Muon anomalous magnetic dipole moment in a low scale type I see-saw model

D. N. Dinh

Institute of Physics, Vietnam Academy of Science and Technology,
10 Dao Tan, Ba Dinh, Hanoi, Vietnam.

###### Abstract

Recent experimental results on muon anomalous magnetic dipole moment have
shown a $4.2\sigma$ tension with the SM prediction, which has blown a fresh
wind into the elementary particle physics community. The problem is believed
to be explained only by physics beyond the standard model. Current work
considers the anomalous moment in a scenario of models with mirror symmetry
and type I see-saw mechanism at low energy scale of electroweak interactions.
After a brief introduction to the model, a detailed numerical analysis of muon
anomalous phenomenology will be carefully performed. Analysis results show
that the model is not successful in explaining the muon moment problem,
however the contributions of channels involving neutral Higgs scalars,
including both the light and heavy ones, might provide sizable corrections to
the discrepancy.

##  1 Introduction

In this letter, we are interested in the class of extended versions of the
standard model with mirror symmetry and light active neutrino masses generated
by the type I see-saw mechanism at the low energy scale of electroweak
interactions [1]. A fermion mirror sector is proposed by introducing a
corresponding mirror partner for each standard model fermion with the same
quantum numbers but opposite chirality. The presence of mirror partners of
left-handed neutrinos, thus the right-handed ones, provides a necessary
condition for the type I see-saw mechanism to operate [2, 3, 4, 5]. In
contrast to the canonical type I see-saw, which operates at ultra-high energy
scale, it is shown in that the new physics scale for the model under
consideration might be as low as 100 GeV, and thus at the electroweak
interaction scale.

We work on an updated version of the class of models that was introduced to
accommodate the 125 GeV SM-like Higgs scalar discovery [6, 7]. In contrast to
the original ones, an additional Higgs doublet has been introduced, so there
are two Higgs doublets, which are respectively responsible for mass generation
in the normal and mirror sectors. Two candidates among the neutral scalars
appearing after spontaneous symmetry breaking are shown to have signals in
agreement with ATLAS and CMS results [7]. Besides, the model also has to
confront the precision measurements of the electroweak processes, especially
the effects of extra chiral doublets. A large parameter space is validated to
be available after being constrained by EW precision data [8]. Moreover, the
production of new fermions, whose masses are connected to the electroweak
scale, and therefore can not exceed TeV, might be observable at the LHC with
maximal running energy reaching 14 TeV. This appealing problem has been
detailed discussed in [9]. The research showed that the $5\sigma$ discovery of
$\nu_{R}$ is possible with as low as $20fb^{-1}$ of integrated luminosity of
the LHC running at 13 TeV for the RH neutrino mass within a range of $200-500$
GeV. For larger $\nu_{R}$ mass, up to $650$ GeV, the required luminosity might
be as high as $100fb^{-1}$. These conclusions are obtained based on an
assumption of large Yukawa coupling $g_{\ell S}$ of the interaction between
the normal lepton, mirror lepton and a light singlet scalar. However, this
coupling should be much smaller due to being constrained by the current upper
bounds on some lepton flavor violation processes [10, 11], and also the muon
anomalous magnetic moment as we will see latter in this research, therefore
the real physics may happen in very different scenario and need a
reconsideration [9].

It is well-known recent time that the combination of Brookhaven E8211 result
[12] and $(g-2)_{\mu}$ experiment at Fermilab [13] for the muon anomalous
magnetic moment has obtained the result

$\Delta a_{\mu}=a_{\mu}^{EXP}-a_{\mu}^{SM}=(251\pm 59)\times 10^{-11},$ (1)

which is a $4.2\sigma$ discrepancy with the SM prediction [14]. Although the
result has not taken into account the recent lattice QCD calculations for
hadronic vacuum polarization [15] and the latest measurement of
$e^{+}e^{-}\to{\rm hadrons}$ [16], it can not be denied to be the strong
evidence of new physics beyond the standard model.

From the theoretical perspective, a large number of researches has
investigated the problem of muon anomalous magnetic dipole moment in various
scenarios of physics beyond the standard model [17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28]. In the class of models with mirror symmetry under
consideration, the muon problem has also been briefly mentioned in some
researches [6, 29], involving the channel with participation of light neutral
scalar. However more detailed analysis should be performed, besides taking
into account the contributions provided by other channels of heavy neutral and
singly charged scalars.

This research discusses the phenomenology of muon anomalous magnetic dipole
moment in the scenario of an extended version of the standard model with
mirror symmetry, accommodating the 125 GeV SM-like scalar discovery. The
contents are arranged as follows: besides the introduction in this section, in
Sect. 2 we briefly introduce the model and required vertices for further
calculations. In Sect. 3, we derive explicit form factors and algebraic
expressions for muon anomalous magnetic dipole moment. Then, numerical
analysis is also performed in this section. Finally, we give the conclusion in
Sect. 4.

##  2  A review of the model

###  2.1 The model content

This under consideration extended version of the EW-scale $\nu_{R}$ model is
constructed based on the symmetric group $SU(2)\times U(1)_{Y}\times
Z^{4}_{SM}\times Z^{4}_{MF}$, in which $SU(2)\times U(1)_{Y}$ is the gauge
group, and $Z^{4}_{SM}\times Z^{4}_{MF}$ is a global symmetry introduced to
forbid some unexpected interactions. The arrangement of scalar and matter
fields under gauge group and their transformations under the global symmetry
are detailed shown in the Table 1. Here, transformation of a given field
$\Psi$ under the $Z^{4}_{SM}\times Z^{4}_{MF}$ is characterized by
$(\omega_{a}^{\alpha},\omega_{b}^{\beta})$, which can be expressed as
$\omega_{a}^{\alpha}\Psi\omega_{b}^{\beta}$, where
$\omega_{a}^{4}$=$\omega_{b}^{4}=1$.

Note that the right-handed neutrinos in this model are components of
$SU(2)\times U(1)_{Y}$ doublets; therefore they are non-sterile and take part
in the weak interaction. Moreover, the heavy right-handed neutrinos naturally
occur in the mirror sector accompanying the light active ones to fulfill the
required conditions for the type I see-saw neutrino mass generation
functioning. Five Higgs scalars (two doublets, two triplets and a singlet) are
introduced to give masses for fermion particles, their roles could be seen in
the later part of the paper.

Multiplets | $SU(2)\times U(1)_{Y}$ | $Z^{4}_{SM}$ | $Z^{4}_{MF}$
---|---|---|---
$\ell_{L}=(\nu_{L},~{}e_{L})^{T}$, $q_{L}=(u_{L},d_{L})^{T}$ |  $(2,-1)$, $(2,1/3)$ | $\omega_{a}^{3}$ | 1
$\ell_{R}^{M}=(\nu_{R},~{}e_{R}^{M})^{T}$, $q_{R}^{M}=(u_{R}^{M},~{}d_{R}^{M})^{T}$ |  $(2,-1)$, $(2,1/3)$ | 1 | $\omega_{b}^{3}$
$e_{R}$, $u_{R}$, $d_{R}$ |  $(1,-2)$, $(1,4/3)$, $(1,-2/3)$ | $\omega_{a}$ | 1
$e_{L}^{M}$, $u_{L}^{M}$, $d_{L}^{M}$ |  $(1,-2)$, $(1,4/3)$, $(1,-2/3)$ | 1 | $\omega_{b}$
$\Phi_{2}=(\phi_{2}^{+},\phi_{2}^{0})$ | (2,1) | $\omega_{a}^{2}$ | 1
$\Phi_{2M}=(\phi_{2M}^{+},\phi_{2M}^{0})$ | (2,1) | 1 | $\omega_{b}^{2}$
$\chi=\left(\chi^{++},\chi^{+},\chi^{0}\right)$ | (3,2) | 1 | $\omega_{b}^{2}$
$\xi=\left(\xi^{+},\xi^{0},\xi^{-}\right)$ | (3,0) | 1 | 1
$\phi_{S}$ | (1,0) | $\omega_{a}^{3}$ | $\omega_{b}$
Table 1: Model’s field content and their transformations under gauge and
global discrete symmetries, where $\omega_{a}^{4}$=$\omega_{b}^{4}=1$.

Before writing down the Yukawa couplings, let us briefly discuss the Higgs
scalar sector. Apparently, the global symmetry defined in the earlier part
only allows $\Phi_{2}$ to couple to SM fermions, while $\Phi_{2M}$ will couple
to the mirror partners. The singlet $\phi_{S}$, which transforms nontrivially
under both $Z^{4}_{SM}$ and $Z^{4}_{MF}$, will couple to a normal and a mirror
fields. Finally, $\chi$ is responsible for introducing two unit lepton number
violation term, which is needed for acquiring Majorana masses for heavy
neutrinos. Detailed expressions of the Yukawa couplings to generate masses for
matter fields are:

$\mathcal{L}_{Y}^{\ell}=g_{\ell}\bar{\ell}_{L}\Phi_{2}e_{R}+g_{\ell}^{M}\bar{\ell}_{R}^{M}\Phi_{2M}e_{L}^{M}+g_{\ell
s}\bar{\ell}_{L}\phi_{s}\ell_{R}^{M}+g_{\ell
s}^{\prime}\bar{e}_{L}^{M}\phi_{s}e_{R}+h.c.,$ (2)
$\displaystyle\mathcal{L}_{Y}^{q}$ $\displaystyle=$ $\displaystyle
g_{u}\bar{q}_{L}\tilde{\Phi}_{2}u_{R}+g_{u}^{M}\bar{q}_{R}^{M}\tilde{\Phi}_{2M}u_{L}^{M}+g_{d}\bar{q}_{L}\Phi_{2}d_{R}+g_{d}^{M}\bar{q}_{R}^{M}\Phi_{2M}d_{L}^{M}$
(3)
$\displaystyle+g_{qs}\bar{q}_{L}\phi_{s}q_{R}^{M}+g_{us}^{\prime}\bar{u}_{L}^{M}\phi_{s}u_{R}+g_{ds}^{\prime}\bar{d}_{L}^{M}\phi_{s}d_{R}+h.c.,$
$\mathcal{L}_{\nu_{R}}=g_{M}\left(\ell_{R}^{M,T}\,\sigma_{2}\right)\,(i\tau_{2}\tilde{\chi})\,\ell_{R}^{M}\,,$
(4)

where $\sigma_{2}$ is the second Pauli matrix,
$\tilde{\Phi}_{2}=i\sigma_{2}\Phi_{2}^{*}$,
$\tilde{\Phi}_{2M}=i\sigma_{2}\Phi_{2M}^{*}$, and $\tilde{\chi}$ form of the
complex Higgs triplet with $Y=2$ is

$\tilde{\chi}=\frac{1}{\sqrt{2}}\vec{\tau}.\vec{\chi}=\left(\begin{array}[]{cc}\frac{1}{\sqrt{2}}\chi^{+}&\chi^{++}\\\
\chi^{0}&-\frac{1}{\sqrt{2}}\chi^{+}\end{array}\right).$ (5)

Note that in the equation (4), $\sigma_{2}$ acts on the space of two-component
Weyl spinors, while $\tau_{2}$ acts on the SU(2) isospin space. We expect to
have Hermetic charged fermion mass matrices, which simply implies $g_{\ell
s}^{\prime}=g_{\ell s}^{\dagger}$, $g_{us}^{\prime}=g_{us}^{\dagger}$ and
$g_{ds}^{\prime}=g_{ds}^{\dagger}$, respectively. Finally, discussion on the
quark sector will not be performed in this letter, because it is not involved
in the phenomenology of physical quantity under consideration of this
research.

###  2.2 Symmetry breaking and mass generations

We discuss in this subsection the mechanism of mass generations for fermion
and scalar particles in this model, when the symmetry is spontaneously
breaking. Let us suppose that Higgs fields develop their vacuum expectation
values (VEV) as the following:
$\langle\Phi_{2}\rangle=(0,v_{2}/\sqrt{2})^{T}$,
$\langle\Phi_{2M}\rangle=(0,v_{2M}/\sqrt{2})^{T}$,
$\langle\chi^{0}\rangle=v_{M}$, and $\langle\phi_{S}\rangle=v_{S}$.

The charged lepton mass matrix that can be easily obtained from eq. (2), is
explicitly expressed as

$M_{\ell}=\left(\begin{array}[]{cc}m_{\ell}&m_{\ell}^{D}\\\
(m_{\ell}^{D})^{\dagger}&m_{\ell M}\end{array}\right)\,,$ (6)

where $m_{\nu}^{D}=m_{\ell}^{D}=g_{\ell s}v_{S}$,
$m_{\ell}=g_{\ell}v_{2}/\sqrt{2}$, and $m_{\ell
M}=g_{\ell}^{M}v_{2M}/\sqrt{2}$. Based on the current experimental status of
searching new fermions beyond the standard model, one expects that masses of
the mirror partners are much heavier than their normal ones, thus it is
reasonable to assume $m_{\ell M}\gg m_{\ell}$ and $m_{\ell M},m_{\ell}\gg
m_{\ell}^{D}$. This assumption allows us to approximately block-diagonalize
$M_{\ell}$ in the same way usually done for the see-saw type I neutrino mass
matrix, then we obtain:

$\displaystyle\tilde{m}_{\ell}=m_{\ell}-\frac{(m_{\ell}^{D})^{2}}{m_{\ell
M}-m_{\ell}}\approx m_{\ell},~{}~{}~{}\tilde{m}_{\ell M}=m_{\ell
M}+\frac{(m_{\ell M}^{D})^{2}}{m_{\ell M}-m_{\ell}}\approx m_{\ell M},$ (7)
$\left(\begin{array}[]{c}\ell_{L(R)}\\\
\ell^{M}_{L(R)}\end{array}\right)\,=\left(\begin{array}[]{cc}U_{\ell
L(R)}&-R_{\ell}U_{\ell L(R)}^{M}\\\ R_{\ell}^{\dagger}U_{\ell L(R)}&U_{\ell
L(R)}^{M}\end{array}\right)\,\left(\begin{array}[]{c}\ell^{\prime}_{L(R)}\\\
\ell^{M^{\prime}}_{L(R)}\end{array}\right),$ (8)

where $\ell^{\prime}_{L(R)}$, $\ell^{M^{\prime}}_{L(R)}$ are respectively the
normal and mirror charged leptons in the mass basis;
$R_{\ell}\approx\frac{m_{\ell}^{D}}{m_{\ell M}}\ll 1$, and
$\tilde{m}_{\ell}=U_{\ell L}m_{\ell}^{d}U_{\ell R}^{\dagger}$,
$\tilde{m}_{\ell M}=U_{\ell L}^{M}m_{\ell M}^{d}{U_{\ell R}^{M}}^{\dagger}$,
in which $m_{\ell}^{d}$ and $m_{\ell M}^{d}$ are diagonal matrices.

After the gauge symmetry is spontaneously broken, the neutral leptons acquire
their masses through a matrix of the canonical form of the type-I see-saw
mechanism. By denoting that $M_{R}=g_{M}v_{M}$, one obtains

$M_{\nu}=\left(\begin{array}[]{cc}0&m_{\nu}^{D}\\\
(m_{\nu}^{D})^{T}&M_{R}\end{array}\right)\,.$ (9)

Here, we have used the fact that the term of form
$\left(\ell_{L}^{T}\,\sigma_{2}\right)\,(i\tau_{2}\tilde{\chi})\,\ell_{L}$ is
forbidden by the discrete symmetry $Z^{4}_{SM}\times Z^{4}_{MF}$ that forbids
to generate Majorana mass term for the light active neutrinos at tree-level.
Moreover, it is shown in [1] that the mass term can arise at one-loop level,
however its magnitude is about two order smaller than that is generated by the
typical see-saw mechanism, therefore the effects are reasonably ignored in
this research.

Approximately block-diagonalizing (9), while keeping in mind that $M_{R}\gg
m_{\nu}^{D}$, the result reads

$\tilde{m}_{\nu}\approx-\frac{(m_{\nu}^{D})^{2}}{M_{R}}=-\frac{(g_{\ell
s}v_{S})^{2}}{g_{M}v_{M}},~{}~{}\tilde{m}_{\nu R}\approx M_{R}.$ (10)

We briefly comment on the light neutrino mass matrix $\tilde{m}_{\nu}$ defined
in (10), which is experimentally constrained to be at sub-eV order. In a
canonical scenario, such a small constraint on $\tilde{m}_{\nu}$ implies
$M_{R}$ should be very heavy $\sim 10^{9}$ GeV or higher. However, in the
current model under consideration, if $(g^{2}_{\ell s}/g_{M})\sim O(1)$ and
$v_{S}\sim O(10^{5}~{}eV)$, $M_{R}$ can be much lower, at the electroweak
scale. Note that this is the most interesting scenario that could be designed
for this model to be testable at the LHC. However, $M_{R}$ is not forbidden to
have larger values, up to about TeV, depending on both the magnitude of
$v_{M}$ and interaction strength $g_{M}$. In fact, $g_{\ell s}$ is constrained
by some rare processes, for instance the $\mu\to e\gamma$ decay which has been
detailed studied in [11], that leads to $(g^{2}_{\ell s}/g_{M})\ll 1$. In this
case $v_{S}$ should be adjusted to be higher (might reach few GeV) to give
correct masses for the light neutrinos if $M_{R}$ is fixed in range of hundred
GeV.

Let $R_{\nu}\approx\frac{m_{\nu}^{D}}{M_{R}}$ be the ratio of the neutrino
Dirac and Majorana mass matrices. Assuming that the light and heavy (mirror)
neutrino mass matrices are diagonalized respectively by
$\tilde{m}_{\nu}=U_{\nu}^{*}m_{\nu}^{d}U_{\nu}^{\dagger}$, $\tilde{m}_{\nu
R}={U_{\nu}^{M}}^{*}m_{\nu M}^{d}{U_{\nu}^{M}}^{\dagger}$, where $m_{\nu}^{d}$
and $m_{\nu M}^{d}$ are diagonal matrices, we can obtain the relations between
the gauge and mass eigenstates of neutrinos as the following

$\left(\begin{array}[]{c}\nu_{L}\\\
(\nu_{R})^{c}\end{array}\right)\,=\left(\begin{array}[]{cc}U_{\nu}&-R_{\nu}U_{\nu}^{M}\\\
R_{\nu}^{\dagger}U_{\nu}&U_{\nu}^{M}\end{array}\right)\,\left(\begin{array}[]{c}\chi_{L}\\\
\chi_{L}^{M}\end{array}\right).$ (11)

Before we introduce the masses and mass states of the new physical scalars
that appear after spontaneous symmetry breaking, let us briefly explain why we
need two Higgs triplets in this model. It is well known that when triplets are
introduced, the tree-level result $\rho=1$, which is precisely measured by
experiment, will be violated. Fortunately, it is also shown in [30] that, in a
scenario of two triplets with appropriate hyper-charges, they can combine to
form a $(3,3)$ representation under the global $SU(2)_{L}\otimes SU(2)_{R}$
symmetry. After symmetry breaking, the custodial $SU(2)$ symmetry is preserved
and $\rho=1$. Thus, along with the triplet $\tilde{\chi}$, we need to add a
real Higgs triplet with $Y=0$, denoted by
$\left(\xi^{+},\xi^{0},\xi^{-}\right)$. The combinations of two triplets
$(3,3)$ and two doublets $(2,2)$ under the global symmetry can be respectively
expressed as:

$\chi=\left(\begin{array}[]{ccc}\chi^{0}&\xi^{+}&\chi^{++}\\\
\chi^{-}&\xi^{0}&\chi^{+}\\\ \chi^{--}&\xi^{-}&\chi^{0*}\end{array}\right)\,,$
(12) $\Phi_{2}=\left(\begin{array}[]{cc}\phi_{2}^{0,*}&\phi_{2}^{+}\\\
\phi_{2}^{-}&\phi_{2}^{0}\end{array}\right),~{}~{}~{}\Phi_{2M}=\left(\begin{array}[]{cc}\phi_{2M}^{0,*}&\phi_{2M}^{+}\\\
\phi_{2M}^{-}&\phi_{2M}^{0}\end{array}\right).$ (13)

Then the proper vacuum alignment for breaking gauge symmetry from
$SU(2)_{L}\times U(1)_{Y}$ to $U(1)_{em}$ can be easily written down:

$\langle\chi\rangle=\left(\begin{array}[]{ccc}v_{M}&0&0\\\ 0&v_{M}&0\\\
0&0&v_{M}\end{array}\right)\,,$ (14)
$\langle\Phi_{2}\rangle=\left(\begin{array}[]{cc}v_{2}/\sqrt{2}&0\\\
0&v_{2}/\sqrt{2}\end{array}\right),~{}~{}~{}\langle\Phi_{2M}\rangle=\left(\begin{array}[]{cc}v_{2M}/\sqrt{2}&0\\\
0&v_{2M}/\sqrt{2}\end{array}\right).$ (15)

Due to the experimental constraint on the $W_{\mu}$ mass, VEVs of the real
components of $\Phi_{2}$, $\Phi_{2M}$ and $\chi$ denoted in (14) and (15),
satisfy the conditions:

$v_{2}^{2}+v_{2M}^{2}+8\,v_{M}^{2}=v^{2}\,,$ (16)

where $v\approx 246~{}GeV$. Related to the above VEVs and for further
discussions, the following notations are used:

$s_{2}=\dfrac{v_{2}}{v};~{}~{}s_{2M}=\dfrac{v_{2M}}{v};~{}~{}s_{M}=\dfrac{2\sqrt{2}\;v_{M}}{v}\,.$
(17)

We analyze the physical scalar spectrum in this model after the gauge symmetry
and the L-R global symmetry of the Higgs potential are spontaneously broken to
the custodial $SU(2)_{D}$. Out of the seventeen degrees of freedom of the two
Higgs triplets (one real and one complex) and two Higgs doublets, three of
them are eaten to give masses to W’s and Z, while the rest are rearranged to
form new physical Higgs bosons. Those that are mass-degenerate are grouped in
the same physical scalar multiplets of the global custodial symmetry. Thus we
have a five-plet (quintet) $(H_{5}^{\pm\pm},\;H_{5}^{\pm},\;H_{5}^{0})$, two
triplets $(H_{3}^{\pm},\;H_{3}^{0})$, $(H_{3M}^{\pm},\;H_{3M}^{0})$ and three
singlets $H_{1}^{0},\;H_{1M}^{0},\;H_{1}^{0\prime}\,$.

We will not introduce any specific case of Higgs potential in this research,
but the above discussion on physical scalars applies to any one of them that
possesses $SU(2)_{L}\otimes SU(2)_{R}$ global symmetry, including the cases
that have been considered in detail in [7]. It is reasonable to assume that
the scalars mentioned earlier have masses at the electroweak scale, in the
range of hundred to few hundred GeVs, because they are remnants of the gauge
symmetry breaking mechanism. Recall that the scalars that are members of a
multiplet (not singlet) have the same masses, while the three singlets
$H_{1}^{0},\;H_{1M}^{0},\;H_{1}^{0\prime}\,$ are not physical states, in
general. These gauge states are linear combinations of some mass eigenstates
$(\tilde{H}_{1}^{0},\;\tilde{H}_{2}^{0},\;\tilde{H}_{3}^{0}\,$) as
$H_{1}^{0}=\sum_{i}^{3}\alpha_{i}\tilde{H}_{i}$,
$H_{1M}^{0}=\sum_{i}^{3}\alpha_{i}^{M}\tilde{H}_{i}$, where
$\sum_{i}^{3}|\alpha_{i}|^{2}=1$ and $\sum_{i}^{3}|\alpha_{i}^{M}|^{2}=1$. The
SM-like Higgs scalar discovered by LHC with mass 125-GeV is one of these three
mass states [7].

Finally, let us discuss $\phi_{s}^{0}$, which is the remaining degree of
freedom of the Higgs sector, originating from $\phi_{S}$. As a singlet,
$\phi_{S}$ does not participate in the gauge symmetry breaking mechanism, so
its VEV $v_{S}$ can have a large range of values from keV to few GeV. In this
research, we will consider $\phi_{s}^{0}$ mass to have the same order of
magnitude as $v_{S}$.

###  2.3 The LFV vertices

In the SM, there is no flavor changing of the neutral current at tree-level in
the lepton sector, so there is no LFV vertex, because the charged lepton mass
matrix and the matrix of Yukawa couplings are simultaneously diagonal, and the
vector gauge bosons only interact with the left-handed components of the
matter fields. However, these properties do not hold in this model, because
the matter content has been enlarged with mirror fermions, and the vector
fields also interact with the right-handed components of the mirror sector.
Therefore, LFV interactions occur at tree-level for both the charged currents
and Yukawa couplings. Their detailed expressions in the gauge basis can be
found in [7].

To facilitate further discussion, we present below the LFV couplings in this
model in the mass eigenstate basis. For consistency with the current
experimental observations and for simplicity, we assume that the charged
lepton and mirror charged lepton mixing matrices are real (so all the involved
complex phases are neglected) and $U_{\ell L}=U_{\ell R}=U_{\ell}$, $U_{\ell
L}^{M}=U_{\ell R}^{M}=U_{\ell}^{M}$. After dropping terms that are subleading
of the second order of $R_{\nu(\ell)}$ and higher, the LFV couplings are given
in table 2 and eqs. from (18) to (23):

Vertices | Couplings
---|---
$(\bar{e}^{\prime}_{L}\gamma^{\mu}\chi_{L})W_{\mu}^{-}$ | $-i\frac{g}{\sqrt{2}}U_{W_{\mu}}^{L}=-i\frac{g}{\sqrt{2}}U_{PMNS}$
$(\bar{e}^{\prime}_{L}\gamma^{\mu}\chi_{L}^{M})W_{\mu}^{-}$ | $-i\frac{g}{\sqrt{2}}U_{W_{\mu}}^{ML}=i\frac{g}{\sqrt{2}}\tilde{R}_{\nu}\left(U_{PMNS}^{M}\right)^{*}$
$(\bar{e}^{\prime}_{R}\gamma^{\mu}\chi_{L}^{c})W_{\mu}^{-}$ | $-i\frac{g}{\sqrt{2}}U_{W_{\mu}}^{R}=-i\frac{g}{\sqrt{2}}\tilde{R}_{\nu}^{T}\left(U_{PMNS}\right)^{*}$
$\bar{e}^{\prime}_{R}\chi_{L}H_{3}^{-}$ | $-i\frac{g}{2}Y_{H_{3}^{-}}^{L}=-i\frac{g~{}s_{M}}{2M_{W}c_{M}}m_{\ell}^{d}U_{PMNS}$
$\bar{e}^{\prime}_{R}\chi_{L}^{M}H_{3}^{-}$ | $-i\frac{g}{2}Y_{H_{3}^{-}}^{ML}=i\frac{g~{}s_{M}}{2M_{W}c_{M}}m_{\ell}^{d}\tilde{R}_{\nu}\left(U_{PMNS}^{M}\right)^{*}$
$\bar{e}^{\prime}_{L}\chi_{L}^{Mc}H_{3}^{-}$ | $-i\frac{g}{2}Y_{H_{3}^{-}}^{MR}=-i\frac{g~{}s_{M}}{2M_{W}c_{M}}\tilde{R}_{\ell}m_{\ell M}^{d}U_{PMNS}^{M}$
$\bar{e}^{\prime}_{R}\chi_{L}H_{3M}^{-}$ | $-i\frac{g}{2}Y_{H_{3M}^{-}}^{L}=-i\frac{g~{}s_{2M}}{2M_{W}s_{2}c_{M}}m_{\ell}^{d}U_{PMNS}$
$\bar{e}^{\prime}_{R}\chi_{L}^{M}H_{3M}^{-}$ | $-i\frac{g}{2}Y_{H_{3M}^{-}}^{ML}=i\frac{g~{}s_{2M}}{2M_{W}s_{2}c_{M}}m_{\ell}^{d}\tilde{R}_{\nu}\left(U_{PMNS}^{M}\right)^{*}$
$\bar{e}^{\prime}_{L}\chi_{L}^{Mc}H_{3M}^{-}$ | $-i\frac{g}{2}Y_{H_{3M}^{-}}^{MR}=-i\frac{g~{}s_{M}}{2M_{W}s_{2M}c_{M}}\tilde{R}_{\ell}m_{\ell M}^{d}U_{PMNS}^{M}$
$\bar{e}^{\prime}_{R}{e_{L}^{M}}^{\prime}\phi^{0}_{s}$ | $-i\frac{g}{2}Y_{\phi^{0}_{s}}^{ML}=-iU^{\dagger}_{\ell R}g_{\ell s}U_{\ell L}^{M}=-i\tilde{g}_{\ell s}$
$\bar{e}^{\prime}_{L}{e_{R}^{M}}^{\prime}\phi^{0}_{s}$ | $-i\frac{g}{2}Y_{\phi^{0}_{s}}^{MR}=-i~{}U^{\dagger}_{\ell L}g_{\ell s}U_{\ell R}^{M}=-i\tilde{g}_{\ell s}$

Table 2: Vertices involving muon anomalous magnetic dipole moment in the mass
eigenstate basis.

$(\bar{e}^{\prime}_{R}{e_{L}^{M}}^{\prime}{\tilde{H}}_{i}^{0})~{}~{}~{}~{}~{}~{}-i\frac{g}{2}Y^{ML}_{{\tilde{H}}_{i}^{0}}=-i\frac{g}{2M_{W}}\left[\frac{\alpha_{i}}{s_{2}}m_{\ell}^{d}\tilde{R}_{\ell}+\frac{\alpha_{i}^{M}}{s_{2M}}\tilde{R}_{\ell}m_{\ell
M}^{d}\right],$ (18)
$(\bar{e}^{\prime}_{L}{e_{R}^{M}}^{\prime}{\tilde{H}}_{i}^{0})~{}~{}~{}~{}~{}~{}-i\frac{g}{2}Y^{MR}_{{\tilde{H}}_{i}^{0}}=-i\frac{g}{2M_{W}}\left[\frac{\alpha_{i}}{s_{2}}m_{\ell}^{d}\tilde{R}_{\ell}+\frac{\alpha_{i}^{M}}{s_{2M}}\tilde{R}_{\ell}m_{\ell
M}^{d}\right],$ (19)

$(\bar{e}^{\prime}_{R}{e_{L}^{M}}^{\prime}{H}_{3}^{0})~{}~{}~{}~{}~{}~{}-i\frac{g}{2}Y^{ML}_{{H}_{3}^{0}}=-i\frac{g}{2M_{W}}\left[\frac{s_{M}}{c_{M}}m_{\ell}^{d}\tilde{R}_{\ell}+\frac{s_{M}}{c_{M}}\tilde{R}_{\ell}m_{\ell
M}^{d}\right],$ (20)
$(\bar{e}^{\prime}_{L}{e_{R}^{M}}^{\prime}{H}_{3}^{0})~{}~{}~{}~{}~{}~{}-i\frac{g}{2}Y^{MR}_{{H}_{3}^{0}}=-i\frac{g}{2M_{W}}\left[-\frac{s_{M}}{c_{M}}m_{\ell}^{d}\tilde{R}_{\ell}-\frac{s_{M}}{c_{M}}\tilde{R}_{\ell}m_{\ell
M}^{d}\right],$ (21)

$(\bar{e}^{\prime}_{R}{e_{L}^{M}}^{\prime}{H}_{3M}^{0})~{}~{}~{}~{}~{}~{}-i\frac{g}{2}Y^{ML}_{{H}_{3M}^{0}}=-i\frac{g}{2M_{W}}\left[-\frac{s_{2M}}{s_{2}}m_{\ell}^{d}\tilde{R}_{\ell}-\frac{s_{2}}{s_{2M}}\tilde{R}_{\ell}m_{\ell
M}^{d}\right],$ (22)
$(\bar{e}^{\prime}_{L}{e_{R}^{M}}^{\prime}{H}_{3M}^{0})~{}~{}~{}~{}~{}~{}-i\frac{g}{2}Y^{MR}_{{H}_{3M}^{0}}=-i\frac{g}{2M_{W}}\left[\frac{s_{2M}}{s_{2}}m_{\ell}^{d}\tilde{R}_{\ell}+\frac{s_{2}}{s_{2M}}\tilde{R}_{\ell}m_{\ell
M}^{d}\right].$ (23)

Here one has used the notations $U_{PMNS}=U_{\ell}^{\dagger}U_{\nu}$, which is
the famous PMNS mixing matrix, $U_{PMNS}^{M}=U_{\ell}^{M\dagger}U_{\nu}^{M}$
and $\tilde{R}_{\ell(\nu)}=U_{\ell}^{\dagger}R_{\ell(\nu)}U_{\ell}^{M}$.

##  3 Phenomenology of muon anomalous magnetic dipole moment

###  3.1 One-loop form-factors and muon anomalous magnetic dipole moment

In this scenario, under the considered model, the muon anomalous magnetic
moment and the $\mu\to e+\gamma$ decay rate are related to the loop integral
factors. In some previous research, one-loop diagrams with various kinds of
internal lines have been calculated [11, 31, 32, 33, 34]. The current work
takes into account the effective charged lepton flavor-changing operators
arising at one-loop, where the virtual particles running inside are either
physical Higgs scalars (which include single and neutral charges, heavy and
light ones) or W gauge bosons accompanied by relevant leptons. One might
wonder on the contribution of doubly charged Higgs scalar channel. However it
can be easily estimated to give sub-leading contribution due to the fact that
the complex Higgs triplet with $Y=2$ can only interact with the lepton mirror
doublet; therefore the channels involving doubly charged Higgs scalar can
provide only high order corrections to the anomalous magnetic dipole moments
of the normal charged leptons. Feynman diagrams of the dominated channels are
shown in Fig.1. The calculation result can be summarized as follows:

Figure 1: Three group of diagrams give leading contributions to the muon
anomalous magnetic dipole moment. Here, $H^{-}$ stands for $H_{3}^{-}$ and
$H_{3M}^{-}$; $H^{0}$ stands for ${\tilde{H}}_{i}^{0}$, $H^{0}_{3}$ and
$H^{0}_{3M}$.

$\mathcal{L}_{eff}=-4\frac{eG_{F}}{\sqrt{2}}\left[(m_{\ell}A_{R}+m_{\ell^{\prime}}A_{L})\bar{\ell^{\prime}}\sigma_{\mu\nu}P_{R}\ell+(m_{\ell}A_{L}+m_{\ell^{\prime}}A_{R})\bar{\ell^{\prime}}\sigma_{\mu\nu}P_{L}\ell\right]F^{\mu\nu}.$
(24)

Here $A_{L,R}$ are the form factors:

$\displaystyle A_{R}=$
$\displaystyle-\sum_{{H^{Q}},k}\frac{M_{W}^{2}}{64\pi^{2}M_{H}^{2}}\left[\left(Y_{H}^{L}\right)_{\mu
k}\left(Y_{H}^{L}\right)_{ek}^{*}G_{H}^{Q}(\lambda_{k})+\frac{m_{k}}{m_{\mu}}\left(Y_{H}^{R}\right)_{\mu
k}\left(Y_{H}^{L}\right)_{ek}^{*}\times R_{H}^{Q}(\lambda_{k})\right]$ (25)
$\displaystyle+\frac{1}{32\pi^{2}}\sum_{k}\left[\left(U_{W_{\mu}}^{L}\right)_{\mu
k}\left(U_{W_{\mu}}^{L}\right)_{ek}^{*}G_{\gamma}(\lambda_{k})-\left(U_{W_{\mu}}^{R}\right)_{\mu
k}\left(U_{W_{\mu}}^{L}\right)_{ek}^{*}\frac{m_{k}}{m_{\mu}}R_{\gamma}(\lambda_{k})\right],$
$\displaystyle A_{L}=$
$\displaystyle-\sum_{{H^{Q}},k}\frac{M_{W}^{2}}{64\pi^{2}M_{H}^{2}}\left[\left(Y_{H}^{R}\right)_{\mu
k}\left(Y_{H}^{R}\right)_{ek}^{*}G_{H}^{Q}(\lambda_{k})+\frac{m_{k}}{m_{\mu}}\left(Y_{H}^{L}\right)_{\mu
k}\left(Y_{H}^{R}\right)_{ek}^{*}R_{H}^{Q}(\lambda_{k})\right]$ (26)
$\displaystyle+\frac{1}{32\pi^{2}}\sum_{k}\left[\left(U_{W_{\mu}}^{R}\right)_{\mu
k}\left(U_{W_{\mu}}^{R}\right)_{ek}^{*}G_{\gamma}(\lambda_{k})-\left(U_{W_{\mu}}^{L}\right)_{\mu
k}\left(U_{W_{\mu}}^{R}\right)_{ek}^{*}\frac{m_{k}}{m_{\mu}}R_{\gamma}(\lambda_{k})\right],$

where $H^{Q}=\phi_{S}^{0},\tilde{H}^{0}_{i}$ $(i=1,2,3)$,
$H^{0}_{3},H^{0}_{3M}$, $H^{+}_{3},H^{+}_{3M}$, and $m_{k}$ are the masses of
associated fermions that accompany with either $H^{Q}$ or $W_{\mu}$ in the
loops. The functions $G_{H}^{Q}(x)$, $R_{H}^{Q}(x)$, $G_{\gamma}(x)$, and
$R_{\gamma}(x)$ appearing in eqs. (25) and (26) are defined as:

$\displaystyle G_{H}^{Q}(x)$ $\displaystyle=$
$\displaystyle-\frac{(3Q-1)x^{2}+5x-3Q+2}{12(x-1)^{3}}+\frac{1}{2}\frac{x(Qx-Q+1)}{2(x-1)^{4}}\log(x),$
(27) $\displaystyle R_{H}^{Q}(x)$ $\displaystyle=$
$\displaystyle\frac{(2Q-1)x^{2}-4(Q-1)x+2Q-3}{2(x-1)^{3}}-\frac{Qx-(Q-1)}{(x-1)^{3}}\log(x),~{}~{}~{}~{}~{}~{}$
(28) $\displaystyle G_{\gamma}(x)$ $\displaystyle=$
$\displaystyle\frac{10-43x+78x^{2}-49x^{3}+4x^{4}+18x^{3}\log(x)}{12(x-1)^{4}},$
(29) $\displaystyle R_{\gamma}(x)$ $\displaystyle=$
$\displaystyle-\frac{x^{2}+x-8}{2(x-1)^{2}}+\frac{3x(x-2)}{(x-1)^{3}}\log(x),$
(30)

where $\lambda_{k}=m_{k}^{2}/M_{{W_{\mu}}({H^{Q}})}^{2}$ has been denoted.

Note that functions introduced in the above equations $G_{H}^{Q}(x)$,
$R_{H}^{Q}(x)$, $G_{\gamma}(x)$, and $R_{\gamma}(x)$ are valid for $x$
variable varying in interval $[0,+\infty)$, and get finite values at the
specific points, such as $x=0,~{}1$ or when $x$ tends to infinity. Compare
with the original expression, introduced in an earlier publication [33],
$G_{\gamma}(x)$ has been divided by $4$ to be consistent to the factor
$1/(32\pi^{2})$ in the definitions of $A_{L}$ and $A_{R}$. The two functions,
$G_{H}^{Q}(x)$ and $R_{H}^{Q}(x)$, to the best of the author’s knowledge, have
not been used in any other discussion on the muon anomalous dipole moment so
far.

The expression for muon anomalous magnetic dipole moment can be easily
extracted from the effective Lagrangian (24), which arrives at

$\Delta
a_{\mu}=\frac{4\pi\alpha_{em}}{\sin^{2}\theta_{w}}\frac{m_{\mu}^{2}}{M_{W}^{2}}(A_{L}+A_{R}),$
(31)

where $\alpha_{em}=1/137$ is the fine-structure constant. Note that formula
(31) should not include the contributions of light neutrino and $W_{\mu}$
loops, which have been taken into account in the standard model.

###  3.2 Numerical analysis of muon anomalous magnetic dipole moment

In this section, we perform numerical analysis of the muon anomalous magnetic
dipole moment expressed by (31) using current experimental data. To better
understand the role of each kind of diagram and for convenience, we separately
consider the contributions of one-loop diagrams with virtual $W$ gauge boson,
neutral and singly charged Higgs scalars to the quantity. Moreover, for
simplicity in further numerical discussions, we assume that three heavy
neutrinos are degenerated in masses, which are denoted as $m_{\chi}^{M}$.
Similarly, we make the same assumption for three mirror charged lepton masses
$m_{\ell}^{M}$.

Before performing a detail discussion, let’s make a rough estimation of the
magnitudes of $R_{\nu(\ell)}$, which have central roles in the phenomenology
of lepton flavour violation and muon anomalous magnetic dipole moment in this
model. Starting from the expression of light neutrino mass matrix in (10) and
its current experimental constraint, we can easily obtain

$\displaystyle R_{\nu}=\frac{m_{\nu}^{D}}{M_{R}}\sim 10^{-5}\sqrt{\frac{1{\rm
GeV}}{M_{R}}}.$ (32)

For $M_{R}\sim 100$ GeV, $|R_{\nu}|$ has value at order of $10^{-6}$. The same
magnitude $|R_{\ell}|\sim 10^{-6}$ is analogously obtained if mirror charged
lepton mass matrix is supposed not to be larger than the EW scale. Note that
it is also relevant to estimate
$|\tilde{R}_{\ell}|=|U_{\ell}^{\dagger}R_{\ell}U_{\ell}^{M}|$, as well as
$|\tilde{R}_{\nu}|=|U_{\ell}^{\dagger}R_{\nu}U_{\ell}^{M}|$, to be at the same
order as $|R_{\ell(\nu)}|\sim 10^{-6}$, due to the basis transformation
matrices $U_{\ell}$ and $U_{\ell}^{M}$ are normalized.

Before carrying on discussions that are distinctive for the current model,
let’s re-obtain the contribution of light active neutrinos and W-boson 1-loop
diagrams by applying eq. (31) for the corresponding interacting couplings
while keeping in mind the unitarity of PMNS matrix. The outcome arrives at

$\Delta a_{\mu}^{\rm
SM}({\chi_{L}})=\frac{\alpha}{8\pi\sin^{2}\theta_{w}}\frac{m_{\mu}^{2}}{M_{W}^{2}}G_{\gamma}(\lambda_{\chi_{L}})=\frac{G_{F}m_{\mu}^{2}}{4\sqrt{2}\pi^{2}}G_{\gamma}(\lambda_{\chi_{L}})\simeq\frac{G_{F}m_{\mu}^{2}}{8\sqrt{2}\pi^{2}}\left(\frac{5}{3}+O(\lambda_{\chi_{L}})\right),$
(33)

which entirely coincides with the result given in the PDG book [35].

Other contributions to the magnetic dipole moment by new physics under the
considered scenario and involving light neutrino and W-boson interactions are
very small and therefore ignorable. This fact is easy to figure out by looking
at these three contributions, which are the last term in (25) and the two last
terms in (26). The interference terms, which contain ratio $m_{k}/m_{\mu}$,
are strongly suppressed by a factor of $m_{k}/m_{\mu}\sim 10^{-6}$, where
$m_{k}<0.1{\rm eV}$ for light active neutrino mass and $m_{\mu}=106{\rm MeV}$
have been used; while the rest gets a tiny value due to being proportional to
$U^{R\dagger}_{W_{\mu}}U^{R}_{W_{\mu}}\sim\tilde{R}_{\nu}^{*}\tilde{R}_{\nu}^{T}\ll
U^{L\dagger}_{W_{\mu}}U^{L}_{W_{\mu}}$.

Figure 2: The correlation between
$\left(U_{W}^{ML\dagger}U_{W}^{ML}\right)_{\mu\mu}$ and heavy neutrino mass
$m_{\chi}^{M}$ when muon anomalous magnetic dipole moment $\Delta a_{\mu}$ is
set at its current best fit value; the channel of virtual W-boson and heavy
neutrinos.

In figure 2, we show the correlation between
$\left(U_{W}^{ML\dagger}U_{W}^{ML}\right)_{\mu\mu}$ and heavy neutrino mass
$m_{\chi}^{M}$ when muon anomalous magnetic dipole moment is set at its
current experimental best fit value $\Delta a_{\mu}=251\times 10^{-11}$, for
the channels with participation of virtual W-boson and heavy neutrinos. The
figure shows that the channel contribution is significant only if
$\left(U_{W}^{ML\dagger}U_{W}^{ML}\right)_{\mu\mu}$ has magnitude about 0.1 or
larger; however the real value is extremely tiny due to
$U_{W}^{ML\dagger}U_{W}^{ML}\sim\tilde{R}_{\nu}^{\dagger}\tilde{R}_{\nu}\sim
10^{-12}$.

We continue next with the contributions of 1-loop diagrams with virtual singly
negative charged Higgs $H^{-}$. In contrast to the previously considered
cases, the interference terms are enhanced by factor $m_{k}/m_{\mu}\sim 1000$,
where $m_{k}\sim 100$ GeV and $m_{\mu}=106$ MeV; therefore strongly dominates
in comparison with the others. To have a more intuitive understanding, we
present the correlations between relevant Yukawa couplings as a function of
neutrino masses of the channels in which virtual singly charged Higgs took
part in the loops when muon anomalous dipole moment is fixed at the best-fit
value. These correlations are presented for the cases if only the first term
of (25) and (26) are taken into account (Fig.3) and all terms are considered
(left-panel, Fig.4). The Yukawa coupling absolute values obtained in Fig.4 are
about three orders smaller than those in Fig.3, which are certainly consistent
with the mentions in earlier parts. The negative sign of the horizontal axis
in Fig.4 shows that the interference terms, thus the channel, would give
subtractions to the muon anomalous dipole moment if the involving Yukawa
couplings are positive.

Figure 3:  $(Y^{\dagger}_{H^{-}}Y_{H^{-}})_{\mu\mu}$ as function of physical singly charged Higgs scalar mass at some specific values of mirror neutrino masses; if only left or right sector is taken into account, $\Delta a_{\mu}$ is set at the present best fit value. |
---|---
Figure 4: The correlations between Yukawa couplings and : i, Singly charged
Higgs mass (left-panel) for $m_{\chi^{M}_{L}}=80~{}(200)~{}{\rm GeV}$, blue
(red) lines; ii, Heavy neutrino masses (right-panel) for
$m_{H^{-}}=70~{}(300)~{}{\rm GeV}$, blue (red) lines, when $\Delta
a_{\mu}=251\times 10^{-11}$ is fixed.

Ignoring the negative sign, the smallest absolute values of Yukawa couplings,
which are easily figured out to be at $m_{H^{-}}=70$ GeV, obtained from Fig.4
left-panel are

$|((Y_{H^{-}}^{ML})^{\dagger}Y_{H^{-}}^{MR}+(Y_{H^{-}}^{MR})^{\dagger}Y_{H^{-}}^{ML})_{\mu\mu}|\simeq
5.29\times 10^{-4}(7.02\times 10^{-4}),$ (34)

for $m_{\chi^{M}_{L}}=80~{}(200)~{}{\rm GeV}$. These results are apparently
available for both cases of $H_{3}^{-}$ and $H_{3M}^{-}$. Magnitude of
$|((Y_{H^{-}}^{ML})^{\dagger}Y_{H^{-}}^{MR}+(Y_{H^{-}}^{MR})^{\dagger}Y_{H^{-}}^{ML})_{\mu\mu}|$,
in fact, might be estimated basing on the model scheme and supposing that
mirror charged lepton masses are about 100GeV, the calculation implies

$|((Y_{H^{-}}^{ML})^{\dagger}Y_{H^{-}}^{MR}+(Y_{H^{-}}^{MR})^{\dagger}Y_{H^{-}}^{ML})_{\mu\mu}|\sim
10^{-12}\times\left(\frac{s_{M}}{s_{2}c_{M}^{2}}\right)\frac{6m_{\mu}m_{\ell}^{M}}{M_{W}^{2}}\sim
7.0\times 10^{-15}\left(\frac{s_{M}}{s_{2}c_{M}^{2}}\right),$ (35)

for the case of $H^{-}_{3M}$; and $\sim 7.0\times
10^{-15}\left(\frac{s_{M}}{c_{M}}\right)^{2}$; if the negative scalar
participating in loops is $H^{-}_{3}$. Thus the real values are extremely tiny
($<10^{-8}$), even if $s_{2}$ and $c_{M}$ of the mixing angles are as small as
0.01. The above analysis figure out a fact that contributions of the singly
charged Higgs scalar channel to the muon anomalous dipole moment are too small
to be able to explain the muon anomalous magnetic dipole moment experimental
results.

|
---|---
Figure 5: The correlations between Yukawa couplings and : i, Heavy neutral
Higgs scalar masses (left-panel); ii, Mirror charged lepton masses (right-
panel) for diagrams, whose particles running inside loops are heavy physical
scalars and mirror charged leptons, $\Delta a_{\mu}=251\times 10^{-11}$ is
fixed.

The correlations between the Yukawa couplings involving the heavy neutral
Higgs scalars (which are ${\tilde{H}}_{i}^{0}$, (i=1,2,3), $H_{3}^{0}$ and
$H_{3M}^{0}$) and either the Higgs or mirror charged lepton masses are
respectively shown in the Fig.5 left (or right) panel. The discussion on light
Higgs channel will be presented in a latter separate part due to the enormous
difference in mass hierarchies. The same as previous considered case of
$H^{-}$ channels, contributions of the diagrams involving neutral scalar
(including both light and heavy ones) diagrams are predominated by the mixing
terms due to the heaviness in masses of mirror charged leptons. To explain the
muon anomalous dipole moment, magnitude of
$(Y_{H^{0}}^{ML\dagger}Y_{H^{0}}^{MR})_{\mu\mu}$ requires a value within
$10^{-4}-10^{-3}$ range, which slightly increase with the increasing of
neutral Higgs scalar mass (Fig.5, left panel). At the initial points of the
lines, corresponding to $m_{H^{0}}\simeq 50$GeV, we have

$(Y_{H^{0}}^{ML\dagger}Y_{H^{0}}^{MR})_{\mu\mu}\simeq 1.28\times
10^{-4}(2.61\times 10^{-4}),$ (36)

for $m_{m_{\ell}^{M}}=80~{}(200)~{}{\rm GeV}$, respectively. Carry on the same
strategy as in earlier part, the theoretical estimation can be performed using
eqs. from (18) to (23). The results arrive at

$(Y_{H^{0}}^{ML\dagger}Y_{H^{0}}^{MR})_{\mu\mu}\sim
2\alpha^{2}\frac{\left(\tilde{R}_{\ell}^{\dagger}(m_{\ell
M}^{d})^{2}\tilde{R}_{\ell}\right)_{\mu\mu}}{M_{W}^{2}}\sim
4.69\alpha^{2}\times 10^{-12},$ (37)

where mirror charged lepton masses are taken about $100$GeV, $\alpha$ denotes
for $\frac{\alpha_{i}}{s_{2}}$, $\frac{s_{M}}{c_{M}}$ or
$\frac{s_{2}}{s_{2M}}$, corresponding to ${\tilde{H}}_{i}^{0}$, (i=1,2,3),
$H_{3}^{0}$ or $H_{3M}^{0}$, respectively. Notes that to obtain eq. (37),
first terms of the Yukawa couplings defined from (18) to (23), which contain
$m_{\ell}^{d}$ thus sub-dominate in comparison to the second ones with
$m_{\ell M}^{d}$, are reasonably neglected. Equation (37) also means the
currently considered channels might provide contributions to the muon
anomalous magnetic dipole moment about three orders higher than those of the
singly charged scalars in magnitude. In fact, real values of charged lepton
masses can be larger at order of several hundreds GeV. For instance, if
$m_{\ell}^{M}=500$ GeV is taken,
$(Y_{H^{0}}^{ML\dagger}Y_{H^{0}}^{MR})_{\mu\mu}\sim 1.17\alpha^{2}\times
10^{-10}\sim 1.12\times 10^{-6}$ for $\alpha=100$, which occurs at
$s_{2}=0.01$, $c_{M}=0.01$, or $s_{2M}=0.01$, corresponding to the case of
${\tilde{H}}_{i}^{0}$, (i=1,2,3), $H_{3}^{0}$ or $H_{3M}^{0}$, respectively.
Therefore, contributions of the heavy neutral Higgs scalar channels can not be
able to explain the muon anomalous magnetic dipole moment, but they might be
possible to provide sizable corrections.

|
---|---
Figure 6: The same content as Fig.5, for the case of diagrams with loops
formed by light Higgs scalar and mirror charged leptons.

The same contents as Fig.5 are presented in Fig.6, in which heavy neutral
Higgs masses are replaced by those of the light one at lower scale from keV to
order of GeV. At a given value of $m_{\ell}^{M}$, magnitude of
$((Y^{L}_{\phi_{s}^{0}})^{\dagger}Y^{R}_{\phi_{s}^{0}})_{\mu\mu}$ does not
change with the increase in light Higgs scalar mass until about 10 GeV, then
slowly increase. From Fig.5 left panel, we easily obtain from the constant
lines

$((Y^{L}_{\phi_{s}^{0}})^{\dagger}Y^{R}_{\phi_{s}^{0}})_{\mu\mu}\simeq
8.75\times 10^{-5}(1.13\times 10^{-4}),$ (38)

corresponding to $m_{m_{\ell}^{M}}=80~{}(200)~{}{\rm GeV}$, respectively.
After a simple computation, the result arrives at relations of Yukawa coupling
matrix

$|\tilde{g}_{\ell s}^{\dagger}\tilde{g}_{\ell s}|_{\mu\mu}\simeq 5.39\times
10^{-5}(6.96\times 10^{-5}),$ (39)

which are five orders larger than the upper constraints obtained by current
experimental bound of $\mu\to e\gamma$ decay [11]111Here we have recast eq.
(51) in [11] to have similar form as (39) for more convenient in comparison.

$|\tilde{g}_{\ell s}^{\dagger}\tilde{g}_{\ell s}|_{\mu e}\simeq 5.29\times
10^{-10}(1.30\times 10^{-9}).$ (40)

The two equations (39) and (40) are easily concurrently fulfilled; if
$\tilde{g}_{\ell s}$ is proportional to a matrix, which is close to unitary
form. Beside being constrained by muon anomalous dipole moment and the rare
lepton flavor violation decay $\mu\to e\gamma$, $g_{\ell s}$ is also
constrained by searching new particles at the LHC, especially new scalars in
the high mass region. The previous publication [7] studied the decay
$H_{3}^{0}\rightarrow\bar{\ell}^{M}\ell^{M}\rightarrow\bar{\ell}\phi_{s}\ell\phi_{s}^{*}$,
where $\phi_{s}$ is invisible and considered as missing transverse energy.
This process is identified with 2 leptons and missing energies, which imitates
the signal of scalar decay $H_{SM}^{0}\rightarrow
W^{-}W^{+}\rightarrow\bar{\ell}\nu\ell\bar{\nu}$. No excess over the
background was detected at both ATLAS and CMS, however an upper constraint
$|g_{\ell s}|^{2}\leq 10^{-6}$ was obtained (see [7]). This limit is smaller
than the required values obtained in (39), therefore it disfavors the
possibility to explain the discrepancy of the muon magnetic dipole moment with
the standard model prediction. For more quantitative view, if $|g_{\ell
s}|^{2}=10^{-6}$ is taken, the channel of light neutral scalar might
contribute to the muon anomalous magnetic dipole moment a deviation as

$\Delta a_{\mu}({\phi_{s}^{0}})\simeq 4.7\times 10^{-11},$ (41)

which is about size of the uncertainty of the two-loop electroweak
contribution and smaller than those of the hadronic effects. Therefore, the
contribution of this model to the muon problem is currently not important.
However, it might have a role in the future when we have more precise
measurements on the muon magnetic dipole moment.

##  4  Conclusion

Recent experimental measurements on muon magnetic dipole moment show
substantial discrepancies with standard model predictions. These differences
might be understood in scenarios of physics beyond the standard model with new
particles and interactions. In this work, we have derived an algebraic formula
and performed numerical analysis for muon anomalous magnetic dipole moment at
1-loop approximation in an extended model with mirror symmetry, in which the
light neutrino masses are generated by the type-I see-saw at low scale of the
electroweak. We have shown that the contributions provided by neutrino and
either W boson or singly charged Higgs scalar channels are too small to be
taken into account. Moreover, the channel of heavy neutrinos and singly
charged Higgs also gives a contribution opposite to that of the rest. For the
case that particles running inside loops are heavy neutral Higgs scalars and
mirror charged leptons, although contribution of this channel is not able to
explain the experimental results, it might provide a sizable amount of
correction to the muon anomalous magnetic dipole moment; if at least one of
the following quantities $s_{2}$, $c_{M}$, or $s_{2M}$ has the value as small
as 0.01 or less. As the most promising case, the channel involving light
neutral Higgs scalar is expected to be able to explain the muon anomalous
problem, however the required magnitude of $g_{\ell s}$ is larger than its
upper limit obtained from experimental search on heavy new scalars. In any
case, the contribution of this channel might become more important in the
future when the uncertainties of the two-loop electroweak and hadronic
contributions are improved.

## Acknowledgments

This research is funded by Vietnam National Foundation for Science and
Technology Development (NAFOSTED) under grant number 103.01-2019.307.

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