Resolving the W-Boson Mass in the Lepton Specific Two Higgs Doublet Model
Ali Çiçia,111E-mail<EMAIL_ADDRESS>Hüseyin Dağb,222E-mail:
<EMAIL_ADDRESS>
a Turgut Özal Anatolian High School, Van, Turkey
bBursa Technical University, Bursa, Turkey

###### Abstract

In this study, the parameter space of the Lepton Specific Two Higgs Doublet
Model (LS-2HDM) is investigated to align the W-boson mass reported by the CDF
experiment with recent theoretical and experimental findings. The Lepton
Specific Two Higgs Doublet Model, a distinguished category within Two Higgs
Doublet Models, contains two CP-even, one CP-odd, and two charged scalar
bosons, which play crucial role in estimating W boson mass. First, constraints
from diverse experimental data, including ATLAS 13 TeV analyses, rare B-meson
decays, and Lepton Flavor Universality in tau-lepton and Z-boson decays, are
determined and imposed on the parameter space of the model. These constraints
are subsequently applied to potential solutions generated through random scans
using SARAH 4.13.0 and analyzed using SPheno 4.0.3. The analysis indicates the
possibility of realizing the CDF-reported W-boson mass up to $1\sigma$ within
the low $\tan\beta$ regime ($2.5\lesssim\tan\beta\lesssim 8.0$). Furthermore,
it establishes mass limits for the additional scalar boson as $164\lesssim
m_{h_{2}}\lesssim 195$ GeV, $330\lesssim m_{A}\lesssim 575$ GeV, and
$345\lesssim m_{H^{\pm}}\lesssim 685$ GeV. Moreover, it is observed that
instead of the masses, mass differences of the new scalars of the model are
more constrained to assure the CDF reported value of the W boson. Finally, all
potential solutions estimating the W-boson mass within a $1\sigma$ vicinity
are rigorously tested using the HiggsTools package. Remarkably, only one
solution remains valid, estimating $M_{W}=80.4103$ GeV within a $2.44\sigma$
vicinity of the CDF-reported value.

##  I Introduction

The Standard Model (SM) has withstood rigorous testing, successfully been
explaining various phenomena in particle physics. However, recent measurements
of the W-boson mass at the Collider Detector at Fermilab (CDF) have revealed
substantial discrepancies between experimental observations and theoretical
predictions within the SM framework. The CDF reported a precise measurement of
the W-boson mass using the $8.8fb^{-1}$ dataset from $p\bar{p}$ collisions
with a center-of-mass energy of 1.96 TeV as [1]

$\displaystyle M_{W}^{\rm CDF}=80.4335\pm 0.0094\ \text{GeV},$ (1)

which deviates from the SM prediction $M_{W}^{\rm SM}=80.357\pm 0.006{\rm
GeV}$ by $7\sigma$ [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Such a notable
discrepancy suggests the potential existence of new physics phenomena,
necessitating a comprehensive exploration of physics Beyond the Standard Model
(BSM). The inclusion of BSM particles might induce quantum corrections
accountable for the deviation observed in the W-boson mass, hence they have
sparked a growing interest in exploring new physics BSM, mostly by altering
the oblique parameters, $S$, $T$, and $U$. These approaches include a broad
range of theoretical frameworks, including effective field theory methods [14,
15, 16, 17, 18, 19, 20], supersymmetric (SUSY) models [21, 22, 23, 24, 25, 26,
27] , leptoquark models [28, 29, 30], gravitational approaches [31, 32],
Little Higgs models [33], and extensions of the SM involving additional scalar
singlets [34, 35, 36, 37] or triplets [38, 39, 40] . Additionally, models
featuring vector-like leptons have been explored [41, 42, 43, 44, 45, 46],
alongside investigations from the point of view of neutrino masses and seesaw
mechanisms [47, 48], and other therotical approaches [49, 50, 51, 52, 53, 54,
55, 56, 57, 58]. Furthermore, Two Higgs Doublet Models (2HDM) have been
attracting a considerable interest by providing a simple extension of SM [59,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78],
while the Higgs bosons in its spectrum can interfere with the SM particle they
can yield some deviations in W-boson mass, Lepton Flavor Universality (LFU)
[79, 80, 81] , muon g-2 [82, 83, 84] through their radiative contributions.
Among these efforts, certain studies specifically target the W boson mass
discrepancy while also addressing the dark matter problem by choosing a
focused approach on both issues within their framework [59, 33, 34, 41, 44,
58, 52, 54]. It appears that this anomaly attracts a lot of attention from
several theoretical approaches, among which 2HDMs is an important one.
However, while 2HDMs are widely used in understanding the W boson mass
anomaly, their parameter space is limited by experimental data from several
experiments [79, 80, 85, 80, 81, 86, 87, 88, 89, 90, 91]. Therefore, in this
study, motivated by these theoretical and experimental efforts within 2HDM
frameworks, an exploratory investigation aimed at reconciling the W boson mass
discrepancy by scanning possible solutions within the 2HDM parameter space.

This study focuses on investigating the impact of the parameter space of 2HDM
on the W-boson mass. Two crucial considerations are taken into account in our
analyses: theoretical limitations and compatibility with the current
experimental data. Theoretical limitations arise from constraints related to
stability of the scalar potential and perturbativity. The predictions of 2HDMs
must also align with the outcomes of various experiments, such as those
involving rare decays of B-meson, Z-boson decay, tau-lepton decay, and
observations from the Large Hadron Collider (LHC). To explore the implications
of different types of 2HDMs within these theoretical and experimental
limitations, the HiggsTools framework was utilized [92]. Among several types
of 2HDMs, the Lepton Specific 2HDM (LS-2HDM) has emerged as the most promising
model, providing numerous solutions that meet the experimental constraints for
additional Higgs bosons. This model also enables to avoid from the current
strong limits determined by recent CMS findings on scalar masses of the 2HDM,
where a Generalized 2HDM was chosen [91]. The following steps are followed in
this work. Firstly, the mass spectra of the scalar sector consistent with both
theoretical and experimental constraints are obtained. After identifying these
solutions, an analysis of the implications for Lepton Flavor Universality is
conducted, particularly by taking into account the processes involving the
tau-lepton and Z-boson. These analyses further constrain the parameter space
of the LS-2HDM. Subsequently, the compatibility of the LS-2HDM with the
observed W-boson mass by CDF is considered together with of theoretical and
experimental restrictions. Finally, it is obtained that the CDF reported
W-boson mass can be successfully accommodated within the framework of the
LS-2HDM, and the remaining parameter space of the model is also determined.

This work is structured as follows: The second section summarizes LS-2HDM,
focusing on its Higgs sector and Yukawa interactions. The third section
outlines the theoretical and experimental constraints used in this analysis,
along with a detailed discussion of their impacts on the parameter space. In
the fourth section, the parameter space of the LS-2HDM is systematically
explored, and potential solutions are provided. Finally, the last section
offers discussions and concluding remarks.

##  II Lepton Specific 2HDM

2HDMs are obtained by extending the scalar sector of the SM with the addition
of a second scalar doublet possessing the same quantum numbers as the SM Higgs
Doublet. The gauge group of the 2HDMs is identical to that of the SM. In
2HDMs, eight scalar fields are introduced and following a spontaneous symmetry
breaking process akin to that of the SM, three of these fields confer masses
of the gauge bosons, while the remaining five fields undergo mixing, resulting
in five distinct physical scalar bosons. These additional degrees of freedom
present in the 2HDM have far-reaching implications for Higgs boson
phenomenology. The scalar doublets of 2HDMs are given as

$\Phi_{1}=\left(\begin{array}[]{c}\phi_{1}+i\phi_{2}\\\
\phi_{3}+i\phi_{4}\end{array}\right)\qquad\text{and}\qquad\Phi_{2}=\left(\begin{array}[]{c}\phi_{5}+i\phi_{6}\\\
\phi_{7}+i\phi_{8}\end{array}\right),$ (2)

where $\phi_{3}$ and $\phi_{7}$ develop non-zero vacuum expectation values
(VEVs) as $<\phi_{3}>=v_{1}/\sqrt{2}$ and $<\phi_{7}>=v_{2}/\sqrt{2}$
satistfying $v_{SM}=\sqrt{v_{1}^{2}+v_{2}^{2}}$. The particle content of 2HDMs
in the scalar sector are two CP-even Higgs ($h_{1,2}$), one CP-odd Higgs
($A$), and two charged Higgs ($H^{\pm}$). For a comprehensive review of 2HDMs,
please refer to Ref. [93] and the citations within there.

In 2HDMs, the addition of a second scalar doublet in the Yukawa sector leads
to Yukawa couplings that lack of flavor-diagonal properties, thereby resulting
in the emergence of tree-level FCNC processes which are severely constrained
from experiments. To resolve this issue, a viable approach is to introduce a
discrete symmetry to the scalar and Yukawa potentials [94]. The application of
this symmetry limits the interactions between the additional scalar doublet
and fermions, suppresses flavor-changing neutral currents at tree-level, and
establishes the model as a viable approach for satisfying experimental
constraints. One important example of this discrete symmetry is the Z2
symmetry [94, 95, 96, 97], which is described as

$\displaystyle\Phi_{1}\rightarrow-\Phi_{1}\;\;\text{and\ \
}\Phi_{2}\rightarrow\Phi_{2}~{},$ (3) $\displaystyle D_{j}\rightarrow
D_{j},\;\;U_{j}\rightarrow U_{j}\;\;\text{and \ \ }E_{j}\rightarrow-E_{j}~{},$

where $E_{j}$ denotes right-handed leptons, $U_{j}$ and $D_{j}$ denote right-
handed up-type and down-type quarks. The 2HDMs with this choice of $Z_{2}$
symmetry are often called as Lepton Specific 2HDM (LS-2HDM). Under $Z_{2}$
symmetry, the tree level potential of the LS-2HDM becomes

$\displaystyle V_{\rm tree}=$ $\displaystyle
m_{1}^{2}|\Phi_{1}|^{2}+m_{2}^{2}|\Phi_{2}|^{2}-\left(m_{3}^{2}\Phi_{1}^{\dagger}\Phi_{2}+h.c.\right)+\frac{\lambda_{1}}{2}|\Phi_{1}|^{4}+\frac{\lambda_{2}}{2}|\Phi_{2}|^{4}$
(4)
$\displaystyle+\lambda_{3}|\Phi_{1}|^{2}|\Phi_{2}|^{2}+\lambda_{4}|\Phi_{1}^{\dagger}\Phi_{2}|^{2}+\frac{\lambda_{5}}{2}\left[(\Phi_{1}^{\dagger}\Phi_{2})^{2}+h.c.\right],$

where $m_{1,2}$ correspond to the mass terms of the scalar potentials, and
$\lambda_{1,..,5}$ refer to self-couplings. In Eqn. 4, the term involving
$m_{3}^{2}$ arises from a combination of scalar doublets and violates the
$Z_{2}$ symmetry, resulting in soft symmetry breaking. The expressions for the
masses of extra scalar bosons at tree level are obtained from Eqn. 4 as

$\displaystyle
m_{H^{\pm}}^{2}=\frac{1}{2}\left(m_{3}^{2}\frac{1}{\sin\beta\cos\beta}-\lambda_{3}v_{\text{SM}}^{2}\right),$
(5) $\displaystyle
m_{h_{1,2}}^{2}=\frac{1}{4}m_{3}^{2}\left(\tan\beta+\cot\beta\mp\frac{2}{\sin
2\alpha}\right)+\lambda_{1}\cos^{2}\beta
v_{\text{SM}}^{2}+\lambda_{2}\sin^{2}\beta
v_{\text{SM}}^{2}\mp\frac{\lambda_{3}\sin\beta\cos\beta
v_{\text{SM}}^{2}}{\sin 2\alpha}\mp\frac{\lambda_{5}\sin\beta\cos\beta
v_{\text{SM}}^{2}}{\sin 2\alpha},$ $\displaystyle
m_{A}^{2}=\frac{1}{2}\left(m_{3}^{2}\frac{1}{\sin\beta\cos\beta}-\lambda_{3}v_{\text{SM}}^{2}+\lambda_{4}v_{\text{SM}}^{2}\right).$

where $\sin\alpha$ is the mixing angle of CP-even Higgs bosons and
$\tan\beta=v_{2}/v_{1}$ is the ratio of VEVs of doublets. In addition to these
expressions in tree-level, radiative corrections to scalar masses must be
taken into account. These corrections can be calculated through one-loop
improved scalar potential described as [98, 99]

$\displaystyle V=V_{\rm tree}+V_{\rm loop},$ (6)

where the loop potential is described as

$V_{\rm
loop}=\dfrac{1}{64\pi^{2}}\displaystyle\sum_{\alpha}n_{\alpha}m_{\alpha}^{4}\left[\log\left(\dfrac{m_{\alpha}^{2}}{\mu^{2}}\right)-\dfrac{3}{2}\right].$
(7)

with

$n_{\alpha}=(-1)^{2s_{\alpha}}Q_{\alpha}C_{\alpha}(2s_{\alpha}+1),$ (8)

where $\mu$ is the renormalization scale, $m_{\alpha}$ denote the masses of
particles contributing at loop-level, $s_{\alpha}$ are the spin of the
particles, $Q_{\alpha}=1(2)$ for neutral (charged) particles and
$C_{\alpha}=3(1)$ for quarks (leptons), and $\alpha$ runs over all the
particles that couple to the scalars at tree-level. Furthermore, The Yukawa
Lagrangian under $Z_{2}$ symmetry can be written as [99]

$\mathcal{L}_{\mathit{Y}}=-Y_{e}^{ij}\bar{L}_{i}\Phi_{1}E_{j}-Y_{u}^{ij}\bar{Q}_{i}\Phi_{2}^{C}U_{j}-Y_{d}^{ij}\bar{Q}_{i}\Phi_{2}D_{j}+h.c.~{},$
(9)

where $Y_{e,u,d}^{ij}$ are the Yukawa couplings, and $L_{i}$ and $Q_{i}$
denote the $SU(2)_{L}$ doublets for leptons and quarks, respectively. Similar
to the Yukawa couplings in the SM, the effective Yukawa couplings for the
LS-2HDM are obtained as shown in Table 1. Since the mass of the heavy top
quark is primarily determined by $\Phi_{2}$, the value of $\tan\beta$ is
constrained as $v_{2}>v_{1}$. If $v_{2}\gg v_{1}$ and $\tan\beta\gg 1$, the
Yukawa coupling of the CP-odd Higgs boson with quarks becomes negligibly
small, while its coupling with leptons increases.

$Y_{u}^{h_{1}},Y_{d}^{h_{1}},Y_{l}^{h_{1}}$ | $\cos\alpha/\sin\beta$ | $\cos\alpha/\sin\beta$ | $-\sin\alpha/\cos\beta$
---|---|---|---
$Y_{u}^{h_{2}},Y_{d}^{h_{2}},Y_{l}^{h_{2}}$ | $\sin\alpha/\sin\beta$ | $\sin\alpha/\sin\beta$ | $\cos\alpha/\cos\beta$
$Y_{u}^{A},Y_{d}^{A},Y_{l}^{A}$ | $\cot\beta$ | $-\cot\beta$ | $\tan\beta$
Table 1: Effective Yukawa couplings for LS-2HDM

##  III Theoretical and Experimental Constraints

In this section, theoretical and experimental constraints on the parameter
space of the LS-2HDM are investigated to determine the allowed parameter
regions for our analysis. First, theroretical limitations on the scalar
potential of the LS-2HDM are considered. The scalar potential given in 4 and 7
should adhere to the vacuum stability and perturbativity conditions which
limits the self couplings as [100]

$\lambda_{i}<4\pi(i=1,\ldots,5).$ (10)

In order to provide unitarity at tree level, self couplings must satisfy the
relations given by

$\displaystyle
3(\lambda_{1}+\lambda_{2})\pm\sqrt{9(\lambda_{1}-\lambda_{2})^{2}+4(2\lambda_{3}+\lambda_{4}|)^{2}}<16\pi,$
(11)
$\displaystyle\lambda_{1}+\lambda_{2}\pm\sqrt{(\lambda_{1}-\lambda_{2})^{2}+4|\lambda_{5}|^{2}}<16\pi,$
$\displaystyle\lambda_{1}+\lambda_{2}\pm\sqrt{(\lambda_{1}-\lambda_{2})^{2}+4|\lambda_{5}|^{2}}<16\pi,$
$\displaystyle\lambda_{3}+2\lambda_{4}\pm 3|\lambda_{5}|<8\pi,$
$\displaystyle\lambda_{3}\pm\lambda_{4}<8\pi,$
$\displaystyle\lambda_{3}\pm|\lambda_{5}|<8\pi.$

Furthermore to ensure that the scalar potential of the LS-2HDM is finite, free
of flat directions, and stable at large field values, following conditions

$\displaystyle\lambda_{1,2}>0,$ (12)
$\displaystyle\lambda_{3}>-\sqrt{\lambda_{1}\lambda_{2}},$
$\displaystyle\lambda_{3}+\lambda_{4}-|\lambda_{5}|>-\sqrt{\lambda_{1}\lambda_{2}},$

are imposed.

Besides the aforementioned theoretical constraints, the LS-2HDM is subject to
stringent experimental constraints, as well. It is evident that the
predictions of LS-2HDM should align with a wide range of experimental
observations, including precision measurements of the electroweak sector and
collider searches for new particles and phenomena. First, the constraint on
the charged scalar boson mass was determined from the Large Electron-Positron
Collider (LEP) data as $m_{H^{\pm}}>80$ GeV [101]. As discussed in the
previous section, due to the significant influence of $m_{h_{1}}$ on $m_{t}$,
${h_{1}}$ is identified as the SM Higgs boson, and its mass is fixed at
$m_{h_{1}}\simeq 125\pm 2$ GeV [102, 103], since the theoretical calculations
of the Higgs boson mass involve about 2 GeV uncertainty [104, 105]. Moreover,
experimental data from electroweak precision measurements to the W-boson mass
in BSM models are used to determine the oblique parameters $S$, $T$ and $U$,
and they are constrained as [106],

$\displaystyle S=-0.02\pm 0.10,$ (13) $\displaystyle T=0.03\pm 0.12,$
$\displaystyle U=0.01\pm 0.11,$

where the expressions of these parameters will be given in Section IV.4.

In addition, limitations from rare B-meson decays, such as
$B_{s}\rightarrow\mu^{+}\mu^{-}$ and $B_{s}\rightarrow X_{s}\gamma$ should be
considered, since they are sensitive probes for extra scalars of BSM. In the
LS-2HDM framework, these decay processes receive contributions from scalar
states via loops, which limits models parameter space. Therefore, the
following bounds on the rare B-meson decay branching ratios

$\displaystyle 1.95\times 10^{-9}\leq BR(B_{s}\rightarrow\mu^{+}\mu^{-})\leq
3.43\times 10^{-9}\;(2\sigma)$ (14) $\displaystyle 2.99\times 10^{-4}\leq
BR(B_{s}\rightarrow X_{s}\gamma)\leq 3.87\times 10^{-4}\;(2\sigma)$

are applied to the parameter space [107, 108, 86]. Finally, parameters of
LS-2HDM which estimate $m_{W}$ within 3$\sigma$ vicinity CDF result given in
Eqn. (1) are considered in this work.

Following these considerations, constraints on the parameters of LS-2HDM can
be summarized in the following four groups.

G1

Theoretical constraints:

  * •

The self couplings should satisfy the ranges given in Eqns. 10, 11, and 12.

G2

Experimental constraints:

  * •

$m_{H^{\pm}}\geq 80$ GeV,

  * •

$-0.12\leq S\leq 0.08$, $-0.09\leq T\leq 0.15$, $-0.1\leq U\ 0.12$,
$(1\sigma)$,

  * •

$1.95\times 10^{-9}\leq BR(B_{s}\rightarrow\mu^{+}\mu^{-})\leq 3.43\times
10^{-9}(2\sigma)$,

  * •

$2.99\times 10^{-4}\leq BR(B_{s}\rightarrow X_{s}\gamma)\leq 3.87\times
10^{-4}(2\sigma)$.

G3

$h_{1}$ is chosen to be SM Higgs boson with

  * •

$m_{h_{1}}=125.0\pm 2$ GeV.

G4

Constraint on W-boson mass reads

  * •

$80.4053\leq M_{W}^{\rm LS-2HDM}\leq 80.4617$ GeV ($3\sigma$).

Labeling of constraints are chosen for brevity in further discussions.

##  IV EXPLORING THE PARAMETER SPACE

In this section, the parameter space of the LS-2HDM is explored by performing
a random scan of potential parameters using SPheno 4.0.3, generated via SARAH
4.13.0 [109, 110]. In these scans, solutions which satisfy the electroweak
symmetry breaking condition, $(v_{1}^{2}+v_{2}^{2}\simeq v_{\rm SM}^{2})$ are
accepted. To ensure that the results of our random parameter scans are
consistent with current measurements of the W-boson mass, the range of the
self couplings are chosen as

$\displaystyle 0$ $\displaystyle\leq\lambda_{1,2}\leq\pi,$ (15) $\displaystyle
0$ $\displaystyle\leq\lambda_{3}\leq 4\pi,$ $\displaystyle-2\pi$
$\displaystyle\leq\lambda_{4,5}\leq 0,$ $\displaystyle-5$ $\displaystyle\leq
m_{3}^{2}\leq 5\;{\rm TeV}^{2},$ $\displaystyle 1.2$
$\displaystyle\leq\tan\beta\leq 40.0\;.$

After successively applying the constraints listed in the previous section,
the resulting number of solutions and their respective color coding in plots,
which will be used throughout the rest of this work, are as follows:

  * •

The green points represent the solutions satisfying G1, and
$N_{green}=275346$.

  * •

The blue points correspond to a subset of green points, representing solutions
satisfy G1 and G2, and $N_{blue}=221017$.

  * •

The yellow points are a further subset of blue points, corresponding to
solutions that satisfy G1, G2 and G3, and $N_{yellow}=2591$.

  * •

The red points represent a subset of the yellow points, corresponding to
solutions that satisfy G1, G2, G3 and G4, and $N_{red}=119$.

###  IV.1 Mass Spectrum of LS-2HDM

In this section, the mass spectrum of the LS-2HDM will be analyzed by imposing
the aforementioned groups of constraints from theoretical and experimental
considerations.

Figure 1:  The relation between $\tan\beta$ and the mass of the LS-2HDM
scalars, for $m_{h_{2}}$ (top-left), $m_{A}$ (top-right), $m_{H^{\pm}}$
(bottom-left) and $m_{h_{1}}$ (bottom-right). Color coding of solutions are
described in detail in previous text and summarized as: Green points satisfy
conditions G1, blue points satisfy conditions G1 and G2, yellow points satisfy
conditions G1, G2 and G3, and red points satisfy conditions G1, G2, G3 and G4.
Solid black lines indicate ATLAS 13 TeV analysis observation limits and gray
shaded areas are excluded by these analysis [87, 88]. In $\tan\beta$ vs
$m_{h_{1}}$ plot (bottom-right), condition G3 is not applied. Dashed vertical
lines represent the SM Higgs mass.

First, the correlation between the masses of the LS-2HDM scalars and
$\tan\beta$ are illustrated in Fig. 1. It is seen from Fig. 1 (top-left) that
solutions satisfying the theoretical constraints specified by G1 do not
introduce further limitations on $m_{h_{2}}$. The blue points in this plot
represent solutions that meet both theoretical and experimental constraints
specified in groups G1 and G2 within the ranges of
$1.6\lesssim\tan\beta\lesssim 40.0$ and $70\lesssim m_{h_{2}}\lesssim 620$
GeV. It’s evident that the application of G2 does not result in a significant
reduction of the parameter space. The yellow points signify solutions that
additionally fulfill the condition $m_{h_{1}}=125\pm 2$ GeV (G1, G2, and G3).
While the application of G3 does not restrict the range of $\tan\beta$, it
does reduce the number of solutions in the parameter space, and imposes the
condition $m_{h_{2}}\gtrsim 126$ GeV. The solutions indicated by the red
points in Fig.1 (top-left) depict the parameter space that fulfills all four
constraints outlined by groups G1, G2, G3, and G4, i.e. additionally
satisfying restrictions coming from $M_{W}$, which limits the mass of the CP-
even scalar boson as $135\lesssim m_{h_{2}}\lesssim 430$ GeV. The region
between the vertical dashed lines in Fig. 1 (top-left) corresponds to the case
where $h_{2}$ is the SM Higgs boson, which does not satisfy G4, i.e. W-boson
mass condition. Therefore, the selection of $h_{1}$ as the SM Higgs boson, as
specified in constraint G3, is clarified.

Additionally, the relationship between $m_{A}$ and $\tan\beta$ is depicted in
Fig. 1 (top-right), together with the experimental constraints from ATLAS 13
TeV results, where $m_{A}$ span the full $60\lesssim m_{A}\lesssim 750$ GeV
range. It is observed that the application of constraints G1 and G2 imposes
limitations as $80\lesssim m_{A}\lesssim 600+4.1\tan\beta$ GeV. Moreover,
adding the condition outlined as G3 reduces the number of solutions within the
aforementioned regions. However, imposing W-mass constraints described as G4
require $100\lesssim m_{A}\lesssim 650$ GeV. The black line indicates the
upper exclusion limit from ATLAS 13 TeV analysis [87], and it eliminates the
majority of the solutions in the parameter space. Consequently, the solutions
satisfying all constraints outlined as G1, G2, G3 and G4, and also imposing
limits from ATLAS 13 TeV model independent scalar mass analysis, the mass of
the CP-odd scalar boson and $\tan\beta$ are restricted as
$330\;\text{GeV}\lesssim m_{A}\lesssim 600\;\text{GeV}~{}\text{and}\ \
\tan\beta\lesssim 8.0$. Based on these findings, low $\tan\beta$ region
described here will be focused in the rest of this study. For the charged
scalars of LS-2HDM, the variation of $m_{H^{\pm}}$ on the $\tan\beta-
m_{H^{\pm}}$ plane is given in Fig. 1 (bottom-left). After applying
constraints G1 and G2, the remaining solutions bounded as $110\
\text{GeV}\lesssim m_{H^{\pm}}\lesssim\ 605+3.8\tan\beta\ \text{GeV}$.
However, after applying G3, the lower bound on $m_{H^{\pm}}$ increases to 150
GeV. After adding cuts from G4, the acceptable solutions satisfy $230\lesssim
m_{H^{\pm}}\lesssim 685$ GeV, and number of valid solutions are reduced. The
exculusion limits from ATLAS 13 TeV [88] requires $\tan\beta\geq 2.5$.

Until now, the CP-even scalar boson $h_{1}$ has been identified as the SM
Higgs boson, and this requirement was described as condition G3. In order to
explore the behavior of $m_{h_{1}}$, we remove condition G3, allowing
$m_{h_{1}}$ to vary without constraints. The relationship between $\tan\beta$
and $m_{h_{1}}$, in this case, is presented in Fig. 1 (bottom-right). It is
observed that $m_{h_{1}}$ exhibits distinct behavior in two specific regions
of $\tan\beta$. For $\tan\beta\lesssim 5$, $m_{h_{1}}$ shows an inverse
proportionality to $\tan\beta$, ranging from $50\lesssim m_{h_{1}}\lesssim
288$ GeV, where upper limit arises from condition G4. Conversely, for
$\tan\beta\geq 5$, $m_{h_{1}}$ demonstrates a direct proportionality to
$\tan\beta$, with values ranging as $50\lesssim m_{h_{1}}\lesssim 450$ GeV.
This contrasting behavior in the low and high $\tan\beta$ regions is evident
from Eqn. 5.

Figure 2: The variation of the cross-sections $\sigma(gg\rightarrow
h_{2}\rightarrow\tau\tau)$ (top-left), $\sigma(gg\rightarrow h_{2}\rightarrow
WW)$ (top-right) and $\sigma(gg\rightarrow h_{2}\rightarrow ZZ)$ (bottom) with
respect to $m_{h_{2}}$. The color scheme is same as Fig. 1, and black shaded
areas are excluded by Refs. [87, 89, 90]

Moreover, an analysis conducted by ATLAS at 13 TeV has established an upper
limit on $m_{h_{2}}$ through various channels, including the decay modes of
$h_{2}$ into two tau-leptons, two W-bosons, or two Z-bosons [87, 89, 90]. To
investigate these limitations, the variation of the cross-sections
$\sigma(gg\rightarrow h_{2}\rightarrow\tau\tau)$, $\sigma(gg\rightarrow
h_{2}\rightarrow WW)$ and $\sigma(gg\rightarrow h_{2}\rightarrow ZZ)$ with
respect to $m_{h_{2}}$ are depicted in Fig. 2. It is observed from Fig. 2
(top-left) that as $m_{h_{2}}$ increases, the production cross-section in
channel $\sigma(gg\rightarrow h_{2}\rightarrow\tau\tau)$ decreases
exponentially. According to the restrictions outlined in Ref. [87], nearly
half of the solutions that satisfy criteria G4 are excluded, and the allowed
parameter range necessitates $m_{h_{2}}\gtrsim 200$ GeV, along with
$\sigma(gg\rightarrow h_{2}\rightarrow\tau\tau)\lesssim 2\times 10^{-1}$ pb.
Observing Fig. 2 (top-right), it’s easily seen that the $\sigma(gg\rightarrow
h_{2}\rightarrow WW)$ value decreases as $m_{h_{2}}$ increases, as expected.
Importantly, the solutions that satisfy G4 remain unaffected by the constraint
reported in Ref. [89]. For $m_{h_{2}}\gtrsim 450$ GeV, solutions with
$\sigma(gg\rightarrow h_{2})\simeq 1.0$ pb and $BR(h_{2}\rightarrow WW)\simeq
1.0$ can be obtained. A similar behavior can be seen in Fig. 2 (bottom) where
the dependence of $\sigma(gg\rightarrow h_{2}\rightarrow WW)$ on $m_{h_{2}}$
is plotted, the production cross-section decreases as $m_{h_{2}}$ increases.
Although some solutions satisfying G4 are excluded by the constraints outlined
in Ref. [90], it’s worth noting that for $m_{h_{2}}$ exceeding 450 GeV, the
cross-section for the decay channel $\sigma(gg\rightarrow h_{2}\rightarrow
ZZ)$ remains above 1 pb. After considering these findings from Fig.2, the mass
of the heavier CP-even scalar boson is restricted as $164\lesssim
m_{h_{2}}\lesssim 195$ GeV.

###  IV.2 Lepton Flavor Universality in Z-Boson Decay

The LS-2HDM makes a notable contribution to lepton universality through the
inclusion of extra scalar bosons in the leptonic Yukawa couplings. Electroweak
precision measurements conducted by LEP experiments, utilizing data from
Z-boson resonances, estimate the ratio of decay widths as [79, 80]

$\displaystyle{\Gamma_{Z\to\mu^{+}\mu^{-}}\over\Gamma_{Z\to e^{+}e^{-}}}$
$\displaystyle=$ $\displaystyle 1.0009\pm 0.0028\,,$ (16)
$\displaystyle{\Gamma_{Z\to\tau^{+}\tau^{-}}\over\Gamma_{Z\to e^{+}e^{-}}}$
$\displaystyle=$ $\displaystyle 1.0019\pm 0.0032\,.$

Following this result, the contributions to the leptonic decays of the Z boson
can be given as

$\displaystyle\delta_{\mu\mu}$ $\displaystyle=$
$\displaystyle{\Gamma_{Z\to\mu^{+}\mu^{-}}\over\Gamma_{Z\to
e^{+}e^{-}}}-1=(9\pm 28)\times 10^{-4},$ (17) $\displaystyle\delta_{\tau\tau}$
$\displaystyle=$ $\displaystyle{\Gamma_{Z\to\tau^{+}\tau^{-}}\over\Gamma_{Z\to
e^{+}e^{-}}}-1=(19\pm 32)\times 10^{-4},$

where $\delta_{\mu\mu}$ and $\delta_{\tau\tau}$ are the parameters defined to
measure the deviation from SM. The couplings of the Z-boson with fermions and
extra scalar bosons are given as [80]

$\displaystyle-{\cal L}$
$\displaystyle={g\over\cos\theta_{W}}Z^{\mu}\big{\\{}\bar{f}\gamma_{\mu}(((T_{3}(f_{L})-Q(f_{L})\sin\theta_{W}^{2})\delta
g_{L}^{\rm LS-2HDM})P_{L}+((T_{3}(f_{R})Q(f_{R})\sin\theta_{W}^{2})+\delta
g_{R}^{\rm LS-2HDM})P_{R})f$ (18) $\displaystyle+i(-{1\over
2}+\sin\theta_{W}^{2})H^{+}\overleftrightarrow{\partial_{\mu}}H^{-}+A\overleftrightarrow{\partial_{\mu}}H\big{\\}}.$

where, the contributions from LS-2HDM arise through $\delta
g_{R,L}^{LS-2HDM}$. In the LS-2HDM, the loop contributions to the Z boson
decay widths are expressed in terms of these parameters, and they are defined
as

$\displaystyle\delta g_{L}^{\rm 2HDM}$
$\displaystyle=k\left[-\dfrac{1}{2}B_{Z}\left(r_{A}\right)-\dfrac{1}{2}B_{Z}\left(r_{h_{2}}\right)-2C_{Z}\left(r_{A},r_{h_{2}}\right)+\sin^{2}\theta_{W}\left(B_{Z}\left(r_{A}\right)+B_{Z}\left(r_{h_{2}}\right)+\tilde{C}_{Z}(r_{A})+\tilde{C}_{Z}(r_{h_{2}})\right)\right],$
(19) $\displaystyle\delta g_{R}^{\rm 2HDM}$
$\displaystyle=k\left[2C_{Z}\left(r_{A},r_{h_{2}}\right)-2C_{Z}\left(r_{H^{\pm}},r_{H^{\pm}}\right)+\tilde{C}_{Z}(r_{H^{\pm}})-\dfrac{1}{2}\tilde{C}_{Z}(r_{A})-\dfrac{1}{2}\tilde{C}_{Z}(r_{h_{2}})\right.,$
$\displaystyle+\left.\sin^{2}\theta_{W}\left(B_{Z}\left(r_{A}\right)+B_{Z}\left(r_{h_{2}}\right)+2B_{Z}\left(r_{H^{\pm}}\right)+\tilde{C}_{Z}(r_{A})+C_{Z}\left(r_{H^{\pm}},r_{H^{\pm}}\right)\right)\right],$

where, $k=m_{f}^{2}\tan^{2}\beta/16\pi^{2}v^{2}$, $m_{f}$ are fermion masses,
$\Lambda$ is the cut-off scale, $\mu$ is the renormalization scale, and
$r_{\phi}=m_{\phi}^{2}/m_{Z}^{2}$ ($\phi=A,h_{2},H^{\pm}$). The loop functions
employed in Eqn. 19 are calculated as

$\displaystyle B_{Z}(r)$
$\displaystyle=-\dfrac{1}{2}\left(\ln\Lambda^{2}+\log(4\pi)\right)-\dfrac{1}{4}+\dfrac{1}{2}\log
r,$ (20) $\displaystyle\tilde{C}_{Z}(r)$
$\displaystyle=-\dfrac{1}{2}\left(\ln\Lambda^{2}+\log(4\pi)\right)-\dfrac{1}{2}-r\left(1+\log
r\right)+r^{2}\left(\log r\log(1+r^{-1})-{\rm dilog}(-r^{-1})\right)$
$\displaystyle-\dfrac{i\pi}{2}\left(1-2r+2r^{2}\log\left(1+r^{-1}\right)\right),$
$\displaystyle C_{Z}\left(r_{1},r_{2}\right)$
$\displaystyle=C_{Z}(\dfrac{m_{\phi_{1}}^{2}}{m_{Z}^{2}},\dfrac{m_{\phi_{2}}^{2}}{m_{Z}^{2}})=C_{00}(0,0,m_{Z}^{2},m_{\phi_{1}}^{2},m_{\phi_{2}}^{2}),$
$\displaystyle C_{00}$
$\displaystyle=\dfrac{1}{4}\left(\ln\Lambda^{2}+\log(4\pi)-\ln(\dfrac{m_{Z}^{2}}{\mu^{2}})\right)+\dfrac{3}{8}-\dfrac{r_{1}^{2}\ln
r_{1}}{4(r_{1}-1)(r_{1}-r_{2})}+\dfrac{r_{2}^{2}\ln
r_{2}}{4(r_{2}-1)(r_{1}-r_{2})}.$

The parameters defined in Eqn. 17 can be related to $\delta g_{R,L}^{LS-2HDM}$
appearing in Eqn. 18 as

$\displaystyle\delta_{ll}={2g_{L}^{e}{\rm Re}(\delta g^{\rm
LS-2HDM}_{L})+2g_{R}^{e}{\rm Re}(\delta g^{\rm
LS-2HDM}_{R})\over{g_{L}^{e}}^{2}+{g_{R}^{e}}^{2}},$ (21)

where $l=\mu,\tau$. Throughout this analysis, lepton universality in the
Z-boson decay are investigated through parameters given in Eqn. 21.

In order to understand the impact of LS-2HDM on lepton universality, such as
the loop-induced contributions to the Z boson decay widths, it’s essential to
explore the precise measurements of lepton universality ratios provided in
Eqn. 16 within the framework of LS-2HDM. To this end, the distribution of
solutions that satisfy the constraints outlined as G1, G2, G3 and G4 are
plotted on $\delta_{ll}$ and $\tan\beta$ plane in left for $\delta_{\mu\mu}$
and in right for $\delta_{\tau\tau}$, in Figure 3.

Figure 3: The variation of $\delta_{\mu\mu}$ with respect to $\tan\beta$
(left) and $\delta_{\tau\tau}$ with respect to $\tan\beta$ (right). Color
coding of the plots is the same as Fig. 1. The dashed, dotted and solid lines
indicate deviations from the current limits by Eqn. 17 within $1\sigma$,
$2\sigma$, and $3\sigma$, respectively. For $\delta_{\mu\mu}$ (left), all
solutions lie within 1$\sigma$ vicinity.

It is seen from Fig. 3 that the dependence of $\delta_{ll}$ on $\tan\beta$ is
quadratic as indicated in Eqn. 19, in both positive or negative directions
depending on the scalar masses of the LS-2HDM. Moreover, Fig. 3 (left)
demonstrates that the deviation of $\delta_{\mu\mu}$ from the SM prediction
lies within $1\sigma$ level and increases with the $\tan\beta$. Notably, to
satisfy the W-boson mass constraint given by G4, the value of
$\delta_{\mu\mu}$ must be negative. Furthermore, Fig. 3 (right) demonstrates
that $\delta_{\tau\tau}$ displays a greater sensitivity to $\tan\beta$, due to
the contribution of $m_{\tau}^{2}$. It is important to note that the
contribution can be either positive or negative, but if the parameter space is
restricted by G4, $\delta_{\tau\tau}$ is typically negative with a few
exceptions. The majority of solutions satisfying all constraints—G1, G2, G3,
and G4—are observed to fall within a $3\sigma$ deviation from the observed
value of $\delta_{\tau\tau}=1.0019$.

To understand the behavior of $\delta_{\tau\tau}$, three solutions represented
by the red points from the $\delta_{\tau\tau}$ \- $\tan\beta$ plane in Fig.
3(right), satisfying all the constraints, are selected. Corresponding scalar
masses, alongside $\Delta M_{W}=M_{W}^{\rm LS-2HDM}-M_{W}^{\rm SM}$
—representing the deviation of the W-boson mass in LS-2HDM from its SM value—
are detailed in Table 2. As can be seen from the Table 2 that $m_{h_{2}}$ and
$m_{H^{\pm}}$ increase with $\delta_{\tau\tau}$. Although there isn’t an
explicit linear correlation between $m_{A}$ and $\delta_{\tau\tau}$, an
increase in $\delta_{\tau\tau}$ can result in higher values of $m_{A}$. It can
also be seen from Eqn. 5 that, when $m_{3}^{2}$ and $\lambda_{3}$ are
negative, $m_{A}$ and $m_{H^{\pm}}$ increase with $\tan\beta$. In Table 2, the
value of $\Delta M_{W}$ at chosen solutions are also given, which does not
exhibit significant dependence on alterations within the parameter space.

$\delta_{\tau\tau}$ | $-0.9929\times 10^{-4}\;(1\sigma)$ | $-26.1088\times 10^{-4}\;(2\sigma)$ | $-61.6004\times 10^{-4}\;(3\sigma)$
---|---|---|---
$\tan\beta$ | $3.3094$ | $18.903$ | $23.623$
$m_{h_{1}}$ | $125.164$ | $123.095$ | $124.39$
$m_{h_{2}}$ | $169.964$ | $253.095$ | $303.058$
$m_{A}$ | $388.266$ | $382.756$ | $479.669$
$m_{H^{\pm}}$ | $416.205$ | $421.037$ | $511.855$
$\Delta M_{W}$ | $0.0533$ | $0.0539$ | $0.0536$
Table 2: Properties of solutions which are selected from Fig. 3 (right). All
masses and $\Delta M_{W}^{\prime}$ are given in GeV unit.
$1\sigma,2\sigma,3\sigma$ are the deviations from value of $\delta_{\tau\tau}$
(1.0019).

###  IV.3 Lepton Flavor Universality in Tau Decay

Following the analysis in previous section, the behaviour of LFU in tau decays
within LS-2HDM is also investigated. At the loop-level, the LFU in tau decay
gets contributions from the LS-2HDM scalars, and these contributions bring
additional constraints on the parameter space of the LS-2HDM. The ratio of
these couplings are defined as

$\left(g_{\tau}\over g_{\mu}\right)=1+\delta_{\rm
loop},\qquad\left(g_{\tau}\over g_{e}\right)=1+\delta_{\rm tree}+\delta_{\rm
loop},\qquad\left(g_{\mu}\over g_{e}\right)=1+\delta_{\rm tree},$ (22)

where $\delta_{\rm loop}$ and $\delta_{\rm tree}$ are the contributions to SM
value from loop and tree level processes. In LS-2HDM, these contributions are
calculated as [85, 80, 81]

$\displaystyle\delta_{\rm tree}$ $\displaystyle={m_{\tau}^{2}m_{\mu}^{2}\over
8m^{4}_{H^{\pm}}}t^{4}_{\beta}-{m_{\mu}^{2}\over
m^{2}_{H^{\pm}}}t^{2}_{\beta}{g(m_{\mu}^{2}/m^{2}_{\tau})\over
f(m_{\mu}^{2}/m_{\tau}^{2})},$ (23) $\displaystyle\delta_{\rm loop}$
$\displaystyle={1\over 16\pi^{2}}{m_{\tau}^{2}\over
v^{2}}t^{2}_{\beta}\left[1+{1\over
4}\left(H(m_{A}^{2}/m_{H^{\pm}}^{2})+s^{2}_{\beta-\alpha}H(m_{h_{2}}^{2}/m_{H^{\pm}}^{2})+c^{2}_{\beta-\alpha}H(m_{h_{1}}^{2}/m_{H^{\pm}}^{2}\right)\right],$

where

$\displaystyle f(x)$ $\displaystyle\equiv 1-8x+8x^{3}-x^{4}-12x^{2}\ln(x),$
$\displaystyle g(x)$ $\displaystyle\equiv 1+9x-9x^{2}-x^{3}+6x(1+x)\ln(x),$
(24) $\displaystyle H(x)$ $\displaystyle\equiv\ln(x)(1+x)/(1-x),$

and $t_{\beta}=\tan\beta$, $s_{\beta-\alpha}=\sin(\beta-\alpha)$,
$c_{\beta-\alpha}=\cos(\beta-\alpha)$ are defined for brevity. Using pure
leptonic processes, HFLAV collaboration obtained the values of the ratios
given in Eqn. 23 as [86]

$\left(g_{\tau}\over g_{\mu}\right)=1.0009\pm 0.0014,\qquad\left(g_{\tau}\over
g_{e}\right)=1.0027\pm 0.0014,\qquad\left(g_{\mu}\over g_{e}\right)=1.0019\pm
0.0014.$ (25)

These averages define the ranges for loop and tree-level contributions as

$\displaystyle\delta_{loop}=0.0009\pm 0.0014~{}\ \ \text{and}\ \ \ \
\delta_{tree}=0.0019\pm 0.0014~{},$ (26)

and these limits will be considered for further analysis.

Figure 4: $\delta_{loop}$ vs $\tan\beta$ (left), and $\delta_{tree}$ vs
$\tan\beta$ (right). Color coding is the same as Fig. 1. All solutions lay
within the $1\sigma$ vicinity of HFLAV results.

To examine the relationship between solutions satisfying all four constraints,
they are plotted in the $\delta_{loop}$ and $\delta_{tree}$ versus $\tan\beta$
planes in Fig. 4 (left and right), respectively. It is observed from Fig. 4
that applying constraints reduces the number of solutions, and restricts them
in negative plane. Furthermore, all solutions shown in red in Fig. 4 (left)
lie within $2\sigma$ vicinity of the HFVAL average value of $\delta_{loop}$
given in Eqn. 26. Remarkably, for solutions with $\tan\beta<20$, this
proximity is confined within the 1$\sigma$ range. On the other hand,
concerning $\delta_{tree}$, solutions meeting all constraints are clustered
just below zero, confined within a $1\sigma$ to $2\sigma$ window of the
average value, as observed in Fig. 4 (right). Finally, it is also apparent
from Fig. 4 that with increasing $\tan\beta$, loop contributions to the ratios
from LS-2HDM become more significant in comparison to those at the tree level.

###  IV.4 W-Boson Mass in LS-2HDM

As previously noted, the CDF collaboration has recently reported a new
measurement of the W-boson mass, deviating by 7$\sigma$ from the prediction of
the SM, as given in Eqn.1. This section aims to investigate whether the
LS-2HDM framework can provide solutions compatible with both experimental and
theoretical limitations outlined as G1 and G2, while allowing for a deviation
of $3\sigma$ from the reported W-boson mass by CDF. Specifically, the behavior
of the W-boson mass within the framework of LS-2HDM will be investigated.

In the context of the LS-2HDM, the W-boson mass can be calculated as

[111]

$\displaystyle M_{W}^{2}={M_{W}^{\rm
SM}}^{2}\left(1+\frac{s_{W}^{2}}{c_{W}^{2}-s_{W}^{2}}\Delta r^{\prime}\right)\
,$ (27)

where

$\displaystyle\Delta
r^{\prime}=\frac{\alpha}{s_{W}^{2}}\left(-\frac{1}{2}S+c_{W}^{2}T+\frac{c_{W}^{2}-s_{W}^{2}}{4s_{W}^{2}}U\right)\
,$ (28)

represents the measure of deviations from SM, and where $s_{W}=\sin\theta_{W}$
and $c_{W}=\cos\theta_{W}$ with $\theta_{W}$ is being the weak angle. In Eqn.
28, $S$, $T$, and $U$ represent the oblique parameters, containing the effects
of incorporation of additional scalar bosons contributions through loops
within the LS-2HDM framework to $M_{W}$. These parameters are defined based on
precision measurements in electroweak physics, and in SM, $S=T=U=0$ serves as
the reference point. The explicit form of the $S$, $T$, $U$ parameters are
given as [112]

$\displaystyle S=$ $\displaystyle\frac{1}{\pi
m_{Z}^{2}}\left\\{-\mathcal{B}_{22}\left(m_{Z}^{2};m_{H^{\pm}}^{2},m_{H^{\pm}}^{2}\right)+\sin^{2}(\beta-\alpha)\mathcal{B}_{22}\left(m_{Z}^{2};m_{h_{2}}^{2},m_{A}^{2}\right)\right.$
(29)
$\displaystyle+\cos^{2}(\beta-\alpha)\left[\mathcal{B}_{22}\left(m_{Z}^{2};m_{h_{1}}^{2},m_{A}^{2}\right)+\mathcal{B}_{22}\left(m_{Z}^{2};m_{Z}^{2},m_{h_{2}}^{2}\right)-\mathcal{B}_{22}\left(m_{Z}^{2};m_{Z}^{2},m_{h_{1}}^{2}\right)\right.$
$\displaystyle\left.\left.-m_{Z}^{2}\mathcal{B}_{0}\left(m_{Z}^{2};m_{Z}^{2},m_{h_{2}}^{2}\right)+m_{Z}^{2}\mathcal{B}_{0}\left(m_{Z}^{2};m_{Z}^{2},m_{h_{1}}^{2}\right)\right]\right\\},$

$\displaystyle T=$ $\displaystyle\frac{1}{16\pi
s_{W}^{2}m_{W}^{2}}\left\\{\mathcal{F}\left(m_{H^{\pm}}^{2},m_{A}^{2}\right)+\sin^{2}(\beta-\alpha)\left[\mathcal{F}\left(m_{H^{\pm}}^{2},m_{h_{2}}^{2}\right)-\mathcal{F}\left(m_{A}^{2},m_{h_{2}}^{2}\right)\right]\right.$
(30)
$\displaystyle+\cos^{2}(\beta-\alpha)\left[\mathcal{F}\left(m_{H^{\pm}}^{2},m_{h_{1}}^{2}\right)-\mathcal{F}\left(m_{A}^{2},m_{h_{1}}^{2}\right)+\mathcal{F}\left(m_{W}^{2},m_{h_{2}}^{2}\right)-\mathcal{F}\left(m_{W}^{2},m_{h_{1}}^{2}\right)\right.$
$\displaystyle-\mathcal{F}\left(m_{Z}^{2},m_{h_{2}}^{2}\right)+\mathcal{F}\left(m_{Z}^{2},m_{h_{1}}^{2}\right)+4m_{Z}^{2}B_{0}\left(0;m_{Z}^{2},m_{h_{2}}^{2}\right)-4m_{Z}^{2}B_{0}\left(0;m_{Z}^{2},m_{h_{1}}^{2}\right)$
$\displaystyle\left.\left.-4m_{W}^{2}B_{0}\left(0;m_{W}^{2},m_{h_{2}}^{2}\right)+4m_{W}^{2}B_{0}\left(0;m_{W}^{2},m_{h_{1}}^{2}\right)\right]\right\\},$

$\displaystyle S+U=$ $\displaystyle\frac{1}{\pi
m_{W}^{2}}\left\\{\mathcal{B}_{22}\left(m_{W}^{2};m_{H^{\pm}}^{2},m_{A}^{2}\right)-2\mathcal{B}_{22}\left(m_{W}^{2};m_{H^{\pm}}^{2},m_{H^{\pm}}^{2}\right)+\sin^{2}(\beta-\alpha)\mathcal{B}_{22}\left(m_{W}^{2};m_{H^{\pm}}^{2},m_{h_{2}}^{2}\right)\right.$
(31)
$\displaystyle+\cos^{2}(\beta-\alpha)\left[\mathcal{B}_{22}\left(m_{W}^{2};m_{h_{1}}^{2},m_{H^{\pm}}^{2}\right)+\mathcal{B}_{22}\left(m_{W}^{2};m_{W}^{2},m_{h_{2}}^{2}\right)-\mathcal{B}_{22}\left(m_{W}^{2};m_{W}^{2},m_{h_{1}}^{2}\right)\right.$
$\displaystyle\left.\left.+m_{W}^{2}\mathcal{B}_{0}\left(m_{W}^{2};m_{W}^{2},m_{h_{2}}^{2}\right)-m_{W}^{2}\mathcal{B}_{0}\left(m_{W}^{2};m_{W}^{2},m_{h_{1}}^{2}\right)\right]\right\\},$

where

$\displaystyle\mathcal{B}_{22}\left(q^{2};m_{1}^{2},m_{2}^{2}\right)\equiv\mathcal{B}_{22}\left(q^{2};m_{1}^{2},m_{2}^{2}\right)-\mathcal{B}_{22}\left(0;m_{1}^{2},m_{2}^{2}\right),$
(32)
$\displaystyle\mathcal{B}_{0}\left(q^{2};m_{1}^{2},m_{2}^{2}\right)\equiv\mathcal{B}_{0}\left(q^{2};m_{1}^{2},m_{2}^{2}\right)-\mathcal{B}_{0}\left(0;m_{1}^{2},m_{2}^{2}\right),$

are the loop functions arising from two-point loop integrals given as

$\displaystyle
B_{22}\left(q^{2};m_{1}^{2},m_{2}^{2}\right)=\frac{1}{4}(\Delta+1)\left[m_{1}^{2}+m_{2}^{2}-\frac{1}{3}q^{2}\right]-\frac{1}{2}\int_{0}^{1}dxX\ln(X-i\epsilon),$
(33) $\displaystyle
B_{0}\left(q^{2};m_{1}^{2},m_{2}^{2}\right)=\Delta-\int_{0}^{1}dx\ln(X-i\epsilon),$

where

$X\equiv
m_{1}^{2}x+m_{2}^{2}(1-x)-q^{2}x(1-x),\quad\Delta\equiv\frac{2}{4-d}+\ln
4\pi-\gamma.$ (34)

Shorthand notation $\mathcal{F}$ in Eqn. 30 is defined as

$\displaystyle\mathcal{F}\left(m_{1}^{2},m_{2}^{2}\right)\equiv\dfrac{1}{2}\left(m_{1}^{2}+m_{2}^{2}\right)-\dfrac{m_{1}^{2}m_{2}^{2}}{m_{1}^{2}-m_{2}^{2}}\ln\left(\dfrac{m_{1}^{2}}{m_{2}^{2}}\right),$
(35)

satisfying the following relations

$\mathcal{F}\left(m_{1}^{2},m_{2}^{2}\right)=\mathcal{F}\left(m_{2}^{2},m_{1}^{2}\right)\;{\rm
and}\;\mathcal{F}\left(m^{2},m^{2}\right)=0.$ (36)

To this end, one-loop contributions to both the oblique parameters and the
W-boson mass in LS-2HDM are computed. To constrain the solutions, $M_{W}^{\rm
LS-2HDM}$ is restricted to fall within a $3\sigma$ range of $M_{W}^{\rm CDF}$,
namely $M_{W}^{\rm LS-2HDM}=80.4335\pm 3\times 0.0094$ GeV, resulting in the
value of $\Delta M_{W}$ as

$\displaystyle\Delta M_{W}$ $\displaystyle=80.4335{\rm GeV}-80.357{\rm
GeV}=0.0765\;{\rm GeV},$ (37)

which provides the intervals given in Table 3.

| $\Delta M_{\rm W}^{\rm min}$ | $\Delta M_{\rm W}^{\rm max}$
---|---|---
$1\sigma$ | 0.0671 | 0.0859
$2\sigma$ | 0.0577 | 0.0953
$3\sigma$ | 0.0483 | 0.1047
Table 3: Limits on $\Delta M_{W}$. All masses are in GeV.

Figure 5: Solutions satisfying conditions G1, G2, G3 and G4 plotted in the
$\Delta M_{W}$ versus $m_{h_{1}}$ (top-left), $m_{h_{2}}$ (top-right), $m_{A}$
(bottom-left), and $m_{H^{\pm}}$ (bottom-right). Color coding of the points is
the same as Fig. 1. The solid and dashed red lines represent the deviations
from SM and CDF-reported value, respectively. The dashed, dotted and solid
lines indicate deviations from $M_{W}^{CDF}$ given in Table 3 within
$1\sigma$, $2\sigma$, and $3\sigma$, respectively. In $\Delta M_{W}$ vs
$m_{h_{1}}$ plot (top-left), constraint G3 is not applied.

Following these considerations, a thorough analysis of the parameters within
the LS-2HDM that meet both the theoretical and experimental constraints
outlined as G1, G2, and G3 is performed. The relations between the masses of
LS-2HDM scalars and $\Delta M_{W}$ are depicted in Fig. 5. In the top-left
panel of Fig. 5, the plot illustrates the relation between $\Delta M_{W}$ and
$m_{h_{1}}$, where $m_{h_{1}}$ is not restricted to be SM for these solutions.
For the new scalars in LS-2HDM, the solutions are plotted on the $\Delta
M_{W}$ vs $m_{h_{2}}$, $m_{A}$, and $m_{H^{\pm}}$ planes in Fig. 5. From these
plots, it can be concluded that assuming $h_{1}$ as SM Higgs boson predicts
$\Delta M_{W}$ within $1\sigma$ to $3\sigma$ uncertainity, hence it is viable,
and the masses of new scalars are restricted to be $140\lesssim
m_{h_{2}}\lesssim 430$ GeV, $230\lesssim m_{H^{\pm}}\lesssim 685$ GeV, and
$105\lesssim m_{A}\lesssim 665$ GeV.

Fig. 6 display the solutions for $\Delta M_{W}$ in correlation with
$\tan\beta$, $m_{h_{2}}-m_{h_{1}}$, $m_{A}-m_{h_{2}}$, and $m_{H^{\pm}}-m_{A}$
planes. Observing Fig. 6 (top-left), it is seen that the viable solutions do
not limit $\tan\beta$, and the contributions from LS-THDM to the W-boson mass
generally exhibit positivity. However, the imposition of constraints grouped
under G1 necessitates that $\tan\beta$ must exceed 2. Furthermore, it is
noticeable that the W-boson mass aligns with predictions within the framework
of the model. However, imposing condition G3, which assumes $h_{1}$ as the SM
Higgs boson, restricts the deviation of $\Delta M_{W}$ to within the $1\sigma$
to $3\sigma$ range of the CDF-reported value but fails to predict results
closer to it. This discrepancy might suggest a potential tension between
$h_{1}$ being SM-like and $M_{W}^{\rm CDF}$ in the LS-2HDM framework.
Furthermore, it is observed from Fig. 6 (top-right) that the W-boson mass can
be obtained in LS-2HDM by imposing the constriant for the mass difference
between the CP-even scalar bosons as $20\lesssim m_{h_{2}}-m_{h_{1}}\lesssim
300$ GeV. Similarly, the mass difference between pseudoscalar and heavy scalar
bosons can take values in between the range $-125\lesssim
M_{A}-m_{h_{2}}\lesssim 425$ GeV. Finally, in Fig. 6 (bottom-right), it is
observed that the restricted range for the mass difference between charged
scalars and pseudoscalar states is $15\lesssim m_{H^{\pm}}-m_{A}\lesssim 127$
GeV. Notably, the relationship between $\Delta M_{W}$ and $m_{H^{\pm}}-m_{A}$
differs due to the dependence of $m_{H^{\pm}}-m_{A}$ on the oblique parameters
$S$, $T$ and $U$.

Figure 6: Solutions satisfying conditions G1, G2, G3 and G4 plotted in the
$\Delta M_{W}$ versus $\tan\beta$ (top-left), $m_{h_{2}}-m_{h_{1}}$ (top-
right), $m_{A}-m_{h_{2}}$ (bottom-left), and $m_{H^{\pm}}-m_{A}$ (bottom-
right). Color coding of the points is the same as Fig. 1. The solid and dashed
red lines represent the deviations from SM and CDF-reported value,
respectively. The dashed, dotted and solid lines indicate deviations from
$M_{W}^{CDF}$ given in Table 3 within $1\sigma$, $2\sigma$, and $3\sigma$,
respectively.

Figure 7: Solutions satisfying all four limitations plotted in $T$ vs
$m_{H^{\pm}}-m_{A}$ (top-left), $S$ vs $m_{H^{\pm}}-m_{A}$ (top-right), and
$U$ vs $m_{H^{\pm}}-m_{A}$ (bottom). Color coding of the points is the same as
Fig. 1. Solid purple lines indicates the values given in Eq. 13. Vertical and
horizontal solid red lines represent zero values.

Fig. 7 illustrates the impacts of the oblique parameters in correlation with
the mass difference in $T$ vs $m_{H^{\pm}}-m_{A}$ (top-left), $S$ vs
$m_{H^{\pm}}-m_{A}$ (top-right), and $U$ vs $m_{H^{\pm}}-m_{A}$ (bottom).
Solutions that satisfy all four sets of constraints estimate the $T$, $S$, and
$U$ oblique parameters within the $1\sigma$ vicinity, as described by Eqn. 13.
Notably, Fig. 7 (right) demonstrates a similar trend to the plots depicting
the behavior of the $S$, $T$, and $U$ parameters. However, the influence of
parameter $T$ on $\Delta M_{W}$ appears to be predominant. Finally, these
solutions specify the following ranges

$\displaystyle 0.1\;$ $\displaystyle\lesssim\;T\;\lesssim 0.15\ ,$
$\displaystyle-0.02\;$ $\displaystyle\lesssim\;S\;\lesssim 0.045\ ,$
$\displaystyle-1.10\times 10^{-3}\;$ $\displaystyle\lesssim\;U\;\lesssim
6.05\times 10^{-3}\ ,$ (38)

within which $M_{W}^{CDF}$ can be obtained within the $1\sigma$ to $3\sigma$
vicinity.

###  IV.5 Further Analysis

In this section, all results discussed in previous sections will be
summarized. In addition, benchmark points and their properties will be given.
Finally, all obtained solutions will be tested in HiggsTools package [92].

Parameter | Minimum Value | Maximum Value
---|---|---
$m_{h_{2}}-m_{h_{1}}$ | $20$ | $300$
$m_{H^{\pm}}-m_{A}$ | $15$ | $127$
$m_{A}-m_{h_{2}}$ | $-125$ | $425$
$m_{h_{2}}$ | $164$ | $195$
$m_{A}$ | $330$ | $575$
$m_{H^{\pm}}$ | $345$ | $685$
$\tan\beta$ | $2.5$ | $8.0$
$T$ | $0.1$ | $0.15$
$S$ | $-0.02$ | $0.045$
$U$ | $-1.1\times 10^{-3}$ | $6.05\times 10^{-3}$
Table 4:  The constraints on the model parameters which satisfy all four
aforementioned group of limitations. All masses are given in GeV unit.

The properties of solutions satisfying all for constraints listed before as
G1, G2, G3 and G4, are given in Table 4. Within these ranges,
$0.0483\lesssim\Delta M_{W}\lesssim 0.0671$ GeV, i.e. LS-2HDM can estimate the
mass of the W boson as

$\displaystyle 80.4053\lesssim M_{W}^{LS-2HDM}\lesssim 80.4147\ \text{GeV}.$
(39)

Among these solutions, three points closely estimating $M_{W}^{LS-2HDM}$
around the Standard Model value, and an additional three points estimating
$M_{W}^{LS-2HDM}$ around the CDF-reported value, have been selected as
benchmarks. Their respective properties are detailed in Table 5.

Parameter | 1\. Point | 2\. Point | 3\. Point | 4\. Point | 5\. Point | 6\. Point
---|---|---|---|---|---|---
$M_{W}^{\rm LS-2HDM}$ | $80.3569$ | $80.3589$ | $80.3514$ | $80.4240$ | $80.4236$ | $80.4234$
Deviation from $M_{W}^{\rm CDF}$ | $8.06\sigma$ | $7.85\sigma$ | $8.64\sigma$ | $1.01\sigma$ | $1.04\sigma$ | $1.05\sigma$
$m_{h_{1}}$ | $126.708$ | $125.883$ | $123.418$ | $124.079$ | $124.096$ | $124.902$
$m_{h_{2}}$ | $176.646$ | $497.022$ | $595.514$ | $281.157$ | $372.323$ | $250.747$
$m_{A}$ | $426.245$ | $292.904$ | $136.475$ | $208.244$ | $190496$ | $530.388$
$m_{H^{\pm}}$ | $426.245$ | $343.396$ | $234.652$ | $293.41$ | $290.731$ | $563.959$
$\tan\beta$ | $22.284$ | $36.243$ | $33.786$ | $9.07$ | $8.23$ | $5.53$
Table 5: Properties of solutions which are selected as benchmark. All masses
are given in GeV unit.

Finally, to assess the feasibility of the regions outlined in Table 4, all
solutions passing these constraints are tested in the HiggsTools package, with
only one solution, whose properties are listed in Table 6, remaining viable.
This unique solution estimates $M_{W}^{LS-2HDM}=80.4103$ GeV, within
$2.44\sigma$ vicinity of CDF-reported value. It is important to mention that
these solution is unique among the ones that are survived from the solutions
provided and filtered within this work, and additional points might appear
with more data.

${\lambda_{1}}_{\rm input}$ | ${\lambda_{2}}_{\rm input}$ | ${\lambda_{3}}_{\rm input}$ | ${\lambda_{4}}_{\rm input}$
---|---|---|---
$1.6203$ | $1.3252\times 10^{-1}$ | $5.5846$ | $-3.0072$
${\lambda_{5}}_{\rm input}$ | $m^{2}_{3\rm input}$ | $\tan\beta$ | $\cos(\beta-\alpha)$
$-2.2657$ | $-3.7084\times 10^{3}$ | $3.3094$ | $0.398$
$m_{h_{1}}$ | $m_{h_{2}}$ | $m_{A}$ | $m_{H^{\pm}}$
$125.16$ | $169.96$ | $388.27$ | $416.20$
$m_{h_{2}}-m_{h_{1}}$ | $m_{H^{\pm}}-m_{A}$ | $m_{A}-m_{h_{2}}$ | $m_{H^{\pm}}-m_{h_{1}}$
$44.8$ | $27.93$ | $219.31$ | $291.04$
$B_{s}\rightarrow\mu^{+}\mu^{-}$ | $B_{s}\rightarrow X_{s}\gamma$ | $\delta_{\mu\mu}$ | $\delta_{\tau\tau}$
$3.145\times 10^{-9}$ | $3.084\times 10^{-4}$ | $-3.512\times 10^{-7}$ | $-9.929\times 10^{-5}$
$g_{\tau}/g_{\mu}$ | $g_{\tau}/g_{e}$ | $g_{\mu}/g_{e}$ | S
$0.999993$ | $0.999992$ | $0.999999$ | $-1.4809\times 10^{-2}$
T | U | $M_{W}^{\rm LS-2HDM}$ | $\Delta a_{\mu}$
$1.1913\times 10^{-1}$ | $1.096\times 10^{-3}$ | $80.4103\;(2.44\sigma)$ | $1.204\times 10^{-13}$
Table 6: Properties of the solution which is not excluded by HiggsTools. All
mass values are given in GeV units.

##  V Conclusions

In this work, the LS-2HDM parameter space is investigated with the aim of
obtaining a parameter space where the W-boson mass measured by CDF experiment
is confronted within the framework of LS-2HDM. To this end, both theoretical
and experimental constraints are applied to the parameter space. Consequently,
only a limited range of parameter values remains viable within the LS-2HDM
framework, most of which are found to be compatible with the reported
measurements of LFU tau-lepton and Z-boson decay. Moreover, the charged Higgs
boson mass, oblique parameters, and constraints from rare B-meson decays were
found to have a weak effect on the parameter space. Furthermore, assuming
$h_{1}$ to be the SM Higgs boson reduced the number of solutions considerably.
Additionally, the primary restriction imposed was that the LS-2HDM estimates
had to align with the CDF-reported value of the W boson mass within a
$3\sigma$ range, among other imposed restrictions. Within the viable
solutions, it is evident that the LS-2HDM can estimate the CDF-reported value
of the W boson mass within a range of $1\sigma$ to $3\sigma$. As a result of
this analysis, the available parameter space of LS-2HDM is summarized in Table
4. Within these ranges, LS-2HDM can be effectively utilized for making
predictions.

It is important to note that imposing the $\Delta M_{W}$ condition directly on
the masses may not always be convenient for determining restrictions on
parameters. Using additional parameters related to masses, such as the mass
differences employed in this analysis, results in stronger limits. Another
important point to note is that within the framework of the LS-2HDM analysis
conducted in this study, $M_{W}^{CDF}$ could not be achieved closer than
$1\sigma$. This limitation arises from assuming $m_{h_{1}}$ as the SM Higgs
boson. This discrepancy between $h_{1}$ being SM-like and $M_{W}^{\rm CDF}$
might be a sign of an inconsistency that requires further investigations.

To ensure a comprehensive analysis, six solutions were selected, with three
estimating $M_{W}^{SM}$ and the remaining three estimating $M_{W}^{CDF}$
within a $1\sigma$ proximity. These chosen solutions were used as benchmarks,
and their predictions were provided. It is observed that $\tan\beta$ plays a
crucial role in estimating W boson mass within LS-2HDM. For higher values of
$\tan\beta$, LS-2HDM successfully predicts $M_{W}^{SM}$, while for lower
values of $\tan\beta$, it predicts $M_{W}^{CDF}$. To comprehensively conclude
this analysis, all solutions satisfying the conditions outlined in Table 4
were tested using the HiggsTools package, which incorporates the most recent
constraints. Among all solutions, only one, whose properties are presented in
Table 6, has survived. It estimates $M_{W}^{LS-2HDM}=80.4103$ GeV, with a
deviation of $\Delta M_{W}=0.0232$. This result indicates that LS-2HDM can
predict the CDF-reported value of the W boson mass within approximately $\sim
2.44\sigma$ proximity, a much closer estimation than that of SM. The most
important feature of this solution is that it has passed the theoretical,
experimental and W-boson-related limitations outlined as G1, G2, G3 and G4,
and finally passed through HiggsTools which imposes the strictest bounds. It
is noteworthy that this solution provides an interesting possibility to test
the LS-2HDM framework and study its potential consequences. However, given the
narrowness of the allowed region and the complexity of the model, it is still
challenging to draw strong conclusions. Further experimental data, as well as
advanced theoretical calculations, will be required to fully explore the
implications of this solution and the LS-2HDM model as a whole.

###### Acknowledgements.

The authors would like to extend their sincere gratitude to Cem Salih Ün for
the invaluable support and expertise provided throughout this study. Part of
the computational work reported in this paper were conducted at the High
Performance and Grid Computing Center (TRUBA Resources) of the National
Academic Network and Information Center (ULAKBIM) of TUBITAK. Additionally,
the authors acknowledge partial support from the Turkish Energy, Nuclear and
Mineral Research Agency under project number 2022TENMAK(CERN)A5.H3.F2-02.

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