# Compatibility of muon $g-2$, $W$ mass anomaly in type-X 2HDM

Jongkuk Kim  Korea Institute for Advanced Study, Seoul 02455, Korea
<EMAIL_ADDRESS>
###### Abstract

Recently CDF II Collaboration reported that they measured W boson mass
precisely. The measurement is deviated from the Standard Model (SM) Prediction
at 7$\sigma$. Also, the recent FNAL measurement of the muon magnetic moment
shows a $4.2\sigma$ deviation. To resolve the W boson as well as the muon
$(g-2)$ anomalies, we explore the type-X two Higgs doublet model (2HDM). We
analyze the $(m_{A},\tan\beta)$ parameter space of the type-X 2HDM consistent
with the muon $(g-2)$, lepton universality test and recent $W$ boson mass
measurments from the CDF II collaboration. We find that the measurement of the
$h\to AA$ branching ratio gives a strong lower mass bound on $m_{H}$.

Two Higgs Doublet Models, Light (Pseudo)Scalar, Muon Anomalous Magnetic
Moment, CDF Anomaly

###### pacs:

††preprint: KIAS-P220xx

##  I Introduction

The CDF experiment presented a high precision measurement of mass of the $W$
boson: $m_{W}^{CDF}=80.4335\pm 0.0094$ GeV, which is about $7\sigma$ away from
the SM prediction CDF:2022hxs . Such deveiation is in conflict expermental
measurements of $m_{W}$ by Tevatron CDF:2013dpa  and ATLAS ATLAS:2017rzl .
Undeniably, we should interpret the deviation with caution as it might
originate from systematic errors.

On the other hand, this result stands with the growing number of experimental
anomalies, including the muon anomalous magnetic moment, $B$-anomalies etc.
Hence it is worthwhile to reexamine the implication of the CDF result
scenarios, which can explain other observed anomalies. The deviation in $W$
mass can originate from the quantum correction due to the presence of BSM
particles which modifies the $S,T$ and $U$ parameters. Several analysises has
been presented in this direction Zhu:2022tpr ; Lu:2022bgw ; Du:2022pbp ;
deBlas:2022hdk ; Yang:2022gvz ; Athron:2022qpo ; Strumia:2022qkt ;
Cacciapaglia:2022xih ; Liu:2022jdq ; Sakurai:2022hwh ; Athron:2022isz ;
Asadi:2022xiy ; Song:2022xts ; Bahl:2022xzi ; Cheng:2022jyi ; Lee:2022nqz ;
DiLuzio:2022xns ; Gu:2022htv ; Babu:2022pdn ; Paul:2022dds ; Biekotter:2022abc
; Balkin:2022glu ; DiLuzio:2022ziu ; Cheung:2022zsb ; Du:2022brr ;
Endo:2022kiw ; Crivellin:2022fdf ; Heo:2022dey ; Ahn:2022xeq ; Zheng:2022irz ;
PerezFileviez:2022gmy ; Ghoshal:2022vzo ; Peli:2022ybi ; Kawamura:2022uft ;
Kanemura:2022ahw ; Nagao:2022oin ; Mondal:2022xdy ; Zhang:2022nnh ;
Carpenter:2022oyg ; Popov:2022ldh ; Arcadi:2022dmt ; Chowdhury:2022moc ;
Borah:2022obi ; Cirigliano:2022qdm ; Zeng:2022lkk ; Du:2022fqv ;
Ghorbani:2022vtv ; Bhaskar:2022vgk ; Baek:2022agi ; Cao:2022mif ;
Borah:2022zim ; Lee:2022gyf ; Almeida:2022lcs ; Cheng:2022aau ; Addazi:2022fbj
; Heeck:2022fvl ; Abouabid:2022lpg ; Tan:2022bip ; Faraggi:2022emm ;
Batra:2022pej ; Benbrik:2022dja ; Cai:2022cti ; Dermisek:2022xal ;
Chen:2022ocr ; Gupta:2022lrt .

Here we are interested in the 2HDM of type-X, which can explain the muon
$(g-2)$ if the pseudoscalar ($A$) is much lighter than the other scalars. A
considerable deviation in the $S$ and $T$ parameters is necessary to
accommodate CDF results. We analyzed the parameter space which can
simultaneously satisfy CDF measurement and $(g-2)_{\mu}$. To explain
$(g-2)_{\mu}$, we require the pseudoscalar to be lighter than $m_{h}/2$ where
$h$ is the SM Higgs bosons. And it opens up the $h\to AA$ decay model, which
has been explored at LHC by CMS and ATLAS collaboration in various four-
fermion final states, including $4\tau,2\mu 2\tau$ ATLAS:2015unc ; CMS:2017dmg
; ATLAS:2018coo ; CMS:2018qvj ; CMS:2019spf ; CMS:2020ffa . We showed that the
LHC limit puts a strong lower limit on the mass spectrum of the type-X 2HDM.
Since the scalars are leptophobic in type-X 2HDM, strong constraint comes from
the measurement of lepton universality in charged and neutral current
processes. We have considered both the leptonic $Z$ decays and leptonic/semi-
leptonic $\tau$ decays to constrain the available parameter space. We showed
that it is possible to explain the $(g-2)_{\mu}$ anomaly after satisfying all
the experimental and theoretical constraints, including the CDF measurement.
We found that the masses of the heavy BSM scaalrs are strongly restricted from
below, and can LHC and ILC can explore the parameters space.

The paper is organized as follows: In Sec. II, we briefly summarize type-X
2HDM, muon $g-2$ contribution, lepton universality test and oblique
parameters. In Sec. III, we describe all of the theoretical and experiment
constraints and find preferred region. We conclude in Sec. IV.

##  II The Model Basics

In this section we briefly describe the necessary elements of the type-X 2HDM.
For a detailed discussions about multi Higgs doublet model please see
Gunion:1989we ; Djouadi:2005gj ; Branco:2011iw .

###  II.1 2HDM with type-X Yuakwa interaction

The model contains two Higgs doublets $\Phi_{1}$ and $\Phi_{2}$ with $Y=1/2$.
Presence of two Higgs doublets where both the doublets couple to the fermions
leads to flavour changing neutral current (FCNC) interaction at tree level. To
avoid this, we impose a $\mathbb{Z}_{2}$ symmetry such that
$\Phi_{1}\rightarrow-\Phi_{1}$ and $\Phi_{2}\rightarrow\Phi_{2}$. The scalar
potential can be written as,

$\displaystyle V_{\mathrm{2HDM}}$ $\displaystyle=$ $\displaystyle-
m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}-m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}-\Big{[}m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}+\mathrm{h.c.}\Big{]}+\frac{1}{2}\lambda_{1}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)^{2}+\frac{1}{2}\lambda_{2}\left(\Phi_{2}^{\dagger}\Phi_{2}\right)^{2}$
$\displaystyle+\lambda_{3}\left(\Phi_{1}^{\dagger}\Phi_{1}\right)\left(\Phi_{2}^{\dagger}\Phi_{2}\right)+\lambda_{4}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)\left(\Phi_{2}^{\dagger}\Phi_{1}\right)+\Big{\\{}\frac{1}{2}\lambda_{5}\left(\Phi_{1}^{\dagger}\Phi_{2}\right)^{2}+\rm{h.c.}\Big{\\}}.$

After the electroweak symmetry breaking, the scalars $\Phi_{1}$ and $\Phi_{2}$
will acquire vacuum expectation values (VEV) $v_{1}$ and $v_{2}$ respectively.
We can parameterize the doublets in the following way,
$\Phi_{j}=(H_{j}^{+},(v_{j}+h_{j}+iA_{j})/\sqrt{2})^{T}$ and obtain the
physical states $A$, $h$, $H$, $H^{\pm}$ by appropriately rotating the gauge
eigenstates:

$\displaystyle\begin{pmatrix}H\\\ h\end{pmatrix}$
$\displaystyle=\begin{pmatrix}c_{\alpha}&&s_{\alpha}\\\
-s_{\alpha}&&c_{\alpha}\end{pmatrix}\begin{pmatrix}h_{1}\\\
h_{2}\end{pmatrix}$ $\displaystyle A(H^{\pm})$
$\displaystyle=-s_{\beta}\;A_{1}(H_{1}^{\pm})+c_{\beta}\;A_{2}(H_{2}^{\pm}),$
(1)

where $s_{\alpha}=\sin\alpha$, $c_{\beta}=\cos\beta$ and
$\tan\beta=v_{2}/v_{1}$. The CP-even state $h$ is identified with the SM-like
Higgs with mass $m_{h}\approx 125$ GeV.

In this article we will consider the Type-X 2HDM where the RH leptons are odd
under $\mathbb{Z}_{2}$ symmetry. The relevant Yukawa Lagrangian is given by,

$-\mathcal{L}_{Y}=Y^{u}\bar{Q_{L}}\widetilde{\Phi_{2}}u_{R}+Y^{d}\bar{Q_{L}}\Phi_{2}d_{R}+Y^{e}\bar{l_{L}}\Phi_{1}e_{R}+h.c.,$
(2)

where $\widetilde{\Phi_{2}}=i\sigma_{2}\Phi_{2}^{*}$. After symmetry breaking
the Yukawa Lagrangian takes the form,

$\displaystyle\mathcal{L}_{\mathrm{Yukawa}}^{\mathrm{Physical}}$
$\displaystyle=$
$\displaystyle-\sum_{f=u,d,\ell}\frac{m_{f}}{v}\left(\xi_{h}^{f}\overline{f}hf+\xi_{H}^{f}\overline{f}Hf-i\xi_{A}^{f}\overline{f}\gamma_{5}Af\right)-\left\\{\frac{\sqrt{2}V_{ud}}{v}\overline{u}\left(\xi_{A}^{u}m_{u}P_{L}+\xi_{A}^{d}m_{d}P_{R}\right)H^{+}d+\frac{\sqrt{2}m_{l}}{v}\xi_{A}^{l}\overline{v}_{L}H^{+}l_{R}+\mathrm{h.c.}\right\\}.$

The Yukawa multiplicative factors, i.e. $\xi_{\phi}^{f}$ are given in Table 1.

| $\xi_{h}^{u}$ | $\xi_{h}^{d}$ | $\xi_{h}^{\ell}$ | $\xi_{H}^{u}$ | $\xi_{H}^{d}$ | $\xi_{H}^{\ell}$ | $\xi_{A}^{u}$ | $\xi_{A}^{d}$ | $\xi_{A}^{\ell}$
---|---|---|---|---|---|---|---|---|---
Type-X | $c_{\alpha}/s_{\beta}$ | $c_{\alpha}/s_{\beta}$ | $-s_{\alpha}/c_{\beta}$ | $s_{\alpha}/s_{\beta}$ | $s_{\alpha}/s_{\beta}$ | $c_{\alpha}/c_{\beta}$ | $\cot\beta$ | $-\cot\beta$ | $\tan\beta$
Table 1: The multiplicative factors of Yukawa interactions in type X 2HDM

The coupling modifier $\xi_{h}^{\ell}=-s_{\alpha}/c_{\beta}$ can be written as
$\sin(\beta-\alpha)-\tan\beta\cos(\beta-\alpha)$.

###  II.2 Relevant Parameter Space for Muon $(g-2)$

In the type-X 2HDM contribution to $(g-2)_{\mu}$ comes at one loop diagram
mediated by $H,A$ and $H^{\pm}$. The contribution from $H$ ix is positive,
whereas the $A$ mediated process contributes negatively. The contribution to
$(g-2)_{\mu}$ coming from $H^{\pm}$ diagram is insignificant. In addition,
two-loop Barr-Zee diagram contributes to the $(g-2)_{\mu}$ and dominated over
the one-loop contribution. At two-loop, $A(H)$ mediated process contributes
positively(negatively). One can identify the parameter region of
$(m_{A},\tan\beta)$ plane consistent with the measurement of
$a_{\mu}=(g-2)_{\mu}/2$ Muong-2:2021ojo :

$a_{\mu}(\mbox{Exp})-a_{\mu}(\mbox{SM})=(251\pm 59)\times 10^{-11}.$ (3)

The pseudoscalar has to be sufficiently light, and a large enough enhancement
on the leptonic coupling is necessary, which requires a large $\tan\beta$. So
for our numerical analysis, we will consider $m_{A}$ to vary within the range
of $10-90$ GeV, and we scan $\tan\beta$ between $20-100$. Lower vaues of
$m_{A}$ are stronly constrained by $B_{S}\to\mu\mu$ processes. Note that limit
on charged Higgs coming from $b\to s\gamma$ limits $\tan\beta>3$
Haller:2018nnx  for type-X 2HDM.

###  II.3 CDF Result and Mass Spectrum

Figure 1:  Preferred contours for the S and T oblique parameters with
2$\sigma$ C.L. The blue contour correspond to PDG data. The red contour
correspond to data recently reported by the CDF II collaboration. The left and
right vertical bars show the $S$ parameter when $m_{H}$ lies between $200$GeV
and $400$GeV respectively. In type-X 2HDM model, $S$ is a negative correction.

The quartic couplings in Eq. II.1 are constrained by the requirement of vacuum
stability, perturbativity and unitarity. These constrains can be satisfied
when Wang:2014sda ,

$\displaystyle m_{H}\simeq m_{H^{\pm}}$ $\displaystyle\leq$
$\displaystyle\sqrt{\lambda_{max}}\ v=\sqrt{4\pi}\ v.$ (4)

Here we assumed the wrong sign scenario i.e $\tan\beta\cos(\beta-\alpha)\sim
2$.

The constrains from precision measurement are usually defined in terms of S,T
and U parameters. As we have discussed in the introduction, the effect of
newly measured $m_{W}$ by CDF can be parametrized in terms of these
parameters. Assuming $\Delta U=0$ the allowed values of S and T parameters are
discussed in Ref. Lu:2022bgw , and the allowed range for S and T parameters
are:

$\displaystyle S$ $\displaystyle=$ $\displaystyle 0.15\pm
0.08,~{}~{}~{}~{}~{}T=0.27\pm 0.06.$ (5)

Figure 2: Allowed range for $T$ parameter in the 2HDM with $m_{H}=200$ GeV
(purple), $300$ GeV (blue), and $400$ GeV (Red).

In the 2HDM, the contributions to oblique parameters induced by the scalar
bosons are given by Toussaint:1978zm

$\displaystyle S$ $\displaystyle=$
$\displaystyle-\frac{1}{4\pi}\left[f_{2}\left(m_{H^{\pm}},m_{H^{\pm}}\right)-\sin^{2}\left(\beta-\alpha\right)f_{2}\left(m_{H},m_{A}\right)-\cos^{2}\left(\beta-\alpha\right)f_{2}\left(m_{h},m_{A}\right)\right],$
$\displaystyle T$ $\displaystyle=$
$\displaystyle-\frac{1}{32\pi^{2}\alpha_{EM}v^{2}}\left[-f_{1}(m_{A},m_{H^{\pm}})+\sin^{2}\left(\beta-\alpha\right)\left(f_{1}\left(m_{H},m_{A}\right)-f_{1}\left(m_{H},m_{H^{\pm}}\right)\right)\right.$
(6)
$\displaystyle\left.\quad\quad\quad\quad\quad\quad+\cos^{2}\left(\beta-\alpha\right)\left(f_{1}\left(m_{h},m_{A}\right)-f_{1}\left(m_{h},m_{H^{\pm}}\right)\right)\right],$

where

$\displaystyle f_{1}\left(m_{1},m_{2}\right)$ $\displaystyle=$
$\displaystyle\frac{m^{2}_{1}+m^{2}_{2}}{2}-\frac{m^{2}_{1}m^{2}_{2}}{m^{2}_{1}-m^{2}_{2}}\log\frac{m^{2}_{1}}{m^{2}_{2}},$
(7) $\displaystyle f_{2}\left(m_{1},m_{2}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\left(\frac{4}{3}-\frac{m^{2}_{1}\log
m^{2}_{1}-m^{2}_{2}\log
m^{2}_{2}}{m^{2}_{1}-m^{2}_{2}}-\frac{m^{2}_{1}+m^{2}_{2}}{\left(m^{2}_{1}-m^{2}_{2}\right)^{2}}f_{1}\left(m_{1},m_{2}\right)\right).$
(8)

The S parameter remains very small, and this strongly constrain the possible
value of T parameter.

We plotted the allowed space in S-T plane in Fig. 1 and the vertical bar shows
the value of S parameter for heavy Higgs mass in the range of 200 to 400 GeV.
As expected, only a small range of T parameter is allowed. We have satisfied
the T parameter and found that the CP even heavy Higgs and the charged Higgs
can not be degenerate. We plotted the mass differnce between $H$ and $H^{\pm}$
in Fig. 2. The mass differnce has a mild dependence on the pseudoscalar mass.

We plotted the allowed space in the S-T plane in Fig. 1 and the vertical bar
shows the value of the S parameter for heavy Higgs mass in the range of 200 to
400 GeV. As expected, only a small range in the T parameter is allowed. We
have satisfied the T parameter and found that the CP even heavy Higgs and the
charged Higgs can not be degenerate. We plotted the mass difference between
$H$ and $H^{\pm}$ in Fig. 2. The mass difference has a mild dependence on the
pseudoscalar mass.

###  II.4 Lepton Universality

The lepton universality has been measured by $Z$ and $\tau$ decays at the
level of $0.1\%$ ALEPH:2005ab ; HFLAV:2016hnz . In the type-X 2HDM, loop
corrections provide huge contributions due to the enhanced leptonic couplings
of extra Higgs bosons with large $\tan\beta$.

Let us first consider the lepton universality from $Z$ boson decays. The $Z$
decay measurements can be converted into the leptonic branching ratios which
reads

$\displaystyle\frac{\Gamma\left(Z\to\mu^{+}\mu^{-}\right)}{\Gamma\left(Z\to
e^{+}e^{-}\right)}$ $\displaystyle=$ $\displaystyle 1.0009\pm 0.0028,$
$\displaystyle\frac{\Gamma\left(Z\to\tau^{+}\tau^{-}\right)}{\Gamma\left(Z\to
e^{+}e^{-}\right)}$ $\displaystyle=$ $\displaystyle 1.0019\pm 0.0032,$ (9)

with a positive correlation of $0.63$. For each lepton flavor, we can evaluate
different quantities which can be parameterized as

$\displaystyle\delta_{\ell\ell}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma\left(Z\to\ell^{+}\ell^{-}\right)}{\Gamma\left(Z\to
e^{+}e^{-}\right)}-1.$ (10)

A sizable contributions can generate from the tau Yukawa coupling of
$m_{\tau}\tan\beta/v$.

The other lepton universality has been evaluated by the pure and semi-hadronic
decay processes which are given by

$\displaystyle\left(g_{\tau}\over g_{\mu}\right)=1.0010\pm
0.0015,\quad\left(g_{\tau}\over g_{e}\right)=1.0029\pm
0.0015,\quad\left(g_{\mu}\over g_{e}\right)=1.0019\pm 0.0014,$
$\displaystyle\left(g_{\tau}\over g_{\mu}\right)_{\pi}=0.9961\pm
0.0027,\quad\left(g_{\tau}\over g_{\mu}\right)_{K}=0.9860\pm 0.0070,$ (11)

where the correlation matrix is

$\displaystyle\left(\begin{array}[]{ccccc}1&+0.53&-0.49&+0.24&+0.11\\\
+0.53&1&+0.48&+0.26&+0.10\\\ -0.49&+0.48&1&+0.02&-0.01\\\
+0.24&+0.26&+0.02&1&+0.06\\\ +0.11&+0.10&-0.01&+0.06&1\end{array}\right).$
(17)

In the large $\tan\beta$ limit, there are two important corrections. One
correction is originated from the heavy charged Higgs boson at the tree-level.
The other one comes from the extra Higgs bosons through loop-level
corrections.

So far in the literature the constraint on 2HDM-X coming from lepton
universality has been calculated Abe:2015oca ; Chun:2016hzs ; Chun:2019oix
assuming that $H$ and $H^{\pm}$ degenerate. However, the CDF result calls for
a non-degenerate mass spectrum, and we have recalculated the lepton
universality limits for such a scenario. We found that unless $H$ is
relatively light ($m_{H}\sim 200$ GeV) the non-degenerate mass spectrum does
not affect the lepton universality limits significantly. This is due to the
fact that the lepton universality mostly depends on the mass gap between
$m_{A}$ and $m_{H}$ when $H^{\pm}$ is heavier. Following the
$\chi^{2}$-analysis in Ref.Chun:2016hzs ; Chun:2018ibr , we obtain the limits
on the lepton universality of $\tau$ and $Z$ decays.

Figure 3: Branching ratio of $h\to AA$ for different values of $m_{A}$ and
$m_{H}$ as a function of the coupling modifier define in Tab. 1.

###  II.5 Constraints from Higss decay

Since the pseudoscalar is lighter than $m_{h}/2$, the SM Higgs can decay to a
pair of pseudoscalars. Since the pseudoscalar couples strongly to leptons, the
relevant signal at the LHC is $pp\to h\to AA\to 4\tau/2\mu 2\tau$. The both
CMS and ATLAS have studied these channels ATLAS:2015unc ; CMS:2017dmg ;
ATLAS:2018coo ; CMS:2018qvj ; CMS:2019spf ; CMS:2020ffa . From these
observations, the limit on $BR(h\to AA)$ is as low as $<2\%$ for $m_{A}=9$
GeV. For most of the $m_{A}$ range, the limit is $BR(h\to AA)<5\%$. We need to
satisfy these constraints.

The relevant coupling for the Higgs decay to $AA$ is given by Gunion:2002zf ,

$\displaystyle\lambda_{T}$ $\displaystyle=$
$\displaystyle\frac{1}{4}s_{2\beta}^{2}(\lambda_{1}+\lambda_{2})+(\lambda_{3}+\lambda_{4}+\lambda_{5})(s_{\beta}^{4}+c_{\beta}^{4})-2\lambda_{5}$
$\displaystyle\lambda_{U}$ $\displaystyle=$
$\displaystyle\frac{1}{2}s_{2\beta}(s_{\beta}^{2}\lambda_{1}-c_{\beta}^{2}\lambda_{2}+c_{2\beta}(\lambda_{3}+\lambda_{4}+\lambda_{5}))$
$\displaystyle\lambda_{hAA}$ $\displaystyle=$
$\displaystyle-v(\lambda_{T}s_{\beta-\alpha}-\lambda_{U}c_{\beta-\alpha})$
(18)

By rewriting $\sin\alpha=\xi_{h}^{\ell}\cos\beta$ and by inverting the quartic
coupling to the relevant mass parameter we can calculate the branching ratio
as a function of $m_{H},m_{A},\xi_{h}^{\ell}$. Figure. 3 depictes the
branching ratios. Evidently, $BR(h\to AA)$ is close to $100\%$ for most of the
parameter space, unless the parameters are fine-tuned to cancel. This
cancellation happens for a particular value of coupling modifier depending on
the mass of heavy Higgs. The supression in the branching ratio does not depend
on the quartic coupling $\lambda_{1}$, another free parameter of the model.
From the Figure, it is evident that the parameter space is very strongly
constrained. Since the coupling modifier dictates the deviation of Higgs
Yukawa coupling, it is constrained by the measurement of Higgs coupling
strength. The current measurement of these coupling strengths by CMS and ATLAS
collaboration allows $\xi_{h}^{\ell}$ to be within 0.7 to 1.15 at 2$\sigma$
CMS:2022kdi ; ATLAS:2022yrq . Thus, the combination of Higgs signal strength
measurement with the measurement of Higgs branching ratio to light
pseudoscalar puts a lower limit on the mass spectrum of 2HDM-X, and the CP
even Higgs has to heavier than 300 GeV.

##  III Numerical Results and Discussions

Figure 4: Allowed parameter space in $(m_{A},\tan\beta)$ plane describing all
the constraints. The brown area is allowed by the muon $g-2$ at $2\sigma$ C.L.
Below blue solid (dashed) lines by the lepton universality test with $\tau$
decays are allowed. Below red solid (dashed) lines by the lepton universality
test with $Z$ decays are preffered. We take $m_{H}=300~{}{\rm(top)}$ and
$m_{H}=400~{}{\rm(bottom)}$GeV. Solid (dashed) lines correspond to the minimal
(maximal) mass difference to resolve the CDF II $W$ boson mass anomaly.

In the previous section we have discussed the relevant parameter space and the
contraines coming from various theoretical as well as experimental
observatios. We scaned the parmater space in the following range:

$m_{A}:10-90~{}\textrm{
GeV},~{}\tan\beta:20-90,~{}m_{H}:300~{}\&~{}400\textrm{ GeV}$ (19)

The allowed parameter and the constraints are shown in Fig. 4 where the
top(bottom) panel corresponds to $m_{H}=300(400)$ GeV. To explain
$(g-2)_{\mu}$, we neet light pseudoscalar and large $\tan\beta$ and the
parameter space is strongly constrained by the measurement of lepton
universality. However, small parameter space with $m_{A}$ between 10 to 50 GeV
is still allowed when $\tan\beta$ is in the range $40-60$ GeV.

Since the BSM scalars are hadrophobic, the present limit on the scalars of
type-X 2HDM is rather weak. The existing LHC limits originate from the
fermionic interactions of the BSM scalars, i.e. the existing search strategy
looks for BSM scalars produced via gluon fusion and decay to $b\bar{b}$ or
$\tau\tau$. Since the quark coupling to the new scalars is proportional to
$\cot\beta$, there is no limit on BSM scalars if $\tan\beta$ is greater than
10. The parameter space of this model is well restricted, and the non-
fermionic decay mode should be studied to probe the allowed parameter space.

##  IV Conclusion

In this work, we explored type-X 2HDM to constrain the $(m_{A},\tan\beta)$
parameter region with the recent measurement obtained by CDF II collaboration.
To resolve the W boson mass discrepancy, the mass difference between $m_{H}$
and $m_{H^{\pm}}$ are the order of $\mathcal{O}(10)$GeV due to enhancement in
the oblique correction, $T\sim 0.1-0.2$. We calculated constraints coming from
the lepton universality test and measurement of Higgs signal strength at the
LHC. We found that the CDF measurement does not constrain the 2HDM-X parameter
space, which can explain the muon $(g-2)$ anomaly. Interestingly, the Higgs
signal strength measurement, specifically the measurement of $BR(h\to AA)$,
put a strong lower limit on the mass of the heavy Higgs boson, which should be
heavier than $200$GeV.

## Acknowledegments

The work is supported in part by KIAS Individual Grant, Grant No. PG074202
(JK) at Korea Institute for Advanced Study. I would like to thank Tanmoy
Mondal for valuable comments and discussion on the related topics.

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