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

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.
Recently, the Fermilab CDF collaboration [54], based on 2002-2011 data with 8.8 fb −1 integrated luminosity, reported a new measurement of W -boson mass: which differs from the SM prediction M SM W = 80.357 ± 0.006 GeV [55] with 7σ significance. It is intriguing to note that the CDF measurement also differs with M W measurements reported by ATLAS [56] and LHCb [57], an issue to be resolved 3 in the future with improved mea-surements. Here, we will take a view that the CDF measurement hints towards NP presence in the M W value.
Here, we show that same set of H + interactions that explain R D ( * ) anomaly unavoidably induce a destructive NP contribution desired to simultaneously explain the tension in b → s + − , while keeping the ratios R K ( * ) unaltered. Our results strengthen the viewpoint that a common NP could be behind the charged-and neutralcurrent B anomalies. The M W anomaly can also be explained by a NP contribution to the Peshkin-Takeuchi T parameter [78], 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. [79] for a comprehensive review). The general H + interactions in the fermion mass basis are given by the Lagrangian [80] where where v 246 GeV, and Λ 4 , Λ 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 and ρ d = 0. The texture as such in eq. (5) is the most economical choice to affect rate of B → D ( * ) τν only: the 4 Here we assume, to accord with the current data [82,83], that there is very little mixing between SM Higgs (h) and H boson.
off-diagonal coupling ρ tc facilitates H + mediated b → c transitions that are not CKM suppressed and diagonal lepton coupling ρ τ τ ensures that only semitauonic modes are affected. With the above choice the Lagrangian in eq. (3) simplifies to (dropping a V td suppressed term) 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. [76,84], based on experimental results of Refs. [85][86][87][88], find that mass range m H + > 400 GeV for an explanation of the R D ( * ) anomaly is likely ruled out due to a constraint from pp → bc → τ ν process. However, the low-mass region m H + < 400 is still not excluded [76]. It was pointed out recently [77] that τ ν search with an additional b-tagged jet (pp → bH ± → bτ ν) 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.

A. RD and RD *
The H + boson mediates b → cτν transition at treelevel (shown in fig. 1a), the effect of which can be parametrized by the following effective Hamiltonian where the coefficient C S,L at scale µ ∼ m H + is given by The contributions of C S,L to ratios The scalar interaction in eq. (7) contributes rather significantly (due to lack of chirality suppression) to the B c → τ ν branching ratio. Numerically, it is given as [89] This decay is not measured yet. However, based on the precisely measured lifetime of B c meson [92], a theoretical constraint on maximally allowed BR(B c → τ ν) can be obtained [93]. Recent estimates [90,94] suggest that BR(B c → τ ν) 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 → D ( * ) τν [3,6], which are known to be sensitive to scalar NP [95][96][97]. Compared to ratios R D ( * ) , the decay distributions are quite sensitive to hadronic form factors and parametric (e.g. V cb ) uncertainties. Furthermore, the corresponding experimental analyses [3,6] 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 [98] will be helpful in overcoming these issues (e.g., see discussion in Ref. [95]).
The H + contributions to b → s + − processes have been discussed in several works (for example, see [69,70,99,100]), most of which have focused on top quark-H + loop diagrams. Such contributions, which are local in the effective field theory at scale µ ∼ m b , are not present in our setup (see eq. (6)). Instead, the typical b → s + − contributions arise from the diagrams involving charm quark in the loop as shown in figs. 1b and 1c.
The leading contribution to b → s + − 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 (c R b L )(s L c R ) mediating b → scc transition. This four-quark operator is generated at tree level via a diagram similar to fig. 1a with theτ νH + vertex replaced byscH + . For convenience we make use of a Fierz identity and define the following b → scc effective Hamiltonian where α, β are the color indices, and the coefficientC V,LR at scale µ ∼ m H + is given asC V,LR = −v 2 |ρ tc | 2 /4m 2 H + . Then, closing the charm loop of the b → scc operator in eq. (12) (diagram shown in fig. 1b with H + integrated out) effects a nonlocal NP contribution to the vector operator (s L γ µ b L )(¯ γ µ ). Adapting the results of Refs. [101,102] to our case, we obtain the following NP contribution to the b → s + − Wilson coefficient 6 where function h(q 2 , m c ) is given in eq. (11) of Ref. [105]. The eq. (13) gives sufficiently accurate result 7 if the coef-ficientC 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 µ ∼ m H + , the renormalization group (RG) running effects are important. Therefore, instead of eq. (13), we use wilson package [106] (which is based on the results of Refs. [107][108][109][110][111][112][113][114]), accounting for one-loop RG evolution ofC V,LR (µ), to evaluate the mixing into C 9 (µ b ). Numerically, taking the NP scale µ high = 200 GeV as an example case, we find C 9 (µ b = 4.8 GeV) = 5.17C V,LR (µ high ). The Z-penguin diagram ( fig. 1b with γ → Z) can be ignored as the corresponding loop function vanishes in the m c → 0 limit. Another contribution to b → s + − comes from the box diagram in fig. 1c which gives [99] The contributions in eq. (13) are lepton flavor universal (due to γ vertex). On the other hand, the contribution in eq. (14) in principle introduces τ vs. e, µ violation in our setup. However, this contribution depends on coupling product |ρ tc ρ τ τ | that, as we will see later, is strongly constrained by the b → cτν processes (and by demanding a solution to the R D ( * ) anomaly), causing contributions in eq. (14) to be completely negligible in the relevant parameter space. Consequently, NP contributions to b → s + − are practically described by eq. (13) and universal to all lepton flavors. As a result, in our setup the ratios R K ( * ) are SM-like in agreement with the observations made by the LHCb [33,34]. Equally important to note is that since NP contributions to Wilson coefficients (C 10 ), related to axial-vector current, are negligible, the rate of the rare decay B s → µ + µ − remain SM-like, which is also consistent with the new CMS result [115] based on 2016-2018 data corresponding to integrated luminosity of 140 fb −1 . The recent global fit to b → s + − data (excluding R K ( * ) and BR(B s → µ + µ − ), which anyway remain unaffected in the considered scenario) shows that the NP scenario [116] C 9 = −0.95 ± 0.13 (15) is strongly favored over the no NP hypothesis, corresponding to 6.1 σ pull away from the SM (for other NP scenarios see Ref. [117,118] 8 ). In our analysis, we will use eq. (15) to explain the current b → s + − discrepancies.
There are few important flavor constraints on ρ tc . The most stringent constraint comes from the mass difference (∆M Bs ) in B s −B s mixing. The H + -induced box diagram (diagram with W + and H + in loop vanishes in the m c → 0 limit), shown in fig. 1d, gives rise to the effective Hamiltonian, H eff = C bs (sγ µ Lb)(sγ µ Lb), where The current value of the mass differences is ∆M Bs = 17.741 ± 0.020 ps −1 [92], which is to be compared with the SM prediction ∆M Bs = 18.4 +0.7 −1.2 ps −1 [129]. Another relevant constraint arises from radiative decay B → X s γ, which gets modified due to the loop diagram shown in fig. 1e. The corresponding dipole coefficient C 7 at scale µ ∼ m H + at the leading order is given by [99] while the coefficient related to b → sg is C 8 (6/7)C 7 . The current experimental value for the branching ratio of B → X s γ is (3.32 ± 0.15) × 10 −4 [2].
There are H + contributions to K −K mixing parameters ε K and ∆M K . The corresponding contributions arise from box diagram shown in fig. 1d after replacing external quarks {bs} → {sd}. We calculate NP contribution to K −K mixing following Ref. [130] and use experimental values from Ref. [92]; the resulting constraints however turn out to be weaker than those from B physics.
As mentioned in Introduction, the CDF value of M W differs from the corresponding SM prediction by 7σ. This difference can be attributed to a NP correction to T parameter in the 2HDM (see, e.g., Refs. [131][132][133][134][135][136][137][138][139][140][141][142][143][144][145] where ∆r contains quantum corrections associated with oblique parameters and renormalization of α e . Within the SM, (∆r) SM 0.038 [133]. Assuming that modifications in ∆r arise from a NP contribution to T parameter, one can parametrize NP effects as ∆r = (∆r) SM − (c 2 W /s 2 W )α e (M Z ) T , where T in 2HDM is given by with loop function Note that F (a, b) vanishes in the limit of a → 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 ( * ) and b → s + − anomalies. The values of m H and m A then can be obtained from eq. (4), with the quartic couplings Λ 4 , Λ 5 varied within perturbative limits. We also include NP contribution arising from S parameter following Ref. [133]; 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 -τ -ν τ correction diagram in fig. 1f gives a constraint on ρ τ τ coupling. This correction is sensitive to the mass splitting of physical scalars in 2HDM and can be parametrized by writing gauge coupling g W τ ν → g W τ ν (1 + δg), where δg is with loop function I(x, y) given by [146,147], Note that the function I(x, y) vanishes in the combined limit x → 1 and y → 1. The correction δg modifies leptonic decay rate of τ as Γ τ → ντν → Γ SM τ → ντν (1+2δg). The LFU test in τ decays is then given by g τ /g e = 1+δg, which is to be compared with the HFLAV value g τ /g e = 1.0029 ± 0.0014 [2]. We note that the ρ τ τ needed in our setup is very small (see next section), rendering δg to be completely negligible ∼ O(10 −5 ) . The smallness of ρ τ τ also guarantees that the NP correction (calculated using the formula given in Ref. [67]) to the partial leptonic width of Z → τ τ is also negligible.

IV. RESULTS
In our numerical analysis, theoretical predictions of the flavor observables are obtained using flavio [148]. Our main results are shown in fig. 2. In the first plot, we show results for R D ( * ) and b → s + − together with relevant constraints in the (|ρ tc |, |ρ τ τ |) plane for m H + = 200 GeV. In the plot, the phase φ (≡ arg(ρ tc ρ τ τ )) 9 is fixed by maximizing the global log-likelihood function 10 in the space of NP parameters. This is performed using iminuit [149,150], which gives the best fit values |ρ tc | = 0.659, ρ τ τ = 0.052, φ 2π/3. The green band shows region consistent (within 1σ) with the current data on R D and R D * . The vertical yellow band corresponds to value C 9 ∼ −1 (1σ range of eq. (15), to be exact). The individual 95% C.L. exclusion bounds from ∆M Bs , BR(B → X s γ), and ε K are also shown as vertical lines. The constraint BR(B c → τν) < 0.63 is shown as dashdotted orange curve (sitting just on top of 1σ solution of R D ( * ) ), which rules out the region above it. We note that the R D ( * ) solution (green band) only constrains the product |ρ tc ρ τ τ |, so the sizes of individual couplings remain unresolved. Including the data on b → s + − decays, which are essentially sensitive to |ρ tc |, a far better constraint on the parameter space is achievable. The contours in magenta color show 1σ and 2σ region where both R D ( * ) and b → s + − data can be explained together. We also show smaller values C 9 = −0.5, −0.3 as solid yellow lines, illustrating the impact of ρ tc variation on NP in b → s + − . In the second plot, we show the results for m H + = 250 GeV. The constraints from B → X s γ and ε K are relaxed and lie outside plot range. The best fit point now reads |ρ tc | = 0.784, ρ τ τ = 0.068, and φ same as before. In this case we note that ∆M Bs constraint (dashdotted blue line) already covers most of the 1σ 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 = −1, but smaller (but appreciable) values such as C 9 ∼ −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 1σ of the CDF measurement (eq. (2)); the light (dark) green points show M W values which are below (above) 1σ range. In our setup, as mentioned earlier, the prediction of M W depends on m H + and quartic couplings Λ 4 , Λ 5 . To obtain scan results, we vary m H + uniformly in the range (180 GeV, 300 GeV) and Λ 4 , Λ 5 in the range (− √ 4π, √ 4π). To select allowed points, we require m 2 H , m 2 A > 0, and reject m H,A ≤ 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 (eq. (19)) is negative, whereas a positive correction is needed to obtain a positive shift in M W value. In the special case Λ 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 Λ 4 , eq. (4) gives m 2 H + −m 2 H = m 2 A −m 2 H + , 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 → c ν data, reinforced further by the recent LHCb result on the combined measurement of R D and R D * . On the other hand, no such effect is seen in b → s + − , and LFU ratios R K ( * ) are now SM-like. However, the discrepancies in the branching fractions and optimized observables related to b → s + − decays still remain. In this article we show that a H + boson, of few hundred GeV mass, can simultaneously explain anomalies in R D ( * ) and b → s + − 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 → D ( * ) τν imply a destructive NP contribution in the Wilson coefficient C 9 . We show that the current constraints allow for C 9 ∼ −1, a preferred solution to address discrepancies in the b → s + − 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 chargedcurrent B decays. I also thank Manu George for reading the paper carefully. For diagrams and figures in paper, I acknowledge using JaxoDraw [151] and Matplotlib [152]. 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 [80] V (H 1 , where parameters M 2 12 , Λ 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 [153][154][155] where only one of the Higgs doublets receives vacuum expectation value, we define doublets H 1 , H 2 as so that H 1 = v/ √ 2 and H 2 = 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 with γ denoting the h-H mixing angle (analogous to (β − α) in type-II 2HDM notation). Minimization of the potential gives conditions: M 2 11 + Λ 1 (v 2 /2) = 0, M 2 12 − Λ 6 (v 2 /2) = 0. The relations between potential parameters and the scalar masses, which we are mainly interested in, are given as [80] m 2 In case of very small mixing angle, i.e., cos γ → 0, which we have taken in this paper, the simplified relations for the scalar masses are m 2 h Λ 1 v 2 , m 2 A = M 2 22 + (v 2 /2)(Λ 3 + Λ 4 − Λ 5 ), and the ones given in eq. (4).