# The Lepton Flavor Changing Decays and One-loop Muon Anomalous Magnetic
Moment in the Extended Mirror Twin Higgs Models

Guo-Li<EMAIL_ADDRESS>Fei<EMAIL_ADDRESS>Wenyu
<EMAIL_ADDRESS>1 School of Physics and Microelectronics,
Zhengzhou University, Zhengzhou, 450001, China
2 Institute of Theoretical Physics, Faculty of Science, Beijing University of
Technology, Beijing 100124, China

###### Abstract

Mirror Twin Higgs(MTH) models always contain heavy gauge bosons and extra
Higgses. Besides, to accommodate tiny neutrino masses via seesaw mechanism,
new heavy neutrinos can also be introduced in MTH extension models. Such new
particles and interactions may lead to new contributions to the lepton flavor
violating (LFV) processes, including $\ell_{i}\to\ell_{j}\gamma$ and
$\ell_{i}\to\ell_{j}\ell_{k}\ell_{l}$. We find that current experimental data
can stringently constrain the parameter spaces and certain LFV processes can
possibly be tested by the next generation colliders. One-loop contributions of
the new particles to the muon anomalous magnetic momentum are also calculated.
Such contributions can still not solve the discrepancy between the experiments
and the prediction of the standard model.

lepton flavor violation, branching ratio, Mirror Twin Higgs models

###### pacs:

13.38.Dg,14.66.-z,12.60.-i

##  I Introduction

In the Standard Model (SM), the conservation of individual lepton flavor
numbers and the presentation of the GIM mechanism GIM-1970  guarantee that the
LFV processes in the SM are tiny. For example, the SM prediction of the branch
ratio $BR_{SM}(l_{j}\to l_{i}\gamma)\sim 10^{-55}$ SMljlig  is very small,
which would never be probed at the colliders. Therefore, it will be an obvious
evidence of new physics beyond the SM if we observe any signature of the LFV
processes in future experiments, so LFV processes are very efficient in
exploring new physics beyond the SM. As the upper bounds of the LFV processes
by current experiments have been given with impressive accuracy, it is
interesting to explore whether theoretical predictions of a typical new
physics model can agree with the experimental results.

Many works have been completed to look for the LFV processes
$\ell_{i}\to\ell_{j}\gamma$, $\ell_{i}\to\ell_{j}\ell_{k}\ell_{l}$ in new
physics models, see e.g. bSM1 ; bSM2 ; bSM3 ; bSM4 ; bSM5 ; bSM6 ; 1701.00947
; 1812.03860 ; 1812.02449 ; 1912.05900 ; 0506070 ; 0307126 ; 0206056 . At the
same time, additional interactions involving the leptons may contribute to the
muon anomalous magnetic momentum. The presence of additional contributions to
$g_{\mu}-2$ are welcome to solve the discrepancy between the theoretical
predictions of SM and the experiments. Therefore, it is well motivated to see
if the anomaly can be solved in various new physics models g-2-review1 ;
g-2-review2 ; g-2-review3 ; g-2-review4 ; g-2-review5 ; 1811.04777 .

It is well known that the neutrino oscillation experiments neutrino1 ;
neutrino2 ; neutrino3 ; neutrino4 ; neutrino5 ; neutrino6 ; neutrino7 ;
neutrino8 ; neutrino9  indicate that neutrinos possess tiny masses and can
convert into each other. So individual lepton flavor numbers
$L_{i}=L_{e},~{}L_{\mu},~{}L_{\tau}$ may be violated at the electroweak scale.
Weinberg’s effective dimension-5 operator is the lowest one that can generate
tiny Majorana-type neutrino masses. Such an operator can be
ultraviolet(UV)-completed to obtain three types of tree-level seesaw
mechanism: type I seesaw type I , involving the exchange of right-handed
neutrinos; type II seesaw type II , involving the exchange of scalar triplet;
type III type III , involving the exchange of fermion triplet. The neutrino
seesaw mechanisms can also be embedded into various well motivated new physics
models to accommodate the neutrino masses, such as the Mirror Twin Higgs (MTH)
model litt-hier1 .

MTH, which extends the SM with a mirror copy, can stabilize the Higgs mass
from quadratic divergent quantum corrections by a discrete mirror symmetry.
The new particles, which are related to the SM particles by this discrete
symmetry, do not carry the SM color charge and can hardly be produced and
detected at the colliders, ameliorating the stringent constraints on the mass
of the top partner by the Large Hadron Collider(LHC) parti-tev . Although MTH
is theoretically appealing, it encounters difficulties by cosmological
considerations. In the simplest realization of MTH litt-hier1  model, the twin
particles will eventually transfer their entropies into twin photons and twin
neutrinos, which behave as extra radiation components and contribute to the
expansion of the Universe. We know that the SM neutrinos will contribute to
the effective degrees of freedom $N_{eff}$ by an amount $N_{eff}\sim 3.046$
neff_sm-1 ; neff_sm-2 . In the MTH model, the Twin photons and Twin neutrinos,
which are the extra relativistic species, increase the SM prediction by an
additional amount $\Delta N_{eff}\sim 5.6$ neff_mth . However, Big bang
nucleosynthesis(BBN), anisotropies in the cosmic microwave background (CMB) by
the Planck 2018 constrained $N_{eff}\sim 3$ neff_exp1 ; neff_exp2 , which
challenges MTH.

Various modifications are proposed to reduce the $N_{eff}$ value and solve the
problems of MTH. For example, the first and second generations of twin
fermions as well as twin photon can be absent modif-mth1 . The twin neutrinos
can also be heavy modif-mth2 ; modif-mth3 . In certain circumstances, even the
Twin photon can be heavy without affecting naturalness modif-mth4 ; modif-mth5
. Asymmetric entropy production can be possible after the Twin and SM sectors
decouple neff_mth ; modif-mth3 ; modif-mth6 ; modif-mth7 .

In the seesaw extension models of MTH, extra massive right-handed neutrinos in
both sectors can be introduced to generate tiny neutrino masses. Typical
lepton-flavor-changing couplings may naturally appear in this mechanism, which
can lead to interesting phenomenological consequences and possibly constrain
some parameters in this model. Besides, as new interactions involving the
leptons may contribute to the muon anomalous magnetic momentum, it is
interesting to see if the discrepancy on the $g_{\mu}-2$ anomaly can be
ameliorated in this model. Therefore, we will study the LFV processes
$\ell_{i}\to\ell_{j}\gamma$, $\ell_{i}\to\ell_{j}\ell_{k}\ell_{l}$ and the
muon anomalous magnetic moment $a_{\mu}$ at the one-loop level in this work.

The paper is organized as follows. In section II, the MTH model will be
extended to include new LFV couplings. We will discuss the new structures and
the effective couplings. One-loop contributions to the muon anomalous magnetic
moment $a_{\mu}$ will be calculated in section III. In section IV, the
amplitudes and the branching ratios of rare LFV processes
$\ell_{i}\to\ell_{j}\gamma$, $\ell_{i}\to\ell_{j}\ell_{k}\ell_{l}$ will be
calculated, respectively. The corresponding numerical results are also given.
Section V summarizes and concludes the whole work.

##  II The MTH models and the lepton flavor changing couplings

###  II.1 The general discussion of the model and the LFV couplings

In the MTH framework, the particle contents include two parts: the SM
particles and its mirror copy of all the SM fields. The SM particles and the
mirror sectors are related by the discrete $Z_{2}$ twin symmetry. The global
symmetry of the Higgs sector in the simplest realization can be taken to be
$SU(4)\times U(1)$. Both the global symmetry and the discrete symmetry are
approximate. The global symmetry contains the SM electroweak gauge symmetries
and twin gauge symmetries. The pseudo-Nambu-Goldstone bosons (pNGBs), which
arises from the spontaneously breaking of the the global $SU(4)\times U(1)$
symmetry to $SU(3)\times U(1)$, contain the SM Higgs doublet. The Higgs mass,
under the joint action of the global symmetry and the discrete twin symmetry,
are protected from one loop quadratic divergence.

As noted previously, additional contributions to $N_{eff}$ of MTH will cause
difficulties in cosmology444In scenarios with multiple dark matter components,
small contributions to $N_{eff}$ can possibly solve the discrepancy between
the value of $H_{0}$ extracted from local measurements versus CMB data.. One
can impose another portal, such as the neutrino portal to reduce the $N_{eff}$
to the acceptable level. Meanwhile, new flavor-changing couplings can be
generated via the mixing between the SM neutrino and the Twin neutrino
1703.06884 ,

$\displaystyle\mathcal{L_{{\rm int}(\nu,\tilde{\nu})}}$ $\displaystyle=$
$\displaystyle\frac{g}{\sqrt{2}}\bar{\ell}\gamma^{\mu}P_{L}(c_{\theta}\nu+s_{\theta}\tilde{\nu})W^{+}_{\mu}+{\rm
h.c.}$ (1)
$\displaystyle+\frac{g}{2c_{w}}(c_{\theta}\bar{\nu}+s_{\theta}\bar{\tilde{\nu}})\gamma^{\mu}P_{L}(c_{\theta}\nu+s_{\theta}\tilde{\nu})Z_{\mu}\,,$

where $c_{\theta}=\cos\theta$, $s_{\theta}=\sin\theta$. Here $\theta$ is the
mixing angle between ordinary neutrino and twin neutrino. The LFV couplings in
Eq.(1) can origin from typical neutrino seesaw extension of MTH model via
introducing the massive right-handed neutrino 1611.07975 ,

${\cal L}\supset-
y_{ij}(L^{i}_{A}H_{A}N^{j}_{A}+L^{i}_{B}H_{B}N^{j}_{B})-\frac{1}{2}(M_{N})_{ij}(N_{A}^{i}N_{A}^{j}+N_{B}^{i}N_{B}^{j})-(M_{AB})_{ij}N_{A}^{i}N_{B}^{j}~{},$
(2)

in both the SM and twin sectors. The Weinberg dimension-5 operator, which was
used in some references ( such as Ref. 1512.05357 ), can be generated by
integrating out the heavy right-handed neutrinos.

###  II.2 The general form of the LFV couplings in the MTH models

The new particles, which induce the LFV couplings, could be new gauge bosons
$Z_{\mu}^{H}$, new scalars $H^{0},H^{\pm}$ and $N_{H}$ species of heavy
neutrinos $\nu_{H,i}$. The general forms of the LFV couplings from the
effective Lagrangian of leptonic sector Emam:2007dy ; 1105.1047  can be
summarized as 555 The gauge couplings involving heavy neutrino $\nu_{H,i}$ can
arise from the couplings of the form
$\bar{\nu}_{i;L}\gamma^{\mu}W_{\mu}l_{j;L}$. The left-handed neutrino, which
is the interaction eigenstate, can be written as the combination of mass
eigenstates involving the light and heavy neutrinos. We do not include in the
Lagrangian the terms involving the SM-like light neutrinos and keep only the
new contributions involving heavy neutrinos from typical neutrino mass
generation mechanisms. Similar discussions can be given for the Yukawa-type
interactions with $\nu_{H,i}$.

$\displaystyle{\cal L}$ $\displaystyle\supseteq$
$\displaystyle\frac{g_{2}}{\sqrt{2}}\sum\limits_{i=1}^{4N_{g}-3}\sum\limits_{j=1}^{3}V^{*}_{ji}{\bar{\nu}}_{H,i}\gamma^{\mu}W_{\mu}P_{L}\ell_{j}+y_{l_{j}}\sum\limits_{i=1}^{N_{h}}\sum\limits_{j=1}^{3}U^{*}_{ji}{\bar{\nu}}_{H,i}H^{+}P_{R}\ell_{j}+h.c.$
$\displaystyle+\lambda_{\ell}\bar{\ell}H^{0}\ell+g_{\ell}\bar{\ell}\gamma^{\mu}Z_{\mu}^{H}P_{L}\ell+\lambda_{ji}\bar{\ell_{i}}H^{0}\ell_{j}+\sum\limits_{i\neq
j}g_{ji}\bar{\ell_{i}}\gamma^{\mu}Z_{\mu}^{H}(f_{v}+f_{a}\gamma^{5})\ell_{j}+h.c.$

Here $g_{\ell},~{}\lambda_{\ell}$ are the couplings of the heavy neutral gauge
boson and the scalar to the charged lepton, respectively. The coefficients
$g_{ji},~{}\lambda_{ji}$ are the couplings between different lepton flavors.
We consider both the vector type($f_{v}$) and the axial-vector ($f_{a}$) type
couplings of $Z_{\mu}^{H}$ to different flavor generations.

The first term of eqn.(II.2) is a generalization of eqn.(1) from typical
neutrino seesaw extension of MTH model. In general, the neutrino mass matrix
$M$ will involve the left-handed neutrinos, the right-handed neutrinos and
their twin neutrino partners, which will be a $4N_{g}\times 4N_{g}$ matrix for
$N_{g}$ generation of neutrino species. The matrix $M$ is arranged so as that
its upper-left $3\times 3$ submatrix of $M$ denotes the ordinary Pontecorvo-
Maki-Nakagawa-Sakata (PMNS) mns-maki-1962 ; 0712.4019  neutrino mixing matrix.
The flavor changing couplings $V_{ji}$, which is just the $M_{nm}$ matrix
element of the most general $4N_{g}\times 4N_{g}$ unitary neutrino mixing
matrix $M$ with $1\leq n\leq 3$ and $4\leq m\leq 4N_{g}$, also characterize
the deviation of the PMNS matrix to unitary. It can be estimated to be
$\sum\limits_{j}|V_{ji}|^{2}\lesssim
1-\sum\limits_{i=1}^{3}|U_{Li}|^{2}\lesssim 0.01$ with $L=e,\mu,\tau$1105.1047
; deviation-to-unitary . As noted in 1703.06884 , the ”two seesaw” scenario
with type-I seesaw in both the SM and mirror sector can not induce large
mixing between SM neutrino and heavy mirror neutrino. The ”SM seesaw, Dirac
twin neutrinos” scenario can possibly induce large mixing to reduce the twin
sector’s contribution to the radiation density. The sum of the heavy neutrinos
in the first term of eqn.(II.2) can be understood to be dominated by the
mirror $\tilde{\nu}_{R}$ parts in this scenario. We will not specify the
nature of the heavy neutrinos in our following discussions and only adopt
their upper bounds in our numerical calculations.

The second term of eqn.(II.2) with extended Higgs sector can appear in certain
composite Twin Higgs models. The coupling can be decomposed as the production
of $y_{i}=m_{l_{i}}/\sqrt{2}v$ and $U_{ij}$, with the matrix $U_{ij}$ contains
$V_{ji}$ and the mixing between ordinary Higgs and exotic Higgs. In this work,
we will also take them as free parameters.

##  III One-loop contribution to the muon anomalous magnetic moment

The Brookheaven National Laboratory bennett  had measured the anomalous
magnetic moment of the muon

$a^{\rm exp}_{\mu}=11659208.0(6.3)\times 10^{-10},$ (4)

which has a $3.3\sigma$ to $3.6\sigma$ deviation from the prediction of the SM
discrepancy1 ; discrepancy2

$\Delta a_{\mu}=a^{\rm exp}_{\mu}-a^{\rm SM}_{\mu}=(28.3\pm 8.7\ {\rm to}\
28.7\pm 8.0)\times 10^{-10}.$ (5)

From the couplings of Eq. (II.2), one can easily calculate the new
contributions to the muon anomalous magnetic moment which are generated by one
loop diagrams involving the exchange of $W_{\mu}$ gauge boson/charged scalar
$H^{\pm}$ and heavy neutrino, or $Z_{\mu}^{H}$ gauge boson and muon, or
$H^{0}$ neutral scalar and muon, as shown in Fig.(1).

Note that the contributions from Fig.(1)(e)(f) contain self energy of the
fermions, which will disappear due to the electromagnetic gauge invariance.

We can define $a_{\mu}^{TH}$ as

$a_{\mu}^{TH}=a^{\nu W}_{\mu}+a^{\nu
H}_{\mu}+a^{Z^{0}}_{\mu}+a^{H^{0}}_{\mu}.$ (6)

The contribution to muon anomalous magnetic moment $a_{\mu}$ involving $W$ and
the massive neutrino loop (see Fig.1a) can be written as

$a^{\nu
W}_{\mu}\simeq\frac{V_{\mu}^{2}em_{\mu}}{64\pi^{2}m_{W}^{4}}\sum_{k=1}^{3}{V^{*}_{\mu
k}V_{\mu k}m_{\nu_{k}}^{2}}.$ (7)

The contribution to $a_{\mu}$ involving the charged Higgs $H^{\pm}$ and the
heavy neutrino loops (see Fig.1(b)) can be written as

$a^{\nu
H}_{\mu}\simeq-\frac{y_{\mu}^{2}em_{\mu}}{32\pi^{2}m_{H^{\pm}}^{4}}\frac{5}{6}\sum_{k=1}^{3}U^{*}_{\mu
k}U_{\mu k}m_{\nu_{k}}^{2}.$ (8)

We should note that the contribution involving the charged Higgs $H^{\pm}$ and
the light neutrino loops are much smaller than this contribution for not very
small $m_{\nu,H}$.

The contribution to $a_{\mu}$ involving $Z_{\mu}^{H}$ (see Fig.1(c)) can be
written as 1105.1047

$a^{Z^{0}}_{\mu}=\frac{g^{2}_{\mu}}{4\pi^{2}}\frac{m^{2}_{\mu}}{M^{2}_{Z_{H}^{0}}}\bullet\frac{1}{3}.$
(9)

Finally, the contribution to $a_{\mu}$ involving the extra Higgs can be given
by 1105.1047

$a^{H^{0}}_{\mu}=\frac{\lambda^{2}_{\mu}}{32\pi^{2}}\frac{2m^{2}_{\mu}}{m^{2}_{H^{0}}}\bullet\frac{2}{3}.$
(10)

Figure 1: The triangle and the penguin type diagrams for the muon anomalous
magnetic moment at the one-loop level(with both the final and the initial
particles being muons) and the decays $\ell_{i}\to\ell_{j}\gamma$( with the
final state fermion line being $\ell_{j}$ and the initial state line being
$\ell_{i}$).

We can estimate the contributions of these four terms. If we assume that the
heavy neutrinos are degenerate with $m_{1}=m_{2}=m_{3}=m_{\nu,H}$, the term
$\sum_{k=1}^{3}V^{*}_{\mu k}V_{\mu k}m_{\nu_{H,k}}^{2}$ in $a^{\nu W}_{\mu}$
can be estimated to be $m_{\nu,H}^{2}\sum_{k=1}^{3}|V_{\mu k}|^{2}\lesssim
0.01m_{H,\nu}^{2}$. Similarly, $\sum_{k=1}^{3}U^{*}_{\mu k}U_{\mu
k}m_{\nu_{k}}^{2}$ can be estimated to be less than
$0.01m_{H,\nu}^{2}\sin^{2}\zeta$, with $\zeta$ the mixing between ordinary
Higgs and exotic Higgs. The couplings can be given as $y_{l}=m_{l}/v$ and
$em_{\mu}/(64\pi^{2})\sim 5.4\times 10^{-5}$, respectively. If we take
$m_{H^{\pm}}\sim 300~{}GeV$ and the mixing angle $\sin\zeta\lesssim 0.01$, the
contributions $a^{\nu W}_{\mu}$ and $a^{\nu H}_{\mu}$ can be estimated to be
$4.6\times 10^{-20}m_{\nu}^{2}$ and $0.57\times 10^{-20}m_{\nu}^{2}$,
respectively. Thus the magnitude of the contributions depend on the mass scale
of the heavy neutrino mass $m_{\nu,H}$, which should be larger than $10^{5}$
TeV for them to account for the discrepancy (see Eq.(5)). However, too large
mass of the heavy (mirror right-handed like) neutrino will induce large mirror
$Z_{2}$ breaking, reintroducing quadratic sensitivity to the cutoff scale into
the Higgs mass term. With very large neutrino mass, the contributions of the
charged gauge boson and the charged Higgs at one-loop level can still not
possibly account for the discrepancy between the experiments and the
theoretical predictions, which can be seen in Fig.2(a). So we will not take
such a heavy neutrino.

We can fix the following mass parameters $m_{\nu}=2000$ GeV, $m_{H^{+}}=300$
GeV and vary the one-loop muon anomalous magnetic moment with the mixing
parameters $V_{\mu}$ and $y_{\mu}$.The results given in Fig.2(b) also indicate
that it is difficult to explain the discrepancy between the experiments and
the theoretical prediction.

Since there is a minus sign before the charged scalar contribution to the muon
anomalous magnetic moment, the curve of Fig.2(b) becomes flat, which can be
explained by the formulae given in Eq.(7) and Eq.(8). Though there seems a
large decline in the curve of $a_{\mu}$ verse mixing $y_{\mu}$, the decline
value of muon anomalous magnetic moment actually is quite small, from
$1.38\times 10^{-11}$ to $1.2\times 10^{-11}$.

(a)

(b)

Figure 2:  The one-loop contributions of the LFV couplings to the the muon
anomalous magnetic moment.

The contributions of the neutral gauge boson and the neutral Higgs to the muon
anomalous magnetic moment are of order $1.02\times 10^{-10}$ and $5.7\times
10^{-10}$, respectively, even with $g\sim{\cal O}(1)$, $\lambda\sim{\cal
O}(1)$, $m_{Z_{H}^{0}}\sim 1TeV$ and $m_{H^{0}}\sim 300~{}GeV$. They are too
small to account for the discrepancy and can be neglected safely. Of course,
the couplings can not be chosen to be ${\cal O}(1)$ with the suppression of
the muon mass.

So we conclude from Fig.(2) that, as long as the heavy neutrino mass is not
too large, the contributions of both the gauge bosons and the scalars to the
one-loop muon anomalous magnetic moment can not account for the discrepancy.

##  IV The processes $\ell_{i}\to\ell_{j}\gamma$ and
$\ell_{i}\to\ell_{j}\ell_{k}\ell_{l}$

In this section, we analyze the branching ratios of certain LFV processes in
the MTH model, including $\mu\to e\gamma$, $\tau\to e(\mu)\gamma$, $\mu\to
3e$, $\tau\to 3e$, $\tau\to 3\mu$.

###  IV.1 The width and the Branching ratios of the rare decay
$\ell_{i}\to\ell_{j}\gamma$

The Feynman diagrams for $\ell_{i}\to\ell_{j}\gamma$, which can be mediated by
the gauge bosons, the neutral Higgs and the charged Higgs, are shown in Fig.1.
The initial and the final leptons are $\ell_{i}$ and $\ell_{j}$, respectively.

The effective amplitude of $\ell_{i}\to\ell_{j}\gamma$, which are mediated by
the $W$ and the massive neutrinos (corresponding to the Fig.1(a)), can be
written as

$M^{\nu
W}(\ell_{i}\to\ell_{j}\gamma)\simeq\frac{ieV_{\mu}V_{e}^{*}m_{i}}{32\pi^{2}}\bullet\bar{u}(p)(C_{12}-2C_{21}+2C_{23})(1+\gamma^{5})[2(p^{\prime}\cdot\epsilon)]u(p^{\prime}).$
(11)

where
$C_{ij}=C_{ij}(p_{\gamma},-p^{\prime},m_{i}^{2},m_{i}^{2},m_{Z^{0}}^{2}(m_{H}^{2}))$
are the one-loop three point functions looptools  and the final state lepton
masses are neglected with $m_{j}=0$.

Similarly, we can obtain the effective amplitude mediated by the charged Higgs
$H^{\pm}$ and the massive neutrinos (corresponding to the Fig.1(b))

$M^{\nu
H}(\ell_{i}\to\ell_{j}\gamma)\simeq\frac{iey_{\mu}y_{e}^{*}m_{i}}{32\pi^{2}}\bullet\bar{u}(p)(C_{21}-C_{23}+C_{11}-C_{12})(1+\gamma^{5})[2(p^{\prime}\cdot\epsilon)]u(p^{\prime}).$
(12)

Similarly, the contribution of the heavy neutral $Z_{H}^{0}$ and the heavy
Higgs $H^{0}$ to $\ell_{i}\to\ell_{j}\gamma$ can be written as

$\displaystyle M^{Z_{H}^{0}}(\ell_{i}\to\ell_{j}\gamma)$ $\displaystyle=$
$\displaystyle\frac{ieg_{\mu}g_{\mu
e}m_{i}}{16\pi^{2}}\bullet\bar{u}(p)[2(f_{a}-f_{v})(C_{22}-C_{23}-C_{12})P_{R}~{},$
$\displaystyle-4(f_{v}+f_{a})C_{12}P_{L}][2(p^{\prime}\cdot\epsilon)]u(p^{\prime})$
$M^{H^{0}}(\ell_{i}\to\ell_{j}\gamma)=\frac{-ie\lambda_{\mu}\lambda_{\mu
e}}{16\pi^{2}}m_{i}(C_{22}-C_{23}-C_{12}-C_{0})\bullet\bar{u}(p)[2(p^{\prime}\cdot\epsilon)]u(p^{\prime})~{},$
(13)

where $f_{v},~{}f_{a}$ are the vector and the axial-vector coupling
coefficients of the heavy neutral gauge boson to leptons, respectively.

So the total amplitude can be given as

$\displaystyle M_{total}$ $\displaystyle=$ $\displaystyle M^{\nu W}+M^{\nu
H}+M^{Z^{0}}+M^{H^{0}}$ (14)
$\displaystyle=\bar{u}(p)\cdot\frac{iem_{i}}{32\pi^{2}}{\large\\{}V_{\mu}V_{\mu
e}^{*}(C_{12}-2C_{21}+2C_{23})(1+\gamma^{5})$ $\displaystyle+y_{\mu}y_{\mu
e}^{*}(C_{21}-C_{23}+C_{11}-C_{12})(1+\gamma^{5})$
$\displaystyle+2g_{\mu}g_{\mu
e}[-2(f_{v}-f_{a})(C_{22}-C_{23}-C_{12})P_{R}-4(f_{v}+f_{a})C_{12}P_{L}]$
$\displaystyle-2\lambda_{\mu}\lambda_{\mu}(C_{22}-C_{23}-C_{12}-C_{0}){\large\\}}\cdot
2(p^{\prime}\cdot\epsilon)u(p^{\prime})$
$\displaystyle=m_{i}\cdot\bar{u}(p)(A+B\gamma^{5})\cdot
2(p^{\prime}\cdot\epsilon)]u(p^{\prime}),$

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{ie}{32\pi^{2}}\\{V_{\mu}V_{\mu
e}^{*}(C_{12}-2C_{21}+2C_{23})+y_{\mu}y_{\mu
e}^{*}(C_{21}-C_{23}+C_{11}-C_{12})$ $\displaystyle+2g_{\mu}g_{\mu
e}[-(f_{v}-f_{a})(C_{22}-C_{23}-C_{12})-2(f_{v}+f_{a})C_{12}]$
$\displaystyle-2\lambda_{\mu}\lambda_{\mu e}(C_{22}-C_{23}-C_{12}-C_{0})~{},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{ie}{32\pi^{2}}\\{V_{\mu}V_{\mu
e}^{*}(C_{12}-2C_{21}+2C_{23})+y_{\mu}y_{\mu
e}^{*}(C_{21}-C_{23}+C_{11}-C_{12})$ $\displaystyle+2g_{\mu}g_{\mu
e}[-(f_{v}-f_{a})(C_{22}-C_{23}-C_{12})+2(f_{v}+f_{a})C_{12}]\\}~{}.$

So, the decay width is given as

$\displaystyle\Gamma(\ell_{i}\to\ell_{j}\gamma)=\frac{m^{3}_{i}}{8\pi}\cdot
m_{i}^{2}(|A|^{2}+|B|^{2})=\frac{m^{5}_{i}}{8\pi}(|A|^{2}+|B|^{2}).$ (17)

####  IV.1.1 The process $\mu\to e\gamma$

The width of the dominant muon-decay mode $\mu\to e\nu\bar{\nu}$ is given as,

$\Gamma^{\mu}=\Gamma(\mu\to e\nu\bar{\nu})\simeq
m^{5}_{\mu}G^{2}_{F}/(192\pi^{3})~{},$ (18)

where $G_{F}=\frac{\sqrt{2}g^{2}}{8m_{W}^{2}}$ is the Fermi coupling constant.
So the branching ratios of the rare decay $\mu\to e\gamma$ is given by

$\displaystyle BR(\mu\to e\gamma)=\frac{\Gamma(\mu\to e\gamma)}{\Gamma(\mu\to
e\nu\bar{\nu})}=\frac{24\pi^{2}}{G_{F}^{2}}(|A|^{2}+|B|^{2}),$ (19)

Figure 3: With $V_{\mu e}=y_{\mu e}=g_{\mu e}=\lambda_{\mu e}=10^{-5}$, the
branching ratios of $\mu\to e\gamma$ versus parameters
$~{}m_{H^{\pm}},~{}m_{Z_{H}^{0}},~{}m_{H^{0}}$ are plotted respectively.
Figure 4: With
$m_{Z^{0}}=1000~{}GeV,~{}m_{\nu}=2000~{}GeV,~{}m_{H^{\pm}}=m_{H^{0}}=300~{}GeV$,
the branching ratios of $\mu\to e\gamma$ versus parameters $V_{\mu
e},~{}y_{\mu e},~{}g_{\mu e},~{}\lambda_{\mu e}$ are plotted respectively.

In Fig.(3), we show the contributions to the branch ratios from the new gauge
bosons and the scalars, respectively. We can see that the dependence of the
branch ratios on the various mass parameters are oscillatory in some ranges.
So in the following discussions, we will take the fixed values for them:
$m_{Z_{H}^{0}}=1000~{}GeV,~{}m_{H^{\pm}}=m_{H^{0}}=300~{}GeV,~{}m_{\nu}=2~{}TeV$.

In Fig.(4), we show the dependence of the branch ratios on the couplings,
which increase monotonously along with the couplings. So the couplings can
play a more important rule in controlling the branching ratios in comparison
with the mass parameters. We will concentrate mainly on the effects of the
couplings in our following discussions.

The experimental upper bound is given by 1605.05081 ; 1107.5547

$BR(\mu\to e\gamma)<4.2\times 10^{-13}.$ (20)

From this upper bound, the approximate range of the couplings can be given as

$V_{\mu e}\geq 0.00145,~{}y_{\mu e}\geq 0.0035,~{}g_{\mu e}\geq
0.00003,~{}\lambda_{\mu e}\geq 0.0026,$ (21)

for each type of contribution of new particle with fixed mass parameters.

Constraints in (21) for the couplings $V_{\mu e}$ and $y_{\mu e}$ are mild,
given the electroweak coupling strengthen $\sim g/(2\sqrt{2})$. Constraints on
the coupling $\lambda_{\mu e}\geq 0.0026$, however, is rather stringent
because of the flavor changing suppression for the $H^{0}\mu\bar{e}$
couplings. As noted in mns-maki-1962 , suppressed flavor mixing between $\mu$
and $e$ is of order $10^{-5}$, which is much smaller than the detectable
limit. So, we will not discuss the contribution from the the neutral Higgs in
our subsequent discussions.

Though $g_{\mu e}$ is also suppressed by the flavor changing mixing, the lower
bound in (21) is mild and acceptable. So, still some parameter space for
$g_{\mu e}$ can survive. Given that the branching ratio is also affected by
the structure factors $f_{v},~{}f_{a}$, we will consider the effects of them
in Fig.6. From the left panel of Fig.6, we can see that the branching ratios
prefer large differences between $f_{v},~{}f_{a}$, which indicates that the
left-handed coupling will give dominant contributions. To verify that, we scan
the parameter space and show the points in the right part of Fig.6 with
$g_{\mu e}=0.00005$. It can be seen that possible ranges of $f_{v}$ and
$f_{a}$ concentrate on different sides of the figures.

Figure 5: With
$m_{Z_{H}^{0}}=1000~{}GeV,~{}m_{\nu}=2000~{}GeV,~{}m_{H^{\pm}}=m_{H^{0}}=300~{}GeV$,
the contour of $V_{\mu e}$ and $y_{\mu e}$ from the constraint of the
branching ratios of $\mu\to e\gamma$.

Figure 6: With $m_{Z_{H}^{0}}=1000~{}GeV$ and $g_{\mu e}=0.00005$. Left panel:
$f_{v}=0.5$, the contribution of the parameter $f_{a}$ to the $\mu\to e\gamma$
branching ratio. Right panel: the scan the possible survivals of
$f_{v},~{}f_{a}$ in $10000$ random points.

Actually, we also take into account the coherent terms and scan the possible
ranges of the couplings. The contours of these couplings are shown in Fig.5.
From the figures, we can see that larger $V_{\mu e}$ or $y_{\mu e}$ is
favorable for the processes to be detected at the colliders.

####  IV.1.2 The processes $\tau\to e\gamma$ and $\tau\to\mu\gamma$

As for the $\tau\to e\gamma$ and $\tau\to\mu\gamma$, the total width is
different from that of the muon since the $\tau$ can decay not only into
$e\nu\bar{\nu}$, but also into $\mu\nu\bar{\nu}$. Besides, it can decay
hadronically since tau is heavier than the light quarks. Taking into account
the unitarity of the quark mixing matrix,
$|V_{ud}|^{2}+|V_{us}|^{2}=1-|V_{ub}|^{2}\simeq 1$, one can easily get the
lowest-order width (see e.g. Ref. 1310.7922  and the Refs. in it),

$\Gamma_{\tau;total}=\Gamma_{\mu}\frac{m_{\tau}^{5}}{m_{\mu}^{5}}[2+N_{C}(|V_{ud}|^{2}+|V_{us}|^{2})]\simeq
5\cdot\Gamma_{\mu}\frac{m_{\tau}^{5}}{m_{\mu}^{5}}$ (22)

So the branching ratios of $\tau\to e\gamma$ and $\tau\to\mu\gamma$ can be
written as,

$\displaystyle BR(\tau\to e/\mu\gamma)=\frac{\Gamma(\tau\to
e/\mu\gamma)}{\Gamma^{\tau}_{total}}=\frac{24\pi^{2}}{5G_{F}^{2}}(|A|^{2}+|B|^{2}),$
(23)

The experimental upper bound are taubound-0908.2381 ,

$BR(\tau\to e\gamma)<3.3\times
10^{-8},~{}~{}~{}~{}~{}~{}BR(\tau\to\mu\gamma)<4.4\times 10^{-8}.$ (24)

Figure 7: With the couplings fixed $V_{\tau e}=y_{\tau e}=0.01,~{}g_{\tau
e}=\lambda_{\tau e}=0.00005$, the branching ratios of $\tau\to e\gamma$ versus
parameters left: $~{}m_{H^{\pm}},~{}m_{Z_{H}^{0}},~{}m_{H^{0}}$; right: with
$m_{Z_{H}^{0}}=1000~{}GeV,~{}m_{\nu}=2000~{}GeV,~{}m_{H^{\pm}}=m_{H^{0}}=300~{}GeV$,
the constraints of the parameters $V_{\tau e},~{}y_{\tau e},~{}g_{\tau
e},~{}\lambda_{\tau e}$ on the branching ratios of $\tau\to e\gamma$.
Figure 8: With
$m_{Z_{H}^{0}}=1000~{}GeV,~{}m_{\nu}=2000~{}GeV,~{}m_{H^{\pm}}=m_{H^{0}}=300~{}GeV$,
the contour of $V_{\tau e}$ and $y_{\tau e}$ from the constraint of the
branching ratios of $\tau\to e\gamma$.

The LFV processes $\tau\to e\gamma$ and $\tau\to\mu\gamma$ can be discussed as
follows. Since the final particle masses are neglected and the couplings are
the same for the two processes, we only need to discuss the case $\tau\to
e\gamma$.

In the left panel of Fig.7, we show the branching ratio of $\tau\to e\gamma$
from the contribution of the new gauge bosons and the scalars, respectively.
It can be seen that the branching ratios are not sensitive to the change of
the mass parameters for the charged gauge boson, the charged Higgs, and the
heavy neutrino, respectively. In the region with lighter $m$, the neutral
gauge boson and the Higgs can give large contributions with little suppression
from the masses. On the other hand, in the heavier $m$ region, the decay rates
are almost independent on the $m$ parameters and the curves are flat in their
tails.

Numerical results indicate that the branching ratios of $\tau$ to $e\gamma$
shown in Fig.7 are smaller than those of $\mu$. It can be understood as the
following reasons. The branching ratios of $\tau$ to $e\gamma$ are not the
same as those of $\mu$ because the $\tau$ lepton is about $17$ times heavier
than the $\mu$ lepton, corresponding to a much large $y_{\tau}$. Moreover, the
total width of $\tau$ lepton is $5$ times larger than that of $\mu$ lepton.

The right panel of Fig.7 shows the dependence of the branching ratios on the
various couplings. When $y_{\tau e}$ closes to $0.1$, the branching ratio can
reach the detectable sensitivity at the collider, as shown in (24). This range
of coupling, however, is too large to be realistic.

So we can crudely draw a conclusion that the decay $\tau\to e\gamma$ (and
$\tau\to\mu\gamma$ ) can hardly be detected at the collider. Even though this
LFV process can hardly be detected, it is instructive to see the constraints
of the branching ratio $\tau\to e\gamma$ on $V_{\tau e}$ and $y_{\tau e}$. We
show in Fig.(8) the allowed values of the two parameters by the constraints in
(24). It is clear from the figure that the branch ratio is not sensitive to
$y_{\tau e}$ and large values of $V_{\tau e}$ can be allowed.

###  IV.2 The decay $\mu\to 3e$ and $\tau\to\ell e^{+}e^{-}$ ($\ell=e,~{}\mu$)

####  IV.2.1 The decay $\mu\to 3e$ and $\tau\to 3e$ mediated by the neutral
gauge bosons and the neutral scalar Higgs

Since the mediator neutral gauge bosons are much heavier than the muon, the
width of the three-body decay $\mu\to 3e$ via the neutral gauge bosons can be
written as

$\displaystyle{\rm Br}(\mu^{-}\to e^{-}e^{+}e^{-})$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\mu^{-}\to e^{-}e^{+}e^{-})}{\Gamma(\mu^{-}\to
e^{-}\nu\nu)}$ (25) $\displaystyle=$ $\displaystyle\frac{\kappa\tilde{g}_{V\mu
e}^{2}\tilde{g}_{Vee}^{2}/M_{V}^{4}}{\kappa g_{W}^{4}/M_{W}^{4}}$
$\displaystyle=$ $\displaystyle\frac{\tilde{g}_{V\mu
e}^{2}\tilde{g}_{Vee}^{2}}{g_{W}^{4}}\frac{M_{W}^{4}}{M_{V}^{4}},$

where $\tilde{g}_{V\mu e}^{2}=|g^{(V)}_{V\mu e}|^{2}+|g^{(A)}_{V\mu e}|^{2}$,
$\tilde{g}_{ee}^{2}=|g^{(V)}_{Vee}|^{2}+|g^{(A)}_{Vee}|^{2}$. $M_{V}$ is the
mass of the gauge boson, $\kappa$ is a kinematic-spin factor common to all
decay modes mediated by the neutral vector boson, while $g_{W}$(
$G_{F}/\sqrt{2}=g_{W}^{2}/(2M_{W}^{2})$) and $M_{W}$ are the electroweak
coupling and the $W$ boson mass, respectively.

So via the same idea, the three-body decay of the lepton $\tau$ involving
vector mesons in the intermediate state can be written as

$\displaystyle{\rm Br}(\tau^{-}\to\ell^{-}e^{+}e^{-})$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\tau^{-}\to\ell^{-}e^{+}e^{-})}{\Gamma(\tau^{-}\to\ell^{-}\nu\nu)}\frac{\Gamma(\tau^{-}\to\ell^{-}\nu\nu)}{\Gamma(\tau^{-}\to
All)}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\tau^{-}\to\ell^{-}e^{+}e^{-})}{\Gamma(\tau^{-}\to\ell^{-}\nu\nu)}\frac{\Gamma(\tau^{-}\to\ell^{-}\nu\nu)}{\Gamma(\mu^{-}\to
e^{-}\nu\nu)}\frac{\Gamma(\mu^{-}\to e^{-}\nu\nu)}{\Gamma(\tau^{-}\to All)}$
$\displaystyle=$
$\displaystyle\frac{\tilde{g}_{V\tau\ell}^{2}\tilde{g}_{Vee}^{2}}{g_{W}^{4}}\frac{M_{W}^{4}}{M_{V}^{4}}\left(\frac{M_{\tau}}{M_{\mu}}\right)^{5}\frac{\Gamma^{\mu}_{total}}{\Gamma^{\tau}_{total}}$

Note that in the above formulas, the final lepton masses have been neglected.

The current 90% C.L. bounds on the above decay modes are pdg-2018 ; tau23e ,

$BR(\mu\to 3e)<1.0\times 10^{-12},\\\ BR(\tau\to 3e(e\mu\mu))<2.7\times
10^{-8},BR(\tau\to 3\mu)<2.1\times 10^{-8}.$ (27)

Figure 9: The neutral boson contributions of the LFV couplings to the three
final leptonic states of the second or the third generation lepton.

The dependence of the branching ratios on the parameters $\tilde{g}$ and the
mediator gauge boson mass $m_{V}$ are given in Fig.10 for $\mu\to 3e$ and
$\tau\to 3e$. We find that the branching ratios with our choice of parameters
are much smaller than the experimental bounds Eq.(27) in most of the allowed
parameter space. We can roughly estimate the upper bounds for the couplings
$\tilde{g}_{\mu e},\tilde{g}_{\tau e}$ to be

$\tilde{g}_{\mu e}\leq 0.0018,~{}~{}\tilde{g}_{\tau e}\leq 0.65~{},$ (28)

with $M_{V}=1000~{}$GeV.

Figure 10: The branching ratios of $\mu\to 3e$(left) and $\tau\to 3e$(right)
versus parameters $\tilde{g}$ for $m_{Z_{0}^{\prime}}=1000,~{}2000,~{}5000$
GeV, respectively.

Clearly, the neutral Higgs scalar mediated decay modes
$\ell_{i}\to\ell_{j}\ell_{j}\ell_{j}$ are much smaller than that of the
neutral $Z_{0}^{\prime}$ gauge boson. The reason is that the Yukawa couplings
are usually highly suppressed, i.e, $m_{\tau}/v,~{}m_{\mu}/v$ are much smaller
than the gauge coupling $g_{W}$.

####  IV.2.2 The decay $\mu\to 3e$ and $\tau\to 3e$ mediated by the charged
scalar Higgs and the right handed neutrino at the one-loop level

Since there are no tree-level contributions to the
$\ell_{i}\to\ell_{j}\ell_{j}\ell_{j}$ decay modes by the charged bosons and
Higgs, some works, such as Refs. 1702.04399 ; 1909.12147 ,consider these
processes at the one-loop level. We will reappear this with the flavor
changing couplings mediated by the charged Higgs and the right-handed
neutrino.

Figure 11:  The one-loop charged scalar contributions of the LFV couplings to
the three final leptonic states of the second or the third generation lepton.

The couplings between the charged Higgs, the charged gauge bosons with the
right-handed neutrinos and the charged leptons can induce the three-body decay
of the massive charged leptons at one-loop level. Typical Feynman diagrams are
shown in Fig.11 and the branching ratio1702.04399 ; 1909.12147  can be written
as

$\displaystyle\mathcal{B}^{(N)}(\ell_{i}\rightarrow\ell_{j}\bar{\ell}_{j}\ell_{j})$
$\displaystyle=\frac{3(4\pi)^{2}\alpha_{e}^{2}}{8G_{F}^{2}}\Bigg{[}|A_{ND}|^{2}+|A_{D}|^{2}\left(\frac{16}{3}\log\left(\frac{m_{i}}{m_{j}}\right)-\frac{22}{3}\right)$
$\displaystyle+\frac{1}{6}|B|^{2}+\frac{1}{3}\frac{m_{i}^{2}m_{j}^{2}\left(3\sin^{4}\theta_{W}-\sin^{2}\theta_{W}+\frac{1}{4}\right)}{m_{W}^{4}\sin^{4}\theta_{W}}\left|A_{D}\right|^{2}$
$\displaystyle+\left(-2A_{ND}A_{D}^{\ast}+\frac{1}{3}A_{ND}B^{\ast}-\frac{2}{3}A_{D}B^{\ast}+\mathrm{h.c.}\right)\Bigg{]}$
$\displaystyle\times\mathcal{B}(\ell_{i}\rightarrow\ell_{j}\nu_{i}\bar{\nu}_{j}),$
(29)

where $\alpha_{e}$ is the fine structure constant. $A_{D}$ is the dipole
contribution,

$A_{D}=\frac{g_{j}^{\ast}g_{i}F(x)}{2(4\pi)^{2}m^{2}_{H^{\pm}}},$ (30)

with $g_{i,j}$ the couplings, $m_{H^{\pm}}$ the mass of heavy Higgs
($H^{\pm}$) and $x=\frac{m^{2}_{vR}}{m^{2}_{H^{\pm}}}$. Expression of $F(x)$
will given shortly. The coefficients $A_{ND}$ and $B$ are the non-dipole
contributions from the penguin and the box diagrams, respectively, which read

$A_{ND}=\frac{g_{j}^{\ast}g_{i}}{6(4\pi)^{2}}\frac{1}{m_{H^{\pm}}^{2}}G\left(x\right),$
(31)

and

$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{1}{(4\pi)^{2}e^{2}m_{H^{\pm}}^{2}}\sum_{i,j=1}^{3}\left[\frac{1}{2}D_{1}\left(x_{i},x_{j}\right)g_{jj}^{\ast}g_{jj}g_{ij}^{\ast}g_{ii}+\sqrt{x_{i}x_{j}}D_{2}\left(x_{i},x_{j}\right)g_{jj}^{\ast}g_{jj}^{\ast}g_{ij}g_{ii}\right]$
(32) $\displaystyle\longrightarrow$
$\displaystyle\frac{1}{(4\pi)^{2}e^{2}m_{H^{\pm}}^{2}}\left[\frac{1}{2}D_{1}(x)g_{j}^{\ast}g_{j}g_{j}^{\ast}g_{i}+x~{}D_{2}(x)g_{j}^{\ast}g_{j}^{\ast}g_{j}g_{i}\right].$

Note that only one flavor of (mediated) neutrino is assumed. So
$D_{1}(x_{i},x_{j})=D_{1}(x)$, $D_{2}(x_{i},x_{j})=D_{2}(x)$, and

$\displaystyle F\left(x\right)$
$\displaystyle=\frac{1-6x+3x^{2}+2x^{3}-6x^{2}\log x}{6\left(1-x\right)^{4}},$
(33) $\displaystyle G\left(x\right)=$
$\displaystyle\frac{2-9x+18x^{2}-11x^{3}+6x^{3}\log
x}{6\left(1-x\right)^{4}},$ (34) $\displaystyle D_{1}(x)$
$\displaystyle=\frac{-1+x^{2}-2x\log x}{\left(1-x\right)^{3}},$ (35)
$\displaystyle D_{2}(x)=$ $\displaystyle\frac{-2+2x-\left(1+x\right)\log
x}{\left(1-x\right)^{3}},$ (36)

and

$F\left(1\right)=\frac{1}{10},\qquad G\left(1\right)=\frac{1}{4},\qquad
D_{1}(1)=\frac{1}{3},\qquad D_{2}\left(1\right)=\frac{1}{6}.$ (37)

Figure 12: The branching ratios of $\mu\to 3e$ at the one-loop level versus
parameters $g_{e}$ with $g_{\mu}=0.1,~{}0.3,~{}1$, respectively.

With fixed masses for the charged scalar Higgs and the heavy neutrinos, the
branching ratio at the one-loop level is shown in Fig.12. We can see that the
decay rate is still below the detection sensitivity in Eq.(27) even if we
choose the couplings to be ${\cal O}(1)$. So new released data can possibly
test our predictions. Similar discussions can be given for the same three-body
decays mediated by the $W$ boson and the heavy neutrinos at one-loop level.
We, however, neglect such contributions since their contributions may be
comparably small for the couplings of same size.

##  V discussion and conclusion

We discuss the possible effects of the new exotic fields $H^{\pm}$ (together
with the right-handed neutrino), $H^{0}$ and $Z_{H}^{0}$ in the MTH models on
the muon anomaly magnetic moment $g-2$ and the LFV processes
$\ell_{i}\to\ell_{j}\gamma$ and $\ell_{i}\to\ell_{j}\ell_{k}\ell_{l}$,
respectively.

From the discussion given above, with the assumption of degenerate heavy
neutrino for simplification, we can draw the following conclusions:

  * •

The new contributions from these new particles at one-loop level can not
account for the discrepancy of the muon anomalous magnetic moment between the
theoretical predictions and the experimental results, since the new
contributions are too small in MTH models.

  * •

In most parameter space, the decay channel $\ell_{i}\to\ell_{j}\gamma$ can
hardly be probed at current colliders. However, the decay is sensitive to the
parameter $V_{e\mu}$, and it can possibly reach the detection sensitivity in
some narrow parameters space.

  * •

Similarly, the decay channel $\ell_{i}\to\ell_{j}\ell_{k}\ell_{l}$ can hardly
be detected at current colliders.

Previous conclusions indicate that the constraints of LFV processes on the
parameters of MTH are quite weak. Updated experimental data on LFV processes
may shed new light on the discovery potential of MTH models.

Acknowledgments

This work was supported by the National Natural Science Foundation of
China(NSFC) under grant 11675147,12075213, 11775012, by the Key Project by the
Education Department of Henan Province under grant number 21A140025, by the
Fundamental Research Cultivation Fund for Young Teachers of Zhengzhou
University(JC202041040) and the Academic Improvement Project of Zhengzhou
University.

## References

  * (1)
  * (2) S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D 2, 1285 (1970).
  * (3) S. Petcov, Sov. J, Nucl. Phys., 25: 340 (1977).
  * (4) J.Chang, K. Cheung, H. Ishida et al, JHEP, 1710: 039 (2017) arXiv: hep-ph/1707.04374.
  * (5) T. A. Chowdhury and S. Nasri, Phys.Rev. D, 97: 075042 (2018) arXiv: hep-ph/1801.07199.
  * (6) J. L. Diaz-Cruz, J. J. Toscano, Phys. Rev. D, 62: 116005 (2000) arXiv:hep-ph/9910233.
  * (7) T. Han, D. Marfatia, Phys. Rev. Lett., 86: 1442 (2001) arXiv: hep-ph/0008141.
  * (8) K. Agashe, R. Contino, Phys. Rev. D, 80: 075016 (2009) arXiv:hep-ph/0906.1542.
  * (9) S. Casagrande et al., JHEP, 10: 094 (2008) arXiv:hep-ph/0807.4937.
  * (10) Guo-Li Liu, Fei Wang, Kuan Xie, Xiao-Fei Guo, Phys. Rev. D96, (2017) 035005, arXiv:1701.00947.
  * (11) Jinlei Yang, T. Feng, Y. Yan, Wei Li, S. Zhao, H. Zhang, Phys. Rev. D 99, (2019) 015002, arXiv:1812.03860.
  * (12) Xiaofang Han, Tianjun Li, Lei Wang, Yang Zhang,Phys. Rev. D 99, (2019) 095034, arXiv:1812.02449.
  * (13) Mario W. Barela, V. Pleitez, Phys. Rev. D 101, (2020) 015024, arXiv:1912.05900.
  * (14) Chongxing Yue, Z. Zong, Li Zhou, Shuo Yang, Phys.Rev.D71, (2005) 115011, arXiv:hep-ph/0506070.
  * (15) Junjie Cao, Zhaohua Xiong, Jin Min Yang, Eur.Phys.J. C32, (2004) 245, arXiv:hep-ph/0307126.
  * (16) Deirdre Black, Tao Han, Hong-Jian He, Marc Sher, Phys.Rev.D66, (2002) 053002, arXiv:hep-ph/0206056.
  * (17) J. P. Miller, E. de Rafael, and B. L. Roberts, Rept. Prog. Phys. 70, 795 (2007), arXiv:hep-ph/0703049.
  * (18) F. Jegerlehner and A. Nyffeler, Phys. Rept. 477, 1 (2009), arXiv:0902.3360.
  * (19) J. P. Miller, E. de Rafael, B. L. Roberts, and D. Stckinger, Annu. Rev. Nucl. Part. Sci. 62, 237 (2012).
  * (20) M. Lindner, M. Platscher, and F. S. Queiroz, Phys. Rept. 731, 1 (2018), arXiv:1610.06587.
  * (21) F. Jegerlehner, Springer Tracts Mod. Phys. 274, pp.1 (2017).
  * (22) Guo-Li Liu, Qing-Guo Zeng, Euro. Phys. J. C79, (2019) 612, arXiv:1811.04777.
  * (23) K. Abe et al (T2K Collab), Phys. Rev. Lett., 107: 041801 (2011).
  * (24) P. Adamson et al (MINOS Collab), Phys. Rev. Lett., 107: 181802 (2011).
  * (25) Y. Abe et al (DOUBLE-CHOOZ Collab), Phys. Rev. Lett., 108: 131801 (2012).
  * (26) F. An et al (DAYA-BAY Collab), Phys. Rev. Lett., 108: 171803 (2012).
  * (27) J.Ahn et al (RENO Collaboration), Phys. Rev. Lett., 108: 191802 (2012).
  * (28) P. Ghosh, S. Roy, JHEP, 0904: 069 (2009).
  * (29) M. C. Gonzalez-Garcia and M. Maltoni, Phys. Rep., 460: 1 (2008).
  * (30) M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado and T. Schwetz, JHEP, 1212: 123 (2012).
  * (31) I. Girardi, S. T. Petcov, A. V. Titov, Nucl. Phys. B, 894: 733-768 (2015).
  * (32) P. Minkowski, Phys. Lett. 67B (1977)421;
R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912;
T. Yanagida, Conf. Proc. C7902131 (1979)95;
M. Gell-Mann, P. Ramond, and R. Slansky, Conf. Proc. C790927 (1979) 315.

  * (33) M. Magg and C. Wetterich, Phys. Lett. B 94 (1980) 61;
J. Schechter and J. W. F. Valle, Phys. Rev. D 22 (1980) 2227;
C. Wetterich, Nucl. Phys. B187 (1981) 343;
G. Lazarides, Q. Shafi, and C. Wetterich, Nucl. Phys. B181 (1981) 287;
R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 23 (1981) 165.

  * (34) R. Foot, H. Lew, X. G. He, and G. C. Joshi, Z. Phys. C44 (1989) 441;
E. Ma, Phys. Rev. Lett. 81 (1998) 1171;
E. Ma and D. P. Roy, Nucl. Phys. B644 (2002) 290.

  * (35) Z. Chacko, H.-S. Goh, and R. Harnik, Phys. Rev. Lett. 96, 231802 (2006), arXiv:hep-ph/0506256 [hep-ph].
  * (36) ATLAS Collaboration, M. Aaboud et al., Phys. Rev. D98 (2018), no. 3, 032008, arXiv:1803.10178; CMS Collaboration, A. M. Sirunyan et al., JHEP 05 (2018) 025, arXiv:1802.02110; ATLAS Collaboration, G. Aad et al., Phys. Lett. B758 (2016) 249-268, arXiv:1602.06034; CMS Collaboration, A. M. Sirunyan et al., JHEP 08 (2018) 177, arXiv:1805.04758.
  * (37) P. F. de Salas and S. Pastor, ”Relic neutrino decoupling with flavour oscillations revisited”, JCAP 1607 (2016), no. 07, 051, arXiv:1606.06986.
  * (38) G. Mangano, G. Miele, S. Pastor, and M. Peloso, ”A Precision calculation of the effective number of cosmological neutrinos”, Phys. Lett. B534 (2002) 8-16, arXiv:astro-ph/0111408.
  * (39) Z. Chacko, N. Craig, P. J. Fox, and R. Harnik, JHEP 07 (2017) 023, arXiv:1611.07975.
  * (40) R. H. Cyburt, B. D. Fields, K. A. Olive, and T.-H. Yeh, Rev. Mod. Phys. 88 (2016) 015004, arXiv:1505.01076. ”Big Bang Nucleosynthesis: 2015”,
  * (41) Planck Collaboration, N. Aghanim et al., arXiv:1807.06209.
  * (42) N. Craig, A. Katz, M. Strassler, and R. Sundrum, JHEP 07 (2015)105, arXiv:1501.05310.
  * (43) R. Barbieri, L. J. Hall, and K. Harigaya, JHEP 11 (2016) 172, arXiv:1609.05589.
  * (44) C. Csaki, E. Kuflik, and S. Lombardo, Phys. Rev. D96 (2017), no. 5, 055013, arXiv:1703.06884.
  * (45) B. Batell and C. B. Verhaaren, arXiv:1904.10468.
  * (46) D. Liu and N. Weiner, arXiv:1905.00861.
  * (47) N. Craig, S. Koren, and T. Trott, JHEP 05 (2017) 038, arXiv:1611.07977.
  * (48) N. Craig, S. Knapen, P. Longhi, and M. Strassler, JHEP 07 (2016) 002, arXiv:1601.07181.
  * (49) Csaba Csaki, Eric Kuflik, Salvator Lombardo, Phys. Rev. D 96, 055013 (2017), arXiv:1703.06884 [hep-ph].
  * (50) Zackaria Chacko, Nathaniel Craig, Patrick J. Fox, Roni Harnik, Report number: FERMILAB-PUB-16-555-T, arXiv:1611.07975 [hep-ph].
  * (51) Yang Bai, Ran Lu, Sida Lu, Jordi Salvado, Ben A. Stefanek, Phys. Rev. D 93, 073004 (2016), arXiv:1512.05357.
  * (52) W. Emam and S. Khalil, Eur. Phys. J. C 55, 625 (2007), [arXiv:0704.1395 [hep-ph]].
  * (53) W. Abdallah, A. Awad, S. Khalil, H. Okada, Eur. Phys. J. C 72(2012)2108, arXiv:1105.1047 [hep-ph].
  * (54) Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28, 870 (1962).
  * (55) A.G. Akeroyd (Taiwan, Natl. Cheng Kung U.), Mayumi Aoki (Tokyo U., ICRR), Hiroaki Sugiyama, Phys.Rev.D77:075010,2008, arXiv:0712.4019 [hep-ph].
  * (56) S. Antusch, C. Biggio, E. Fernandez-Martinez, M. B. Gavela and J. Lopez-Pavon, JHEP 0610, 084 (2006) [arXiv:hep-ph/0607020]; M. Malinsky, T. Ohlsson, Z. z. Xing and H. Zhang, Phys. Lett. B 679, 242 (2009) [arXiv:0905.2889 [hep-ph]]; A. Ibarra, E. Molinaro and S. T. Petcov, arXiv:1103.6217 [hep-ph].
  * (57) G. W. Bennett et al, [Muon g-2 Collaboration], Phys. Rev. D 73 (2006)072003.
  * (58) F. Jegerlehner and R. Szafron, Eur.Phys.J.C71:1515 (2011)[arXiv:1101.2872 [hep-ph]].
  * (59) M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, Eur.Phys.J.C71:1515 (2011)[arXiv:1010.4180 [hep-ph]].
  * (60) T. Hahn, M. Perez-Victoria, Comput. Phys. Commun. 118 (1999) 153 [hep-ph/9807565]. T. Hahn, Nucl. Phys. Proc. Suppl. 89 (2000) 231, hep-ph/0005029; ibid.135 (2004) 333, hep-ph/0406288; hep-ph/0408247. G. ’t Hooft and M. Veltman, Nucl. Phys. B153 (1979) 365.
  * (61) A. M. Baldini et al., MEG Collaboration, Eur. Phys. J. C76,(2016) 434, arXiv:1605.05081 [hep-ex].
  * (62) J. Adam et al., MEG Collaboration, Phys. Rev. Lett. 107, 171801 (2011), arXiv:1107.5547 [hep-ex].
  * (63) Antonio Pich, Report. No: IFIC/13-79, FTUV/13-1028, arXiv:1310.7922 [hep-ph].
  * (64) [21] B. Aubert et al. [ BABAR Collaboration ], Phys. Rev. Lett.104, (2010) 021802. [arXiv:0908.2381 [hep-ex]].
  * (65) M. Tanabashi et al., , Particle Data Group Collaboration, Phys. Rev. D98,(2018) 030001.
  * (66) K. Hayasaka, et al.(Belle collaboration), Phys. Lett. B687, 139 (2010) [arXiv:1001.3221 [hepex]].
  * (67) Meziane Chekkal, Amine Ahriche, Amine Bouziane Hammou, Salah Nasri, Phys. Rev. D95, (2017) 095025, arXiv:1702.04399 [hep-ph].
  * (68) Subhasmita Mishra, Anjan Giri, arXiv:1909.12147 [hep-ph].