Relations between $b\rightarrow c\tau \nu$ Decay Modes in Scalar Models

As a consequence of the Ward identity for hadronic matrix elements, we find relations between the differential decay rates of semileptonic decay modes with the underlying quark-level transition $b\rightarrow c\tau \nu$, which are valid in scalar models. The decay-mode dependent scalar form factor is the only necessary theoretical ingredient for the relations. Otherwise, they combine measurable decay rates as a function of the invariant mass-squared of the lepton pair $q^2$ in such a way that a universal decay-mode independent function is found for decays to vector and pseudoscalar mesons, respectively. This can be applied to the decays $B\rightarrow D^{*}\tau\nu$, $B_s\rightarrow D_s^*\tau\nu$, $B_c\rightarrow J/\psi\tau\nu$ and $B\rightarrow D\tau\nu$, $B_s\rightarrow D_s\tau\nu$, $B_c\rightarrow \eta_c\tau\nu$, with implications for $R(D^{(*)})$, $R(D_s^{(*)})$, $R(J/\psi)$, $R(\eta_c)$, and $\mathcal{B}(B_c\rightarrow \tau\nu)$. The slope and curvature of the characteristic $q^2$-dependence is proportional to scalar new physics parameters, facilitating their straight forward extraction, complementary to global fits.

A lot of progress has been made in the research of the ability of new physics (NP) models, including from the beginning scalar models, to explain the data [12,43,.
An important way to probe for NP are relations between different decay modes. In nonleptonic decays this is a tool which is known for a long time, and there based on SU(3) F methods, see for example Refs. [148][149][150][151][152][153][154][155].
For semileptonic b → cτ ν decays, model-specific relations that connect different decay modes are known for left-handed vector models as the relation [94-96, 103, 121, 138] left-handed vector models: which is e.g. also found in the R-parity violating SUSY model considered in Refs. [117,126].
No matter which decay channel is considered on the left-hand side, the same expression is obtained on the right-hand side. In this paper, we present similar relations between differential decay rates of different decay modes in scalar models. They can be found in Eqs. (48)- (52) and Fig. 1. The resulting decay-mode independent functions of the invariant lepton mass-squared q 2 are a finger print of the model: Its slope and curvature are directly proportional to NP parameters which can thus be readily extracted. A departure from that characteristic function would be a sign of NP beyond scalar models.
In contrast to non-leptonic sum rules, which are based on the approximate flavor symmetry of QCD, the relations that we consider here are based on ones between hadronic form factors which follow from the Ward identity, and are therefore exact. We do not use flavor symmetries to derive these relations.
On the other hand, translating the allowed region of the model independent two-dimensional scalar global fit to b → cτ ν observables performed in Refs. [86,87] into allowed tan β and m H ± values in the 2HDM-II, we obtain very small values of order tan β 1 and m H ± 2 GeV. That means the current measurements of b → cτ ν observables can only be explained simultaneously for parameter values clearly excluded by other bounds, e.g.
B(B → X s γ). This observation agrees with Fig. 6 in Ref. [175] where the allowed parameter space for explaining R(D * ) and R(D) also converges only for very small tan β and m H ± , excluded by other data. Applying the bounds from Ref. [157] to the 2HDM-I, the resulting Wilson coefficients [84] are of the order |C R | ≡ cot 2 βm b m τ /m 2 H ± 2 · 10 −5 , far too small in order to account for either one of R(D ( * ) ) [175]. However, examples of more general 2HDMs with flavor-alignment exist that indeed can explain R(D ( * ) ) [84,167,168].
For b → sll LFU ratios R(K ( * ) ), the Wilson coefficients C 9,10 play an important role, see for recent fits Ref. [178]. However, in the 2HDM-I or II the contributions to C 9,10 are suppressed by cot 2 β, which would only have an impact for tan β 1, i.e. they can also not account for R(K ( * ) ) [179,180].
Therefore, both charged and neutral current anomalies are challenging for the 2HDM of types I and II. If the anomalies turn out to be true, other forms of 2HDMs with more freedom to account for the data will be needed. In any case the exploration of the parameter space of 2HDMs, with their important interplay of different observables from quark and lepton flavor physics as well as high-p T measurements, will remain a cornerstone of NP studies.
Note that in order to probe NP in b → cτ ν, it has to be accounted for the additional complication that the measurements e.g. of R(D ( * ) ) itself also depend on the specific model, see Ref. [55] for details.
We follow here a model-independent way of presenting our results. In Sec. II we introduce the notation for differential b → cτ ν decay rates in the SM and scalar models, including rates for fixed V -polarization and fixed τ -polarization, respectively. We make explicit how these decay rates are related to b → clν decay rates to light leptons l = e, µ. In Sec. III we present the relations between different decay modes and derive implications for bin-wise integrated rates as well as the LFU observables R(V ) and R(P ). In Sec. IV we give numerical results for current and hypothetical future data, after which we conclude in Sec. V.

A. SM Decay Rates
For the Standard Model (SM) expressions of B q → {V, P }τ ν decays like B → D * τ ν, B s → D * s τ ν, B c → J/ψτ ν and B → Dτ ν, B s → D s τ ν, B c → η c τ ν we employ the notation of Refs. [11,13,51] dΓ where Here we use furthermore and equivalently, the dimensionless variable The corresponding physical ranges of these are given as Note that dΓ/dw and dΓ/dq 2 are connected by the Jacobian τ,2,TH depends on the form factors P 1 and f 0 , respectively, which can be provided by Lattice QCD or Heavy Quark Effective Theory (HQET). They are related as follows to the convention of Ref. [50] (BGL), see Table I in Ref. [13], where [11,50] Note that dΓ {V,P } τ,1,EXP /dw contains only information from decays to light leptons dΓ {V,P } EXP /dw, see Eq. (11). The latter is given in terms of helicity amplitudes as [51,83,84,181,182] The analogous expressions for heavy final lepton states are i.e. dΓ {V,P } τ,2 /dw is proportional to the additional longitudinal helicity amplitude H 2 {V,P },0t . One can measure the decay rates with a fixed D * helicity, and thereby measure each squared helicity amplitude in Eq. (27) separately. We write the corresponding decay rates as They fulfill by definition The corresponding decay rates to τ -leptons are related to those for light leptons as Similarly, for the decay rates with polarized τ -leptons of helicity ±1/2 we write and where the expressions in terms of helicity amplitudes can be found in Refs. [83,84].
From these we can read off that

B. Scalar Model Decay Rates
For the NP part of the effective theory of a charged scalar that contributes to b → cτ ν, we adapt the notation of Ref. [84], where we implicitly use the Wilson coefficients at the m b -scale. We consider only additional scalar couplings to heavy leptons. For sum and difference of these couplings we use the For scalar models it is known that the only modification that enters B q → V τ ν and B q → P τ ν contribute to the longitudinal helicity amplitudes and are proportional to the form factors P 1 and f 0 , respectively [83,84].
The reason is that from applying the Ward identity one obtains [83] V Furthermore, for B → Dτ ν it follows [183] P Therefore, in scalar models [83,84] where on the left hand side are only quantities that can be measured directly, whereas on the right hand side are theoretical parameters only. We have furthermore [84,133,184] where is the SM expression for B(B c → τ ν) and we write

A. Relations for Differential Rates
We present now a method to differentiate between the SM and scalar models and compare different hadronic b → cτ ν decay modes in a very direct way. In order to do so, only theory input on the respective mode-dependent dΓ {V,P } τ,2,TH /dq 2 is necessary. Using the decay rate expressions introduced in Sec. II, that knowledge makes it possible to isolate q 2 -dependent functions which do not depend on the concrete decay channel anymore, thereby in turn connecting different decay channels: and with the functions and in the SM, trivially The slope and curvature of S ∆C (q 2 ) and S ΣC (q 2 ) are directly related to scalar NP parameters.
The notation "∀ B q → V τ ν" and "∀ B q → P τ ν" implies that the relations hold equally for all decay channels like B → D * τ ν, B s → D * s τ ν, B c → J/ψτ ν, and B → Dτ ν, B s → D s τ ν, B c → η c τ ν, respectively, with the same respective q 2 -dependent left hand side.
Eqs.  Table II of Ref. [87], and that we RGE-evolve [86,185] down to the m b -scale. It is understood, that for a given decay channel the shown curve is only valid between the endpoints m 2 τ < q 2 < q 2 max ({V, P }), see Eq. (56).
The region q 2 < m 2 τ is unphysical. From the curvature and slope of S {∆C, ΣC} (q 2 ) one can directly extract the NP parameters.
relations is a hint for scalar models. In the SM-limit Eqs. (48) and (51) which is decay-mode dependent. Note further, that the relations hold as a function of q 2 .
For different decay channels, a given q 2 point corresponds to different w values, see Eq. (16).
That is why we employ dΓ {V,P } τ /dq 2 here, rather than dΓ {V,P } τ /dw. In Fig. 1, we show the q 2 -dependence of Eqs. (48)-(52) for example values of C L,R which correspond to minima that are found in global fits to the available b → cτ ν decay data [86,87].
On top of the above relations that allow the differentiation between SM and scalar models by measuring the characteristic q 2 -dependence, we have additional relations between the decays to τ leptons and light leptons that do not allow the differentiation between SM and scalar models, but only the one of other models from the SM and scalar models. These are Again, vector and tensor models would violate these relations.
Eqs. (48)-(52) can also be used in order to test form factor calculations. In the ratios scalar NP cancels out, i.e. we can check the ratios P V 1 1 (q 2 ) 2 /P V 2 1 (q 2 ) 2 and f P 1 0 (q 2 ) 2 /f P 2 0 (q 2 ) 2 directly from data, relying not anymore on the SM, but on the weaker assumption that at most scalar NP is present. Of course, more general NP would invalidate this test. However, this would then also be seen in the violation of Eqs. (57), (58).
Comparing to results present in the literature, the analytic relations found here are different from the numerical sum rule for the integrated observables R(D * ), R(D) and R(Λ c ) in Eqs. (28), (29) of Ref. [86], see also Ref. [87]. While Eqs. (48)-(52) are model-specific, i.e. can be used to differentiate between models, the sum rule in Refs. [86,87] is valid for any NP model and can thus be used as a consistency check of the data.

Bin-wise Relations
In practice, only binned measurements of the q 2 -dependent decay rates are performed.
The integration of the relations Eqs.
and completely analogous equations for the decay rates with fixed D * -and τ -polarization, respectively. We stress that it is implied that Eqs. (60) and (61)  In practice it is challenging to obtain the integrals in Eqs. (60) and (61) and again analogous equations for the decay rates with fixed D * -or τ -polarization.
If the above equations are applied to multiple bins of one or several decay modes, it can be directly solved for the NP parameters. Furthermore, it can in principle be solved for the NP parameters multiple times, generating additional relations. In the next section we make this explicit for the case where the bin is the complete q 2 -range.

Relations for R(V ) and R(P )
We discuss now the special case of Eqs. (62), (63) when the bin that we integrate over is the complete q 2 -range. To that end we define with Γ {V,P } being the integrated decay rate for decays to light leptons, so that R(V ) and R(P ) are defined as usual.

Approximate Relations
In the limit of a small NP contribution, i.e. in case that we find approximate relations that are simpler than the ones derived in Sec. III B 2. From Eqs. (69), (71) we have in this case and When |∆C| 2 is not known, a check of Eqs. (81) and (82) is not available. However, the conditions Eqs. (81), (82) also imply the weaker inequalities which can be used for a consistency check after the extraction of Re(∆C) through Eq. (84).
Analogously, for semileptonic decays to pseudoscalars we have the approximate relations 2 Re (ΣC) = ∆R(P ) Eq. (84) for V = D * and V = J/ψ. Note that no direct measurement of R EXP τ,1 (D * ) is available, so that we use its SM value, see Eq. (67). We use the fit results for B → D * τ ν from Ref. [16], including R(D * ) SM as given in Eq. 6, which employs recent data on decays to light leptons [1,15,49], as well as HQET input for P 1 , see Ref. [16] for details. For the needed integrals we obtain For B c → J/ψτ ν we use Eqs. (7), (8) and the fit results provided in Ref. [26]. We obtain for the needed integrals Therein, we also take into account the correlations between the z-expansion coefficients of the form factors of B c → J/ψτ ν provided in Ref. [26], however, we do not take into account further correlations like with the form factor coefficients of B → D * τ ν. Note that our input from Refs. [16,26] takes into account statistical and systematic errors, and so do consequently also our numerical results. We use furthermore [186] f Bc = (0.434 ± 0.015) GeV .
As input for B → D * τ ν and B c → J/ψτ ν we employ again the fit results from Refs. [16,26].
Furthermore, we vary B(B c → τ ν) in the conservative region 0 ≤ B(B c → τ ν) ≤ 0.6 [86,87], see also Refs. [121,133,184,[187][188][189]. We use the fit results Eqs. (90)- (93) in Eq. (98) and for simplicity use Gaussian error propagation without correlations to calculate the error of R(J/ψ). We obtain thereby the scalar model prediction which has a 1.7σ tension with the current measurement Eq. (7). In order to further explore the implications of Eq. (98), we consider a hypothetical future data set given in Table I, and motivated from prospects at Belle II and LHCb. At 50 ab −1 Belle II expects a relative error on R(D * ) of (±1.0 ± 2.0)%, see Table 50 in Ref. [61]. At 50 fb −1 LHCb expects an absolute precision, combining statistical and systematical errors, for R(D * ) of ∼ 0.006 and for R(J/ψ) of ∼ 0.05, see Fig. 55 in Ref. [60]. With the input of R(D * ) from Table I, we find the prediction Eq. (99) almost unchanged, This highlights the importance of a future improvement of the theory uncertainty of the scalar form factors. However, the deviation of R(J/ψ) EXP as given in Table I would amount in this scenario to an exclusion of scalar models by 7.2σ.
Note that with future data of course many more opportunities arise to apply the methods presented above, when the spectrum of b → cτ ν decays is measured. This will further enhance the possible significances for the exclusion of models, as well as the ability to detect NP.

V. CONCLUSIONS
We find relations between differential decay rates of different b → cτ ν decay modes in scalar models. The relations are given in Eqs. We note that a generalization of these results to b → uτ ν decays seems straight forward.
Note that in case the anomalies turn out to be a statistical fluctuation, the 2HDM type II would again be a very important and viable candidate for further studies. In that case, and disregarding the flipped sign solution, Higgs data shows that we are close to the alignment limit cos(β − α) = 0, see the constraints on the parameter space of tan β vs. cos(β − α) from ATLAS and CMS Run I+II in Fig. 11 of Ref. [190]. However, without imposing a symmetry it would actually be unnatural if the alignment limit was fulfilled exactly, which raises the interest in the parameter space with 1% cos(β − α) 10% and the corresponding more stringent bounds in that region, roughly overall about tan β 15.
Future experimental results will show if the charged current anomalies are indeed true.
With future theoretical results on the scalar form factors from lattice QCD as well as experimental measurements of the q 2 -dependence of b → cτ ν decays, using the above methodology we will then be able to improve the probes for new physics in a very direct and clear way.

ACKNOWLEDGMENTS
We thank Paolo Gambino, Martin Jung, Henry Lamm and Richard Lebed for discussions.
The work of A.S. is supported in part by the US DOE Contract No. de-sc 0012704. S.S. is