Search for more sensitive observables to charged scalars in $B \rightarrow D^{(*)}\tau\nu_{\tau}$ decays

It has been known that $B \rightarrow D^{(*)} \tau \nu_{\tau}$ are good observables in the search for the charged Higgs. The recent obervation of deviation from standard-model by almost 4$\sigma$ by Babar, Belle and LHCb in $R(D^{(*)})$ revived the interest in possible signal of presence of charged Higgs in these modes. But such a large deviation in the rates, where standard-model has tree level contribution, coming from a charged Higgs alone is highly unlikely. However these decay modes are good probes to search for small charged Higgs signal if we can construct sensitive observables in these modes. In this work we would like to propose four new observables which shows much more sensitivity to the presence of charged Higgs than the usual observables such as $A_{\lambda}^{D^{(*)}}$ and $A_{\theta}^{D^{(*)}}$. These four observable are (1) $\frac{1}{A_{\lambda}^{D}}$,\ (2) $Y_{1}(q^{2}) = \frac{A^{D}_{\theta}}{A^{D}_{\lambda}}$,\ (3) $Y_{2}(q^{2}) = \frac{d\Gamma(B \rightarrow D^{*}\tau\nu_{\tau})}{d\Gamma_{D}(\lambda_{\tau}=+1/2) - d\Gamma_{D}(\lambda_{\tau}=-1/2)}$ and (4) $Y_{3}(q^{2}) = (\frac{q^{2}}{m^{2}_{\tau}})(A^{D}_{\lambda} + 1)\frac{1}{A^{D}_{\lambda}}$.

The LHC discovery of a scalar behaving like the standard-model (SM) Higgs boson [1] marks the tentative experimental completion of SM with all the particles it predicted observed experimentally.
But even after the LHC discovery of SM like Higgs, still its clear that it is not complete because in SM there is no explanation of Dark Matter and Dark Energy, CP violation due to KM weak phase is turn out to be too small to account for the observed baryon asymmetry of the universe, then there is the strong CP problem and also the fine tunning problem in renormalization of Higgs mass etc. Hence it is pretty evident that we require new-physics (NP) at some scale above about 200 GeV. The absence of clear cut NP signal from both flavor and collider experiments till date may indicate that the scale of NP is much higher than the electro-weak scale. However there are many loophole for low mass NP in current direct search by LHC due to sensitivity limits of LHC to light weakly coupled particles.
Babar [2] and Belle [3] have ruled out 2HDM type-II at 99.8% CL from disagreement of its prediction with data as an explanation of the anomalies in R(D) and R(D * ). From the on set it is very easy to see that this anomalies can be explained by a non universal left handed vector particle but a simple non-universally interacting heavier gauge boson (W ′ ± ) is highly constrained by null results from LHC search for W ′ → tb signals [13] [14], and also by precision measurements in µ [15] and τ [16]. Therefore as of now it is very difficult to built a non-universal gauge model that can fit all the constrains and so in this paper we will mostly strick to a model-independent analysis only. It has been shown first in reference [17] that the observed excess in R(D ( * ) ) can be explained with baryon and lepton number conserving Lepto-quark (LQ) models and followed in with many special cases and variations of the LQ models has been proposed to explain not only R(D ( * ) ) but also observed deviations in R K = Br(B→Kµµ) Br(B→Kee) and the so called P ′ 5 anomalies. But some of these LQ models turn out to be not viable when all precision data till date are taken into account, for details see the recent review in [18]. In any case as of now the experimental inputs seems to be too few and far apart to build a complete and consistent NP model if at all NP shows up at the reach of the upgraded LHC and Belle-II. In following sections we will give a general model-independent analysis of possible contribution from charged scalar to these deviations and observables sensitive to their presence. This paper is organized as follows: In Section II we present the general formulism of the analysis and lay the theoretical framework of the paper. Section III contains an introduction to observable sensitive to NP. Section IV contains the core of this work and it deals with new and more sensitive observables to the presence of charged scalar NP. In section V we conclude the paper.

Theoretical Framework.
We assume that all the neutrinos is are left handed, then the most general effective Hamiltonian that contains all possible four-fermion operators of dimention four for the decay process b → clν l , where l = τ , µ or e here, is given as with the operators define as In Eqs. (5) we have explicitly shown the relative negative sign between effective four fermion operators due to exchange of heavy scalar particles and heavy vector particles. This is due to sign difference between a scalar propagator and a vector propagator 1 . In many analysis the relative sign is implicitly absorbed into the effective coefficients, but if the relative sign between the vector four current operators and the scalar four current operators are explicitly shown will help us rule out few models from the on set given that we expect NP contribution is less than the SM contribution. Fore instance, in 2HDM of type-I and type-II, the effective coupling are positive and so these type of models will interferes destructively with SM, due to the relative negative sign, and so 2HDM of type-I and type-II can only reduce the values of R(D ( * ) ) instead of increasing it as required by experiments in all the parameter spaces where the NP part is less than SM part. So it is clear from this that the relative sign can actually help us in ruling out all the models of new scalar particles whose effective coupling are nonnegative for the most parts of the parameter space where NP part is less than the SM part. In this work we will not deal with new vector and tensor terms. Now with only scalar and vector (SM) type operators remaining we can express the effective Hamiltonian in Eqs. (5) as where The most stringent B physics constrains on the scalar NP explanation of R(D ( * ) ) comes from the decay rates Br(B c → τ ν τ ) or Br(B u → τ ν τ ) depending on the particularities of the NP model. So in what follows we will take these observables and their measured bounds as additional constrains, wherever applicable, when fixing the coefficients of the effective operators to R(D ( * ) ) data. Assuming all hadronization are due to strong interaction, due to parity conservation of strong force, only scalar and vector current can contribute in R(D) and so it only constrains the ǫ s l and in the case of Br(B c → τ ν τ ) and Br(B u → τ ν τ ), only pseudo-scalar and axial-vector current can contribute and so these observables only constrain ǫ p l . However to R(D * ), both vector and axial-vector currents can contribute but only pseudo-scalar current can contribute and so R(D * ) constrains ǫ p l . In presence of charged scalar particle, the differential decay rate of B → D ( * ) τ ν τ can be expressed as (9) and For details of relation between vector, axial-vector, scalar, psuedo-scalar and tensor currents and their respective form factors see [19] [17]. For numerical values of the parameters in the form factors, we will use those given in [17] with exception that we will use R 3 (1) = 0.97 instead of R 3 (1) = 1.22 of that reference.
3 Observables sensitive to NP.
With lack of any persistent sign of NP from direct searches at LHC, the precision physics is becoming more and more important to at-least sense the direction of the possible nature of NP. So it has become crucial to find sensitive observables to NP that can be tested in flavor precision machines such as Belle II and LHCb etc. The remaining part of this work is concern with finding more sensitive observables than the usual ones like tau spin asymmetry, A D ( * ) λ , which is already defined in Eqs. (14) and forwardbackward asymmetries, A D ( * ) θ , which will be defined in the following sections. We will be mainly concerned with charged scalar NP and define four very sensitive new observables to charged scalar NP in this work.

Observables sensitive to non-scalar NP.
In case of new vector particles with substantial couplings to vector and axial-vector currents, since only vector current will contribute to hadronization in R(D), R(D) constrains only the vector coupling (1+ǫ v NP ). Where we will denote by ǫ v NP and ǫ a NP , the effective couplings of new vector particles to vector and axial-vector effective four currents respectively. Now since ) has some contribution due to new vector particles, then as pointed out in [19], the observable is independent from effects due to presence of any new scalar particles, and so this observable also should show excess similar to R(D * ), where R(D * L ) refers to the ratio for the longitudinally polarized D * . Another observable which can be used to check the presence of new non-scalar particles contributing to R(D ( * ) ) are define as [19] where the A D λ and A D * λ are the τ spin-asymmetry defined as and Similar to X 1 (q 2 ), any deviation in X D

Observables sensitive to charged scalar.
Besides R(D)(q 2 ) and R(D * )(q 2 ), we can define many more observables that are sensitive to the presence of new charged scalar particles in the B → D ( * ) τ ν τ decay distributions. One such observable is tau spin asymmetry (A D ( * ) τ ) which is already defined in Eqs. (14) and Eqs.(15) of section 3.1. Another is the forward-backward asymmetries define as which can be expresses as and where the bar over H t and H 0t refers to H t (1 − where the upper sign is for the τ lepton and lower sign is for the e and µ leptons.
Comparing Eqs. (21,22) to Eqs.(1), we can see that the model prediction fits the combine R(D ( * ) ) data within 1σ of the experimental values.

4.1
One of the the most sensitive new observable to charged scalars that we can construct turn out to be 1 A D τ given as This 1 A D τ observable has two key features that makes it a better observable than A D τ . First as seen where as for the A D λ we have From Eqs. (24) and Eqs. (25)  not that sensitive and so we will not use these observables in this work.
Another sensitive observable to charged scalar NP can be defined as In Figure 2 we have shown the plot of Y 1 (q 2 ) (left) and 1 Y 1 (q 2 ) (Right). 2 As seen form that Figure, besides showing prominent difference between SM and scalar NP, in the two plots, the Y 1 (q 2 ) is more sensitive towards the low q 2 region where as the 1 Y 1 (q 2 ) is more sensitive towards the high q 2 region. And one of the most important feature of these observables turn out to be the difference between the position of SM maxima and the scalar NP maxima in Y 1 (q 2 ). The SM (Blue) maxima occurs at q 2 = S = 6.038 with the maxima value of 1.589 where as the scalar NP (Red) maxima occurs 2 one may think that actually 2q 2 3m 2 τ Y1(q 2 ) will be more sensitive but it turns out that is not the case, in fact the opposite case turn out to be true!
so there is a difference of 0.221 between the q 2 integrated value of observable Y 1 in SM compared to the q 2 integrated value of observable Y 1 in scalar NP. This means that in the q 2 integrated value of Y 1 , we only require SM plus Experimental combine error to reduce below 0.044 to have a 5σ discovery potential of scalar NP with ǫ τ as small as -0.072.
We can also define another sensitive observable to charged scalar NP as and a plot of the the observable Y 2 (q 2 ) is shown in the Figure 3. One of the key feature of this observable is the gap between the SM maxima and NP maxima, which is 2.167. In this observable, the contrast between SM and scalar NP comes out more prominently than any of the other new observables for ǫ as small as -0.072. But in this observable the shift between the position of SM maxima and scalar NP maxima is very small, about 0.165. Y 2 (q 2 ) is very suitable to test especially models where both ǫ s and ǫ p gets substantial contribution.
where as so there is a difference of 1.359 between the q 2 integrated value of observable Y 2 in SM compared to the q 2 integrated value of observable Y 2 in scalar NP 3 . This implies that in q 2 integrated value of Y 2 , we only require SM plus Experimental combine error to reduce below 0.698 to have a 5σ discovery potential with ǫ τ as small as -0.072.
Yet another sensitive observable to charged scalar NP can be defined as Here ǫ τ s = ǫ τ p = −0.072 ± 0.028 is used from Eqs. (19).
Comparing the q 2 integrated value of about 1.216 for Y 3 above to the q 2 integrated value of about 0.551 for the 1 , it is clear that the observable Y 3 is much more sensitive observable to charged scalar . Also from Eqs.(32) we see that for observable Y 3 , we only require SM plus Experimental combine error to reduce below 0.243 to have a 5σ discovery potential of scalar NP with ǫ τ as small as -0.072. Similar observable can be defined from A D * λ , however this observable is not that sensitive to charged scalars due to supression of charged scalar coupling in D * final state by a factor of m b −mc

Conclusions.
In this work we have given four new observables which are very sensitive to the presence of charged scalars in the B → D ( * ) τ ν τ decays. All the new observables shows substantial deviation from SM values in two main features of them i.e (1) in presence of charged scalar, they all show substantial shift in the position of the maxima from that of the SM value and (2) in presence of charged scalar, they also show substantial deviation in their q 2 integrated value from that of the SM one. In the following we will enumerate the key results for each new observables from the preceding analysis. in this observable turn out to be 0.201. This implies that to have a 5 σ discovery potential