P3H–23–025, TTP23–013

Current Status of the Muon $\bm{g-2}$ Interpretations
within Two-Higgs-Doublet Models

Syuhei Iguro,(a,b) Teppei Kitahara,(c,d) Martin S. Lang,(a)
and Michihisa Takeuchi(e,f)

(a) | Institute for Theoretical Particle Physics (TTP), Karlsruhe Institute of Technology
---|---
|  (KIT), Wolfgang-Gaede-Str. 1, 76131 Karlsruhe, Germany
(b) | Institute for Astroparticle Physics (IAP), KIT, Hermann-von-Helmholtz-Platz 1,
|  76344 Eggenstein-Leopoldshafen, Germany
(c) | Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
|  Nagoya University, Nagoya 464–8602, Japan
(d) |  CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
|  Chinese Academy of Sciences, Beijing 100190, China
(e) |  School of Physics and Astronomy, Sun Yat-sen University, 519082 Zhuhai, China
(f) | Graduate School of Information Science and Technology, Osaka University,
|  Suita, Osaka 565-0871, Japan

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

In this article, we review and update implications of the muon anomalous
magnetic moment (muon $g-2$) anomaly for two-Higgs-doublet models (2HDMs),
which are classified according to imposed symmetries and their resulting
Yukawa sector. In the minimal setup, the muon $g-2$ anomaly can be
accommodated by the type-X (lepto-philic) 2HDM, flavor-aligned 2HDM (FA2HDM),
muon-specific 2HDM ($\mu$2HDM), and $\mu\tau$-flavor violating 2HDM. We
summarize all relevant experimental constraints from high-energy collider
experiments and flavor experiments, as well as the theoretical constraints
from the perturbative unitarity and vacuum stability bounds, to these 2HDMs in
light of the muon $g-2$ anomaly. We clarify the available parameter spaces of
these 2HDMs and investigate how to probe the remaining parameter regions in
future experiments. In particular, we find that, due to the updated
$B_{s}\to\mu^{+}\mu^{-}$ measurement, the remaining parameter region of the
FA2HDM is almost equivalent to the one of the type-X 2HDM. Furthermore, based
on collider simulations, we find that the type-X 2HDM is excluded and the
$\mu$2HDM scenario will be covered with the upcoming Run 3 data.

Keywords: Muon $g-2$, 2HDM, Large Hadron Collider, Flavor Physics

###### Contents

  1. 1 Introduction
  2. 2 Two-Higgs-doublet models
    1. 2.1 Type-X 2HDM
    2. 2.2 Flavor-aligned 2HDM
    3. 2.3 Muon-specific 2HDM
    4. 2.4 $\mu\tau$-flavor violating 2HDM
  3. 3 Current status and prospects for muon $g-2$ interpretations
    1. 3.1 Type-X 2HDM
    2. 3.2 Flavor-aligned 2HDM
    3. 3.3 Muon-specific 2HDM
    4. 3.4 $\mu\tau$-flavor violating 2HDM
  4. 4 Summary and discussion
  5. A Explicit formulae
    1. A.1 Muon $g-2$
    2. A.2 Tau decays
    3. A.3 $Z$ decays
    4. A.4 $B_{s}\to\mu^{+}\mu^{-}$
    5. A.5 Perturbative unitarity and vacuum stability
    6. A.6 Renormalization group equations

##  1 Introduction

The muon anomalous magnetic moment (the muon $g-2$), defined by
$a_{\mu}=(g_{\mu}-2)/2$, is known as a long-standing anomaly in the Standard
Model (SM). Based on the white paper recommended values [1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], the SM prediction
deviates from the experimental values measured at Brookhaven [22, 23, 24] and
at Fermilab [25],

$\displaystyle\Delta
a_{\mu}=a_{\mu}^{\rm{exp}}-a_{\mu}^{\rm{SM}}=\left(25.1\pm 5.9\right)\times
10^{-10}\,,$ (1.1)

at a significance of 4.2$\,\sigma$. The measured value will be more accurately
(and independently) checked by further runs of the Fermilab experiment and by
the upcoming J-PARC experiment, which is based on a different measurement
technique [26, 27].

One should note that the recent evaluation of the leading-order hadronic
vacuum polarization (HVP) contribution to the muon $g-2$, based on a lattice
QCD$+$QED simulation [28], casts doubt on the SM prediction of the white paper
in Eq. (1.1), giving $\Delta a_{\mu}=(10.7\pm 7.0)\times 10^{-10}$, which is
consistent with the measured data $a_{\mu}^{\rm exp}$. This lattice evaluation
has been partially confirmed by different lattice collaborations [29, 30, 31,
32], using window observables for the HVP [33, 34]. The lattice average based
on these window observables provides $\Delta a_{\mu}=(18.3\pm 5.9)\times
10^{-10}$, corresponding to a significance of $3.1\,\sigma$ [35]. However,
these lattice results are incompatible with $e^{+}e^{-}\to\pi^{+}\pi^{-}$ data
[36, 37, 38] at the $3.8\,\sigma$ level [35].#1#1#1Very recently, the CMD-3
collaboration reported new data for the $e^{+}e^{-}\to\pi^{+}\pi^{-}$ cross
section [39]. The result is consistent with the lattice simulation [28].
However, the cross section is incompatible with other precision data by the
KLOE [40] and BaBar [41] collaborations, as well as the previous data of the
prior CMD-2 collaboration [42] without sufficient explanation.

In this article, we assume that the muon $g-2$ anomaly in Eq. (1.1) is a hint
of new physics (NP) beyond the SM and suppose $a_{\mu}^{\rm NP}=(25.1\pm
5.9)\times 10^{-10}$. It is known that many kinds of NP models can accommodate
the muon $g-2$ anomaly, see the recent review paper [43].#2#2#2See Ref. [44]
for a previous review. We will discuss NP interpretations in the context of
two-Higgs-doublet models (2HDMs) in this review article.

In the last decade, our understanding of the Higgs sector has significantly
been renewed after the discovery of a 125 GeV scalar boson, with null results
in searches for additional Higgs bosons at the large hadron collider (LHC). In
the meantime, precision measurements have been improved greatly. For instance,
the Yukawa couplings to third-generation fermions have been determined at the
$10\%$ level, and furthermore the muon and charm Yukawa couplings were started
to be measured. With the larger statistics in the high-luminosity phase of the
LHC (HL-LHC), the LHC will gain access to the triple-Higgs coupling and thus
will determine the shape of the Higgs potential.

However, the fundamental question of whether the discovered 125 GeV Higgs is
the only scalar boson or just one of several scalars remains unanswered. For
example, one of the minimal extensions of the SM features an additional Higgs
doublet, the so-called 2HDM [45], which naturally accommodates electroweak
(EW) precision tests and has rich phenomenology. The 2HDM appears as a low-
energy effective scalar sector of many UV-complete theories, e.g., the minimal
supersymmetric standard model (MSSM) [46, 47], left-right model [48], Pati-
Salam model [49], little Higgs model [50, 51], and so on.

It is known that the muon $g-2$ anomaly in Eq. (1.1) is the same size as the
EW contributions, which implies that an additional $\mathcal{O}$(100) GeV
electroweakly-interacting particle could possibly explain the discrepancy. The
number of variants of the 2HDM that can accommodate the muon $g-2$ anomaly
whilst evading existing experimental constraints is limited, and hence we can
take a bottom-up approach in this review article.

Within the 2HDMs, the suppression of flavor changing neutral currents (FCNCs)
is achieved once a discrete symmetry is imposed under which the two Higgs
doublets and fermions carry suitable charges [45].#3#3#3The suppression of the
FCNCs can also be realized by using an additional $U(1)$ gauge symmetry
instead of a discrete symmetry, see e.g., Refs. [52, 53]. A famous example is
a discrete $\mathbb{Z}_{2}$ symmetry. With the ad-hoc $\mathbb{Z}_{2}$
symmetry, depending on the assignment of the $\mathbb{Z}_{2}$ charges to the
SM fermions, four different types of Yukawa interactions are allowed [54, 55],
known as type-I, type-II, type-X and type-Y 2HDMs. Among the four scenarios,
only the type-X 2HDM can explain the $g-2$ anomaly without conflicting with
experimental constraints [56, 57]. In this model, the two-loop Barr-Zee
diagram, in which a light CP-odd Higgs and the tau lepton are in the loop,
gives the dominant contribution to the muon $g-2$ [58, 59, 60, 61, 62, 63, 64,
65, 66, 67, 68, 69, 70, 71].

An alternative method to eliminate the FCNCs is assuming flavor alignment of
the Yukawa matrices for each type of right-handed fermions. Such a model is
called the flavor-aligned 2HDM (FA2HDM). In the FA2HDM, all Yukawa
interactions are proportional to the corresponding fermion mass matrix [72].
Since the model contains the type-X 2HDM as a certain limit, it is also an
interesting candidate for explaining the muon $g-2$ anomaly. In addition to
the tau-lepton Barr-Zee diagram, a top-quark Barr-Zee diagram can also
contribute to the muon $g-2$ in the FA2HDM [73, 63].

More complicated discrete symmetries, $\mathbb{Z}_{4}$, are also discussed. In
the muon-specific 2HDM ($\mu$2HDM) based on a $\mathbb{Z}_{4}$ symmetry, the
additional Higgs doublet strongly couples only to the muon, without the FCNCs
[74]. Another viable method introduces a lepton-flavor-violating scalar
particle. The magnetic dipole operator requires a chirality flip, which
corresponds to the muon mass within the flavor-conserving scenarios. On the
other hand, if a model contains a $\mu\tau$-flavor violating vertex with a
neutral particle, the chirality of the virtual tau lepton can be flipped
instead and the one-loop contribution is enhanced by a factor of
$m_{\tau}/m_{\mu}\simeq 17$ compared to the flavor-conserving scenarios [75,
76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94,
95]. A $\mathbb{Z}_{4}$ symmetry can naturally realize such a model [85], and
we call it the $\mu\tau$-flavor violating 2HDM ($\mu\tau$2HDM). In this review
article, in light of the muon $g-2$ anomaly, we update the status of these
four possibilities: type-X 2HDM, FA2HDM, $\mu$2HDM, and $\mu\tau$2HDM.
Variants of the 2HDM with even larger discrete symmetries can also be
conceived; however, their implications for the muon $g-2$ anomaly are mostly
the same as in the $\mu$2HDM and $\mu\tau$2HDM.

In this paper we present the following updates and findings:

  * •

Reinterpreting the latest charginos and neutralinos search based on the Run 2
full data, we find that the tau-rich signature at the LHC excludes the muon
$g-2$ explanation within the type-X 2HDM.

  * •

Including the recent CMS $B_{s}\to\mu^{+}\mu^{-}$ measurement, which is
consistent with the SM prediction, we find that the parameter space of the
FA2HDM is similar to the one in the type-X 2HDM in terms of the muon $g-2$.
Therefore the FA2HDM explanation of the anomaly is also excluded.

  * •

We update the collider constraint on $\mu$2HDM by using results of the latest
multi-lepton search. By incorporating also the theoretical constraint from the
Landau pole, we find that the current central value of the muon $g-2$ anomaly
indicates that the Landau pole scale is less than $\simeq 5\,$TeV in the
model. Moreover, we obtain a projection for future sensitivity and find that
500 fb-1 of data will cover the 1 $\sigma$ region of the muon $g-2$ anomaly
once we impose the scale to be larger than $\simeq 5\,$TeV. We also point out
that a muon-flavor exclusive multi-lepton search can improve the sensitivity
to the model.

The outline of the paper is as follows. In Sec. 2, all kinds of minimal 2HDMs
which can resolve the muon $g-2$ anomaly are introduced. In Sec. 3, we examine
various experimental and theoretical constraints on their parameter spaces and
discuss the future experimental prospects for them. Finally, conclusions are
drawn in Sec. 4. In the Appendix, we collect all relevant formulae for the
analyses of this article.

##  2 Two-Higgs-doublet models

First, we introduce two different bases in the various 2HDMs: the so-called
$\mathbb{Z}_{2}$ basis and the Higgs basis, which are mathematically
equivalent.#4#4#4For the parameter conversions between the two bases, see
Appendices of Refs. [96, 97] for the Higgs potential and Ref. [98] for the
Yukawa sector. The $\mathbb{Z}_{2}$ basis respects charge assignments of the
discrete symmetry. One can straightforwardly track the free parameters of the
model in the $\mathbb{Z}_{2}$ basis. On the other hand, the Higgs basis which
will be used in this article can parametrize more general 2HDMs. In
particular, if one assumes the alignment of the SM Higgs boson, any
calculations become significantly simpler in the Higgs basis.

Within the 2HDMs, when the Higgs potential is minimized at the EW symmetry
breaking vacuum, both neutral components of Higgs doublets acquire vacuum
expectation values (VEVs) in general. In the $\mathbb{Z}_{2}$ basis, the Higgs
potential is given by

$\displaystyle V(\Phi_{1},\Phi_{2})$
$\displaystyle=m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}-\left(m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}+{\rm
h.c.}\right)$
$\displaystyle\quad+\frac{\lambda_{1}^{Z_{2}}}{2}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\frac{\lambda_{2}^{Z_{2}}}{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}^{Z_{2}}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{4}^{Z_{2}}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})$
$\displaystyle\quad+\frac{1}{2}\left(\lambda_{5}^{Z_{2}}(\Phi_{1}^{\dagger}\Phi_{2})^{2}+{\rm
h.c.}\right)\,,$ (2.1)

where in general $m^{2}_{11},\,m^{2}_{22}$ and $\lambda_{1-4}^{Z_{2}}$ are
real parameters, while $m_{12}^{2}$ and $\lambda_{5}^{Z_{2}}$ are complex
ones. In this article, we assume absence of CP violation in the Higgs
potential for simplicity, so that $m_{12}^{2}$ and $\lambda_{5}^{Z_{2}}$ are
treated as real parameters as well. The two Higgs doublets in the
$\mathbb{Z}_{2}$ basis are defined as

$\displaystyle\Phi_{1}=\left(\begin{array}[]{c}\omega_{1}^{+}\\\
\frac{v_{1}+h_{1}+\mathrm{i}z_{1}}{\sqrt{2}}\end{array}\right)\,,\quad\Phi_{2}=\left(\begin{array}[]{c}\omega_{2}^{+}\\\
\frac{v_{2}+h_{2}+\mathrm{i}z_{2}}{\sqrt{2}}\end{array}\right)\,.$ (2.6)

The VEVs $v_{1},\,v_{2}$ can be taken to be real and positive and need to
satisfy $v=\sqrt{v_{1}^{2}+v_{2}^{2}}\simeq 246\,$GeV in order to reproduce
the masses of the weak gauge bosons. The ratio of the VEVs is represented by
$\tan\beta=v_{2}/v_{1}~{}(0\leq\beta\leq\pi/2)$.

By taking a certain linear combination of $\Phi_{1,2}$, one can always work in
the Higgs basis [99, 100, 101] where only one Higgs doublet obtains a VEV as

$\displaystyle\left(\begin{array}[]{c}H_{1}\\\
H_{2}\end{array}\right)=\left(\begin{array}[]{cc}\cos\beta&\sin\beta\\\
-\sin\beta&\cos\beta\end{array}\right)\left(\begin{array}[]{c}\Phi_{1}\\\
\Phi_{2}\end{array}\right)\,.$ (2.13)

In the Higgs basis, the doublets can be decomposed as

$\displaystyle H_{1}=\left(\begin{array}[]{c}G^{+}\\\
\frac{v+h+\mathrm{i}G}{\sqrt{2}}\end{array}\right),~{}~{}~{}H_{2}=\left(\begin{array}[]{c}H^{+}\\\
\frac{H+\mathrm{i}A}{\sqrt{2}}\end{array}\right)\,,$ (2.18)

where the fields $G^{+}$, $H^{+}$, $h$, $H$, $G$, and $A$ are linear
combinations of $\omega_{1,2}^{+},h_{1,2}$, and $z_{1,2}$. The Nambu-Goldstone
bosons of the spontaneously broken EW gauge symmetry are denoted by $G^{\pm}$
and $G$, and $H^{\pm}$ denotes an additional charged Higgs boson, while $A$ is
a neutral CP-odd Higgs boson. In principle, the CP-even scalars $h$ and $H$ in
the doublets mix with an angle $\alpha$ to constitute the mass eigenstates.
However, since the LHC found that the interactions of the observed scalar
boson are currently consistent with the SM Higgs expectations, we consider the
case where the mixing between CP-even scalars is negligible corresponding to a
conservative choice, i.e., $\sin\left(\beta-\alpha\right)\simeq 1$, such that
$h$ and $H$ are promoted to mass eigenstates.#5#5#5If non-zero mixing is
considered, we have more stringent constraints on the model from, e.g.,
$h\to\tau^{+}\tau^{-}$. Since experimental constraints are commonly weakened
in the Higgs alignment limit, we expect the study in this article to be
conservative.

In the Higgs basis, the scalar potential is given by

$\displaystyle V(H_{1},H_{2})$
$\displaystyle=M_{11}^{2}H_{1}^{\dagger}H_{1}+M_{22}^{2}H_{2}^{\dagger}H_{2}-\left(M_{12}^{2}H_{1}^{\dagger}H_{2}+{\rm
h.c.}\right)$
$\displaystyle\quad+\frac{\lambda_{1}}{2}(H_{1}^{\dagger}H_{1})^{2}+\frac{\lambda_{2}}{2}(H_{2}^{\dagger}H_{2})^{2}+\lambda_{3}(H_{1}^{\dagger}H_{1})(H_{2}^{\dagger}H_{2})+\lambda_{4}(H_{1}^{\dagger}H_{2})(H_{2}^{\dagger}H_{1})$
$\displaystyle\quad+\left\\{\frac{\lambda_{5}}{2}(H_{1}^{\dagger}H_{2})^{2}+\left[\lambda_{6}(H_{1}^{\dagger}H_{1})+\lambda_{7}(H_{2}^{\dagger}H_{2})\right](H_{1}^{\dagger}H_{2})+{\rm
h.c.}\right\\}\,.$ (2.19)

By matching to the Higgs potential in the $\mathbb{Z}_{2}$ basis, one obtains

$\displaystyle\lambda_{6}$
$\displaystyle=-\frac{1}{2}\sin{2\beta}\left(\cos^{2}{\beta}\lambda^{Z_{2}}_{1}-\sin^{2}{\beta}\lambda^{Z_{2}}_{2}-\cos{2\beta}\lambda^{Z_{2}}_{345}\right)\,,$
(2.20) $\displaystyle\lambda_{7}$
$\displaystyle=-\frac{1}{2}\sin{2\beta}\left(\sin^{2}{\beta}\lambda_{1}^{Z_{2}}-\cos^{2}{\beta}\lambda_{2}^{Z_{2}}+\cos{2\beta}\lambda_{345}^{Z_{2}}\right)\,,$
(2.21)

with
$\lambda_{345}^{Z_{2}}=\lambda_{3}^{Z_{2}}+\lambda_{4}^{Z_{2}}+\lambda_{5}^{Z_{2}}$.
We note that the Higgs alignment condition corresponds to $\lambda_{6}=0$ at
the renormalization scale $\mu\approx m_{W}$ in the Higgs basis, which leads
to $M_{12}^{2}=0$ under the stationary condition.

The scalar mass spectrum is important for our discussion and hence it is
useful to show the mass relations in the Higgs alignment limit in the Higgs
basis:

$\displaystyle\begin{aligned} m_{h}^{2}&=\lambda_{1}v^{2}\,,\quad
m_{A}^{2}=M_{22}^{2}+\frac{\lambda_{3}+\lambda_{4}-\lambda_{5}}{2}v^{2}\,,\\\
m_{H}^{2}&=m_{A}^{2}+\lambda_{5}v^{2}\,,\quad
m_{H^{\pm}}^{2}=m_{A}^{2}-\frac{\lambda_{4}-\lambda_{5}}{2}v^{2}\,.\end{aligned}$
(2.22)

The mass differences among the neutral scalars are crucial parameters to
discuss the muon $g-2$ anomaly and constraints from collider physics. The
numerical relation is given as

$\displaystyle\Delta_{H-A}$ $\displaystyle=m_{H}-m_{A}\,\simeq
60\left(\frac{\lambda_{5}}{1.0}\right)\left(\frac{1000\,{\rm{GeV}}}{m_{H}+m_{A}}\right){\rm{GeV}}\,.$
(2.23)

We also define

$\displaystyle\lambda_{hAA}=\lambda_{3}+\lambda_{4}-\lambda_{5}\,,$ (2.24)

for later convenience, which corresponds to the $hAA$ coupling in the Higgs
alignment limit.

The most general Yukawa sector of the 2HDM in the fermion mass eigenbasis is
given as

$\displaystyle-\mathcal{L}$ $\displaystyle=$
$\displaystyle\overline{Q}_{L}^{i}H_{1}y_{d}^{i}d_{R}^{i}+\overline{Q}_{L}^{i}H_{2}\rho_{d}^{ij}d_{R}^{j}+\overline{Q}_{L}^{i}(V^{\dagger})^{ij}\widetilde{H}_{1}y_{u}^{j}u_{R}^{j}+\overline{Q}_{L}^{i}(V^{\dagger})^{ij}\widetilde{H}_{2}\rho_{u}^{jk}u_{R}^{k}$
(2.25)
$\displaystyle+\overline{L}_{L}^{i}H_{1}y^{i}_{e}e_{R}^{i}+\overline{L}_{L}^{i}H_{2}\rho^{ij}_{e}e_{R}^{j}+\rm{h.c.}\,.$

Here $i,j$ denote flavor indices and $Q=(V^{\dagger}u_{L},d_{L})^{T}$ and
$L=(U\nu_{L},e_{L})^{T}$ are $\rm{SU}(2)_{L}$ doublets, where $V$ and $U$ are
the Cabbibo-Kobayashi-Maskawa (CKM) [102, 103] and the Pontecorvo-Maki-
Nakagawa-Sakata (PMNS) [104, 105] matrices, respectively. In writing Eq.
(2.25) we have assumed that neutrino masses are explained by the seesaw
mechanism introducing heavy right-handed neutrinos, so that in the low-energy
effective theory the left-handed neutrinos have a $3\times 3$ Majorana mass
matrix. Note that the Yukawa couplings $y_{f}$ are expressed in terms of the
fermion masses $m_{f}$, $y_{f}=\sqrt{2}m_{f}/v$. On the other hand, the Yukawa
couplings $\rho_{f}^{ij}$ are a priori arbitrary $3\times 3$ complex matrices
and can in general be sources of flavor violation mediated by additional Higgs
bosons at tree level. In the Higgs alignment limit, the interactions of
$H_{1}$ are exactly the same as the ones of the SM Higgs doublet.

Following the notation of Ref. [98], in terms of the mass eigenstates of the
Higgs bosons, the Yukawa interactions are represented by

$\displaystyle-\mathcal{L}$
$\displaystyle=\sum_{f=u,d,e}\sum_{\phi=h,H,A}y^{f}_{\phi
ij}\overline{f}_{Li}\phi f_{Rj}+{\rm h.c.}$
$\displaystyle\quad+\overline{\nu}_{Li}(U^{\dagger}\rho_{e})^{ij}H^{+}e_{Rj}+\overline{u}_{i}(V\rho_{d}P_{R}-\rho_{u}^{\dagger}VP_{L})^{ij}H^{+}d_{j}+{\rm
h.c.}\,,$ (2.26)

where

$\displaystyle y^{f}_{hij}$
$\displaystyle=\frac{1}{\sqrt{2}}y_{f}^{i}\delta_{ij}=\frac{m_{f_{i}}}{v}\delta_{ij}\,,\quad
y^{f}_{Hij}=\frac{1}{\sqrt{2}}\rho_{f}^{ij}\,,\quad
y^{f}_{Aij}=\left\\{\begin{array}[]{c}-\frac{\mathrm{i}}{\sqrt{2}}\rho_{f}^{ij}~{}~{}~{}{\rm
for}~{}f=u\,,\\\ \frac{\mathrm{i}}{\sqrt{2}}\rho_{f}^{ij}~{}~{}~{}{\rm
for}~{}f=d,e\,.\end{array}\right.$ (2.29)

The off-diagonal components of the Yukawa couplings $\rho_{f}^{ij}$ ($i\neq
j$) induce FCNCs from decays of the scalar bosons and Higgs-mediated
processes. The absence of such FCNCs at experiments#6#6#6The recent ATLAS and
CMS results both show hints of Higgs lepton-flavor violating decays [106, 107,
108]. In order to address these slight deviations, one needs a small neutral-
scalar mixing. motivates us to impose a discrete symmetry which distinguishes
the two Higgs doublets $\Phi_{1,2}$. The Yukawa structure of $\rho_{f}^{ij}$
depends on the charge assignment of the Higgs doublets and fermions as well,
which we will classify in the following sections.

###  2.1 Type-X 2HDM

We start with the type-X 2HDM. This is one of the four types of 2HDMs with
softly-broken $\mathbb{Z}_{2}$ symmetry and thus naturally suppresses FCNCs.
The $\mathbb{Z}_{2}$ assignment is summarized in Table 1.

|  $\Phi_{1}$ |  $\Phi_{2}$ |  $Q$ |  $u_{R}$ |  $d_{R}$ | $(L_{e},L_{\mu},L_{\tau})$ | $(e_{R},\mu_{R},\tau_{R})$
---|---|---|---|---|---|---|---
type-I |  $0$ |  $1$ |  $0$ |  $1$ |  $1$ | $0$ | $1$
type-II |  $0$ |  $1$ |  $0$ |  $1$ |  $0$ | $0$ | $0$
type-X |  $0$ |  $1$ |  $0$ |  $1$ |  $1$ | $0$ | $0$
type-Y |  $0$ |  $1$ |  $0$ |  $1$ |  $0$ | $0$ | $1$
$\mu$2HDM |  $2$ |  $0$ |  $0$ |  $0$ |  $0$ | $(0,-1,0)$ | $(0,1,0)$
$\mu\tau$2HDM |  $2$ |  $0$ |  $0$ |  $0$ |  $0$ | $(0,1,-1)$ | $(0,1,-1)$
Table 1: The charge assignments under the discrete symmetry: the matter fields
are transformed as $\psi\to\exp(2\pi q_{\psi}{\rm{i}}/N)\,\psi$ under the
discrete $\mathbb{Z}_{N}$ symmetry. We show the $\mathbb{Z}_{2}$ charge
assignments for the type-I, II, X, and Y 2HDMs ($0$ is $\mathbb{Z}_{2}$-even,
while $1$ is $\mathbb{Z}_{2}$-odd) [109] and the $\mathbb{Z}_{4}$ charge
assignments for the $\mu$2HDM [74] and the $\mu\tau$2HDM [85].

Out of the four $\mathbb{Z}_{2}$-symmetric 2HDMs, only the type-X 2HDM can
resolve the muon $g-2$ anomaly because in all three other 2HDMs a large
contribution to $a_{\mu}^{\rm NP}$ is not allowed by flavor and collider
constraints and the perturbative unitarity bound. More explicitly, in the
type-I and type-Y 2HDMs, both $\rho_{u}^{ii}$ and $\rho_{e}^{ii}$ are
proportional to $\cot\beta$. Therefore, the upper limit on $\rho_{u}^{tt}$
from perturbativity indirectly sets an upper limit on $\rho_{e}^{\mu\mu}$,
suppressing any potential contributions to $a_{\mu}^{\rm NP}$.#7#7#7See also
Ref. [110]. On the other hand, in the type-II and type-X 2HDMs,
$\rho_{e}^{ii}$ is proportional to $\tan\beta$ while $\rho_{u}^{ii}$ remains
proportional to $\cot\beta$. It is noted that the type-II and type-X 2HDMs
cannot accommodate the muon $g-2$ anomaly at one-loop level with a light CP-
even Higgs.#8#8#8See also Ref. [111]. If one tries to explain $\Delta a_{\mu}$
with $\rho_{e}^{\mu\mu}=\mathcal{O}(1)$, $\rho_{e}^{\tau\tau}$ immediately
becomes non-perturbative because $\rho_{e}^{\tau\tau}/\rho_{e}^{\mu\mu}$ is
fixed by $m_{\tau}/m_{\mu}$. Instead, the contributions to $a_{\mu}^{\rm NP}$
can be dominated by the so-called Barr-Zee diagram [112] at two-loop level
with a light CP-odd Higgs [58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70,
71]. We exhibit the Feynman diagrams in Fig. 1 and the formula in Appendix
A.1.

The difference between the type-II and type-X 2HDMs is the down-type quark
Yukawa couplings to the extra Higgs bosons. In the type-II 2HDM, both the
down-type quark and charged lepton couplings are enhanced by $\tan\beta$ at
the same time. Therefore, the model is severely constrained by $B$-meson
flavor physics and direct searches for extra Higgs bosons. The rare radiative
decay $b\to s\gamma$ gives a lower mass limit for the charged Higgs boson of
$m_{H^{\pm}}\geq 800$ GeV [113]. For the muon $g-2$ anomaly, we need a
$\mathcal{O}$(10–100) GeV light scalar at the same time. Such a large mass
difference is troublesome since the theory will be non-perturbative at less
than 1 TeV.

On the other hand, only the lepton Yukawa couplings are enhanced by
$\tan\beta$ in the type-X 2HDM. Therefore, the constraints from $B$-meson
decays are weaker compared to the type-II ones. The Yukawa structure in the
type-X 2HDM is given by

$\displaystyle\rho_{u}=\frac{\sqrt{2}m_{u_{i}}}{v}\xi^{-1}\,,\quad\rho_{d}=\frac{\sqrt{2}m_{d_{i}}}{v}\xi^{-1}\,,\quad\rho_{e}=-\frac{\sqrt{2}m_{e_{i}}}{v}\xi\,,$
(2.30)

with $\xi=\tan\beta$ and all non-diagonal Yukawa couplings vanishing. Although
$\tan\beta$ is conventionally used, we will use the notation $\xi$ in this
article in order to allow for an easy comparison with the other 2HDM
scenarios.

Figure 1:  The diagram of leading contribution to the muon $g-2$ in the
$\mu$2HDM (left), $\mu\tau$2HDM (right), and type-X 2HDM and FA2HDM (bottom).
The bottom diagram is the two-loop Barr-Zee diagram where $f=\tau,\,t$.

###  2.2 Flavor-aligned 2HDM

In the flavor-aligned 2HDM (FA2HDM), it is assumed that the Yukawa
interactions of the additional scalars are proportional to mass matrices
($\rho_{f}\propto y_{f}$); both the $\rho_{f}$ and $y_{f}$ matrices are
simultaneously diagonalized in the Lagrangian of Eq. (2.25) [72]. Therefore,
tree-level FCNCs are absent in this model.#9#9#9 Even if one imposes the
flavor alignment at tree-level, FCNCs are in general induced radiatively, in
particularly by the renormalization group evolutions (RGEs). However, these
RGE-induced FCNCs are shown to be too small to be probed at current
experiments [114, 115]. Viable UV models for the flavor alignment condition
are discussed in Refs. [116, 117].

There are three free parameters in the Yukawa interactions, $\xi_{u}$,
$\xi_{d}$, and $\xi_{e}$. In other words, the type-X scenario is a special
case of the FA2HDM, in which $\xi_{u}=\xi_{d}=\xi^{-1}$ and $\xi_{e}=-\xi$.
The additional Yukawa couplings are given as

$\displaystyle\rho_{u}=\frac{\sqrt{2}m_{u_{i}}}{v}\xi_{u}\,,\quad\rho_{d}=\frac{\sqrt{2}m_{d_{i}}}{v}\xi_{d}\,,\quad\rho_{e}=\frac{\sqrt{2}m_{e_{i}}}{v}\xi_{e}\,.$
(2.31)

In this model, the dominant contribution to $a_{\mu}^{\rm NP}$ comes from the
two-loop Barr-Zee diagram with a light CP-odd Higgs and a tau lepton, while
the Barr-Zee diagram with a top-quark loop can also contribute [73, 63].

###  2.3 Muon-specific 2HDM

Another interesting scenario is the so-called muon-specific 2HDM ($\mu$2HDM).
In this model, the additional scalars dominantly couple to muons due to a
$\mathbb{Z}_{4}$ charge assignment as summarized in Table 1. This coupling
structure helps to avoid the constraints from $\tau$ decays and loop-induced
$Z$ decays [74]. The Yukawa coupling structure of this model is given as

$\displaystyle\rho_{f}=\frac{\sqrt{2}m_{f}}{v}\xi^{-1}\,,\quad\rho_{e}^{\mu\mu}=-\frac{\sqrt{2}m_{\mu}}{v}\xi\,,$
(2.32)

with $\xi=\tan\beta$, where $f$ denotes all fermions except for $\mu$. The
dominant contribution to $a_{\mu}^{\rm NP}$ stems from one-loop diagrams, see
Fig. 1 (top-left), and the formula is given in Appendix A.1. $m_{H}<m_{A}$ or
equivalently $\lambda_{5}<0$ can yield $\Delta a_{\mu}>0$. The
phenomenologically interesting parameter region is $\xi\gg 1$.

###  2.4 $\mu\tau$-flavor violating 2HDM

In the $\mu\tau$-flavor violating 2HDM ($\mu\tau$2HDM), a discrete
$\mathbb{Z}_{4}$ symmetry with charge assignments shown in Table 1 is imposed.
Due to the unique charge assignment, after the diagonalization of the charged
lepton mass, the only flavor-violating interactions of the additional neutral
scalars in the Higgs alignment limit are the
$\overline{\mu}_{L/R}\tau_{R/L}H(A)$ interactions. The Yukawa coupling
structure of this model is given as

$\displaystyle\rho_{f}^{\rm
diag}=\frac{\sqrt{2}m_{f}}{v}\xi^{-1}\,,\quad\rho_{e}^{\mu\tau}\neq
0\,,\quad\rho_{e}^{\tau\mu}\neq 0\,,$ (2.33)

with $\xi\approx\tan\beta$. Here, $f$ denotes all fermions. The dominant
contribution to $a_{\mu}^{\rm NP}$ comes from the one-loop diagram with a
virtual tau lepton in Fig. 1, and the formula is given in Appendix A.1.

However, even in the case of the Higgs alignment limit, this model also
predicts $\tau\to\mu$ lepton-flavor violating transitions at one-loop level.
To avoid their experimental bounds, the limit $\tan\beta\to\infty$ is a
natural solution, corresponding to the original discrete $\mathbb{Z}_{4}$
symmetry (a variant of the inert doublet model) proposed in Ref. [85]. Since
$a_{\mu}^{\rm NP}$ is insensitive to $\tan\beta$ in this model, we consider
the following Yukawa coupling structure

$\displaystyle\rho_{f}^{\rm diag}=0\,,\quad\rho_{e}^{\mu\tau}\neq
0\,,\quad\rho_{e}^{\tau\mu}\neq 0\,.$ (2.34)

Then, we can safely focus on the phenomenology of the additional scalars. Note
that for realistic neutrino masses and mixing, breaking of the
$\mathbb{Z}_{4}$ symmetry is necessary [95].

##  3 Current status and prospects for muon $g-2$ interpretations

In this main section, we discuss explanations of the muon $g-2$ anomaly based
on the models introduced in Sec. 2, along with relevant flavor and collider
constraints. The future prospects at the (HL-)LHC are also discussed. Table 2
summarizes the interesting mass range and relevant processes.

| $\Delta a_{\mu}$ | mass range | precision | LHC | life time
---|---|---|---|---|---
Type-X 2HDM | 2 loop | $m_{A}=\mathcal{O}(10)\,{\rm{GeV}}\ll m_{H}=m_{H^{\pm}}$ |  $h\to AA$, $Z,\tau$ decays | multi-$\tau$ | Run 2
FA2HDM | 2 loop | $m_{A}=\mathcal{O}(10)\,{\rm{GeV}}\ll m_{H}=m_{H^{\pm}}$ |  $B_{s}\to\mu^{+}\mu^{-}$, $h\to AA$ | multi-$\tau$ | Run 2
$\mu$2HDM | 1 loop | 900 GeV $\leq m_{A,\,H}\leq 1000\,$GeV |  $Z$ decay | multi-$\mu$ | Run 3
$\mu\tau$2HDM | 1 loop | 500 GeV $\leq m_{A,\,H}\leq 1600\,$GeV | $\tau\to\mu\nu\overline{\nu}$ | $\mu^{\pm}\mu^{\pm}\tau^{\mp}\tau^{\mp}$ | HE-LHC

Table 2: Summary table for all 2HDM scenarios which can accommodate the muon
$g-2$ anomaly. The second column shows the loop order of the dominant
contribution to $a_{\mu}^{\rm NP}$. The third column lists the available mass
range of scalars. The fourth and fifth ones show the relevant constraints from
precision measurements and important processes at LHC, respectively. The last
column summarizes how much data is needed to fully explore the parameter space
where the muon $g-2$ anomaly can be solved at the 1 $\sigma$ level.

###  3.1 Type-X 2HDM

The type-X model explains the muon $g-2$ with a two-loop Barr-Zee diagram in
which a CP-odd scalar $A$ of $\mathcal{O}$(20–40) GeV propagates. In the large
$\xi$ limit, the two-loop Barr-Zee correction with a light $A$ and tau
internal loop can generate a large positive contribution to $a_{\mu}^{\rm NP}$
since it is enhanced by $m_{\tau}^{2}/m_{\mu}^{2}$. The Barr-Zee contribution
with $H$ in the loop gives a negative contribution and thus a heavier $H$ is
preferred to enhance $a_{\mu}^{\rm NP}$.

In this mass range, the SM Higgs boson can decay into a pair of light CP-odd
scalars, which modifies the Higgs total width. This additional decay channel
opens for $m_{h}\geq 2m_{A}$ and the tree-level $h\to AA$ partial decay width
is given as

$\displaystyle\Gamma(h\to AA)=\frac{\lambda_{hAA}^{2}v^{2}}{32\pi
m_{h}}\sqrt{1-\frac{4m_{A}^{2}}{m_{h}^{2}}}\,.$ (3.1)

Recent Higgs width measurements restrict the trilinear Higgs coupling to [118,
119]

$\displaystyle|\lambda_{hAA}|\leq 0.03\,.$ (3.2)

A more stringent limit $|\lambda_{hAA}|\lesssim 0.01$ is obtained for
$m_{A}\leq 21\,$GeV based on searches for $h\to
AA\to\mu^{+}\mu^{-}\tau^{+}\tau^{-}$ decays [120]. Since the non-discovery of
$H^{\pm}$ prefers large mass differences between the CP-odd and charged
scalars, $\mathcal{O}(1)$ couplings in the Higgs potential are necessary, see
Eq. (2.22). Therefore, Eq. (3.2) requires parameter tuning at the 1$\%$ level.

Figure 2:  The parameter plane of $\xi$ and $m_{A}$ in the type-X 2HDM. The
cyan and blue regions can accommodate the muon $g-2$ anomaly at the
1$\,\sigma$ and 2$\,\sigma$ levels, respectively. The region to the left of
the green, purple, and brown lines is excluded by the $\tau$ decay, $Z$ decay,
and scalar bremsstrahlung constraints, respectively. The
$B_{s}\to\mu^{+}\mu^{-}$ constraint excludes the region to the left of the red
line. We take $m_{H}=m_{H^{\pm}}=250\,(300)\,$GeV on the left (right) panel.

In Fig. 2, we show the parameter regions where the muon $g-2$ anomaly can be
explained at the 1$\,\sigma$ and 2$\,\sigma$ level by the cyan and blue
regions, respectively. We take $m_{H}=m_{H^{\pm}}=250\,(300)\,$GeV on the left
(right) panel.

In the regime of large lepton couplings in the type-X 2HDM, the tau (leptonic
and hadronic) decays are modified by tree-level and one-loop corrections from
the additional scalars. The tree-level and one-loop corrections have been
calculated in Refs. [121, 59]. Relevant formulae and the current experimental
data are summarized in Appendix A.2. Since the corrections from the additional
scalars are suppressed by $1/m_{H^{\pm}}^{2}$, the $\tau$-decay bound
uniformly becomes weaker, the heavier the additional scalar becomes. Since the
one-loop correction involving two tau-Yukawa couplings is larger than the one
involving a single muon-Yukawa coupling, the box diagram is less important.
The excluded regions from tau decays are shown by the green lines in Fig. 2.

In addition, the one-loop correction to the fermion coupling of the $Z$ boson
provides an important cross-check in this scenario. Thanks to the hierarchical
structure of the Yukawa couplings, the $Z$-boson interaction with the tau
leptons is most sensitive to the additional scalars. For the partial
cancellation of systematic uncertainties, taking the ratio of leptonic decay
widths improves the sensitivity. The LEP averages are given as
[122],#10#10#10It is noted that the recent result of
$\frac{\Gamma(Z\to\mu^{+}\mu^{-})}{\Gamma(Z\to e^{+}e^{-})}$ from ATLAS [123],
which has a twice larger uncertainty, is consistent with the LEP result. In
addition, the uncertainty from LHCb is much larger than that of the LEP [124].
Furthermore, the correlation matrix is not available in the PDG [125].
Therefore we use the LEP average.

$\displaystyle\frac{\Gamma(Z\to\tau^{+}\tau^{-})}{\Gamma(Z\to
e^{+}e^{-})}=1.0019\pm
0.0032\,,\quad\frac{\Gamma(Z\to\mu^{+}\mu^{-})}{\Gamma(Z\to
e^{+}e^{-})}=1.0009\pm 0.0028\,,$ (3.3)

with a correlation of 0.63. A larger mass difference of $m_{H}$–$m_{A}$ yields
a larger deviation from the SM prediction of the $Z$-boson interaction.
Furthermore, the tau polarization asymmetry in $Z\to\tau^{+}\tau^{-}$ is also
precisely measured at the LEP and the average is [125]

$\displaystyle A_{\tau}=0.143\pm 0.004\,.$ (3.4)

The corresponding corrections involving additional scalars are summarized in
Appendix A.3. The excluded regions from $Z$-boson decays are shown by the
purple lines in Fig. 2. Contrary to the conventional decoupling behaviour, it
is known that the additional scalar contributions are enhanced if $m_{A}\ll
m_{Z}\ll m_{H}\simeq m_{H^{\pm}}$ is considered [64]. Therefore, the $Z$-boson
bound becomes stricter for heavier additional scalars, which provides an
exclusion region complementary to the $\tau$-decay bound.

Furthermore, the rare leptonic meson decay $B_{s}\to\mu^{+}\mu^{-}$ gives a
lower bound on $m_{A}$, which is independent of $\xi$, since in the diagram
with a neutral scalar the $\xi$ dependence is cancelled. In the past, the
experimental world average of $\text{BR}(B_{s}\to\mu^{+}\mu^{-})$ had deviated
from the SM prediction [126, 127, 128] by about $2\,\sigma$ [129, 130].
However, the CMS collaboration recently reported the Run 2 full analysis and
found the branching ratio to be consistent with the SM prediction [131]. As a
result, the latest experimental world average is well consistent with the SM
prediction [132]. The dominant contribution to the $B_{s}\to\mu^{+}\mu^{-}$
comes from the light CP-odd scalar mediated diagram at one-loop, see Appendix
A.4 for more details. The new world average leads to a relaxed lower bound on
$m_{A}$ compared to the previous world average. We obtain $m_{A}\gtrsim 27$
GeV in the type-X 2HDM, which is shown by the red lines in Fig. 2.

The LEP probed electroweak $AH$ production in $4\tau$ final states
($e^{+}e^{-}\to Z^{\ast}\to AH\to 2\tau^{+}2\tau^{-}$) without finding a
significant excess over the SM expectation. As a result a lower limit on the
sum of the neutral scalar masses, $m_{H}+m_{A}\geq 190~{}(155)\,$GeV, has been
obtained for the case where
$\text{BR}(A\to\tau^{+}\tau^{-})\times\text{BR}(H\to\tau^{+}\tau^{-})=1(0.1)$
[133, 134]. Lighter scalars have been also searched for in the scalar
bremsstrahlung process ($e^{+}e^{-}\to\tau^{+}\tau^{-}A\to 2\tau^{+}2\tau$)
[133]. This constraint is especially stringent for a very light $A$. An upper
bound of $\xi\leq 34~{}(83)$ was obtained for $m_{A}=10~{}(25)\,$GeV with
BR$(A\to\tau^{+}\tau^{-})\simeq 1$. Furthermore, there have been searches for
a pair of $H^{\pm}$ in $\tau\nu$, $cs$ and $W^{\pm}A$ decay modes. Again the
absence of events exceeding the SM expectation allows us to set the lower mass
limit as $m_{H^{\pm}}\geq 84$–$94\,$GeV, depending on $m_{A}$[135]. The LEP
exclusions, which come from the scalar bremsstrahlung process, are shown by
the brown lines in Fig. 2.

The LHC is also a powerful tool to search for additional scalars. Due to the
nature of a hadron collider, only partial information of a collision is
accessible. Thanks to the large statistics and good control of tau-lepton
identification, low-mass charged Higgs bosons have recently been excluded
[136, 137]. The Run 2 full result of the search for left-handed staus can be
directly adapted to draw exclusion plots because the production cross section
is the same as for charged-Higgs pair production. The ATLAS measurement
excluded $120\,{\rm GeV}\leq m_{H^{\pm}}\leq 390\,{\rm GeV}$ [138] while the
CMS one excluded $115\,{\rm GeV}\leq m_{H^{\pm}}\leq 340\,{\rm GeV}$ [139],
assuming BR$(H^{\pm}\to\tau\nu)=1$. Since the SM background (BG) stems from
$W^{\pm}$ pair production, it is difficult to probe lower values of
$m_{H^{\pm}}$ at the LHC. The low mass window $m_{H^{\pm}}\simeq 100\,$GeV
will close in the near future [140] once the systematic error scales as
$1/\sqrt{L}$. If a large mass difference between the neutral scalars is
assumed, searches for same-sign $H^{\pm}$ become relevant [141]. However, a
more dedicated experimental analysis is necessary to estimate the sensitivity.
It would be worth mentioning that the decay $h\to AZ$ is not possible in the
alignment limit considered throughout this review.

In order to precisely interpret the constraints by the LHC searches, the decay
properties of the additional scalars are important. In addition to
BR$(A\to\tau^{+}\tau^{-})\simeq 1$, we summarize here the relevant parameter
dependence as follows:

$\displaystyle{\rm{BR}}(H\to\tau^{+}\tau^{-})$
$\displaystyle\simeq\frac{\Gamma(H\to\tau^{+}\tau^{-})}{\Gamma(H\to\tau^{+}\tau^{-})+\Gamma(H\to
AZ)}\,,$ (3.5) $\displaystyle{\rm{BR}}(H^{\pm}\to\tau\nu)$
$\displaystyle\simeq\frac{\Gamma(H^{\pm}\to\tau\nu)}{\Gamma(H^{\pm}\to\tau\nu)+\Gamma(H^{\pm}\to
AW^{\pm})}\,,$ (3.6) $\displaystyle\Gamma(H\to\tau^{+}\tau^{-})$
$\displaystyle\simeq\Gamma(H^{\pm}\to\tau\nu)=\frac{m_{H}}{8\pi}\frac{m_{\tau}^{2}}{v^{2}}\xi^{2}\,,$
(3.7) $\displaystyle\Gamma(H\to AZ)$ $\displaystyle=\frac{m_{H}^{3}}{16\pi
v^{2}}\lambda^{3/2}\left(\frac{m_{A}^{2}}{m_{H}^{2}},\frac{m_{Z}^{2}}{m_{H}^{2}}\right)\,,$
(3.8) $\displaystyle\Gamma(H^{\pm}\to AW^{\pm})$
$\displaystyle=\frac{m_{H^{\pm}}^{3}}{16\pi
v^{2}}\lambda^{3/2}\left(\frac{m_{A}^{2}}{m_{H^{\pm}}^{2}},\frac{m_{W}^{2}}{m_{H^{\pm}}^{2}}\right)\,,$
(3.9)

where $\lambda(x_{1},x_{2})=(1-x_{1}-x_{2})^{2}-4x_{1}x_{2}$ and $m_{\tau}$ is
neglected in the phase-space factor. Thus, for large $\xi$, tauonic scalar
decays can make up a significant fraction of the total decay width. For the
relevant mass scale, $\xi\simeq 100$ yields branching ratios $\simeq{\cal
O}(50\%)$, which means that the branching ratios and the resulting LHC
constraints are sensitive in this interesting parameter region.

For the type-X 2HDM interpretation of the muon $g-2$ anomaly, the favored
parameter regions are at large $\xi$ and very light $m_{A}\simeq 30$ GeV, see
Fig. 2. In this case, $\tau$-rich signatures at the LHC become relevant, as
discussed in Ref. [62]. The importance of the $\tau$-rich signatures is also
discussed at the ILC [142], where it is shown that the favored parameter
regions will be fully covered. Although a type-X specific search has not yet
been performed, we can recast the current experimental searches for charginos
and neutralinos in the MSSM, based on these $\tau$-rich signatures [143, 144,
145, 146, 147]. In particular, we consider the ATLAS analysis [147] which
provides detailed kinematic cuts. In the analysis, eight signal regions (SRs)
are defined, designed such that the sensitivity to
$\tilde{\chi}_{1}^{+}\tilde{\chi}_{1}^{-}$ and
$\tilde{\chi}^{\pm}\tilde{\chi}^{0}_{2}$ events is enhanced. As common
features of all the SRs, at least two $\tau$-leptons, a veto on bottom quark
jets in order to reject SM top-quark processes, and large stransverse mass
[148, 149] $m_{\rm T2}>70$–$100$ GeV are required. Some of the SRs aiming for
$\tilde{\chi}_{1}^{+}\tilde{\chi}_{1}^{-}$ and
$\tilde{\chi}_{1}^{+}\tilde{\chi}_{2}^{0}$ require opposite-sign (OS) $\tau$
leptons, while the rest requires same-sign taus for
$\tilde{\chi}_{1}^{+}\tilde{\chi}_{2}^{0}$. As for the SRs with OS taus, a
$Z/h$ veto ($m_{\tau\tau}>120$ GeV) is required to capture the taus from stau
decays, otherwise $m_{\tau\tau}\simeq m_{h}$ is required to capture the
$h\to\tau^{+}\tau^{-}$ in the decay chain. Regarding the missing momentum,
both possibilities of $E_{\rm miss}<150$ GeV and $E_{\rm miss}>150$ GeV are
considered to capture the low-mass and the high-mass spectra, respectively.
Additional selection cuts are imposed depending on the SRs, for details see
Ref. [147]. As a result, $1$–$14$ events were observed in each SR for 139 fb-1
at $\sqrt{s}=13$ TeV, without significant excess. This results in 95%
confidence level (CL) upper limits on the non-SM fiducial cross section
($\sigma_{\rm vis}^{95\%}$) of 0.03–0.1 fb, which put severe bounds even on EW
production processes. This stringent constraint compared to the previous one
is achieved by the very strong selection cut optimized for the chargino and
neutralino searches and due to the large integrated luminosity of 139 fb-1.

Figure 3:  The exclusion factor ($\sigma_{\rm vis}^{\rm type-X}/\sigma_{\rm
vis}^{95\%}$) obtained by the MSSM $\tilde{\chi}^{\pm}\tilde{\chi}^{0}$
searches at the LHC and BR($H\to\tau^{+}\tau^{-}$) are shown in the solid and
dashed contours, respectively, in the type-X 2HDM. The parameter $\xi$ is
fixed to explain the muon $g-2$ anomaly at the $0\,\sigma$ (left panel) and
$-2\,\sigma$ (right panel) levels, corresponding to $a_{\mu}^{\rm
NP}=25.1\times 10^{-10}$ and $13.3\times 10^{-10}$, respectively. The pale
blue region in the bottom-right corner corresponds to $m_{H}<m_{A}$.

We generate EW pair-production events of additional scalars,

$\displaystyle pp\to
HA,\,H^{\pm}A,\,H^{\pm}H,\,H^{\pm}H^{\mp}\,(\to\text{multi-}\tau)\,,$ (3.10)

in the type-X 2HDM using MadGraph5_aMC@NLO [150] $+$ PYTHIA [151] $+$ DELPHES
[152] and apply the selection cuts defined for the above SRs [147]. We
consider the cases where $m_{A}\leq m_{H}=m_{H^{\pm}}$, and $10$ GeV
$<m_{A}<200$ GeV, $100$ GeV $<m_{H}<500$ GeV. The total production cross
section ranges from $5$ fb to $4$ pb. For each model point, we generate 100K
signal events. Figure 3 shows the contour plots of the ratio $\sigma_{\rm
vis}^{\rm type-X}/\sigma_{\rm vis}^{95\%}$ (which we call exclusion factor),
that is, how large an event number is expected relative to the 95% CL upper
limit. The maximal exclusion factors are mainly from the SRs C1C1-LM and
C1N2SS, which were designed to capture chargino pair-production events and
chargino-neutralino pair-production events with same-sign $\tau$ signatures,
respectively. We only take the largest of these SRs to be conservative. We fix
$\xi\,(=\tan\beta)$ as to reproduce $\Delta a_{\mu}$ at the $0\,\sigma$ (left
panel) and $-2\,\sigma$ levels (right panel), corresponding to $a_{\mu}^{\rm
NP}=25.1\times 10^{-10}$ and $13.3\times 10^{-10}$, respectively. The contours
in dotted lines show the expected value of BR$(H\to\tau^{+}\tau^{-})$. The
results show that the interesting regions for the muon $g-2$ anomaly are
completely excluded and it is difficult to save this model unless new decay
modes are introduced. We also checked that selecting any values of
${\rm{BR}}(H\to\tau^{+}\tau^{-})$ and ${\rm{BR}}(H^{\pm}\to\tau\nu)$ results
in the whole region of the depicted plane being excluded as long as
${\rm{BR}}(H\to\tau^{+}\tau^{-})+{\rm{BR}}(H\to ZA)=1$ and
${\rm{BR}}(H^{\pm}\to\tau\nu)+{\rm{BR}}(H^{+}\to W^{+}A)=1$. Even if setting
${\rm BR}(H\to\tau^{+}\tau^{-})={\rm BR}(H^{\pm}\to\tau\nu)=0$, the lowest
value of the exclusion factor is about 1.6. These results are the updated
plots of Fig. 7 in Ref. [62].

Figure 4:  The contour plot of the exclusion factor ($\sigma_{\rm vis}^{\rm
type-X}/\sigma_{\rm vis}^{95\%}$) with fixed $m_{A}=30$ GeV, in the type-X
2HDM. The cyan and blue regions can explain the muon $g-2$ anomaly at the
$1\,\sigma$ and $2\,\sigma$ levels, respectively.

Finally, we show the same exclusion factor defined above in the
$\xi$–$m_{H}(=m_{H^{\pm}})$ plane with fixed $m_{A}=30\,$GeV in Fig. 4. We
note that the figure does not change drastically even if we set
$m_{A}=50\,$GeV as one can infer from contours of the exclusion factor in Fig.
3. In general, if $(m_{H}-m_{A})>m_{Z},m_{W}$, the exclusion factor becomes
larger as the ratio $\xi/m_{H}$ increases. This is due to the larger
$\tau$-branching ratio where the acceptance in the relevant SRs is larger in
the tau modes. If $(m_{H}-m_{A})<m_{Z},m_{W}$, specifically at $m_{H}=100$ GeV
in this plot, the exclusion factor becomes smaller although still larger than
ten. The cyan and blue regions can explain the muon $g-2$ anomaly at the
$1\,\sigma$ and $2\,\sigma$ levels, respectively. One can also see explicitly
that even for
${\rm{BR}}(H\to\tau^{+}\tau^{-})={\rm{BR}}(H^{\pm}\to\tau\nu)=0$, which is
equivalent to the $\xi\to 0$ limit, the exclusion factor is always more than
1, with the minimum around 2 in the bottom-right corner.

In conclusion, the type-X 2HDM interpretation of the muon $g-2$ anomaly in the
mass regions we considered is completely excluded by the chargino pair-
production and chargino-neutralino pair-production searches at the LHC Run 2,
as shown in Figs. 3 and 4. It is impressive that these searches using the
tremendous luminosity accumulated at the LHC already provide such high
sensitivities for EW production processes.

###  3.2 Flavor-aligned 2HDM

In the FA2HDM, the Yukawa couplings $\rho_{f}$ are proportional to the mass
matrix, but the overall constants of up-type and down-type quarks and leptons
are independent free parameters. Therefore, the model includes the type-X 2HDM
and thus has a broader parameter space.

In addition to the Barr-Zee diagram with the tau-lepton in the loop, the one
with a top quark in the loop could contribute to the muon $g-2$. However, the
constraint from $B_{s}\to\mu^{+}\mu^{-}$ sets an upper limit on the top-loop
contribution to the muon $g-2$. This implies a stringent constraint for the
light CP-odd scalar scenario. It is worth mentioning that
$B_{s}\to\mu^{+}\mu^{-}$ can also receive “type-II 2HDM”-like contributions
proportional to $\xi_{d}\,\xi_{e}$. If $\xi_{d}$ is of order
$\xi_{u}m_{t}/m_{b}$, these contributions can become significant despite the
much lighter bottom-quark mass. This can weaken the top-quark loop constraints
from $B_{s}\to\mu^{+}\mu^{-}$, in particular if a fine-tuned cancellation
between $\xi_{d}$ and $\xi_{u}$ occurs. However, we do not consider such
cancellations here and thus neglect contributions involving $\xi_{d}$.

Figure 5:  The parameter plane spanned by $\xi_{e}$ and $\xi_{u}$ in the
FA2HDM. The cyan and blue regions accommodate the muon $g-2$ anomaly at the
$1\,\sigma$ and $2\,\sigma$ levels, respectively. The regions above the red
lines are excluded by $B_{s}\to\mu^{+}\mu^{-}$ at the 95% CL. The black
contours show the size of the top-loop Barr-Zee contribution in units of
$\Delta a_{\mu}$. The regions to the right of the green, purple and brown
lines are excluded by $\tau$ decay, $Z$ decay, and scalar bremsstrahlung
constraints, respectively. We take $m_{A}=20\,(30)\,$GeV and
$m_{H}=m_{H^{\pm}}=250\,(300)\,$GeV on the left (right) panel. The gray dashed
lines represent the type-X 2HDM limit.

In Fig. 5, we show the muon $g-2$ favored region in the $\xi_{e}$ vs.
$\xi_{u}$ plane where $m_{A}=20\,$GeV and $m_{H}=m_{H^{\pm}}=250\,$GeV, and
$m_{A}=30\,$GeV and $m_{H}=m_{H^{\pm}}=300\,$GeV are fixed on the left and
right panels, respectively. The red, green, purple and brown lines are
excluded by $B_{s}\to\mu^{+}\mu^{-}$, $\tau$ decay, $Z$ decay, and scalar
bremsstrahlung constraints, respectively. The black contours show the size of
the top-loop Barr-Zee contribution in units of $\Delta a_{\mu}$. It is found
that the top-quark loop constitutes up to about $1\%$ of the total deviation
for $m_{A}=20\,$GeV. For $m_{A}=30\,$GeV, the contribution is still less than
$2\%$. The constraint becomes more stringent when $A$ gets lighter. We found
that even with $m_{A}=50\,$GeV the contribution can only be up to $5\,\%$. In
order to evade the stringent bound from $B_{s}\to\mu^{+}\mu^{-}$, $\xi_{u}$
needs to be as small as $\mathcal{O}(0.01)$, implying that this scenario is
almost identical to the type-X 2HDM in the context of the muon $g-2$ anomaly.
Note that the gray dashed lines in Fig. 5 represent the type-X 2HDM limit.

If the Yukawa coupling $\xi_{u}$ is small, other flavor observables and
collider processes are less affected, e.g., the single-scalar production via
gluon fusion and di-Higgs production cross section are too small to be probed
[97]. Therefore, the key probe is the same as in the type-X 2HDM case and we
refer the reader to the discussion in the previous section. As a result,
multi-$\tau$ signatures at the LHC exclude the explanation of the muon $g-2$
anomaly within the FA2HDM. We emphasize that the presence of $\xi_{d}$ cannot
significantly reduce the $A\to\tau^{+}\tau^{-}$ branching ratio since
$\xi_{e}$ is already large in any attempt to explain the muon $g-2$ anomaly.
However, the coupling $\xi_{d}$ could also contribute to the production of
scalars and $B_{s}\to\mu^{+}\mu^{-}$.

###  3.3 Muon-specific 2HDM

In the $\mu$2HDM, the one-loop contribution to $a_{\mu}^{\rm NP}$ can explain
the muon $g-2$ anomaly, while the two-loop Barr-Zee contribution is suppressed
by the mass of the heavy additional scalars and the electromagnetic coupling
constant $\alpha$. It is known that $H$ should be lighter than $A$ in order to
have the correct sign of $a_{\mu}^{\rm NP}$. In this case $m_{H}\simeq
m_{H^{\pm}}$ is favored to satisfy both the vacuum stability condition and
$T$-parameter constraint [153]. The size of the mass difference is controlled
by the Higgs quartic couplings and thus constrained by the RGE-based
perturbative unitarity constraint, see Appendices A.5 and A.6 for details. The
black lines of Fig. 6 show the values of $\rho_{e}^{\mu\mu}$ required to
explain the central value of the muon $g-2$ anomaly (top), $-1\,\sigma$
(middle), and $-2\,\sigma$ (bottom) levels, corresponding to $a_{\mu}^{\rm
NP}=25.1\times 10^{-10}$, $19.2\times 10^{-10}$, and $13.3\times 10^{-10}$,
respectively. The orange region requires $\rho_{e}^{\mu\mu}\geq\sqrt{4\pi}$
(corresponding to $\xi_{e}\simeq 5900$), violating perturbativity, and thus we
do not consider it.

Figure 6:  The $\lambda_{5}$–$m_{A}$ parameter plane in the $\mu$2HDM. The
black contours correspond to the value of $\rho_{e}^{\mu\mu}$ that explains
the muon $g-2$ anomaly at the $0\,\sigma$ (upper), $-1\,\sigma$ (middle), and
$-2\,\sigma$ (bottom) levels. The orange regions violate the perturbative
unitarity bound. The purple regions are excluded by the high-$p_{T}$ multi-
lepton searches based on our simulation. The dashed and dotted purple lines
correspond to the extended future prospect with an integrated luminosity of
500 fb-1 and 3 ab-1. The solid and dashed colored (blue, green, and red) lines
in each panel correspond to the parameter regions where the cutoff scale are
$5\,$TeV and $30\,$TeV, respectively.

Since the additional scalars mainly couple to muons, direct searches at the
LHC, e.g., searches for smuons or multi-lepton final states, give a lower
bound on the masses of the additional scalars. Previously, the authors of Ref.
[74] found that the multi-lepton search performed with the 35.9 fb-1 data set
at $\sqrt{s}=13$ TeV [154] excludes Higgs masses below $m_{H}\simeq 620$
GeV.#11#11#11$m_{A}=m_{H^{\pm}}=m_{H}+90$ GeV is assumed to derive the bound.
We updated the analysis with the Run 2 full data [155]. Note that there is a
similar search for NP in multi-lepton final states [156], originally motivated
by heavy vector-like leptons in the type-III seesaw model. However, we cannot
directly use the result since additional jets are required.

Figure 7:  Representative Feynman diagrams for the collider searches in the
$\mu$2HDM (top-left) and $\mu\tau$2HDM (the others).

We generated 100K signal events for the process $pp\to\phi H^{\pm}\to
3\mu+\nu_{\mu}$, shown in Fig. 7 (top-left), with MadGraph5_aMC@NLO [150] for
a given set of $H^{\pm}$ and $\phi$ masses at $\sqrt{s}=13\,$TeV, where $\phi$
denotes $A$ or $H$. Then the scalar sum of charged-lepton $p_{T}$ (LT) and
missing transverse energy (MET) is calculated and compared to the result in
Fig. 3-c of Ref. [155] where both muon and electron are considered. If muon-
exclusive data become available, the signal-to-background ratio will be
amplified, yielding improved sensitivity. We evaluated the Run 2 exclusion
region from the multi-lepton search, which is depicted by the purple shaded
region in Fig. 6. Compared to the previous study [74], the lower mass limit is
increased by about 200 GeV.

The future sensitivity is estimated by assuming the significance scales as the
square root of the luminosity.#12#12#12Since the high-$p_{T}$ lepton signal
region is currently statistically limited, this treatment is justified. We
point out that better sensitivity would be obtained with larger LT+MET bins;
however, this would entail a more complicated experimental analysis.
Therefore, our procedure gives a conservative estimate regarding sensitivity.
We note that smuon searches at the LHC give less stringent constraints [157,
158].

Furthermore, the Landau pole scale $\Lambda_{\rm{LP}}$ is shown as a colored
contour in Fig. 6, where we use solid and dashed lines to illustrate
$\Lambda_{\rm{LP}}=5\,$TeV and $30\,$TeV, respectively. There is still a small
region that can explain the central value of the muon $g-2$ anomaly if one
requires the theory to be perturbative at $5\,$TeV. On the other hand, if
$\Lambda_{\rm{LP}}\geq 30\,$TeV is required, the model cannot explain the muon
$g-2$ anomaly in any region of the parameter space.

We stress that a future 500 fb-1 data set, which approximately corresponds to
the integrated luminosity of the Run 3 full data, can probe the whole
$1\,\sigma$ region once we require that perturbative theory holds at least up
to 5 TeV. The current $2\,\sigma$ region can be covered with a data set of 3
ab-1.

It is noted that our analysis can be readily applicable to the flavor
conserving scenario of general two Higgs doublet model discussed in Refs.
[159, 160].

###  3.4 $\mu\tau$-flavor violating 2HDM

In the $\mu\tau$2HDM, the dominant contribution to $a_{\mu}^{\rm NP}$ arises
from the one-loop diagram involving a scalar and a tau lepton, see Fig. 1
(right). The contribution receives an $m_{\tau}/m_{\mu}$ enhancement factor
compared to the $\mu$2HDM, due to the chirality flip on the internal tau
propagator. Thanks to this enhancement, heavier scalars can serve as an
explanation for the muon $g-2$ anomaly, in contrast to the $\mu$2HDM.

Again the mass difference $m_{H}$–$m_{A}$ needs to be large to explain the
muon $g-2$, see Appendix A.1. Note that the mass difference is determined by
$\lambda_{5}$ in the Higgs alignment limit, see Eq. (2.23). This implies large
scalar couplings in the Higgs potential, and thus the RGEs become important,
see Appendices A.5 and A.6 for details. Once we require that the model remains
perturbative up to 30 TeV (5 TeV), we obtain an upper limit on the scalar mass
scale of 1250 GeV (1650 GeV).

For this model, direct searches at the LHC can effectively put constraints on
the available parameter space. The additional scalars are quark-phobic (see
Eq. (2.34)) and thus the main production mechanism at hadron colliders is EW
pair production. The heavy neutral scalar dominantly decays into $\mu\tau$.
The unique double $\mu\tau$-flavor violating resonances can result in two
same-sign muons and two oppositely charged same-sign tau leptons in the final
state, shown in Fig. 7 (top-right), while the SM background can be neglected
to a good approximation. There is so far no experimental analysis for this
channel, however theoretical sensitivity studies are available [87, 161]. In
these studies, single-scalar production diagrams via Yukawa interactions are
taken into account, shown in Fig. 7 (bottom diagrams), in addition to EW pair
production, enhancing the sensitivity in the scenario of large masses.

Figure 8:  The $\lambda_{5}$–$m_{A}$ parameter plane in the $\mu\tau$2HDM. The
black contour corresponds to the value of
$\rho_{e}^{\mu\tau}\rho_{e}^{\tau\mu}$ that explains
$a_{\mu}^{\rm{NP}}=25.1\times 10^{-10}$. The cutoff scale is depicted by the
purple contour. The blue shaded region is excluded by the tau decay
constraint. The dashed magenta line corresponds to the future prospect of the
HL-LHC, the region to the left of which will be covered. A future HE-LHC will
cover the complete parameter region on the plane. The underlying figure is
taken from Ref. [161].

The black contour in Fig. 8 shows the value of
$\rho_{e}^{\mu\tau}\rho_{e}^{\tau\mu}$ required to explain the central value
of the muon $g-2$ anomaly with the assumption of
$|\rho_{e}^{\mu\tau}|=|\rho_{e}^{\tau\mu}|$. For given masses $m_{H,A}$, the
NP effect in $a_{\mu}$ is large if both couplings $\rho_{e}^{\mu\tau}$ and
$\rho_{e}^{\tau\mu}$ are large while the Landau pole resides at a high-energy
scale. As we are interested in the heaviest scenario, we thus set
$|\rho_{e}^{\mu\tau}|=|\rho_{e}^{\tau\mu}|$. The $\tau$ decay constraint,
which mainly comes from the charged-scalar tree-level correction, is depicted
by the blue region. It is known that the systematic uncertainty is already the
dominant one [162]. The Belle II experiment will improve the sensitivity of
the $\tau$ decays in the future while its detailed prospect is not available.
It should be added that the constraint from $Z$-boson decays is weaker in this
scenario [81].

The dashed magenta contours in Fig. 8 show the HL-LHC reach, where the region
to the left of the contour can be probed. A future high energy (HE)-LHC taking
data at $\sqrt{s}=27$ TeV with 3 ab-1 can cover the entire region in the plane
depicted in Fig. 8, see Ref. [161] for more details. Since the main production
mechanism is via EW processes which are insensitive to $\rho_{e}^{\mu\tau}$
and $\rho_{e}^{\tau\mu}$, the result does not change drastically even if the
anomaly needs to be explained at the $-1\,\sigma$ level.

We note that decays into gauge bosons such as $H^{+}\to W^{+}H$ are not
kinematically allowed in the $\mathcal{O}$(1) TeV scenario since the
difference of the squared masses is $\mathcal{O}(v^{2})$, see Eq. (2.23). Even
if $\Delta a_{\mu}$ decreases, the distinctive signal cross section is
controlled by the gauge coupling and thus the proposed
$\mu^{\pm}\mu^{\pm}\tau^{\mp}\tau^{\mp}$ final state would serve as a smoking
gun signal.

##  4 Summary and discussion

The current deviation in the muon anomalous magnetic moment could be a long-
awaited hint of new physics. In this review article, we revisited the muon
$g-2$ anomaly within two-Higgs-doublet models. Despite the fact that the 2HDM
is one of the simplest extensions of the SM, the model has very rich
phenomenology and constitutes the scalar sector of several UV-completing
models. In addition to the $\mathbb{Z}_{2}$-based type-X 2HDM, a flavor-
aligned 2HDM and the $\mathbb{Z}_{4}$-based muon-specific 2HDM and
$\mu\tau$-flavor violating 2HDM were considered. These models have been known
to be a possible solution to the muon $g-2$ anomaly. We updated the collider
constraints which give crucial bounds and clarified the available parameter
space.

We found that due to the updated constraint from $B_{s}\to\mu^{+}\mu^{-}$ the
up-type Yukawa coupling in the FA2HDM cannot be large unless the contribution
is cancelled by a down-type Yukawa coupling. If we do not rely on this tuning,
the Barr-Zee contribution with a top quark cannot explain more than $5\%$ of
the discrepancy. Therefore, the scenario is effectively the same as the type-X
2HDM in this case.

Although tau-rich signatures at the LHC provide a distinctive test of the
type-X 2HDM and also the FA2HDM interpretation, a detailed Run 2 analysis has
not yet been performed for this signature. Based on the latest chargino-
neutralino searches with the Run 2 full data, we found that the muon $g-2$
anomaly favored parameter region is certainly excluded in the type-X 2HDM and
FA2HDM. We emphasize that even if we employ the cancellation in
$B_{s}\to\mu^{+}\mu^{-}$, as long as tauonic and bosonic scalar decays are
dominant, the interesting region is excluded. Therefore the conventional
$\mathbb{Z}_{2}$-based scenario is no longer viable for the explanation of the
muon $g-2$ anomaly.

We also revisited the $\mu$2HDM and found that the Run 2 data pushed up the
lower mass bound by 200 GeV compared to the previous analysis and the model
encounters the Landau pole at less than 5 TeV if the central value of the muon
$g-2$ anomaly is required. We also found that the complete $1\,\sigma$ region
satisfying $\Lambda_{\rm{LP}}\geq 5\,$TeV can be probed with a near future
$500\,\rm{fb}^{-1}$ data set, which approximately corresponds to the
integrated luminosity of the Run 3 full data. The status and prospect of the
$\mu\tau$2HDM was also summarized. In this model a distinctive
$\mu^{\pm}\mu^{\pm}\tau^{\mp}\tau^{\mp}$ final state at the LHC is a key
prediction. We found that the HL-LHC can probe scenarios with scalars of up to
1.1 TeV. Together with the Landau pole constraint, a future upgrade of the HL-
LHC to energies of $27\,$TeV could cover the complete parameter space relevant
for the muon $g-2$ anomaly. The summary of the relevant flavor and collider
constraints for those four kinds of 2HDMs is shown in Table 2.

In this review, we focused on a simple extension that features only one
additional Higgs doublet. It is known that a further extension, e.g.,
2HDM$+$vector-like lepton, can explain the muon $g-2$ anomaly (see Ref. [163]
for instance), since the heavier vector-like lepton mass can be used to flip
the chirality, leading to a much larger mass enhancement
factor.#13#13#13However, it is known that with only vector-like leptons one
can explain the muon $g-2$ anomaly even without introducing an additional
scalar doublet. Further collider searches for the vector-like lepton scenario
will shed light on a possible realization of these extended models [164, 165,
166].

## Acknowledgements

S. I. appreciates the warm hospitality of Yuji Omura and Kindai University and
where he started this project. The research of S. I. and M. L. is supported by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
grant 396021762-TRR 257. T. K. is supported by the Grant-in-Aid for Scientific
Research from the Ministry of Education, Culture, Sports, Science, and
Technology (MEXT), Japan, No. 21K03572. The work of T. K. is supported by the
Japan Society for the Promotion of Science (JSPS) Core-to-Core Program, No.
JPJSCCA20200002. M.T. is supported by the Fundamental Research Funds for the
Central Universities, the One Hundred Talent Program of Sun Yat-sen
University, China, and by the JSPS KAKENHI Grant, the Grant-in-Aid for
Scientific Research C, Grant No. 18K03611.

##  Appendix A Explicit formulae

In this appendix, we collect all relevant formulae required in the analyses of
this article.

###  A.1 Muon $g-2$

The leading-order Feynman diagrams contributing to possible solutions of the
muon $g-2$ anomaly are shown in Fig. 1.

The one-loop flavor conserving contribution is given as [74]

$\displaystyle\delta
a_{\mu}^{\mu}\simeq\frac{G_{F}v^{2}m_{\mu}^{2}}{8\sqrt{2}\pi^{2}}(\rho_{e}^{\mu\mu})^{2}\left[\frac{1}{m_{A}^{2}}\left(\frac{11}{6}+\log\frac{m_{\mu}^{2}}{m_{A}^{2}}\right)-\frac{1}{m_{H}^{2}}\left(\frac{7}{6}+\log\frac{m_{\mu}^{2}}{m_{H}^{2}}\right)-\frac{1}{6m_{H^{\pm}}^{2}}\right]\,.$
(A.1)

The contributions from $A$ and $H$ have opposite sign and $H^{\pm}$ always
gives a negative yet tiny contribution to the muon $g-2$. A positive shift is
realized for $m_{H}\lesssim m_{A}$.

Next, the one-loop $\mu\tau$-flavor violating contribution receives the tau-
mass chirality enhancement factor as [81],

$\displaystyle\delta
a_{\mu}^{\tau}\simeq\frac{m_{\mu}^{2}}{16\pi^{2}}\rho_{e}^{\mu\tau}\rho_{e}^{\tau\mu}\frac{m_{\tau}}{m_{\mu}}\left(\frac{\log{\frac{m_{H}^{2}}{m_{\tau}^{2}}}-\frac{3}{2}}{m_{H}^{2}}-\frac{\log{\frac{m_{A}^{2}}{m_{\tau}^{2}}}-\frac{3}{2}}{m_{A}^{2}}\right)\,,$
(A.2)

where the $H^{\pm}$-loop contribution does not have the tau-mass enhancement
and can thus be neglected. Again the contributions from $A$ and $H$ have
opposite signs. A positive contribution to the muon $g-2$ requires $m_{H}\leq
m_{A}$ for $\rho_{e}^{\mu\tau}\rho_{e}^{\tau\mu}>0$ and $m_{H}\geq m_{A}$ for
$\rho_{e}^{\mu\tau}\rho_{e}^{\tau\mu}<0$.

Finally, the contribution from the two-loop Barr-Zee diagram is given as [73]

$\displaystyle\delta a_{\mu}^{\rm BZ}\simeq\frac{\alpha
m_{\mu}}{16\pi^{3}}\left\\{\frac{4\rho_{u}^{tt}\rho_{e}^{\mu\mu}}{3m_{t}}\left[F_{1}(x_{tH})-F_{2}(x_{tA})\right]+\frac{\rho_{e}^{\tau\tau}\rho_{e}^{\mu\mu}}{m_{\tau}}\left[F_{1}(x_{\tau
H})+F_{2}(x_{\tau A})\right]\right\\}\,,$ (A.3)

where $x_{f\phi}=m_{f}^{2}/m_{\phi}^{2}$ and the loop functions are defined as

$\displaystyle F_{1}(x)=$ $\displaystyle
x\int^{1}_{0}dy\frac{2y(1-y)-1}{x-y(1-y)}\log\left[\frac{x}{y(1-y)}\right]\,,$
(A.4) $\displaystyle F_{2}(x)=$ $\displaystyle
x\int^{1}_{0}dy\frac{1}{x-y(1-y)}\log\left[\frac{x}{y(1-y)}\right]\,.$ (A.5)

###  A.2 Tau decays

The treatment of the constraint arising from tau-lepton decays is crucial in
order to judge the type-X 2HDM interpretation. There are five precision
observables, the correlations among which should be taken into account. The
HFLAV constraints on the tau effective couplings are summarized as [167],

$\displaystyle\biggl{(}\frac{g_{\tau}}{g_{\mu}}\biggl{)}_{\tau}$
$\displaystyle=1.0009\pm
0.0014\,,~{}~{}\biggl{(}\frac{g_{\tau}}{g_{e}}\biggl{)}_{\tau}=1.00027\pm
0.0014\,,~{}~{}$ $\displaystyle\biggl{(}\frac{g_{\mu}}{g_{e}}\biggl{)}_{\tau}$
$\displaystyle=1.0019\pm
0.0014\,,~{}~{}\biggl{(}\frac{g_{\tau}}{g_{\mu}}\biggl{)}_{\pi}=0.9959\pm
0.0038\,,$ $\displaystyle\biggl{(}\frac{g_{\tau}}{g_{\mu}}\biggl{)}_{K}$
$\displaystyle=0.9855\pm 0.0075\,,$ (A.6)

where the symmetric correlation matrix is given by

$\displaystyle\begin{pmatrix}1&&&&\\\ 0.51&1&&&\\\ -0.50&0.49&1&&\\\
0.16&0.18&0.01&1&\\\ 0.12&0.11&-0.01&0.07&1\\\ \end{pmatrix}\,.$ (A.7)

The effective couplings for leptonic tau decays are defined as

$\displaystyle\left(\frac{g_{\tau}}{g_{\mu}}\right)^{2}_{\tau}\propto\frac{\Gamma(\tau\to
e\nu_{\tau}\overline{\nu}_{e})}{\Gamma(\mu\to
e\nu_{\mu}\overline{\nu}_{e})}\,,\quad\left(\frac{g_{\tau}}{g_{e}}\right)^{2}_{\tau}\propto\frac{\Gamma(\tau\to\mu\nu_{\tau}\overline{\nu}_{\mu})}{\Gamma(\mu\to
e\nu_{\mu}\overline{\nu}_{e})}\,,\quad\left(\frac{g_{\mu}}{g_{e}}\right)^{2}_{\tau}\propto\frac{\Gamma(\tau\to\mu\nu_{\tau}\overline{\nu}_{\mu})}{\Gamma(\tau\to
e\nu_{\tau}\overline{\nu}_{e})}\,,$ (A.8)

while the ones for hadronic tau decays ($h=\pi,K$) are defined as

$\displaystyle\left(\frac{g_{\tau}}{g_{\mu}}\right)^{2}_{h}\propto\frac{\Gamma(\tau\to
h\nu_{\tau})}{\Gamma(h\to\mu\overline{\nu}_{\mu})}\,.$ (A.9)

These ratios are normalized by the phase spaces so that the SM predictions are
1.

Tree-level and one-loop corrections to the tau effective couplings are
calculated in Refs. [121, 59]. The contributions are given by

$\displaystyle\left(\frac{g_{\tau}}{g_{\mu}}\right)_{\tau,\,\pi,\,K}$
$\displaystyle\simeq 1+\delta_{\tau}^{\rm{loop}}-\delta_{\mu}^{\rm{loop}},$
(A.10) $\displaystyle\left(\frac{g_{\tau}}{g_{e}}\right)_{\tau}$
$\displaystyle\simeq
1+\delta_{\tau}^{\rm{tree}}+\delta_{\tau}^{\rm{loop}}-\delta_{\mu}^{\rm{loop}},$
(A.11) $\displaystyle\left(\frac{g_{\mu}}{g_{e}}\right)_{\tau}$
$\displaystyle\simeq 1+\delta_{\tau}^{\rm{tree}},$ (A.12)

with

$\displaystyle\delta_{\tau}^{\rm{tree}}$
$\displaystyle=\frac{1}{2}\left[-\frac{v^{2}}{m_{H^{\pm}}^{2}}\rho_{e}^{\mu\mu}\rho_{e}^{\tau\tau}\frac{m_{\mu}g(m_{\mu}^{2}/m_{\tau}^{2})}{m_{\tau}f(m_{\mu}^{2}/m_{\tau}^{2})}+\frac{v^{4}}{16m_{H^{\pm}}^{4}}\left(\rho_{e}^{\mu\mu}\rho_{e}^{\tau\tau}\right)^{2}\right]\,,$
(A.13) $\displaystyle\delta_{\ell}^{\rm{loop}}$
$\displaystyle=\frac{(\rho_{e}^{\ell\ell})^{2}}{32\pi^{2}}\left\\{1+\frac{1}{4}\left[h\left(\frac{m_{A}^{2}}{m_{H^{\pm}}^{2}}\right)+h\left(\frac{m_{H}^{2}}{m_{H^{\pm}}^{2}}\right)\right]\right\\}\,.$
(A.14)

A contribution from the tree-level $H^{\pm}$ exchange in
$\tau\to\mu\nu_{\tau}\overline{\nu}_{\mu}$ is represented by
$\delta_{\tau}^{\rm tree}$, while radiative corrections to the
$W$-$\ell$-$\nu_{\ell}$ couplings are denoted by $\delta_{\ell}^{\rm loop}$.
The loop functions are given by

$\displaystyle f(x)$ $\displaystyle=1-8x+8x^{3}-x^{4}-12x^{2}\log{x}\,,$
(A.15) $\displaystyle g(x)$
$\displaystyle=1+9x-9x^{2}-x^{3}+6x(1+x)\log{x}\,,$ (A.16) $\displaystyle
h(x)$ $\displaystyle=\frac{1+x}{1-x}\log{x}\,.$ (A.17)

To investigate the exclusion region from tau decays, the $\chi^{2}$ is
constructed based on these five observables including their correlations. For
the $\mu\tau$2HDM, the tau decay constraints are less significant due to the
heavy additional scalar masses, even if one includes the Michel parameters
[168].

###  A.3 $Z$ decays

Sizable lepton Yukawa couplings change the $Z$-boson vertices at the one-loop
level [59, 85, 64]. The corrections to the ratios of the leptonic decay widths
are given by

$\displaystyle\frac{\Gamma(Z\to\tau^{+}\tau^{-})}{\Gamma(Z\to e^{+}e^{-})}$
$\displaystyle\simeq 1+2\frac{g_{L}^{e}{\rm{Re}}(\delta
g_{L})+g_{R}^{e}{\rm{Re}}(\delta g_{R})}{(g_{L}^{e})^{2}+(g_{R}^{e})^{2}}\,,$
(A.18) $\displaystyle\frac{\Gamma(Z\to\mu^{+}\mu^{-})}{\Gamma(Z\to
e^{+}e^{-})}$ $\displaystyle\simeq 1+2\frac{g_{L}^{e}{\rm{Re}}(\delta
g_{L})+g_{R}^{e}{\rm{Re}}(\delta
g_{R})}{(g_{L}^{e})^{2}+(g_{R}^{e})^{2}}\left(\frac{\rho_{e}^{\mu\mu}}{\rho_{e}^{\tau\tau}}\right)^{2}\,,$
(A.19)

with the vertex corrections

$\displaystyle\delta g_{L}$
$\displaystyle=\frac{(\rho_{e}^{\tau\tau})^{2}}{32\pi^{2}}\biggl{\\{}-\frac{1}{2}\left[B_{Z}(x_{A})+B_{Z}(x_{H})+4C_{Z}(x_{A},x_{H})\right]$
$\displaystyle\quad+s_{W}^{2}\left[B_{Z}(x_{A})+B_{Z}(x_{H})+\tilde{C}_{Z}(x_{A})+\tilde{C}_{Z}(x_{H})\right]\biggl{\\}}\,,$
(A.20) $\displaystyle\delta g_{R}$
$\displaystyle=\frac{(\rho_{e}^{\tau\tau})^{2}}{32\pi^{2}}\biggl{\\{}\frac{1}{2}\left[4C_{Z}(x_{A},x_{H})-4C_{Z}(x_{H^{\pm}},x_{H^{\pm}})+2\tilde{C}_{Z}(x_{H^{\pm}})-\tilde{C}_{Z}(x_{A})-\tilde{C}_{Z}(x_{H})\right]$
$\displaystyle\quad+s_{W}^{2}\left[B_{Z}(x_{A})+B_{Z}(x_{H})+2B_{Z}(x_{H^{\pm}})+4C_{Z}(x_{H^{\pm}},x_{H^{\pm}})+\tilde{C}_{Z}(x_{A})+\tilde{C}_{Z}(x_{H})\right]\biggl{\\}}\,,$
(A.21)

where $x_{\phi}=m_{\phi}^{2}/m_{Z}^{2}$, $s_{W}\equiv\sin\theta_{W}$, and
$g_{L,R}=T_{3}-Qs_{W}^{2}$ ($g^{e}_{L}\simeq-0.27$ and $g_{R}^{e}\simeq
0.23$). The loop functions are defined as

$\displaystyle B_{Z}(x)$
$\displaystyle=-\frac{1}{2\bar{\epsilon}}-\frac{1}{4}+\frac{1}{2}\log x\,,$
(A.22) $\displaystyle C_{Z}(x,y)$
$\displaystyle=\frac{1}{4\bar{\epsilon}}-\frac{1}{2}\int^{1}_{0}dz_{1}\int^{z_{1}}_{0}dz_{2}\,\log\left[z_{1}z_{2}+y(1-z_{1})+(x-1)z_{2}\right]\,,$
(A.23) $\displaystyle\tilde{C}_{Z}(x)$
$\displaystyle=\frac{1}{2\bar{\epsilon}}+\frac{1}{2}-x\left(1+\log
x\right)+x^{2}\left[\log x\log(1+x^{-1})-\rm{Li}_{2}(-x^{-1})\right]$
$\displaystyle\quad-\frac{{\rm{i}}\pi}{2}\left[1-2x+2x^{2}\log(1+x^{-1})\right]\,,$
(A.24)

where $\rm{Li}_{2}$ denotes the di-logarithm function and the
1/$\bar{\epsilon}$ poles cancel in the sum. We confirm that the forward-
backward asymmetry and the tau polarization asymmetry in
$Z\to\tau^{+}\tau^{-}$ give less stringent constraints.

For $\mu\tau$2HDM, $(\rho_{e}^{\tau\tau})^{2}$ is replaced by
$(\rho^{\mu\tau}_{e})^{2}$ in $\delta g_{L}$ and by $(\rho^{\tau\mu}_{e})^{2}$
in $\delta g_{R}$, respectively [85]. For
$|\rho^{\mu\tau}_{e}|=|\rho^{\tau\mu}_{e}|$ the constraint is less stringent.

###  A.4 $B_{s}\to\mu^{+}\mu^{-}$

In this section, we discuss the constraint from $B_{s}\to\mu^{+}\mu^{-}$. In
Ref. [169], the calculation of the full one-loop Wilson coefficients
contributing to $B_{s}\to\mu^{+}\mu^{-}$ has been performed within the FA2HDM.
The recent CMS result [131] is consistent with the SM prediction [126, 127,
128]. Since the type-X 2HDM increases the branching ratio of
$B_{s}\to\mu^{+}\mu^{-}$, the recent shift of the experimental world average
[132] weakens the $m_{A}$ bound stemming from $B_{s}\to\mu^{+}\mu^{-}$
compared to the previous world average [129, 130]. This bound is relevant for
the type-X 2HDM and FA2HDM, since the dominant contribution comes from the
one-loop diagram with the light CP-odd scalar mediation, shown in Fig. 9. We
adopted the formulae from Ref. [169] and derived the constraint.

Since $\rho_{e}^{\tau\tau}$ is larger than $\rho_{e}^{\mu\mu}$, one might
think that $B_{s}\to\tau^{+}\tau^{-}$ could be a good decay process in order
to probe the 2HDMs. However, this does not hold and the NP sensitivity would
be the same as $B_{s}\to\mu^{+}\mu^{-}$, because the SM amplitude ($W$-box and
$Z$-penguin diagrams) is also proportional to $m_{\tau}$. Moreover,
$B_{s}\to\tau^{+}\tau^{-}$ has not yet been observed.

Figure 9:  The dominant Feynman diagram for $B_{s}\to\mu^{+}\mu^{-}$ in the
type-X 2HDM and FA2HDM.

###  A.5 Perturbative unitarity and vacuum stability

In this section we discuss theoretical constraints imposed on the couplings in
the scalar potential by the requirement of perturbative unitarity and the
vacuum stability. The constraints from perturbativity can be derived from the
consideration of scattering amplitudes of the Higgs bosons. Following Ref.
[170] where longitudinally polarized gauge bosons are replaced with the
corresponding Nambu-Goldstone bosons, we only consider the scattering
processes involving scalars and gauge bosons. The full set of scattering
amplitudes is expressed as a $22\times 22$ matrix, which falls apart into four
decoupled sub-matrices [171, 172, 173, 174, 175, 176, 59]. The perturbative
unitarity bound is imposed on the $12$ distinct eigenvalues of the matrix as

$|e_{j}|<8\pi\quad(j=1,\dots,12)\,,$ (A.25)

where

$\displaystyle e_{1,2}$ $\displaystyle=\lambda_{3}\pm\lambda_{4}\,,\
e_{3,4}=\lambda_{3}\pm\lambda_{5}\,,\ e_{5,6}=\lambda_{3}+2\lambda_{4}\pm
3\lambda_{5}\,,$ (A.26) $\displaystyle e_{7,8}$
$\displaystyle=\frac{1}{2}\left[(\lambda_{1}+\lambda_{2})\pm\sqrt{(\lambda_{1}-\lambda_{2})^{2}+4\lambda_{4}^{2}}\right]\,,$
(A.27) $\displaystyle e_{9,10}$
$\displaystyle=\frac{1}{2}\left[3(\lambda_{1}+\lambda_{2})\pm\sqrt{9(\lambda_{1}-\lambda_{2})^{2}+4(2\lambda_{3}+\lambda_{4})^{2}}\right]\,,$
(A.28) $\displaystyle e_{11,12}$
$\displaystyle=\frac{1}{2}\left[(\lambda_{1}+\lambda_{2})\pm\sqrt{(\lambda_{1}-\lambda_{2})^{2}+4\lambda_{5}^{2}}\right]\,,$
(A.29)

where all $\lambda_{j}$ are running couplings. Here, the contributions from
$\lambda_{6}$ and $\lambda_{7}$ are discarded; $\lambda_{6}=0$ is fixed by the
Higgs alignment condition and $\lambda_{7}$ is suppressed by large
$\tan\beta$, see Eq. (2.21). We define our cutoff scale $\Lambda_{\rm{LP}}$ as
the minimum scale at which either the vacuum stability or perturbative
unitarity condition breaks down when evolving the couplings with the RGEs from
an input scale to a high-energy scale. It should be noted that the quartic
couplings are also bounded from below by the conditions [177, 178],

$\lambda_{1},\lambda_{2}\geq
0\,,\quad\sqrt{\lambda_{1}\lambda_{2}}+\lambda_{3}\geq
0\,,\quad\sqrt{\lambda_{1}\lambda_{2}}+\lambda_{3}+\lambda_{4}-|\lambda_{5}|\geq
0\,,$ (A.30)

but these conditions are always satisfied in the parameter region of our
interest.

###  A.6 Renormalization group equations

The RGEs of the scalar quartic couplings in the $\mu$2HDM and $\mu\tau$2HDM
are given in the form of

$\frac{d\lambda_{j}}{d\log\mu}=\frac{\beta_{\lambda_{j}}}{(4\pi)^{2}}\,,$
(A.31)

where $\mu$ is the renormalization scale. The RGE running effect is important
for these two 2HDMs since the scalars are heavy and thus the Yukawa couplings
are large.

At the one-loop level, the beta functions $\beta_{\lambda_{j}}$ of the scalar
potential in Eq. (2.19) are given by [178, 179]

$\displaystyle\beta_{\lambda_{1}}$
$\displaystyle=12\lambda_{1}^{2}+4\lambda_{3}^{2}+4\lambda_{3}\lambda_{4}+2\lambda_{4}^{2}+2\lambda_{5}^{2}+24\lambda_{6}^{2}+\frac{3}{4}(3g^{4}+g^{\prime
4}+2g^{2}g^{\prime 2})$ $\displaystyle\quad-3\lambda_{1}(3g^{2}+g^{\prime
2})+12\lambda_{1}y_{t}^{2}-12y_{t}^{4}\,,$ (A.32)
$\displaystyle\beta_{\lambda_{2}}$
$\displaystyle=12\lambda_{2}^{2}+4\lambda_{3}^{2}+4\lambda_{3}\lambda_{4}+2\lambda_{4}^{2}+2\lambda_{5}^{2}+24\lambda_{7}^{2}+\frac{3}{4}(3g^{4}+g^{\prime
4}+2g^{2}g^{\prime 2})-3\lambda_{2}(3g^{2}+g^{\prime 2})$
$\displaystyle\quad+4\lambda_{2}[(\rho_{e}^{\ell\ell})^{2}+(\rho_{e}^{\mu\tau})^{2}+(\rho_{e}^{\tau\mu})^{2}]-4[(\rho_{e}^{\ell\ell})^{4}+(\rho_{e}^{\mu\tau})^{4}+(\rho_{e}^{\tau\mu})^{4}]\,,$
(A.33) $\displaystyle\beta_{\lambda_{3}}$
$\displaystyle=2(\lambda_{1}+\lambda_{2})(3\lambda_{3}+\lambda_{4})+4\lambda_{3}^{2}+2\lambda_{4}^{2}+2\lambda_{5}^{2}+4\lambda^{2}_{6}+16\lambda_{6}\lambda_{7}+4\lambda_{7}^{2}-3\lambda_{3}(3g^{2}+g^{\prime
2})$ $\displaystyle\quad+\frac{3}{4}(3g^{4}+g^{\prime 4}-2g^{2}g^{\prime
2})+2\lambda_{3}[3y_{t}^{2}+(\rho_{e}^{\ell\ell})^{2}+(\rho_{e}^{\tau\mu})^{2}+(\rho_{e}^{\mu\tau})^{2}]\,,$
(A.34) $\displaystyle\beta_{\lambda_{4}}$
$\displaystyle=2\lambda_{4}(\lambda_{1}+\lambda_{2}+4\lambda_{3}+2\lambda_{4})+8\lambda_{5}^{2}+10\lambda_{6}^{2}+4\lambda_{6}\lambda_{7}+10\lambda_{7}^{2}+3g^{2}g^{\prime
2}$ $\displaystyle\quad-3\lambda_{4}(3g^{2}+g^{\prime
2})+2\lambda_{4}[3y_{t}^{2}+(\rho_{e}^{\ell\ell})^{2}+(\rho_{e}^{\tau\mu})^{2}+(\rho_{e}^{\mu\tau})^{2}]\,,$
(A.35) $\displaystyle\beta_{\lambda_{5}}$
$\displaystyle=2\lambda_{5}(\lambda_{1}+\lambda_{2}+4\lambda_{3}+6\lambda_{4})+10\lambda_{6}^{2}+4\lambda_{6}\lambda_{7}+10\lambda_{7}^{2}-3\lambda_{5}(3g^{2}+g^{\prime
2})$
$\displaystyle\quad+2\lambda_{5}[3y_{t}^{2}+(\rho_{e}^{\ell\ell})^{2}+(\rho_{e}^{\tau\mu})^{2}+(\rho_{e}^{\mu\tau})^{2}]\,,$
(A.36) $\displaystyle\beta_{\lambda_{6}}$
$\displaystyle=12\lambda_{1}\lambda_{6}+6\lambda_{3}(\lambda_{6}+\lambda_{7})+8\lambda_{4}\lambda_{6}+4\lambda_{4}\lambda_{7}+10\lambda_{5}\lambda_{6}+2\lambda_{5}\lambda_{7}-3\lambda_{6}(3g^{2}+g^{\prime
2})$
$\displaystyle\quad+3\lambda_{6}[3y_{t}^{2}+(\rho_{e}^{\ell\ell})^{2}+(\rho_{e}^{\tau\mu})^{2}+(\rho_{e}^{\mu\tau})^{2}]\,,$
(A.37) $\displaystyle\beta_{\lambda_{7}}$
$\displaystyle=12\lambda_{2}\lambda_{7}+6\lambda_{3}(\lambda_{6}+\lambda_{7})+4\lambda_{4}\lambda_{6}+8\lambda_{4}\lambda_{7}+2\lambda_{5}\lambda_{6}+10\lambda_{5}\lambda_{7}-3\lambda_{7}(3g^{2}+g^{\prime
2})$
$\displaystyle\quad+\lambda_{7}[3y_{t}^{2}+(\rho_{e}^{\ell\ell})^{2}+(\rho_{e}^{\tau\mu})^{2}+(\rho_{e}^{\mu\tau})^{2}]\,,$
(A.38)

where the Yukawa couplings are defined by $y_{f}=\sqrt{2}m_{f}/v$. The RGEs of
the gauge and Yukawa couplings are defined in the same way, with the beta
functions given by

$\displaystyle\beta_{g_{j}}$ $\displaystyle=b_{j}g_{j}^{3}\,,\quad
b_{j}=\\{7,-3,-7\\}\quad(g_{j}=\\{g^{\prime},g,g_{s}\\})\,,$ (A.39)
$\displaystyle\beta_{y_{t}}$ $\displaystyle=y_{t}\left(-\frac{17}{12}g^{\prime
2}-\frac{9}{4}g^{2}-8g_{s}^{2}+\frac{9}{2}y_{t}^{2}\right)\,,$ (A.40)
$\displaystyle\beta_{\rho_{e}^{\ell\ell}}$
$\displaystyle=\rho_{e}^{\ell\ell}\left[\frac{5}{2}(\rho_{e}^{\ell\ell})^{2}-\frac{9}{4}\left(\frac{5}{3}g^{\prime\,2}+g^{2}\right)\right]\,,$
(A.41) $\displaystyle\beta_{\rho_{e}^{\tau\mu}}$
$\displaystyle=\rho_{e}^{\tau\mu}\left[(\rho_{e}^{\mu\tau})^{2}+\frac{5}{2}(\rho_{e}^{\tau\mu})^{2}-\frac{9}{4}\left(\frac{5}{3}g^{\prime\,2}+g^{2}\right)\right]\,,$
(A.42) $\displaystyle\beta_{\rho_{e}^{\mu\tau}}$
$\displaystyle=\rho_{e}^{\mu\tau}\left[(\rho_{e}^{\tau\mu})^{2}+\frac{5}{2}(\rho_{e}^{\mu\tau})^{2}-\frac{9}{4}\left(\frac{5}{3}g^{\prime\,2}+g^{2}\right)\right]\,.$
(A.43)

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