Test of the $R(D^{(*)})$ anomaly at the LHC

There are discrepancies between the experimental results and the Standard Model predictions, in the lepton flavor universality of the semileptonic $B$ decays: $B \to D^{(*)} \ell \nu$. As the new physics interpretations, new charged vector and charged scalar fields, that dominantly couple to the second and third generations, have been widely discussed. In this paper, we study the signals of the new particles at the LHC, and test the interpretations via the direct search for the new resonances. In particular, we see that the $\tau \nu$ resonance search at the LHC has already covered most of the parameter regions favored by the Belle and BaBar experiments. We find that the bound is already stronger than the one from the $B_c$ decay depending on the mass of charged scalar.


Introduction
In 2012, the BaBar collaboration has reported that there are large discrepancies in the lepton flavor universalities (LFUs) of the semileptonic B decays: B → D ν and B → D * ν ( = e, µ, τ ). The observables to measure the LFUs are defined as R(D ( * ) ) = Br(B → D ( * ) τ ν)/Br(B → D ( * ) l ν) (l = e, µ), (1) and the experimental results are R(D) = 0.440 ± 0.072 and R(D * ) = 0.332 ± 0.030 [1,2]. They are largely deviated from the Standard Model (SM) predictions: R(D) SM = 0.299 ± 0.003 and R(D * ) SM = 0.258 ± 0.005 [3]. * The B decays associated with the light leptons are measured with good accuracy, so that the branching ratios of B → D ( * ) τ ν are larger than the SM predictions. Interestingly, the Belle collaboration has also reported the excesses in R(D ( * ) ) [11][12][13], although the discrepancies are milder than the BaBar results. Thus, it is expected that those excesses are the new physics signals and there are new particles that couple to the SM fermions flavor-dependently. We note that the LHCb collaboration has also investigated only R(D * ) and the latest result is consistent with the SM prediction at the 1 σ level [14,15]. Recently, the Belle experiment also reported the data of the D * polarization in the B → D * τ ν process, and the result is slightly deviated from the SM prediction at 1.5 σ level [16]. Motivated by the excesses, several new physics interpretations have been proposed. One simple way to violate the LFU in the B decay is to introduce a field that couples to τ lepton. The field needs to couple to the heavy quarks, bottom and charm quarks, as well. One good candidate for such a field is a charged scalar, H ± , that has Yukawa couplings with the heavy quarks and heavy leptons [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. The Yukawa couplings are, in general, flavor-dependent, so that we can assume that the couplings with bottom, charm and τ leptons are relatively large, compared to the other elements. Then, the charged scalar exchange at the tree-level induces the violation of the LFU. This simple scenario has been proposed just after the announcement of the BaBar result, and the way to prove the new physics directly/indirectly has been widely discussed.
Instead of the charged scalar, we can discuss a charged vector, W ± , that dominantly couples to the second and third generations [33][34][35][36][37][38][39]. In order to introduce such a vector field, additional gauge symmetry is required and the SM gauge symmetry may be extended. In addition, a non-trivial setup would be necessary to make the W ± couplings flavor-dependent. For instance, we can discuss a gauged flavor symmetry or we can expect that some heavy fermions effectively induce the flavorful couplings according to the mass mixing with the SM fermions.
In this paper, we focus on those two new physics interpretations and discuss the consistency with the direct search for the new phenomena at the LHC in the each setup. In particular, it is recently claimed that the charged scalar explanation is in tension with the B c decay [46][47][48]. We study the τ ν resonance search at the LHC and see that the bound is stronger than the one from the B c decay.
We summarize the each explanation in Sec. 2 2 The explanation of the R(D ( * ) ) anomaly There are large discrepancies between the experimental results and the SM predictions in the LFUs of the semileptonic B decays: B → D ( * ) ν. In the SM, the processes are given by the tree-level diagrams. Then, relatively large new interaction is required to compensate the SM contribution. If there is a heavy charged particle that couples to quarks and leptons flavor-dependently, the following operators could be generated by the heavy particle exchange: here C V SM expresses a SM contribution generated by W boson, with The two terms in the fist line can be generated by the W exchange. † The last two terms can be from the H ± exchange. In this paper, we focus on these two scenarios with the SU(3) c -singlet mediators. In the following subsections, we review the each new physics scenario and estimate the size of coefficient required by the excesses.

Charged scalar case
To begin with, we discuss a possibility that charged scalar, H ± , resides behind the R(D ( * ) ) anomalies. The charged scalar can be introduced by adding extra Higgs SU(2) L doublets. The Yukawa couplings between H ± and the SM fermions depend on the setup, but in general the scalar couples to all of the SM fermions. Most of the Yukawa couplings are strongly constrained by the flavor physics, so that we have to assume a specific alignment of the couplings. Assuming such a specific parameter choice, we can focus on the b → c transition induced by the Yukawa coupling of charged scalar, i.e., As mentioned above, H ± is originated from SU (2) L -doublet scalars, and the neutral components of the doublet also appear as physical fields, after the electroweak symmetry breaking. The masses of the neutral scalars are expected to be around the charged scalar mass (M H ). The Yukawa couplings involving the neutral ones are also evaluated as Y L and Y R , approximately. When one neutral scalar is denoted as H 0 , the couplings between H 0 and the down-type (up-type) quarks are described as Then, we find that Y R is also strongly constrained by the B s -B s mixing, taking into account the neutral Higgs exchange at the tree level. As a result Y R is not useful to improve R(D ( * ) ) [26]. We assume |Y R | |Y L | in our analysis below. Integrating out H ± , we obtain where M H is the charged scalar mass. If the charged scalar mass is less than a few TeV, these operators largely contribute to the rare B decay. The numerical descriptions of R(D) and R(D * ) are given by [19] where C S I (I = L, R) is a normalized coefficient given as C S The required value to achieve the excess of the world average within 1 σ is estimated as where, we select the phase of Y L Y * τ to minimize the χ 2 . The explanation of R(D * ) is, however, constrained indirectly by the B c decay [46,47]. The B c meson decay is easily enhanced by the scalar-type operator. The leptonic decay, B c → τ ν, is still not observed, but the total decay width and the hadronic decay have been measured and the results are consistent with the SM predictions, although the observables suffer from the large theoretical uncertainty. The authors of Refs. [46,47] derive the upper bound on the leptonic decay, taking into account the uncertainty, and obtain the upper bound R(D * ) 0.27 in the charged scalar scenario. The LHCb collaboration has recently reported the consistent R(D * ) results with the SM prediction. The enhancement of R(D) while keeping the R(D * ) consistent to the SM prediction would be only achieved by tuning

W case
We can discuss the possibility that the coefficients in Eq.
where, I denotes the chirality: I = L, R. The couplings g I and g I τ depend on the detail of the setup, and the other couplings involving light SM fermions may arise at the low energy. Assuming the third-generation couplings are dominant, we expect the following operators induced: where M W I denotes the W I mass. R(D) and R(D * ) are numerically evaluated as [38] R where In order to accommodate the R(D ( * ) ) excesses within 1 σ of the world average, the required value of the each coefficient is estimated as 3 Test of the new physics at the LHC In this section, we study the signal of the each scenario at the LHC based on the above discussion. In our models, the charged resonances (V + = W + I , H + ) are produced in association with the third-generation quark and decay to τ ν and bc as follows: The searches for heavy τ ν resonances have been performed at the LHC experiments [40][41][42][43], and severely constrain various models. ‡ We found the analysis reported by the CMS collaboration using the data at the LHC Run II with 35.9 fb −1 [43] sets the most stringent bound on our models, where they focus on the W heavier than 400 GeV with the universal couplings to quarks of all generations. Every release of the new data would improve our bound.
Since the heavy resonances are only couples to the third generation in our model and the spin structures of the H ± and the W are different, the efficiency and the acceptance of the selection cut should be estimated by the simulation. We calculate the acceptances for the case of H ± and W using MadGraph5 [49] and PYTHIA8 [50]. The generated events are interfaced to DELPHES3 [51] for the fast detector simulation.
We follow the event selection cuts exploited in [43] as follows: • exactly one τ -tagged jet, satisfying p T,τ ≥ 80GeV and |η τ | ≤ 2.4, ‡ Our model focus on the heavy resonance decaying into τ ν and it is different from the search for leptoquark using b + τ ν in the final state discussed in Ref. [44]. See also Refs. [26,45].
After the above event selection cuts, we plot the m T distribution and performed the binned log-likelihood analysis using the background m T distribution [43]. We show the resulting 95 % CL upper bound on the signal cross section times its branching ratio to τ ν mode as a function of the resonance mass m V for each model in Figure 1. The difference of the spin structure provides different upper bounds, for charged scalar (red solid), W L (blue solid), and W R (green solid). The constraint on the charged scalar case is more stringent than the other cases because of the harder m T distribution. We also overlay the expected signal cross sections in the same plot for the three cases, in the red hatched region (H ± ), in the blue hatched region (W L ) and in the green hatched region (W R ), assuming the couplings are compatible to accommodate the R(D ( * ) ) observation in 1σ level. In our models, we can parametrize the cross section with (M, g, g τ ), where g is the c-b-V coupling and can be taken real without loss of generality; e.g., (M, g, g τ ) = (M H , Y L , Y τ ), and (M, g, g τ ) = (M W I , g I , g I τ ), respectively. The cross section for the above process is given as follows: Here, we define the variables,ḡ 2 = |g g τ |, which is related to the R(D ( * ) ) observation, and r = |g τ /g|. We find that the signal cross section is maximized at r = √ 3 when we fix g. We also impose the perturbativity of the couplings, i.e. |g|, |g τ | ≤ 1, and r would be constrained as well. We assumeḡ 2 = a V M 2 V to accommodate the R(D ( * ) ) observation at 1 σ, that is a H ± = 1.36 × 10 −6 GeV −2 , and a W L = 1.07 × 10 −7 GeV −2 . Compared with the required a W L , a H ± needs to be large to accomodate the excess. This is stem from small coefficients in Eq. (6). Then we expect the more stringent bound can be obtained for a charged scalar scenario. Since we imposeḡ 2 ≤ 1 due to the perturbativity, M V exhibits an upper bound 850 GeV for H + and 3 TeV for W L . The upper boundary of the region is given by r . The lower boundary of the region is given by r =ḡ 2 . Note that the lower boundary depends on the upper bound of the allowed couplings.
In the following, we discuss the current status in more detail and the compatibility with the R(D ( * ) ) enhancements for each case.

Charged scalar case
First, we investigate the charged scalar signal at the LHC. As we discuss in Sec. 2.1, the R(D ( * ) ) anomaly requires the sizable interaction between the heavy quarks and the τ lepton. In our study of the charged scalar case, there are four free parameters: The explanation of the R(D ( * ) ) anomaly fixes one combination of them, as shown in Eq. (7). Tuning the phase of Y τ , the bound from the B c decay can be evaded, although the enhancement of R(D * ) is suppressed. Note that the magnitudes of the Yukawa couplings are not so large and the total decay width is small enough in our study. The production cross section of H ± only depends on Y L , while |Y τ | contributes to the branching ratio of H ± only. The production cross section times the branching ratio is given by substituting (M, g, g τ ) = (M H , Y L , Y τ ) in Eq. (13). The exclusion line and the predicted region of the charged scalar case are shown in Fig. 1. As we see the red line and the red hatched region in Fig. 1, the region to accommodate the experimental results on R(D ( * ) ) within the 1σ level is already excluded by this τ ν resonance search. The light charged scalar region, e.g. M H 400 GeV, could achive the experimental results within the 2σ level. Fig. 2 shows our predictions on the R(D) and R(D * ) plane for the fixed charged scalar mass, 400 GeV, 500 GeV, 750 GeV and 1000 GeV. The gray region is out of our prediction and the region inside of the black lines is realized by taking Y L and Y τ appropriately. Note that the blue, green and red ellipses correspond to the 1 σ regions reported by the Belle, BaBar and the HFLAV collaborations. The SM prediction for R(D) and R(D * ) is also marked with the asterisk. In Fig. 2, we also draw the constraints from the B c decay by the thick magenta lines. Above the magenta line, the leptonic decay of B c is larger than 30 % and 10 %, so that the region above the line is excluded indirectly. The cyan dashed lines correspond to the predictions by taking |Y L | = π. § § We draw them for an illustration although such a relatively large coupling does not respect the narrow The bound from the heavy resonance search at the LHC is our main topic in this paper. We study the excluded parameter region on the R(D) and R(D * ) plane based on Fig. 1. As the signal cross section depends both on Y L and Y τ , we have one additional degree of freedom than the product |Y L Y τ |. The charged higgs scenario is totally excluded by the τ ν resonance search in the light purple region while it would be allowed by tuning the ratio Y τ /Y L in the light blue region. Interestingly, we find that most of parameter region is already excluded by the LHC searches. Only for the light charged scalar, m H 450 GeV, we can find a parameter region consistent at 1σ by allowing a large |Y L | = π. This is because the LHC τ ν resonance constraints set more stringent constraints for the heavier resonance. We also stress that the bound by the search at the LHC is stronger than the one by the B c decay for the considered range m H ≥ 400 GeV.

W case
Next, we study the W scenario. In the Eq. (8), there are two types of the gauge couplings with the fermions: one is the coupling with right-handed fermions and the other is the one with left-handed fermions. It depends on the charge assignment of the extra nonabelian gauge symmetry. If the extra SU(2) symmetry is assigned to the left-handed fields like the SM SU(2) L , W L only couples to the left-handed fields as well. In a model with W R , SU(2) R symmetry would be assigned to the right-handed fermions. In those cases, W I originated from SU(2) I may mix with W from SU(2) L,SM . The mixing is strongly constrained by the electroweak precision observables, so the mixing should be very tiny. We mainly discuss the case with W L in the following for the demonstration. The one with W R can be treated in the same way.
In the same way to the charged scalar case, we discuss the τ ν resonance originated from the on-shell W L . Our relevant parameters are M W , g L , Re (g L τ ) , Im (g L τ ) .
In the same way as the H ± case, the cross section is given by substituting (M, g, g τ ) = (M W , g L , g L τ ) in Eq. (13). The exclusion line and the predicted region of the W L case are shown in Fig. 1. As we see the blue line and the blue hatched region in Fig. 1, the constraint is not so tight. The exclusion line and the prediction of the W R case are also shown as the green line and the green hatched region. As far as the gauge coupling is less than one, the W R case is almost excluded unless m W R is lighter than 420 GeV, since the gauge coupling coupling required to accommodate R(D ( * ) ) anomaly is relatively large compared to the one in the W L case. Note that there is no bound from the B c decay, since the operator is given by the vector-vector coupling in this scenario.
In HFLAV collaborations are depicted by the same color as in Fig. 2. The SM prediction for R(D) and R(D * ) is marked with the asterisk.
We show the possibly surviving region taking the allowed range of the couplings |g I | ≤ 1 into account, in the magenta arrows (I = L) and in the green arrows (I = R). On the magenta thick line, the W L scenario is surviving by taking g L and g Lτ appropriately within |g L | ≤ 1. Parameter region on the magenta dashed line are completely excluded when we require |g L | ≤ 1, while it can be survived when we allow |g L | ≤ π. The green arrows shows the corresponding region for W R .

Summary
The discrepancies of R(D) and R(D * ) may be the evidence of new physics behind the SM. The excesses suggest that there are extra fields that couple to quarks and leptons flavor-dependently and the size of the coupling is not so small compared to the W boson coupling in the SM. Motivated by this issue, many new physics interpretations have been proposed, and we find that some of good candidates for the extra fields are charged scalar and charged vector fields. Those extra charged fields are predicted by many new physics models beyond the SM. The charged scalar is, for instance, predicted by many extended SMs with extra SU(2) L -doublet scalar fields. Extending the SM gauge group is prophetic of some extra massive gauge bosons. The charged vector can be originated from some extra non-abelian gauge symmetry.
Those good candidates have been studied in flavor physics and collider physics. The simple setups, where the heavy charged scalars couple to the light quarks and leptons, have been already excluded, so that some specific structures of the extended SMs are required to achieve the explanations. Such a specific setup makes it difficult to test indirectly and directly by the experiments.
In this paper, we investigate one observable that does not depend on the detail of the setup, that is, the τ ν resonance originated from the charged particle. We simply consider each minimal setup in the scalar case and in the vector case, and discuss the consistency between the explanation of R(D ( * ) ) and the latest experimental result at the LHC. Interestingly, we found most of the parameter region required by the excesses has been already covered by the resonance search. If the excesses come from the new interactions, the heavy resonances would be relatively light as shown in Fig. 2 and Fig. 3.
Our analysis assumes that the heavy resonances only decay into bc and/or τ ν. If they decay into the other SM fermions or other particles, the bounds obtained in this paper can be relaxed. Such a case, however, faces stronger bounds from flavor physics, since the couplings with light quarks and leptons are large. Therefore, our results would be applicable as long as the decays to bc and/or τ ν are dominant.
It may be interesting to study the bound from the bc resonance search. In the final state, one or two b quarks appear in association with one c quark. Currently, the bound from the di-jet resonance is weaker than that of the τ ν resonance search by 1 or 2 order of magnitude [53,54]. If we can identify the b/c-flavor jet efficiently, we can set stronger exclusion lines [55,56].
Finally, we comment on the polarization of D * in the B → D * τ ν process. Recently, the Belle collaboration has reported the result on the polarization with high accuracy [16]. This measurement may be useful to test the new physics scenarios motivated by the R(D ( * ) ) anomaly. In fact, the possible observables concerned with this problem are recently discussed [58][59][60]. The more general analysis for the D * polarization and R(D ( * ) ), based on the latest experimental result, will be given near future [61]. Figure 2: R(D) vs. R(D * ) with the constraints from the B c decay (magenta lines) and from the charged heavy resonance search at the LHC. The charged scalar masses are fixed at 400 GeV, 500 GeV, 750 GeV and 1000 GeV, as denoted on the each panel. The gray region is out of our prediction and the region inside of the black lines is realized by taking appropriate Y L and Y τ . Note that the blue, green and red ellipses correspond to the 1 σ regions reported by the Belle, BaBar and the HFLAV collaborations. The SM prediction for R(D) and R(D * ) is marked with the asterisk. The cyan dashed lines correspond to the predictions by taking |Y L | = π. In the light purple region, our model is totally excluded by the τ ν resonance search. In the light blue region, our model can be allowed by tuning Y L and Y τ .  Fig. 2. The SM prediction for R(D) and R(D * ) is marked with the asterisk. On the magenta thick arrows, the W L scenario is excluded depending on g L and g Lτ , assuming |g L | ≤ 1. The points on the magenta dashed arrows are excluded as far as |g L | ≤ 1 is required but can be survived allowing |g L | ≤ π. The green arrows show the corresponding cases for the W R .