R ð J = ψ Þ and B − c → J = ψ l − ¯ ν l Lepton Flavor Universality Violating Observables from Lattice QCD

We use our lattice QCD computation of the B c → J= ψ form factors to determine the differential decay rate for the semitauonic decay channel and construct the ratio of branching fractions R ð J= ψ Þ ¼ B ð B − c → J= ψτ − ¯ ν τ Þ = B ð B − c → J= ψμ − ¯ ν μ Þ . We find R ð J= ψ Þ ¼ 0 . 2582 ð 38 Þ and give an error budget. We also extend the relevant angular observables, which were recently suggested for the study of lepton flavor universality violating effects in B → D (cid:2) l ν , to B c → J= ψ l ν and make predictions for their values under different new physics scenarios.

We use our lattice QCD computation of the B c → J=ψ form factors to determine the differential decay rate for the semitauonic decay channel and construct the ratio of branching fractions RðJ=ψÞ ¼ BðB − c → J=ψτ −ν τ Þ=BðB − c → J=ψμ −ν μ Þ. We find RðJ=ψÞ ¼ 0.2582ð38Þ and give an error budget. We also extend the relevant angular observables, which were recently suggested for the study of lepton flavor universality violating effects in B → D Ã lν, to B c → J=ψlν and make predictions for their values under different new physics scenarios. DOI: 10.1103/PhysRevLett.125.222003 Introduction.-B-meson exclusive semileptonic decay processes are powerful tests of the Standard model (SM) in cases where both experimental and theoretical uncertainties can be brought under control. They allow for the determination of elements of the Cabibbo-Kobayashi-Maskawa matrix and tests of its unitarity but are also more detailed probes for new physics scenarios. An exciting possibility is that of nonuniversality of the couplings of charged leptons in electroweak interactions, hinted at in measurements of the ratio, R, of the branching fraction to τ in the final state to that of μ=e for B → D ðÃÞ [1][2][3][4][5][6][7][8][9][10], The combination of experimental results for RðDÞ and RðD Ã Þ is in tension with the SM at 3.1σ [11,12]. This includes a recent result from Belle [10] using semileptonic tagging, which by itself agrees with the SM within 1.6σ.
Most of the pull comes from RðD Ã Þ for which the SM results being used [13][14][15] rely on arguments from sum rules and heavy quark effective theory because lattice QCD results, which provide the best ab initio determination of the form factors, are as yet only available for the A 1 form factor at a zero recoil [16,17]. Extending the lattice QCD calculation to cover more of the q 2 range is underway [18,19]. This motivates a study of R for other b → c semileptonic decay modes, and LHCb recently gave the first results for RðJ=ψÞ from a B c → J=ψ decay [20]. They found RðJ=ψÞ ¼ 0.71 AE 0.17 stat AE 0.18 sys and compared it to SM results from a variety of model calculations. Since it is difficult to quantify uncertainties from individual models, the spread of results between models must be taken as a measure of this. LHCb quote a range of SM values from 0.25 to 0.28 from [21][22][23][24], giving a 10% spread. Since this is the same level as the discrepancy seen in RðD Ã Þ currently, it is unlikely that this accuracy would be good enough to distinguish experiment (given improved uncertainties there) and the SM. More recent results see a similar spread [25][26][27]. In addition, the dominant systematic error for the experiment comes from the uncertainty in input form factors for B c → J=ψ from these models.
In [28], we have given results for the B c → J=ψ form factors from lattice QCD for the first time. We work on gluon field configurations that include u, d, s, and c quarks in the sea and use a discretization of the Dirac equation, which was developed by the HPQCD Collaboration to have particularly small discretization errors [29]. This makes it suitable for handling heavy b and c quarks where discretization errors are relatively large. It also enables us to use a relativistic formalism in which the quark currents that couple to the W boson can be normalized in a fully nonperturbative way. We use HPQCD's "heavy-HISQ" technique, which has been successful in determining the decay constants of heavy-light mesons [30][31][32][33] and is now being applied to B semileptonic form factors [34,35], where it enables the full q 2 range of the decay to be covered. Further details can be found in [28], where we give the form factors and derive the total rate and branching fraction for the decay with μν μ in the final state. Here, we focus on the τν τ mode and determine RðJ=ψÞ in the SM with quantified uncertainties. We also give values for asymmetries, which show sensitivity to new physics.
Theoretical background.-We give below the full differential rate [36] for B − c → J=ψl −ν , where l is a lepton with mass m l , with respect to squared four-momentum transfer, q 2 , and angles defined in Fig. 1. We assume that the J=ψ is identified through its (purely electromagnetic) decay to μ þ μ − , defining the angle θ J=ψ , and we sum over the μ þ μ − helicities, Here, j ⃗ p 0 j is the magnitude of the J=ψ spatial momentum in the B c rest frame, and η EW is the same structure-independent electroweak correction as in [28], 1.0062(16) [37]. The k i and H i are given in Table I. We include in the expressions terms with factors of m 2 l =q 2 that were dropped in [28]. These are significant for the case when l ¼ τ and include the helicity amplitude H t which does not otherwise appear. Integrating over angles, the differential rate in q 2 is then given by The helicity amplitudes are defined in terms of standard Lorentz-invariant form factors [38] as Γ and RðJ=ψÞ.-The form factors were computed across the full physical q 2 range in [28] using lattice QCD. They are given in terms of a polynomial in z, with coefficients a F n , and a pole term corresponding to B c states with the quantum numbers of each current, The meson and subthreshold resonance masses, M pole , that need to be used in reconstructing the form factors are given in [28]. We assemble the helicity amplitudes using Eq. (5); these are plotted as a function of q 2 in Fig. 2 (Fig. 10 of [28]). Differential and total decay rates are then calculated. Where an integration over q 2 is necessary, we use a simple trapezoidal interpolation in order to ensure covariances are carried through correctly, taking sufficiently many points that the results are insensitive to the addition of any further points.
The differential rate dΓ=dq 2 is plotted in Fig. 3, comparing SM rates for l ¼ μ and l ¼ τ. We also compute the total decay rates, and from these, and their ratio, The error budget for these results is given in Table II. The largest contributions for both Γðl ¼ τÞ and Γðl ¼ μÞ are the discretization effects from the heavy quark mass, the statistical uncertainty in the lattice data, and the quark mass mistunings effects. These errors and their potential improvement are discussed in [28]. There is significant cancellation of these correlated errors in R, resulting in a factor of ≈5 reduction in uncertainty compared to Γ, and leaving the dominant error that from lattice statistics. The value for RðJ=ψÞ is very close to that expected in the SM for RðD Ã Þ [12]. RðJ=ψÞ is given here as the ratio of the rates to τ and μ; we showed in [28] that the decay rates to μ and e differ by 0.4%. R NP ðJ=ψÞ, angular observables and tests of lepton flavor universality.-The effects of new physics (NP) may be considered through the inclusion of complexvalued NP couplings, g i , i ∈ S; P; V; A; T; T5, in the effective Hamiltonian describing b → clν decays [39]. Reference [40] takes new physics to appear in the l ¼ τ channel only and fits the g i individually against the experimental average values of RðDÞ and RðD Ã Þ. Angular observables sensitive to the different NP scenarios are then constructed. Here, we use the values of g for  left-handed and right-handed vector couplings given in [40], which we reproduce here in Eq. (10), and examine their impact on RðJ=ψÞ and the angular observables for B c → J=ψ. In Fig. 4 (left-hand plot), we see that the semitauonic differential rate increases very markedly for the best fit value of g V L or g V R inferred from RðD ðÃÞ Þ [40]. This results in a corresponding 10σ increase of RðJ=ψÞ, to give the values below, The difference between RðJ=ψÞ and R g V R=L ðJ=ψÞ is then close to the difference between the SM and current experimental average value of RðD Ã Þ that g V L and g V R were designed to reproduce. Note, however, that our values for R g V R=L ðJ=ψÞ are still in tension with the experimental result.
We also compute the angular observables defined in [40], which are relevant for B − c → J=ψl −ν l . These are the forward-backward asymmetry A FB for the lepton l (note that the forward direction is that of the J=ψ in Fig. 1), the lepton polarization asymmetry A λ l , and the longitudinal polarization fraction for the J=ψ, F J=ψ L . Writing the observables are defined as A FB and A λ τ are plotted as a function of q 2 in Fig. 4, showing both the behavior in the SM and the impact of the possible NP couplings g V R and g V L . Note the very different shape of the SM curves for A FB and A λ τ in the SM compared to those for l ¼ e or μ given in [28]. The helicity -1 virtual W will throw a helicity −1=2 lepton predominantly in the W direction (i.e., backwards) but the mass of the τ changes the lepton helicity mixture. g V R accentuates this effect by boosting the contribution of jH þ j 2 but without changing the τ helicity mixture. For an observable O l i ðq 2 Þ ¼ N l i ðq 2 Þ=D l i ðq 2 Þ, the integrated quantities are defined as We give results for hA λ τ i, hA FB i, and hF J=ψ L i in Table III for B − c → J=ψτ −ν τ in the SM and for the NP couplings g V R and FIG. 4. dΓ=dq 2 , A FB , and A λ τ for B − c → J=ψτ −ν τ in the SM and for the values of g V R and g V L given in Eq. (10) from [40]. dΓ=dq 2 is normalized to the total rate in the l ¼ μ case, Γ μ , and the g V L and g V R curves overlap. For A F;B , the SM and g V L curves overlap, and for A λ τ , all three curves overlap. PHYSICAL REVIEW LETTERS 125, 222003 (2020) g V L from Eq. (10). Our SM results agree within 2σ with those from a covariant light-front quark model that included information from our preliminary lattice QCD results [41]. We also construct the lepton flavor universality violating (LFUV) ratios, These are given in Table IV, where we see that RðA FB Þ can distinguish between a NP right-handed vector coupling and a left-handed one. None of the other LFUV ratios change significantly from their SM values under the addition of either g V R or g V L . This is consistent with what was seen for B → D Ã lν l in [40].
Conclusion.-We give the first computation in lattice QCD of the branching fraction ratio RðJ=ψÞ that tests for lepton flavor universality in B c → J=ψ semileptonic decay. Our value in the SM is RðJ=ψÞ ¼ 0.2582ð38Þ, with an error budget in Table II. This is in tension with the LHCb result at 1.8σ, where σ is the experimental uncertainty. The B c → J=ψ form factors that we have calculated [28] to do this should enable the dominant systematic error in the experimental determination of RðJ=ψÞ to be reduced, allowing progress towards an accurate test of the SM.
We illustrate how NP might show up in B c → J=ψ decay with predictions for a variety of angular observables and τ to e=μ ratios both in the SM and with additional NP couplings consistent with the current average of experimental measurements of B → D ðÃÞ decay. We have shown that RðJ=ψÞ is close to the value of RðD Ã Þ both in the SM and in the presence of NP entering through either g V R or g V L . The resultant value of R g V R=L ðJ=ψÞ is still in tension with the experimental value.
We are grateful to the MILC Collaboration for the use of their configurations and code. We thank C. Bouchard, B.    (12), (13), and (14) [40]. The second column gives results in the SM and then the further two columns give results for NP couplings, in the τ channel only, of g V R and g V L [Eq. (10)