Coup de grace to the charged Higgs solution of $P_5^\prime$ and $R_{D^{(*)}}$ discrepancies

We consider a general two Higgs doublet model which can simultaneously solve discrepancies in neutral B meson decay ($b\to s\ell \overline \ell$ distribution) and charged B meson decay ($b\to c\tau\overline\nu$) with a charged Higgs. The model contains two additional neutral scalars at the same mass scale and predicts distinctive signals at the LHC. Based on the recent same-sign top search by the ATLAS collaboration, we found the constraint on the scalar mass spectrum. To probe the remaining mass window, we propose a novel $cg\to t\tau\overline\tau$ process at the LHC.

Interestingly a successful charm penguin contribution to the flavor universal vector operator of b → s and tree level b → cτ ν transition are both controlled by the common b L c R H − interaction where the corresponding Yukawa coupling is denoted as ρ tc u . In the G2HDM, the coupling ρ tc u induces t L c R φ interaction where φ = H, A denotes additional neutral scalars which are SU(2) L partners of the charged Higgs. It is noted that the additional doublet with sizable ρ tc u is discussed with the spontaneous CP violating scenario [64,65] and the electroweak baryogenesis [66]. #3 The available mass range of the charged scalar for the simultaneous explanation is bounded from the above based on the τ ν resonance searches at the LHC [68] as m H + ≤ 400 GeV [69]. Although in Ref. [59] we theoretically showed that the b+τ ν resonance search is a powerful tool to probe the remaining parameters, the corresponding experimental search has not been performed.
Different from recent studies which mainly focus on the charged scalar collider phenomenology in light of deviations in B meson decays [58,59,70], we consider the collider signal of additional neutral scalars. Although connection between sizable ρ tc u and neutral scalars mediated multi-top final states at the LHC has been discussed in Refs. [54,69,[71][72][73][74][75], #4 last summer, the ATLAS collaboration reported the game changing result [80]. They searched for the G2HDM in top-associated processes and directly set the upper limit on ρ tc u . In this letter, we reinterpret the constraint in light of the simultaneous explanation and propose an additional process to cover the remaining parameter space thorough the neutral scalars. Thanks to the electroweak precision data even after the controversial CDF result [81], the mass of those additional scalars (m φ ) should be similar to m H + up to O(v) where v = 246 GeV denotes the vacuum expectation value. Therefore it would be natural to consider the LHC phenomenology to fully probe the interesting parameter space.
The outline of the letter is given as follows. In Sec. II we introduce the model setup and explain the relevant parameters. The favored region and upper limit on the additional scalars are summarized in Sec. III. In Sec. IV we investigate the model prediction of top-associated processes. Summary and discussion will be given in Sec. V.
#3 They used the closed time path formalism [67] to evaluate the produced baryon number. #4 See, also Refs. [76][77][78][79] for the earlier works to probe ρ tc u in a flavor changing top decay. We consider a two Higgs doublet model (2HDM) where an additional scalar doublet is introduced to the SM. The general scalar potential of the model is given as Here, we work in the Higgs basis where only one doublet takes the VEV [82,83]: where G + and G 0 denotes the NG bosons. It is noted that alignment where the SM h lives in H 1 is considered to avoid the constraint from t → ch [84][85][86]. For simplicity, we further assume the CP-conserving scalar potential and then one can define the CP-even and -odd scalar mass eigenstates. The SM-like Higgs is h and H and A correspond to additional the CP-even and -odd neutral scalars. Masses differences among additional scalars are given as, It is noted that other potential couplings does not affect the following discussion. When the both doublets couple to all fermions, the Higgs bosons have flavor violating interactions in general. In this letter we take the bottom-up approach and introduce the interaction Lagrangian of the heavy scalars relevant to b → s and b → cτ ν, (4) where P L/R = (1 ∓ γ 5 )/2 and V are a chirality projection operator and Cabbibo-Kobayashi-Maskawa matrix [87,88], respectively. The neutral scalar interaction and the charged scalar interaction are related by the SU(2) L rotation. We assume that other Yukawa coupling to be small ( O(10 −2 )) for simplicity. For the more detailed phenomenological analysis with other Yukawa couplings, see Refs. [50,54,89]. We will also discuss this point in Sec. V.
For the later convenience we show the approximate formulae for the partial decay width,

III. SUMMARY OF THE AVAILABLE PARAMETER REGION
First we consider the charged Higgs contribution to flavor universal b → s . Since the coupling dependence is different among b → s (induced by the charm penguin ∝ |ρ tc u | 2 ) and the most constraining flavor process, B s − B s mixing (charged Higgs box ∝ |ρ tc u | 4 ), we can set an upper limit on the charged Higgs mass [37,38]. The relevant Hamiltonian for b → s in our model is given as where l = e, µ and τ . We note that contribution from Z penguin is small enough to neglect. We follow the prescription in Ref. [38] and use the following numerical formula, This should be compared with the recent global fit to b → s data of C l 9 (µ b ) = −0.95 ± 0.13 [90]. #6 In Fig. 1, we show 1 (2) σ favored region in green (yellow) on the m H + vs. ρ tc u plane. Since we also has the upper limit on the mass as m H + ≤ 400 GeV and the lower limit form #5 In this letter we neglect light fermion masses, though, one can trivially include the effect. #6 This fit does not include Bs → µµ and lepton flavor universality observables e.g. R K ( * ) . Since C 9 operator does not contribute to Bs → µµ, the result will be unchanged, though. The similar result is also reported in Ref. [13]. the LEP experiment [91], we focus on 100 GeV ≤ m H + ≤ 400 GeV. As mentioned above B s meson mixing puts the most stringent flavor constraint [92] which is shown in magenta.
In this mass region, di-jet resonance searches at the LHC are able to set the upper limit on ρ tc u [58]. We overlay the constraint from the (bottom flavored) di-jet searches in blue [93], purple [94] and cyan [95] where BR(H + → bc)=1 is assumed. It is noted that as we will see soon later, we need a hierarchy of |ρ tc u | |ρ τ τ e | for the simultaneous explanation. As a result H + → bc is the dominant decay mode in the minimal set up of Eq. (4) and hence the exclusion discussed above is unaffected. #7 We see that di-jet constraints touch the interesting parameter region. Run 2 full data would be possible to improve the constraint further.
We move onto the explanation of the R D ( * ) discrepancy. The relevant interaction Hamiltonian is given as The charged Higgs contribution including renormalization group running corrections [99][100][101][102], is approximately given as Adopting the analytic formulae of R D ( * ) in Ref. [58] #8 latest 1 σ explanation is realized with 0.68 |C τ S L (µ b )| 1.13. #9 By combining Eqs. (7,9), one can see that the simultaneous explanation requires the large magnitude difference in ρ tc u and ρ τ τ e . So far we focused on the charged Higgs phenomenology, however, neutral scalar mass spectrum is constrained with the LHC data and electroweak precision observables. The last summer the ATLAS collaboration reported the result of the G2HDM search in top-associated #7 The stau search constraint [96,97] on the charged Higgs is very weak due to BR(H + → bc) 1. See, Fig. 4 of Ref. [98]. #8 Those analytic formulae used in Ref. [58] are consistent with the recent result [103] within the uncertainty. #9 To fit the R D ( * ) data ρ tc * u ρ τ τ e needs to have a complex phase, however, this does not change the following discussion.  Fig. 2. In both diagrams, due to the different CP nature of H and A, the amplitude cancels in the mass degenerate limit. The destructive interference for the dominant s-channel approximately happens up to the width difference [73]. For the simultaneous explanation, ρ tc u needs to be as large as 0.7 (0.8) for m H + = 200 (250) GeV and hence the total width of Γ φ = 0.8 (3.5) GeV is predicted. This indicates that |λ 5 | ≤ O(10 −2 ) is necessary for the simultaneous explanation with m φ ≥ 200 GeV. To simplify the analysis and evade the constraint we set m A = m H in the following.
On the other hand, additional neutral scalars dominantly decay to τ τ for m φ ≤ m t . In that case, the electroweak pair production of neutral scalars results in multiple τ final state. Such a region is studied in Ref. [104] and even only with Run 1 data [105] we can exclude our scenario of m φ ≤ m t . Furthermore do not have an explicit new physics signal with the Run 2 full data [106,107] and hence the exclusion is robust.
Besides, electroweak precision observables are helpful to further constrain the mass spectrum. We consider S and T parameter constraint #12 [108,109] both excluding #10 To adopt the experimental data and extend the constraint down to m φ mt, detailed distribution data is necessary. Although this data is not available in Ref.
[80] and thus beyond scope of this letter. #11 In this analysis they only considered H to be present and ignore A for simplicity. If there is a mild mass difference of O(10) GeV, the constraint will be more stringent by a factor of √ 2. #12 Since the deviation in U parameter is suppressed in this model and including recent controversial CDF result [81]. More concretely we use S = 0.00 ± 0.07, T = 0.05 ± 0.06, (10) with the correlation of ρ = 0.92 [110] (denoted as 2021 fit) and with the correlation of ρ = 0.89 based on the global fit [111] (denoted as 2023 fit). In short section summary, for the simultaneous explanation we need to set m t ≤ m φ ≤ 200 GeV or O(1) GeV level mass degeneracy among neutral scalars.

IV. EXOTIC TOP PROCESSES
In order to fully probe the remaining mass window of m φ we propose another top-associated process, namely gc → c → tφ → tτ τ where the relevant diagram is shown in Fig. 4. #13 In the mass window, even with the hierarchical coupling structure, BR(φ → τ τ ) could be sizable due to the phase space suppression in φ → tc decay. The production cross section is calculated using Mad-Graph5 aMC@NLO [113] using NNPDF2.3 [114] at the leading order in the five flavor scheme with √ s = 13 TeV. Fig. 5 shows the cross section in pb as a function of m φ . The prediction of the 1 σ simultaneous explanation was obtained by fixing the charged Higgs mass m H + = 150 GeV (blue), 200 GeV (orange), 250 GeV (green) and m φ (black). It is observed that bands are overlapping and the cross section is as large as 30 fb∼10 pb for the mass window. #14 A heavier charged scalar predicts the larger signal rate since it requires larger couplings. and the uncertainty in S and T parameters will be reduced considerably, we set U = 0. #13 It would be worthwhile to mention that tt inclusive cross section measurement still has an uncertainty of 70 pb [112] and does not exclude the scenario with gc → c → tφ → ttc channel. #14 For the numerical analysis we include φ → H ± W ∓ if the phase space is available. Estimating the size of the electroweak SM back ground (BG) is not difficult even for our mass range. For instance, tZq and thq production contribute to t + τ τ + q final state with cross section of 50 fb [115] and 5 fb [116] where τ τ comes from Z and h decay for each. Therefore the contribution from those processes are expected to be moderate. On the other hand, it is not easy to estimate the precise amount of the miss-tag associated BG e.g. from tW − q → tτ ν +/ j and tt → tW − j → tτ ν +/ j where slashed final state will be miss-tagged as a hadronically decaying τ (τ h ). For the precise determination we need a considerable help from the experimental side and thus investigating the sensitivity of this channel is beyond the scope of this letter. #15 Actually Ref. [117] searched for the thq production with h → τ τ with Run 2 full data. They set the upper limit of µ = 8.1 +8.2 −7.5 where µ denotes a signal strength. This approximately leads to the upper limit on σ(thq → tτ τ q) 100 fb for m τ τ = 125 GeV. Since the invariant mass of our signal is larger, the corresponding SMBG would be smaller and thus we can expect the better sensitivity.

V. SUMMARY AND DISCUSSION
Recently the charged Higgs solution to B anomalies became more interesting than ever. The charged Higgs need to interact with left-handed bottom quark and thus can be a part of an additional doublet. Hence a two Higgs doublet model is a minimal model and there are also two additional neutral scalars. The Yukawa interaction of those scalars are related by SU(2) L rotation and the simultaneous explanation predicts distinctive signal at the LHC. The theoretical proposals to probe the solution via charged Higgs mediated processes was made last year, however, the crucial process has not been tested experimentally yet. Although, in the meantime, the ATLAS #15 The charge asymmetry of the top quark would help to improve the sensitivity since the SM single top has the production asymmetry, while our signal does not have this feature. experiment reported the game changing constraint on the neutral scalars. In this letter we reinterpret the ATLAS constraint and obtained the condition for the mass spectrum of the additional neutral scalars: O(1) GeV mass degeneracy among H and A or m t ≤ m φ ≤ 200 GeV where φ denotes H and A. We also pointed out that the signal cross section of gc → tφ → tτ τ could be as large as 10 fb∼10 pb for the mass window. Imposing a U(1) Peccei-Quinn symmetry [118], {H 1 , H 2 } → {H 1 , H 2 e iα } can prohibit λ 5 and realize the mass degeneracy of additional neutral scalars [119]. Although this symmetry should be broken since we also need Yukawa couplings, ρ tc u and ρ τ τ e and therefore the more complicated setup is necessary [120][121][122][123].
In general, other couplings e.g. di-bottom quark coupling, namely ρ bb d would be non-negligible. For instance, one would think that O(10 −2 ) of ρ bb d could reduce the branching ratio of φ → τ τ thanks to the color factor and revive the scenario with m φ ≤ m t . #16 Although this is difficult since the ATLAS collaboration searched additional particles in flavor changing top decays set O(10 −4 ) upper bound on BR(t → qX)×BR(X → bb) very recently [86]. Therefore an additional coupling to bottom quarks, does not save the scenario. Since c → b miss tagging rate, c→b is about 15 ∼ 20 % [124], even if neutral scalars decay into charm quarks, the scenario is difficult to survive the constraint. On the other hand, ρ bb d would be able to reduce signal rate of gc → tτ τ process.
It would to worthwhile to emphasize that the ATLAS bound [80] does not necessarily kill the solo R D ( * ) solution even without mass degeneracy. This is because that the contribution to C S L is proportional to the coupling product of ρ tc * u ρ τ τ e (see, Eq. (9)) and hence the larger ρ τ τ e allows the smaller ρ tc u . If we want to avoid the ATLAS bound on ρ tc u by setting m A , m H ≤ 200 GeV instead, electroweak precision parameters at 2 σ give the upper limit on the charged Higgs mass as m H + ≤ 270 GeV (290 GeV) for Eq. (10) (Eq. (11)). In this case, t + τ τ would provide a key test since BR(φ → τ τ ) will be amplified compared to the scenario for the simultaneous explanation.