Signature of Lepton flavor universality violation in $B_s \to D_s\tau\nu$ semileptonic decays

Deviation from the standard model prediction is observed in many semileptonic $B$ decays mediated via $b \to c$ charged current interactions. In particular, current experimental measurements of the ratio of branching ratio $R_D$ and $R_{D^{\ast}}$ in $B \rightarrow D^{(*)}l \nu$ decays disagree with standard model expectations at the level of about $4.1\sigma$. Moreover, recent measurement of the ratio of branching ratio $R_{J/\Psi}$ by LHCb, where $R_{J/\Psi} = \mathcal B(B_c \to J/\Psi\,\tau\nu)/\mathcal B(B_c \to J/\Psi\,\mu\nu)$, is more than $2\sigma$ away from the standard model prediction. In this context, we consider an effective Lagrangian in the presence of vector and scalar new physics couplings to study the implications of $R_D$ and $R_{D^{\ast}}$ anomalies in $B_s \to D_s\,\tau\nu$ decays. We give prediction of several observables such as branching ratio, ratio of branching ratio, forward backward asymmetry parameter, $\tau$ polarization fraction, and the convexity parameter for the $B_s \to D_s\,\tau\nu$ decays within the standard model and within various new physics scenarios.


I. INTRODUCTION
There are several reasons to believe that standard model (SM) of particle physics is not a complete theory, and thus there must be physics beyond the SM. It is therefore crucial to find the pattern of the New Physics (NP) that is responsible for various long standing anomalies. The underlying framework of SM assumes that the charge and neutral leptons are universal in the weak interaction. However, various recent studies on semileptonic B decays such as B → D ( * ) lν, with l either e, µ, or τ , challenged the lepton flavor universality [1]. From past few years, many experiments such as B−factories have reported observables that are deviating from the SM prediction. In particular, the ratio of branching ratio R D and R D * in B → D ( * ) lν are measured to have large discrepancy with respect to its SM counterpart.
A very precise SM prediction of the ratio of branching ratio R D in B → D l ν using the form factors obtained in lattice quantum chromodynamics (QCD) is reported to be 0.300 ± 0.008 [2][3][4][5]. Similarly for R D * , it was reported to be 0.252 ± 0.003 [6]. Comparing with the current world average of R D = 0.403 ± 0.040 ± 0.024 and R D * = 0.310 ± 0.015 ± 0.008 from BABAR [7], Belle [8][9][10], and LHCb [11], the combined deviation currently stands at about 4.1σ. For definiteness, we report in Table-I the current status of experimentally measured ratio of branching ratio R D and R D * [12]. Various studies in explaining the observed anomalies in B meson decays can be found in . Very recently, LHCb has measured the ratio of branching ratio R J/ψ to be 0.71 ± 0.17 ± 0.18 [42]. Comparing with the SM prediction [43][44][45], we find the deviation to be at more than 2σ. Although a precise calculation of B c → J/Ψ form factors is not available at present, a preliminary results for the form factors are provided by HPQCD collaboration using Lattice QCD [46].
Inspired by the anomalies present in B → (D, D * )τ ν decays, we study the corresponding B s → D s τ ν semileptonic

II. PHENOMENOLOGY
The natural way to introduce NP effects in a model independent approach is to construct a effective Lagrangian for the weak decays that includes both SM and the beyond the SM physics. We follow Refs. [64,65] and write the effective weak Lagrangian for the b → c τ ν quark level transition decays in the presence of vector and scalar type NP interactions as where, G F is the Fermi coupling constant and |V cb | is the Cabibbo-Kobayashi-Mashkawa (CKM) matrix element. The effective Lagrangian of Eq. 1 is considered at renormalization scale µ = m b . The NP Wilson coefficients (WCs) denoted by V L , V R , S L , and S R involve left-handed neutrinos, whereas, the WCs denoted byṼ L ,Ṽ R ,S L , andS R involve right-handed neutrinos, respectively. Assuming all NP WCs to be real in the present analysis we rewrite the above equation as [13], where, The B s → D s l ν decay amplitude depends on non perturbative hadronic matrix element which can be parametrized in terms of B s → D s transition form factors as follows.
where, q µ = P µ Bs − P µ Ds refers to the momentum transfer. It should be mentioned that we use the equation of motion to find the scalar matrix element. We follow Ref. [59] for the relevant form factors f 0 (q 2 ) and f + (q 2 ). The expressions pertinent for our discussion are [59] Here, P 0,+ are Blaschke factors and M 0 = 6.42 (10) GeV and M + = 6.330 (9) GeV are the resonance masses. We refer to Ref. [59] for all the omitted details. The differential decay distribution for the B s → D s l ν decays can be expressed as where | P Ds | = λ(m 2 Bs , m 2 Ds , q 2 )/2 m Bs is the three momentum vector of the outgoing meson and λ(a, b, c) = a 2 + b 2 + c 2 − 2 (a b + b c + c a). Note that θ denotes the angle between the D s meson and the lepton three momentum vector in the (l ν) rest frame. The covariant contraction L µν H µν can be calculated using the helicity techniques of Refs. [66,67]. For completeness, we present here the final expression for the differential decay distribution of three body B s → D s l ν decays [13]. where By performing the cos θ integration in Eq. 8, we get Setting G V = G A = 1 and all other NP couplings to zero, we obtain We define several q 2 dependent observables such as differential branching ratio DBR(q 2 ), ratio of branching ratio R(q 2 ), lepton side forward backward asymmetry A l F B (q 2 ), polarization fraction of the charged lepton P l (q 2 ), and convexity parameter C l F (q 2 ) for the B s → D s l ν decays. Those are where dΓ(+)/dq 2 and dΓ(−)/dq 2 represent differential decay width of positive and negative helicity leptons, respectively. In the presence of various NP, the explicit expressions for A l F B , dΓ(+)/dq 2 , dΓ(−)/dq 2 , and C l F are The SM expressions are obtained by setting all the NP couplings to zero.
The average values of the forward-backward asymmetry of the charged lepton < A l F B >, the longitudinal polarization fraction of the lepton < P l >, and the convexity parameter < C l F > are obtained by separately integrating the numerators and denominators over q 2 .

A. Input parameters
Before proceeding for the analysis, we report in Table-II all the input parameters that are relevant for our numerical computation. For the mass and lifetime parameter, we use the latest values reported in Ref. [68]. Similarly for the CKM matrix element |V cb | and the Fermi coupling constant G F , we use Ref. [68]. The lepton masses (m e , m τ ) and meson masses (B s , D s ) are in GeV units, whereas Fermi coupling constant G F is in GeV −2 units. The lifetime of B s meson (τ Bs ) is in seconds. The quark masses m b (m b ) and m c (m b ) evaluated at renormalization scale µ = m b are in GeV units. For the relevant B s → D s form factor parameters, we follow the most recent Lattice QCD calculation of Ref. [59]. The uncertainties associated with |V cb |, and the form factors parameters are written within parenthesis. We do not report the uncertainties associated with other input parameters as they do not play an important role in our analysis.
We wish to determine the consequences of various NP couplings on various observables for the B s → D s τ ν decays in a model independent way. It is, therefore, crucial to determine the size of the SM uncertainties in each observable that may come from various input parameters. Uncertainties in the theoretical prediction of the observables mainly come from two sources. First, it may come from not very well known CKM matrix element |V cb | and second, it may come from the non perturbative hadronic inputs such as decay constant and meson to meson form factors. To gauge the effect of above mentioned uncertainties on various observables, we use a random number generator and vary these input parameters within 1σ of their central values.  First, we wish to give prediction of various observables for both the e and τ mode within the SM. We report in Table.-III the SM central values and the 1σ ranges of each observable for the e and the τ modes. Here, the central values are obtained by considering only the central values of theory input parameters whereas, the 1σ ranges of each observable is obtained by performing a random scan of the hadronic parameters and the CKM matrix element within 1σ of their central values. The value of ratio of branching ratio R Ds in Table.-III is quite similar to the value reported in Ref. [59]. The slight difference may come from different choices of input parameters.  We notice that the SM prediction for the e mode is quite different from the τ mode. There is even a sign change in the polarization fraction P l while going from the e to the τ mode. Again, the forward backward asymmetry parameter for the e mode is zero, whereas, it is non zero positive for the τ mode. Similarly, the convexity parameter C l F for the e mode is much larger in magnitude than for the τ mode. It is worth mentioning that the mass of the charged lepton plays an important role. In Fig. 1, we show the q 2 dependence of each observable for the e and the τ modes, respectively. We notice that the A l F B (q 2 ), P l (q 2 ) and the C l F (q 2 ) observables remain constant in the entire q 2 region for the e mode. This could be very well understood from Eq. 14. In the massless limit, i.e, in the m l → 0 limit, the q 2 dependence gets cancelled in the ratio for the A l F B (q 2 ), P l (q 2 ) and the C l F (q 2 ) observables. Now we proceed to discuss various NP effects in B s → D s τ ν decays.

C. New physics in Bs → Ds τ ν decays
Study of B s → D s τ ν decays both theoretically and experimentally is well motivated because of the long standing anomalies present in R D and R D * . We wish to study the implication of these existing anomalies on the B s → D s τ ν decays in a model independent way. We consider four different NP scenarios based on NP contributions from two different operators. In order to determine the allowed NP parameter space, we impose 1σ constraint coming from the measured ratio of branching ratios R D and R D * . We use the average values of R D and R D * reported in Table. I in our analysis. For the uncertainties we added the statistical and systematic uncertainties in quadrature. Again, we assume that only the third generation leptons get contribution from NP.

Scenario I: only VL and VR NP couplings
In this scenario, we vary V L and V R and set all other NP couplings to zero. This is to ensure that NP contribution to the B s → D s τ ν decay mode is coming only from vector type NP couplings that involves left handed neutrinos. In the presence of such NP, the dΓ/dq 2 , R(q 2 ), A τ F B (q 2 ), P τ (q 2 ), and C τ F (q 2 ) can be expressed as dΓ dq 2 It is evident from Eq. 15 that dΓ/dq 2 and R(q 2 ) depend on V L and V R NP couplings and are proportional to G 2 V , whereas, P τ (q 2 ), A τ F B (q 2 ), and C τ F (q 2 ) do not depend on these NP couplings since the contribution coming from V L and V R NP couplings gets canceled in the ratio. The allowed ranges of V L and V R after imposing 1σ constraint coming from R D and R D * are shown in the left panel of Fig. 2. In the right panel we show the corresponding ranges in B(B c → τ ν) and B(B s → D s τ ν). From the right panel of Fig. 2, we notice that the B(B c → τ ν) obtained in this scenario lies in the 2% − 3% range. This is consistent with the SM calculation. We report in Table-IV  ranges of each observable for the B s → D s τ ν decays with (V L , V R ) NP couplings of Fig. 2. We see significant deviation in B(B s → D s τ ν) and R Ds from the SM prediction. As expected, the ranges of P τ Ds , A τ F B , and C τ F do not vary at all with such NP couplings.
We show in Fig. 3 the q 2 dependence of various observables with the allowed values of V L and V R NP couplings of Fig. 2. The SM 1σ range is shown with blue band, whereas, the allowed range with V L and V R NP couplings is  shown with green band. It is evident from Fig. 3 that the differential branching ratio DBR(q 2 ) and ratio of branching ratio R(q 2 ) deviate considerably from the SM expectation. Again, as expected, we do not observe any deviation of A τ F B (q 2 ), P τ (q 2 ) and C τ F (q 2 ) from the SM expectation in this NP scenario.

Scenario II: only SL and SR NP couplings
In this scenario, we consider the effect of new scalar couplings only, i.e,(S L , S R ) = 0 and all the other NP couplings are zero. In the presence of S L and S R NP couplings, the differential decay width, ratio of branching ratio, forward backward asymmetry, polarization fraction of the τ lepton, and the convexity parameter can be expressed as , We impose 1σ constraint coming from experimental values of R D and R D * to determine the allowed values of S L and S R NP couplings. The resulting (S L , S R ) allowed ranges are shown in the left panel of Fig. 4.. In the right panel, we show the corresponding ranges in B(B c → τ ν) and B(B s → D s τ ν) obtained using the allowed values of (S L , S R ) NP couplings. It should be noted that the B(B c → τ ν) obtained in this scenario is rather large; more than 30%. Thus, although (S L , S R ) NP couplings can simultaneously explain the anomalies present in R D and R D * , it, however, fails to satisfy the B(B c → τ ν) ≤ 30% constraint obtained in the SM. Although this particular scenario is ruled out by   the B(B c → τ ν) constraint, nevertheless, we report in Table. V the allowed ranges of all the observables obtained using the allowed values of S L and S R NP couplings of Fig. 4. The deviation from the SM prediction observed in this scenario is quite significant. We notice that the forward backward asymmetry parameter can assume negative values within this scenario, which is quite distinct from SM expectation. We show the effect of NP on various q 2 dependent observables in Fig. 5. We show with blue the SM 1σ band, whereas, we show with green the allowed band once the NP is switched on. The deviation observed in this scenario is rather large and it is, indeed, more pronounced that the deviation obtained with (V L , V R ) NP couplings. Unlike scenario I, there is no cancellation of NP effects in A τ F B (q 2 ), P τ (q 2 ), and C τ F (q 2 ). We notice that, although there is no zero crossing in the SM for the A τ F B (q 2 ) parameter, we may observe zero crossing depending on the values of S L and S R NP couplings. Similar conclusion can be drawn for the τ polarization fraction P τ (q 2 ) as well. Moreover, depending on the values of the NP couplings, shape of the q 2 distribution curve of each observable can be quite different from its SM counterpart.

Scenario III: only VL and VR NP couplings
To study the effect of new vector type NP couplings associated with right handed neutrino interactions, we consider ( V L , V R ) to be nonzero while all other NP couplings to be zero. In this scenario, the differential decay width, ratio of branching ratio, forward backward asymmetry, τ polarization fraction, and the convexity parameter take the following simple form: In the left panel of Fig. 6 we show the allowed region of ( V L , V R ) NP couplings that is obtained once 1σ R D and R D * experimental constraint is imposed. Similarly, in the right panel we show the corresponding ranges of B(B c → τ ν) and B(B s → D s τ ν) obtained with the ( V L , V R ) NP couplings. Similar to Scenario I, we notice that B(B c → τ ν) obtained in this scenario lies within (2 − 3)% range. This is, again, consistent with the SM prediction. In Table. VI, we report the possible ranges of all the observables for the B s → D s τ ν decays. The deviation from the SM prediction observed in this scenario is quite similar to the deviation observed in scenario I. However, there is one subtle difference. Unlike scenario I, a significant deviation from the SM prediction for the τ polarization fraction is observed in this scenario. This is evident from Eq. 17 that the NP effect does not get cancelled for the τ polarization fraction P τ . In Fig. 7, we show the variation of various observables such as differential branching ratio, ratio of branching ratio, forward-backward asymmetry, τ polarization fraction, and convexity parameter as a function of q 2 . The deviation from the SM prediction observed in this scenario is quite similar to scenario I. As expected, we observe a significant deviation in τ polarization parameter in this scenario. All the above mentioned analysis for the observed deviations are clearly reflected in Eq. 17.   In this scenario, we wish to see the effect of new scalar type NP couplings on various observables. To this end, we consider ( S L , S R ) to be non zero and all other NP couplings to be zero. In this scenario, the differential decay width, ratio of branching ratio, forward backward asymmetry, τ polarization fraction, and the convexity parameter take the following form: In order to determine the allowed ranges of ( S L , S R ) NP couplings, we impose 1σ constraint coming from experimentally measured values of R D and R D * . The resulting NP parameter space, shown in the left panel of Fig. 8, can simultaneously explain the anomalies present in R D and R D * . We show in the right panel the allowed ranges in B(B c → τ ν) and B s → D s τ ν with such NP. We notice that the B(B c → τ ν) obtained in this scenario is not compatible with the upper bound of B(B c → τ ν) ≤ 30% obtained in the SM. The numerical values written in the square brackets of Table-VII represent the allowed ranges of observables obtained with the allowed values of ( S L , S R ) of Fig. 8. Similar to scenario II, we see significant deviation of all the observables from the SM expectation. We show in Fig. 9 the q 2 distribution of various observables for the B s → D s τ ν decays. The blue band corresponds to the 1σ SM range, whereas, the green band corresponds to the range of the observables once the ( S L , S R ) NP couplings are switched on. The deviation observed in this scenario is rather large. We notice that although, in the SM, there is no zero crossing in the τ polarization parameter, there may or may not be a zero crossing depending on the values of the NP couplings. For the differential branching ratio, the peak of the q 2 distribution may shift towards high q 2 region.

IV. CONCLUSION
In view of the long standing anomalies in R D and R D * , we study the corresponding B s → D s τ ν semileptonic decays in a model independent framework. We use the helicity formalism to study the B s → D s τ ν semileptonic decays within the context of an effective Lagrangian in the presence of NP and explore four different NP scenarios based on contributions coming from two different NP operators. We give prediction on various observables such as branching ratio, ratio of branching ratio, forward backward asymmetry, longitudinal polarization fraction of the charged lepton, and the convexity parameter for this decay mode within SM and within four different NP scenarios.
We first report the central values and the 1σ ranges of each observable within the SM for both the e and the τ modes. We notice that all the observables change considerably while going from the e mode to the τ mode. The value of R Ds is quite similar to the value reported in Ref. [59]. We also give the first prediction of the longitudinal polarization fraction of the charged lepton, lepton side forward backward asymmetry, and the convexity parameter for the B s → D s l ν decays.
For our NP analysis, we assume that NP effects are coming from vector and scalar type NP couplings only. We notice that NP scenarios with (V L , V R ) and ( V L , V R ) NP couplings are compatible with the B(B c → τ ν) constraint. However, NP scenarios with (S L , S R ) and ( S L , S R ) NP couplings are ruled out due to the constraint coming from the lifetime of B c meson.
Study of B s → D s τ ν decays both theoretically and experimentally is crucial because it may provide new insights into the R D and R D * anomaly as this decay mode is mediated via the same b → c charged current interaction. Moreover, a precise determination of the branching fractions of this decay mode will allow an accurate determination of the CKM matrix element |V cb |.