Impact of vector new physics couplings on $B_s \to (K,\,K^{\ast})\tau\nu$ and $B \to \pi\tau\nu$ decays

Experimental measurements of $R_{D}$, $R_{D^*}$ and $R_{J/\Psi}$ in $B \to (D,\,D^{\ast})l\nu$ and $B_c \to J/\Psi l \nu$ decays mediated via $b \to c\,l\,\nu$ charged current interactions deviate from standard model prediction by $2.3\sigma$, $3.4\sigma$ and $2\sigma$, respectively. In addition, a deviation of $1.5\sigma$ from the standard model prediction has been witnessed in $\mathcal B(B \to \tau \nu)$ mediated via $b \to u\,l\,\nu$ charged current interactions as well. Motivated by the anomalies present in $B$ and $B_c$ meson decays, we analyze the corresponding $B_s \to (K,\,K^{\ast})\,\tau\,\nu$ and $B \to \pi\tau\nu$ semileptonic decays within the standard model and beyond. We use an effective field theory formalism in which $b \to c$ and $b \to u$ semileptonic decays are assumed to exhibit similar new physics patterns. We give the predictions of various observables such as the branching fractions, ratio of branching ratios, lepton side forward backward asymmetry, lepton polarization fraction and convexity parameter for $B_s \to (K,\,K^{\ast})\tau \nu$ and $B \to \pi\tau\nu$ decay channels within the standard model and within various NP scenarios.


I. INTRODUCTION
The electroweak interactions which are mediated via Z 0 and W ± bosons are categorized into flavor changing neutral current and charged current interactions.Deviations from the standard model (SM) predictions are observed not only in decays mediated via the b → (c, u) charged current processes but also in decays mediated via the b → s neutral current process.The precise SM predictions of the ratio of branching ratios R D , R D * and R J/Ψ , where are 0.300 ± 0.008 [1][2][3][4], 0.252 ± 0.003 [5] and [0.25, 0.29] [ [6][7][8] for R D , R D * and R J/Ψ , respectively.On the other hand, the average experimental values reported by HFLAG are 0.407 ± 0.039 ± 0.024 and 0.304 ± 0.013 ± 0.007 for R D and R D * measured from BABAR [9], BELLE [10][11][12], LHCb [13] and 0.71 ± 0.17 ± 0.18 for R J/Ψ from LHCb [14] measurement.This amounts to a combined deviation of 4.1σ in R D and R D * [15] and around 2σ in R J/Ψ from the SM expectations.Similarly, discrepancy between the measured value and the SM value has been observed in the b → u quark level transition decays as well.Average value of the branching ratio B(B → τ ν) = (10.9± 2.4) × 10 −5 reported in Ref. [16] from BABAR [17,18] and Belle [19,20] measurements is not in good agreement with the SM expectations [21][22][23].However, the measured value of B(B → πlν) = (14.5 ± 0.5) × 10 −5 from BELLE [24][25][26] and BABAR [27][28][29][30][31] is consistent with its SM counterpart.The SM prediction, however, depends on not very well known CKM matrix element |V ub | and various meson to meson transition form factors.We define an observable in which the |V ub | dependency cancels in the ratio.That is where τ B 0 and τ B − are the lifetime of B 0 and B − mesons.Using the measured values of B(B → τ ν), B(B → π l ν) and the direct measurement of the ratio τ B 0 /τ B − = 1.076 ± 0.004 [16], we get R l π = 0.698 ± 0.155.In the SM, we obtain R l π = 0.566.This clearly shows a mild deviation from SM prediction.We also consider another useful observable which is potentially sensitive to NP, i.e, In the SM, we obtain R π = 0.641.Again, a naive estimate would give R π < 1.784 using the present world average of B(B → π l ν) = (1.45 ± 0.05) × 10 −4 [16] and the upper limit on B(B → πτ ν) < 2.5 × 10 −4 reported by Belle Collaboration [32].Similarly, by considering the branching fraction of B → πτ ν [32] and B(B → π l ν) [16], R π = 1.05 ± 0.51 was obtained in Ref. [33].These indirect hints of existence of NP led the physics community to look for various NP scenarios.There exists various model-dependent and model-independent analysis in the literature in order to explain these anomalies details of which can be found in Refs. .
In this paper, we are mainly interested to discuss the NP effects in B s → (K, K * )τ ν and B → πτ ν semileptonic decays mediated via b → uτ ν charged current interactions.Within the SM, the branching ratio and ratio of branching ratios of B s → (K, K * )τ ν and B → πτ ν decays have been studied extensively by various authors [84][85][86][87][88][89][90].Very recently, in Ref. [91], the authors have performed a model independent analysis of NP effects in B s → (K, K * )τ ν decays using the experimental constraints coming from B → τ ν channel.Our main aim is to study the implication of R D , R D * , R J/Ψ , and R l π anomalies in B s → (K, K * )τ ν and B → πτ ν semileptonic decays in a model dependent way.To this end, we use an effective theory formalism in the presence of NP and perform a combined analysis of b → u and b → c semileptonic decays.This is where we differ significantly from Ref. [91].Again, for various meson to meson transition form factors, we use very recent lattice QCD results of Refs.[88][89][90].More importantly, we give the first prediction of various observables such as τ polarization fraction and convexity parameter for B s → (K, K * )τ ν and B → πτ ν decays within the SM and within various NP scenarios.
The present discussion in this paper will proceed as follows.In section II, we first report the most general effective Lagrangian governing the b → (u, c) l ν weak decays in the presence of NP.We also report all the relevant formulas corresponding to the various meson to meson form factors in this section.The relevant expressions for all the observables in the presence of vector NP couplings obtained using helicity formalism are reported in section II.In section III, we report the results pertaining to all the observables within the SM and within various NP scenarios.Finally we conclude with a brief summary of our results in section IV.

II. METHODOLOGY
The most general effective Lagrangian for b → u l ν transition decays which includes both SM and beyond SM contributions is of the form [92,93] where the four fermion operators O W and O are defined as Here i = L, R and σ µν = i[γ µ , γ ν ]/2.The left and right projection are defined by P L, R = (1 ∓ γ 5 )/2.We note that W and W represent the complex Wilson coefficients (WCs) of NP contribution due to left handed and right handed neutrino interactions, respectively.The δ τ l restricts the NP effects only to the τ mode.Assuming the WCs to be real and considering NP contributions from the vector type NP couplings alone, the effective Lagrangian can be written as [56] L where, where q = p − p ′ is the momentum transfer.For the B s → (K, K * ) and B → π transition form factors we use the formulas and the input values reported in Ref [88][89][90].The final expressions of f 0 (q 2 ) and f + (q 2 ) for B s → K l ν decays are [88] P 0 (q 2 )f 0 (q 2 ) = Similarly, for the B → π transition form factors, the relevant expressions are [90] f + (q 2 ) = 1 where N z = 4 and Here M P refers to the mass of K or π meson, M 0 = m B * = 5.6794 (10)GeV and M + = 5.32520 (48)GeV represent the resonance masses.Again, for B s → K * form factors, the relevant expressions pertinent for our numerical analysis are [89] where t = q 2 and F (t) refers to the form factors V , A 0 , A 1 and A 12 , respectively.Here and where t 0 = 12 GeV and t ± = (M Bs ± M K * ) 2 .We refer to Refs.[88][89][90] for all the omitted details.
Using the effective Lagrangian of Eq. 6, the three body differential decay distribution for the B → (P, V ) l ν decays can be written as where L µν and H µν are the leptonic and hadronic current tensors.Here | P (P,V ) | = λ(m 2 B , m 2 (P,V ) , q 2 )/2m B with λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + bc + ca) represent the three momentum vector of the outgoing meson.One can use the helicity techniques for the covariant contraction of L µν and H µν details of which can be found in Refs.[94,95].We follow Ref. [56] and write the expression for differential decay distribution for B → (P, V ) l ν decays in terms of the helicity amplitudes H's and A's as follows: where θ is the angle between the P P, V and lepton three momentum vector in the l − ν rest frame and By performing the cos θ integration in Eq. 17, we get The SM equations can be obtained by setting Explicit expressions of the helicity amplitudes H's and A's are presented in Ref. [56].
The ratio of branching ratio is defined as where M = K, K * , π and l = µ.We also define various q 2 dependent observables such as differential branching ratio DBR(q 2 ), ratio of branching ratio R(q 2 ), 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 decay modes as follows: where dΓ (P,V ) (+)/dq 2 and dΓ (P,V ) (−)/dq 2 represents differential branching ratio of positive and negative helicity leptons, respectively.We also give predictions for the average values of the forward-backward asymmetry of the charged lepton < A l F B >, the convexity parameter < C l F >, and the longitudinal polarization fraction of the charged lepton < P l > which are calculated by separately integrating the numerator and the denominator over q 2 .It is worth mentioning that for the B q → (P, V )τ ν decays, the forward backward asymmetry parameter A τ F B (q 2 ), the τ polarization fraction P τ (q 2 ), and the convexity parameter C τ F (q 2 ) do not depend on V L NP coupling if we assume that the NP effect is coming from new vector interactions V L only.The NP dependency gets canceled in the ratio.On the other hand, although A τ F B (q 2 ) and C τ F (q 2 ) do not depend on V L NP coupling, the τ polarization fraction P τ (q 2 ), however, does depend on this NP coupling.Measurement of the τ polarization fraction P τ for these decay modes in future will be crucial to determine the exact nature of NP.Now let us proceed to the results and discussion.

A. Input parameters
We first list out the theory input parameters in Table I that II represents the respective form factor inputs for B s → Klν [88], B s → K * lν [89] and B → πlν [90] decays.For our analysis, we consider the uncertainties pertaining only to CKM matrix elements and form factor inputs.The number written within the parenthesis refers to the corresponding 1σ uncertainties.We also report the experimental input parameters R D , R D * , R J/Ψ and R l π with their uncertainties measured by various B factory experiments such as BABAR, BELLE and LHCb in Table III.In our analysis, we added the statistical and systematic uncertainties in quadrature.The 2σ range of each of the experimental input parameters is also reported in Table III.[14] and R l π .Second row reports the 2σ range of the respective ratio of branching ratios.

B. Standard model predictions
We first report in Table .IV the SM predictions of various observables such as branching ratio (BR), ratio of branching ratio (R), forward backward asymmetry parameter (< A l F B >), the polarization fraction of the charged lepton (< P l >), and the convexity parameter (< C l F >) for the B s → K l ν, B s → K * lν and B → πlν decay modes, where l is either a µ lepton or a τ lepton, respectively.We find the branching ratio of all the decay modes to be of the order of 10 −4 .We also give first prediction of various observables such as < P l > and < C l F > for these decay modes.The central values reported in Table .IV are calculated by considering the central values of the input parameters reported in Table I and Table II, whereas, for the 1σ ranges, we perform a random scan over the theoretical inputs such as CKM matrix elements and the form factor inputs within 1σ of their central values.We observe that all the observables differs significantly while going from the µ mode to the τ mode.The forward backward asymmetry parameter < A µ F B > for the B s → K µ ν and B → πµν decays is vanishingly small, whereas, < P µ > and < C µ F > are nearly equal to 1 and −1.5, respectively.Although, < P µ > for the B s → K * µν decays is quite similar to B s → K µ ν and B → πµν decays, the < A µ F B > and < C µ F > for the B s → K * µν decays are quite different from the B s → K µ ν and B → πµν decays.In Fig. 1, we show the q 2 dependency of all the observables for the µ mode and  the τ mode, respectively.We notice that the q 2 behavior of all the observables for the µ mode is quite different from the corresponding τ mode.Again, the forward backward asymmetry parameter A l F B (q 2 ), the τ polarization fraction P l (q 2 ), and the convexity parameter C l F (q 2 ) for the B s → K µν and B → πµν remain constant throughout whole q 2 region.Similarly, for the B s → K * µν decays, we observe that the τ polarization fraction P l (q 2 ) remains constant in the whole q 2 region.This is quite obvious as m l → 0 the q 2 dependency cancels in the ratio for these parameters.There is a zero crossing in the A τ F B (q 2 ) parameter for the B s → K * τ ν decays.However, we do not observe any zero crossing in the A τ F B (q 2 ) parameter for B s → Kτ ν and B → πτ ν decays.Similarly, we observe a zero crossing in the P τ (q 2 ) observable for all the decay modes.We now proceed to discuss various NP scenarios.

C. Beyond the SM predictions
We wish to determine the impact of NP on various observables pertaining to B s → (K, K * )τ ν and B → πτ ν decays.To this end, we use an effective theory formalism in the presence of vector type NP couplings and perform a model dependent analysis based on anomalies present in R D , R D * , R J/Ψ , and R l π as well as the requirement FIG. 1: q 2 dependent observables of Bs → K l ν (first column), Bs → K * l ν (second column) and B → π l ν (third column) decays in the SM for the µ (violet) and τ (green) modes.
B(B c → τ ν) ≤ 10% obtained from the LEP1 data [96].The branching ratio of taunic B c decays put a severe constraint on the scalar NP couplings [61].Hence we do not consider scalar NP couplings in our present analysis.We consider only two different NP scenarios based on NP contribution coming from V L and V L NP couplings.We consider only one NP WC at a time.We impose 2σ constraint coming from the measured value of R D , R D * , R J/Ψ , and R l π to determine the allowed NP parameter space.It should be mentioned that the NP contribution coming from V R NP couplings can not simultaneously explain the anomalies present in R D , R D * , R J/Ψ , and R l π within 2σ.Similarly, the NP contribution from V R NP coupling is exactly same as the contribution coming from V L NP coupling.Hence, we omit the discussion related to these NP couplings.

Scenario I: For VL NP coupling
In this scenario, we assume that NP contribution is coming only from V L NP couplings.We vary V L while keeping all other NP couplings to be zero.We show in the left panel of Fig. 2 the allowed range of V L NP coupling once the 2σ constraints from the measured values of R D , R D * , R J/Ψ , and R l π are imposed.In the right panel, we show the ranges in B(B → πτ ν) and R π obtained using the allowed ranges of V L NP coupling.Allowed ranges of B(B → πτ ν) and R π obtained in this scenario are compatible with the upper bound reported by Belle Collaboration.We also report the allowed ranges in the branching ratio and the ratio of branching ratios for the B s → (K, K * )τ ν and B → πτ ν decays in Table .V. We see a significant deviation from the SM prediction in the branching ratios and the ratio of branching ratios with V L NP couplings.Since the forward backward asymmetry parameter A τ F B , the τ polarization fraction P τ , and the convexity parameter C τ F do not depend on V L NP coupling, we do not observe any deviation from the SM prediction for these observables.
We show the q 2 dependence of differential branching ratio (DBR(q 2 )) and ratio of branching ratio R(q 2 ) for the B s → Kτ ν, B s → K * τ ν and B → πτ ν decays in Fig. 3.The SM range is shown with green band, whereas, the NP band obtained using the allowed values of V L NP coupling from Fig. 2 is shown with violet band.Again, as expected the remaining observables such as A τ F B (q 2 ), P τ (q 2 ) and C τ F (q 2 ) exhibit no deviations from SM expectation as the V L dependency cancels in the ratio.

Scenario II: For VL NP coupling
In this scenario, we vary only V L and set all other NP couplings to zero.This is to ensure that NP contribution is coming only from vector NP operator that involves right handed neutrinos.The allowed NP parameter space is obtained by using a 2σ constraint coming from the measured values of R D , R D * , R J/Ψ , and R l π .This is to ensure that the resulting NP parameter space can simultaneously explain the anomalies present is R D , R D * , R J/Ψ , and R l π .We show in the left panel of Fig. 4 the allowed range of V L in this scenario.The corresponding ranges in B(B → πτ ν) and R π , shown in the right panel of Fig. 4, are compatible with the upper bound reported by Belle Collaboration.We also report the ranges of the branching ratio, ratio of branching ratios and the τ polarization fraction for the B s → Kτ ν, B s → K * τ ν and B → πτ ν decays in Table.VI.We do not report the range of the forward backward asymmetry parameter A τ F B and C τ F since they do not depend on V L NP coupling.We see significant deviation from the SM expectation in the branching ratio, ratio of branching ratio, and the τ polarization fraction for these decay modes in this scenario.Although, the deviation observed in this scenario is quite similar to the deviation observed with V L FIG. 3: Differential ratios R(q 2 ) and differential branching ratios DBR(q 2 ) for Bs → Kτ ν (first column), Bs → K * τ ν (second column) and B → πτ ν (third column) decays using the VL NP coupling of Fig. 2 are shown with violet band, whereas, the corresponding SM ranges are shown with green band.The omitted plots such as A τ F B (q 2 ), P τ (q 2 ) and C τ F (q 2 ) are not affected by VL NP coupling.NP coupling, there is one subtle difference.Unlike scenario I, the τ polarization fraction P τ does depend on V L NP coupling.Measurement of P τ can, in principle, rule out either of these two scenarios.
In Fig. 5, we show the q 2 dependence of the ratio of branching ratio R(q 2 ), differential branching ratio DBR(q 2 ) and τ polarization fraction P τ (q 2 ) for the B s → Kτ ν, B s → K * τ ν and B → πτ ν decays, respectively.The remaining observables such as forward-backward asymmetry and convexity parameter are not affected by the V L NP coupling and hence we omit these results.The SM range is shown with green band, whereas the band obtained by using the allowed V L NP coupling is shown with violet.It is evident that we do observe deviations in R(q 2 ), DBR(q 2 ) and P τ (q 2 ) from the SM predictions in the presence of V L NP coupling.It is worth mentioning that measurement of τ polarization fraction will play a crucial role in distinguishing between these two scenarios.FIG.5: Differential ratios R(q 2 ), differential branching ratios DBR(q 2 ) and the τ polarization fraction P τ (q 2 ) for Bs → Kτ ν (first column), Bs → K * τ ν (second column) and B → πτ ν (third column) decays using the VL NP coupling of Fig. 4 are shown with violet band, whereas, the corresponding SM ranges are shown with green band.The omitted plots such as A τ F B (q 2 ) and C τ F (q 2 ) are not affected by VL NP coupling.

IV. CONCLUSION
Motivated by the anomalies present in R D , R D * , R J/Ψ , and R l π , we report the SM and beyond the SM predictions of various observables in B s → K τ ν, B s → K * τ ν and B → πτ ν decays in a model dependent way.We perform a combined analysis of the b → c and b → u charged current interactions using an effective field theory approach in the presence of vector NP couplings alone.We start our analysis with the SM predictions by providing the central values and 1σ ranges of each observable for B s → K l ν, B s → K * l ν and B → π l ν decay modes.We give the predictions for both µ and τ modes, respectively.Considerable changes are observed while going from µ mode to τ mode.The branching ratio for each decay mode is of the order of 10 −4 .We give the first prediction of various observables such as < A l F B >, < P l >, and < C l F > within the SM and within various NP scenarios.It is also evident that the q 2 dependence of all the observables for the µ mode is quite different from that of the τ mode.We observe that some observables for the µ mode remain constant throughout the whole q 2 region.
For the NP analysis, we consider two NP scenarios with new vector type operators that involve left handed as well as right handed neutrinos.We impose 2σ experimental constraints from the measured values of the ratio of branching ratios R D , R D * , R J/Ψ and R l π and obtain the allowed ranges in the NP couplings that can simultaneously explain all these anomalies.We give prediction of various physical observables such as the branching ratio, ratio of branching ratios, forward backward asymmetry, lepton polarization and convexity parameter for the B s → (K, K * )τ ν and B → πτ ν decay modes in each scenario.The deviation from the SM prediction with V L NP coupling is quite similar to the deviation observed with V L NP coupling.However, with V L NP coupling, there is deviation from the SM prediction in the τ polarization fraction P τ for all the decay modes.
Although there is hint of NP in semileptonic B decays mediated via charged current interactions, it is not yet confirmed.Study of B s → K l ν, B s → K * lν and B → π lν decay modes theoretically as well as experimentally is very well motivated as these can provide complementary information regarding NP.Again, it will have the direct consequence on predictions or the measurements of the CKM matrix element |V ub |.The precise value of |V ub | will are relevant for our numerical analysis.The theory inputs such as mass of pseudoscalar mesons (K, π), vector meson (K * ), leptons (m µ , m τ ), and quarks (m b , m c ) are in GeV units.m b (µ) and m c (µ) refer to the masses of b and c quarks evaluated at µ = m b renormalization scale.|V cb | and |V ub | are the corresponding CKM matrix elements for b → c and b → u transition decays.The Fermi coupling constant G F and the lifetime of B 0 (τ B 0 ) and B s (τ Bs ) mesons are in the units of GeV −2 and seconds, respectively.The entries in the Table

4 FIG. 2 :
FIG. 2: In the left panel we show the allowed ranges in VL NP coupling and the corresponding ranges in RD (violet), RD * (green), R J/Ψ (blue), and R l π (yellow) once 2σ experimental constraint is imposed.The corresponding ranges in B(B → πτ ν) and Rπ are shown in the right panel.

4 FIG. 4 :
FIG. 4: In the left panel we show the allowed ranges in VL NP coupling and the corresponding ranges in RD (violet), RD * (green), R J/Ψ (blue), and R l π (yellow) once 2σ experimental constraint is imposed.The corresponding ranges in B(B → πτ ν) and Rπ are shown in the right panel.

TABLE II
: Form factor inputs for Bs → Klν, Bs → K * lν and B → πlν

TABLE III :
First row reports the average values of the experimental inputs RD, RD * [15], R J/Ψ

TABLE IV :
The central values and 1σ ranges of each observable for both µ and τ modes in SM are reported for Bs → Klν, Bs → K * lν and B → πlν decays.

TABLE VI :
Allowed ranges of each observable in the presence of VL NP coupling of Fig.4.