Lepton universality violation and right-handed currents in $b \to c \tau \nu$

We consider the recent LHCb result for $B_c\to J/\psi \tau \nu$ in conjunction with the existing anomalies in $R(D)$ and $R(D^\star)$ within the framework of a right-handed current with enhanced couplings to the third generation. The model predicts a linear relation between the observables and their SM values in terms of two combinations of parameters. The strong constraints from $b\to s \gamma$ on $W-W^\prime$ mixing effectively remove one of the combinations of parameters resulting in an approximate proportionality between all three observables and their SM values. To accommodate the current averages for $R(D)$ and $R(D^\star)$, the $W^\prime$ mass should be near 1 TeV, and possibly accessible to direct searches at the LHC. In this scenario we find that $R(J/\psi)$ is enhanced by about 20\% with respect to its SM value and about 1.5$\sigma$ below the central value of the LHCb measurement. The predicted $d\Gamma/dq^2$ distribution for $B\to D(D^\star) \tau \nu$ is in agreement with the measurement and the model satisfies the constraint from the $B_c$ lifetime.

Different models for the form factors produce a SM result in the range 0.25 to 0.28 [13][14][15][16] which is about 2σ lower. For definiteness, we will use as SM value the most recent result [17] R(J/ψ) = 0.283 ± 0.048 .
Not surprisingly, these anomalies have generated a large number of possible new physics explanations including additional Higgs doublets, gauge bosons and leptoquarks .
In Ref. [26] we have studied R(D) and R(D ) in the context of a right-handed W with enhanced couplings to the third generation [48,49]. Here, we revisit this possibility motivated by the new measurement of R(J/ψ), and to address additional constraints from the dΓ/dq 2 distributions [34] and the B ± c lifetime [50].
To separate the symmetry breaking scales of SU (2) L and SU (2) R , we introduce the two Higgs multiplets H L (1, 2, 1)(−1) and H R (1, 1, 2)(−1) with respective vevs v L and v R . An additional bi-doublet φ (1, 2, 2)(0) scalar with vevs v 1,2 is needed to provide mass to the fermions. Since both v 1 and v 2 are required to be non-zero for fermion mass generation, the W L and W R gauge bosons of S(2) L and SU (2) R will mix with each other. In terms of the mass eigenstates W and W , the mixing can be parametrized as In the mass eigenstate basis the quark-gauge-boson interactions are given by, where Rij the unitary matrices which rotate the right handed quarks u Ri and d Ri from the weak eigenstate basis to the mass eigenstate basis.
The model has a different neutrino spectrum than the SM: three left-handed neutrinos ν L i and one right-handed neutrino ν R 3 . Additional scalars ∆ L (1, 3, 1)(2) and ∆ R (1, 1, 3)(2) with vevs v L,R ∆ are needed to generate neutrino masses. In order for the possibly enhanced SU (2) R interaction with the third generation to explain the B decay anomalies, we need the right-handed neutrino to be light, which requires v L,R ∆ to be small. In this model, the neutrinos will receive Majorana masses from the vevs of ∆ L,R and Dirac masses from φ. The mass eigenstates (ν m L , (ν m R 3 ) c ) are related by a unitary transformation to the weak eigenstates as In our model U L = (U Lij ), U RL = (U RLi3 ) and U LR = (U LR3i ) and U R = (U R33 ) are 3 × 3, 3 × 1, 1 × 3 and 1 × 1 matrices, respectively. Writing the rotation of charged lepton weak eigenstates L,R into mass eigenstates m L,R as L,R = V L,R m L,R , the lepton interaction with W and W becomes where U is approximately the PMNS matrix. From Eqs. 6 and 8 we see that a large g R /g L will enhance the third generation interactions with W . The final neutrino flavor is not identified in B meson decays so it must be summed. For the processes involving left-and right-handed charged leptons, neglecting neutrino masses compared with the charged lepton masses, the final decay rates into a charged lepton j , when summed over the different neutrino final states, are proportional to To obtain these results we used the unitarity of U : The starting point for our calculations is the differential decay rate dΓ/dq 2 with q 2 = (p B − p D ( ) ) 2 for the SM. We use the notation, parameterization and values of Ref. [2] for all the relevant form factors. This will be sufficient for a comparison to the experimentally determined shape of this distribution as well as the total decay rate. Of course, dΓ/dq 2 is obtained after integrating over angles and summing over polarizations. Other observables, such as angular correlations, can also be used to discriminate between the SM and new physics scenarios as well, but we will not consider that possibility in this paper as they have not been measured yet.
In the type of model we consider, in addition to the SM diagram, there is a W mediated diagram as well as interference between these two. When all the neutrino masses can be neglected, there is no interference between the left and right-handed lepton currents and this allows us to write simple formulas for both dΓ/dq 2 and the decay rate Γ in terms of the corresponding SM results and the following two combinations of constants: The first term arises from the separate W and W contributions, whereas the second term is induced by W − W mixing. The superscript bc is used to denote the b → c quark transition and is useful in order to generalize the notation to other cases. Our results for the different modes are then: • Leptonic decay B ± c → ± ν . The hadronic transition in this case proceeds only through the axial vector form factor thus yielding • Semileptonic decay B → Dτ ν. In this case only the vector form factor contributes to the hadronic transition resulting in This implies that the normalized distribution 1/ΓdΓ/dq 2 for this mode is identical to the SM. Upon integration, • Semileptonic decay B → D τ ν. This mode is more complicated in that both the vector and axial vector form factors contribute and they behave differently. Defining we can write the differential decay rate as Using the values of the form factors given in Ref. [2], the SM is dominated by the axialvector hadronic form factor as can be seen from Figure 1. A transition through a purely axial-vector form factor (blue curve) has a spectrum shape almost indistinguishable from the SM case (black curve). On the other hand, a transition through a purely vector hadronic form factor produces a spectrum shifted towards lower q 2 , as shown by the red curve in the same figure. The vector and axial-vector form factors do not interfere in this distribution, and their respective contributions to the total decay rate in the SM are This implies that the vector form factor contributes only 5.6% of the SM rate. Integrating Eq. 16 we find very close to the result one would obtain from a pure axial-vector transition.
• Semileptonic decay B c → J/ψτ ν. This is also a pseudoscalar to vector transition so it has the same behaviour as the previous mode.
where now In this case we use the values for the form factors as given in Ref. [17] to display the SM differential distribution on the right panel of Figure 1 as the black curve. Once again the blue (almost indistinguishable from the black one) and red curves illustrate transitions mediated by a purely axial-vector or vector form factors respectively, normalized to the total decay rate. We see that in this case the axial-vector hadronic form factor is even more dominant than in B → D τ ν. Their respective contributions to the total decay rate in the SM in this mode are the vector form factor barely contributes 1.5% of the SM rate. It follows that for our model, These parameters result in a differential distribution dΓ/dq 2 for B → D τ ν that is dominated by the axial-form factor and is thus very similar to the SM one as shown in Figure 2. It should be clear from this result, in conjunction with the BaBar comparison of the SM vs observed distribution [2], that dΓ/dq 2 is in agreement with the results of our model. This is shown in the right panel of Figure 2 where the normalized distributions are compared with the BaBar data. The SM and our model are a good fit to the data and they are almost indistinguishable. This is due to the dominance of the F bc dir term over the mixing contribution. Including the latest result, R(J/ψ), does not change the fit significantly due to its large uncertainty. The prediction for this quantity, R(J/ψ) = 0.34, is thus on the low side of the central value by about 1.5 standard deviations. The differential distribution dΓ/dq 2 in this case is also very similar to the SM one, as seen in Figure 3, and would not serve to distinguish this model. The rate Γ(B − c → τ − ν) is predicted from Eq. 12 to be about 20% larger than its SM value, which is well within the bound from the B c lifetime discussed in Ref. [50].

IV. DISCUSSION
We now examine the parameter values of Eq. 23 in the context of the model with RH currents as shown in Eq. 11. The anomalies suggest that only the τ lepton is affected as the experiment sees no difference between muons and electrons. The model must then single out only the third family for enhancement with |V R3τ | ∼ 1. Now V Rcb ≡ V u Rtc V d Rbb and V d Rbb ∼ 1 and V u Rtc is of the same order as V cb , as discussed in our global analysis of the model in Ref. [51]. This requires g R /g L M W /M W ∼ 0.7 to reproduce the first result of Eq. 11.
Requiring the model to be perturbative implies that g R /g L < ∼ 10. If we take this ratio to be in the range (5 − 10) we find in turn that M W is in the range 574 − 1150 GeV, well within the direct reach of LHC.
The second combination of parameters, F bc mix , requires that we re-examine bounds on W − W mixing, in particular the combination ξ ef f = ξ W g R /g L . This combination is constrained by b → sγ [48,52,53] and we applied this constraint to our model in Ref. [26]. We can update that result using the most recent HFLAV collaboration average: B(b → X s γ) = (3.32 ± 0.15) × 10 −4 [8] combined with the NNLL SM calculation B(b → X s γ) = (3.15 ± 0.23) × 10 −4 [54]. Assuming that the new physics interferes constructively with the SM, at the 3σ level the allowed range becomes −1.
This range severely restricts the possible size of F bc mix . For example, if we use F bc dir = 1.28 in Eq. 11 in combination with the above constraint on ξ ef f , we find and it is not possible to reach the value F bc mix = 0.04 in the fit, Eq. 11. Under these conditions the three observables R(D), R(D ), and R(J/ψ) become approximately proportional to the SM results, It is interesting to notice that this universal enhancement of the three asymmetries reproduces what is found in models with an additional SU (2) L symmetry [35,36]. We illustrate the situation in Figure 4 which takes into account these constraints. The very narrow width of the model prediction range is due to the tight constraint on mixing and comparison with data implies that the model needs a W mass very close to 1 TeV to successfully explain these anomalies. The best direct limits on such a W come from 19.7 fb −1 of CMS data at √ s = 8 TeV [55]. The first result presented in that paper excludes an SSM W 1 with mass below 2.7 TeV. Since the production couplings for W are model dependent it is more useful to quantify the constraint as σ × B(W → τ ν) < ∼ 3 fb. In the SSM model considered by CMS B(W → τ ν) ≈ 8.5% for W masses of order a TeV, where decay into top-bottom is allowed. In the non-universal model discussed here, this branching fraction approaches 25% when g R >> g L and W couples almost exclusively to the third generation. At the same time the production cross-section at LHC for our W would be very suppressed due to its negligible couplings to the light fermions. Roughly then, For the first term in the last bracket, corresponding to a direct coupling of the W to the light quarks, we have: [56]; and fitting the existing body of FCNC constraints implies that V u Rtu ∼ 10 −3 [51]. For the second term in the bracket we already saw that ξ W is at most 10 −3 in this scenario and we conclude that the corresponding σ × B(W → τ ν) in our model is more than 6 orders of magnitude smaller than that of an SSM W and the CMS data does not place any significant constraint.
The CMS paper also quantifies their result using a type of non-universal W that also singles out the third generation dubbed 'NUGIM' [57,58]. In this case the CMS data excludes a W with mass below 2.0 − 2.7 TeV. Comparing the relevant figure of merit, σ × B(W → τ ν), of this model to ours we see that B(W → τ ν) can be quite similar but Whereas V Rud can (and in fact is constrained to be) very small, the parameter s E /c E of NUGIM is of order one (for this reason the W − W mixing in the NUGIM model is not important in σ(pp → W )). The net result is that the CMS limits do not directly apply to our model. A separate study is needed for an accurate comparison of our model to LHC results, taking into account production from heavier quarks. As mentioned before, our model relies on the existence of an additional light neutrino to explain these anomalies and this can have other observable consequences. In Ref. [26] we have already seen that there are no significant constraints from the invisible Z width. At the same time the model can provide an enhancement to the rare K → πνν modes [59] where new results are expected from NA62 and KOTO.
The existence of a light right-handed neutrino contributes to the effective neutrino number ∆N ef f which is also constrained by cosmological considerations and this may affect the viability of our model. There is some uncertainty as to the value of this constraint, but commonly used numbers are, for example [60], 1 For SSM it is assumed that the W is a heavy copy of the SM W , with its same couplings to fermions.
As we saw above, our model requires (29) to explain the R D ( * ) anomalies and this is only slightly weaker than the usual weak interaction. At the same time, the exchange of a W can bring the new ν R into thermal equilibrium with the SM particles through scattering of right-handed neutrinos with tauons at a rate proportional to (g R M W /g L M W ) 4 |V R3τ | 4 relative to the usual weak interaction. In fact, with V Rcb /V cb ∼ 1 this would result in ∆N ef f ∼ 1 bringing into question the viability of our model. The mixing induced interaction, proportional to ξ W , is smaller and does not lead to large contributions to ∆N ef f . However, the aforementioned scattering of right-handed neutrinos with tauons is only effective for temperatures T R above T τ ∼ m τ . At the time of big bang nucleosynthesis (BBN), the temperature is about T BBN ∼ 1 MeV, implying that ∆N ef f is suppressed by a factor where g * (T ) is the effective number of relativistic degrees of freedom at temperature T and g * (T BBN ) = 10.75. In addition, g * (T R ) is larger than g * (T QCD ) ∼ 58 since the QCD phase transition temperature T QCD is of order a few hundred MeV [61]. All this implies that the contribution to ∆N ef f from our additional neutrino is less than 0.1 and safely within the BBN constraint. Similarly, τ decay processes into ν R plus other SM particles are also suppressed by the same factor r, but one might worry about additional processes without this suppression. For example, ν R scattering off an electron or a muon. However, these are proportional to the additional mixing parameters |V R3e(µ) | 4 and can be made sufficiently small by lowering V R3e(µ) .
Another potentially worrisome process is the exchange of a Z in the scattering of a ν R off an electron or SM neutrino ν L . In this case the interaction strength is proportional to (g 2 Y /M 2 Z ) 2 [49], and when compared to Z exchange induced ν L scattering off an electron or ν L , it is suppressed by a factor of (M Z /M Z ) 4 . The constraint on ∆N ef f becomes in this case a lower bound on the Z mass, M Z > ∼ 200 GeV.
In conclusion we find that new right handed currents affect the semi-tauonic B decay anomalies in a way that is consistent with current bounds, including those on the effective number of neutrino species from BBN. A confirmation of a high value for R(J/ψ) would exclude them as a viable explanation and would also exclude new left-handed currents. The most promising way to rule out this explanation of the anomalies is the exclusion of a W in the τ -channel at LHC in the mass range 1 − 1.4 TeV. The suppression of our W couplings to light fermions significantly complicates this comparison.
of PRC (Grant Nos. 11575111 and 11735010), and partially supported by a grant from Science and Technology Commission of Shanghai Municipality (Grants No.16DZ2260200) and National Natural Science Foundation of China (Grants No.11655002).