ (28)
$\displaystyle=$
$\displaystyle\dfrac{1}{2}\tan^{-1}\left[\dfrac{\sqrt{2}\,{\mu
v}}{\mu^{2}_{4}-\mu^{2}_{2}+(\lambda_{14}-\lambda_{12}-\lambda^{\prime}_{12})\frac{v^{2}}{2}+(\lambda_{34}-\lambda_{23})\frac{v_{\mu\tau}}{2}}\right]$

and the masses corresponding to the physical states $\phi_{2}$ and $\phi_{4}$
are

$\displaystyle M^{2}_{\phi_{2}}$ $\displaystyle=$
$\displaystyle\left[\mu^{2}_{2}+(\lambda_{12}+\lambda^{\prime}_{12})\frac{v^{2}}{2}+\lambda_{23}\frac{v_{\mu\tau}}{2}\right]\cos^{2}\theta_{D}+\left[\mu^{2}_{4}+\lambda_{14}\frac{v^{2}}{2}+\lambda_{34}\frac{v_{\mu\tau}}{2}\right]\sin^{2}\theta_{D}-\frac{\mu
v}{\sqrt{2}}\sin 2\theta_{D}\,,$ $\displaystyle M^{2}_{\phi_{4}}$
$\displaystyle=$
$\displaystyle\left[\mu^{2}_{2}+(\lambda_{12}+\lambda^{\prime}_{12})\frac{v^{2}}{2}+\lambda_{23}\frac{v_{\mu\tau}}{2}\right]\sin^{2}\theta_{D}+\left[\mu^{2}_{4}+\lambda_{14}\frac{v^{2}}{2}+\lambda_{34}\frac{v_{\mu\tau}}{2}\right]\cos^{2}\theta_{D}+\frac{\mu
v}{\sqrt{2}}\sin 2\theta_{D}\,.$

In our work, we have considered that $\phi_{4}$ is the lightest state and is a
suitable for dark matter candidate. Similar to the
$\phi^{\prime}_{2}-\Phi^{\prime}_{4}$ mixing, there is another mass mixing
between the CP even neutral components
$(\phi^{\prime}_{1},\,\,\phi^{\prime}_{3})$ of the Higgs doublet
$\Phi^{\prime}_{1}$ and the $L_{\mu}-L_{\tau}$ breaking scalar
$\Phi^{\prime}_{3}$ respectively. This mixing is due to the presence of a
quartic interaction term with coefficient $\lambda_{13}$. Therefore, after
spontaneous symmetry breaking this quartic interaction generates off-diagonal
terms in the $2\times 2$ mass matrix which is given by

$\displaystyle
M^{2}_{\phi^{\prime}_{1}-\phi^{\prime}_{3}}=\begin{pmatrix}2\lambda_{1}v^{2}&\lambda_{13}vv_{\mu\tau}\\\
\lambda_{13}vv_{\mu\tau}&2\lambda_{3}v^{2}_{\mu\tau}\end{pmatrix}\,.$ (30)

This symmetric mass matrix can be diagonalised in a similar manner as above
and as a result, we get two physical states $h_{1}$ and $h_{3}$ with a mixing
angle $\theta$ which can be expressed as,

$\displaystyle\theta=\dfrac{1}{2}\tan^{-1}\left[\frac{\lambda_{13}vv_{\mu\tau}}{\lambda_{3}v_{\mu\tau}^{2}-\lambda^{2}_{1}v^{2}}\right].$
(31)

In this work we have considered $h_{1}$ as the SM like Higgs boson which was
discovered by the ATLAS ATLAS:2012yve  and the CMS CMS:2012qbp  collaborations
in 2012 and having mass $M_{h_{1}}=125.5$ GeV. We would like to mention in
passing that the CP odd neutral components of $\Phi^{\prime}_{1}$ and
$\Phi^{\prime}_{3}$ turn into the Goldstone bosons after both $Z$ and
$Z_{\mu\tau}$ become massive. Therefore, in the scalar sector of the extended
model, apart from the SM like Higgs boson $h_{1}$, we have one CP even scalar
($h_{2}$), one charged scalar (part of the doublet $\Phi^{\prime}_{2}$) and
two complex scalars ($\phi_{2}$, $\phi_{4}$). The latter take part in one loop
diagram contributing to $(g-2)_{e}$ while $\phi_{4}$ plays the role of dark
matter with enhanced detection possibilities directly due to $\phi_{2}$.

##  IV Anomalous magnetic moment in the extended model

As mentioned in the previous section, besides the contribution coming from
$Z_{\mu\tau}$ through kinetic mixing, the new Yukawa interactions (defined in
Eq. (23) involving $e$, $\chi$ and $\phi_{i}$ ($i=2,\,4$) provide additional
contribution to $(g-2)_{e}$. The one loop Feynman diagrams for $\phi_{2}$ and
$\phi_{4}$ are shown in Fig. 3.

Figure 3: Feynman diagrams contributing to the anomalous magnetic moment.

The expression of $\Delta{a}_{e}$ due to these scalar mediated diagrams is
given by Leveille:1977rc

$\displaystyle\Delta{a}^{scalar}_{e}$ $\displaystyle=$
$\displaystyle-\dfrac{Q^{\chi}_{em}\,m_{e}}{8\pi^{2}}\left[(g^{s}_{\phi_{2}})^{2}\,\mathcal{I}(m_{e},\,M_{\chi},\,M_{\phi_{2}})+(g^{p}_{\phi_{2}})^{2}\,\mathcal{I}(m_{e},\,-M_{\chi},\,M_{\phi_{2}})\right.$
(32)
$\displaystyle\left.(g^{s}_{\phi_{4}})^{2}\,\mathcal{I}(m_{e},\,M_{\chi},\,M_{\phi_{4}})+(g^{p}_{\phi_{4}})^{2}\,\mathcal{I}(m_{e},\,-M_{\chi},\,M_{\phi_{4}})\right]\,,$

where the loop integral function $\mathcal{I}(m_{1},m_{2},m_{3})$ is defined
as

$\displaystyle\mathcal{I}(m_{1},\,m_{2},\,m_{3})=\int_{0}^{1}dx\dfrac{x^{2}-x^{3}+\frac{m_{2}}{m_{1}}x^{2}}{m_{1}^{2}x^{2}+(m_{2}^{2}-m_{1}^{2})x+m^{2}_{3}(1-x)}\,$
(33)

and the associated coefficients $g^{(s,\,p)}_{({\phi_{2}},\,{\phi_{4}})}$ have
the following expressions

$\displaystyle g^{s}_{\phi_{2}}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{2}\left(\beta^{R}_{e\chi}\cos\theta_{D}-\beta_{e\chi}^{L}\sin\theta_{D}\right),$
$\displaystyle g^{p}_{\phi_{2}}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{2}\left(\beta^{R}_{e\chi}\cos\theta_{D}+\beta_{e\chi}^{L}\sin\theta_{D}\right),$
$\displaystyle g^{s}_{\phi_{4}}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{2}\left(\beta^{R}_{e\chi}\sin\theta_{D}+\beta_{e\chi}^{L}\cos\theta_{D}\right),$
$\displaystyle g^{p}_{\phi_{4}}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{2}\left(\beta^{R}_{e\chi}\sin\theta_{D}-\beta_{e\chi}^{L}\cos\theta_{D}\right).$
(34)

These are scalar and pseudo scalar couplings of $\phi_{i}$ with $e$ and $\chi$
i.e. we have an interaction like
$\overline{e}\left(g^{s}_{\phi_{i}}+\gamma_{5}\,g^{p}_{\phi_{i}}\right)\chi\phi_{i}$.
The total BSM contribution in $(g-2)_{e}$ in the extended model is
$\Delta{a}_{e}=\Delta{a}^{scalar}_{e}+\Delta{a}^{Z_{\mu\tau}}_{e}$, where
$\Delta{a}^{Z_{\mu\tau}}_{e}$, the contribution due to $Z_{\mu\tau}$, is given
in Eq. (18). However, since there is no new Yukawa coupling for muon, the BSM
effect in the anomalous magnetic moment of muon is solely due to $Z_{\mu\tau}$
as also in the minimal $L_{\mu}-L_{\tau}$ model.

Figure 4: Parameter space in $\beta^{L}_{e\chi}-M_{\phi_{2}}$ plane
reproducing correct value of $\Delta{a}_{e}$ in 2$\sigma$ range. This plot is
generated for $\beta^{R}_{e\chi}=\beta^{L}_{e\chi}$ and other relevant
parameters have varied according to Eq. (67).

In Fig. 4, we have shown the allowed parameter space in
$\beta^{L}_{e\chi}-M_{\phi_{2}}$ plane for which $\Delta{a}_{e}$ lies within
the $2\sigma$ range of experimental measurement. In this plot, we have
considered the Yukawa coupling $\beta^{R}_{e\chi}=\beta^{L}_{e\chi}$. For each
$\beta^{L}_{e\chi}$ and $M_{\phi_{2}}$, the corresponding value of
$\theta_{\rm D}$ is indicated by the colour bar. One thing we would like to
emphasize here that we have adopted the result $\Delta{a_{e}}=(-8.8\pm
3.6)\times 10^{-13}$ corresponding to $\alpha_{em}$ measured at Berkeley with
137Cs atom primarily due to large statistical deviation $2.4\sigma$ from the
SM prediction. As a result, we require the new physics to contribute
destructively. The origin of this negative contribution is coming from the
fact that the loop function $\mathcal{I}$ (see Eq. (33) for the scalar part
and the pseudo scalar part of interaction are equal222In general, the
magnitudes of $\mathcal{I}(m_{e},\,M_{\chi},\,M_{\phi_{i}})$ and
$\mathcal{I}(m_{e},\,-M_{\chi},\,M_{\phi_{i}})$ are not equal. However, both
become identical for the range of $M_{\chi}$ ($\sim\mathcal{O}$(TeV)) we have
considered. and opposite. In this case, the coefficients
($g^{(s,\,p)}_{\phi_{2},\,\phi_{4}}$) and the mass difference between
$\phi_{2}$ and $\phi_{4}$ with $M_{\phi_{2}}>M_{\phi_{4}}$ are playing the
pivotal role for obtaining a negative $\Delta{a^{scalar}_{e}}$. On the other
hand, for the positive $\Delta{a_{e}}$, which was predicted using the fine
structure constant measured at LKB with 87Rb atom, the deviation reported from
the SM result is still $+1.6\sigma$ only. Nevertheless, we can easily
accommodate that result also in the present model by flipping the sign of the
mixing angle $\theta_{D}$ (by changing the sign of the trilinear coupling
$\mu$) so that the coefficients for the scalar and the pseudo scalar parts of
interaction interchange among themselves.

##  V Neutrino mass

As described in Eq. (24), the Lagrangian associated with the right handed
neutrinos generate light neutrino masses by the Type-I seesaw mechanism
Minkowski:1977sc ; Yanagida:1979as ; Mohapatra:1979ia ; Schechter:1980gr . A
similar technique for generating neutrino masses in the context of ${\rm
U(1)}_{L_{\mu}-L_{\tau}}$ symmetry has already been explored in detail by the
present authors in Biswas:2016yan . We get both Dirac and Majorana masses when
the SM Higgs doublet $\Phi^{\prime}_{1}$ and the singlet scalar
$\Phi^{\prime}_{3}$ acquire VEVs. The Dirac mass matrix $M_{D}$ has the
following form once the electroweak symmetry breaks,

$\displaystyle
M_{D}=\left(\begin{array}[]{ccc}\frac{y_{ee}v}{\sqrt{2}}{}{}&~{}~{}0{}{}&~{}~{}0\\\
{}{}\hfil&{}{}\hfil\\\ 0{}{}&~{}~{}\frac{y_{\mu\mu}v}{\sqrt{2}}{}{}&~{}~{}0\\\
{}{}\hfil&{}{}\hfil\\\
0{}{}&~{}~{}0{}{}&~{}~{}\frac{y_{\tau\tau}v}{\sqrt{2}}\\\
\end{array}\right)\,,$ (40)

and the Majorana mass matrix $M_{R}$ takes the following form when ${\rm
U(1)}_{L_{\mu}-L_{\tau}}$ symmetry gets broken,

$\displaystyle\mathcal{M}_{N}=\left(\begin{array}[]{ccc}M_{ee}{}{}&~{}~{}\dfrac{v_{\mu\tau}}{\sqrt{2}}h_{e\mu}{}{}&~{}~{}\dfrac{v_{\mu\tau}}{\sqrt{2}}h_{e\tau}\\\
{}{}\hfil&{}{}\hfil\\\
\dfrac{v_{\mu\tau}}{\sqrt{2}}h_{e\mu}{}{}&~{}~{}0{}{}&~{}~{}M_{\mu\tau}\,e^{i\xi}\\\
{}{}\hfil&{}{}\hfil\\\
\dfrac{v_{\mu\tau}}{\sqrt{2}}h_{e\tau}{}{}&~{}~{}M_{\mu\tau}\,e^{i\xi}{}{}&~{}~{}0\\\
\end{array}\right)\,.$ (46)

In the Dirac mass matrix, we can rotate away the phases by redefining the left
handed neutrinos on the other hand we cannot get rid of all the phases
associated with the Majorana mass matrix and one phase remains which we have
considered in (2,3) position. We can write down the neutrino mass matrix in
the basis $({\nu_{L}}_{\alpha},{N_{R}}^{c}_{\alpha})$ ($\alpha=e,\mu,\tau$) as
follows,

$\displaystyle M_{\nu}=\begin{pmatrix}0&M_{D}\\\
M^{T}_{D}&M_{N}\end{pmatrix}\,.$ (47)

As demanded by the oscillation experiments and cosmology, we have the
following allowed range of the oscillation parameters,

  * •

bound on the sum of all three light neutrinos from cosmology,
$\sum_{i}m_{i}<0.23$ eV at $2\sigma$ C.L. Ade:2015xua ,

  * •

$3\sigma$ range of mass squared differences, $6.93<\dfrac{\Delta
m^{2}_{21}}{10^{-5}}\,{\text{eV}^{2}}<7.97$ and $2.37<\dfrac{\Delta
m^{2}_{31}}{10^{-3}}\,{\text{eV}^{2}}<2.63$ Capozzi:2016rtj ,

  * •

$3\sigma$ bound on the mixing angles
$30^{\circ}<\,\theta_{12}\,<36.51^{\circ}$,
$37.99^{\circ}<\,\theta_{23}\,<51.71^{\circ}$ and
$7.82^{\circ}<\,\theta_{13}\,<9.02^{\circ}$ Capozzi:2016rtj .

Moreover, we also have the bound on the effective number of relativistic d.o.f
($N_{eff}$) allowed from cosmology which is $N_{eff}=2.99\pm 0.17$
Planck:2018vyg . Therefore, to be consistent with all these observations and
experimental measurements we can have three neutrinos in the sub-eV scale and
other three are in the higher scale. This is indeed possible if we go to the
seesaw regime where $M_{N}\gg M_{D}$. Then we can diagonalize the $M_{\nu}$
matrix as follows,

$\displaystyle m^{light}_{\nu}\simeq-M^{T}_{D}M^{-1}_{N}M_{D}\,,\,\,\,{\rm
and}\,\,\,m^{heavy}_{N}\simeq M_{N}$ (48)

where $m^{light}_{\nu}$ corresponds to three sub-eV scale Majorana neutrinos
while $m^{heavy}_{N}$ corresponds to three heavier Majorana neutrinos. The
detailed numerical analysis and the ranges of associated parameters are given
in Biswas:2016yan .

##  VI SIMP dark matter

We have seen in Section III that there are two neutral scalars $\phi_{2}$,
$\phi_{4}$. The $L_{\mu}-L_{\tau}$ charge assignment (see Table 3) among the
scalar fields is extremely crucial not only to get a stable dark matter
candidate but also it dictates the nature of dark matter As a result, the
lightest scalar $\phi_{4}$ is naturally stable and has a cubic self-
interaction term when $\Phi^{\prime}_{3}$ gets a VEV. The latter is
responsible for number changing interactions occurring through higher order
scattering like $3\rightarrow 2$ processes. When the coupling $\xi$ of the
cubic term is large enough so that the main number changing processes are
$3\rightarrow 2$ scatterings rather than the usual $2\rightarrow 2$ pair
annihilations of dark matter into the SM fields, the freeze-out of $\phi_{4}$
is primarily determined by the condition $\Gamma_{3\rightarrow
2}\lesssim\mathcal{H}$. Moreover, the dark matter $\phi_{4}$ maintains kinetic
equilibrium with the SM bath by virtue of elastic scattering with
$Z_{\mu\tau}$ where the latter remains thermally connected with the SM
leptons. This is known as the SIMP paradigm Carlson:1992fn ; Hochberg:2014dra
. In this work, we have explored the phenomenology of SIMP dark matter in the
context of U(1)${}_{L_{\mu}-L_{\tau}}$ gauge extension. The Boltzmann equation
expressing the evolution of comoving number density of dark matter is given
by333see Biswas:2020ubd  for a detailed derivation of the Boltzmann equation

$\displaystyle\frac{dY_{\rm DM}}{dx}$ $\displaystyle=$
$\displaystyle\dfrac{{\rm s}^{2}}{\mathcal{H}\,x}\langle\sigma_{3\rightarrow
2}^{\rm tot}{\rm v}^{2}\rangle\,Y^{2}_{\rm DM}\left(Y_{\rm DM}-Y^{\rm eq}_{\rm
DM}\right)-\dfrac{\rm s}{\mathcal{H}\,x}\langle\sigma_{2\rightarrow 2}^{\rm
tot}{\rm v}\rangle\left(Y^{2}_{\rm DM}-(Y^{\rm eq}_{\rm DM})^{2}\right).$

Where $Y_{\rm DM}$ is the total comoving number densities of both $\phi_{4}$
and $\phi^{\dagger}_{4}$ respectively while $x=M_{\phi_{4}}/T$ with $T$ being
the photon temperature. Here we have assumed that there is no asymmetry
between the number densities of particle and anti-particle of dark matter. The
entropy density and the Hubble parameters are denoted by ${\rm s}$ and
$\mathcal{H}$. The quantity $\langle\sigma_{3\rightarrow 2}^{\rm tot}{\rm
v}^{2}\rangle$ is the thermal average of total scattering cross section for
all relevant number changing $3\rightarrow 2$ processes for $\phi_{4}$ taking
into account all symmetry factors

$\displaystyle\sigma_{3\rightarrow 2}^{\rm
tot}=-\dfrac{2}{3!}\,\sigma_{\phi_{4}\phi_{4}\phi_{4}\rightarrow\phi^{\dagger}_{4}\phi_{4}}-\dfrac{2}{2!}\,\sigma_{\phi^{\dagger}_{4}\phi_{4}\phi_{4}\rightarrow\phi^{\dagger}_{4}\phi^{\dagger}_{4}}+\dfrac{1}{3!}\,\sigma_{\phi^{\dagger}_{4}\phi^{\dagger}_{4}\phi^{\dagger}_{4}\rightarrow\phi^{\dagger}_{4}\phi_{4}}+\dfrac{1}{2!}\,\sigma_{\phi_{4}\phi^{\dagger}_{4}\phi^{\dagger}_{4}\rightarrow\phi_{4}\phi_{4}}\,.$
(50)

Here, $+(-)$ represents increase(decrease) of $\phi_{4}$ due to a particular
scattering, e.g. the first term is the scattering cross section for a
$3\rightarrow 2$ process like
$\phi_{4}\phi_{4}\phi_{4}\rightarrow\phi_{4}\phi^{\dagger}_{4}$ which reduces
the comoving number density of $\phi_{4}$ by 2 unit per scattering. Moreover,
the factor $3!$ in the denominator is due to three identical particles in the
initial state. As we have computed these $3\rightarrow 2$ processes in the
non-relativistic limit of dark matter following Berlin:2016gtr , the thermal
average $\langle\sigma_{3\rightarrow 2}^{\rm tot}{\rm v^{2}}\rangle$ is
identical to $\sigma_{3\rightarrow 2}^{\rm tot}{\rm v^{2}}$. The matrix
amplitude square ($|\mathcal{M}|^{2}$) for a particular $3\rightarrow 2$
process is related to $\sigma{\rm v^{2}}$, in the non-relativistic limit, as
Berlin:2016gtr

$\displaystyle\langle\sigma{\rm v}^{2}\rangle_{3\rightarrow
2}=\frac{\sqrt{5}}{S\times 384\pi
M^{3}_{\phi_{4}}}\int_{-1}^{+1}d\cos\theta\,|\mathcal{M}_{3\rightarrow
2}|^{2}\,,$ (51)

where initial and final state particles are either $\phi_{4}$ or
$\phi_{4}^{\dagger}$ or both. The symmetry factor $S$ depends on number of
identical particles in the final state. We have determined these amplitudes
using CalcHEP Belyaev:2012qa  while the necessary model files have been
generated by the Mathematica based package FeynRules Alloul:2013bka . Relevant
diagrams for the scattering
$\phi_{4}\phi_{4}\phi_{4}\rightarrow\phi^{\dagger}_{4}\phi_{4}$ are shown in
Fig. 5 and the Feynman diagrams for other processes can be generated easily
following these diagrams. The necessary vertex factors are listed in Appendix
A.

Figure 5: Feynman diagrams for the $3\rightarrow 2$ scattering
$\phi_{4}\phi_{4}\phi_{4}\rightarrow\phi_{4}\phi^{\dagger}_{4}$, where
comoving number density of $\phi_{4}$ decreases by 2 unit per scattering.

The second term in Eq. (LABEL:Boltz) is coming from $2\rightarrow 2$
scatterings. These can be proceed through mediation of scalars ($h_{1}$,
$h_{2}$) and gauge bosons ($Z$, $Z_{\mu\tau}$). In this work, since we want to
explore the phenomenon of freeze-out within the dark sector only, we have
chosen feeble scalar couplings. Therefore, the pair-annihilations of
$\phi_{4}$ and $\phi^{\dagger}_{4}$ into light SM fermions are possible
through exchange of $L_{\mu}-L_{\tau}$ gauge boson. Additionally, we have
annihilation channel like $\phi^{\dagger}_{4}\phi_{4}\rightarrow
Z_{\mu\tau}Z_{\mu\tau}$. Once we obtain these scattering cross sections using
CalcHEP, the thermal average $\langle\sigma^{\rm tot}_{2\rightarrow 2}{\rm
v}\rangle$ can be computed as

$\displaystyle\langle\sigma^{\rm tot}_{2\rightarrow 2}{\rm
v}\rangle=\dfrac{1}{4}\dfrac{1}{r^{4}_{\phi_{4}}x^{4}\,{\rm
K^{2}_{2}}(r_{\phi_{4}}x)}\int_{2\,r_{\phi_{4}}x}^{\infty}dZ\,\left(\sum_{Y}\sigma_{\phi_{4}\phi^{\dagger}_{4}\rightarrow
Y\overline{Y}}\right)\,Z^{2}\left(Z^{2}-4r^{2}_{\phi_{4}}x^{2}\right){\rm
K_{1}}(Z)\,.$ (52)

The sum is over all possible final states $Y$ and $Z=\dfrac{\sqrt{s}}{T}$
where $s$ is one of the Mandelstam variables. The variables
$r_{\phi_{4}}=\dfrac{M_{\phi_{4}}}{M_{0}}$ and $x=\dfrac{M_{0}}{T}$ where
$M_{0}$ is an arbitrary mass scale. In this work we have chosen
$M_{0}=M_{\phi_{4}}$, hence $r_{\phi_{4}}=1$. We would like to note here that
we need sufficient elastic scatterings between dark matter and light
$L_{\mu}-L_{\tau}$ charged leptons to keep the dark sector thermally connected
with the SM bath. We have discussed our numerical results later and we now
discuss the relevant bounds that we have considered in the present work. These
are listed below.

Figure 6: Feynman diagrams for the self interactions between
$\phi_{4}\phi_{4}$ and $\phi_{4}\phi^{\dagger}_{4}$. The diagrams for the
process
$\phi^{\dagger}_{4}\phi^{\dagger}_{4}\rightarrow\phi^{\dagger}_{4}\phi^{\dagger}_{4}$
are same as those for $\phi_{4}\phi_{4}\rightarrow\phi_{4}\phi_{4}$.

  * •

Self Interaction: Dark matter $\phi_{4}$ and $\phi^{\dagger}_{4}$ in our
present model have the following self scatterings which conserve their
individual numbers

$\displaystyle\phi_{4}\phi_{4}\rightarrow\phi_{4}\phi_{4}\,,\,\,\,\phi_{4}\phi^{\dagger}_{4}\rightarrow\phi_{4}\phi^{\dagger}_{4}\,,\,\,\,\phi^{\dagger}_{4}\phi^{\dagger}_{4}\rightarrow\phi^{\dagger}_{4}\phi^{\dagger}_{4}\,.$
(53)

The effective self interaction cross section, considering both $\phi_{4}$ and
$\phi^{\dagger}_{4}$ contribute an equal amount to the relic density, is given
by

$\displaystyle\sigma_{\rm
self}=\dfrac{1}{4}\left(\sigma_{\phi_{4}\phi_{4}}+\sigma_{\phi_{4}\phi^{\dagger}_{4}}+\sigma_{\phi^{\dagger}_{4}\phi^{\dagger}_{4}}\right)\,.$
(54)

The individual scattering cross sections are obtained using the package
CalcHEP and the corresponding Feynman diagrams are shown in Fig. 6. Thereafter
we have enforced the non-relativistic limit by putting $s\simeq
4\,M^{2}_{\phi_{4}}+M^{2}_{\phi_{4}}\,{\rm v}^{2}$ and taking ${\rm
v}\rightarrow 0$ where ${\rm v}$ is the relative velocity in the centre of
mass frame. From the observation of the bullet cluster Clowe:2003tk ;
Markevitch:2003at  there is a bound on the ratio $\frac{\sigma_{\rm
self}}{M_{\phi_{4}}}\lesssim 1\,\,{\rm cm^{2}}/{\rm g}$. The bound depends on
the relative velocity of dark matter particles in a particular galaxy.
Similarly, the Abell 520 cluster merger predicts $\frac{\sigma_{\rm
self}}{M_{\phi_{4}}}\sim 1\,\,{\rm cm^{2}}/{\rm g}$ . Moreover, there is a
wide range of the allowed values of $\frac{\sigma_{\rm self}}{M_{\phi_{4}}}$
from various astrophysical observations and N-body simulations Tulin:2017ara .
To be consistent with maximum number of observations, in this work, we have
considered $0.1\,\,{\rm cm^{2}/g}\leq\frac{\sigma_{\rm
self}}{M_{\phi_{4}}}\leq 10\,\,{\rm cm^{2}}/{\rm g}$.

  * •

Perturbativity and Unitarity: We have considered perturbative limit ($<4\pi$)
on all the quartic couplings in Eq. (25) so that the vacuum does not become
unbounded from below for the large value of scalar fields. Moreover,
decomposing the matrix amplitude of a scattering process into partial waves,
the requirement of unitarity of S-matrix demands

$\displaystyle|\mathcal{M}|\leq 16\pi\,.$ (55)

  * •

Direct Detection bound: The dark matter candidates $\phi_{4}$ and
$\phi^{\dagger}_{4}$ in the present model can be detected at the direct
detection experiments by scattering with heavy nuclei and electrons as well.
Instead of being predominantly an SU(2)L singlet like state, the mixing of
$\Phi^{\prime}_{4}$ with the neutral component of inter doublet
$\Phi^{\prime}_{2}$ generates $\phi_{4}\phi_{4}^{\dagger}Z$ vertex. This is a
vectorial interaction (proportional to $\gamma_{\mu}$) only. On the other
hand, the $\bar{f}fZ$ ($f$ is any SM fermion) vertex factor has both the
vectorial as well as the axial vectorial (proportional to
$\gamma_{\mu}\gamma_{5}$) parts. While the vectorial part is responsible for
spin independent scattering, spin dependent scattering is possible due to the
axial vectorial part. As a result, we have both spin independent as well as
spin dependent scatterings when dark matter scatters off through $Z$ boson.
The spin independent elastic scattering cross sections is given by

$\displaystyle\sigma_{\rm SI}$ $\displaystyle=$
$\displaystyle\left(\dfrac{g_{2}\,\sin^{2}\theta_{D}}{2\,\cos\theta_{W}}\right)^{2}\dfrac{\mu^{2}_{\phi_{4}\,N}}{\pi\,M^{4}_{Z}}\times$
(56)
$\displaystyle\left[\dfrac{\mathbb{Z}\left\\{2a_{q}(\frac{1}{2},\frac{2}{3})+a_{q}(-\frac{1}{2},-\frac{1}{3})\right\\}+(\mathbb{A-Z})\left\\{a_{q}(\frac{1}{2},\frac{2}{3})+2a_{q}(-\frac{1}{2},-\frac{1}{3})\right\\}}{\mathbb{A}}\right]^{2},$

where,

$\displaystyle
a_{q}(T_{3},Q^{q}_{em})=-\dfrac{g}{2\,\cos\theta_{W}}\left\\{\left(T_{3}-2\,Q^{q}_{em}\sin^{2}\theta_{W}\right)\cos\theta_{\mu\tau}+\epsilon\sin\theta_{W}\left(T_{3}-2\,Q^{q}_{em}\right)\sin\theta_{\mu\tau}\right\\}$

is the vectorial part of $\bar{q}qZ$ coupling while $\mathbb{Z}$, $\mathbb{A}$
are atomic number and mass number of the detector nucleus respectively. The
reduced mass between dark matter matter and nucleon $N$ is denoted by
$\mu_{\phi_{4}\,N}$. The spin dependent scattering cross section is given by

$\displaystyle\sigma_{\rm
SD}=\left(\dfrac{g_{2}\,\sin^{2}\theta_{D}}{2\,\cos^{2}\theta_{W}}\right)^{2}\dfrac{\mu^{2}_{\phi_{4}\,N}}{\pi\,M^{4}_{Z}}{\rm
v^{2}_{lab}}\,F_{q}^{2}\,,$ (57)

the quantity $F_{q}$ is given by

$\displaystyle F_{q}=\dfrac{b_{q}(\frac{1}{2})\left\\{\Delta^{p}_{u}\,\langle
S_{p}\rangle+\Delta^{n}_{u}\,\langle
S_{n}\rangle\right\\}+b_{q}(-\frac{1}{2})\left\\{\Delta^{p}_{d}\,\langle
S_{p}\rangle+\Delta^{n}_{d}\,\langle
S_{n}\rangle\right\\}+b_{q}(-\frac{1}{2})\left\\{\Delta^{p}_{s}\,\langle
S_{p}\rangle+\Delta^{n}_{s}\,\langle S_{n}\rangle\right\\}}{\langle
S_{p}\rangle+\langle S_{n}\rangle}$

where, $\Delta^{p(n)}_{q}$ represents the spin content of quark $q$ in
proton(neutron). The recent values of $\Delta$’s are
$\Delta^{p}_{u}=\Delta^{n}_{d}=0.84$, $\Delta^{p}_{d}=\Delta^{n}_{u}=-0.43$
and $\Delta^{p}_{s}=\Delta^{n}_{s}=-0.09$ Cheng:2012qr . The contributions of
proton and neutron to nuclear spin are denoted by $\langle S_{p}\rangle$ and
$\langle S_{n}\rangle$ respectively. For 129Xe isotope $\langle
S_{p}\rangle=0.010$ and $\langle S_{n}\rangle=0.329$ XENON:2019rxp . The
function
$b_{q}(T_{3})=\dfrac{g_{2}}{2\,\cos\theta_{W}}\left(\cos\theta_{\mu\tau}+\epsilon\sin\theta_{W}\sin\theta_{\mu\tau}\right)T_{3}$
is the axial vectorial part of $\bar{q}qZ$ coupling and ${\rm v_{lab}}\simeq
10^{-3}$ is the local velocity of dark matter with respect to the laboratory
frame. Moreover, in the present work since the dark mass range is in sub-GeV
range, the elastic scatterings with electron also transfer energy efficiently
1108.5383 . As shown in Essig:2015cda , MeV scale DM can excite electron from
valence band to conduction band and give rise to ionisation excitation.
Therefore, our dark matter can also be detected through elastic scatterings
with electron. We have calculated $\phi_{4}-e$ elastic scattering for the
range of parameters we needed for the phenomenology and we have found that it
is well below the current bound. The cross section for $\phi_{4}-e$ elastic
scattering has the following form,

$\displaystyle\sigma_{elec}$ $\displaystyle=$
$\displaystyle\left(\dfrac{g_{2}\,\sin^{2}\theta_{D}}{2\,\cos\theta_{W}}\right)^{2}\dfrac{\mu^{2}_{\phi_{4}\,e}}{\pi\,M^{4}_{Z}}\left\\{a^{2}_{e}(-\frac{1}{2},-1)+b^{2}_{e}(-\frac{1}{2}){\rm
v^{2}_{lab}}\right\\}\,,$ (58)

where the functions $a_{e}(T_{3},Q^{e}_{em})$ and $b_{e}(T_{3})$ are identical
with $a_{q}$ and $b_{q}$ for the quarks. We can easily notice that the axial
vector part (proportional to $b_{e}$) of $\bar{e}eZ$ interaction gives a
velocity suppressed contribution to $\sigma_{elec}$ as in the case for spin
dependent scattering with nuclei (Eq. (57)).

  * •

Kinetic equilibrium: In this work, although the freeze-out occurs after the
chemical imbalance for $3\rightarrow 2$ scatterings is created within the dark
sector, the kinetic equilibrium between the two sectors continues and it is
primarily possible through elastic scatterings of dark matter with $\nu_{\mu}$
and $\nu_{\tau}$ where light gauge boson $Z_{\mu\tau}$ plays an important
role.

Figure 7: The range of $g_{\mu\tau}$ and $M_{Z_{\mu\tau}}$ for which dark
matter maintains kinetic equilibrium with the SM bath through elastic
scatterings with $\nu_{\mu}$ and $\nu_{\tau}$ respectively.

In Fig 7, we show the region in $g_{\mu\tau}-M_{Z_{\mu\tau}}$ plane for four
different values of $M_{\phi_{4}}$, where kinetic equilibrium is maintained
for the dark and the visible sectors. For that we have used the condition
$\dfrac{1}{n_{\rm scatt}}\dfrac{\Gamma_{\rm el}}{\mathcal{H}}\bigg{|}_{T\simeq
M_{\phi_{4}}/20}>1$ Gorbunov:2011zz ; Gondolo:2012vh ; Biswas:2021kio , where
$\Gamma_{\rm el}=\sum_{\alpha=\mu,\,\tau}n^{\rm
eq}_{\nu_{\alpha}}\langle\sigma_{\phi_{4}\nu_{\alpha}\rightarrow\phi_{4}\nu_{\alpha}}{\rm
v}\rangle$ is the total scattering rate per dark matter and $n_{\rm
scatt}=M_{\phi_{4}}/T$ is the number of scatterings needed to transfer energy
$\sim T$ between the SM bath and $\phi_{4}$. We have compared the effective
interaction rate $\Gamma_{\rm el}/n_{\rm scatt}$ with the Hubble parameter
($\mathcal{H}$) around the freeze-out era of $\phi_{4}$ which is $\sim
M_{\phi_{4}}/20$. For completeness, in the same $g_{\mu\tau}-M_{Z_{\mu\tau}}$
plane, we have shown the allowed parameter space satisfying $(g-2)_{\mu}$ in
$2\sigma$ range by the black dots.

  * •

Relic density bound: The abundance of dark matter has been determined quite
precisely by satellite borne CMB experiments particularly the Planck
experiment. The current value of dark matter relic density is Planck:2018vyg

$\displaystyle\Omega_{\rm DM}h^{2}=0.120\pm 0.001$ (59)

  * •

Invisible Higgs Decay: We are considering sub-GeV dark matter in the present
case. As a result, the SM like Higgs boson $h_{1}$ can decay into a pair of
$\phi_{4}$ and $\phi^{\dagger}_{4}$. Moreover, $h_{1}$ can decay into a pair
of light gauge boson $Z_{\mu\tau}$ also. These additional decay modes
contribute to the invisible decay of $h_{1}$. The LHC has placed an upper
bound on the branching ratio of total invisible decay of the Higgs boson,
which is CMS:2018yfx ,

$\displaystyle{\rm Br}(h_{1}\rightarrow{\rm invisible\,\,channels})<0.19\,.$
(60)

However, the bound is easily satisfied in our model as we have considered
feeble scalar portal couplings while the other decay mode $h_{1}\rightarrow
Z_{\mu\tau}Z_{\mu\tau}$ is suppressed by $\sin^{4}\theta_{\mu\tau}$.

###  VI.1 Numerical Results

(a) $Y_{\rm DM}$ vs $x$ plot for two different values of $\xi$ (b) $Y_{\rm
DM}$ vs $x$ plot for two different values of $\theta_{D}$ (c) $Y_{\rm DM}$ vs
$x$ plot for two different $M_{\phi_{2}}$ (d) $Y_{\rm DM}$ vs $x$ plot for two
values of $M_{\phi_{4}}$

(e) $Y_{\rm DM}$ vs $x$ plot for two values of $M_{Z_{\mu\tau}}$ (f) $Y_{\rm
DM}$ vs $x$ plot for two different values of $g_{\mu\tau}$ (g) Comparison on
the effect of $3\rightarrow 2$ and $2\rightarrow 2$ scatterings in dark matter
freeze-out

Figure 8: Numerical results: Evolution of $Y_{\rm DM}$ with $x$ for various
model parameters like $\xi$, $\theta_{D}$, $M_{\phi_{2}}$, $M_{\phi_{4}}$,
$M_{Z_{\mu\tau}}$ and $g_{\mu\tau}$.

In this section we have shown our results which we have obtained by solving
the Boltzmann equation (Eq (LABEL:Boltz)) numerically. The solution of Eq.
(LABEL:Boltz) is shown in Fig. 8 where we have demonstrated the evolution of
$Y_{\rm DM}$ with $x$ for different model parameters. In all these plots of
Fig. 8, the red solid line is the solution of the Boltzmann equation for the
following set of model parameters $\xi=0.501\times 10^{-2}$,
$\theta_{D}=-0.071$, $M_{\phi_{2}}=2.27$ TeV, $M_{\phi_{4}}=0.182$ GeV,
$g_{\mu\tau}=6.27\times 10^{-4}$ and $M_{Z_{\mu\tau}}=58.5$ MeV, which
reproduces the correct relic density. In plot (a) of Fig. 8, we have shown how
the era of freeze-out and the final abundance both change when we increase
$\xi$ from $0.501\times 10^{-2}$ to 0.0501 and it is indicated by the green
solid line. As the $3\rightarrow 2$ scattering cross section increases with
$\xi$ this results in a delayed freeze-out with a reduced final abundance. In
plot (b), we have demonstrated the effect of increasing $\theta_{D}$ on
$Y_{\rm DM}$. We have found that the change in mixing angle $\theta_{D}$ has a
similar effect on $Y_{\rm DM}$ as it is shown in plot (a) for the parameter
$\xi$. However, in this case $Y_{\rm DM}$ does not decrease as much as it is
for the parameter $\xi$ for one order increase in magnitude of $\theta_{D}$.

In plot (c) we have shown the impact of $M_{\phi_{2}}$ on $Y_{\rm DM}$. Here
we have increased the value of $M_{\phi_{2}}$ from $2.27$ TeV to 22.7 TeV and
corresponding $Y_{\rm DM}$ has been indicated by the green solid line. Any
increase in $M_{\phi_{2}}$ enhances the magnitude of cubic coupling444Since
$\theta_{D}$ is -ve, which represents a rotation in the reverse direction than
in Eq. (27), the coefficient $\mu$ is a +ve number. $\mu$ as
$\mu=-\dfrac{\left(M^{2}_{\phi_{2}}-M^{2}_{\phi_{4}}\right)\sin
2\theta_{D}}{\sqrt{2}v}$ with $M_{\phi_{2}}>>M_{\phi_{4}}$, which eventually
increases the trilinear interaction among $\phi_{1}$, $\phi_{1}$ and
$h_{1}(h_{3})$ and hence the scattering cross section $\sigma^{\rm
tot}_{3\rightarrow 2}$. The effect of the mass of dark matter on $Y_{\rm DM}$
has been demonstrated in plot (d) where we have considered
$M_{\phi_{4}}=0.182$ GeV (red solid line) and 1.82 GeV (green solid line)
respectively. Plots (e) and (f) show the dependence of $L_{\mu}-L_{\tau}$
gauge coupling $g_{\mu\tau}$ and gauge boson mass $M_{Z_{\mu\tau}}$ on $Y_{\rm
DM}$. It is seen from both the plots that any increase in $g_{\mu\tau}$ and
$M_{Z_{\mu\tau}}$ has opposite effect on $Y_{\rm DM}$ respectively and it is
determined by the corresponding change in $v_{\mu\tau}$ that appears in the
couplings. Finally, in plot (g) we have shown the effect of $2\rightarrow 2$
and $3\rightarrow 2$ scatterings on the freeze-out of dark matter. Here the
red solid line represents a situation when both $3\rightarrow 2$ as well as
$2\rightarrow 2$ scatterings are present and the dark matter freezes-out
around $x\simeq 20$. However, if we switch off the $3\rightarrow 2$
interactions, the freeze-out of dark matter occurs a lot earlier ($x\simeq
2$). It is due to the reason that the cross sections of $2\rightarrow 2$
scatterings are not as large as that of the $3\rightarrow 2$ scatterings which
are predominantly responsible for the number changing processes.

(a) Variation of $\Omega_{\phi_{4}}h^{2}$ with $\xi$ for three different
values of $M_{\phi_{4}}$. (b) Variation of $\Omega_{\phi_{4}}h^{2}$ with
$M_{\phi_{4}}$ for four different values of $\theta_{D}$. (c)
$\Omega_{\phi_{4}}h^{2}$ vs $\theta_{D}$ for three different values
$M_{\phi_{4}}$.

Figure 9: Variation of relic density with relevant model parameters

The variation of relic density with three most relevant parameters $\xi$,
$M_{\phi_{2}}$ and $\theta_{D}$ is depicted in Fig. 9. In plot (a), dependence
of $\Omega_{\phi_{4}}h^{2}$ with $\xi$ has been shown for three different
values of $M_{\phi_{4}}$. Analogous to the previous plot in Fig. 8(a), here
also we have noticed a similar behaviour except for $\xi<10^{-3}$, as the
relic density decreases sharply with the increase of quartic coupling $\xi$
due to enhancement of $\sigma^{\rm tot}_{3\rightarrow 2}$. In plot (b), the
effect of $M_{\phi_{2}}$ on $\Omega_{\phi_{4}}h^{2}$ is shown for different
values of $\theta_{D}$. We can see that for low value of mixing angle
($|\theta_{D}|=\pi/180$ rad), the relic density is almost insensitive to the
mass of $\phi_{2}$. However, as we increase the magnitude of $\theta_{D}$,
$\Omega_{\phi_{4}}h^{2}$ decreases with $M_{\phi_{2}}$ replicating the
situation shown in Fig. 8(c). The last figure in plot (c) demonstrates
$\Omega_{\phi_{4}}h^{2}$ as a function of $\theta_{D}$. Here, three lines are
for three different values of $M_{\phi_{4}}$ and the nature of all three lines
are exactly identical to each other, i.e. the relic density is independent of
the dark sector mixing angle for $|\theta_{D}|\lesssim 0.1$ rad and thereafter
it starts decreasing with the increase of magnitude of $\theta_{D}$. The
difference in magnitude of $\Omega_{\phi_{4}}h^{2}$ in these three lines for
different $M_{\phi_{4}}$ originates from two factors. The relic density is
proportional to both $M_{\phi_{4}}$ and $Y_{\rm DM}$ where the latter also
gets enhanced with $M_{\phi_{4}}$ as shown in Fig. 8(d).

Figure 10: Left panel: Allowed parameter space is $\xi-M_{\phi_{4}}$ plane for
different constraints. Right pane: Variation of $\sigma_{\rm
self}/M_{\phi_{4}}$ with $M_{\phi_{4}}$.

In Fig. 10, we show our allow parameter space in $\xi-M_{\phi_{4}}$ plane. In
order to obtain this we have scanned over the parameters in the following
range

$\displaystyle\begin{array}[]{cccccc}10^{-5}&\leq&\xi&\leq&1.0\,\,,\\\
10^{-3}\,\,{\rm rad}&\leq&|\theta_{D}|&\leq&0.1\,\,{\rm rad}\,\,,\\\
10^{-4}&\leq&g_{\mu\tau}&\leq&10^{-2}\,\,,\\\ 10^{-3}\,\,{\rm
GeV}&\leq&{M_{Z_{\mu\tau}}}&\leq&1.0\,\,{\rm GeV}\,\,,\\\ 10^{3}\,\,{\rm
GeV}&\leq&{M_{\phi_{4}}}&\leq&10^{4}\,\,{\rm GeV}\,\,,\\\ 10^{-2}\,\,{\rm
GeV}&\leq&{M_{\phi_{2}}}&\leq&10.0\,\,{\rm GeV}\,\,,\end{array}$ (67)

and have imposed necessary constraints one by one. The result is shown in the
left panel of Fig. 10. In this plot, the blue dots describe a region in $\xi-
M_{\phi_{4}}$ plane that reproduces the correct dark matter relic density in
$3\sigma$ range as determined by the Planck experiment. On top of that, we
have imposed bound from dark matter self-interaction $0.1\,\,{\rm
cm^{2}/g}\leq\frac{\sigma_{\rm self}}{M_{\phi_{4}}}\leq 10\,\,{\rm
cm^{2}}/{\rm g}$. The resultant parameter space is indicated by the red square
shaped points. Finally, we have introduced another constraint coming from the
unitarity limit of scattering amplitudes as mentioned in Eq. (55). The
parameter space satisfying all three constraints is shown by the green diamond
shaped points. We can notice that in order to satisfy these three constraints
we need $M_{\phi_{4}}\lesssim 200$ MeV while the corresponding quartic
coupling is restricted to be $\lesssim 1$. The similar result has also been
presented in a different manner in the right panel of Fig. 10. Here we have
shown the variation of $\sigma_{\rm self}/M_{\phi_{4}}$ with $M_{\phi_{4}}$
and the corresponding value of the parameter $\xi$ has been indicated by the
colour bar. The only constraint applied in this plot is that each and every
point in $\sigma_{\rm self}/M_{\phi_{4}}-M_{\phi_{4}}$ plane satisfies the
relic density bound i.e. $0.117\leq\Omega_{\phi_{4}}h^{2}\leq 0.123$.

Figure 11: Spin independent (left panel) and spin dependent (right panel)
scattering cross sections with exclusion limits on $\sigma_{\rm SI}$ from
various ongoing as well as future experiments.

The spin independent and spin dependent elastic scattering cross sections are
calculated using Eqs. (56, 57) for the parameter range given in Eq. (67). The
results are shown in both panels of Fig. 11. In the left panel, we demonstrate
$\sigma_{\rm SI}$ as a function of $M_{\phi_{4}}$ while the corresponding spin
dependent cross sections $\sigma_{\rm SD}$ are given in the right panel. In
each plot, the colour bar provides the value of dark mixing angle
$\theta_{D}$. We can easily notice that $\sigma_{\rm SD}$ for a particular
value of $M_{\phi_{4}}$ is almost $10^{-6}$ times smaller than the
corresponding $\sigma_{\rm SI}$ and it is primarily due to the reason that
$\sigma_{\rm SD}$ is suppressed by ${\rm v^{2}_{\rm lab}}$ (Eq. (57)) where
${\rm v}_{\rm lab}\simeq 10^{-3}$ is the local velocity of dark matter
particles with respect to the laboratory frame. In the left panel, we have
shown existing and future bounds from various direct detection experiments for
comparison. In the low mass region of $M_{\phi_{4}}\leq 100$ MeV, we have
exclusion limit on $\sigma_{\rm SI}$ from NEWS-G Durnford:2021mzg  and it has
been indicated by the black dashed dot dot line. The current bounds from two
other low mass dark matter experiments namely CDMSlite SuperCDMS:2017nns  and
DarkSide-50 DarkSide:2018bpj  are shown by the dashed and the dashed dot lines
respectively. The upper limit on $\sigma_{\rm SI}$ from “GeV-TeV” scale
experiment like Xenon1T XENON:2018voc , which has a very small overlap with
our considered range of dark matter mass, has also been depicted by the black
solid line. Moreover, the future prediction from DARWIN DARWIN:2016hyl  is
shown by the dashed line. On the other hand, for the spin dependent
scattering, the most recent exclusion limit is from XENON1T experiment
XENON:2019rxp . However, we have not shown that in the right panel as the
largest value of $\sigma_{\rm SD}\sim 10^{-49}$ cm2 for 1 GeV$\lesssim
M_{\phi_{4}}\lesssim$ 10 GeV lies several orders of magnitude below than the
current limit from XENON1T where the maximum exclusion is $6.3\times 10^{-42}$
cm2 for the dark matter mass of 30 GeV at 90% C.L. XENON:2019rxp .

##  VII Collider Signature

Figure 12: Charged particle production at the $e^{+}e^{-}$ collider with dark
matter $\phi_{4}$ as one of the intermediate states.

In this work, we have studied the pair production of charged particles
($\chi^{+}\chi^{-}$) defined as $\chi=\chi_{L}\oplus\chi_{R}$. The produced
$\chi^{\pm}$ subsequently decays into lepton and SIMP dark matter i.e.
$e^{+}e^{-}\rightarrow\chi^{+}(e^{+}\phi_{4})\chi^{-}(e^{-}\phi_{4})\rightarrow
e^{+}e^{-}\cancel{E}_{T}$. The Feynman diagrams contributing in the signal are
mediated by $\gamma$, $Z$ and $\phi_{4}$ respectively and have been displayed
in Fig. 12. We have studied the signal at two different $e^{+}e^{-}$ colliders
namely Compact Linear Collider (CLIC) CLICPhysicsWorkingGroup:2004qvu ;
Dannheim:2012rn ; Linssen:2012hp ; CLICDetector:2013tfe ; CLICdp:2017vju ;
Aicheler:2018arh  and International Linear Collider (ILC) Behnke:2013xla ;
Adolphsen:2013kya ; Behnke:2013lya ; Baer:2013cma ; Adolphsen:2013jya . In the
former case, we have considered centre of mass (c.o.m) energy $\sqrt{s}=380$
GeV and $3000$ GeV while $\sqrt{s}=500$ GeV and $1000$ GeV for the latter at
the time of the pair production of vector like fermion. Depending on the c.o.m
energy of the collider, we have an upper bound on the mass of $\chi^{\pm}$
upto which it can be produced. In particular, in the present work we have
investigated the signal at the detector level for c.o.m energy $\sqrt{s}=1000$
GeV and $3000$ GeV of the $e^{+}e^{-}$ linear collider. Although there is no
dedicated search for the present model at the CMS or ATLAS detector, still the
same kind of signal can be produced at the hadron collider. We have produced
the $\chi^{+}\chi^{-}$ final state at the $pp$ collider using MadGraph
Alwall:2014hca ; Alwall:2011uj  for $\sqrt{s}=13$ TeV and find that this is
lower than the current exclusion limit given by the CMS collaboration for the
13 TeV run of LHC with 35.9 $fb^{-1}$ integrated luminosity CMS:2018xqw . This
has been displayed in Fig. 13 where the red points correspond to the upper
limit on the pair production cross section of the singly charged fermion
coming from the study of 13 TeV run of LHC by CMS whereas the blue cross
points correspond to the cross section for the present model at the $pp$
collider mediated by gauge bosons like $\gamma$ and $Z$. Therefore, we
conclude that the charge particle mass range we have considered in the present
model is safe from the LHC bound. In contrary to the $pp$ collider, at
$e^{+}e^{-}$ collider the signal $e^{+}e^{-}\cancel{E}_{T}$ has an additional
$t-$ channel diagram mediated by the MeV scale SIMP dark matter $\phi_{4}$.
This $t-$channel diagram enhances the cross section by an order of magnitude
larger than the $s$-channel diagrams mediated by $\gamma$ and $Z$ gauge
bosons.

Figure 13: Production of $\chi^{+}\chi^{-}$ at the 13 TeV pp collider. Red
line corresponds the bound for the production of singly charged fermion when
we look for the $l^{+}l^{-}+$ MET signal at the final state. Any model which
predicts the production rate above the red line is already ruled out.

In accomplishing the collider analysis for the present model, we have
considered cut based analysis using a series of sophisticated packages
available for the collider study. In particular, we have used FeynRules
Alloul:2013bka  for implementing the present model and generated the UFO model
files which fed into the MadGraph Alwall:2014hca ; Alwall:2011uj
subsequently. We have then used MadGraph for generating the parton level
process. For showering we have used inbuilt PYTHIA Sjostrand:2006za  in the
MadGraph and finally for the event analysis we use DELPHES package
deFavereau:2013fsa ; Selvaggi:2014mya ; Mertens:2015kba .

Figure 14: Left panel shows the pair production of $\chi^{+}$ and $\chi^{-}$
for different c.o.m energy for two values of $\chi^{\pm}$ mass at the
$e^{+}e^{-}$ lepton collider. Right panel shows the variation of $\chi^{+}$
and $\chi^{-}$ production rate with $\chi^{\pm}$ mass $M_{\chi^{\pm}}$ for
different values of c.o.m energies.

In the left and right panels of Fig. (14), we have shown the variation of
$\chi^{+}\chi^{-}$ production cross section at $e^{+}e^{-}$ collider with the
centre of mass energy of the collider for two different values of $\chi^{\pm}$
mass and the mass of the charged particle for the different centre of mass
energies respectively. In the left panel, blue dashed line corresponds to
$M_{\chi}=300$ GeV and the red dashed-dot line corresponds to $M_{\chi}=600$
GeV and both the line varies inversely with the center of mass energy
$\sqrt{s}$. For $M_{\chi^{+}}=600$ GeV (red dash-dotted line), we can see that
$\chi^{+}\chi^{-}$ starts producing when the c.o.m energy of the $e^{+}e^{-}$
reached a minimum threshold energy which is $\sqrt{s}=1200$ GeV. For the
higher values of the c.o.m energy, we can see that blue dashed and red dashed-
dot lines coincide with each other implies that the production cross section
does not depend on mass for higher c.o.m values. Whereas the right panel shows
the variation of $\chi^{+}\chi^{-}$ production cross section with the mass
$M_{\chi^{+}}$ for different values of the c.o.m energy. One can easily see
that the production cross section decrease with the increasing value of the
c.o.m energy which is consistent with the discussion of the left panel. The
different c.o.m energies are the proposed experimental set up for CLIC and ILC
colliders. When the $\chi^{+}$ mass reached the threshold value i.e.
$M_{\chi^{\pm}}\sim\frac{\sqrt{s}}{2}$, there is a sharp fall in the
production cross section. In the later part of the draft, we have done the
collider analysis for $\sqrt{s}=1000$ GeV and $\sqrt{s}=3000$ GeV c.o.m
energy.

###  VII.1 Analysis

In the present work, we are interested in the signal which contains opposite
sign di-electrons ($e^{+}e^{-}$) and transverse missing energy
($\cancel{E}_{T}$) in the final state. In discovering the signal from the
present model, same kind of signal morphology can appear in the final state
from the known SM backgrounds. The dominant backgrounds which can mimic the
signal are as follows,

  1. 1.

At the electron positron collider, the dominant background which can mimic the
signal is $e^{+}e^{-}\rightarrow e^{+}e^{-}Z$, where $Z$ can decay to
$\nu_{l}\bar{\nu}_{l}$. Therefore, finally it gives $e^{+}e^{-}\cancel{E}_{T}$
which exactly resembles the signal. This kind of background also includes $ZZ$
production channel which subsequently decays to electrons ($e^{+}e^{-}$) and
neutrinos ($\nu_{l}\bar{\nu}_{l}$).

  2. 2.

Another dominant background can come from the pair production of $W^{+}W^{-}$
mode at the $e^{+}e^{-}$ collider. The W-boson subsequently decays
leptonically to leptons and neutrinos and can replicate the signal as
$e^{+}e^{-}\rightarrow W^{+}(\rightarrow l^{+}\nu_{l})W^{-}(\rightarrow
l^{-}\bar{\nu}_{l})\rightarrow e^{+}e^{-}\cancel{E}_{T}$ .

  3. 3.

Another potentially relevant background is the pair production of $t\bar{t}$
mode which can also mimic the signal when $t$ quark decays leptonically
associated with two b-quarks. This background can mock the signal at the
electron positron collider as $e^{+}e^{-}\rightarrow t(\rightarrow
b\,l^{+}\nu_{l})\bar{t}(\rightarrow\bar{b}\,l^{-}\bar{\nu}_{l})\rightarrow
b\bar{b}\,e^{+}e^{-}\cancel{E}_{T}$. As we will see, this kind of signal is
easy to avoid with the $b$-tagging.

At the time of generating the events, we have not put any veto to forbid the
processes contains other particles than the leptons and missing energy. We can
put b-veto which will reduce $t\bar{t}$ background. In Figs. (15, 16), we have
shown the variation of backgrounds and signals about the different kinematical
variables namely transverse momentum of the leading electron ($P^{e_{1}}_{T}$)
and second leading electron ($P^{e_{2}}_{T}$), pseudorapidity of the leading
electron ($\eta_{e_{1}}$) and transverse missing energy ($\cancel{E}_{T}$).
From the figures, we can choose the values of the kinematical variables which
will prefer the signal over backgrounds. In this work, we have used the
following kind of cuts on the kinematical variables in order to reduce the
backgrounds without affecting the signal much. The details of the cuts on the
kinematical variables are as follows,

  1. A0.

We have considered the events which contain opposite sign di-electron
($e^{+}e^{-}$) and transverse missing energy ($\cancel{E}_{T}$). We have also
put the minimal cut on the transverse momentum of the electrons which is
$p^{min}_{T,e}\geq 10$ GeV. We have collected the events which satisfy the
pseudorapidity of the electrons $\eta_{e}<2.5$. These cuts have been
implemented at the time of the partonic generation of the events.

  2. A1.

We consider events that have opposite sign di-electron pair ($e^{+}e^{-}$) in
the final state.

  3. A2.

From the left panel of Fig. (15), we can see that if we put strong cut on the
leading electron $P^{e_{1}}_{T}\geq 130$ GeV then we can reduce the $t\bar{t}$
background. We have chosen relatively soft cut on the second leading electron
which is $P^{e_{2}}_{T}\geq 60$ GeV .

  4. A3.

To reduce the background which comes from $ZZ$ mode, we have used $Z-veto$.
$Z-veto$ means, we have accepted the the events which violate the condition
$|m_{ee}-91.2|<10$ GeV, where $m_{ee}$ is the di-electron invariant mass.

  5. A4.

In order to eliminate the $t\bar{t}$ background, we have implemented $b-$veto.
This implies we have rejected the events which contains $b-$quarks in the
final state.

  6. A5.

From the left panel of Fig. (16), we can see that background and signal peak
at different values of pseudorapidity. Therefore, to reduce the background
without affecting the signal much we have considered the events which have
pseudorapidity in the range, $|\eta^{e}|<1.5$.

  7. A6.

From the right panel of Fig. (16), we can see that if we implement a cut on
the transverse missing energy then background can be reduced significantly. We
have adapted the missing energy cut which is $\cancel{E}_{T}>160$ GeV.

Figure 15: Left panel shows the variation of fraction of events distribution
for backgrounds and signal with the transverse momentum of leading electron
whereas the right panel shows the variation with respect to the rapidity of
the leading electron.

Figure 16: Variation of fraction of events for backgrounds and signal with respect to transverse missing energy and invariant mass of the di-electron in the left and right panels respectively. | SM Backgrounds at 1 TeV ILC
---
Channels | Cross section (fb)
$e^{+}e^{-}Z\,(\rightarrow\nu_{l}\bar{\nu_{l}})$ | $15.59$
$W^{+}(\rightarrow e^{+}\nu_{l})\,W^{-}(\rightarrow e^{-}\bar{\nu_{l}})$ | $13.24$
$t(\rightarrow be^{+}\nu_{l})\,\bar{t}(\rightarrow\bar{b}e^{-}\bar{\nu_{l}})$ | $1.61$
Total Backgrounds |
| Effective Cross section after applying cuts (fb)
---
A0 + A1 |  A2 |  A3 |  A4 |  A5 |  A6
$10.07$ | $6.07$ | $5.88$ | $5.88$ | $3.92$ | $2.43$
$8.88$ | $6.25$ | $6.25$ | $6.25$ | $4.28$ | $1.57$
$0.73$ | $0.20$ | $0.20$ | $0.05$ | $0.04$ | $0.02$
|  |  |  |  | 4.02
Table 4: Cut-flow table for the obtained cross sections corresponding to the different SM backgrounds. See the text for the details of the cuts A0-A6. The c.m.energy is $\sqrt{s}=1$ TeV, relevant for ILC. | SM Backgrounds at $3$ TeV CLIC
---
Channels | Cross section (fb)
$e^{+}e^{-}Z\,(\rightarrow\nu_{l}\bar{\nu_{l}})$ | $6.18$
$W^{+}(\rightarrow e^{+}\nu_{l})\,W^{-}(\rightarrow e^{-}\bar{\nu_{l}})$ | $1.44$
$t(\rightarrow be^{+}\nu_{l})\,\bar{t}(\rightarrow\bar{b}e^{-}\bar{\nu_{l}})$ | $0.19$
Total Backgrounds |
| Effective Cross section after applying cuts (fb)
---
A0 + A1 |  A2 |  A3 |  A4 |  A5 |  A6
$2.95$ | $2.85$ | $2.83$ | $2.83$ | $1.06$ | $0.82$
$0.74$ | $0.73$ | $0.73$ | $0.73$ | $0.36$ | $0.27$
$0.01$ | $0.004$ | $0.004$ | 0.001 | 0.0006 | 0.0005
|  |  |  |  | 1.09
Table 5: Cut-flow table for the obtained cross sections corresponding to the
SM backgrounds. The details of the cuts A0-A6 are mentioned in the text. We
perform the simulation for $3$ TeV CLIC.

In Tables (4,5), we have shown the survival of the backgrounds for ILC
($\sqrt{s}=1$ TeV) and CLIC ($\sqrt{s}=3$ TeV) colliders, respectively. In
Table (6), we have shown the response of the signal production cross section
after applying different cuts, A0 to A6. We can see that the cuts are
effective in lowering the backgrounds and at the same time cuts reduce the
signal cross section less significantly than the backgrounds. In determining
the statistical significance of signal over background, we have used Eq. (68).
in Eq. (68), $s$ corresponds to number of events 555We determine the number of
events by multiplying the cross section with the luminosity. for signal after
applying all the cuts and $b$ is the number of background events after
applying all the cuts,

$\mathcal{S}=\sqrt{2\times\left[(s+b){\rm ln}(1+\frac{s}{b})-s\right]}.$ (68)

Signal at $e^{+}e^{-}$ Collider | Effective CS after cuts (fb) | Stat Significance ($\mathcal{S}$)
---|---|---
Experiment | Mass (GeV) | $\beta_{e\chi}$ | CS (fb) | A0+A1 | A2 | A3 | A4 | A5 | A6 | $\mathcal{L}=1~{}{\rm fb^{-1}}$ |  $\mathcal{L}=10$ ${\rm fb^{-1}}$
1 TeV ILC | 350.0 | 0.1 | 41.20 | 29.60 | 24.03 | 23.73 | 23.73 | 22.57 | 19.53 | 6.65 | 21.03
450.0 | 0.1 | 29.56 | 21.30 | 19.83 | 19.60 | 19.60 | 19.50 | 17.38 | 6.11 | 19.32
3 TeV CLIC | 600.0 | 0.1 | 5.13 | 2.91 | 2.78 | 2.78 | 2.78 | 2.16 | 2.05 | 1.60 | 5.04
700.0 | 0.1 | 5.47 | 3.08 | 2.99 | 2.99 | 2.99 | 2.45 | 2.34 | 1.78 | 5.64
Table 6: Cut-flow table of signal cross section at ILC ($\sqrt{s}=1$ TeV) and
CLIC ($\sqrt{s}=3$ TeV). We have considered
$\beta^{L}_{e\chi}=\beta^{R}_{e\chi}=\beta_{e\chi}=0.1$ and different values
of $\chi^{\pm}$ mass.

The last two columns in Table 6 corresponds to the statistical significance of
the signal. For $1$ TeV collider, we can see that the present model can have
more than $6\sigma$ significance for $1\,\,fb^{-1}$ luminosity. For $3$ TeV
collider, we need $10\,\,fb^{-1}$ luminosity in order to get the $5\sigma$
statistical significance for the signal. We can see that the current model can
be tested at the very early run of the ILC and CLIC colliders.

Figure 17: Left and right panels show the allowed regions (by cyan colour
points) after satisfying the $(g-2)_{e}$ in $\beta^{L}_{e\chi}-M_{\chi}$ plane
for 1 TeV and 3 TeV lepton colliders, respectively. The solid and dashed lines
correspond to the variation of $3\sigma$ and $5\sigma$ statistical
significance lines of $e^{+}e^{-}\cancel{E}_{T}$ signal with the charged
singlet fermion mass for the luminosity 1 $fb^{-1}$ and 10 $fb^{-1}$,
respectively. The other parameters kept fixed at
$\beta^{R}_{e\chi}=\beta^{L}_{e\chi}$, $M_{\phi_{4}}=100$ MeV,
$\theta_{D}=0.1$ and $M_{\phi_{2}}$ has been varied between 1 to 10 TeV.

In Fig. (17), we have shown the scatter plots in the
$M_{\phi_{4}}-\beta^{L}_{e\chi}$ plane after satisfying $(g-2)_{e}$ by the
cyan colour points. In the figure, the left panel corresponds to the 1 TeV ILC
collider whereas the right panel is for the 3 TeV CLIC collider. The variation
among the cyan colour points are due to the variation of $\phi_{2}$ mass,
$M_{\phi_{2}}$, which we have considered in the range $1-10$ TeV. We have kept
fixed the other parameters as mentioned in the caption of the figure. We see
from both the panel that a sharp correlation between $M_{\chi}$ and
$\beta^{L}_{e\chi}$ after satisfying the $(g-2)_{e}$ in the range $\Delta
a_{e}=(-8.7\pm 3.6)\times 10^{-13}$. Moreover, the parameters which we have
varied for $(g-2)_{e}$ also affect the production cross section of
$\chi^{+}\chi^{-}$ channel at $e^{+}e^{-}$ collider. We have displayed the
$1\sigma$ and $3\sigma$ statistical significance lines of the signal,
$e^{+}e^{-}\rightarrow e^{+}e^{-}\cancel{E}_{T}$, by the solid and dashed
lines, respectively, whereas the red and blue colors on them imply the
$1\,fb^{-1}$ and $10\,fb^{-1}$ luminosity. It is easily understood from the
figure that if we increased the luminosity keeping statistical significance
fixed then we can access the lower values of $\beta^{L}_{e\chi}$ as well. As
exhibited in the right panel of Fig. (14), the production cross section of
$\chi^{+}\chi^{-}$ with the charged fermion mass, $M_{\chi}$, is flat except
at the kinematical threshold limit $M_{\chi}\lesssim\frac{\sqrt{s}}{2}$ where
the production cross section falls abruptly. In Table 6, we have shown the
signal strength remains after applying different cuts $A0-A6$ for two
benchmark points. Therefore, from the table, we can determine the average
value of cut efficiency for the signal which is the ratio of the signal cross
section after applying the A6 cut and the production cross section without any
cuts. We find the average value of the cut efficiency for 1 TeV collider is
0.53 whereas for 3 TeV, it is 0.43. We used these values in finding the
$1\sigma$ and $3\sigma$ isocontours of statistical significance for the signal
in the $M_{\chi}-\beta^{L}_{e\chi}$ plane. We see that we need constant value
of $\beta^{L}_{e\chi}$ for wide range of $\chi^{\pm}$ mass. Moreover, one can
notice that we need higher values of $\beta^{L}_{e\chi}$ in the kinematical
threshold region ($M_{\chi}\simeq\frac{\sqrt{s}}{2}$, $s$ is c.o.m energy)
limit because the production cross section of $\chi^{+}\chi^{-}$ sharply falls
there. Additionally, the most appealing thing to be noticed here is that we
have overlap region in the $M_{\chi}-\beta^{L}_{e\chi}$ plane which gives us
the correct value of $(g-2)_{e}$ and the $\geq 3\sigma$ statistical
significance of the signal over background.

##  VIII Summary and conclusion

In this work, we have extended the minimal U(1)${}_{L_{\mu}-L_{\tau}}$ model
by a scalar doublet ($\Phi^{\prime}_{2}$), a singlet scalar
($\Phi^{\prime}_{4}$) and a vector like singlet fermion $\chi$ to address the
deviations found in experiments from the theoretical predictions of anomalous
magnetic moments for both muon and electron. All these new fields have
specific ${L_{\mu}-L_{\tau}}$ charges as required by the new Yukawa
interactions. We have shown that in the minimal model, considering the present
bounds on $Z_{\mu\tau}$ from various experiments particularly from Borexino,
it is not possible to address both these anomalies simultaneously. Apart from
the one loop contribution due to the neutral gauge boson $Z_{\mu\tau}$ similar
to the minimal ${L_{\mu}-L_{\tau}}$ model, the additional contribution coming
from one loop diagrams involving $\chi$ and $\phi_{2}$ or $\phi_{4}$ provide
the deficit in $(g-2)_{e}$ as required by the experimental data when the
relevant parameters remain within the following range
$\beta^{L}_{e\chi}\gtrsim 5\times 10^{-2}$, $10^{-3}\,\,{\rm
rad}\lesssim|\theta_{D}|\lesssim 0.1$ rad and $1\,\,{\rm TeV}\leq
M_{\phi_{2}}\leq$ 10 TeV. Interestingly, in order to achieve this, we also
have a natural SIMP dark matter candidate $\phi_{4}$, an admixture of
$\Phi^{\prime}_{4}$ and $\phi^{\prime}_{2}$ (neutral part of
$\Phi^{\prime}_{2}$), the signature of which can be found as missing energy at
the upcoming $e^{+}e^{-}$ linear colliders like ILC and CLIC. In order to
explore the dynamics of dark matter in detail we have considered all possible
theoretical and experimental constraints arising from dark matter self
interaction, perturbativity and unitarity, spin dependent and spin independent
elastic scatterings, kinetic equilibrium between dark and visible sectors,
Higgs invisible decay branching and the relic density bound. The kinetic
equilibrium of the SIMP dark matter with the SM bath is possible due to the
elastic scatterings with $\nu_{\mu}$ and $\nu_{\tau}$ where $Z_{\mu\tau}$
plays the role of mediator. We have shown that the parameter space in
$g_{\mu\tau}-M_{Z_{\mu\tau}}$ plane satisfying $(g-2)$ is fully consistent
with the range of $g_{\mu\tau}$ and $M_{Z_{\mu\tau}}$ necessary for
maintaining kinetic equilibrium of dark matter with the SM bath.

The characteristics of the SIMP dark matter is achieved through the number
changing $3\rightarrow 2$ processes like
$\phi_{4}\phi_{4}\phi_{4}\rightarrow\phi^{\dagger}_{4}\phi_{4}$,
$\phi^{\dagger}_{4}\phi_{4}\phi_{4}\rightarrow\phi^{\dagger}_{4}\phi^{\dagger}_{4}$
which supersede the contribution coming from $2\rightarrow 2$ processes
($\phi_{4}\phi_{4}\rightarrow f\bar{f}$) due to the appearance of the
$\phi^{3}_{4}$ term when $\Phi^{\prime}_{3}$ gets a VEV $v_{\mu\tau}$ and
breaks the $L_{\mu}-L_{\tau}$ symmetry. Therefore, the symmetry breaking scale
is involved in the freeze-out dynamics of dark matter and thereby determining
the final abundance of $\phi_{4}$. We have found that we need an MeV scale
dark matter with $M_{\phi_{4}}\lesssim 200$ MeV to satisfy the unitarity bound
which at the same time being consistent with the self interaction limit
$0.1\,\,{\rm cm^{2}/g}\leq\frac{\sigma_{\rm self}}{M_{\phi_{4}}}\leq
10\,\,{\rm cm^{2}}/{\rm g}$ and the relic density bound
$0.117\leq\Omega_{\phi_{4}}h^{2}\leq 0.123$ considered in this work. Moreover,
we have also searched for the collider signature of the charged fermion
($\chi^{\pm}$) at the $e^{+}e^{-}$ linear colliders. For the present model, at
$e^{+}e^{-}$ collider, we have an additional $t$-channel diagram, in
comparison to the hadron collider, mediated by the MeV scale SIMP dark matter,
which enhances $\chi^{+}\chi^{-}$ pair production cross section. The produced
$\chi^{\pm}$ can decay as $\chi^{\pm}\rightarrow e^{\pm}\phi_{4}$. Therefore,
we have studied an opposite sign di-electron and missing energy
($e^{+}e^{-}\cancel{E}_{T}$) signal at the final state. After investigating
the relevant backgrounds which can mimic the present signal and performing the
cut based analysis of signal and backgrounds we find that TeV scale
$\chi^{\pm}$ can be detected at the early run of $e^{+}e^{-}$ collider with
$\geq 3\sigma$ statistical significance for luminosity as low as 10 $fb^{-1}$.
We have also discussed the compatible region in the
$\beta^{L}_{e\chi}-M_{\chi}$ parameter space which can simultaneously explain
the $(g-2)_{e}$ and also demands $\geq 3\sigma$ statistical significance for
the signal. Therefore, upon conclusion, our model can accommodate both
$(g-2)_{e,\mu}$, neutrino masses and mixings, a natural SIMP dark matter and
also an interesting collider imprint of the dark sector including the charged
fermion $\chi^{\pm}$ at the upcoming $e^{+}e^{-}$ linear colliders.

## Acknowledgement

The authors would like to thank Jinsu Kim for the useful discussion. This work
used the Scientific Compute Cluster at GWDG, the joint data center of Max
Planck Society for the Advancement of Science (MPG) and University of
Göttingen.

## Appendix

##  Appendix A Necessary Vertex factors

In this appendix we have listed the necessary vertex factors.

$\displaystyle\phi_{4}\phi_{4}\phi_{4}:-3\sqrt{2}\,\xi
v_{\mu\tau}\cos^{3}\theta_{D}\,,$ (69)
$\displaystyle\phi^{\dagger}_{4}\phi^{\dagger}_{4}\phi^{\dagger}_{4}:-3\sqrt{2}\,\xi
v_{\mu\tau}\cos^{3}\theta_{D}\,,$ (70)
$\displaystyle\phi_{4}\phi_{4}\phi^{\dagger}_{4}\phi^{\dagger}_{4}:-4\left(\lambda_{4}\,\cos^{4}\theta_{D}+\lambda_{2}\,\sin^{4}\theta_{D}+\lambda_{24}\sin^{2}\theta_{D}\cos^{2}\theta_{D}\right)\,,$
(71) $\displaystyle
h_{1}\phi^{\dagger}_{2}\phi_{4}:-\dfrac{1}{2}\cos\theta\left(\sqrt{2}\,\mu\cos
2\theta_{D}+v(\lambda_{12}+\lambda^{\prime}_{12}-\lambda_{14})\sin
2\theta_{D}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+v_{\mu\tau}(\lambda_{23}-\lambda_{34})\cos\theta_{D}\sin\theta_{D}\sin\theta\,,$
(72) $\displaystyle
h_{1}\phi_{2}\phi^{\dagger}_{4}:-\dfrac{1}{2}\cos\theta\left(\sqrt{2}\,\mu\cos
2\theta_{D}+v(\lambda_{12}+\lambda^{\prime}_{12}-\lambda_{14})\sin
2\theta_{D}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+v_{\mu\tau}(\lambda_{23}-\lambda_{34})\cos\theta_{D}\sin\theta_{D}\sin\theta\,,$
(73) $\displaystyle
h_{3}\phi^{\dagger}_{2}\phi_{2}:-v_{\mu\tau}\cos\theta\left(\lambda_{23}\cos^{2}\theta_{D}+\lambda_{34}\sin^{2}\theta_{D}\right)-\sin\theta\Big{(}-\sqrt{2}\mu\sin\theta_{D}\cos\theta_{D}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+v\cos^{2}\theta_{D}(\lambda_{12}+\lambda^{\prime}_{12})+\lambda_{14}v\sin^{2}\theta_{D}\Big{)}\,,$
(74) $\displaystyle
h_{3}\phi^{\dagger}_{4}\phi_{4}:-v_{\mu\tau}\cos\theta\left(\lambda_{34}\cos^{2}\theta_{D}+\lambda_{23}\sin^{2}\theta_{D}\right)-\sin\theta\Big{(}+\sqrt{2}\mu\sin\theta_{D}\cos\theta_{D}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+v\sin^{2}\theta_{D}(\lambda_{12}+\lambda^{\prime}_{12})+\lambda_{14}v\cos^{2}\theta_{D}\Big{)}\,,$
(75) $\displaystyle
h_{3}\phi_{4}\phi_{4}\phi_{4}:-3\sqrt{2}\,\xi\cos^{3}\theta_{D}\cos\theta\,,$
(76) $\displaystyle
h_{3}\phi^{\dagger}_{4}\phi^{\dagger}_{4}\phi^{\dagger}_{4}:-3\sqrt{2}\,\xi\cos^{3}\theta_{D}\cos\theta\,,$
(77) $\displaystyle
h_{1}\phi_{4}\phi_{4}\phi_{4}:3\sqrt{2}\,\xi\cos^{3}\theta_{D}\sin\theta\,,$
(78) $\displaystyle
h_{1}\phi^{\dagger}_{4}\phi^{\dagger}_{4}\phi^{\dagger}_{4}:3\sqrt{2}\,\xi\cos^{3}\theta_{D}\sin\theta\,,$
(79)
$\displaystyle\phi_{2}\phi_{4}\phi_{4}:3\sqrt{2}\,v_{\mu\tau}\,\xi\cos^{2}\theta_{D}\sin\theta_{D}\,,$
(80)
$\displaystyle\phi^{\dagger}_{2}\phi^{\dagger}_{4}\phi^{\dagger}_{4}:3\sqrt{2}\,v_{\mu\tau}\,\xi\cos^{2}\theta_{D}\sin\theta_{D}\,,$
(81) $\displaystyle
h\phi^{\dagger}_{4}\phi_{4}:-\sin\theta\Big{(}v\lambda_{14}\cos^{2}\theta_{D}+\sqrt{2}\mu\cos\theta_{D}\sin\theta_{D}+v(\lambda_{12}+\lambda_{12}^{\prime})\sin^{2}\theta_{D}\Big{)}+$
$\displaystyle
v_{\mu\tau}\cos\theta\Big{(}\lambda_{34}\cos^{2}\theta_{D}+\lambda_{23}\sin^{2}\theta_{D}\Big{)}$
(82)
$\displaystyle\phi^{\dagger}_{4}(p_{1})\phi_{4}(p_{2})Z:\dfrac{g_{2}}{2\,\cos\theta_{W}}\sin^{2}\theta_{D}(p_{1}-p_{2})^{\alpha}\,,$
(83)
$\displaystyle\phi^{\dagger}_{4}(p_{1})\phi_{4}(p_{2})Z_{\mu\tau}:\dfrac{g_{\mu\tau}}{3}(p_{1}-p_{2})^{\alpha}\,.$
(84)

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