New Physics effect on $B_c \to J/\psi \tau\bar\nu$ in relation to the $R_{D^{(*)}}$ anomaly

We study possible new physics (NP) effects on $B_c \to J/\psi \tau\bar\nu$, which has been recently measured at LHCb as the ratio of $R_{J/\psi} = \mathcal B(B_c \to J/\psi \tau\bar\nu)/\mathcal B(B_c \to J/\psi \mu\bar\nu)$. Combining it with the long-standing $R_{D^{(*)}}$ measurements, in which the discrepancy with the prediction of the standard model is present, we find possible solutions to the anomaly by several NP types. Then, we see that adding the $R_{J/\psi}$ measurement does not improve NP fit to data, but the NP scenarios still give better $\chi^2$ than the SM. We also investigate indirect NP constraints from the lifetime of $B_c$ and NP predictions on the $\tau$ longitudinal polarization in $\bar B \to D^* \tau\bar\nu$.

Some leptoquark (LQ) models contribute toB → D ( * ) τν with scalar-tensor operators so that C S2 ±7.8C T at the m b scale 2 . Then, they also explain the R D ( * ) anomaly. For a dedicated study, see Ref. [8].
In Ref. [57], this anomaly has been investigated by looking at the lifetime of B c meson along with the decay B c → τν. As C X = 0 (for X = T ) also contributes to B c → τν, it is necessary that the contribution does not exceed the fraction of the total decay width of B c , which has been experimentally measured and theoretically calculated. Indeed, this could allow us to exclude a large contribution from C Si = 0. In Ref. [58], a stronger limit on the scalar contribution has been suggested with using LEP1 data for B c → τν.
In September 2017, the LHCb collaboration reported a new measurement regarding b → cτ ν in B c . To be specific, the ratio has been obtained with dataset of run 1 (3 fb −1 ) [59, 60]. Thus, this new measurement enables us to develop explanations for the anomaly with the above NP scenarios, which will be shown in this paper. We will also revisit the constraints with use of the lifetime of B c and put some predictions on the τ longitudinal polarization.
This letter is then organized as follows. In Sec. 2, we obtain a formula for the decay rate of B c → J/ψτν in the presence of the NP operators. A description of form factors for the B c → J/ψ transition is also given. In Sec. 3, we proceed to numerical analysis and obtain possible solutions to the R D , R D * , and R J/ψ measurements by the NP scenarios. We also investigate NP effect on the lifetime of B c , associated with B c → τν, and the τ longitudinal polarization inB → D * τν. The Sec. 4 is devoted to summary.

DESCRIPTION OF HADRONIC AMPLITUDE AND FORM FACTORS
The hadronic transition of B c → J/ψ can be written in analogy with that ofB → D * . Namely, we can obtain the formula for the decay rate of B c → J/ψτν as follows [8], where Hs are hadronic helicity amplitudes given by and The functions V c , A c i , and T c i are form factors (FFs) for the B c → J/ψ transition whose definitions are given in Appendix A. The scalar hadronic amplitude is obtained as in (8) using the quark-level equation of motion.
The FFs for the vector and axial-vector currents have been investigated in Ref. [61] with the use of perturbative FIG. 1: Correlation between RD * and R J/ψ in the presence of one NP operator, V1, V2, S2, or T (left) and of LQ specific operators with CS 2 = ±7.8CT (right). The red dot shows the SM predictions with the error bar for R J/ψ . Note that S1 and S2 have the same contribution.

NUMERICAL ANALYSIS
For numerical evaluation on R J/ψ , we take the following values for input; m Bc = 6.275 GeV, m J/ψ = 3.096 GeV, m τ = 1.777 GeV, m b + m c = 6.2 GeV, and m b − m c = 3.45 GeV [64]. Then, the SM predicts where the uncertainty comes from the inputs of V c (0), A c 0 (0), A c 1 (0), and A c 2 (0). The result is consistent with Refs. [65,66]. This is compared with (3) and thus, one finds that there exists a 1.7σ deviation from the SM, i.e., [χ 2 ] SM J/ψ 2.9. Note that the R J/ψ measurement still include a large uncertainty. Combined with the R D and R D * measurements [60,67], it turns out [χ 2 ] SM J/ψ+D+D *

22.
In fig. 1, we show correlation between R D * and R J/ψ in the presence of one NP operator (V 1 , V 2 , S 2 , or T ) and LQ specific operators (LQ ± : C S2 = ±7.8C T ), where the NP type is denoted in the plot and the dashed lines show the  present experimental results at 1σ. We see that single NP operators and LQs cannot simultaneously accommodate the present experimental results of R D * and R J/ψ within 1σ. Remind that C V1 ∼ 0.15 explains the present central values of the R D and R D * measurements, (for example, see Ref. [9].) If this is the case, one expects R J/ψ ∼ 0.37. This is one possible way to probe and/or distinguish NP in the b → cτ ν process. As briefly explained in Sec. 1, the lifetime of B c is a significant tool to constrain NP in b → cτν, which has been pointed out in Ref. [57]. The idea was as follows. The B c lifetime has been measured as τ exp Bc = (0.507 ± 0.008) ps [64], whereas it has been theoretically calculated as τ th Bc = (0.52 +0.18 −0.12 ) ps [57] within the SM by using an operator product expansion [68]. As for the latter, the branching fractions have also been obtained and then pure/semi-tauonic modes could have 5 % for the central value (τ th;cent Bc ) and 30 % for the 1σ upper limit (τ th;+1σ Bc ), comparing it with τ exp;cent Bc . Thus, when the branching fraction of B c → τν becomes large as a consequence of explaining the R D ( * ) anomaly, it would be constrained. Since B c → τν is sensitive to the scalar operators S 1,2 , the limit from the B c lifetime disfavors the possible solution to the R D ( * ) anomaly by the S 1,2 operator.
This approach can be further developed by including B c → J/ψτν. For simplicity, we take the difference δΓ tot = 1/τ exp;cent Bc − 1/τ th;+1σ Bc = 0.544ps −1 and demand that the tauonic decay rates do not exceed the difference, namely, δΓ tot > Γ(B c → τν) + Γ(B c → J/ψτν). This condition would give an additional constraint on the NP effect in the b → cτ ν process. Recently, Ref. [58] pointed out that LEP1 data taken at the Z peak can give a stronger constraint on the branching fraction of B c → τν. The conservative limit is then given as 10 %. We will also take this limit in the numerical study 3 . The branching fraction of B c → τν is written as For the analysis, we take f Bc = 434 MeV and |V cb | = 4.09 × 10 −2 . We also obtain the favored regions on C X , derived from the R D , R D * , and R J/ψ measurements by simply evaluating χ 2 .
In fig. 2, we show NP bounds in the complex plane of C X for the V 1 , V 2 , S 2 , T , LQ + , and LQ − scenarios. Favored regions from the R D , R D * , and R J/ψ measurements, allowed at 95% confidence level (CL), are shown in red color. On the other hand, regions in gray with black solid boundaries are disfavored by the limit from the B c lifetime, obtained by the aforementioned method. The black dashed curves are then the limit obtained by the condition B(B c → τν) 10%. We see that the S 2 solution is totally excluded by the B c lifetime, which is consistent with Ref. [57], even though the S 2 solution has the better fit result to the R D , R D * , and R J/ψ measurements ([χ 2 ] S2;min J/ψ+D+D * ∼ 3) than the SM ([χ 2 ] SM J/ψ+D+D * 22). One also finds that the constraint from the B c lifetime is significant for the LQ scenarios. When we consider the limit on B(B c → τν) from the LEP1 data, the LQ + solution to the R D ( * ) anomaly is severely constrained. The minimum value of χ 2 for each NP scenario is obtained as exhibited in the plot. The V 1 , V 2 , T , and LQ − scenarios have better fit results to the anomaly than the SM, and are consistent with the B c lifetime and the B(B c → τν) limit.
Additional measurements, relevant for b → cτ ν, would improve the investigation to probe NP. Indeed, the τ , where Γ ± D * is the partial decay rate for the tau helicity to be ±1/2 -has been measured by the Belle experiment [5] and thus it would give an additional hint for the NP effect in b → cτ ν. When we take the best fitted value of C X , obtained from the above study, we can predict P τ D * for each NP scenario. The result is shown in Table I. We see that the V 1,2 and LQ ± scenarios predict consistent values with the present Belle result. The prediction for the T scenario deviates at 1σ from data, which has been pointed out in Ref. [57]. Since the present data includes a large uncertainty, these results are not conclusive yet. This study will be improved in the upcoming Belle II experiment [69]. FIG. 2: Favored regions from the RD, RD * , and R J/ψ measurements at 95% CL (red) and disfavored regions by the limit from the Bc lifetime (gray with solid boundaries) and from the Bc → τν branching ratio (dashed curves), in the complex plane of CX for the V1, V2, S1, S2, T , LQ + , and LQ − scenarios. A minimal value of χ 2 for each NP scenario is also shown in the legend.

SUMMARY
We have studied possible NP effects on B c → J/ψτν in terms of the effective field theory. We provided analytic formula for the decay rate of B c → J/ψτν in the presence of all type of NP operators. Given the recently reported data of R J/ψ = B(B c → J/ψτν)/B(B c → J/ψµν) together with the present data of R D ( * ) , the discrepancy with the SM prediction reaches ∼ 4.5σ. Then it has turned out that the NP scenarios with the V 1 , V 2 , S 2 , T , LQ + , and LQ − operators have better fit to the R J/ψ+D+D * anomaly, although a consistent explanation within 1σ is not available.
On the other hand, the lifetime of B c , considering the NP effect in B c → τν and B c → J/ψτν, gives the useful constraint so that the S 2 solution to the R J/ψ+D+D * anomaly is disfavored. The LQ ± solutions are still consistent with, but close to, the limit of the B c lifetime. When we consider the limit on the branching fraction of B c → τν obtained from the LEP1 data, given as 10%, the LQ + solution is severely constrained. The V 1 , V 2 , and T solutions are still free from the limits of the B c lifetime and the B(B c → τν).
We have also shown the predictions on the τ longitudinal polarization obtained by taking the best fit to the R J/ψ+D+D * measurements for the NP scenarios. This is compared with the Belle result and then the predictions for V 1 , V 2 , LQ + , and LQ − are still consistent due to the large experimental uncertainty, whereas that for T stands at 1σ. We expect that these studies will be improved at the Belle II experiment and by using run 2 data of LHCb. In particular, precise measurement of the three ratios (R J/ψ , R D , R D * ) would enable us to test the hypothesis of single operator dominance for NP. Further additional measurements regarding the b → cτ ν process, such as q 2 distributions ofB → D ( * ) τν [9], definitely give us significant hint for the NP solutions.
Note added: (19) in the published version was accidentally replaced with the incorrect form. To be precise,