Lepton Flavor Universality tests through angular observables of B ¯ ¯ ¯ ¯ → D (∗) ℓ ν ¯ ¯ ¯ decay modes
Damir Becirevic, Marco Fedele, Ivan Nisandzic, Andrey Tayduganov
DDO-TH 19/10LPT-Orsay-19-26QFET-2019-06TTP19-022
Lepton Flavor Universality tests through angularobservables of B → D ( ∗ ) (cid:96)ν decay modes Damir Beˇcirevi´c a , Marco Fedele b , Ivan Niˇsandˇzi´c c , Andrey Tayduganov d a Laboratoire de Physique Th´eorique, Bˆat. 210 (UMR 8627)Universit´e Paris Sud, Universit´e Paris-Saclay, 91405 Orsay cedex, France. b Departament de F´ısica Qu`antica i Astrof´ısica (FQA), Institut de Ci`encies del Cosmos(ICCUB), Universitat de Barcelona, Mart´ı i Franqu`es 1, E08028 Barcelona, Spain. c Institut f¨ur Theoretische Teilchenphysik,Karlsruher Institut f¨ur Technologie, Engesser-Str.7, D-76128 Karlsruhe, Germany. d Fakult¨at Physik, TU Dortmund, Otto-Hahn-Str.4, D-44221 Dortmund, Germany.
Abstract
We discuss the possibility of using the observables deduced from the angular dis-tribution of the B → D ( ∗ ) (cid:96) ¯ ν decays to test the effects of lepton flavor universalityviolation (LFUV). We show that the measurement of even a subset of these ob-servables could be very helpful in distinguishing the Lorentz structure of the NewPhysics contributions to these decays. To do so we use the low energy effective the-ory in which besides the Standard Model contribution we add all possible Lorentzstructures with the couplings (Wilson coefficients) that are determined by matchingtheory with the measured ratios R ( D ( ∗ ) ) exp . We argue that even in the situation inwhich the measured R ( D ( ∗ ) ) exp becomes fully compatible with the Standard Model,one can still have significant New Physics contributions the size of which could beprobed by measuring the observables discussed in this paper and comparing themwith their Standard Model predictions.PACS: 13.20.He, 12.60.-i, 12.38.Qk a r X i v : . [ h e p - ph ] J u l Introduction
Ever since the BaBar Collaboration reported on the first experimental indication of thelepton flavor universality violation (LFUV) in B -meson decays, after measuring the ra-tios [1, 2] R ( D ( ∗ ) ) = B ( B → D ( ∗ ) τ ¯ ν ) B ( B → D ( ∗ ) l ¯ ν ) (cid:12)(cid:12)(cid:12)(cid:12) l ∈{ e,µ } , (1)we witnessed an intense activity in trying to understand why R ( D ( ∗ ) ) SM < R ( D ( ∗ ) ) exp ,where SM stands for the theoretically established value in the Standard Model. Sincethen, the experimentalists at Belle and LHCb corroborated the tendency that indeed R ( D ( ∗ ) ) SM < R ( D ( ∗ ) ) exp [3–6], while the theorists worked on scrutinizing the theoreti-cal uncertainties within the SM [7–10] and proposed various models of New Physics (NP)that could accommodate the observed discrepancies [11–42]. The current values, includingthe most recent Belle result [43], are [44]: R ( D ) exp = 0 . , R ( D ) SM = 0 . , (2) R ( D ∗ ) exp = 0 . , R ( D ∗ ) SM = 0 . , (3)which amounts to a combined 3 . σ disagreement between the measurements and the SMpredictions. Measuring the ratios of similar decay rates is convenient because they are in-dependent on the Cabibbo–Kobayashi-Maskawa (CKM) matrix element | V cb | , and becauseof the cancellation of a significant amount of hadronic uncertainties. One should, however,be careful in assessing the uncertainties regarding the remaining non-perturbative QCDeffects, especially in the case of B → D ∗ (cid:96) ¯ ν (cid:96) ( (cid:96) = e, µ, τ ) for which the information con-cerning the shapes of hadronic form factors has never been deduced from the results basedon numerical simulations of QCD on the lattice. In the case of B → D(cid:96) ¯ ν (cid:96) , instead, boththe vector and scalar form factors have been computed on the lattice at several different q = ( p (cid:96) + p ¯ ν (cid:96) ) values [47–49]. Moreover, the theoretical studies reported in Ref. [50, 51]suggest that the soft photon radiation B → D(cid:96) ¯ ν (cid:96) γ could be an important source of uncer-tainty in R ( D ) SM , unaccounted for thus far. Further research in this direction is necessaryand the associated theoretical uncertainty should be included in the overall error budgetbefore the 5 σ (or larger) discrepancy between theory (SM) and experiment is claimed.Notice also that the LHCb Collaboration recently made another LFUV test based on b → c(cid:96) ¯ ν (cid:96) . They reported [52] R ( J/ψ ) = B ( B c → J/ψτ ¯ ν ) B ( B c → J/ψµ ¯ ν ) = 0 . ± . , (4) Very recently the MILC collaboration presented preliminary results for the shapes of the B → D ∗ form factors [45, 46]. Numerical results, that could be used for phenomenology, are yet to be reported. σ larger than predicted in the SM, R ( J/ψ ) SM < R ( J/ψ ) exp .The above-mentioned experimental indications of LFUV have motivated many theoriststo propose a scenario beyond the SM which could accommodate the measured valuesof R ( D ( ∗ ) ) and R ( J/ψ ). In terms of a general low energy effective theory, which willbe described in the next Section, the NP effects can show up at low energies throughan enhancement of the contribution arising from either the (axial-)vector current-currentoperators, the (pseudo-)scalar operators, the tensor one, or from a combination of those [53–61]. Another important aspect in describing the LFUV effects in most of the modelsproposed so far is the assumption that the NP couplings involving τ are the source ofLFUV, whereas those related to l = e, µ are much smaller and can be neglected. Arationale for that assumption is that the LFUV effects have not been observed in othersemileptonic decay modes (which involve lighter mesons and light leptons). In addition tothat, the results of a detailed angular analysis of B → D ( ∗ ) l ¯ ν l [ l ∈ ( e, µ )] by BaBar and byBelle were found to be fully consistent with the SM predictions. We will follow the samepractice in this paper and assume that the deviations from the lepton flavor universalityarise from the NP couplings to the third generation of leptons. We will formulate a generaleffective theory scenario for b → c(cid:96) ¯ ν (cid:96) decays by adding all operators allowed by the CP and the Lorentz symmetries. In doing so we will not account for a possibility of the lightright-handed neutrinos. Such models have been already proposed, but the problem theyoften encounter is that the NP amplitude does not interfere with the (dominant) SM one,and therefore getting R ( D ( ∗ ) ) > R ( D ( ∗ ) ) SM compatible with R ( D ( ∗ ) ) exp requires large NPcouplings which could be in conflict with the bounds on these couplings that could bededuced from direct searches at the LHC.Starting from the general effective theory we will provide the expressions for the fullangular distribution of both decay processes, and then combine the coefficients involvingthe NP couplings in a number of observables. Some of these observables could be studiedat the LHC, but most of them could be tested at Belle II. It is therefore reasonable toexplore the possibilities of testing the effects of LFUV not only via the ratios of branchingfractions [such as R ( D ( ∗ ) )], but also by using the ratios of these newly defined observables.Since various observables are sensitive to different NP operators, we will argue that theexperimental analysis of the ratios we propose to study could indeed help disentangling thebasic features of NP. We will then also provide a phenomenological analysis to illustrateour claim.Before we embark on the details of this work we should emphasize that for the phe-nomenological analysis we need a full set of form factors (including the scalar, pseudoscalarand the tensor ones) computed by means of lattice QCD, which is not the case right now.A dedicated lattice computation of all the form factors relevant to B → D ( ∗ ) (cid:96) ¯ ν (cid:96) is of greatimportance for a reliable assessment of LFUV. Since we have to make (reasonable) as-sumptions about the form factors, our phenomenological results should be understood asa diagnostic tool to distinguish among various Lorentz structures of the NP contributions,2hile the accurate analysis will be possible only when the lattice QCD results becomeavailable.The remainder of this paper is organized as follows: In Sec. 2 we provide the fullangular distributions for both types of decays considered in this paper, and define all theobservables which can be used to better test the effects of LFUV in these decays. In Sec. 3we make the sensitivity study of the observables defined in Sec. 2 to the effects of LFUV,solely based on the deviations of the measured R ( D ( ∗ ) ) with respect to the SM predictions.We also made a short comment on the recently measured F D ∗ L . We finally summarize inSec. 4. Several definitions and technical details are collected in the Appendices. B → D ( ∗ ) (cid:96)ν In this Section we define the effective Hamiltonian for a generic NP scenario and then derivethe expressions for the full angular distribution of the differential decay rate of B → D(cid:96) ¯ ν (cid:96) ,and of B → D ∗ ( → Dπ ) (cid:96) ¯ ν (cid:96) . All angular coefficients will be expressed in terms of helicityamplitudes which are properly defined in terms of kinematical variables and hadronic formfactors. For the reader’s convenience the decomposition of all the matrix elements in termsof form factors is provided in Appendix A of the present paper. With explicit expressionsfor the angular coefficients in hands, we will then proceed and define a set of observablesthat can be used to study the effects of LFUV. Assuming the neutrinos to be left-handed, the most general effective Hamiltonian describ-ing the b → c(cid:96)ν (cid:96) decays, containing all possible parity-conserving four-fermion dimension-6operators, can be written as H eff = √ G F V cb (cid:2) (1 + g V )( cγ µ b )( (cid:96) L γ µ ν L ) + ( − g A )( cγ µ γ b )( (cid:96) L γ µ ν L )+ g S ( cb )( (cid:96) R ν L ) + g P ( cγ b )( (cid:96) R ν L )+ g T ( cσ µν b )( (cid:96) R σ µν ν L ) + g T ( cσ µν γ b )( (cid:96) R σ µν ν L ) (cid:3) + h . c . (5)Note that we use the convention such that in the SM g i = 0, ∀ i ∈ { S, P, V, A, T, T } .It is often more convenient to write the above Hamiltonian in the chiral basis of oper-ators, H eff = 4 G F √ V cb (cid:2) (1 + g V L )( c L γ µ b L )( (cid:96) L γ µ ν L ) + g V R ( c R γ µ b R )( (cid:96) L γ µ ν L )+ g S L ( c R b L )( (cid:96) R ν L ) + g S R ( c L b R )( (cid:96) R ν L )+ g T L ( c R σ µν b L )( (cid:96) R σ µν ν L ) (cid:3) + h . c . , (6)which is obviously equivalent to Eq. (5), with the corresponding effective coefficients relatedas g V, A = g V R ± g V L , g S, P = g S R ± g S L , g T = − g T = g T L . (7)3he last relation is also an easy way to see that ( c L σ µν b R )( (cid:96) R σ µν ν L ), the right-handedtensor operator, cannot contribute to the decay amplitude. B → D(cid:96)ν decay
We first focus on the decay to a pseudoscalar meson and write the differential decay rateas d Γ dq d cos θ (cid:96) = (cid:112) λ BD ( q )64(2 π ) m B (cid:18) − m (cid:96) q (cid:19) (cid:88) λ (cid:96) |M λ (cid:96) ( B → D(cid:96)ν ) | , (8)where we made use of the definition λ BD ( q ) = m B + m D + q − m B m D + m B q + m D q ). Itis convenient to separate the angular from the q dependence and write the above expressionas d Γ dq d cos θ (cid:96) = a θ (cid:96) ( q ) + b θ (cid:96) ( q ) cos θ (cid:96) + c θ (cid:96) ( q ) cos θ (cid:96) , (9)where θ (cid:96) is the polar angle of the lepton momentum in the rest frame of the (cid:96)ν -pair withrespect to the z axis which is defined by the D -momentum in the rest frame of B . Thecorresponding angular coefficients can be written as a θ (cid:96) ( q ) = G F | V cb | π m B q (cid:112) λ BD ( q ) (cid:18) − m (cid:96) q (cid:19) (cid:20) | (cid:101) h − | + m (cid:96) q | (cid:101) h t | (cid:21) , (10a) b θ (cid:96) ( q ) = G F | V cb | π m B q (cid:112) λ BD ( q ) (cid:18) − m (cid:96) q (cid:19) m (cid:96) q R e [ (cid:101) h +0 (cid:101) h ∗ t ] , (10b) c θ (cid:96) ( q ) = − G F | V cb | π m B q (cid:112) λ BD ( q ) (cid:18) − m (cid:96) q (cid:19) (cid:20) | (cid:101) h − | − m (cid:96) q | (cid:101) h +0 | (cid:21) , (10c)where (cid:101) h λ (cid:96) λ are the linear combinations of the “ standard ” hadronic helicity amplitudes whichare defined and computed in Appendix B of the present paper.Integration over the polar angle θ (cid:96) leads to the expression for the differential decay rate, d Γ dq = 2 a θ (cid:96) ( q ) + 23 c θ (cid:96) ( q )= G F | V cb | π m B q (cid:112) λ BD ( q ) (cid:18) − m (cid:96) q (cid:19) × (cid:26) | (cid:101) h − | + m (cid:96) q | (cid:101) h +0 | + 32 m (cid:96) q | (cid:101) h t | (cid:27) . (11)We see that the linear dependence on cos θ (cid:96) in Eq. (9) is lost after integration in θ (cid:96) , but itcan be recovered by considering the forward-backward asymmetry, A FB ( q ) = (cid:90) d Γ dq d cos θ (cid:96) d cos θ (cid:96) − (cid:90) − d Γ dq d cos θ (cid:96) d cos θ (cid:96) d Γ /dq = b θ (cid:96) ( q ) d Γ /dq . (12)4 ( ~p D ∗ ) ‘ν Dπ χ θ D θ ‘ xy Figure 1:
Kinematics of the B → D ∗ ( → Dπ ) (cid:96)ν decay. Another interesting observable for the study of the NP effects is the lepton polarizationasymmetry defined from differential decay rates with definite lepton helicity: d Γ λ (cid:96) =+1 / dq = G F | V cb | π m B q (cid:112) λ BD ( q ) (cid:18) − m (cid:96) q (cid:19) m (cid:96) q (cid:20) | (cid:101) h +0 | + 3 | (cid:101) h t | (cid:21) , (13a) d Γ λ (cid:96) = − / dq = G F | V cb | π m B q (cid:112) λ BD ( q ) (cid:18) − m (cid:96) q (cid:19) | (cid:101) h − | . (13b)The corresponding polarization asymmetry reads A λ (cid:96) ( q ) = d Γ λ (cid:96) = − / /dq − d Γ λ (cid:96) =+1 / /dq d Γ /dq = 1 − d Γ λ (cid:96) =+1 / /dq d Γ /dq . (14) B → D ∗ ( → Dπ ) (cid:96)ν decay We now discuss the case of the ¯ B -meson decaying to the vector meson in the final state.In this work, for the computation of the helicity amplitudes and differential decay rates,we define the system of coordinates as depicted in Fig. 1: the z -axis is set along the D ∗ momentum in the B rest frame, and the x -axis is chosen in a way that the D momentumin the D ∗ rest frame lies in the x − z plane and has the positive x -component.The full angular distribution reads, d Γ dq dm Dπ d cos θ D d cos θ (cid:96) dχ = (cid:112) λ BD ∗ ( q )256(2 π ) m B (cid:18) − m (cid:96) q (cid:19) | ˆ p D | m Dπ (cid:88) λ (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:88) λ D ∗ M λ D ∗ , λ (cid:96) ( B → Dπ(cid:96)ν (cid:96) ) (cid:12)(cid:12)(cid:12)(cid:12) , (15)5here again λ BD ∗ ( q ) = m B + m D ∗ + q − m B m D ∗ + m B q + m D ∗ q ), and | ˆ p D | = (cid:112) λ ( m Dπ , m D , m π )2 m Dπ . (16)After integrating over m Dπ around the D ∗ -resonance, the above distribution becomes d Γ dq d cos θ D d cos θ (cid:96) dχ = 932 π (cid:26) I c cos θ D + I s sin θ D + (cid:2) I c cos θ D + I s sin θ D (cid:3) cos 2 θ (cid:96) + (cid:2) I c cos θ D + I s sin θ D (cid:3) cos θ (cid:96) + (cid:2) I cos 2 χ + I sin 2 χ (cid:3) sin θ (cid:96) sin θ D + (cid:2) I cos χ + I sin χ (cid:3) sin 2 θ (cid:96) sin 2 θ D + (cid:2) I cos χ + I sin χ (cid:3) sin θ (cid:96) sin 2 θ D (cid:27) , (17)where the angular coefficients I i ≡ I i ( q ) are given by: I c = 2 N (cid:20) | (cid:101) H − | + m (cid:96) q | (cid:101) H +0 | + 2 m (cid:96) q | (cid:101) H t | (cid:21) , (18a) I s = N (cid:20) (cid:0) | (cid:101) H − + | + | (cid:101) H −− | (cid:1) + m (cid:96) q (cid:0) | (cid:101) H ++ | + | (cid:101) H + − | (cid:1)(cid:21) , (18b) I c = 2 N (cid:20) −| (cid:101) H − | + m (cid:96) q | (cid:101) H +0 | (cid:21) , (18c) I s = N (cid:20) | (cid:101) H − + | + | (cid:101) H −− | − m (cid:96) q (cid:0) | (cid:101) H ++ | + | (cid:101) H + − | (cid:1)(cid:21) , (18d) I = − N R e (cid:20) (cid:101) H − + (cid:101) H −∗− − m (cid:96) q (cid:101) H ++ (cid:101) H + ∗− (cid:21) = − N β (cid:96) R e (cid:2) H + H ∗− − H T, + H ∗ T, − (cid:3) , (18e) I = N R e (cid:20) ( (cid:101) H − + + (cid:101) H −− ) (cid:101) H −∗ − m (cid:96) q ( (cid:101) H ++ + (cid:101) H + − ) (cid:101) H + ∗ (cid:21) = N β (cid:96) R e (cid:2) ( H + + H − ) H ∗ − H T, + + H T, − ) H ∗ T, (cid:3) , (18f) I = 2 N R e (cid:20) ( (cid:101) H − + − (cid:101) H −− ) (cid:101) H −∗ − m (cid:96) q ( (cid:101) H ++ + (cid:101) H + − ) (cid:101) H ∗ t (cid:21) , (18g)6 c = 8 N m (cid:96) q R e (cid:2) (cid:101) H +0 (cid:101) H ∗ t (cid:3) , (18h) I s = 2 N (cid:0) | (cid:101) H − + | − | (cid:101) H −− | (cid:1) , (18i) I = 2 N I m (cid:20) ( (cid:101) H − + + (cid:101) H −− ) (cid:101) H −∗ − m (cid:96) q ( (cid:101) H ++ − (cid:101) H + − ) (cid:101) H ∗ t (cid:21) , (18j) I = N I m (cid:20) ( (cid:101) H − + − (cid:101) H −− ) (cid:101) H −∗ − m (cid:96) q ( (cid:101) H ++ − (cid:101) H + − ) (cid:101) H + ∗ (cid:21) = N β (cid:96) I m (cid:2) ( H + − H − ) H ∗ − H T, + − H T, − ) H ∗ T, (cid:3) , (18k) I = − N I m (cid:20) (cid:101) H − + (cid:101) H −∗− − m (cid:96) q (cid:101) H ++ (cid:101) H + ∗− (cid:21) = − N β (cid:96) I m (cid:2) H + H ∗− − H T, + H ∗ T, − (cid:3) , (18l)with N ≡ N ( q ) = B ( D ∗ → Dπ ) G F | V cb | π ) m B q (cid:112) λ BD ∗ ( q ) (cid:18) − m (cid:96) q (cid:19) . (19)The explicit expressions for the hadronic helicity amplitudes H λ , as well as their linearcombinations (cid:101) H λ (cid:96) λ , are given in Appendix B.It is now possible to write down three partially integrated decay distributions, integrat-ing all but one angle at a time. • θ (cid:96) distribution : d Γ dq d cos θ (cid:96) = a θ (cid:96) ( q ) + b θ (cid:96) ( q ) cos θ (cid:96) + c θ (cid:96) ( q ) cos θ (cid:96) ,a θ (cid:96) ( q ) = 38 ( I c + 2 I s − I c − I s ) ,b θ (cid:96) ( q ) = 38 ( I c + 2 I s ) ,c θ (cid:96) ( q ) = 34 ( I c + 2 I s ) . (20) • θ D distribution : d Γ dq d cos θ D = a θ D ( q ) + c θ D ( q ) cos θ D ,a θ D ( q ) = 38 (3 I s − I s ) ,c θ D ( q ) = 38 (3 I c − I s − I c + I s ) . (21)7 χ distribution : d Γ dq dχ = a χ ( q ) + c cχ ( q ) cos 2 χ + c sχ ( q ) sin 2 χ ,a χ ( q ) = 18 π (3 I c + 6 I s − I c − I s ) c cχ ( q ) = 12 π I ,c sχ ( q ) = 12 π I . (22)Illustration of the q -dependence of the above observables, for fixed values of g i ’s, is pro-vided in Fig. 2.It is interesting to point out that the terms proportional to cos θ D , cos χ and sin χ areabsent in the distributions written above. Such terms can arise if a non-zero interferencewith the S -wave contribution, B → ( Dπ ) S (cid:96)ν , is included in the distribution, as discussedin Ref. [56]. From the full angular distribution (17) we isolate various coefficients and combine theminto various quantities normalized to the differential decay rate. In the following we define12 such quantities with a goal to use them in order to scrutinize the effects of LFUV thatwe propose to study. When necessary, before defining an observable, we will indicate howthe corresponding coefficients can be isolated from the full angular distribution. • Differential decay rate d Γ dq = 14 (3 I c + 6 I s − I c − I s ) , (23) • Forward-backward asymmetry A FB ( q ) = b θ (cid:96) ( q ) d Γ /dq = 38 ( I c + 2 I s ) d Γ /dq , (24) • Lepton polarization asymmetry A λ (cid:96) ( q ) = d Γ λ (cid:96) = − / /dq − d Γ λ (cid:96) =+1 / /dq d Γ /dq , (25)with d Γ λ (cid:96) =+1 / dq = 14 ( I c + 2 I s + I c − I s ) + I n d Γ λ (cid:96) = − / dq = 12 ( I c + 2 I s − I c + 2 I s ) − I n , (26)8here we introduced an additional coefficient I n = 2 N m (cid:96) q | (cid:101) H t | , (27)which is not present in Eq. (17) because in the full angular distribution we havesummed over the lepton polarization states. • D ∗ polarization fraction R L,T ( q ) = d Γ L /dq d Γ T /dq , (28)where Γ L and Γ T represent the longitudinal and transverse D ∗ polarization decayrates, d Γ L dq = 23 (cid:2) a θ D ( q ) + c θ D ( q ) (cid:3) = 14 (3 I c − I c ) ,d Γ T dq = 43 a θ D ( q ) = 12 (3 I s − I s ) . (29)Alternatively, one can define the quantity F D ∗ L which is a measure of the longitudi-nally polarized D ∗ ’s in the whole ensemble of B → D ∗ (cid:96)ν decays, which is related to R L,T ( q ) as: F D ∗ L ( q ) = R L,T ( q )1 + R L,T ( q ) = 12 3 I c − I c I c + I s ) − I c − I s . (30) F D ∗ L is often referred to as F D ∗ L ( q ) integrated over the available phase space. • R A,B R A,B ( q ) = d Γ A /dq d Γ B /dq , (31) d Γ A dq = 23 (cid:2) a θ (cid:96) ( q ) − c θ (cid:96) ( q ) (cid:3) = 14 ( I c + 2 I s − I c − I s ) ,d Γ B dq = 43 (cid:2) a θ (cid:96) ( q ) + c θ (cid:96) ( q ) (cid:3) = 12 ( I c + 2 I s + I c + 2 I s ) = d Γ dq − d Γ A dq . (32) • A and A A ( q ) = c cχ ( q ) d Γ /dq = 12 π I d Γ /dq ,A ( q ) = c sχ ( q ) d Γ /dq = 12 π I d Γ /dq . (33)9 A and A If, from the full angular distribution, we first define the auxiliary quantities:Φ ( q , χ, θ (cid:96) ) = (cid:20)(cid:90) − − (cid:90) (cid:21) d Γ dq dχd cos θ (cid:96) d cos θ D d cos θ D (cid:101) Φ ( q , χ ) = (cid:20)(cid:90) − − (cid:90) (cid:21) Φ ( q , χ, θ (cid:96) ) d cos θ (cid:96) , (34)then we can extract another two observables as, A ( q ) = (cid:34)(cid:90) π/ π/ − (cid:90) π/ − (cid:90) π π/ (cid:35) (cid:101) Φ ( q , χ ) dχd Γ /dq = − π I d Γ /dq .A ( q ) = (cid:20)(cid:90) π − (cid:90) ππ (cid:21) (cid:101) Φ ( q , χ ) dχd Γ /dq = 2 π I d Γ /dq . (35) • A and A Similarly, if one first defines the asymmetry of the full angular distribution withrespect to θ D , namely,Φ ( q , χ ) = (cid:20)(cid:90) − − (cid:90) (cid:21) d Γ dq dχd cos θ D d cos θ D , (36)one can define two additional observables as follows: A ( q ) = − (cid:34)(cid:90) π/ π/ − (cid:90) π/ − (cid:90) π π/ (cid:35) Φ ( q , χ ) dχd Γ /dq = − I d Γ /dq ,A ( q ) = (cid:20)(cid:90) π − (cid:90) ππ (cid:21) Φ ( q , χ ) dχd Γ /dq = − I d Γ /dq . (37) • A s Finally from the asymmetry with respect to θ (cid:96) .Φ ( q , θ D ) = (cid:20)(cid:90) − − (cid:90) (cid:21) d Γ dq d cos θ D d cos θ (cid:96) d cos θ (cid:96) , (38)we can isolate a term proportional to I s as A s ( q ) = (cid:34) (cid:90) / − / − (cid:90) / − (cid:90) − / − (cid:35) Φ ( q , θ D ) d cos θ D d Γ /dq = − I s d Γ /dq . (39)10n the above definitions we played with Eq. (17) to isolate each of the angular coefficients I i ≡ I i ( q ). Alternatively, with a large enough sample one can fit the full data set toEq. (17) and extract each of the coefficients with respect to the full (differential) decayrate. The SM values of all (cid:104) I i (cid:105) (cid:96) = 1Γ( B → D ∗ (cid:96) ¯ ν (cid:96) ) × ( m B − m D ∗ ) (cid:90) m (cid:96) I (cid:96)i ( q ) dq (40)for each of the leptons in the final state, are given in Tab. 1. (cid:96) (cid:104) I c (cid:105) (cid:96) (cid:104) I s (cid:105) (cid:96) (cid:104) I c (cid:105) (cid:96) (cid:104) I s (cid:105) (cid:96) (cid:104) I (cid:105) (cid:96) (cid:104) I (cid:105) (cid:96) (cid:104) I (cid:105) (cid:96) (cid:104) I c (cid:105) (cid:96) (cid:104) I s (cid:105) (cid:96) (cid:104) I (cid:105) (cid:96) (cid:104) I (cid:105) (cid:96) (cid:104) I (cid:105) (cid:96) e µ τ Table 1:
Standard Model values of the coefficients appearing in the angular distribution (17) ,integrated over the full available phase space, as indicated in Eq. (40) . The values are obtainedby using the form factors that are in the text referred to as CLN+HQET.
The example plots of q -dependent observables for different benchmark values of NPcouplings are depicted in Fig. 2. One can now make the ratios between the above observables with the τ -lepton in the finalstate and the same observables extracted from the decay to l [ l ∈ ( e, µ )]. Of course thiscan only be done for the quantities which are nonzero in the SM. For the observablesproportional to I , , , which are zero in the SM, we consider the differences, namely D ( A , , ) ≡ (cid:104) A τ , , (cid:105) − (cid:18) (cid:104) A e , , (cid:105) + (cid:104) A µ , , (cid:105) (cid:19) . (41)For all the other observables, defined in Eqs. (12), (14) and (24)–(39), we define the LFUVobservables as: R ( O i ) ≡ (cid:104) O τi (cid:105) ( (cid:104) O ei (cid:105) + (cid:104) O µi (cid:105) ) , (42)where each O (cid:96)i is integrated over the available phase space. The only exception to thisdefinition is made in the case of A D FB which we divide by m (cid:96) in order to make the effect ofthe presence of NP more pronounced. Since most of the observables are written in terms In other words, R ( A D FB ) = (cid:104)A D τ FB /m τ (cid:105) / (cid:16) / × (cid:104)A D µ FB /m µ (cid:105) + 1 / × (cid:104)A D e FB /m e (cid:105) (cid:17) . � � �������������������� � � [ ��� � ] � � � ( � ) � � = ���� - ���� ⅈ � � = ���� + ���� ⅈ �� � � � �� - ��� - ��� - ��� - ��� - ��������� � � [ ��� � ] � λ τ ( � ) � � = ���� - ���� ⅈ � � = ���� + ���� ⅈ �� � � � � � � � �� - ������������������ � � [ ��� � ] � � � ( � * ) � � = ���� - ���� ⅈ � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ �� � � � � � � � �� - ��� - ��� - ������������������ � � [ ��� � ] � λ τ ( � * ) � � = ���� - ���� ⅈ � � = ���� + ���� ⅈ �� � � � � � � � ������� � � [ ��� � ] � � � � � � = ���� - ���� ⅈ � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ �� � � � � � � � ������������������������������ � � [ ��� � ] � � � � � � = ���� - ���� ⅈ � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ �� � � � � � � � �� - ���� - ���� - ���� - ���������������� � � [ ��� � ] � � � � = ���� - ���� ⅈ � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ �� � � � � � � � �� - ���� - ������������������������ � � [ ��� � ] � � � � = ���� - ���� ⅈ � � = ���� + ���� ⅈ �� � � � � � � � �� - ��� - ��� - ��� - ��������� � � [ ��� � ] � � � � = ���� - ���� ⅈ � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ �� � � � � � � � �� - ��� - ��� - ��������������� � � [ ��� � ] � � � � � = ���� - ���� ⅈ � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ �� � � � � � � � �� - ���� - ���������������������������� � � [ ��� � ] � � � � = ���� - ���� ⅈ � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ �� � � � � � � � �� - ������������������������ � � [ ��� � ] � � � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ �� � � � � � � � �� - ������������������������������ � � [ ��� � ] � � � � = ���� + ���� ⅈ � � = ���� + ���� ⅈ �� Figure 2: A FB , A λ τ observables relevant to B → Dτ ν and A FB , A λ τ , R L,T , R A,B and A − relevantto B → D ∗ τ ν are displayed for various values of NP couplings and as functions of q . The width of eachcurve comes from the theoretical uncertainties in hadronic form factors and quark masses. The benchmark g i ’s are chosen to be the best fit values, as discussed in the text.
12f the ratios, O (cid:96)i ( q ) = N (cid:96)i ( q ) / D (cid:96)i ( q ) with N and D being generically a numerator and adenominator, the integrated quantities are then defined as (cid:104) O (cid:96)i (cid:105) = (cid:90) q m (cid:96) N (cid:96)i ( q ) dq (cid:90) q m (cid:96) D (cid:96)i ( q ) dq . (43)As it can be seen from the previous subsections, most of the observables are normalizedto the differential decay rate which makes it difficult to monitor a dependence on g V,A,V L (cid:54) =0, since the dependence on these couplings would cancel in these observables. Since inmany specific models it is proposed to accommodate R ( D ( ∗ ) ) exp > R ( D ( ∗ ) ) SM by switchingon g V L (cid:54) = 0, the only departure from the SM value would indeed be in R ( D ( ∗ ) ), while allthe other observables would remain compatible with the SM predictions. To study the sensitivity of the LFUV observables defined in the previous Section on thenon-zero values of g i ’s, we determine possible values of g i ’s from the fit to the measured R ( D ) and R ( D ∗ ). To do so we use the publicly available code HEPfit [62] in which theBayesian statistical approach is adopted.Since the hadronic uncertainties are the main source of the theoretical error, we haveto be careful regarding the dependence of observables on the choice of form factors. Inthat respect we first used the set of form factors obtained in the constituent quark modelof Ref. [63] which contains all of the form factors needed for this study. The uncertain-ties of the results obtained by the quark models are, however, unclear and attributing10% uncertainty to each of the form factors is just an educated guess derived from thecomparison between the predicted and measured decay rates. For that reason the resultsobtained by using the quark model form factors could only be considered as qualitativeand the departures from the SM predictions only as indication (diagnostic) of the presenceof LFUV.Another choice of form factors consists in relying on the experimental analyses of theangular distribution of B → D ∗ lν [ l ∈ ( e, µ )] decays from which the ratios of form factorscan be extracted if one assumes the so called CLN parametrization of the dominant formfactor A ( q ) [64]. The experimental averages of the corresponding parameters [ ρ , R (1), R (1)] and the information about their correlations is provided by HFLAV [44] and we useit in this work. As for the remaining form factors we used the expressions derived in Notice that the HFLAV results are obtained by using the CLN parametrization which fit well thedata. Recent studies [10] show that the so called BGL parametrization of Ref. [65] should be preferredbecause the slopes of the ratios R , ( w ) are not fixed, but left as free parameters. In Ref. [66] it was shown, α s and the leading powercorrections have been included to compute A /A , f T /f + , T , , /A to each of which weattribute 10% of error (considerably larger than those quoted in Ref. [8]). Finally, for theform factors f + and f we use the results obtained in numerical simulations of QCD onthe lattice [68]. We have checked that our final results obtained by using the form factorscomputed in the constituent quark model of Ref. [63] are practically indistinguishable fromthose obtained with the form factors chosen as explained in this paragraph to which we willrefer as “CLN+HQET+LATT”. Since less assumptions are needed in the error estimateof the form factors in the latter case, in the following all our results will be obtained bychoosing CLN+HQET+LATT form factors [8, 44, 64, 68].We now proceed and allow for one complex valued coefficient g i to be non-zero ata time, i.e. we add 2 extra parameters (compared to the SM case) in every fit. Weemphasize once again that we assume the NP to affect only the decay modes with τ in thefinal state, namely, B → D ( ∗ ) τ ¯ ν τ . The joint probability distribution functions ( p.d.f. ’s) forthe coefficients g i are therefore obtained by using R ( D ) exp and R ( D ∗ ) exp as constraint, andthen they are employed to predict all of the LFUV observables discussed in the previousSection. From the comparison with the SM results we can see which quantity is moresensitive to the considered g i (cid:54) = 0. Notice again that the B → Dτ ν τ observables will beaffected by NP effects in g V,S,T , while g V,A,P,T (cid:54) = 0 will modify the B → D ∗ τ ν τ ones. In Fig. 3 and 4 we show the allowed regions for the NP couplings g i relevant to the basis (5)and (6), respectively. We reiterate that the allowed regions for NP couplings are obtainedby letting one (complex valued) coupling g i to be non-zero at a time. Any value of the g i derived in this way is plausible. We also include the limit derived from the B c -mesonlifetime as discussed in Refs. [69, 70] which is particularly restrictive to the pseudoscalarNP contribution g P , as well as in g S L ,S R . In this paper we take the conservative limit B ( B c → τ ¯ ν ) (cid:46)
30% and use the expression B ( B c → τ ¯ ν ) = τ B c m B c f B c G F | V cb | π m τ (cid:18) − m τ m B c (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g V L + ( g S R − g S L ) m B c m τ ( m b + m c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (44)where the B c decay constant f B c = 427(6) MeV has been determined in the lattice QCDstudy of Ref. [71]. however, that both parametrizations provide good fit with data. The resulting | V cb | values, as obtainedfrom fitting the data to these two parametrizations, were different. Since we are not interested in assessingthe value of | V cb | , the choice of parametrization is immaterial. Notice, however, that the most recent studyby Belle [67] in which a larger sample of data has been used, showed that (a) the form factor shapes arefully consistent with the results reported by HFLAV [44], and (b) the values of | V cb | inferred from thefits to two parametrizations (CLN and BGL) are consistent with each other, both being lower than | V cb | extracted from the inclusive decays. - � - � - � � � - � - ���� �� [ � � ] � � [ � � ] � σ � σ � σ ★ - � � � � � - � - ���� �� [ � � ] � � [ � � ] � σ � σ � σ ★ - � - � - � � � - � - ���� �� [ � � ] � � [ � � ] � σ � σ � σ ★ - � - � - � � � - � - ���� �� [ � � ] � � [ � � ] � σ � σ � σ � � � � � � � � � ( � � → τ ν ) ★ - � - � - � � � - � - ���� �� [ � � ] � � [ � � ] � σ � σ � σ Figure 3:
The allowed values for the NP couplings in Eq. (5) as obtained from the the fit with R ( D ) exp and R ( D ∗ ) exp , and by switching one coupling g i at the time. Red stars denote the best fit values. Regardingthe form factors we used those to which we refer in the text as CLN+HQET+LATT. The benchmark values, denoted by red stars in Figs. 3-4, correspond to the best fitvalues. We get g V = 0 .
20 + i . , g A = 0 .
69 + i . ,g S = 0 .
17 + i . , g P = 0 .
58 + i . , g T − g T = 0 . − i . , (45)and in terms of couplings introduced in Eq. (6) we find g V L = 0 . − i . , g V R = − . − i . ,g S L = − . − i . , g S R = 0 .
19 + i . , g T L = 0 . − i . . (46)In Fig. 2 we show the q -dependence of each observable relevant to ¯ B → D ∗ τ ¯ ν discussedin the previous Section, both in the SM and with g i given in Eq. (45).After integrating over q ’s as described in Eq. (43), and by sweeping over the entirerange of g i allowed by R ( D ) exp and R ( D ∗ ) exp , we obtain the results listed in Tab. 2 andTab. 3. The results are presented along with the SM ones in order to make a comparisonsimpler. Notice that the SM values we obtain are fully compatible with those quoted inEq. (2), obtained by HFLAV, even though the central values are slightly different owing to15 - � - � - � � � - � - ���� �� [ � � � ] � � [ � � � ] � σ � σ � σ ★ - � � � � � - � - ���� �� [ � � � ] � � [ � � � ] � σ � σ � σ ★ - � - � - � � � - � - ���� �� [ � � � ] � � [ � � � ] ★ - ��������������������� - ��� - ��� - ��������������� � σ � σ � σ � � � � � � � � � ( � � → τ ν ) ★ - � - � - � � � - � - ���� �� [ � � � ] � � [ � � � ] � σ � σ � σ � � � � � � � � � ( � � → τ ν ) ★ - � - � - � � � - � - ���� �� [ � � � ] � � [ � � � ] � σ � σ � σ Figure 4:
Same as in Fig. 3 but for the NP couplings appearing in Eq. (6) . our choice of form factors. For each g i we then compute R ( D ) and R ( D ∗ ), which are nowobviously compatible with experimental values, and the full list of the LFUV observablesdiscussed in the previous Section. For the observables showing gaussian shapes we quotethe mean values with the corresponding standard deviations. Some of the observables,however, are hardly gaussian for the reasons explained below. For such observables inTabs. 2 and 3 we present the interval of values within 2 σ . The interested reader can alsofind in the tables of Appendix E the predictions for all the LFUV observables obtainedassuming for the benchmark values for the NP couplings given in Eqs. (45,46).Let us now comment on the results we obtain, starting from the ones shown in Tab. 2. • g V : Assuming NP affecting only the vector current, the constraints arising from R ( D ) exp and R ( D ∗ ) exp are such that either no or very small deviation of the LFUVobservables with respect to their SM values is predicted. The only exception is R ( A )which, for allowed g V (cid:54) = 0, becomes lower than its SM counterpart.Notice that for several observables we give only the intervals due to their non-gaussianbehaviour. They can be divided into two classes: those only sensitive to R e ( g V ), suchas R ( A D ∗ F B ), R ( A ) and R ( A ), and those only sensitive to I m ( g V ), namely D ( A ), D ( A ) and D ( A ). Let us focus on one observable from the first class, say R ( A ).As stated above, its value would be indistinguishable from the SM for g V purely16 ( A ) R ( A ) Figure 5:
Left panel: predicted p.d.f. for the observable R ( A ) , assuming a complex g V . Right panel:predicted p.d.f. for the observable R ( A ) , assuming that g V is equal to the best fit value reported inEq. (45) . imaginary. From the plot shown in the first panel of Fig. 3 it is apparent that thereare 2 distinct allowed solutions for R e ( g V ) when I m ( g V ) = 0, and therefore thereare 2 distinct predictions of R ( A ). After sweeping through various I m ( g V ) we fillup the gap between the two distinct real solutions in the case of I m ( g V ) = 0, henceproducing the anticipated non-gaussian prediction for R ( A ), shown in the left panelof Fig. 5.At first sight, given the broad range of predicted values, this observable might notseem of particular use. However, a careful reader will realize that a given value of g V will correspond to a point (and not a disk) in the plane depicted in the first panel ofFig. 3. Therefore, for a given value of Re ( g V ), one gets a very sharp prediction for R ( A ) as shown in the right panel of Fig. 5, where we assumed that g V is equal toits best fit value reported in Eq. (45). Conversely, the measurement of R ( A ) wouldproduce a sharp bound on the real part of g V , corresponding to a vertical stripe in the Re ( g V )- Im ( g V ) plane that intersected with the disk produced by the measurementsof R ( D ) and R ( D ∗ ), therefore severely reducing the allowed region for that coupling.As stated above, a similar reasoning can be performed for R ( A D ∗ F B ), R ( A ), D ( A ), D ( A ) and D ( A ), hence making all these observable extremely interesting. • g A : If NP appears only in the axial current, only the B → D ∗ τ ν τ channel is affected.Similarly to the previous scenario the obtained bounds are such that no apprecia-ble difference is observed in the observables R ( A D ∗ λ (cid:96) ), R ( R L,T ), R ( R A,B ), R ( A ) and R ( A ). On the other hand, analogously to the g V scenario, for R ( A D ∗ F B ), R ( A ), R ( A ), D ( A ), D ( A ) and D ( A ) we get broad ranges of values, with the LFUV17atios affected by Re ( g A ) and the LFUV differences by I m ( g A ). Like in the vectorscenario, the measurement of these observables would highly constrain the allowedregion for g A . • g S : If the NP effects come with the contribution arising solely from the scalar opera-tor, then only B → Dτ ν τ would be affected. In particular, R ( A Dλ (cid:96) ) is sensibly shiftedcompared to the SM prediction, while R ( A DF B ) displays once again a broad predic-tion, due to its sensitivity to Re ( g S ). Therefore, its measurement would provide thestrongest bound on g S . • g P : In contrast to the previous case, the NP giving a nonzero contribution to a termproportional to the pseudoscalar current would result in the changes in the B → D ∗ τ ν τ channel only. The unaffected observables by this choice would be R ( R A,B ), D ( A ) and D ( A ). On the other hand, R ( A D ∗ λ (cid:96) ), R ( R L,T ), R ( A ), R ( A ) and R ( A )are all predicted with good precision, and sensibly different from the values predictedin the SM. Moreover, we again obtain broad ranges for R ( A D ∗ F B ), R ( A ) and D ( A ),with R e ( g P ) affecting the LFUV ratios, while I m ( g P ) (cid:54) = 0 would particularly affectthe LFUV differences. • g T : If the NP effects give a nonzero contribution to the term proportional to thetensor current then both channels are affected. Furthermore, all the LFUV ratios arepredicted with fairly good precision and with sensible discrepancies when comparedwith the SM predictions. Regarding the LFUV differences, D ( A ) is the only onemildly affected by g T (cid:54) = 0, even if perfectly compatible with the SM prediction, whilethe remaining ones are both unaffected by this kind of NP.Similar observations can be obtained analyzing the results from Tab. 3, where we havenot listed the results relative to g T L since they are the same as the ones obtained for g T ,owing to Eq.(7). • g V L : If the effects come with the left-handed vector current alone then the presentexperimental bounds are such that no sensible deviation can be appreciated in anyof the LFUV quantities defined above. • g V R : In contrast to the previous case, if the coupling to the right-handed vectorcurrent is preferred then some LFUV observables in the B → D ∗ τ ν τ channel exhibita sensitivity to its effects, namely R ( A D ∗ F B ), R ( A ), R ( A ), D ( A ), D ( A ) and D ( A ).Moreover, given that the LFUV differences depend on I m ( g V R ) and that, as can beobserved from the second panel of Fig. 4, the fit allows for two distinct solutionsfor Im ( g V R ), we obtain two separate predicted regions for each of these observables.Notice that the range of allowed g V R is far more restricted by the data which isexpected on the basis of the SM gauge invariance.18 g S L and g S R : Assuming NP effects in the scalar current, we obtain both in theleft-handed and in the right-handed cases a similar outcome: three observables areunaffected, namely R ( R A,B ), D ( A ) and D ( A ), while R ( A Dλ (cid:96) ) is found to be sensiblydifferent with respect to its SM value. All the remaining observables display broadprediction ranges, once again with the real (imaginary) part affecting the LFUVratios (differences).To further explore the possibilities of using the angular observables as tests for LFUVwe performed an extra test, motivated by the following observation: as can be seen fromEq. (23), only a small subset of the angular coefficients defined at Eq. (18) appear inthe definition of the decay rate and hence of R ( D ∗ ), i.e. I c, s and I c, s . Therefore,the measurement of the branching fraction can constrain only a part of the coefficientsappearing in the full angular distribution defined in Eq. (17), with the remaining ones stillpotentially affected by the NP effects. In order to test such effects we performed a new setof fits where we assumed that R ( D ) and R ( D ∗ ) would be measured with a central valueequal to the SM predictions with a 10% error. The allowed regions for the couplings arefound to be similar to the ones shown in Figs. 3 and 4, even if reduced in size and thickness.Therefore, the LFUV observables that were previously showing a broad prediction due totheir non-trivial interplay with the real and the imaginary parts of the NP couplings willcontinue to display such a behavior. In other words, even if R ( D ) and R ( D ∗ ) are measuredto agree with the SM predictions, there are 7 angular observables that might still display abehavior unambiguously related to NP effects: four LFUV ratios [ R ( A D,D ∗ F B ), R ( A , )], andthree LFUV differences [ D ( A , , )].All of the above observations show the impact that the measurement of even a smallsubset of the observables presented in Sec. 2.4 would have on our understanding of theNP contribution to the b → c transitions. In particular, the measurement of any of theobservables between R ( A D,D ∗ F B ), R ( A , ) and D ( A , , ) would be of great interest, given theirtwofold power: they would help deciphering the Lorentz structure of the NP contributions,and they would severely constrain the presently allowed region for the NP couplings. Weemphasize once again that a measurement of these observables would be of great interesteven in the case of R ( D ) and R ( D ∗ ) being fully compatible with their SM values. F D ∗ L Very recently the Belle Collaboration presented the results of their first study of the fractionof the longitudinally polarized D ∗ ’s in their full sample of B → D ( ∗ ) τ ¯ ν and found F D ∗ L =0 . σ larger than predicted in the SM, which we find to be ( F D ∗ L ) SM =0 . F D ∗ L > ( F D ∗ L ) SM . Only a marginal enhancement is allowed by switching on g S L ,S R ,P while all the other non-zero NP coefficients would make F D ∗ L < ( F D ∗ L ) SM . We hope19his quantity will be further scrutinized by the other experimental groups and its precisionwill be improved to the point that it can play an important role in discriminating amongvarious models.Since we consistently assume that NP arises from coupling to τ , measuring this quantityin the case of µ or e in the final state would be a very helpful check of our assumption.In other words, if our assumption is right then the measured F D ∗ L ( µ, e ) should be equal to (cid:0) F D ∗ L ( µ, e ) (cid:1) SM = 0 . In this paper we discussed a possibility of using a set of observables that can be extractedfrom the angular distribution of the B → D ( ∗ ) (cid:96) ¯ ν decays in order to study the effects ofLFUV. In particular, we define 3 such observables that can be obtained from the angulardistribution of the B → D(cid:96) ¯ ν decay, and 12 from B → D ∗ (cid:96) ¯ ν .NP contribution to B → D ( ∗ ) τ ¯ ν can be parametrized by the couplings g V L ,V R ,S L ,S R ,T L (defined in the text), the values of which can be constrained by the experimentally mea-sured R ( D ( ∗ ) ), found to be larger than its SM prediction. We explored the possibility offeeding R ( D ( ∗ ) ) exp − R ( D ( ∗ ) ) SM by turning on one coupling at the time and found that themeasurement even of a subset of observables can indeed help disentangling among variouspossibilities. In other words, their measurement can considerably help in constraining thecomplex valued couplings g i ’s and thereby help us understanding the Lorentz structure ofthe NP contribution(s). For that purpose, like in the case of R ( D ( ∗ ) ) we point out thatfor the observables for which the SM prediction is non-zero it is convenient to considerthe ratios between the value extracted from the B → D ( ∗ ) τ ¯ ν mode and the one extractedfrom B → D ( ∗ ) l ¯ ν with l ∈ ( e, µ ). Instead, for the quantities which are zero in the SM weconsider the differences.Since many of the proposed observables provide information of the NP couplings thatcannot be accessed through the measurement of the branching fraction, we show that evenin the case in which R ( D ( ∗ ) ) exp = R ( D ( ∗ ) ) SM one can still have non-zero NP couplingswhich can be checked by measuring the ratios/differences of the observables deduced fromthe angular distributions of the considered decay modes.Even though the observables discussed in this paper are most interesting in the caseof τ -lepton in the final state, their experimental measurements in the case of e and/or µ in the final state are very important too. They would help us checking on the assumptionthat is mostly made in the literature, namely that he NP affects only the couplings to τ and not to e or µ . In that case the measurements would coincide with the SM predictions.Furthermore, in some models of NP the coupling to µ can be large whereas the oneto the electron negligibly small [73]. In that situation it is very important to check alsoon the ratios of the observables discussed here in the case of B → D ( ∗ ) µ ¯ ν with respect tothose measured in the case of B → D ( ∗ ) e ¯ ν . 20inally, in order to make this study quantitatively sound, beside the experimental input,one also needs the lattice QCD information concerning the shapes of hadronic form factorsrelevant to the B → D ∗ (cid:96) ¯ ν decay, which are still missing. Acknowledgements
We wish to thank Gudrun Hiller and Donal Hill for useful discussions during the completion ofthe manuscript. M.F. thanks the Fondazione Della Riccia for partial financial support during thecompletion of this work, and acknowledges the financial support from MINECO grant FPA2016-76005-C2-1-P, Maria de Maetzu program grant MDM-2014-0367 of ICCUB and 2017 SGR 929.The work of I.N. is supported by BMBF under grant no. 05H18VKKB1. The work of A.T.has been supported by the DFG Research Unit FOR 1873 “Quark Flavour Physics and EffectiveField Theories”. This project has also received support from the European Union’s Horizon2020 research and innovation programme under the Marie Sklodowska-Curie grant agreementN. 690575 and 674896. b s . E x p . S M g V g A g S g P g T R D . ± . . ± . . ± . − . ± . − . ± . R D ∗ . ± . . ± . . ± . . ± . − . ± . . ± . R ( A D F B ) − . ± . . ± . − [ − . , . ] − . ± . R ( A D λ (cid:96) ) −− . ± . − . ± . −− . ± . −− . ± . R ( A D ∗ λ (cid:96) ) − . ± . . ± . . ± . − . ± . . ± . R ( R L , T ) − . ± . . ± . . ± . − . ± . . ± . R ( R A , B ) − . ± . . ± . . ± . − . ± . . ± . R ( A D ∗ F B ) − . ± . [ − . , . ][ − . , . ] − . ± . − . ± . R ( A ) − . ± . . ± . . ± . − . ± . . ± . R ( A ) − . ± . . ± . . ± . − . ± . . ± . R ( A ) − . ± . [ − . , . ][ − . , . ] − . ± . . ± . R ( A ) − . ± . [ − . , . ][ − . , . ] − . ± . . ± . D ( A ) − [ − . , . ][ − . , . ] − . ± . . ± . D ( A ) − [ − . , . ][ − . , . ] − D ( A ) − [ − . , . ][ − . , . ] − F D ∗ L − . ± . . ± . . ± . − . ± . . ± . T a b l e : T h e v a l u e s o f L F UV r a t i o s / d i ff e r e n ce s . B e s i d e s t h e S M r e s u l t s i n e a c h f o ll o w i n g c o l u m n w e s h o w t h e r e s u l t s o b t a i n e db y s w i t c h i n go n t h e N P c o up li n g g i [ i ∈ { V , A , S , P , T } ] c o rr e s p o nd i n g t o t h a t c o l u m n i n s u c h a w a y t h a t w e s w ee p t h r o u g h t h e v a l u e s a ll o w e db y R ( D ( ∗ ) ) e x p a ss h o w n i n F i g . . F o r t h e q u a n t i t i e s w h i c hb a r e l y e x h i b i t aga u ss i a nb e h a v i o r ( f o r t h e r e a s o n s d i s c u ss e d i n t h e t e x t) w e p r o v i d e t h e σ r a n g e s . I n t h e l a s t li n e w e a l s og i v e t h ec o rr e s p o nd i n g v a l u e o f t h e f r a c t i o n o f l o n g i t ud i n a ll y p o l a r i ze d D ∗ i n t h e s a m p l e o f B → D ( ∗ ) τ ¯ ν . b s . E x p . S M g V L g V R g S L g S R R D . ± . . ± . . ± . . ± . . ± . . ± . R D ∗ . ± . . ± . . ± . . ± . . ± . . ± . R ( A D F B ) − . ± . . ± . . ± . . ± . . ± . R ( A D λ (cid:96) ) −− . ± . − . ± . − . ± . − . ± . − . ± . R ( A D ∗ λ (cid:96) ) − . ± . . ± . . ± . . ± . . ± . R ( R L , T ) − . ± . . ± . . ± . . ± . . ± . R ( R A , B ) − . ± . . ± . . ± . . ± . . ± . R ( A D ∗ F B ) − . ± . . ± . − . ± . . ± . . ± . R ( A ) − . ± . . ± . . ± . . ± . . ± . R ( A ) − . ± . . ± . . ± . . ± . . ± . R ( A ) − . ± . . ± . . ± . . ± . . ± . R ( A ) − . ± . . ± . . ± . . ± . . ± . D ( A ) − − . ± .
005 0 . ± . [ − . , . ][ − . , . ] D ( A ) − − . ± .
003 0 . ± . D ( A ) − − . ± .
01 0 . ± . F D ∗ − . ± . . ± . . ± . . ± . . ± . T a b l e : S a m e a s i n T a b . bu t i n t h e b a s i s d e fin e d i n t h ee ff ec t i v e H a m il t o n i a n ( ) . W e s tr e ss t h a t f o rt h e f o r m f a c t o r s w e u s e C L N + H Q E T + L A TT , a s d i s c u ss e d i n t h e t e x t . Form factors
The hadronic matrix element are parametrized as follows: (cid:104) D ( k ) | cγ µ b | B ( p ) (cid:105) = (cid:20) ( p + k ) µ − m B − m D q q µ (cid:21) f + ( q ) + q µ m B − m D q f ( q ) , (47) (cid:104) D ( k ) | cb | B ( p ) (cid:105) = 1 m b − m c q µ (cid:104) D ( k ) | cγ µ b | B ( p ) (cid:105) = m B − m D m b − m c f ( q ) , (cid:104) D ( k ) | cγ b | B ( p ) (cid:105) =0 , (48) (cid:104) D ( k ) | cσ µν b | B ( p ) (cid:105) = − i ( p µ k ν − k µ p ν ) 2 f T ( q ) m B + m D , (cid:104) D ( k ) | cσ µν γ b | B ( p ) (cid:105) = − i (cid:15) µναβ (cid:104) D ( k ) | cσ αβ b | B ( p ) (cid:105) = − (cid:15) µναβ p α k β f T ( q ) m B + m D . (49) (cid:104) D ∗ ( k, ε ) | cγ µ b | B ( p ) (cid:105) = − i(cid:15) µναβ ε ∗ ν p α k β V ( q ) m B + m D ∗ , (cid:104) D ∗ ( k, ε ) | cγ µ γ b | B ( p ) (cid:105) = ε ∗ µ ( m B + m D ∗ ) A ( q ) − ( p + k ) µ ( ε ∗ q ) A ( q ) m B + m D ∗ − q µ ( ε ∗ q ) 2 m D ∗ q (cid:104) A ( q ) − A ( q ) (cid:105) , (50)with A ( q ) = m B + m D ∗ m D ∗ A ( q ) − m B − m D ∗ m D ∗ A ( q ) , (51) (cid:104) D ∗ ( k, ε ) | cb | B ( p ) (cid:105) =0 , (cid:104) D ∗ ( k, ε ) | cγ b | B ( p ) (cid:105) = − m b + m c q µ (cid:104) D ∗ ( k, ε ) | cγ µ γ b | B ( p ) (cid:105) = − ( ε ∗ q ) 2 m D ∗ m b + m c A ( q ) , (52)The tensor contribution can be parametrized as (cid:104) D ∗ ( k, ε ) | cσ µν b | B ( p ) (cid:105) = (cid:15) µναβ (cid:2) ε ∗ α ( p + k ) β g + ( q ) + ε ∗ α q β g − ( q )+( ε ∗ q ) p α k β g ( q ) (cid:3) , (cid:104) D ∗ ( k, ε ) | cσ µν γ b | B ( p ) (cid:105) = − i (cid:15) µναβ (cid:104) D ∗ ( k, ε ) | cσ αβ b | B ( p ) (cid:105) = i (cid:8)(cid:2) ε ∗ µ ( p + k ) ν − ( p + k ) µ ε ∗ ν (cid:3) g + ( q )+ (cid:2) ε ∗ µ q ν − q µ ε ∗ ν (cid:3) g − ( q ) + ( ε ∗ q ) [ p µ k ν − k µ p ν ] g ( q ) (cid:9) , (53)24here g ± , can be related to the “standard” T − form factors as g + ( q ) = − T ( q ) ,g − ( q ) = m B − m D ∗ q [ T ( q ) − T ( q )] ,g ( q ) = 2 q (cid:20) T ( q ) − T ( q ) − q m B − m D ∗ T ( q ) (cid:21) . (54)The additional form factors commonly used in the literature are defined as A ( q ) = 116 m B m D ∗ (cid:20) ( m B − m D ∗ − q )( m B + m D ∗ ) A ( q ) − λ BD ∗ ( q ) m B + m D ∗ A ( q ) (cid:21) ,T ( q ) = 18 m B m D ∗ (cid:20) ( m B + 3 m D ∗ − q )( m B + m D ∗ ) T ( q ) − λ BD ∗ ( q ) m B − m D ∗ T ( q ) (cid:21) . (55)In order to cancel the divergence at q = 0, the following conditions must be imposed f + (0) = f (0) , A (0) = A (0) , T (0) = T (0) , A (0) = m B − m D ∗ m B m D ∗ A (0) . (56)In this work we use the convention (cid:15) = 1 (or equivalently (cid:15) = − (cid:15) = −
1, the pseudo-tensor matrix elements in Eqs.(49),(53)change the sign since σ µν γ = − sgn[ (cid:15) ]( i/ (cid:15) µναβ σ αβ . B Helicity amplitude formalism
Using the property of the off-shell vector boson V ∗ polarization vectors, (cid:88) λ η ∗ µ ( λ ) η ν ( λ ) δ λ = g µν , δ , ± = − δ t = − , (57)one can write the B → M V ∗ → M (cid:96)ν ( M = D, D ∗ ) amplitudes of general vector and tensorcurrents as M λ M , λ (cid:96) V ( A ) ∝ (cid:104) M ( λ M ) | J µ had | B (cid:105)(cid:104) (cid:96) ( λ (cid:96) ) ν | J lep , µ | (cid:105) = (cid:88) λ η ∗ µ ( λ ) (cid:104) M ( λ M ) | J µ had | B (cid:105) η ν ( λ ) (cid:104) (cid:96) ( λ (cid:96) ) ν | J ν lep | (cid:105) δ λ = (cid:88) λ δ λ H λ M V ( A ) , λ L λ (cid:96) V − A, λ , M λ M , λ (cid:96) T ( T ∝ (cid:104) M ( λ M ) | J µν had | B (cid:105)(cid:104) (cid:96) ( λ (cid:96) ) ν | J lep , µν | (cid:105) = (cid:88) λ,λ (cid:48) iη ∗ µ ( λ ) η ∗ ν ( λ (cid:48) ) (cid:104) M ( λ M ) | J µν had | B (cid:105) ( − i ) η α ( λ ) η β ( λ (cid:48) ) (cid:104) (cid:96) ( λ (cid:96) ) ν | J αβ lep | (cid:105) δ λ δ λ (cid:48) = (cid:88) λ,λ (cid:48) δ λ δ λ (cid:48) H λ M T ( T , λλ (cid:48) L λ (cid:96) T − T , λλ (cid:48) , (58)25here H and L denote the leptonic and hadronic helicity amplitudes defined in Eqs. (63),(67).The ± i factors in M T ( T are introduced for convenience, in order to make all hadronic B → M V ∗ and leptonic amplitudes real if g i ∈ R and when χ → M λ (cid:96) S ( P ) ∝ (cid:104) M ( λ M = 0) | J had | B (cid:105)(cid:104) (cid:96) ( λ (cid:96) ) ν | J lep | (cid:105) = H S ( P ) L λ (cid:96) S − P . (59)Here λ M and λ ( (cid:48) ) denote the meson and the virtual boson helicities in the B referenceframe. The lepton helicity λ (cid:96) is defined in the (cid:96)ν rest frame.For the four-body final state B → D ∗ V ∗ → Dπ(cid:96)ν decay the total amplitude has theform M λ M , λ (cid:96) X ∝ (cid:104) Dπ | D ∗ ( λ D ∗ ) (cid:105)(cid:104) D ∗ ( λ D ∗ ) | J X had | B (cid:105)(cid:104) (cid:96) ( λ (cid:96) ) ν | J lep , X | (cid:105) BW D ∗ , (60)where the propagation of the intermediate resonant state is parametrized by the Breit-Wigner function, BW D ∗ ( m Dπ ) = 1 m Dπ − m D ∗ + im D ∗ Γ D ∗ . (61)Since the width of D ∗ is very small, one can use the narrow width approximation,1( m Dπ − m D ∗ ) + m D ∗ Γ D ∗ Γ D ∗ (cid:28) m D ∗ −−−−−−→ πm D ∗ Γ D ∗ δ ( m Dπ − m D ∗ ) , (62)and integrate out the m Dπ dependence in the phase space. B.1 Leptonic amplitudes
The leptonic amplitudes are defined as L λ (cid:96) V − A, λ ( q , χ, θ (cid:96) ) = η µ ( λ ) (cid:104) (cid:96) ( λ (cid:96) ) ν | (cid:96)γ µ (1 − γ ) ν | (cid:105) ,L λ (cid:96) S − P ( q , χ, θ (cid:96) ) = (cid:104) (cid:96) ( λ (cid:96) ) ν | (cid:96) (1 − γ ) ν | (cid:105) ,L λ (cid:96) T − T , λλ (cid:48) ( q , χ, θ (cid:96) ) = − L λ (cid:96) T − T , λ (cid:48) λ = − iη µ ( λ ) η ν ( λ (cid:48) ) (cid:104) (cid:96) ( λ (cid:96) ) ν | (cid:96)σ µν (1 − γ ) ν | (cid:105) . (63)Using the polarization vectors and Dirac spinors, given in the Appendix C, and gamma-matrices in the Weyl (chiral) representation, one can obtain the explicit formulas for thevector type amplitudes: L + V − A, + ( q , χ, θ (cid:96) ) = ±√ m (cid:96) β (cid:96) sin θ (cid:96) e − iχ ,L + V − A, − ( q , χ, θ (cid:96) ) = ±√ m (cid:96) β (cid:96) sin θ (cid:96) ,L + V − A, ( q , χ, θ (cid:96) ) = 2 m (cid:96) β (cid:96) cos θ (cid:96) e − iχ ,L + V − A, t ( q , χ, θ (cid:96) ) = − m (cid:96) β (cid:96) e − iχ ,L − V − A, ± ( q , χ, θ (cid:96) ) = (cid:112) q β (cid:96) (1 ± cos θ (cid:96) ) e ∓ iχ ,L − V − A, ( q , χ, θ (cid:96) ) = − (cid:112) q β (cid:96) sin θ (cid:96) ,L − V − A, t ( q , χ, θ (cid:96) ) = 0 , (64)26ith β (cid:96) = (cid:112) − m (cid:96) /q . Expressions for the scalar leptonic amplitudes read L + S − P ( q , χ, θ (cid:96) ) = − (cid:112) q β (cid:96) e − iχ ,L − S − P ( q , χ, θ (cid:96) ) = 0 . (65)Finally, the tensor type amplitudes are given by L + T − T , +0 ( q , χ, θ (cid:96) ) = (cid:112) q β (cid:96) sin θ (cid:96) e − iχ ,L + T − T , − ( q , χ, θ (cid:96) ) = (cid:112) q β (cid:96) sin θ (cid:96) ,L + T − T , + − ( q , χ, θ (cid:96) ) = − L + T − T , t = 2 (cid:112) q β (cid:96) cos θ (cid:96) e − iχ ,L + T − T , + t ( q , χ, θ (cid:96) ) = ∓ (cid:112) q β (cid:96) sin θ (cid:96) e − iχ ,L + T − T , − t ( q , χ, θ (cid:96) ) = ∓ (cid:112) q β (cid:96) sin θ (cid:96) ,L − T − T , ± ( q , χ, θ (cid:96) ) = ±√ m (cid:96) β (cid:96) (1 ± cos θ (cid:96) ) e ∓ iχ ,L − T − T , + − ( q , χ, θ (cid:96) ) = − L − T − T , t = − m (cid:96) β (cid:96) sin θ (cid:96) ,L − T − T , ± t ( q , χ, θ (cid:96) ) = −√ m (cid:96) β (cid:96) (1 ± cos θ (cid:96) ) e ∓ iχ . (66)By setting χ → x − z plane is defined by the lepton momentum) one ends upwith the expressions coinciding with those given in Refs. [41, 74]. B.2 Hadronic amplitudes
The general helicity amplitudes, in the operator basis (5), read: H λ M V ( A ) , λ ( q ) = ( g V ( A ) ± η ∗ µ ( λ ) (cid:104) M ( λ M ) | cγ µ ( γ ) b | B (cid:105) ,H S ( P ) , λ ( q ) = g S ( P ) (cid:104) M ( λ M = 0) | c ( γ ) b | B (cid:105) ,H λ M T ( T , λλ (cid:48) ( q ) = − H λ M T, λ (cid:48) λ ( q ) = i g T ( T η ∗ µ ( λ ) η ∗ ν ( λ (cid:48) ) (cid:104) M ( λ M ) | cσ µν ( γ ) b | B (cid:105) , (67)which can be written explicitly for the B → D case as h ( q ) ≡ h V, ( q ) = ( g V + 1) (cid:115) λ BD ( q ) q f + ( q ) , (68a) h t ( q ) ≡ h V, t ( q ) = ( g V + 1) m B − m D (cid:112) q f ( q ) , (68b) h S ( q ) (cid:39) g S m B − m D m b − m c f ( q ) , (68c) To avoid confusion and simplify notation, we denote the B → D amplitudes by h and omit thesuper-index λ D = 0. T, t ( q ) = − g T g T h T , + − ( q ) = − g T (cid:112) λ BD ( q ) m B + m D f T ( q ) , (68d) h A, λ ( q ) = h P ( q ) = h T, + − ( q ) = h T , t ( q ) = 0 , (68e)and for the B → D ∗ case as H ± V, ± ( q ) = ∓ ( g V + 1) (cid:112) λ BD ∗ ( q ) m B + m D ∗ V ( q ) , (69a) H ± A, ± ( q ) = − ( g A − m B + m D ∗ ) A ( q ) , (69b) H ( q ) ≡ H A, ( q ) = ( g A −
1) 8 m B m D ∗ (cid:112) q A ( q ) , (69c) H t ( q ) ≡ H A, t ( q ) = ( g A − (cid:115) λ BD ∗ ( q ) q A ( q ) , (69d) H P ( q ) (cid:39) − g P (cid:112) λ BD ∗ ( q ) m b + m c A ( q ) , (69e) H ± T, ± ( q ) = ∓ g T g T H ± T , ± t ( q ) = ± g T m B − m D ∗ (cid:112) q T ( q ) , (69f) H ± T, ± t ( q ) = ∓ g T g T H ± T , ± ( q ) = ± g T (cid:115) λ BD ∗ ( q ) q T ( q ) , (69g) H T, + − ( q ) = − g T g T H T , t ( q ) = − g T m B m D ∗ m B + m D ∗ T ( q ) , (69h) H V, ( q ) = H V, t ( q ) = H S ( q ) = H T, t ( q ) = H T , + − ( q ) = 0 , (69i)where again λ BM ( q ) = m B + m M + q − m B m M + m B q + m M q ). In deriving theexpressions for the above amplitudes we used the decomposition of the hadronic matrixelements in terms of form factors listed in Appendix A.28 .3 D ∗ → Dπ amplitude The D ∗ → Dπ amplitude can be parametrized as (cid:104) Dπ | D ∗ ( λ D ∗ ) (cid:105) = g D ∗ Dπ ε µ ( λ D ∗ ) p µD , (70)where the coupling g D ∗ Dπ parameterizes the physical D ∗ → Dπ decay and can be de-termined from the numerical simulations of QCD on the lattice or extracted from themeasured width of D ∗ + , Γ( D ∗ → Dπ ) = C πm D ∗ g D ∗ Dπ | ˆ p D | , (71)where C = 1 if the outgoing pion is charged, and C = 1 / p D is the D three-momentum in the D ∗ rest frame. It must be stressed that g D ∗ Dπ is m Dπ -independent,and the entire dependence of the amplitude (60) on m Dπ is assumed to be described bythe Breit-Wigner function.The amplitudes are computed in the D ∗ reference frame, where the D, π momenta arein the x − z plane, and are given by (cid:104) Dπ | D ∗ ( ± (cid:105) = ± √ g D ∗ Dπ | ˆ p D | sin θ D , (cid:104) Dπ | D ∗ (0) (cid:105) = − g D ∗ Dπ | ˆ p D | cos θ D . (72) B.4 Relations between amplitudes
One can notice from Eqs. (64) and (65) that L λ (cid:96) S − P = (cid:112) q m (cid:96) L λ (cid:96) V − A, t . (73)Therefore, in order to further simplify the expressions, one can absorb the hadronic S/P amplitudes into the
V /A time-like ones and redefine, (cid:101) H λ M (=0) V/A, t ≡ H V/A, t + (cid:112) q m (cid:96) H S/P . (74)29oreover, from Eqs. (64) and (66) it can be seen that L + T − T , ± = ± (cid:112) q m (cid:96) L + V − A, ± ,L + T − T , + − = − L + T − T , t = (cid:112) q m (cid:96) L + V − A, ,L + T − T , ± t = − (cid:112) q m (cid:96) L + V − A, ± ,L − T − T , ± = ± m (cid:96) (cid:112) q L − V − A, ± ,L − T − T , + − = − L − T − T , t = m (cid:96) (cid:112) q L − V − A, ,L − T − T , ± t = − m (cid:96) (cid:112) q L − V − A, ± . (75)In this way, summing over the off-shell boson polarizations λ, λ (cid:48) and taking into accountthe δ λ factors in Eq. (58), one can write the general total amplitude as M λ M , λ (cid:96) ∝ − (cid:88) λ = ± , ( H λ M V, λ + H λ M A, λ ) L λ (cid:96) V − A, λ + ( (cid:101) H λ M V, t + (cid:101) H λ M A, t ) L λ (cid:96) V − A, t + 2 (cid:88) λ = ± (cid:20) ( H λ M T, λ + H λ M T , λ ) L λ (cid:96) T − T , λ − ( H λ M T, λt + H λ M T , λt ) L λ (cid:96) T − T , λt (cid:21) + 2( H λ M T, + − + H λ M T , + − ) L λ (cid:96) T − T , + − − H λ M T, t + H λ M T , t ) L λ (cid:96) T − T , t . (76)Using the relations (75) one can write it in a more compact form : M λ M , λ (cid:96) ∝ − (cid:88) λ = ± , (cid:101) H λ M , λ (cid:96) λ L λ (cid:96) V − A, λ + (cid:101) H λ M t L λ (cid:96) V − A, t , (77)where the redefined amplitudes are given as, (cid:101) H λ M , + ± ≡ H λ M V, ± + H λ M A, ± − (cid:112) q m (cid:96) (cid:0) H λ M T, ± t ± H λ M T, ± + H λ M T , ± t ± H λ M T , ± (cid:1) , (cid:101) H λ M , +0 ≡ H λ M V, + H λ M A, − (cid:112) q m (cid:96) (cid:0) H λ M T, + − + H λ M T, t + H λ M T , + − + H λ M T , t (cid:1) , (cid:101) H λ M , −± ≡ H λ M V, ± + H λ M A, ± − m (cid:96) (cid:112) q (cid:0) H λ M T, ± t ± H λ M T, ± + H λ M T , ± t ± H λ M T , ± (cid:1) , (cid:101) H λ M , − ≡ H λ M V, + H λ M A, − m (cid:96) (cid:112) q (cid:0) H λ M T, + − + H λ M T, t + H λ M T , + − + H λ M T , t (cid:1) , (cid:101) H λ M t ≡ (cid:101) H λ M V, t + (cid:101) H λ M A, t . (78)Absorbing the scalar and tensor amplitudes into (cid:101) h and (cid:101) H allows to significantly simplifythe calculations and to write the expressions for the angular coefficients of the differential30ecay rate in a much more compact way. To be more specific, we introduce the followinglinear combinations : (cid:101) h +0 ( q ) ≡ h ( q ) − (cid:112) q m (cid:96) h T ( q ) , (cid:101) h − ( q ) ≡ h ( q ) − m (cid:96) (cid:112) q h T ( q ) , (cid:101) h t ( q ) ≡ h t ( q ) + (cid:112) q m (cid:96) h S ( q ) , (79)and (cid:101) H + ± ( q ) ≡ H ± ( q ) − (cid:112) q m (cid:96) H T, ± ( q ) , (cid:101) H +0 ( q ) ≡ H ( q ) − (cid:112) q m (cid:96) H T, ( q ) , (cid:101) H −± ( q ) ≡ H ± ( q ) − m (cid:96) (cid:112) q H T, ± ( q ) , (cid:101) H − ( q ) ≡ H ( q ) − m (cid:96) (cid:112) q H T, ( q ) , (cid:101) H t ( q ) ≡ H t ( q ) + (cid:112) q m (cid:96) H P ( q ) , (80)with h T ( q ) ≡ h T, t ( q ) + h T , + − ( q ) ,H ± ( q ) ≡ H ± V, ± ( q ) + H ± A, ± ( q ) ,H T, ± ( q ) ≡ H ± T, ± t ( q ) ± H ± T, ± ( q ) + H ± T , ± t ( q ) ± H ± T , ± ( q ) ,H T, ( q ) ≡ H T, + − ( q ) + H T , t ( q ) . (81)To simplify and shorten the final expressions of angular observables, we omit the super-index λ M in (cid:101) h and (cid:101) H .Note that using the relations between various hadronic tensor amplitudes [Eqs. (68,69)],the tensor contribution to (cid:101) h and (cid:101) H in Eqs. (79,80) vanishes if g T = g T . This is reason-able since the operator cσ µν (1 + γ ) b (cid:96)σ µν (1 − γ ) ν identically vanishes due to the Fierztransformations. C Polarization vectors and spinors
In the B rest frame the four-momenta of B ( p ), D ( ∗ ) ( k ) and q are p µ = m B , k µ = E D ( ∗ ) | q | , q µ = q −| q | . (82)31he polarization vectors of D ∗ ( ε ) and the virtual vector boson ( η ) are defined in the B rest frame as in Ref. [74]: ε µ ( ± ) = 1 √ ∓ − i , ε µ (0) = 1 m D ∗ | q | E D ∗ , (83)and η µ ( ± ) = 1 √ ∓ i , η µ (0) = 1 (cid:112) q | q | − q , η µ ( t ) = 1 (cid:112) q q −| q | , (84)with | q | = (cid:112) λ BD ( ∗ ) ( q )2 m B , q = m B − m D ( ∗ ) + q m B , E D ( ∗ ) = m B + m D ( ∗ ) − q m B . (85)Other alternative parametrization can be found in the seminal paper [75], in which the z -axis is also along D ( ∗ ) momentum. Note that in Ref. [75] all four-vectors are defined ascovariant, while in this work we define all vectors as contravariant.The Dirac spinors, used in the leptonic helicity amplitudes (63) calculation, are definedas [74] u ( λ = ± /
2) = (cid:32) (cid:112) E ∓ | p | ξ ± (cid:112) E ± | p | ξ ± (cid:33) , v ( λ = ± /
2) = (cid:32) − (cid:112) E ± | p | ξ ∓ (cid:112) E ∓ | p | ξ ∓ (cid:33) , (86)where the helicity eigenspinors ξ + = (cid:18) cos θ sin θ e iφ (cid:19) , ξ − = (cid:18) − sin θ e − iφ cos θ (cid:19) , (87)describe either particles of helicity ± / ∓ / λ ν = 1 /
2. The ν spinor, v ( λ ν = 1 / θ ν = π − θ (cid:96) and φ ν = φ (cid:96) + π . In our chosensystem of coordinates the leptonic azimuthal angle φ (cid:96) ≡ χ . D Four-body phase space
The four-body phase space can be reduced to the product of the two-body phase spaces: d Φ =(2 π ) (cid:90) (cid:89) i =1 d p i (2 π ) E i δ (cid:18) P − (cid:88) j =1 p j (cid:19) = dm π dm π d Φ ( p , p ) d Φ (ˆ p , ˆ p ) d Φ (ˆ p , ˆ p ) , (88)32here m ij = p ij = ( p i + p j ) . The two-body phase space is given by the standard expression d Φ (ˆ p i , ˆ p j ) = 116 π | ˆ p i | m ij d cos θ i dφ i , (89)with three-momentum ˆ p i defined in the ij rest frame.Using Eq. (88) one can write the phase space for the B → D ∗ ( → Dπ ) (cid:96)ν (cid:96) as, d Φ = 164(2 π ) dm Dπ dq | ˆ p Dπ | m B d cos θ Dπ dφ Dπ | ˆ p D | m Dπ d cos θ D dφ D | ˆ p (cid:96) | (cid:112) q d cos θ (cid:96) dφ (cid:96) , (90)where ˆ p Dπ (= − q ), ˆ p D , ˆ p (cid:96) and the corresponding angles are defined in the B , Dπ and (cid:96)ν rest frames respectively, namely, | ˆ p Dπ | = (cid:112) λ ( m B , m Dπ , q )2 m B , | ˆ p D | = (cid:112) λ ( m Dπ , m D , m π )2 m Dπ , | ˆ p (cid:96) | = q − m (cid:96) (cid:112) q , (91)where, as before, λ ( a, b, c ) = a + b + c − ab + ac + bc ).Integrating over the polar and azimuthal angles of the D ∗ momentum ( θ Dπ , φ Dπ ) andover the azimuthal angle of the D momentum ( φ D ), one obtains d Φ = 164(2 π ) | q | m B | ˆ p D | m Dπ (cid:18) − m (cid:96) q (cid:19) dq dm Dπ d cos θ D d cos θ (cid:96) dχ . (92)Here we defined the angle φ (cid:96) = χ with respect to the Dπ rest frame.Similarly, one can obtain the three-body phase space for the B → D(cid:96)ν decay : d Φ = dq π d Φ ( p D , q ) d Φ (ˆ p (cid:96) , ˆ p ν ) → π ) | q | m B (cid:18) − m (cid:96) q (cid:19) dq d cos θ (cid:96) . (93) E Fit results with NP couplings at best fit values
In here we show the fit results obtained following the same procedure as the one explainedin Sec. 3.1, but fixing for each scenario the NP coupling at the best fit value, which canbe found at Eqs. (45,46). Similarly to Tab. 2, these results have been obtained enforcingthe experimental results for R ( D ) and R ( D ∗ ) and allowing for NP effects in one coefficientat a time. In particular, we show in Tab. 4 the results for the NP coefficients employed inEq. (5), while we report in Tab. 5 the results for the ones introduced in Eq. (6). Since allthe predicted observables behave in a Gaussian manner, we write for all of them the meanvalues and the standard deviations. 33 b s . E x p . S M g V g A g S g P g T R D . ± . . ± . . ± . − . ± . − . ± . R D ∗ . ± . . ± . . ± . . ± . − . ± . . ± . R ( A D F B ) − . ± . . ± . − . ± . − . ± . R ( A D λ (cid:96) ) −− . ± . − . ± . −− . ± . −− . ± . R ( A D ∗ λ (cid:96) ) − . ± . . ± . . ± . − . ± . . ± . R ( R L , T ) − . ± . . ± . . ± . − . ± . . ± . R ( R A , B ) − . ± . . ± . . ± . − . ± . . ± . R ( A D ∗ F B ) − . ± . . ± . − . ± . −− . ± . − . ± . R ( A ) − . ± . . ± . . ± . − . ± . . ± . R ( A ) − . ± . . ± . . ± . − . ± . . ± . R ( A ) − . ± . . ± . . ± . − . ± . . ± . R ( A ) − . ± . . ± . . ± . − . ± . . ± . D ( A ) − ( . ± . ) · − . ± . − ( − . ± . ) · − − . ± . D ( A ) − ( . ± . ) · − . ± . − D ( A ) − ( . ± . ) · − . ± . − T a b l e : A ll t h e o b s e r v a b l e s o b t a i n e db y u s i n g t h e v a l u e s o f t h e N P c o up li n g s g i v e n i n E q . . b s . E x p . S M g V L g V R g S L g S R R D . ± . . ± . . ± . . ± . . ± . . ± . R D ∗ . ± . . ± . . ± . . ± . . ± . . ± . R ( A D F B ) − . ± . . ± . . ± . . ± . . ± . R ( A D λ (cid:96) ) −− . ± . − . ± . − . ± . − . ± . − . ± . R ( A D ∗ λ (cid:96) ) − . ± . . ± . . ± . . ± . . ± . R ( R L , T ) − . ± . . ± . . ± . . ± . . ± . R ( R A , B ) − . ± . . ± . . ± . . ± . . ± . R ( A D ∗ F B ) − . ± . . ± . − . ± . . ± . . ± . R ( A ) − . ± . . ± . . ± . . ± . . ± . R ( A ) − . ± . . ± . . ± . . ± . . ± . R ( A ) − . ± . . ± . . ± . . ± . . ± . R ( A ) − . ± . . ± . . ± . . ± . . ± . D ( A ) − − . ± . . ± . ( − . ± . ) · − D ( A ) − − . ± . D ( A ) − − . ± . T a b l e : S a m e a s i n T a b . bu t i nd i ff e r e n t b a s i s o f o p e r a t o r s a nd f o r v a l u e s o f N P c o up li n g s g i v e n i n E q . . eferences [1] BaBar
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