Beyond $\mathcal{R}(D^{(*)})$ with the general 2HDM-III for $b\to c\tau\nu$

We review the parameter regions allowed by measurements of $\mathcal{R}(D^{(*)})$ and by a theoretical limit on ${\cal B}(B_{c}\to\tau\nu)$ in terms of generic scalar and pseudoscalar new physics couplings, $g_s$ and $g_p$. We then use these regions as constraints to predict the ranges for additional observables in $b\to c\tau\nu$ including the differential decay distributions $d\Gamma/dq^{2}$; the ratios $\mathcal{R}(J/\psi)$ and $\mathcal{R}(\Lambda_{c})$; and the tau-lepton polarisation in $B\to D^{(\star)}\tau\nu$, with emphasis on the CP violating normal polarisation. Finally we map the allowed regions in $g_s$ and $g_p$ into the parameters of four versions of the Yukawa couplings of the general 2HDM-III model. We find that the model is still viable but could be ruled out by a confirmation of a large $\mathcal{R}(J/\psi)$.


Introduction
Amongst the most interesting current results in B physics, the searches for lepton universality in semileptonic B decays stand out. On the experimental side, hints at deviations from the standard model (SM) in some of these modes have existed for several years, withB → Dτ ν being measured by BaBar [1,2] and Belle [3]; and withB → D τ ν being measured by BaBar [1,2], Belle [3][4][5] and LHCb [6,7]. On the theoretical side, many extensions of the SM violate lepton universality whereas the SM does not. The tests involve comparing semileptonic B decays into tau-leptons to those with muons and electrons through ratios such as, where l represents either e or µ. The current values for these quantities hint to the existence of new physics, as can be seen when comparing the current HFLAV averages [8], to the current SM predictions from the lattice for R(D) [9,10] or from a range of models for R(D ) [11,12], For our new calculations in this paper, we will use the CCQM model for form factors which yields somewhat lower values for these quantities albeit with larger errors, R SM (D) = 0.27 ± 0.03 and R SM (D ) = 0.24 ± 0.02. A related measurement, B + c → J/ψτ + ν τ , has been reported by LHCb [13] and also hints to disagreement with the SM, although the errors are too large at present to reach a definitive conclusion, Different predictions for the SM arising from different models for form factors produce a range 0.24 to 0.28 [14][15][16][17][18] which is about 2σ lower that the LHCb result. With the CCQM form factors we obtain R(J/ψ) SM = 0.24 ± 0.02 , which we use as the SM prediction in our numerical analysis. Not surprisingly, these anomalies have generated enormous interest in the community. From the experimental side, we expect a measurement of the corresponding ratio for semileptonic Λ b → Λ c τ ν, R(Λ c ) to be reported soon. From the theory side there have been several proposals for additional observables to be studied in connection with these modes such as the tau-lepton polarisation [19][20][21][22][23][24]. In fact, the Belle collaboration has already reported a result for the longitudinal tau polarisation inB → D * τ −ν τ [5] P τ L (D * ) = −0.38 ± 0.51 +0.21 −0.16 , a result in agreement with the SM prediction [19] P τ L (D * ) SM = −0.497 ± 0.013, albeit with large uncertainty.
There have also been a large number of theory papers interpreting these results in the context of specific models, including additional Higgs doublets, gauge bosons and leptoquarks [19,23,. One of the first possibilities considered was the 2HDM type II, where BaBar [2] determined it was not possible to simultaneously fit R(D) and R(D ). However, a charged Higgs with couplings proportional to fermion masses is an obvious candidate to explain non-universality in semitauonic decays, prompting consideration of the more general 2HDM-III. Several authors have examined the flavour phenomenology of the 2HDM-III in the context of the anomalies mentioned above. Refs. [19,27,57] concluded that it is possible to explain R(D) and R(D ) in this way after considering existing flavour physics constraints. More recently, Ref. [23,58], add an analysis of the longitudinal tau-lepton polarisation and forward-backward asymmetries in b → c/u τ ν decays within the 2HDM-III.
In this paper we revisit the b → cτ ν modes in the presence of new (pseudo)-scalar operators to include several new results. We begin in Section II with a review of the constraints imposed by the measurements of R(D ( * ) ) and the theoretical limit on B(B c → τ ν) [59,60]. We then use these constraints to obtain the predicted ranges for R(J/ψ), the tau polarisation in B → D ( * ) τ ν decays, the differential decay rates and the ratio R(Λ c ) in Section III. We pay particular attention to the transverse tau polarisation which is T -odd [20,21,[61][62][63][64][65] as the 2HDM-III model allows for CP violation and would naturally give rise to this effect. We also consider the dΓ/dq 2 distributions [40] in B → D ( * ) τ ν but find that they offer no discriminating power in this case. They do serve to illustrate the CCQM model for the form factors. In Section IV we review the basics of the general two Higgs doublet model and the four different parameterizations for its Yukawa couplings. We then map this parameter space into the generic allowed regions obtained in Section II, finding they are completely accessible to this model. Finally, in Section V we conclude.

b → cτ ν constraints on new (pseudo)-scalar couplings
The effective Hamiltonian responsible for b → cτ ν transitions that results from the SM plus the 2HDM-III can be written in terms of the SM plus generic scalar operators in the form, where C cb SM = 4G F V cb / √ 2 and the operators are given by As the existing constraints will apply separately to the scalar and the pseudoscalar couplings, it is convenient to define The effect of the effective Hamiltonian, Eq. 8, on the ratios R(D ( * ) ) is known in the literature [11,27,57] and can be written as ratios r D ( * ) = R(D ( * ) )/R SM (D ( * ) ), r D = 1 + 1.5 Re (g S ) + 1.0 |g S | 2 , r D * = 1 + 0.12 Re (g P ) + 0.05 |g P | 2 .
A few remarks are in order. First, Refs. [30,57] observe that the coefficient of |g S | 2 can be changed from 1.0 to 1.5 to approximate some detector effects in BaBar. As we use the HFLAV average value for r D from both BaBar and Belle results, we will not include this correction in our numerics. Second, the CCQM model we use for the form factors leads to the slightly different expression r D * = 1 + 0.1 Re(g P ) + 0.03 |g P | 2 , but with larger theoretical errors. We will discuss the effect of this below. It is also known that there are values of C cb L and C cb R that can explain both of these ratios, and that the possible solutions become tightly constrained when one also requires that B(B c → τ ν) ≤ 30% [59], which for NP given by scalar operators implies that the ratio be smaller than around 14.6. An even tighter constraint, by a factor of three, is advocated in Ref. [60]. We summarize these results in Figure 1. On the left panel we consider the constraint on g S which arises solely from satisfying R(D) at the 2σ level and appears as the blue ring. The black ring shows the effect of approximating the BaBar detector effects as suggested by Refs. [30,57]. The central panel shows the constraints on g P : the red ring arising from satisfying r D * at the 2σ level and the green circle from B(B c → τ ν) ≤ 30%.
The small combined allowed region shows the tension between these two requirements. On the right panel we illustrate these combined constraints on g P as the red crescent shape. If one adopts the condition B(B c → τ ν) ≤ 10% [60] instead, there is no allowed region that also satisfies r D * at the 2σ level, but there is one at the 3σ level and we show this in black. As mentioned above, the expression for r D * with the CCQM form factors is slightly different but with larger errors which allow a larger overlap with B(B c → τ ν) ≤ 30% and this is shown as the orange crescent. For our predictions in the next section we will use the blue ring in the left panel and the red crescent in the right panel. Some, but not all, of these results have appeared before in the literature. For example Refs. [66,67] do not include a constraint from B(B c → τ ν) in their results.  The small crescent region where these two intersect is the constraint of g P that we use for our predictions. This region is magnified as the red crescent on the right panel where it is also compared with the larger orange region which uses the CCQM form factors for R(D * ), and with the black region which shows the intersection between R(D * ) at the 3σ level and B(B c → τ ν) ≤ 10%.

Differential decay distributions for B → D ( ) τ ν.
In Figure 2 we compare the distributions dΓ/dq 2 for B → D ( ) τ ν using the CCQM form factors with parameter values from Ref. [66]. The results indicate that the predicted spectrum is in good agreement with the measurements within the CCQM uncertainties (which the authors of Ref. [66] estimate at about 10%). The modifications to these predictions from g P and g S as constrained above are indistinguishable from the SM within this level of accuracy.

R(J/ψ)
As already mentioned, there is also a more recent measurement of R(J/ψ) given in Eq. 4, which can be used as an additional test of the model. Using the form factors shown in the appendix with CCQM values from Ref. [68], this can be written in terms of generic scalar coefficients as Note that this result is almost identical to that for r D when the CCQM form factors are used for that case as well. The differential distribution dΓ/dq 2 for B c → J/ψτ ν receives tiny corrections from g S,P as constrained above, making it indistinguishable form the SM one. In Figure 3 we show the prediction for R(J/ψ) that is consistent with the measured R(D ) at 2σ as well as B(B c → τ ν) ≤ 30%. The largest prediction (∼ 1.075) is about 1.5σ away from the LHCb measurement thanks to its present large uncertainty, which in terms of this ratio is r J/ψ = 2.5 ± 1.0. A confirmation of a large value for r J/ψ can potentially rule out (pseudo)-scalar explanations of these anomalies.

Im(gP)
r J/ψ Figure 3: Predictions for r J/ψ compatible with the measured R(D ) at 2σ as well as

Polarisations
In general, we can define normal, longitudinal and transverse polarisations of the τ lepton as a function of q 2 in terms of the vectors [64], Of particular interest is the normal polarisation, P τ N (D ( ) ), which is generated by CP violating phases that arise from extended scalar sectors or Yukawa flavour changing couplings 1 . This observable is very small in the SM, where it can only arise due to unitarity phases in electroweak loop corrections [20,21,[61][62][63][64][65].
With the numerical CCQM form factors of Ref. [64], we find that new (pseudo)-scalar complex couplings lead to (15) in the allowed parameter regions obtained above. In particular we see that P τ L (D ) as measured by Belle [5] is consistent with all the predictions given the current large uncertainty. The figures also indicate that a large CP violating P τ N (D) polarisation is possible.

Λ b → Λ c lν decays
As mentioned in the introduction, there is one more ratio in the b → cτ ν family that is expected to be measured soon by LHCb, namely R(Λ c ). In terms of the CCQM form factors we show the differential decay rate [67,69] in the appendix. From the partial decay width Eq.(52), we first obtain the Λ b → Λ c µν µ normalised spectral distribution for the SM and compare it with the one measured by LHCb [70] in Figure 6. The green and yellow shaded areas indicate the estimated 10% and 20% errors in the prediction according to [64]. Once again, this figure serves to calibrate the performance of the CCQM form factors in this case. We also find in this case that the spectral distribution with new g S,P couplings constrained as above, cannot differentiate between the models. We turn to a prediction for R(Λ c ) which is defined analogously to the previous ratios, With the form factors in the appendix this leads to It also leads to R(Λ c ) SM = 0.295, which compares well with other values found in the literature R(Λ c ) SM = 0.33 ± 0.01 [69]. Figure 7 shows R(Λ c ) with new contributions from g S or g P in their allowed ranges. As Eq. 17 shows no interference between g S and g P , the two new contributions simply add.

General two Higgs doublet model
The most general 2HDM-III, unlike the type I and type II more common versions, allows flavour changing neutral currents (FCNC) at tree-level which are then suppressed with family symmetries, minimal flavour violation, or specific patterns for the Yukawa couplings, for example. The most general renormaliseable quartic scalar potential is commonly written as [71], where the two scalar doublets are Discrete symmetries in 2HDM type I and II force the parameters µ 12 and λ 6,7 to vanish. The charged Higgs bosons that appear in the mass eigenstate basis correspond to the combinations with the rotation angle given by tan β = t β = s β c β = υ 2 υ 1 and υ 2 2 + υ 2 1 = υ 2 with υ = 246 GeV. There are three neutral scalars that are not CP eigenstates as the parameters µ 12 and λ 6,7 can be complex and violate CP. These, however, will not play any role in our discussion beyond the occasional use of existing constraints on the mixing amongst the neutral scalars.
The most general Yukawa Lagrangian in the 2HDM-III without discrete symmetries is given by where Φ 1,2 = iσ 2 Φ * 1,2 , Q L and L L denote the left-handed quark and lepton doublets, u R , d R and l R the right-handed quark and lepton singlets and Y u,d,l After spontaneous EWSB and in the fermion mass basis, the charged Higgs couplings to fermions can be written as: where f (x) = √ 1 + x 2 . This form follows the notation of Refs. [72][73][74] in which the first term in each line in Eq. 22 is the coupling in one of the four 2HDM without FCNC and the second term is a flavour changing correction that makes it a type III model. Furthermore, the Cheng-Sher ansatz [75] has been implemented to control the size of the FCNC, but also allowing a CP violating phase: The additional parameters that occur as a consequence of allowing flavour changing couplings areχ q.l ij . The parameters X, Y and Z given in Table 1 are the ones that occur in each of the four types of 2HDM with natural flavour conservation.
We now turn to the question of the scalar coefficients in Eq. 8 within the. context of the 2HDM-III considered here. Tree-level exchange of the charged Higgs produces 2HDM-III Assuming that the parametersχ u i,j are of the same order and that theχ d i,j are also of the same order, as we expect in the context of the Cheng-Sher ansatz, the contributions from the heaviest fermions dominate the sums and Eq. 25 reduces to The allowed parameter regions of Figure 1 then imply constraints on the parameters m H ± , tan β,χ u,d,l ij which we discuss next in some detail. The general result is that it is possible to reach the allowed regions in Figure 1 with parameters of the model. Ref. [76] finds solutions for generalised models which can be written in terms of our Eqs. 26 with the factorsX,Ỹ ,Z being arbitrary parameters, independent of tan β. The solutions they find occur for points withX ∼ O(10),Ỹ ∼ O(100),Z ∼ O(100) and m H < 550 GeV.
Once we allow for FCNC, all four cases of 2HDM-III can be mapped into the allowed regions in Figure 1. In Figures 8-11 we illustrate the results in two dimensional projections of parameter space. In all cases we present three figures. In the first one we consider the plane tan β − m H ± and scan over all the real and imaginary parts of the χ parameters looking for points that satisfy the primary constraints R(D * ) at 2σ and B(B c → τ ν) ≤ 30%. In the second and third plots we illustrate regions of the parameter space of theχ's where the constraints are satisfied, in particular we specifically show solutions in the vicinity of g s = −0.5 + 0.7i and g P = 0.63 as these two points lie well inside the allowed regions of Figure 1. With solutions in this region of parameter space we findX,Ỹ ,Z are O(10) to O(1000). It is important to emphasise, however, that these are only illustrations and that there are infinitely many solutions. Looking at the four models then, • Model I. We present numerical results for this case in Figure 8. On the left panel we illustrate the region where solutions exist in the tan β − m H ± plane. We see that a lower value of tan β and/or m H ± is needed to obtain solutions with smaller values of |χ u,d,l |. The region shown is dominated by low values of tan β which are compatible with constraints from LHC and LEP on the flavour conserving version of this model as seen in Figure 4 of Ref. [76]. Figure 9 of the same reference indicates that values of tan β 2 are ruled out by B decay constraints. These constraints, however, can be significantly modified by flavour changing parameters such asχ d bs . We are not aware of any global fit to the full set of parameters in the general 2HDM.  Figure 1 shown in green.
• Model II. We present numerical results for this case in Figure 9. On the left panel we illustrate the region where solutions exist in the tan β − m H ± plane. We see that in this case a higher value of tan β and/or a lower value of m H ± is needed to obtain solutions with smaller values of |χ u,d,l |. The tan β − m H ± region of solutions in this case is consistent with the constraints on the corresponding flavour conserving version of this model in Ref. [76]. The centre and right panels illustrate that solutions consistent with the Cheng-Sher ansatz exist in this case.  Figure 1 shown in green.
• Model X. We present numerical results for this case in Figure 10. On the left panel we illustrate the region where solutions exist in the tan β − m H ± plane. We see that in this case a higher value of tan β and/or a lower value of m H ± is needed to obtain solutions with smaller values of |χ u,d,l |. This scenario is similar to Model II in that the tan β − m H ± region of solutions is consistent with the constraints on its corresponding flavour conserving version as per Ref. [76] (called type IV in that reference). The region illustrated on the centre and right panels needs |χ d,l | values larger than what the Cheng-Sher ansatz would suggest are natural. However, the left panel indicates that there are other solutions which are also consistent with this ansatz.
• Model Y. Finally, we present numerical results for this case in Figure 11. On the left panel we illustrate the region where solutions exist in the tan β − m H ± plane. We see that in this case a higher value of tan β and/or a lower value of m H ± is needed to obtain solutions with smaller values of |χ u,d,l |. The tan β − m H ± region of solutions is once again consistent with its corresponding flavour conserving version [76] (called type III in that reference), although the overlap region mostly lies in the upper range of both tan β and m H ± shown in the left |χ  |≤20, |χ  |≤8 panel. This panel also suggests that in this case, theχ parameters are required to be larger than expected in the Cheng-Sher ansatz.
Additional considerations that may restrict the parameters in the general model arise form Yukawa couplings to the neutral (SM-like) Higgs defined as Once we introduce non-zero couplingsχ u ct ,χ u ct ,χ l τ τ as in Eq. 26, they also appear in g hτ τ , g hct and g hsb , and are given by These expressions simplify in the alignment limit, defined as cos(β − α) → 0, in which case the couplings of h tend to the SM Higgs couplings. To linear order in cos(β − α) we obtain for models II, X The first constraint arises from the process h → τ + τ − , for which the measured signal strength is [77] leads to −0.32 |h l τ τ | 2 − 1 0.58 (31) at the 95% confidence level. In addition, if the flavour changing couplings get too large they will conflict with the non-observation of t → ch and with indirect limits on h → bs. For B(t → hc) < 0.22% at 95% c.l. [78] one finds cos(β − α) sin βχ u ct The process h → bs has not been constrained yet, but it has been argued in the literature that a branching ratio as large as B(h → bs) ∼ 36% can remain consistent with other flavour results in these types of models [79]. Adopting this number and with the 95% c. l. Γ H < 0.013 GeV [78] we find, The constraints in Eqs. 31-33 depend on cos(β −α) and disappear in the alignment limit.
Ref. [80] presents upper bounds on cos(β − α) of O(0.1) that depend on tan β for the four types of flavor conserving models, so they do not automatically extend to our case.

Summary Conclusions
We have revisited the 2HDM-III as a possible explanation for the R(D ( * ) ) anomalies. We first summarised the constraints known in the literature in terms of generic (pseudo)scalar couplings and discussed the possible conflict between R(D * ) and B(B c → τ ν) ≤ 30%. We found that the parameter space that can explain these two anomalies at the two-sigma level is limited to the region B(B c → τ ν) > 23%. The bound B(B c → τ ν) > 10% advocated in Ref. [60] in turn restricts the possible explanation of R(D ( * ) ) to the 3σ level within these models. Armed with these constraints we predicted the ranges of other observables in b → cτ ν reactions, including R(J/ψ) and R(Λ c ). We find that the large central value in the current measurement of R(J/ψ) is consistent with this model at about the 2σ level with the currently large experimental error, but that a more precise measurement of this quantity could place it in conflict with R(D ( * ) ).
We found that the distributions dΓ/dq 2 in B → Dτ ν, B → D ( * ) τ ν or B c → J/ψτ ν cannot distinguish between the SM or models with new (pseudo)-scalar couplings.
We presented predictions for the tau-lepton polarisation in B → D ( * ) τ ν in the presently allowed region of parameter space. In particular we find that phases in the Yukawa couplings can produce substantial T-odd normal polarisations.
We considered four versions of the 2HDM-III which are constructed by extending the four flavour conserving 2HDM with the addition of flavour changing couplings that we have limited in size with the Sher-Cheng ansatz. We mapped the allowed regions in g P − g S into the parameter space of these four models. We found that the allowed (m H ± , tan β) ranges also satisfy the LHC and LEP constraints found in the literature for the flavour conserving versions of these models. We also found that the allowed regions of parameter space are not further constrained by h → τ τ , t → hc, h → bs.

A Helicity Amplitudes
The invariant form factors describing the hadronic transitionsB → D and B → D * are defined as usual where P = p 1 + p 2 , q = p 1 − p 2 , and 2 is the polarization vector of the D * meson which satisfies † 2 · p 2 = 0. The particles are on their mass shells: p 2 1 = m 2 B and p 2 2 = m 2 D ( * ) . All the expressions are written in terms of helicity form factors, which are related to those in Eq. 34 for the B → D transition by [64] where |p 2 | = λ 1/2 (m 2 B , m 2 D ( * ) , q 2 )/2m B is the momentum of the daughter meson with For our numerical estimates we use the helicity amplitudes calculated in the covariant confined quark model (CCQM) with the double-pole parameterisation of Refs. [64,66]: Similarly, for the Λ b decay we need the vector and axial current form factors [69,81] which satisfy the parity relations, where λ 2 and λ W denote the helicities of the daughter baryon Λ c and the virtual W boson respectively. In the SM the helicity amplitudes H V (A) λ 2 ,λ W are given by [81] H with M ± = M Λ b ± M Λc and Q ± = M 2 ± − q 2 . The helicity amplitudes for scalar and pseudo-scalar operators needed for 2HDM are [67] H SP λ 2 ,0 =H S λ 2 ,0 − H P λ 2 ,0 , In this way, the partial decay width of the Λ b → Λ c lν process is given by [67] dΓ where δ l = m 2 l /2q 2 , ij = (−

B Polarisations
Following Ref. [64], the ratios R(D ( * ) ) are given by with and g S ≡ C cb L + C cb R /C cb SM and g P ≡ C cb L − C cb R /C cb SM . In terms of Eq. 55, the longitudinal differential polarisation will be, Similarly, the transverse polarisation is given by Finally, in the presence of CP-violating phases in the NP Higgs exchange amplitude, there is a normal differential polarisation that reads 2 dP τ N (D) To calculate the integrated, or q 2 averaged polarisations, one has to include the q 2 -dependent phase-space factor C(q 2 ) = |p 2 |(q 2 − m 2 τ ) 2 /q 2 [64], .
2 Note that there is a typo in Eq. 41 of Ref. [64] where the denominator of P (D) N (q 2 ) should have a 4 instead of a 2. We thank C. T. Tran for confirming this.