# Interplay of the charged Higgs effects in $R_{D^{(\ast)}}$, $b\to
s\ell^{+}\ell^{-}$ and $W$-mass

Girish Kumar  Department of Physics, National Taiwan University, Taipei 10617,
Taiwan

###### Abstract

Current data on semileptonic charged- and neutral-current $B$ decays show
deviations from the predictions of the Standard Model. It is well known that a
charged Higgs boson, belonging to the two-Higgs doublet model without $Z_{2}$
symmetry, offers one of the simplest solution to the charged-current $B$
decays. We show that this solution naturally induces a negative shift of
$\mathcal{O}(1)$ in the Wilson coefficient ($C_{9\ell}$) of operator
$(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{\ell}\gamma^{\mu}\ell)$, potentially
resolving the tension in neutral-current $B$ decays as well. Interestingly,
the lepton universality ratios in $b\to s\ell^{+}\ell^{-}$ decays, in tune
with the recent LHCb result, remain SM-like. Precision constraints from
neutral $B$ and $K$ meson mixing, decays $B_{c}\to\tau\bar{\nu}$, $B\to
X_{s}\gamma$, and leptonic decays of $\tau$ and $Z$ can be satisfied.
Furthermore, a positive shift in $W$-boson mass, nicely in agreement with the
CDF measurement, is also possible, requiring the neutral scalars to be heavier
than the charged Higgs but within the sub-TeV region.

##  I Introduction

It is remarkable that though there already exists irrefutable experimental
evidence (e.g., baryon asymmetry of the Universe, neutrino masses) and
persuasive theoretical reasons (e.g., naturalness problem, flavor problem) for
physics beyond the Standard Model (SM), no new physics (NP) particle has
turned up so far at the LHC. One reason could be that the NP scale is very
heavy and beyond LHC reach. However, in recent years a number of measurements,
especially those associated with semileptonic decays of $B$ mesons, have been
significantly at odds with the SM predictions and could be telltale sign of
sub-TeV scale NP accessible at the LHC. We discuss one example of such NP—a
charged Higgs boson $(H^{+})$ of a few hundred GeV mass, which can help in
alleviating the tension between theory and the current data.

In semileptonic $B$ decays, one set of prominent anomalies, persisting for
many years and strengthened further by the recent LHCb measurement LHC , are
in the lepton flavor universality (LFU) ratios

$\displaystyle R_{D^{(\ast)}}=\frac{\mathrm{BR}(B\to
D^{(\ast)}\tau\bar{\nu})}{\mathrm{BR}(B\to
D^{(\ast)}\ell\bar{\nu})};~{}~{}\ell=\\{e,\,\mu\\}.$ (1)

The current world average by HFLAV Amhis  _et al._ (2021), based on
measurements by $B$ factories Lees  _et al._ (2013, 2012); Hirose  _et al._
(2017); Huschle  _et al._ (2015); Caria  _et al._ (2020) and LHCb Aaij  _et
al._ (2015a, 2018a, 2018b); LHC , gives ${R_{D}}=0.358\pm 0.028$ and
${R_{D^{\ast}}}=0.285\pm 0.013$. Individually, these values disagree with the
SM expectation Bigi and Gambino (2016); Gambino  _et al._ (2019); Bordone  _et
al._ (2020); Bernlochner  _et al._ (2017); Jaiswal  _et al._ (2017);
Martinelli  _et al._ (2022); Bigi  _et al._ (2017); Bazavov  _et al._ (2021)
at the significance level of $2.2\sigma$ and $2.3\sigma$, respectively. Taken
together (the correlation coefficient is -$0.29$), the disagreement increases
to $3.2\sigma$.111The trend of surplus in tauonic modes is also observed in
the measurement of $R_{J/\psi}$ Aaij  _et al._ (2018c), the LFU ratio defined
similar to eq. 1 for $B_{c}\to J/\psi$ transition, while the ratio
$R_{\Lambda_{c}}$ (related to $\Lambda_{b}\to\Lambda_{c}$ transition) shows a
relative deficit Aaij  _et al._ (2022a). These deviations, however, are
relatively mild in significance as the current associated experimental
uncertainties are large. We therefore do not include these two ratios in our
analysis. We refer to a recent paper Fedele  _et al._ (2022) analyzing the
impact of $R_{\Lambda_{c}}$ inclusion in NP analysis. On the other hand,
measurements of analogous muon vs electron LFU ratios are in agreement with
the SM predictions Glattauer  _et al._ (2016); Abdesselam  _et al._ (2017);
Waheed  _et al._ (2019).

Another set of anomalies are observed in $b\to s\mu^{+}\mu^{-}$ decays. One of
these is the $\sim 3\sigma$ anomaly in the measurement of angular observable
$P_{5}^{\prime}$ in the decay $B\to K^{\ast}\mu^{+}\mu^{-}$ Aaij  _et al._
(2013, 2016, 2020). Another sizable tension, of $3.6\sigma$ significance, is
reported by LHCb Aaij  _et al._ (2015b, 2021a, 2021b) in the measurement of
branching fraction of the $B_{s}\to\phi\mu^{+}\mu^{-}$ decay, finding it below
the SM expectation. The data on $B\to K\mu^{+}\mu^{-}$ Aaij  _et al._ (2014)
and $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ Aaij  _et al._ (2015c) also show a
deficit in branching fractions with respect to the SM predictions. The recent
update from LHCb LHC (2022a, b) however finds no evidence of LFU breaking in
$b\to s\ell^{+}\ell^{-}$ decays: the ratios $R_{K^{(\ast)}}={\mathrm{BR}(B\to
K^{(\ast)}\mu^{+}\mu^{-})}/{\mathrm{BR}(B\to K^{(\ast)}e^{+}e^{-})}$ Hiller
and Kruger (2004) measured in the dilepton invariant mass bins
$0.1<q^{2}<1.1~{}\mathrm{GeV^{2}}$ and $1.1<q^{2}<6~{}\mathrm{GeV^{2}}$ are in
complete agreement222It is worth noting that rates of both the $B^{+}\to
K^{+}\mu^{+}\mu^{-}$ Aaij  _et al._ (2014) and $B^{+}\to K^{+}e^{+}e^{-}$
decays in the central $q^{2}$ bin are now low compared to the corresponding SM
predictions while their ratio ($R_{K}$) remains SM-like LHC (2022b). with the
SM predictions known with percent-level accuracy Bordone  _et al._ (2016);
Isidori  _et al._ (2020, 2022). These new findings overturn previous results
Aaij  _et al._ (2017, 2019, 2022b) that reported a deficit in
$R_{K^{(\ast)}}$. Although the predictions of individual branching fractions
and optimized angular observables such as $P_{5}^{\prime}$ are subject to
significant hadronic uncertainties Matias  _et al._ (2012); Das and Sinha
(2012); Descotes-Genon  _et al._ (2013); Horgan  _et al._ (2014); Beaujean
_et al._ (2014); Mandal  _et al._ (2014); Lyon and Zwicky (2014); Jäger and
Martin Camalich (2016); Ciuchini  _et al._ (2016, 2021a); Gubernari  _et al._
(2021), at present it is contestable whether the long-distance effects can
fully account for the tension in $b\to s\ell^{+}\ell^{-}$, and NP may be
warranted to explain the data (e.g., see Ref. Gubernari  _et al._ (2022)).

Recently, the Fermilab CDF collaboration Aaltonen  _et al._ (2022), based on
2002-2011 data with $8.8~{}\text{fb}^{-1}$ integrated luminosity, reported a
new measurement of $W$-boson mass:

$\displaystyle M_{W}=80.4335\pm 0.0094\,\text{GeV},$ (2)

which differs from the SM prediction $M_{W}^{\mathrm{SM}}=80.357\pm 0.006$ GeV
Awramik  _et al._ (2004) with $7\sigma$ significance. It is intriguing to note
that the CDF measurement also differs with $M_{W}$ measurements reported by
ATLAS Aaboud  _et al._ (2018) and LHCb Aaij  _et al._ (2022c), an issue to be
resolved333Recently it has been suggested Bandyopadhyay  _et al._ (2022) that
discrepancies between different $M_{W}$ measurements could be due to a light
NP particle modifying the missing transverse momentum in the detector. in the
future with improved measurements. Here, we will take a view that the CDF
measurement hints towards NP presence in the $M_{W}$ value.

In this article we show that a $H^{+}$ boson, naturally present in simple
extensions of the SM such as two-Higgs doublet model (2HDM) Lee (1973), can
account for the above-mentioned anomalies. The fact that $H^{+}$ can explain
${R_{D}}$ and ${R_{D^{\ast}}}$ anomalies is well known in literature Crivellin
_et al._ (2012); Celis  _et al._ (2013); Tanaka and Watanabe (2013); Ko  _et
al._ (2013); Crivellin  _et al._ (2013, 2016); Kim  _et al._ (2015); Cline
(2016); Lee (2017); Iguro and Tobe (2017); Iguro and Omura (2018); Chen and
Nomura (2018); Martinez  _et al._ (2018); Li  _et al._ (2018); Cardozo  _et
al._ (2021); Athron  _et al._ (2022a); Iguro (2022); Blanke  _et al._ (2022).
Here, we show that same set of $H^{+}$ interactions that explain
$R_{D^{(\ast)}}$ anomaly unavoidably induce a destructive NP contribution
desired to simultaneously explain the tension in $b\to s\ell^{+}\ell^{-}$,
while keeping the ratios $R_{K^{(\ast)}}$ unaltered. Our results strengthen
the viewpoint that a common NP could be behind the charged- and neutral-
current $B$ anomalies. The $M_{W}$ anomaly can also be explained by a NP
contribution to the Peshkin-Takeuchi $T$ parameter Peskin and Takeuchi (1992),
which helps determine the allowed mass range of the physical scalars in the
2HDM.

##  II $H^{+}$ interactions and relevant NP parameters

The $H^{+}$ boson we consider belongs to a 2HDM without any special discrete
symmetry (see Ref. Branco  _et al._ (2012) for a comprehensive review). The
general $H^{+}$ interactions in the fermion mass basis are given by the
Lagrangian Davidson and Haber (2005)

$\displaystyle{\cal
L}_{H^{+}}=-\bar{u}(V\rho^{d}R-\rho^{u\dagger}\,VL)d\,H^{+}-\bar{\nu}\,\rho^{e}R\,e\,H^{+}+\text{H.c.},$
(3)

where $\rho^{f}$ ($f=u,d,e$) are $3\times 3$ NP Yukawa matrices, $V$ denotes
the CKM matrix, and $L/R\equiv(1\mp\gamma_{5})/2$ are the chirality
projectors. In addition to the $H^{+}$ boson, 2HDM also has CP-even/odd scalar
bosons $H$, $A$. Their Yukawa interactions are not important for our analysis;
we refer to Refs. Davidson and Haber (2005); Mahmoudi and Stal (2010) for
their details. However we need to define the masses of $H$, $A$, as those
would be required in the computation of $M_{W}$ in the model. The masses of
$H$, $A$ are related to mass of $H^{+}$ by the relations444Here we assume, to
accord with the current data Sirunyan  _et al._ (2019a); Aad  _et al._ (2020),
that there is very little mixing between SM Higgs ($h$) and $H$ boson.

$\displaystyle
m_{H^{+}}^{2}=m_{H}^{2}-\frac{v^{2}}{2}(\Lambda_{4}+\Lambda_{5}),\quad
m_{A}^{2}=m_{H}^{2}-v^{2}\Lambda_{5}\,,$ (4)

where $v\simeq 246$ GeV, and $\Lambda_{4}$, $\Lambda_{5}$ are the quartic
couplings in the Higgs potential (see the Appendix for details).

To explain the anomalies with a minimal set of NP parameters, we make the
ansatz that NP Yukawa matrices have the following simple structure

$\displaystyle\rho^{u}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 0&\rho_{tc}&0\\\
\end{pmatrix},\quad\rho^{e}=\begin{pmatrix}0&0&0\\\ 0&0&0\\\
0&0&\rho_{\tau\tau}\\\ \end{pmatrix},$ (5)

and $\rho^{d}=0$. The texture as such in eq. 5 is the most economical choice
to affect rate of $B\to D^{(\ast)}\tau\bar{\nu}$ only: the off-diagonal
coupling $\rho_{tc}$ facilitates $H^{+}$ mediated $b\to c$ transitions that
are not CKM suppressed and diagonal lepton coupling $\rho_{\tau\tau}$ ensures
that only semitauonic modes are affected. With the above choice the Lagrangian
in eq. 3 simplifies to (dropping a $V_{td}$ suppressed term)

$\displaystyle{\cal
L}_{H^{+}}=(\rho_{tc}^{\ast}V_{tb}\,\bar{c}_{R}b_{L}+\rho_{tc}^{\ast}V_{ts}\bar{c}_{R}s_{L}-\rho_{\tau\tau}\,\bar{\nu}_{\tau
L}\tau_{R})H^{+}+\mathrm{H.c.},$ (6)

which together with eq. 4 defines all the Yukawa interactions and NP
parameters relevant in our setup.

Concerning direct search constraints on $H^{+}$, analyses in Refs. Iguro  _et
al._ (2019a); Iguro (2022), based on experimental results of Refs. Sirunyan
_et al._ (2019b, 2018a); Aaboud  _et al._ (2019); Sirunyan  _et al._ (2018b),
find that mass range $m_{H^{+}}>400$ GeV for an explanation of the
$R_{D^{(\ast)}}$ anomaly is likely ruled out due to a constraint from $pp\to
bc\to\tau\nu$ process. However, the low-mass region $m_{H^{+}}<400$ is still
not excluded Iguro (2022). It was pointed out recently Blanke  _et al._ (2022)
that $\tau\nu$ search with an additional $b$-tagged jet ($pp\to bH^{\pm}\to
b\tau\nu$) could be useful in probing this low-mass region of $H^{+}$. In this
article we therefore focus on the $m_{H^{+}}<400$ GeV region.

##  III Observables

In this section we discuss $H^{+}$ contributions to the anomalous observables
together with the relevant constraint on our setup.

###  III.1 ${R_{D}}$ and ${R_{D^{\ast}}}$

The $H^{+}$ boson mediates $b\to c\tau\bar{\nu}$ transition at tree-level
(shown in fig. 1a), the effect of which can be parametrized by the following
effective Hamiltonian

$\displaystyle{\cal
H}_{\text{eff}}=2\sqrt{2}G_{F}V_{cb}\,C_{S,L}(\bar{c}_{R}b_{L})(\bar{\tau}_{R}\nu_{\tau
L}),$ (7)

where the coefficient $C_{S,L}$ at scale $\mu\sim m_{H^{+}}$ is given by

$\displaystyle
C_{S,L}=\frac{\rho_{tc}^{\ast}\rho_{\tau\tau}^{\ast}}{2\sqrt{2}G_{F}V_{cb}\,m_{H^{+}}^{2}}.$
(8)

The contributions of $C_{S,L}$ to ratios ${R_{D}}$ and ${R_{D^{\ast}}}$ are
numerically parametrized (at scale $\mu\sim m_{b}$) as Iguro  _et al._
(2022)555Similar formulas are also given in Refs. Blanke  _et al._ (2019);
Iguro  _et al._ (2019b).

$\displaystyle{R_{D}}$
$\displaystyle\simeq({R_{D}})_{\mathrm{SM}}[1+1.49\operatorname{Re}(C_{S,L})+1.01\,|C_{S,L}|^{2}],$
(9) $\displaystyle{R_{D^{\ast}}}$
$\displaystyle\simeq({R_{D^{\ast}}})_{\mathrm{SM}}[1-0.11\operatorname{Re}(C_{S,L})+0.04|C_{S,L}|^{2}].$
(10)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1: Feynman diagrams for $b\to c\tau\bar{\nu}$ (1a), $b\to
s\ell^{+}\ell^{-}$ (1b, 1c), $B_{s}-\bar{B}_{s}$ mixing (1d), $b\to s\gamma$
(1e), and $W$-$\tau$-$\nu$ vertex (1f).

The scalar interaction in eq. 7 contributes rather significantly (due to lack
of chirality suppression) to the $B_{c}\to\tau\nu$ branching ratio.
Numerically, it is given as Iguro  _et al._ (2022)

$\displaystyle\text{BR}(B_{c}\to\tau\nu)\simeq 0.02\,|1-4.35\,C_{S,L}|^{2}.$
(11)

This decay is not measured yet. However, based on the precisely measured
lifetime of $B_{c}$ meson Workman  _et al._ (2022), a theoretical constraint
on maximally allowed $\text{BR}(B_{c}\to\tau\nu)$ can be obtained Alonso  _et
al._ (2017). Recent estimates Blanke  _et al._ (2019); Aebischer and Grinstein
(2021) suggest that $\text{BR}(B_{c}\to\tau\nu)$ as large as $60\%$ to $63\%$
is still possible.

In our analysis we have not considered the constraint from the differential
decay distributions of $B\to D^{(\ast)}\tau\bar{\nu}$ Lees  _et al._ (2013);
Huschle  _et al._ (2015), which are known to be sensitive to scalar NP Sakaki
_et al._ (2015); Freytsis  _et al._ (2015); Celis  _et al._ (2017a). Compared
to ratios $R_{D^{(\ast)}}$, the decay distributions are quite sensitive to
hadronic form factors and parametric (e.g. $V_{cb}$) uncertainties.
Furthermore, the corresponding experimental analyses Lees  _et al._ (2013);
Huschle  _et al._ (2015) are model dependent and require the NP model’s
contributions to the background and the signal efficiency in order to obtain
the data. Also, since the correlations among different data bins are not
available, a combined data analysis is difficult. The improved measurements at
Belle II Altmannshofer  _et al._ (2019) will be helpful in overcoming these
issues (e.g., see discussion in Ref. Sakaki  _et al._ (2015)).

###  III.2 $b\to s\ell^{+}\ell^{-}$

The $H^{+}$ contributions to $b\to s\ell^{+}\ell^{-}$ processes have been
discussed in several works (for example, see Iguro and Tobe (2017); Iguro and
Omura (2018); Crivellin  _et al._ (2019); Athron  _et al._ (2022b)), most of
which have focused on top quark-$H^{+}$ loop diagrams. Such contributions,
which are local in the effective field theory at scale $\mu\sim m_{b}$, are
not present in our setup (see eq. 6). Instead, the typical $b\to
s\ell^{+}\ell^{-}$ contributions arise from the diagrams involving charm quark
in the loop as shown in figs. 1b and 1c.

The leading contribution to $b\to s\ell^{+}\ell^{-}$ comes from the penguin
diagram in fig. 1b. This contribution in the effective field theory can be
obtained via the penguin insertion of the four-quark operator
$(\bar{c}_{R}\,b_{L})(\bar{s}_{L}\,c_{R})$ mediating $b\to sc\bar{c}$
transition. This four-quark operator is generated at tree level via a diagram
similar to fig. 1a with the $\bar{\tau}\nu H^{+}$ vertex replaced by
$\bar{s}cH^{+}$. For convenience we make use of a Fierz identity and define
the following $b\to sc\bar{c}$ effective Hamiltonian

$\displaystyle-{\cal
H}_{\text{eff}}=\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{\ast}\,\tilde{C}_{V,LR}\,(\bar{s}_{L}^{\alpha}\gamma^{\mu}b_{L}^{\beta})(\bar{c}_{R}^{\beta}\gamma_{\mu}c_{R}^{\alpha}),$
(12)

where $\alpha,\beta$ are the color indices, and the coefficient
$\tilde{C}_{V,LR}$ at scale $\mu\sim m_{H^{+}}$ is given as
$\tilde{C}_{V,LR}=-{v^{2}}|\rho_{tc}|^{2}/{4m_{H^{+}}^{2}}$. Then, closing the
charm loop of the $b\to sc\bar{c}$ operator in eq. 12 (diagram shown in fig.
1b with $H^{+}$ integrated out) effects a nonlocal NP contribution to the
vector operator $(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{\ell}\gamma^{\mu}\ell)$.
Adapting the results of Refs. Jäger  _et al._ (2020, 2018) to our case, we
obtain the following NP contribution to the $b\to s\ell^{+}\ell^{-}$ Wilson
coefficient666We follow notation of Ref. London and Matias (2022) for the
$b\to s$ operator basis Buchalla  _et al._ (1996).

$\displaystyle
C_{9\ell}(q^{2},\mu)=\left[\frac{4}{9}+h(q^{2},m_{c},\mu)\right]\tilde{C}_{V,LR},$
(13)

where function $h(q^{2},m_{c})$ is given in eq. (11) of Ref. Beneke  _et al._
(2001). The eq. 13 gives sufficiently accurate result777The dominant
contribution comes from $q^{2}$-independent terms in $h(q^{2},m_{c},\mu)$
(see, e.g., Ref. Jäger  _et al._ (2020)). if the coefficient
$\tilde{C}_{V,LR}$ arise at a scale close to the $B$-meson scale. However,
since in our model the four-quark operator is generated at higher scale
$\mu\sim m_{H^{+}}$, the renormalization group (RG) running effects are
important. Therefore, instead of eq. 13, we use wilson package Aebischer  _et
al._ (2018) (which is based on the results of Refs. Jenkins  _et al._ (2013,
2014); Alonso  _et al._ (2014); Jenkins  _et al._ (2018a); Aebischer  _et al._
(2017); Jenkins  _et al._ (2018b); Celis  _et al._ (2017b); Herren and
Steinhauser (2018)), accounting for one-loop RG evolution of
$\tilde{C}_{V,LR}(\mu)$, to evaluate the mixing into $C_{9\ell}(\mu_{b})$.
Numerically, taking the NP scale $\mu_{\mathrm{high}}=200$ GeV as an example
case, we find
$C_{9\ell}(\mu_{b}=4.8\,\mathrm{GeV})=5.17\,\tilde{C}_{V,LR}(\mu_{\mathrm{high}})$.
The $Z$-penguin diagram (fig. 1b with $\gamma\to Z$) can be ignored as the
corresponding loop function vanishes in the $m_{c}\to 0$ limit.

Another contribution to $b\to s\ell^{+}\ell^{-}$ comes from the box diagram in
fig. 1c which gives Crivellin  _et al._ (2019)

$\displaystyle
C_{9\tau}^{\text{NP}}=C_{10\tau}^{\text{NP}}=-\frac{v^{2}}{64\pi\alpha_{e}\,m_{H^{+}}^{2}}|\rho_{tc}\,\rho_{\tau\tau}|^{2}.$
(14)

The contributions in eq. 13 are lepton flavor universal (due to
$\gamma\ell\ell$ vertex). On the other hand, the contribution in eq. 14 in
principle introduces $\tau$ vs. $e,\mu$ violation in our setup. However, this
contribution depends on coupling product $|\rho_{tc}\,\rho_{\tau\tau}|$ that,
as we will see later, is strongly constrained by the $b\to c\tau\bar{\nu}$
processes (and by demanding a solution to the $R_{D^{(\ast)}}$ anomaly),
causing contributions in eq. 14 to be completely negligible in the relevant
parameter space. Consequently, NP contributions to $b\to s\ell^{+}\ell^{-}$
are practically described by eq. 13 and universal to all lepton flavors. As a
result, in our setup the ratios $R_{K^{(\ast)}}$ are SM-like in agreement with
the observations made by the LHCb LHC (2022a, b). Equally important to note is
that since NP contributions to Wilson coefficients ($C_{10\ell}$), related to
axial-vector current, are negligible, the rate of the rare decay
$B_{s}\to\mu^{+}\mu^{-}$ remain SM-like, which is also consistent with the new
CMS result CMS (2022) based on 2016-2018 data corresponding to integrated
luminosity of $140~{}\text{fb}^{-1}$. The recent global fit to $b\to
s\ell^{+}\ell^{-}$ data (excluding $R_{K^{(\ast)}}$ and
$\mathrm{BR}(B_{s}\to\mu^{+}\mu^{-})$, which anyway remain unaffected in the
considered scenario) shows that the NP scenario Hurth  _et al._ (2022a)

$\displaystyle C_{9\ell}=-0.95\pm 0.13$ (15)

is strongly favored over the no NP hypothesis, corresponding to $6.1\,\sigma$
pull away from the SM (for other NP scenarios see Ref. Greljo  _et al._
(2022); Ciuchini  _et al._ (2022)888For fits before the new $R_{K^{(\ast)}}$
measurements, see Refs. Alok  _et al._ (2019); Datta  _et al._ (2019); Hurth
_et al._ (2022b); Algueró  _et al._ (2022); Altmannshofer and Stangl (2021);
Ciuchini  _et al._ (2021b); Geng  _et al._ (2021); Alok  _et al._ (2022);
Singh Chundawat (2022); Biswas  _et al._ (2021).). In our analysis, we will
use eq. 15 to explain the current $b\to s\ell^{+}\ell^{-}$ discrepancies.

There are few important flavor constraints on $\rho_{tc}$. The most stringent
constraint comes from the mass difference ($\Delta M_{B_{s}}$) in
$B_{s}-\bar{B}_{s}$ mixing. The $H^{+}$-induced box diagram (diagram with
$W^{+}$ and $H^{+}$ in loop vanishes in the $m_{c}\to 0$ limit), shown in fig.
1d, gives rise to the effective Hamiltonian, ${\cal
H}_{\text{eff}}=C_{bs}\,(\bar{s}\gamma^{\mu}Lb)(\bar{s}\gamma^{\mu}Lb)$, where

$\displaystyle C_{bs}=\frac{V_{ts}^{\ast
2}V_{tb}^{2}\,|\rho_{tc}|^{4}}{128\pi^{2}m_{H^{+}}^{2}}.$ (16)

The current value of the mass differences is $\Delta M_{B_{s}}=17.741\pm
0.020~{}\text{ps}^{-1}$ Workman  _et al._ (2022), which is to be compared with
the SM prediction $\Delta M_{B_{s}}=18.4^{+0.7}_{-1.2}~{}\text{ps}^{-1}$ Di
Luzio  _et al._ (2019).

Another relevant constraint arises from radiative decay $B\to X_{s}\gamma$,
which gets modified due to the loop diagram shown in fig. 1e. The
corresponding dipole coefficient $C_{7}$ at scale $\mu\sim m_{H^{+}}$ at the
leading order is given by Crivellin  _et al._ (2019)

$\displaystyle
C_{7}=-\frac{7}{36}\frac{v^{2}}{4m_{H^{+}}^{2}}|\rho_{tc}|^{2},$ (17)

while the coefficient related to $b\to sg$ is $C_{8}\simeq(6/7)C_{7}$. The
current experimental value for the branching ratio of $B\to X_{s}\gamma$ is
$(3.32\pm 0.15)\times 10^{-4}$ Amhis  _et al._ (2021).

There are $H^{+}$ contributions to $K-\bar{K}$ mixing parameters
$\varepsilon_{K}$ and $\Delta M_{K}$. The corresponding contributions arise
from box diagram shown in fig. 1d after replacing external quarks
$\\{bs\\}\to\\{sd\\}$. We calculate NP contribution to $K-\bar{K}$ mixing
following Ref. Hou and Kumar (2022) and use experimental values from Ref.
Workman  _et al._ (2022); the resulting constraints however turn out to be
weaker than those from $B$ physics.

Figure 2: First and second plots show results of fits to ${R_{D}}$ and
${R_{D^{\ast}}}$ ($1\sigma$) and $b\to s\ell^{+}\ell^{-}$ ($1\sigma$) for
$m_{H^{+}}=200$ GeV and $250$ GeV, respectively. The dark and light magenta
color contours are the global $1\sigma$ and $2\sigma$ allowed regions. The
third plot shows results of the parameter scan where the red points
corresponds to $M_{W}$ values that are within $1\sigma$ of eq. 2. See text for
other details.

###  III.3 Shift in $M_{W}$

As mentioned in Introduction, the CDF value of $M_{W}$ differs from the
corresponding SM prediction by $7\sigma$. This difference can be attributed to
a NP correction to $T$ parameter in the 2HDM (see, e.g., Refs. Bahl  _et al._
(2022); Song  _et al._ (2022); Babu  _et al._ (2022); Arco  _et al._ (2022);
Lee  _et al._ (2022); Lu  _et al._ (2022); Ahn  _et al._ (2022); Sakurai  _et
al._ (2022); Tran  _et al._ (2022); Arcadi and Djouadi (2022); Ghorbani and
Ghorbani (2022); Abouabid  _et al._ (2022); Botella  _et al._ (2022); Kim  _et
al._ (2022); Hessenberger  _et al._ (2022); Arco  _et al._ (2022)). The SM
value of $M_{W}$ is calculable as Babu  _et al._ (2022)

$\displaystyle
M_{W}^{2}=\frac{M_{Z}^{2}}{2}\left(1+\sqrt{1-\frac{4\pi\alpha_{e}\,(1+\Delta
r)}{\sqrt{2}G_{F}M_{Z}^{2}}}\,\right),$ (18)

where $\Delta r$ contains quantum corrections associated with oblique
parameters and renormalization of $\alpha_{e}$. Within the SM, $(\Delta
r)_{\mathrm{SM}}\simeq 0.038$ Babu  _et al._ (2022). Assuming that
modifications in $\Delta r$ arise from a NP contribution to $T$ parameter, one
can parametrize NP effects as $\Delta r=(\Delta
r)_{\mathrm{SM}}-(c_{W}^{2}/s_{W}^{2})\alpha_{e}(M_{Z})\,T$, where $T$ in 2HDM
is given by

$\displaystyle
T=\frac{1}{16\pi^{2}\alpha_{e}(M_{Z})v^{2}}\\{F(m_{H^{+}}^{2},m_{H}^{2})+F(m_{H^{+}}^{2},m_{A}^{2})$
$\displaystyle-F(m_{H}^{2},m_{A}^{2})\\},$ (19)

with loop function

$\displaystyle F(a,b)=\frac{a+b}{2}-\frac{ab}{a-b}\log\frac{a}{b}.$ (20)

Note that $F(a,b)$ vanishes in the limit of $a\to b$, indicating that at least
two of the scalar states should have different masses in order to obtain a
nonzero contribution to the $T$ parameter. In our setup, the allowed range of
$m_{H^{+}}$ is fixed from seeking solution to $R_{D^{(\ast)}}$ and $b\to
s\ell^{+}\ell^{-}$ anomalies. The values of $m_{H}$ and $m_{A}$ then can be
obtained from eq. 4, with the quartic couplings $\Lambda_{4}$, $\Lambda_{5}$
varied within perturbative limits. We also include NP contribution arising
from $S$ parameter following Ref. Babu  _et al._ (2022); however these
contributions are subdominant.

If $m_{H}$, $m_{A}$, and $m_{H^{+}}$ are not equal, which is the case to
obtain a finite $T$ parameter as discussed above, then the vertex
$W$-$\tau$-$\nu_{\tau}$ correction diagram in fig. 1f gives a constraint on
$\rho_{\tau\tau}$ coupling. This correction is sensitive to the mass splitting
of physical scalars in 2HDM and can be parametrized by writing gauge coupling
$g_{W\tau\nu}\to g_{W\tau\nu}(1+\delta g)$, where $\delta g$ is

$\displaystyle\delta
g=\frac{|\rho_{\tau\tau}|^{2}}{32\pi^{2}}I(m_{H}^{2}/m_{H^{\pm}}^{2},m_{A}^{2}/m_{H^{\pm}}^{2}),$
(21)

with loop function $I(x,y)$ given by Abe  _et al._ (2015, 2019),

$I(x,y)=1+\frac{1}{4}\frac{1+x}{1-x}\log x+\frac{1}{4}\frac{1+y}{1-y}\log y.$
(22)

Note that the function $I(x,y)$ vanishes in the combined limit $x\to 1$ and
$y\to 1$. The correction $\delta g$ modifies leptonic decay rate of $\tau$ as
$\Gamma_{\tau\to\ell\nu_{\tau}\bar{\nu}_{\ell}}\to\Gamma^{\mathrm{SM}}_{\tau\to\ell\nu_{\tau}\bar{\nu}_{\ell}}(1+2\delta
g)$. The LFU test in $\tau$ decays is then given by
${g_{\tau}}/{g_{e}}=1+\delta g$, which is to be compared with the HFLAV value
$g_{\tau}/g_{e}=1.0029\pm 0.0014$ Amhis  _et al._ (2021). We note that the
$\rho_{\tau\tau}$ needed in our setup is very small (see next section),
rendering $\delta g$ to be completely negligible $\sim\mathcal{O}(10^{-5})$ .
The smallness of $\rho_{\tau\tau}$ also guarantees that the NP correction
(calculated using the formula given in Ref. Cline (2016)) to the partial
leptonic width of $Z\to\tau\tau$ is also negligible.

##  IV Results

In our numerical analysis, theoretical predictions of the flavor observables
are obtained using flavio Straub (2018). Our main results are shown in fig. 2.
In the first plot, we show results for $R_{D^{(\ast)}}$ and $b\to
s\ell^{+}\ell^{-}$ together with relevant constraints in the ($|\rho_{tc}|$,
$|\rho_{\tau\tau}|$) plane for $m_{H^{+}}=200$ GeV. In the plot, the phase
$\phi$ ($\equiv\operatorname{arg}(\rho_{tc}\rho_{\tau\tau})$)999We take
$\rho_{tc}$ real so that $\phi$ corresponds to the phase of $\rho_{\tau\tau}$.
is fixed by maximizing the global log-likelihood function101010We assume that
experimental uncertainties follow gaussian distribution and for
$\mathrm{BR}(B_{c}\to\tau\bar{\nu})$ constraint we assume that it has a
uniform probability within limits $[0,0.63]$ and zero elsewhere. in the space
of NP parameters. This is performed using iminuit James and Roos (1975);
Dembinski and et al. (2020), which gives the best fit values
$|\rho_{tc}|=0.659$, $\rho_{\tau\tau}=0.052$, $\phi\simeq 2\pi/3$. The green
band shows region consistent (within $1\sigma$) with the current data on
${R_{D}}$ and ${R_{D^{\ast}}}$. The vertical yellow band corresponds to value
$C_{9\ell}\sim-1$ ($1\sigma$ range of eq. 15, to be exact). The individual
$95\%$ C.L. exclusion bounds from $\Delta M_{B_{s}}$, $\text{BR}(B\to
X_{s}\gamma)$, and $\varepsilon_{K}$ are also shown as vertical lines. The
constraint $\text{BR}(B_{c}\to\tau\bar{\nu})<0.63$ is shown as dash-dotted
orange curve (sitting just on top of $1\sigma$ solution of $R_{D^{(\ast)}}$),
which rules out the region above it. We note that the $R_{D^{(\ast)}}$
solution (green band) only constrains the product
$|\rho_{tc}\rho_{\tau\tau}|$, so the sizes of individual couplings remain
unresolved. Including the data on $b\to s\ell^{+}\ell^{-}$ decays, which are
essentially sensitive to $|\rho_{tc}|$, a far better constraint on the
parameter space is achievable. The contours in magenta color show $1\sigma$
and $2\sigma$ region where both $R_{D^{(\ast)}}$ and $b\to s\ell^{+}\ell^{-}$
data can be explained together. We also show smaller values
$C_{9\ell}=-0.5,-0.3$ as solid yellow lines, illustrating the impact of
$\rho_{tc}$ variation on NP in $b\to s\ell^{+}\ell^{-}$. In the second plot,
we show the results for $m_{H^{+}}=250$ GeV. The constraints from $B\to
X_{s}\gamma$ and $\varepsilon_{K}$ are relaxed and lie outside plot range. The
best fit point now reads $|\rho_{tc}|=0.784$, $\rho_{\tau\tau}=0.068$, and
$\phi$ same as before. In this case we note that $\Delta M_{B_{s}}$ constraint
(dash-dotted blue line) already covers most of the $1\sigma$ range of eq. 15,
but there is still some allowed region left. Our results therefore indicate
that for $m_{H^{+}}>250$ GeV it becomes difficult to obtain $C_{9\ell}=-1$,
but smaller (but appreciable) values such as $C_{9\ell}\sim-0.5$ are still
possible.

In the third plot, we show parameter scan in the plane of mass-differences
$m_{H^{+}}-m_{H}$ and $m_{H^{+}}-m_{A}$, where the red points corresponds to
$M_{W}$ values within $1\sigma$ of the CDF measurement (eq. 2); the light
(dark) green points show $M_{W}$ values which are below (above) $1\sigma$
range. In our setup, as mentioned earlier, the prediction of $M_{W}$ depends
on $m_{H^{+}}$ and quartic couplings $\Lambda_{4},\Lambda_{5}$. To obtain scan
results, we vary $m_{H^{+}}$ uniformly in the range ($180$ GeV, $300$ GeV) and
$\Lambda_{4},\Lambda_{5}$ in the range ($-\sqrt{4\pi},\sqrt{4\pi}$). To select
allowed points, we require $m_{H}^{2}$, $m_{A}^{2}>0$, and reject $m_{H,A}\leq
100$ GeV. We note that significant population of red points is when both $H,A$
are heavier than $H^{+}$. There are a few red points in the region when
$H^{+}$ is heavier than both $H,A$. However, we do not find any solution when
only one of the $H,A$ is heavier or lighter than $H^{+}$; this is because in
these corners of parameter space, the NP correction to $T$ parameter (section
III.3) is negative, whereas a positive correction is needed to obtain a
positive shift in $M_{W}$ value. In the special case $\Lambda_{5}=0$, the
second relation in eq. 4 implies $m_{H}=m_{A}$, which in fig. 2 (right)
corresponds to the positive diagonal. So, even though the parameter space is
reduced a lot, $W$-boson mass consistent with the CDF measurement can still be
obtained. On the other hand, in case of vanishing $\Lambda_{4}$, eq. 4 gives
$m_{H^{+}}^{2}-m_{H}^{2}=m_{A}^{2}-m_{H^{+}}^{2}$, from which one can deduce
that both $H$, $A$ cannot be simultaneously heavier or lighter than $H^{+}$,
and therefore from the arguments presented above we find that the CDF value of
$M_{W}$ will not be explained in this case.

##  V Conclusions

At present there are hints of LFU violation in $b\to c\ell\bar{\nu}$ data,
reinforced further by the recent LHCb result on the combined measurement of
${R_{D}}$ and ${R_{D^{\ast}}}$. On the other hand, no such effect is seen in
$b\to s\ell^{+}\ell^{-}$, and LFU ratios $R_{K^{(\ast)}}$ are now SM-like.
However, the discrepancies in the branching fractions and optimized
observables related to $b\to s\ell^{+}\ell^{-}$ decays still remain. In this
article we show that a $H^{+}$ boson, of few hundred GeV mass, can
simultaneously explain anomalies in $R_{D^{(\ast)}}$ and $b\to
s\ell^{+}\ell^{-}$ data. That a $H^{+}$ boson can explain the former is
already known in literature. Here we uncover a nice correlation between
$H^{+}$ effects in the charged- and neutral-current $B$ decays: the enhanced
rates of $B\to D^{(\ast)}\tau\bar{\nu}$ imply a destructive NP contribution in
the Wilson coefficient $C_{9\ell}$. We show that the current constraints allow
for $C_{9\ell}\sim-1$, a preferred solution to address discrepancies in the
$b\to s\ell^{+}\ell^{-}$ decays. Additionally, we also show that the
discrepancy observed in $M_{W}$ by CDF II can also be explained by allowing
splitting among the physical scalars masses; the solution prefers the neutral
states $H,A$ heavier than $H^{+}$.

Acknowledgments I would like to thank Namit Mahajan for a useful conversation
and his inputs on the manuscript, and Monika Blanke and Teppei Kitahara for
helpful communication regarding their work on charged-current $B$ decays. I
also thank Manu George for reading the paper carefully. For diagrams and
figures in paper, I acknowledge using JaxoDraw Binosi and Theussl (2004) and
Matplotlib Hunter (2007). This research work is supported by NSTC
111-2639-M-002-002-ASP of Taiwan.

##  Appendix A Scalar potential and mass-spectrum

Here we provide details about the scalar potential of a general 2HDM and the
relations of its parameters with the masses of the scalars in the model. With
$H_{1}$, $H_{2}$ denoting Higgs doublets, the scalar potential is given by
Davidson and Haber (2005)

$\displaystyle
V(H_{1},H_{2})=M_{11}^{2}|H_{1}|^{2}+M_{22}^{2}|H_{2}|^{2}-\left(M_{12}^{2}H_{1}^{\dagger}H_{2}+\mathrm{H.c.}\right)+\frac{\Lambda_{1}}{2}|H_{1}|^{4}+\frac{\Lambda_{2}}{2}|H_{2}|^{4}+\Lambda_{3}|H_{1}|^{2}|H_{2}|^{2}$
$\displaystyle+~{}\Lambda_{4}|H_{1}^{\dagger}H_{2}|^{2}+\left\\{\frac{\Lambda_{5}}{2}\left(H_{1}^{\dagger}H_{2}\right)^{2}\right.+\left[\Lambda_{6}|H_{1}|^{2}+\Lambda_{7}\left|H_{2}\right|^{2}\right]H_{1}^{\dagger}H_{2}\biggl{.}+\,\mathrm{H.c.}\biggr{\\}},$
(23)

where parameters $M_{12}^{2}$, $\Lambda_{i}$ $(i=5,6,7)$ in general can have
complex phases. In this paper, we have taken $V(H_{1},H_{2})$ to be CP-
invariant for simplicity, and therefore all the potential parameters are real.

Working in the Higgs basis Georgi and Nanopoulos (1979); Lavoura and Silva
(1994); Botella and Silva (1995) where only one of the Higgs doublets receives
vacuum expectation value, we define doublets $H_{1}$, $H_{2}$ as

$\displaystyle H_{1}=\left(\begin{matrix}G^{+}\\\
\frac{1}{\sqrt{2}}{(v+H_{1}^{0}+iG^{0})}\\\ \end{matrix}\right),\quad
H_{2}=\left(\begin{matrix}H^{+}\\\ \frac{1}{\sqrt{2}}{(H_{2}^{0}+iA)}\\\
\end{matrix}\right),$ (24)

so that $\langle H_{1}\rangle=v/\sqrt{2}$ and $\langle H_{2}\rangle=0$. In the
above notation, $G^{+}$, $G^{0}$ are the goldstone bosons, $H^{+}$ and $A$ are
the charged scalar and CP-odd scalar, respectively, while the physical CP-even
neutral scalars $h$ and $H$ are given by

$\displaystyle h=H_{1}^{0}\sin\gamma+H_{2}^{0}\cos\gamma,\quad
H=H_{1}^{0}\cos\gamma-H_{2}^{0}\sin\gamma,$ (25)

with $\gamma$ denoting the $h$-$H$ mixing angle (analogous to $(\beta-\alpha)$
in type-II 2HDM notation).

Minimization of the potential gives conditions:
$M_{11}^{2}+\Lambda_{1}(v^{2}/2)=0$, $M_{12}^{2}-\Lambda_{6}(v^{2}/2)=0$. The
relations between potential parameters and the scalar masses, which we are
mainly interested in, are given as Davidson and Haber (2005)

$\displaystyle m_{H+}^{2}$
$\displaystyle=M_{22}^{2}+\frac{v^{2}}{2}\Lambda_{3}\,,$ (26) $\displaystyle
m_{A}^{2}-m_{H+}^{2}$
$\displaystyle=-\frac{v^{2}}{2}\left(\Lambda_{5}-\Lambda_{4}\right)\,,$ (27)
$\displaystyle m_{H}^{2}+m_{h}^{2}-m_{A}^{2}$
$\displaystyle=v^{2}\left(\Lambda_{1}+\Lambda_{5}\right)\,,$ (28)
$\displaystyle\left(m_{H}^{2}-m_{h}^{2}\right)^{2}$
$\displaystyle=\left[m_{A}^{2}+\left(\Lambda_{5}-\Lambda_{1}\right)v^{2}\right]^{2}+4\Lambda_{6}^{2}v^{4}\,,$
(29) $\displaystyle\sin\gamma\cos\gamma$
$\displaystyle=-\frac{\Lambda_{6}v^{2}}{m_{H}^{2}-m_{h}^{2}}.$ (30)

In case of very small mixing angle, i.e., $\cos\gamma\to 0$, which we have
taken in this paper, the simplified relations for the scalar masses are
$m_{h}^{2}\simeq\Lambda_{1}v^{2}$,
$m_{A}^{2}=M_{22}^{2}+(v^{2}/2)(\Lambda_{3}+\Lambda_{4}-\Lambda_{5})$, and the
ones given in eq. 4.

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