Further tests of lepton flavour universality from the charged lepton energy distribution in b → c semileptonic decays : The case of Λ b → Λ c ` ν̄ `

In a general framework, valid for any H → H `ν̄` semileptonic decay, we analyze the dΓ/(dωd cos θ`) and d Γ/(dωdE`) distributions, with ω being the product of the hadron fourvelocities, θ` the angle made by the three-momenta of the charged lepton and the final hadron in the W− center of mass frame and E` the charged lepton energy in the decaying hadron rest frame. Within the Standard Model (SM), dΓ/(dωdE`) ∝ ( c0(ω) + c1(ω)E`/M + c2(ω)E 2 ` /M 2 ) , with M the initial hadron mass. We find that c2(ω) is independent of the lepton flavor and thus it is an ideal candidate to look for lepton flavor universality (LFU) violations. We also find a correlation between the a2(ω) structure function, that governs the (cos θ`) 2 dependence of dΓ/(dωd cos θ`), and c2(ω). Apart from trivial kinematical and mass factors, the ratio of a2(ω)/c2(ω) is a universal function that can be measured in any semileptonic decay, involving not only b → c transitions. These two SM predictions can be used as stringent tests in the present search for signatures of LFU violations. Finally, we calculate the dΓ/(dωdE`) distribution for the case of Λb → Λc`ν̄` decay, using a set of form-factors obtained from an unquenched lattice QCD simulation, and show the lepton-helicity decomposition of dΓ/dω for the τ−mode.

New preliminary measurements by the Belle collaboration [13] reduce however this tension with the SM predictions to 1.2σ. A general model-independent analysis of different b → clν l charged current (CC) transition operators has been addressed in Ref. [14] within an effective field theory approach. The main conclusion of this study is that the anomaly is still present and can be solved by NP, in agreement with previous works (see for instance the pioneering work of Ref. [15]).
This anomaly can be corroborated in Λ b → Λ c lν l decays, which are also governed by the b → c transition. The ω-shape of the differential width for muons has been recently measured by the LHCb collaboration [16], and there exist prospects [17] that the level of precision in the R Λ c ¼ ΓðΛ b →Λ c τν τ Þ ΓðΛ b →Λ c μν μ Þ ratio might reach that obtained for R D ðÃÞ. The form factors relevant for this transition are strongly constrained by heavy quark spin symmetry (HQSS), since no subleading Isgur-Wise (IW) function occurs at order OðΛ QCD =m b;c Þ and only two subleading functions enter at the next order [18,19]. Precise results for the form factors were obtained in Ref. [20] using lattice QCD (LQCD) with 2 þ 1 flavors of dynamical domain-wall fermions. Leading and subleading HQSS IW functions are simultaneously fitted to LQCD results and LHCb data, and are used to accurately predict the R Λ c ratio in the SM [19]. Therefore, this reaction is, from the theoretical point of view, as appropriate as the B → D ðÃÞ processes for the study of b → c LFU violations. A sum rule relating R Λ c to R D ðÃÞ , independent of any NP scenario up to small corrections, was found in Refs. [21,22]. There it is shown that R Λ c does not provide additional information on the Lorentz structure of NP but provides an important consistency check of the R D ðÃÞ measurements. The full four-differential angular distribution of the Λ b → Λ c ð→ Λ 0 π þ Þl −ν l decay has been recently studied in Ref. [23] with the finding that the full set of angular observables analyzed is sensitive to more combinations of NP couplings than the R D ðÃÞ ratios. In this latter reference some discrepancies with the results of the previous study of Ref. [24] are pointed out. NP corrections to R Λ c have also been examined in other works [14,[25][26][27][28][29][30]. Some of them pay also attention to the double differential rate, d 2 Γ=ðdωd cos θ l Þ, in addition to R Λ c or the ω spectrum [ω being the product of the hadron fourvelocities and θ l the angle made by the three-momenta of the charged lepton and the final hadron in the center of mass (c.m.) of the two final leptons]. Thus, forwardbackward asymmetry has been calculated for this baryon decay [25][26][27][28], while the full c.m. charged lepton angular dependence has also been analyzed for B-meson reactions, see for instance [14,15]. However, to our knowledge, the charged lepton energy (E l ) distribution in the decaying hadron rest frame has never been considered neither for Λ b → Λ c nor B → D ðÃÞ semileptonic decays.
Finally, we should also mention that the Λ b → Λ c lν l decay provides an alternative method to determine the Cabibbo-Kobayashi-Maskawa matrix element jV cb j and to study the unitarity triangle within the SM.
In this work we introduce a general framework to study any baryon/meson semileptonic decay for unpolarized hadrons, though we refer explicitly here to those induced by the b → c transition. Within this scheme we find general expressions for the d 2 Γ=ðdωd cos θ l Þ and d 2 Γ=ðdωdE l Þ differential decay widths (see below), each of them expressible in terms of three different structure functions (SFs). Proceeding in this way, we have uncovered two new observables that can be measured and used as model independent tests for LFU violation analyses. This is discussed in the next section, and it constitutes the most relevant result of this work. Indeed, we identify two contributions, one in the E l spectrum and a second one in the cos θ l distribution, which are independent of the lepton flavor in the SM. They provide novel, modelindependent and clean tests of LFU. Moreover, we show that the ratio of both of them within the SM should be a universal function, which could be measured in all types of hadron (baryon or meson) semileptonic decays governed by CC c → s, c → d, s → u, b → u transitions. This charged lepton energy-angle correlation should be experimentally accessible and, if violated, it would be a clear  indication of NP, also eliminating some possible Lorentz  structures for the new terms as we will discuss. We also generalize the formalism to account for left and right scalar and vector NP contributions using the scheme of Ref. [14].
To illustrate our findings, we will apply the general framework to the analysis of the Λ b → Λ c decay. Using the state of the art LQCD form factors of Ref. [20], we evaluate the six SM SFs and the dΓ=dω differential rate. For the case of a final τ lepton we give explicitly the contributions coming from positive and negative helicities measured, both in the c.m. of the W − boson and in the Λ b rest frame (LAB). As mentioned above, the relevance of the LAB lepton energy spectrum for LFU violation has never been studied for Λ b or for B decays. Indeed, we also discuss how some features of that spectrum can be used to distinguish between different NP scenarios that otherwise lead to the same total and differential semileptonic decay widths.

II. DECAY WIDTH
We consider the semileptonic decay of a bottomed hadron (H b ) into a charmed one (H c ) and lν l , driven by the CC b → c transition. In the SM, the differential decay width for massless neutrinos reads [31] with G F ¼ 1.166 × 10 −5 GeV −2 the Fermi coupling constant, M (M 0 ) the mass of the initial (final) hadron and W and L, the hadron and lepton tensors. The latter one, after summing over all lepton polarizations is given by (ϵ 0123 ¼ þ1) with k 0 (k) the outgoing charged lepton (neutrino) fourmomentum. In addition, the product of the two hadron four-velocities ω and q 2 ¼ ðk þ k 0 Þ 2 are related via q 2 ¼ M 2 þ M 02 − 2MM 0 ω and s 13 ¼ ðp − kÞ 2 , with p the four-momentum of the decaying H b particle. Finally, the dimensionless hadron tensor is constructed from the nonleptonic CC vertex j μ cc ¼cð0Þγ μ ð1 − γ 5 Þbð0Þ as with p 0 ¼ p − q. The sum is done over initial (averaged) and final hadron spins, and the states are normalized as h ⃗p; rj ⃗p 0 ; si ¼ ð2πÞ 3 ðE=MÞδ 3 ð ⃗p − ⃗p 0 Þδ rs , with r, s spin indexes. Lorentz covariance leads to the general decomposition actually valid for any H → H 0 CC transition with unpolarized hadrons. 1 The W i SFs are scalar functions of q 2 or equivalently of ω. The double differential decay width can be rewritten introducing the angle (θ l ) made by the charged lepton (l) and the final hadron (H c ) in the W − boson c.m. frame as with M ω ¼ ðM − M 0 ωÞ and m l the mass of the charged lepton. The variable ω varies from 1 to ω max ¼ ðM 2 þ M 02 − m 2 l Þ=ð2MM 0 Þ and cos θ l between −1 and 1. The terms proportional to m 2 l in each of the coefficients a i¼0;1;2 account for the contributions from positive helicity of the outgoing l. This follows from the expression of the lepton tensor for a charged lepton with well-defined helicity (h ¼ AE1), where We are also interested in the double differential decay width with respect to ω and the energy (E l ) of the charged lepton in the LAB frame The relevance of this distribution is in the fact that within the SM, and up to small electroweak corrections, the c 2 ðωÞ SF does not depend on the lepton mass. Therefore, this function determined in l ¼ e, μ decays should be the same as that seen in τ decays. This is a clear test for LFU in all types of semileptonic b → c decays that to our knowledge has not been considered so far. A similar comment holds for a 2 ðωÞ entering in the c.m. angular distribution of Eq. (5), after accounting for the trivial ð1 − m 2 l =q 2 Þ kinematical factor. Furthermore, the ratio is a universal function that should be found in all types of q → q 0 lν l transitions, since in that ratio the SF W 2 cancels out. This is a test of the predictions of the SM and, in principle, this ratio can be measured in any semileptonic decay: Notice that a lefthanded vector current that couples exclusively to the τ lepton, which is the so far preferred NP explanation of the anomalies [14], will have no effect on the ratio in Eq. (10). However, in that case both a 2 ðωÞ=ð1 − m 2 l =q 2 Þ and c 2 ðωÞ will change for a final τ by a factor j1 þ C V L j 2 , where we follow here the notation in Ref. [14]. A right-handed vector current [the C V R term in Eq. (2.1) in Ref. [14]] will affect the ω-dependence of both a 2 ðωÞ=ð1 − m 2 l =q 2 Þ and c 2 ðωÞ while the ratio in Eq. (10) will still go unaffected. In this case one would expect c 2 and a 2 to change differently than the total decay width τ=μ ratios. Further Lorentz dependencies, like the scalar and tensor ones, that include modifications in the lepton vertexes, would in principle modify all three quantities. We will further illustrate this point below in Sec. III C. In any case, any violation of Eq. (10) will be a clear indication of NP beyond the SM, not driven by left-or right-handed vector current operators.
Using Eq. (7) in the LAB frame, we obtain d 2 ΓðhÞ= ðdωdE l Þ for a charged lepton with a well-defined helicity (h ¼ AE1) the charged lepton three-momentum. For a massless charged lepton the h ¼ þ1 contribution vanishes, as expected from conservation of chirality.
The individual contributions to d 2 Γ=ðdωdE l Þ from τ leptons with positive and negative helicity in the LAB frame cannot be obtained from the depolarized l ¼ μ, e and l ¼ τ data alone. In contrast, neglecting the electron or muon masses, the angular distribution of Eq. (5) can be used, together with measurements of the l ¼ μ, e and l ¼ τ d 2 Γ=ðdωd cos θ l Þ differential decay width, to separate the individual contributions of positive and negative τ helicities in the c.m. frame. This is to say, with great accuracy, ð1 − m 2 τ =q 2 Þ −2 × d 2 Γ=ðdωd cos θ τ Þ for a τ with negative helicity can be determined from the unpolarized d 2 Γ=ðdωd cos θ l Þ measured for muons or electrons.
In this section we apply the above-described general formalism to the study of the semileptonic Λ b → Λ c decay. We present first SM results, and later we also discuss the effect of some NP contributions to the a 2 and c 2 coefficients.

A. Form factors
The hadronic matrix element can be parametrized in terms of three vector (F i ) and three axial (G i ) form factors, which are functions of ω and that are greatly constrained by HQSS near zero recoil (ω ¼ 1) [18,19] From this equation one can obtain the W i SFs, and hence the a i , c i coefficients, in terms of F i and G i . The explicit expressions are given in Appendix A. These form factors [Eq. (15)] are easily related to those used in the LQCD calculation of Ref. [20] (see also Appendix A), which were given in terms of the Bourrely-Caprini-Lellouch parametrization [34] [see Eq. (79) of [20]]. A different determination of the form factors within QCD sum rules in full theory is done in Ref. [35]. Taking into account the experimental and theoretical uncertainties, the LQCD form factors describe well the Λ 0 b → Λ þ c μ −ν μ normalized spectrum ðdΓ=dq 2 Þ=Γ recently measured by the LHCb collaboration [16] (see Fig. 5 in that reference). From the integrated distribution given in Ref. [20] and using the Λ 0 b lifetime (1.471 AE 0.009 ps) and the Λ 0 b → Λ þ c μ −ν μ branching fraction [ð6.2 AE 1.4Þ%] quoted in [31], one obtains jV cb j ¼ 0.044 AE 0.005 which is compatible with the values reported by the HFLAV [2].
For numerical calculations we use here the 11 parameters and statistical correlations given in Tables VIII and IX of Ref. [20].

B. SM results
The results obtained for the a i , c i SFs, both for m l ¼ 0 (appropriate for l ¼ e, μ) and for m l ¼ m τ , are shown in Figs. 1 and 2. We also display the 68% confident level (C.L.) bands that we Monte Carlo derive from the correlation matrix reported in [20].
As mentioned above, within the SM, the c 2 ðωÞ SF is the same for all charged leptons, providing a new testing ground for LFU violation studies in b → c decays. We also observe that finite lepton mass corrections are quite small for c 1 , while become more sizable for the rest of the SFs, which are given here for the very first time using the realistic LQCD results of Ref. [20].
For completeness, in Fig. 3 we show the dΓ=dω differential decay width and its corresponding uncertainty band inherited from the statistical correlated fluctuations of the LQCD form factors. For the τ case, we show explicitly the SM predictions for the contributions from tau leptons with positive and negative helicities, both in the c.m. and LAB frames.
C. c 2 and a 2 sensitivity to NP In this section we shall investigate the effect of NP on the c 2 and a 2 SFs for the Λ b → Λ c semileptonic decay. The derived formulas are general and not specific to this transition. We shall consider the effective Hamiltonian taken from Ref. [14]. The Wilson coefficients, C i , parametrize possible deviations from the SM, i.e., C SM i ¼ 0, and could be in general, lepton and flavor dependent, though in [14] are assumed to be present only in the third generation of leptons. Moreover, these coefficients are taken to be real (CP-symmetry conserving limit). Complex Wilson  FIG. 1. Angular SFs a 0 , a 1 and a 2 [Eq. (5)] for the Λ 0 b → Λ þ c l −ν l decay obtained using the LQCD form factors of Ref. [20]. Bands account for 68% C.L. intervals deduced from the correlation matrix given in [20]. coefficients can explain the b → cτν τ anomalies as well as real ones, but they do not offer any clear advantages regarding the fit quality, so they have not been considered in the effective low-energy Hamiltonian approach of Ref. [14] for simplicity. In Table VI of that reference, the authors provide four different fits (4, 5, 6 and 7) that include all the above terms. Of these four fits, we shall only consider the last two. The reason being that for fits 4 and 5 the SM coefficient is almost canceled and its effect is replaced by NP contributions, what seems to be an unlikely situation from a physical point of view. In fits 6 and 7 the C T Wilson coefficient is very small (0.01 þ0.09 −0.07 and −0.02 þ0.08 −0.07 , respectively) and here for simplicity we shall make it zero. With these approximations, the amplitude changes from the original current-current J Hrr 0 μ ðp; p 0 ÞJ μ Lh ðk; k 0 Þ term to  with e W μν ðp; qÞ constructed with the vector and axial form factors modified by using the multiplicative factors C V and C A introduced above. In the above expression, h ¼ AE1 is the charged lepton helicity, with the new lepton terms given by 2 where we have introduced three new real scalar SFs, W SP , W I1 and W I2 , that depend on q 2 alone. We readily obtain the NP corrections to the double differential decay width where the differentã j are given by Eq. (6) where thec j are given by Eq. where F S and F P are the scalar and pseudoscalar form factors that are directly related [see Eqs. (2.12) and (2.13) of Ref. [26]] to the f 0 vector and g 0 axial ones given in [20]. We thus have while the interference hadron tensor reads Expressions for W SP , W I1 and W I2 in terms of e F S;P ¼ C S;P F S;P [Eq. (28)], e F i ¼ C V F i and e G i ¼ C A G i are given in Appendix B.
As mentioned above, c 2 and a 2 are not affected by the left and right scalar NP terms and these SFs are only modified by the left and right vector Wilson coefficients C V L ;V R . This turns out to be very relevant. Fits 6 and 7 in Ref. [14] provide very different values for C V L and C V R which implies different NP changes in c 2 and a 2 SFs. However, the two fits produce very similar results for the R Λ c ratio (roughly 0.42, to be compared to the SM prediction of 0.33 AE 0.02). In this situation the c 2 or a 2 SFs are observables that could differentiate one fit from the other. In the left panel of Fig. 4 we show the ratio ðc 2 Þ NP =ðc 2 Þ SM ¼ ða 2 Þ NP =ða 2 Þ SM as a function of ω for the two fits under consideration. The ω-dependence is hardly visible (for an explanation see the discussion below) but, as seen in the figure, the changes in magnitude of the NP corrections are significantly different in the two fits, and are not accounted for by errors. Hence, a measurement of c 2 for τ decay would not only be a direct measurement of the possible existence of NP, but it would also allow to distinguish from fits that otherwise give the same total and differential dΓ=dω decay widths (see the right panel of Fig. 4). It would thus provide information on the type of NP that is needed to explain the data.
For completeness in Fig. 5, we show the c.m. angular a 0 , a 1 and a 2 (top) and LAB τ-energy c 0 , c 1 and c 2 (bottom) SFs as functions of ω. In addition to c 2 and a 2 , we find that a 1 , and both c 0 and c 1 , can also be used to distinguish between the two NP scenarios related to the minima 6 and 7 of Ref. [14]. We recall here that the NP parameters were obtained in that reference from a general model-independent analysis of b → cτν τ transitions, including measurements of R D , R D Ã , their q 2 differential distributions, the recently measured longitudinal D Ã polarization F D Ã L , and constraints from the B c → τν τ lifetime. We would like to stress that all c 0 , c 1 and c 2 SFs, which determine the LAB d 2 Γ=ðdωdE l Þ distribution, are quite differently affected by the two NP settings analyzed here, even though both give rise to indistinguishable d 2 Γ=dω differential decay widths.
Moreover, we also see that the ratio ða 1 Þ NP =ða 1 Þ SM would exhibit some sizable ω-dependence, in particular in the case of fit 7. This is in contrast to the case of the ðc 2 Þ NP =ðc 2 Þ SM ratio, depicted in the left panel of Fig. 4, which turned out to be practically flat. This is because only linear C V R terms could induce a nonzero ω-dependence for ðc 2 Þ NP =ðc 2 Þ SM , but however, to a high degree of approximation (it would be exact in the heavy quark limit), e W 2 is FIG. 4. Left: ðc 2 Þ NP =ðc 2 Þ SM ratio for the Λ b → Λ c semileptonic transition obtained with the parameters of fits 6 and 7 in Ref. [14]. Right: SM and NP predictions for the dΓ=dω distribution for the τ decay mode. As in other figures, the LQCD form factors of Ref. [20] have been used and the uncertainty bands account for 68% C.L. intervals.
given by M Λ b ð e F 2 1 ðωÞ þ e G 2 1 ðωÞÞ=M Λ c with F 1 ðωÞ ∼ G 1 ðωÞ and, in this approximation linear effects on C V R cancel exactly.
In the discussions on Figs. 4 and 5 above, we have assumed uncorrelated Gaussian distributions for the Wilson coefficients C V L , C V R , C S L and C S R and have averaged the asymmetric errors quoted in Ref. [14], since correlation matrices are not provided in that reference. This should be sufficient for the illustrative purposes of this subsection.
Nevertheless, in what follows we will estimate the effects produced by the correlations between the Wilson coefficients in the ðc 2 Þ NP =ðc 2 Þ SM ratio.
Note that in Ref. [14], the uncertainties of a given parameter y i were determined as the shifts Δy i around the best-fit value y min i of that parameter, such that the minimization of χ 2 j y i ¼y min i þΔy i varying all remaining parameters in the vicinity of the minimum leads to an increase Δχ 2 ¼ 1. This procedure leads, in general, to asymmetric errors, and to non-Gaussian correlations that cannot be accounted for by a single matrix. The effects of these correlations on ðc 2 Þ NP =ðc 2 Þ SM are shown in Fig. 6. We have chosen this ratio because it hardly depends on ω, and thus in Fig. 6 we have fixed it to the intermediate value of 1.15. In the left (fit 6) and middle (fit 7) panels of this figure, we depict ðc 2 Þ NP =ðc 2 Þ SM j ω¼1. 15 and R Λ c for several sets of Wilson coefficients, which give rise to the R D and R D Ã values given in the bottom and top X-axes.
The ratios R D , R D Ã and R Λ c (black dashed curves in the bottom plots), and the χ 2 shown in the right panel of Fig. 6 have been computed as described in [14] 3 and have been obtained from the authors of that reference [36]. In Fig. 6, the Wilson-coefficients space is scanned starting from fit 6 and 7 minima, through successive small steps in the multiparameter space leading to moderate merit-function enhancements and R Λ c variations (see the right plot of Fig. 6). There exist one-to-one relations between each set of Wilson coefficients (sWC) used in the left (fit 6) and middle (fit 7) panels of Fig. 6 and the chi-square values or the variations ΔR Λ c ð¼ R sWC Λ c − R min Λ c Þ shown in the right plot of the figure. Note that at some point for ΔR Λ c < −0.02, the local fit 7 collapses into fit 6. We see that fits 6 and 7 chi squares grow from their local minimum values, and the Δχ 2 ¼ 1 and Δχ 2 ¼ 2.71 increments can be used [14] to determine the 68% (1σ) and 90% (2σ) C.L. intervals of the NP predictions for R D , R D Ã and R Λ c . We also use here these χ 2 variations to estimate the uncertainties on the results for ðc 2 Þ NP =ðc 2 Þ SM j ω¼1. 15 . In addition, the 68% C.L. errors induced from the Λ b → Λ c LQCD form factors [20] are very small for this ratio, and are shown by the shaded bands in both plots. Comparing the results depicted in Fig. 4 with the variation of ðc 2 Þ NP =ðc 2 Þ SM j ω¼1.15 ½¼ ða 2 Þ NP =ða 2 Þ SM j ω¼1. 15 between the 1σ-vertical lines shown in Fig. 6, we observe that the inclusion of the Wilson-coefficients statistical correlations reduces the uncertainties on this ratio by factors of 5 and 3 for the predictions obtained from fits 6 and 7, FIG. 5. The c.m. angular a 0 , a 1 and a 2 (top) and LAB τ-energy c 0 , c 1 and c 2 (bottom) SFs for the Λ 0 b → Λ þ c τ −ν τ decay obtained using the LQCD form factors of Ref. [20]. As in Fig. 4, we show the SM predictions and the NP results obtained from fits 6 and 7 of Ref. [14]. 3 The χ 2 function is defined in Eq. (3.1) of that reference, and it is constructed using the meson inputs collected in Sec. 2.3. respectively. Thus, now we find that these two NP scenarios give rise to results for this latter observable separated by more than 5σ, 1.40 AE 0.04 versus 2.06 AE 0.09, despite predicting fully compatible R D , R D Ã and R Λ c integrated ratios. This discussion strongly reinforces our previous conclusions from Fig. 4.
A final remark concerns the errors induced by neglecting the tensor NP contribution. In the bottom plots of the first two panels of Fig. 6, we compare for different sets of Wilson coefficients, the predictions for R Λ c obtained from the full model of Ref. [14] (MPJP black dashed curves) with those obtained in this work (magenta and cyan dashed lines), where C T has been set to zero. For the latter predictions, we also display the errors (68% C.L. bands) inherited from the Λ b → Λ c LQCD form factors. We see that within the 1σ intervals, both for fit 6 and fit 7, the effects of the NP tensor term on R Λ c are moderately small, and are partially accounted for the uncertainties of the LQCD inputs. This continues to be the case for all sets of fit 7 Wilson coefficients considered in the χ 2 -plot of Fig. 6, while for fit 6 and in regions above 1σ, jC T j appreciably grows and its effects become sizable.

IV. SUMMARY
We have introduced a general framework, valid for any H → H 0 lν l semileptonic decay, to study the lepton polarized c.m. d 2 Γ=ðdωd cos θ l Þ and LAB d 2 Γ=ðdωdE l Þ differential decay widths. To our knowledge, this is the first time that the relevance of the d 2 Γ=ðdωdE l Þ differential decay width has been put forward as a candidate for LFU violation studies in b → c decays. Specifically, within the SM the c 2 ðωÞ SF appearing in that distribution is the same for all charged leptons. That makes it a perfect quantity for LFU violation studies. We have also found a correlation between the a 2 ðωÞ SF related to the ðcos θ l Þ 2 dependence in d 2 Γ=ðdωd cos θ l Þ and c 2 ðωÞ. This correlation is shown in Eq. (10) and states that the ratio a 2 ðωÞ=c 2 ðωÞ, corrected by trivial kinematical and mass factors, gives a universal function valid for any H → H 0 semileptonic decay. Again, this is a clear prediction of the SM that can be checked against experiment. These two results could play a relevant role as further tests of the SM and LFU.
We have also generalized the formalism to account for some NP terms, and shown that neither c 2 nor a 2 are modified by left and right scalar NP terms, being however sensitive to left and right vector corrections. We also found that the relation of Eq. (10) for the a 2 =c 2 ratio is not modified by these latter NP contributions.
Finally, we have presented SM and NP predictions for the Λ b → Λ c transition. We have shown that a measurement of c 2 (or a 2 ) for τ decay would not only be a direct measurement of the possible existence of NP, but it would also allow to distinguish from NP fits to b → cτν τ anomalies in the meson sector, which otherwise give the same total and differential dΓ=dω widths. The same applies to the other two SFs, c 0 and c 1 , which appear in the LAB d 2 Γ=ðdωdE l Þ differential width, and for the a 1 coefficient (cos θ l linear term) in the c.m. angular distribution.

ACKNOWLEDGMENTS
We warmly thank F. J. Botella, C. Murgui, A. Peñuelas and A. Pich for useful discussions. This research has been supported by the Spanish Ministerio de Economía y FIG. 6. Left and middle panels: Effects of Wilson-coefficient statistical correlations on ðc 2 Þ NP =ðc 2 Þ SM j ω¼1.15 for the NP fits 6 and 7 of Ref. [14]. Each set of Wilson coefficients is identified by its predictions for R D and R D Ã (X-axes) and R Λ c (black dashed curve, labeled as MPJP, in the bottom plots) [14,36]. SM predictions for the R SM D ¼ 0.300 AE 0.05 and R SM D Ã ¼ 0.251 AE 0.004 ratios are below the ranges considered, while R SM Λ c ¼ 0.33 AE 0.02. We also show R Λ c computed after neglecting the NP tensor contribution (magenta and cyan dashed lines for fits 6 and 7, respectively). Shaded bands in our baryon results stand for 68% C.L. uncertainties inherited from LQCD inputs [20]. Right panel: Fits 6 and 7 chi-square values [14,36] for each set of Wilson coefficients (sWC) used in the left and middle panels, and represented in this plot by ΔR Λ c ¼ R sWC Λ c − R min Λ c , with R min Λ c ¼ 0.405 and 0.415 for fits 6 and 7, respectively. See the text for more details. and In the numerical calculations, we use m b ¼ 4.18 AE 0.04 GeV and m c ¼ 1.27 AE 0.03 GeV as in Ref. [26].