B-meson charged current anomalies: the post-Moriond status

In this note, we discuss the impact of the recent Belle result on the various theoretical explanations of the $R_D$ and $R_{D^*}$ anomalies. The pure tensor explanation, which was strongly disfavoured by the measurements of $F_L^{D^*}$ and high-$p_T$ $p \, p \to \tau \, \nu$ searches before Moriond, is now completely allowed because of reduction of the experimental world-average. Moreover, the pure right-chiral vector solution (involving right-chiral neutrinos) has now moved into the $2\sigma$ allowed range of the LHC $p \, p \to \tau \, \nu$ searches. We also critically re-examine the bound on $\mathcal{B}(B_c^- \to \tau^- \bar{\nu}_\tau)$ from LEP data and show that the bound is considerably weaker than the number $10\%$ often used in the recent literature.

In this note, we discuss the impact of the recent Belle result on the various theoretical explanations of the RD and RD * anomalies. The pure tensor explanation, which was strongly disfavoured by the measurements of F D * L and high-pT p p → τ ν searches before Moriond, is now completely allowed because of reduction of the experimental world-average. Moreover, the pure right-chiral vector solution (involving right-chiral neutrinos) has now moved into the 2σ allowed range of the LHC p p → τ ν searches. We also critically re-examine the bound on B(B − c → τ −ν τ ) from LEP data and show that the bound is considerably weaker than the number 10% often used in the recent literature.
The Belle collaboration has recently published results for R D and R D * with a semileptonic tag [1,2], and their result is consistent with the Standard Model (SM) expectation within 1.2σ. Consequently, the experimental world average has moved towards the SM. However, the tension between the experimental world average and the SM expectation is still more than 3σ, and thus, it is interesting to re-examine the status of the various New Physics (NP) explanations in view of the new worldaverage. In Table. I below, we collect all the experimental results related to this anomaly.
The most general effective Lagrangian for the decay b → c τ −ν τ involving mass dimension-6 operators and only left-chiral neutrinos can be written as c. If one uses power-counting rules arising from linearlyrealised SU(2) × U(1) gauge invariance, it turns out that the Wilson Coefficient (WC) C RL V , with the possibility of lepton non-universality, is only generated at the mass dimension-8 level [20]. Thus, it is expected to be suppressed compared to the other WCs as long as the scale * bardhan@post.bgu.ac.il † diptimoy.ghosh@iiserpune.ac.in of NP is not too close to the Higgs vacuum expectation value, thus we will ignore it in this analysis. If one also assumes the existence of light right-chiral neutrino(s), as was first done in [21] to solve the R D anomaly, five additional operators can be constructed by the replacement P L → P R in the leptonic currents of Eq. 1. In particular, a pure-right chiral vector current namely, was considered by several authors [22][23][24] , and we will include it in our analysis. As the experimental situation for R D and R D * is far from clear, we do not try to perform a fit to the WCs; for an early global fit, see [25]. Instead, we show how R D and R D * vary with respect to the WCs, and overlay the current 1σ experimental world-average and the corresponding currently allowed values of the WCs.
In new experimental world-average for R D is now consistent with the SM expectation at ∼ 1σ level. So the anomaly is mostly driven by R D * . In order to be consistent with both R D and R D * simultaneously at the 1σ level, C LL V has to be in the range C  [26] 1 (bound from LHC p p → τ ν + X searches was also studied in [27,28]). Note that, both the WCs C LL V and C RR V can be generated by a single U 1 (3, 1, 2/3) Leptoquark mediator [24,[29][30][31][32].
Variations of R D and R D * with respect to C LL T and C LL S = −8C LL T are shown in Fig. 2. It can be seen from the left panel of Fig. 2 that a simultaneous solution of R D and R D * is possible for C LL T in the range We remind the readers that the corresponding value of C LL T before the recent Belle result was C LL T ∼ 0.35 [20,33] which was strongly disfavoured both by the LHC p p → τ ν searches [26,34,35]  the m b scale) shown on the right panel is interesting because it is generated by a single S 1 (3, 1, 1/3) Leptoquark mediator [37]. The allowed range of the WC in this case is [0.113, 0.170] which, as can be seen from Fig. 3, produces B(B − c → τ −ν τ ) less then its SM value, and thus is completely safe. 1 Note, however, that for |C RR V | = 0.305, the value of R D ( * ) is at the lower edge of the experimental 1σ allowed region. Moreover, the sensitivity of the current high-p T measurements is not enough to constrain the left-handed scenario C LL V ≈ 1.05. Thus, the right-handed scenario is statistically worse than the C LL V solution.
Another single mediator solution that has been discussed in the literature is the so-called R 2 (3, 2, 7/6) Leptoquark [38,39]. which, contrary to the S 1 (3, 1, 1/3) Leptoquark mediator, generates C LL S ≈ +8C LL T (see the sign difference) at the m b scale 2 . In the left panel of Fig. 4 A much better description of the data is possible if imaginary WCs are assumed as shown in the right panel of Fig. 4. The case of imaginary WCs in this context was first discussed in [40], and later also in [39,[41][42][43][44]. In this case,  Fig. 3. However, the authors of Ref. [45] claimed an upper bound of 10% on this branching ratio, arising from the LEP data taken on the Z peak. Thus, the Im bound is taken at face value. While some authors [41] expressed concerns about the validity of this bound, not much effort was made to estimate as to how much this bound can be relaxed. We will discuss this in detail in the next section.
As the operator C RL S alone cannot explain R D and R D * simultaneously, we do not discuss it anymore.
Before concluding this section, we would like to make a couple of comments on the impact of F D * L and P D * τ on the various scenarios. In all the scenarios explaining the R D and R * D anomalies, the variation of P D * τ is less than ∼ 2.5% from the SM prediction. Unfortunately, this is also true about F D * L , the only exception being the Im[C LL S ] = 8Im[C LL T ] solution in which case the variation can be 5 − 10% below the SM. Thus, distinguishing the various explanations by either P D * τ or F D * L looks difficult at the moment.
As mentioned in the previous section, the authors of [45] used the LEP data [46] collected at the Z peak to put an upper bound on the branching fraction of B − c → τ −ν τ . As this constraint has potentially interesting consequences for the R D and R D * anomalies, in this section we will revisit it in detail.
In Ref. [46], the L3 collaboration obtained an upper bound on the number of B − → τ −ν τ events, N (B − → τ −ν τ ) < 3.8. Based on this, they provided an upper bound B(B − → τν τ ) < 5.7 × 10 −4 at 90% C.L. As c or a B − u meson, and Ref. [46] uses a value f b→B − = 0.382 ± 0.025, the bound in Eq. 3 can be translated into the following bound Separating the total number of events into those coming from B − u and B − c decays, we get This gives, The quantities B(B − u → τ −ν τ ) and f b→B − u are known experimentally: [4,47] (7) f b→B − u = 0.412 ± 0.008 [4,47](LEP) f b→B − u = 0.340 ± 0.021 [4,47](Tevatron) (9) Note that, the hadronization fractions in Z decays do not necessarily need to be identical to those in pp collisions because of the different momentum distributions of the b-quark in these processes; in pp collisions, the b quarks have momenta close to m b , rather than ∼ m Z /2 in Z decays. In fact, CDF and LHCb collaborations have reported evidence for a strong p T dependence of he Λ 0 b fraction [48][49][50][51]. The LHCb and the ATLAS collaborations have also studied the p T dependence of f b→Bs /f b→B d [52, 53], but the results are not conclusive yet.
Therefore, we use the measurement of f b→B − u from LEP only and plot the upper bound on Fig. 8. The upper bound In order to find a real upper bound on B(B − c → τ −ν τ ) we need to know the value of f b→B − u /f b→B − c , or at least a lower bound on f b→B − u /f b→B − c . Moreover, we need to know f b→B − u /f b→B − c at LEP, and with the exact kinematical cuts used in [46].
Ref. [45] tries to find the ratio f b→B − u /f b→B − c from measurements of R π + /K + and R π + /µ + defined as It then follows that we get, As the LHCb and CMS measurements of R π + /K + are about 2.5σ away from each other, we consider them separately and do not use their average. Moreover, while the LHCb Collaboration uses the cuts 0 < p T (B + c ), p T (B + u ) < 20 GeV and 2.0 < η < 4.5 in their analysis (at √ s = 8 TeV), the CMS Collaboration uses p T (B + c ), p T (B + u ) > 15 GeV and |η| < 1.6 (at √ s = 7 TeV). Thus the discrepancy could be due to the dependence of fb →B + c /fb →B + u on kinematics. Plugging Eqs. 18 and 19 into Eq. 6, one can obtain a bound on B(B − c → τ −ν τ ) directly as a function of B (B + c → J/ψ µ + ν µ ). This is shown in the right panel of Fig. 6. Using B (B + c → J/ψ µ + ν µ ) ≤ 2.5 × 10 −2 , as used in [45], we get fb →B + c /fb →B + u 3 × 10 −3 and B(B − c → τ −ν τ ) 14% from the CMS data, the latter being similar but slightly weaker than [45].
We would like to make two comments at this stage: has not yet been measured, a model independent bound is not possible. Moreover, even the SM calculation, and in particular the uncertainty, is not fully under control at the moment. Thus, a precise bound on B(B − c → τ −ν τ ) cannot be obtained currently. • Even in the presence of better information on B (B + c → J/ψ µ + ν µ ), Eqs. (18) and (19) provide values of fb →B + c /fb →B + u at the LHC and for the specific kinematic regions used in [55] and [56]. As discussed before, the value of fb →B + c /fb →B + u at LEP may be different from the above because of 1) larger average p T of the b-mesons produced at LEP 2) bb pairs produced at LEP are in the colour singlet state contrary to most of the bb pairs produced at the LHC which are in the colour octet state.
In view of the above, we try to estimate the ratio fb →B + c /fb →B + u at LEP using the event generator Pythia8 [57,58] which has Hadronization model tuned to provide a good description of the available experimental data. The results are shown in Table. II. In each of the cases presented in Table. II, we have generated 1 million events in order to reduce the statistical uncertainty. In Case-I, we have used the same p T and η cuts as in [56], and we get a value Fig. 6) which is much larger than the values considered in [45]. In the third row of Table. II, we  changed the p T cut to p T < 15GeV in order to check the p T dependence of the Hadronization fractions. In this case, we get f b→B − c /f b→B − u = 1.89 × 10 −3 which is considerably larger than that in Case-I. This is consistent with the general findings in [48][49][50][51][52][53] and confirms that the measurement of f b→B − c /f b→B − u from LHCb (Eq. 15 and 18) which uses p T (B + c ), p T (B + u ) < 20 GeV is indeed not expected to be the same as that measured in CMS (Eq. 16 and 19) which used p T (B + c ), p T (B + u ) > 15 GeV. In rows 4 and 5 of Table. II, we considered bb production through only Z boson (produced bb are in QCD singlet state) and through only QCD interactions (produced bb are in QCD triplet state) respectively. We observed only ∼ 10% variation in the f b→B − c /f b→B − u between these two cases.
Finally, at the Z peak, we obtain f b→B − u = 0.42, fb →Bs = 0.094 (not shown in the table), and f b→B − c /f b→B − u = 1.07×10 −3 , the first two numbers being consistent with their experimental measurements [4,47].
Using the number f b→B − c /f b→B − u = 1.07 × 10 −3 , from Fig. 5, we get We warn the readers that this bound should only be taken as an estimate because, after all, Pythia only uses a Hadronization model adjusted to describe a large amount of available experimental data well (as we saw, indeed it reproduced the correct values for f b→B − u and fb →Bs ), and the value of f b→B − c obtained from Pythia is neither based on any first principle calculation nor on direct experimental data.
To summarise, in this short note, we have shown that • the recent Belle results on R D and R D * have interesting implications on the various possible EFT explanations of the data. The most important being that the pure tensor explanation is now completely allowed both by the measurement of F D * L and the high-p T p p → τ ν searches by ATLAS and CMS.
• the solution in terms of a pure right-chiral vector current (involving right-chiral neutrinos) has now moved into the 2σ allowed range of the LHC p p → τ ν searches.
• the upper bound on the branching fraction of B − c → τ −ν τ from the LEP data is much weaker than the bound 10% used in the recent literature. Our estimate of this bound, based on the Hadronization model implemented in Pythia8, is approximately 40%. This bound, while being independently important, may also have interesting implications on the various scalar-pseudoscalar explanations of the R D and R D * data.