Bayesian fit analysis to full distribution data of B ¯ → D (∗) ℓ ν ¯ : | V cb | determination and New Physics constraints
BBayesian fit analysis to full distribution data of ¯ B → D ( ∗ ) (cid:96) ¯ ν : | V cb | determination and New Physics constraints Syuhei Iguro (a) and Ryoutaro Watanabe (b)(a)
Department of Physics, Nagoya University, Nagoya 464-8602, Japan (b)
INFN, Sezione di Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy
May 6, 2020
Abstract
We investigate the semi-leptonic decays of ¯ B → D ( ∗ ) (cid:96) ¯ ν in terms of the Heavy-Quark-Effective-Theory (HQET) parameterization for the form factors, which is described withthe heavy quark expansion up to O (1 /m c ) beyond the simple approximation consideredin the original CLN parameterization. An analysis with this setup was first given in theliterature, and then we extend it to the comprehensive analyses including (i) simultaneousfit of | V cb | and the HQET parameters to available experimental full distribution data andtheory constraints, and (ii) New Physics (NP) contributions of the V and T types, suchas ( cγ µ P R b )( (cid:96)γ µ P L ν (cid:96) ) and ( cσ µν P L b )( (cid:96)σ µν P L ν (cid:96) ), to the decay distributions and rates. Forthis purpose, we perform Bayesian fit analyses by using Stan program, a state-of-the-artpublic platform for statistical computation. Then, we show that our | V cb | fit results forthe SM scenarios are close to the PDG combined average from the exclusive mode, andindicate significance of the angular distribution data. In turn, for the SM + NP scenarios,our fit analyses find that non-zero NP contribution is favored at the best fit point for bothSM + V and SM + T depending on the HQET parameterization model. A key featureis then realized in the ¯ B → D ( ∗ ) τ ¯ ν observables. Our fit result of the HQET parametersin the SM(+ T ) produces a consistent value for R D while smaller for R D ∗ , compared withthe previous SM prediction in the HFLAV report. On the other hand, SM + V pointsto smaller and larger values for R D and R D ∗ than the SM predictions. In particular,the R D ∗ deviation from the experimental measurement becomes smaller, which could beinteresting for future improvement on measurements at the Belle II experiment. a r X i v : . [ h e p - ph ] M a y ontents ¯ B → D(cid:96) ¯ ν : w distribution . . . . . . . . . . . . . . . . . . . . . . . 62.3 Formula for ¯ B → D ∗ (cid:96) ¯ ν : full angular distribution . . . . . . . . . . . . . . . . . 6 | V cb | determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4.2 NP scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4.3 Observables for ¯ B → D ( ∗ ) τ ¯ ν . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.4 Theoretical uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 α s and /m Q Corrections 18B Angular dependence 21C Constraints from QCDSR 22D Fit results with some details 24 Introduction
The semi-leptonic processes ¯ B → D ( ∗ ) (cid:96) ¯ ν for (cid:96) = e, µ have been studied from various per-spectives. In particular, the decay rates are of great interest as it determines the Cabibbo-Kobayashi-Maskawa [1, 2] (CKM) matrix element | V cb | in the Standard Model (SM). Kineticdistributions of the processes are also important, for instance, to experimentally measure theratios with the semi-tauonic modes, R D ( ∗ ) = B ( ¯ B → D ( ∗ ) τ ¯ ν ) / B ( ¯ B → D ( ∗ ) (cid:96) ¯ ν ), in which discrep-ancies between the experimental measurements and the SM predictions have been reported.To investigate these issues, however, we need a sufficient knowledge on the hadron transi-tions ¯ B → D ( ∗ ) . In the literature, there are several theoretical descriptions on the form factors(FFs). The CLN parameterization [3], applying heavy quark symmetry to FFs based on theHeavy-Quark-Effective-Theory [4, 5] (HQET), has been used for this purpose. The BGL pa-rameterization [6] is an alternative that relies only on QCD dispersion relations, which impliesthe model independent one.An advantage of the former is that it describes the ¯ B → D and ¯ B → D ∗ FFs with a fewcommon parameters, and thus a combined analysis is possible, e.g. , see Ref. [7]. The latter,on the other hand, includes larger number of independent parameters so that a flexible fitanalysis is given, although it needs experimental data with higher statistics. Then, the | V cb | determinations from these two approaches have been in the spotlight since their results are notconsistent with each other, see discussions in Refs. [8, 9, 10, 11, 12, 13, 14].In the recent studies of Refs. [15, 16], the authors have revisited the HQET parameterizationby adopting a setup beyond the CLN approximation and taking 1 /m c corrections into accountfor the heavy quark expansion. This approach introduces 23 free HQET parameters whichhave to be determined from experiments and/or theoretical constraints (as also reviewed anddiscussed in this paper.) At the expense of such a large number of parameter set, it has beenfound [16] that the SM fit result of | V cb | is in good agreement with the one obtained from theBGL parameterization.In this paper, we investigate ¯ B → D ( ∗ ) (cid:96) ¯ ν with the use of this HQET parameterization byconcerning the following points: • We include all the available full distribution data of ¯ B → D ( ∗ ) (cid:96) ¯ ν from the Belle measure-ments [17, 18, 19] in our fit analysis to simultaneously determine | V cb | and the HQETparameters. Indeed, this is not the case for the reference as will be explained later. • We consider New Physics (NP) effects on ¯ B → D ( ∗ ) (cid:96) ¯ ν that could affect both branchingratios and decay distributions. Here, a simultaneous fit for the size of the NP contribu-tions, | V cb | , and the HQET parameters is performed in our analysis. Then it is shownthat a non-negligible NP contribution is still allowed and it satisfies the experimentaldata. We also provide complete formulae on the decay distributions and the FFs in thepresence of NP. • We perform Bayesian fit analysis with the use of
Stan [21], a public platform for statisticalcomputation, which has been widely known in the statistical science community andthus could give independent check to the previous studies. We also obtain quantitativeevaluations on our fit results in various parameterization scenarios with/without NP bylooking at information criterion [22].In addition, we put some comparison with the CLN parameterization, and also see our predic-tions on the ¯ B → D ( ∗ ) τ ¯ ν observables. Then, we see that NP predictions on R D ( ∗ ) obtained from The BaBar experimental analysis is given in Ref. [20], but they do not provide detailed information ondistribution data. B → D ( ∗ ) τ ¯ ν observables. We would like to stress that this is a comprehensivefit analysis for the HQET parameterization with/without the NP contributions.This paper is organized as follows. In Sec. 2, we describe our theory setup for the HQETparameterization and formulae for the decay distributions in the presence of NP. In Sec. 3, wedetail our fit procedure along with summary of theory constraints and experimental measure-ments to be taken in our analysis. Then we discuss our results in the various scenarios. Finally,a summary is put in Sec. 4. Details of our fit results, distribution formulae, and some theoryconstraints are given in Appendices. In this work, we start with the effective Hamiltonian that affects ¯ B → D ( ∗ ) (cid:96) ¯ ν , given as H eff = 4 G F √ V cb (cid:20) ( cγ µ P L b )( (cid:96)γ µ P L ν (cid:96) ) + C V ( cγ µ P R b )( (cid:96)γ µ P L ν (cid:96) ) + C T ( cσ µν P L b )( (cid:96)σ µν P L ν (cid:96) ) (cid:21) , (1)where P L/R = (1 ∓ γ ) / C T ( V ) (cid:54) = 0 indicates existence of a tensor ( V + A vector in ¯ cb )type NP. The SM-like NP always rescales V cb and then we do not consider this case since itseffect has to be examined by indirect or combined approaches. As long as the light lepton mode( (cid:96) = e, µ ) is concerned, note that the scalar type operators, ( cP R b )( (cid:96)P L ν (cid:96) ) and ( cP L b )( (cid:96)P L ν (cid:96) ),do not affect the present processes due to the light lepton mass suppression. We assume thatNP has e - µ universal ( C eX = C µX ≡ C X ) and C X is real. This is a conservative choice since B ( ¯ B → D ( ∗ ) µ ¯ ν ) / B ( ¯ B → D ( ∗ ) e ¯ ν ) ≈ ± O (%) has been reported [17, 18, 19]. Also the neutrinois always taken as left-handed.In the following part of this section, we will present theory descriptions and formulae nec-essary for our fit analysis. In the HQET basis, all possible types of the B → D ( ∗ ) current are defined as (cid:104) D | ¯ cγ µ b | B (cid:105) HQET = √ m B m D (cid:2) h + ( v + v (cid:48) ) µ + h − ( v − v (cid:48) ) µ (cid:3) , (2) (cid:104) D | ¯ cb | B (cid:105) HQET = √ m B m D ( w + 1) h S , (3) (cid:104) D | ¯ cσ µν b | B (cid:105) HQET = − i √ m B m D h T (cid:2) v µ v (cid:48) ν − v (cid:48) µ v ν (cid:3) , (4) (cid:104) D ∗ | ¯ cγ µ b | B (cid:105) HQET = i √ m B m D ∗ h V ε µνρσ (cid:15) ∗ ν v (cid:48) ρ v σ , (5) (cid:104) D ∗ | ¯ cγ µ γ b | B (cid:105) HQET = √ m B m D ∗ (cid:2) h A ( w + 1) (cid:15) ∗ µ − ( (cid:15) ∗ · v ) ( h A v µ + h A v (cid:48) µ ) (cid:3) , (6) (cid:104) D ∗ | ¯ cγ b | B (cid:105) HQET = −√ m B m D ∗ ( (cid:15) ∗ · v ) h P , (7) (cid:104) D ∗ | ¯ cσ µν b | B (cid:105) HQET = −√ m B m D ∗ ε µνρσ (cid:2) h T (cid:15) ∗ ρ ( v + v (cid:48) ) σ + h T (cid:15) ∗ ρ ( v − v (cid:48) ) σ + h T ( (cid:15) ∗ · v )( v + v (cid:48) ) ρ ( v − v (cid:48) ) σ (cid:3) , (8)where v µ = p µB /m B , v (cid:48) µ = p µD ( ∗ ) /m D ( ∗ ) , w = v · v (cid:48) = ( m B + m D ( ∗ ) − q ) / (2 m B m D ( ∗ ) ), and h X ≡ h X ( w ) are the HQET form factors in terms of w . Then, h X can be represented by the4eading Isgur-Wise [4] (IW) function ξ and its correction, defined as h X ( w ) = ξ ( w )ˆ h X ( w ). Inthis work, we considerˆ h X = ˆ h X, + α s π δ ˆ h X,α s + ¯Λ2 m b δ ˆ h X,m b + ¯Λ2 m c δ ˆ h X,m c + (cid:18) ¯Λ2 m c (cid:19) δ ˆ h X,m c , (9)whereˆ h X, = (cid:40) X = + , A , A , S, P, T, T X = − , A , T , T , (10)and others indicate higher order corrections in α s and 1 /m b,c expansions. In this work, the aboveHQET expansion is given at the matching scale µ b = 4 . (cid:15) a = α s /π = 0 . (cid:15) b = ¯Λ / (2 m b ) = 0 . (cid:15) c = ¯Λ / (2 m c ) =0 . δ ˆ h X,f . Thus we neglect those uncertainties hereafter. Thecomplete expressions for δ ˆ h X,f are summarized in Appendix A.The 1 /m Q correction consists of three unknown sub-leading IW functions defined as ξ ( w ), χ ( w ), and χ ( w ) [7], whereas 1 /m Q of six subsub-leading IW functions (cid:96) ( w ) [23]. Thus wehave in total ten IW functions that are in principle unknown and then have to be fitted. Wealso employ the notation such as η ( w ) = ξ ( w ) ξ ( w ) , ˆ χ i ( w ) = χ i ( w ) ξ ( w ) , ˆ (cid:96) i ( w ) = (cid:96) i ( w ) ξ ( w ) . (11)Then, we can express any of the IW functions by means of series expansion around w = 1.Namely, we take f ( w ) = (cid:88) n =0 f ( n ) n ! ( w − n , (12)for f = ξ , η , ˆ χ i , and ˆ (cid:96) i . Here, f ( n ) ≡ ∂ n f ( w ) ∂w n (cid:12)(cid:12)(cid:12) w =1 are free parameters to be fitted by theoreticaland/or experimental analysis. Analytic properties of the matrix elements indicate that theabove expansion can be represented by w ( z ) = 2 (cid:18) z − z (cid:19) − , (13)up to the order of interest. For instance, we have f ( w ) = f (0) + 8 f (1) z + 16 (cid:0) f (1) + 2 f (2) (cid:1) z + 83 (cid:0) f (1) + 48 f (2) + 32 f (3) (cid:1) z + O ( z ) . (14)Note that ξ (0) = 1 and ˆ χ (0)3 = 0 in the HQET description. Following Ref. [16], the cases ofNNLO(3 / /
1) : ξ ( w ) up to z , ˆ χ , ( w ) and η ( w ) up to z , ˆ (cid:96) ( w ) up to z , (15)NNLO(2 / /
0) : ξ ( w ) up to z , ˆ χ , ( w ) and η ( w ) up to z , ˆ (cid:96) ( w ) up to z , (16)are investigated in our analysis. In addition, we considerNLO(3 / / -) : ξ ( w ) up to z , ˆ χ , ( w ) and η ( w ) up to z , ˆ (cid:96) ( w ) = 0 , (17)5ust for comparison to see how ˆ (cid:96) ( w ) improves the parameter fit.A final remark is that we have two kinds of expansion, namely, by (cid:15) a,b,c and z in the formfactor ˆ h X . A significant point is that their highest orders, as assumed above, have to be keptin observables even though it is obtained by multiplying ˆ h X s. Otherwise, higher order termsthan what we take are included unfairly. Schematically, a proper expansion for any observableis written asObs. = O ( (cid:15) z ) + O ( (cid:15) z ) + O ( (cid:15) z ) + O ( (cid:15) z ) + O ( (cid:15) a z ) + O ( (cid:15) a z ) + O ( (cid:15) a z ) + O ( (cid:15) a z )+ O ( (cid:15) b,c z ) + O ( (cid:15) b,c z ) + O ( (cid:15) b,c z ) + O ( (cid:15) c z ) + O ( (cid:15) c z ) , (18)before the w integration, where (cid:15) a = α s /π and (cid:15) b,c = ¯Λ / (2 m b,c ). B → D(cid:96) ¯ ν : w distribution The differential decay rate of ¯ B → D(cid:96) ¯ ν with respect to w is written as d Γ D dw = G F | V cb | m B m D η π (1 − r D w + r D ) √ w − (cid:104) (1 + C V ) H s ( w ) + 2 | C T | H Ts ( w ) (cid:105) , (19)where r D = m D /m B and η EW = 1 . ± . H s ( w ) = m B √ r D √ w − (cid:112) − r D w + r D [(1 + r D ) h + ( w ) − (1 − r D ) h − ( w )] ,H Ts ( w ) = − m B √ r D √ w − h T ( w ) . (20)Note that the tensor NP do not interfere with the SM since the (cid:96) helicity is flipped in themassless limit of the light lepton due to spin structure of ¯ (cid:96)σ µν ν . One can see that G ( w ) = h + ( w ) − − r D r D h − ( w ) , (21)is the usual normalization factor for the SM. In the CLN parameterization, it is approximatedwith a single parameter such as G ( w ) ≈ G (1) [1 − ρ z + (51 ρ − z − (252 ρ − z ]. Com-paring it with the present forms of h ± ( w ) given in Appendix A, we obtain G (1) (cid:39) . − . η (0) + 0 . (cid:96) (0)1 − . (cid:96) (0)4 , (22) − ρ G (1) (cid:39) . . ξ (1) − . χ (0)2 + 22 . χ (1)3 − . (cid:0) η (1) + η (0) ξ (1) (cid:1) + 0 . (cid:0) ˆ (cid:96) (1)1 + ˆ (cid:96) (0)1 ξ (1) (cid:1) − . (cid:0) ˆ (cid:96) (1)4 + ˆ (cid:96) (0)4 ξ (1) (cid:1) , (23)in our setup. We can see that the NNLO parameters ˆ (cid:96) ( n )1 , affect these quantities. Note that the z and z terms in the CLN parameterization are approximated with the single parameter ρ and hence it is not applicable for our case. B → D ∗ (cid:96) ¯ ν : full angular distribution Concerning the available experimental data, we show the full differential decay rate for B → D ∗− ( → ¯ D π − ) (cid:96) ¯ ν in the presence of the NP contributions: d Γ full D ∗ dw d cos θ (cid:96) d cos θ V dχ = B ( D ∗− → ¯ D π − ) G F | V cb | m B m D ∗ η π ) (24)6 (1 − r D ∗ w + r D ∗ ) √ w − (cid:88) i =1 J i ( θ (cid:96) , θ V , χ ) H i ( w ) , where J i include angular dependences obtained as J = (1 − cos θ (cid:96) ) sin θ V , J = (1 + cos θ (cid:96) ) sin θ V , J = 4 sin θ (cid:96) cos θ V J = − θ (cid:96) sin θ V cos 2 χ , J = − θ (cid:96) (1 − cos θ (cid:96) ) sin θ V cos θ V cos χ , J = +4 sin θ (cid:96) (1 + cos θ (cid:96) ) sin θ V cos θ V cos χ , J = 2 sin θ (cid:96) sin θ V , J = 8 cos θ (cid:96) cos θ V , (25)and H i indicate hadronic parts described as H ( w ) = (cid:16) H + ( w ) − C V H − ( w ) (cid:17) , H ( w ) = (cid:16) H − ( w ) − C V H + ( w ) (cid:17) , H ( w ) = (1 − C V ) H ( w ) , H ( w ) = (cid:16) H + ( w ) − C V H − ( w ) (cid:17)(cid:16) H − ( w ) − C V H + ( w ) (cid:17) + 16 | C T | H T − ( w ) H T + ( w ) , H ( w ) = (1 − C V ) H ( w ) (cid:16) H + ( w ) − C V H − ( w ) (cid:17) + 8 | C T | (cid:16) H T ( w ) H T − ( w ) − H T ( w ) H T + ( w ) (cid:17) , H ( w ) = (1 − C V ) H ( w ) (cid:16) H − ( w ) − C V H + ( w ) (cid:17) + 8 | C T | (cid:16) H T ( w ) H T − ( w ) − H T ( w ) H T + ( w ) (cid:17) , H ( w ) = 8 | C T | (cid:16) H T + ( w ) + H T − ( w ) (cid:17) , H ( w ) = 8 | C T | H T ( w ) . (26)Note again that there is no interference term between the vector and tensor currents. Then,we can write the Hadronic Amplitudes H ( T ) n from Refs. [26, 27, 28] as H ± ( w ) = m B √ r D ∗ (cid:104) ( w + 1) h A ( w ) ∓ √ w − h V ( w ) (cid:105) ,H ( w ) = m B √ r D ∗ √ w − (cid:112) − r D ∗ w + r D ∗ (cid:104) ( r D ∗ − w ) h A ( w ) + ( w − r D ∗ h A ( w ) + h A ( w )) (cid:105) ,H T ± ( w ) = m B √ r D ∗ − r D ∗ ( w ∓ √ w − (cid:112) − r D ∗ w + r D ∗ × (cid:104) h T ( w ) + h T ( w ) + ( w ± √ w − h T ( w ) − h T ( w )) (cid:105) ,H T ( w ) = − m B √ r D ∗ (cid:104) ( w + 1) h T ( w ) + ( w − h T ( w ) + 2( w − h T ( w ) (cid:105) . (27)The angular dependence of Eq. (25) can be derived as explained in Appendix B. The normal-ization factor is given as F (1) = h A (1) and then our setup leads to F (1) (cid:39) . . (cid:96) (0)2 . (28) Note that the definition of θ (cid:96) here is not the same as θ τ in Ref. [26], but related as θ (cid:96) = π − θ τ .
7n the CLN parameterization, w dependence on F ( w ) is approximated by using the follow-ing functions: h A ( w ) with the slope ρ D ∗ similar to G ( w ), R ( w ) = h V ( w ) /h A ( w ), and R ( w ) = (cid:0) h A ( w ) + r D ∗ h A ( w ) (cid:1) /h A ( w ), where the latter two are expanded in ( w − (cid:15) a,b,c and z expansions. We will get back to this point later. There are theoretical studies to evaluate the form factors at specific points of w with respectto the following quantities: f B → D + ( w ) = 12 √ r D (cid:104) (1 + r D ) h + ( w ) − (1 − r D ) h − ( w ) (cid:105) , (29) f B → D ( w ) = √ r D (cid:20) w + 11 + r D h + ( w ) + w − − r D h − ( w ) (cid:21) , (30) f B → DT ( w ) = 1 + r D √ r D h T ( w ) , (31) A B → D ∗ ( w ) = √ r D ∗ (1 + w )1 + r D ∗ h A ( w ) , (32) A B → D ∗ ( w ) = 12 √ r D ∗ (cid:104) ( w + 1) h A ( w ) + ( w r D ∗ − h A ( w ) + ( r D ∗ − w ) h A ( w ) (cid:105) , (33) V B → D ∗ ( w ) = 1 + r D ∗ √ r D ∗ h V ( w ) , (34) T B → D ∗ ( w ) = 12 √ r D ∗ (cid:104) (1 + r D ∗ ) h T ( w ) − (1 − r D ∗ ) h T ( w ) (cid:105) , (35) T B → D ∗ ( w ) = √ r D ∗ (cid:20) w + 11 + r D ∗ h T ( w ) − w − − r D ∗ h T ( w ) (cid:21) , (36) T B → D ∗ ( w ) = 1 + r D ∗ √ r D ∗ (cid:104) ( w + 1) h T ( w ) + ( w − h T ( w ) − ( w − h T ( w ) (cid:105) . (37)Then, the lattice studies [29, 30, 31] provide the following evaluations f B → D + ( { , . , . } ) = { . , . , . } , (38) f B → D ( { , . , . } ) = { . , . , . } , (39) h A (1) = 0 . ± . . (40)In Ref. [32], the form factors at q = { , − , − , − } GeV have been evaluated by a light-conesum rule (LCSR) approach. The result can be summarized as f B → D + ( { . , . , . , . } ) = { . , . , . , . } , (41) f B → D ( { . , . , . } ) = { . , . , . } , (42) f B → DT ( { . , . , . , . } ) = { . , . , . , . } , (43) A B → D ∗ ( { . , . , . , . } ) = { . , . , . , . } , (44) A B → D ∗ ( { . , . , . } ) = { . , . , . } , (45) V B → D ∗ ( { . , . , . , . } ) = { . , . , . , . } , (46)8 B → D ∗ ( { . , . , . , . } ) = { . , . , . , . } , (47) T B → D ∗ ( { . , . , . } ) = { . , . , . } , (48) T B → D ∗ ( { . , . , . , . } ) = { . , . , . , . } , (49)where w = { . , · · · } and w = { . , · · · } correspond to q = { , · · · } for B → D and B → D ∗ , respectively. Thanks to this comprehensive work, for instance, a fit analysis to“theory constraints only” is even possible.In addition, QCD sum rule (QCDSR) can evaluate the sub-leading IW functions as inRefs. [33, 34, 35]. By using formulae in the literature with updated QCD input data, we derivethe following constraints − . < ˆ χ (0)2 < − . , − . < ˆ χ (1)2 < +0 . , − . < ˆ χ (2)2 < +0 . , (50)+0 . < ˆ χ (1)3 < +0 . , − . < ˆ χ (2)3 < +0 . , (51)+0 . < η (0) < +0 . , +0 . < η (1) < +0 . , − . < η (2) < − . . (52)We show a detail of these constraints in Appendix C.In addition, we need to take care of Unitarity Bound (UB) for the case of the HQETparameterization. Following Eqs.(5) – (20) of Ref. [3], we obtain the functions U J P in terms ofthe present HQET parameters to be constrained by U + < χ + (0) ≈ (5 . ± . × − , (53) U − < ˜ χ − (0) ≈ (1 . ± . × − , (54) U + < m B ∗ m D ∗ χ + (0) ≈ (3 . ± . × − , (55) U − < m B ∗ m D ∗ ˜ χ − (0) ≈ (3 . ± . × − , (56)where the explicit forms of U J P are a bit lengthy and thus we put a Mathematica file in thesource of the ArXiv version. The above numerical bounds are obtained by using recent data of(excited) B c states [36, 37] and quark masses [38], instead of the original one [3].For now, we leave discussion on how we take the uncertainties of these theoretical constraintsin our fit analysis. It will be explained later. The kinetic distributions of ¯ B → D ( ∗ ) (cid:96) ¯ ν have been measured by the Belle collaboration inRefs. [17, 18, 19]. Available experimental data are then w distributions of ¯ B → D(cid:96) ¯ ν [17](denoted as Belle15), and full kinetic ( w, θ (cid:96) , θ V , χ ) distributions of ¯ B → D ∗ (cid:96) ¯ ν with the successivedecay D ∗ → Dπ [18, 19]. The latter includes two independent measurements; one with hadronictagging [18] (Belle17) and with untagged approach [19] for each e and µ mode (Belle18- e andBelle18- µ ).The Belle15 data correspond to the binned decay rate with respect to w , where the fourprocesses, ¯ B → D + e − ¯ ν , ¯ B → D + µ − ¯ ν , ¯ B − → D e − ¯ ν , and ¯ B − → D µ − ¯ ν , are combined. TheBelle17 data are given in terms of the unfolded decay rate of ¯ B → D ∗ + (cid:96) − ¯ ν for a correspondingbin ∆Γ∆ x . This is derived from Eq. (24) as∆Γ∆ x = 1 B ( D ∗ + → D π + ) (cid:90) ∆ x d Γ full D ∗ dx , (57)for x = ( w, cos θ (cid:96) , cos θ V , χ ). On the other hand, the Belle18 data are shown in terms of binnedsignal event ∆ N ∆ x (cid:12)(cid:12) i (for i -th bin) in which the folded effect is presented by Response Matrix R ame Object DescriptionBelle15 [17] ¯ B → D(cid:96) ¯ ν w distribution (10)Belle17 [18] ¯ B → D ∗ + (cid:96) − ¯ ν ( w, θ (cid:96) , θ V , χ ) distributions (40)Belle18- e [19] ¯ B → D ∗ + e − ¯ ν ( w, θ (cid:96) , θ V , χ ) distributions (40)Belle18- µ [19] ¯ B → D ∗ + µ − ¯ ν ( w, θ (cid:96) , θ V , χ ) distributions (40)BR [38] ¯ B → D ( ∗ ) (cid:96) ¯ ν branching ratios (2)Lattice [29, 30, 31] FFs ( f B → D + , , h A ) Eqs. (38)–(40) constraints (7 ∗ )LCSR [32] FFs ( f B → D + , ,T , A B → D ∗ , , V B → D ∗ , T B → D ∗ , , ) Eqs. (41)–(49) constraints (33)QCDSR [33, 34, 35] FFs ( ˆ χ ( n )2 , , η ( n ) ) Eqs. (50)–(52) constraints (8)UB [3] U J P Eqs. (53)–(56) constraints (4)
Table 1: Summary of the experimental data and the theory constraints used in our fit analysis.Numbers of independent data points are also exhibited in brackets. ( ∗ ) The relation f + ( q =0) = f ( q = 0) implies that the lattice result has only 6 independent observables.together with efficiency ε among the bins. This is obtained as∆ N ∆ x (cid:12)(cid:12)(cid:12)(cid:12) i = N B τ B B ( ¯ D → K − π + ) R ij ε j (cid:90) ∆ x j d Γ full D ∗ dx , (58)where N B , R ij , and ε j are provided in Ref. [19] for each e and µ modes.Furthermore, we also take the world averages of the branching ratios (BR) of ¯ B → D ( ∗ ) (cid:96) ¯ ν [38]in our fit. A short summary for the experimental data and the theory constraints is shown inTable 1. Correlations among the bins for each measurement are also taken into account in ourfit analysis, (see corresponding references.) In this work, a Bayesian fit analysis is applied to obtain allowed ranges of the HQET param-eters and | V cb | with the use of Markov-Chain-Monte-Carlo (MCMC) method by Stan [21], astate-of-the-art platform for statistical modeling and high-performance statistical computation ,implemented in
MathematicaStan [39]. The analysis is performed by MCMC runs involving10 chains with Hamiltonian Monte Carlo algorithm giving 10 sampling points for every fit.Although Stan is widely known in the statistical science community, it has not often beenused in particle physics analysis. This enables us to give independent check of fit resultsobtained from public/private codes developed by particle physicists.Our fit procedure is briefly exhibited as follows. We basically take into account the fullexperimental data points of ¯ B → D ( ∗ ) (cid:96) ¯ ν and the applicable theoretical constraints on thespecific FFs, as summarized in Table 1. Namely, 184 data points are used to fit the freeparameters. Regarding the theory constraints, we need to declare ways of treating uncertainties.First, we simply take them as normal distributions in order to obtain mean values and variancesfrom the sampling points of the fitted parameters. As for the UBs, it is assumed such as, e.g. ,( U + − /χ + (0) . This means that 1 σ deviation is the threshold for UB which should have tobe satisfied in a final result. We will check this point later. After then, we will also discuss theQCDSR bounds on ˆ χ ( n )2 , and η ( n ) since they include special input of T and ω (see Appendix C10or more detail) that have no fair description of “central value”. For comparison and later discussion, we also consider the following case where limited datapoints are taken into account for a fit analysis: w +theory – only the w distributions alongwith the theory constraints and the branching ratios.As for the phenomenological mode, we investigate SM, SM + V , and SM + T as described inEq. (1) with the HQET parameterization for the FFs. Then we evaluate Information Criterion that offers the predictive accuracy of the model. To be precise, we employ cAIC defined as [22]cAIC = − L + 2 k ( k + 1) n − k − , (59)where L is the maximum likelihood and n ( k ) denotes the number of data points (the modelparameters to be fitted). The second term gives a penalty for overestimate of increasing numberof model parameters. In our case, k = 23 + 1(+1) in the SM(+NP) for NNLO (3 / / k = 13 + 1(+1) for (2 / / n = 184. A preferred model has a smaller cAIC. First, we show our fit results of the HQET parameters and | V cb | for the SM(+NP) scenarios atthe NNLO heavy quark expansion in Table 2. We also evaluate how the present phenomeno-logical models improve fit to data points by looking at difference in Information Criterion froma reference model. We define ∆IC model = cAIC − cAIC model , where cAIC = 987 . / / -). We remark that a larger value of ∆IC implies abetter improvement from the reference model. As seen from the result, all the present modelsimprove the fit compared with the reference model in which ˆ (cid:96) ( n ) i = 0 is taken. This illustratessignificance of non-zero values (beyond variances) for the NNLO parameters.On the other hand, one finds that SM (2 / /
0) is more preferred than SM (3 / / / /
1) are surplus to the present available experimental/theory 184 data points, and then 13in (2 / /
0) are sufficient to explain the available data points at present. However, we believethat this is not conclusive since it could vary as additional measurements become available inthe future, e.g. , by the Belle II experiment. Therefore, we still continue to examine the bothcases of (2 / /
0) and (3 / /
1) in the following part of this work. For more details of our fitresults, such as correlation matrix, see Appendix D.As a consistency check with integrated observables, we generate the branching ratios andthe D ∗ polarization ( e mode) that result in B ( ¯ B → D + (cid:96) − ¯ ν ) SM = (cid:104) (2 . ± . . ± . (cid:105) , (60) B ( ¯ B → D ∗ + (cid:96) − ¯ ν ) SM = (cid:104) (5 . ± . . ± . (cid:105) , (61) F D ∗ L ( ¯ B → D ∗ + e − ¯ ν ) SM = (cid:104) . ± .
002 ; 0 . ± . (cid:105) , (62)in the SM for the cases of (cid:2) (2 / / / / (cid:3) , respectively. This is compared with the experi-mental measurements of B ( ¯ B → D + (cid:96) − ¯ ν ) exp = (2 . ± . ± . B ( ¯ B → D ∗ + (cid:96) − ¯ ν ) exp =(5 . ± . ± . F D ∗ L ( ¯ B → D ∗ + e − ¯ ν ) exp = 0 . ± . F D ∗ L ( ¯ B → D ∗ + e − ¯ ν ) exp is still preliminary (and thus we did not takeit in our fit.) We can see that they are in good agreements within uncertainties, but the bestfit point for the D mode is a bit smaller than data. This issue might be similar for the LCSR bounds, but it is beyond the scope of the present work. M (2 / /
0) SM (3 / /
1) SM + V (2 / /
0) SM + V (3 / /
1) SM + T (2 / / | V cb | × . ± . . ± . . ± . . ± . . ± . C NP - - 0 . ± .
01 0 . ± . | . ± . | ξ (1) − . ± . − . ± . − . ± . − . ± . − . ± . ξ (2) +1 . ± .
10 +1 . ± .
26 +1 . ± .
10 +1 . ± .
25 +1 . ± . ξ (3) - − . ± .
75 - − . ± .
73 -ˆ χ (0)2 − . ± . − . ± . − . ± . − . ± . − . ± . χ (1)2 +0 . ± .
02 +0 . ± .
02 +0 . ± .
02 +0 . ± .
02 +0 . ± . χ (2)2 - − . ± .
02 - − . ± .
02 -ˆ χ (1)3 − . ± . − . ± . − . ± . − . ± . − . ± . χ (2)3 - − . ± .
03 - +0 . ± .
03 - η (0) +0 . ± .
06 +0 . ± .
11 +0 . ± .
06 +0 . ± .
11 +0 . ± . η (1) +0 . ± .
03 +0 . ± .
03 +0 . ± .
03 +0 . ± .
03 +0 . ± . η (2) - − . ± .
05 - − . ± .
05 -ˆ (cid:96) (0)1 +0 . ± .
16 +0 . ± .
18 +0 . ± .
16 +0 . ± .
18 +0 . ± . (cid:96) (1)1 - +1 . ± .
09 - − . ± .
92 -ˆ (cid:96) (0)2 − . ± . − . ± . − . ± . − . ± . − . ± . (cid:96) (1)2 - − . ± .
56 - − . ± .
55 -ˆ (cid:96) (0)3 − . ± . − . ± . − . ± . − . ± . − . ± . (cid:96) (1)3 - +3 . ± .
35 - +4 . ± .
31 -ˆ (cid:96) (0)4 +0 . ± . − . ± .
94 +0 . ± . − . ± .
94 +0 . ± . (cid:96) (1)4 - +1 . ± .
93 - +1 . ± .
91 -ˆ (cid:96) (0)5 +1 . ± .
43 +3 . ± .
17 +2 . ± .
59 +6 . ± .
32 +1 . ± . (cid:96) (1)5 - +2 . ± .
47 - +2 . ± .
51 -ˆ (cid:96) (0)6 +0 . ± .
15 +4 . ± .
76 +0 . ± .
23 +7 . ± .
87 +0 . ± . (cid:96) (1)6 - +5 . ± .
97 - +5 . ± .
04 -∆IC 162 . . . . . Table 2: Fit results of the simultaneous determinations for the HQET parameters and | V cb | in several phenomenological models at NNLO with/without NP. Larger value of ∆IC indicatesbetter improvement of the fit from the reference model of NLO(3 / / -). | V cb | determination Our fit results for | V cb | in the SM (2 / /
0) and (3 / /
1) scenarios are both close to the PDGcombined average, (39 . ± . × − , from the exclusive mode [38]. In Table 3, we put summaryfor the recent | V cb | determinations along with the normalization factors G (1) and F (1).Here we would like to discuss difference in the | V cb | determination between our results andone from Ref. [16]. In their work, | V cb | has been extracted by using the fit result of the HQETparameters, and after then, by taking the integrated branching ratios of ¯ B → D ( ∗ ) (cid:96) ¯ ν . Althoughthe former fit analysis includes the experimental w distributions, it is utilized only to fit theHQET parameters. Indeed, we find that their result can be reproduced when we perform thefit analysis with the data set of w +theory as shown in Table 3. Therefore, we emphasize thatthe angular distributions are also significant for the | V cb | determination.We also provide a fit result for G ( w ) and F ( w ) comparing them with those in the CLNparameterization. The traditional form of G ( w ) is expanded by z , with the coefficients by12 ll (2 / / all (3 / /
1) PDG/HFLAV [38, 40] w +theory (3 / /
1) Ref. [16] (3 / / | V cb | × . ± . . ± . . ± . . ± . . ± . G (1) 1 . ± .
006 1 . ± .
006 1 . ± .
009 1 . ± .
006 - F (1) 0 . ± .
009 0 . ± .
011 0 . ± .
012 0 . ± .
011 -
Table 3: Comparison of the | V cb | determinations along with the normalization factors F (1)and G (1). In our work, these factors are simultaneously produced by the fit analysis.means of the slope parameter ρ , and with the assumption estimated by UB as in Ref. [3]. Inour study, we can directly produce the coefficients in z expansion, defined as G ( w ) ≡ G (1) (cid:88) n =0 g n z n , (63)with g = 1. Our result is then g = − . ± . , g = 24 . ± . , g = − . ± . , (64)and G (1) = 1 . ± .
006 for SM (3 / / g = − ρ , g = 51 ρ − , g = − ρ + 84 , (65)for ρ = 1 . ± .
033 [40]. One can see that our result has ∼ G ( w ) is calculated for the evaluationof the decay rate.The CLN form for F ( w ) is constructed with h A ( w ), R ( w ), and R ( w ). As already ex-plained, its CLN approximation is not appropriate for analyses with recent precise data. In-stead, we provide the z expanded F ( w ) squared such as F ( w ) ≡ F (1) (cid:88) n =0 f n z n , (66)with F (1) = 0 . ± . , f = − . ± . , f = 55 . ± . , f = − ± . (67) We have seen the fit results including the NP contributions in Table 2. In the SM + T scenarios, our fit result indicates that the T contribution is constrained as | C T | < .
025 at95% confidence level, which means zero-consistent, for the case of (3 / / | C T | = 0 . ± .
01 is obtained for (2 / /
0) as seen from the table, which implies that the bestfit point favors non-zero T contribution although the uncertainty is still large. For both cases,the HQET parameters and | V cb | are then all consistent with those in the SM scenarios. Thiscould be very interesting since the HQET parameterization model affects the fit result of theNP effect, and also the fit analysis has the NP sensitivity at the level of O (%).13 - C V | V c b | × B ! X c ` ⌫
0) scenarios,respectively. The yellow band indicates the allowed region from the inclusive mode. The contourlines correspond to ∆ χ = 1 ,
4. The red bars are the SM results for | V cb | . [Bottom] contour plotfor predictions on R D and R D ∗ in the SM(+NP) scenarios, where the regions for SM(3 / / V (3 / / V (2 / /
0) are shown in red, blue, and green, respectively. Thecombined experimental result (gray solid curves that correspond to ∆ χ = 1 , ,
9) and the SMprediction in the literature (red bar) are taken from Ref. [40].The SM + V scenarios also have the non-zero preferred value with the large uncertainty, C V = 0 . ± .
01 for (3 / /
1) and C V = 0 . ± .
01 for (2 / / | V cb | than those in the SM, which would be interesting as it is different from the case forSM + T . Indeed, these changes improve the fit to the branching ratios. We obtain B ( ¯ B → D + (cid:96) − ¯ ν ) SM+ V = (cid:104) (2 . ± . . ± . (cid:105) , (68) B ( ¯ B → D ∗ + (cid:96) − ¯ ν ) SM+ V = (cid:104) (5 . ± . . ± . (cid:105) , (69)for SM + V (cid:2) (2 / / / / (cid:3) from the fit result. Thus we can see that the branching ratiosare in perfect agreements with the experimental measurements.In Fig. 1 (top), we show preferred regions of | V cb | and C X in the SM + NP scenarios, wherethe regions in blue, green, and gray are favored in the SM + V (3 / / V (2 / / T (2 / /
0) scenarios, respectively. We also include the allowed region from the inclusiveprocess for SM + NP as depicted in the yellow region. It can be derived with the use ofRefs. [15, 42, 43], in which discrepancy of the | V cb | determination among the exclusive and14 D R D ∗ P Dτ P D ∗ τ F D ∗ L SM (2 / /
0) 0 . ± .
004 0 . ± .
001 0 . ± . − . ± .
007 0 . ± . / /
1) 0 . ± .
006 0 . ± .
004 0 . ± . − . ± .
020 0 . ± . . ± .
003 0 . ± .
005 - - -SM (Ref. [16]) 0 . ± .
003 0 . ± .
003 0 . ± . − . ± .
015 0 . ± . V (2 / /
0) 0 . ± .
006 0 . ± .
005 0 . ± . − . ± .
007 0 . ± . V (3 / /
1) 0 . ± .
006 0 . ± .
007 0 . ± . − . ± .
020 0 . ± . Table 4: Predictions of the ¯ B → D ( ∗ ) τ ¯ ν observables.inclusive processes has been investigated. Then, it is found that our fit result loosens thedeviation in the SM + V and SM + T scenarios, but it is still not in a sufficient agreement.A corresponding LHC bound on C X is naively obtained by the following discussion. Thesedays LHC constraints on NP effects have been getting severer. In Ref. [44], the authors haveshown that τ searches with high p T at 36fb − [45, 46] give an upper limit on the WCs for the b → cτ ν current. (See, also Refs.[47, 48, 49, 50, 51] in the context of NP interpretations of the R D ( ∗ ) anomaly.) Similarly, e and µ searches with high p T at 139fb − [52] give an upper limiton C X defined as in Eq. (1). Comparing those experimental constraints in looking at a tail ofthe m T plane ( ∼ . | C V | (cid:46) . | C T | (cid:46) .
05. Therefore, our fit result of C X ∼ O (0 .
01) is in the region of interest also for theLHC search. A further study of the LHC bound in higher p T ranges is work in progress. ¯ B → D ( ∗ ) τ ¯ ν With the CLN parameterization, SM predictions and/or NP investigations have been providedwith respect to ¯ B → D ( ∗ ) τ ¯ ν in the literature, ( e.g. , see Refs. [53, 54] for recent works), sincethe experimental results have shown significant deviations from the SM predictions in themeasurements of the ratios: R D = 0 . ± . ± .
013 and R D ∗ = 0 . ± . ± . τ and D ∗ polarizations in ¯ B → D ∗ τ ¯ ν have also been reported as P D ∗ τ = − . ± . +0 . − . [55] and F D ∗ L = 0 . ± . ± .
04 [41].Here, we also investigate these ¯ B → D ( ∗ ) τ ¯ ν observables with the use of our fit results inthe SM(+NP) scenarios of the HQET parameterization. Note that, in this work, we considerNP contributions only to the ( e, µ ) modes. Namely, the denominators of the ratios R D ( ∗ ) areonly affected by the NP contribution. In this sense, our NP investigation has a different viewfrom numerous previous studies for the R D ( ∗ ) anomaly, e.g. , see Refs. [56, 57, 58] for the caseof the HQET parameterization. Also note that we take the proper (cid:15) a,b,c expansion for theseobservables.In Table 4, we list our predictions on the ¯ B → D ( ∗ ) τ ¯ ν observables in the present models,along with those from Refs. [40, 16]. Our analysis shows that the SM(2 / /
0) predicts smallervalues for both R D and R D ∗ than those of the HFLAV report. On the other hand, the SM(3 / / R D while smaller for R D ∗ . This is a similar behavior with thatobtained in Ref. [16]. Then, it is also found that the polarizations for (3 / /
1) are consistentwith the reference. We obtain the same results for the cases of SM + T .The SM + V models give R D ( ∗ ) different from the SM predictions, which could be NPsignals at the Belle II experiment with large statistics. To be precise, both cases point to the R D and R D ∗ values smaller and larger than the SM predictions, respectively. In particular, it is15 ��� (cid:1) �� SM ( / /-) SM ( / / ) SM + V ( / / ) SM ( / / ) SM + V ( / / ) Breakdown of χ ( min ) from 184 data points Br ( ) QCDSR ( ) Lattice ( ) LCSR ( ) UB ( ) Belle15 ( ) Belle17 ( ) Belle18 ( ) Figure 2: The breakdown of the χ deviations at our best fit results from the 184 data points.interesting that the R D ∗ deviation from the measurement becomes smaller for SM + V (3 / / R D ( ∗ ) is analyzed by means of both τ and ( e, µ ) distributions subtracted frombackground. The present measurement is then done with the assumption that the ( e, µ ) modesobey the SM. Thus, in the presence of NP in the ( e, µ ) modes, re-analysis is needed by takingthe NP effect. Although such a NP effect of O (%) is negligible for the present analysis, it couldbecome significant as larger number of events are accumulated at the Belle II experiment.Finally, we show a summary plot for the predictions on R D and R D ∗ in the SM(+NP)scenarios in Fig. 1 (bottom), where the allowed regions for SM(3 / / V (3 / / V (2 / /
0) are shown in red, blue, and green, respectively. The combined experimentalresult (gray curves) and the referred SM prediction (red bar) are taken from Ref. [40].
For the present analyses so far, we have treated the theory constraints as being normallydistributed for simplicity and in order to obtain the applicable outputs. Thanks to it, we candisplay the breakdown of the χ deviations for our fit results as in Fig. 2.The UBs are taken as the Gaussian distribution assuming the central value as zero whilethe standard deviation as the calculated upper limit given in Eqs. (53)–(56). We have checkedthe breakdown of the χ deviation from the UBs for all the present models considered in ouranalysis and then have confirmed that those for the UBs are all within 1 σ .As for the bounds from QCDSR, we have derived the constraints of Eqs. (50)–(52) and againtaken as normal distributions. Our fit results, however, show that some of the NLO parametersare deviated from these constraints as seen in Table 2. In particular, our MCMC run findsthe best fit point of ˆ χ (1)3 ∼ O ( − .
01) that has a large deviation from the QCDSR constraint ∼ O (+0 . χ breakdown for the QCDSR constraints is χ ∼ / /
1) (SM (2 / / χ ( n )2 , and η ( n ) , are restricted as in Eqs. (50)–(52). Then we find that theoutputs of | V cb | and the LO parameters ξ ( n ) are not much affected while those of the NNLOparameters ˆ (cid:96) ( n ) i are shifted, compared with the results obtained in Table 4. In this case, however,the NLO parameter fits have bad convergences and their distributions are far away from thenormal distributions. Also, we have checked that the observables such as the branching ratiosare all consistent. In this sense, we can say that our main conclusion is not affected by thisissue. 16 Summary
We have investigated the semi-leptonic decays of ¯ B → D ( ∗ ) (cid:96) ¯ ν in terms of the HQET param-eterization for the form factors, with the heavy quark expansion up to O (1 /m c ), and beyondthe simple approximation considered in the original CLN parameterization. It is given with the z = ( √ w + 1 − √ / ( √ w + 1 + √
2) expanded form, and then the highest order for the expan-sion is in principle arbitrary. In our work, we have followed the models from Ref. [16] denotedas (2 / /
0) and (3 / /
1) for the z expansions in the (leading/sub-leading/subsub-leading) IWfunctions.The analysis with this setup was first given in Ref. [16], and then we have extended it tothe comprehensive analyses including (i) simultaneous fit of | V cb | and the HQET parametersto the available experimental full distribution data and the theory constraints, and (ii) NPcontributions of the V and T types, such as ( cγ µ P R b )( (cid:96)γ µ P L ν (cid:96) ) and ( cσ µν P L b )( (cid:96)σ µν P L ν (cid:96) ), tothe decay distributions and rates. For this purpose, we have performed the Bayesian fit analysesby using Stan program, a state-of-the-art public platform for statistical computation, in whichMCMC runs with various algorithms are possible.Then it has been shown that our | V cb | fit results for the SM (2 / /
0) and (3 / /
1) scenar-ios are both close to the PDG combined average from the exclusive mode [38] as summarizedin Table 3. We have also found that the fit to the w distribution data with the theory con-straints ( w +theory ) reproduce the larger | V cb | value completely consistent with that reportedin Ref. [16]. This could imply significance of the angular distribution data for ¯ B → D ∗ (cid:96) ¯ ν .Besides, we have evaluated Information Criterion to see how the inclusion of the O (1 /m c )parameters improve the fit. Then we see that the 23 HQET parameters of (3 / /
1) are surpluswhile 13 of (2 / /
0) are sufficient for the statistical modeling to explain the present availabledata points.The SM+NP scenarios have been studied with the same manner. At first, we have confirmedthat SM + T (3 / /
1) is constrained as | C T | < .
025 at 95% confidence level and the best fitpoint is zero-consistent. On the other hand, it has turned out that SM + T (2 / /
0) is allowed tohave non-zero contribution, | C T | = 0 . ± .
01, to the processes. This could be very interestingsince the HQET parameterization model affects the fit result of the NP effect. Furthermore, asignificant point is that the fit analysis has the NP sensitivity at the level of O (%).Then, we have also obtained non-zero preferred values in the SM + V scenarios as C V =0 . ± .
01 for (3 / /
1) and C V = 0 . ± .
01 for (2 / / Information Criterion also suggeststhat SM + V is favored at the same level with the SM scenarios. In addition, both cases givelarger | V cb | than those in the SM, but they are still not in a sufficient agreement with the | V cb | determination from the inclusive process. This is summarized in Fig. 1 (top). The applicableLHC bound is naively given as | C V | (cid:46) . | C T | (cid:46) .
05 estimated from the m T plane at ∼ . B → D ( ∗ ) τ ¯ ν observables in the presentmodels. It is summarized in Table 4 and Fig. 1 (bottom). Our prediction in SM(2 / /
0) hassmaller values for both R D and R D ∗ comapred with those in the HFLAV report. On the otherhand, SM(3 / /
1) predicts the consistent value for R D while smaller for R D ∗ . In the SM + V scenarios, NP only contributes to the light-lepton modes and then it results in the R D and R D ∗ values smaller and larger than the SM predictions, respectively. It is also seen that the R D ∗ deviation from the experimental measurement becomes milder ( ∼ . σ ) than the one in theSM. This is a key feature for this model derived from our fit analysis. Quantitatively, we have4 . σ and 3 . σ significances for the R D ( ∗ ) deviation in SM (2 / /
0) and (3 / / . σ in SM (HFLAV) assuming no correlation. As for SM + V , we see3 . σ (2 / /
0) and 2 . σ (3 / / HEPfit package [59]. We leave it for our future work.We conclude from this work that the available full distribution data of ¯ B → D ( ∗ ) (cid:96) ¯ ν haspotential to fit a large number of the parameters in the HQET parameterization together with | V cb | , and a further improvement is expected at the Belle II experiment. The fit analysis also hasthe NP sensitivity with the O (%) level of the SM contribution, and then it could be examinedwith the ¯ B → D ( ∗ ) τ ¯ ν observables in future. Interesting directions of future work are, forexample, CP violation [60] and QED corrections [61] in the ¯ B → D ( ∗ ) (cid:96) ¯ ν distributions withrespect to the HQET parameterization. Acknowledgements
We are grateful to Martin Jung for useful comments on the HQET parameterization. Wethank Minoru Tanaka for discussion about QCDSR. RW thanks Vincent Picaud for helpfulsuggestions on the usage of
MathematicaStan . RW also thanks Marco Ciuchini for discussionon MCMC. SI is grateful to Kazuhiro Tobe for discussion about various aspects of this work. SIalso thanks Kodai Matsuoka and Tsuzuki Noritsugu for useful comments on the experimentalmeasurements of ¯ B → D ( ∗ ) (cid:96) ¯ ν and ¯ B → D ( ∗ ) τ ¯ ν . We also thank Nagoya University TheoreticalElementary Particle Physics Laboratory for providing computational resources. The work of SIis supported by the Japan Society for the Promotion of Science (JSPS) Research Fellowshipsfor Young Scientists, No. 19J10980 and Core to Core Program, No. JPJSCCA20200002. AppendixA α s and /m Q Corrections
Here we list functions for the α s and 1 /m b,c corrections. We have followed the analytic resultfrom Ref. [7]. The α s corrections, δ ˆ h X,α s , are given as δ ˆ h + ,α s = 16 z cb ( w − w cb ) (cid:2) z cb ( w − w cb ) Ω w ( w ) + ( w + 1) (cid:0) − w + w + 2 w cb )( z cb + 1) z cb + 2( w − w (2 + 3 w cb ) + w cb ) z cb − z cb (cid:1) r w ( w )+ ( w cb − w ) (cid:0) w + w (10 + z cb ) z cb + ( − − w cb + z cb ) z cb (cid:1) z cb − (1 + w cb − w − w )(1 − z cb ) z cb log z cb (cid:3) + V ( µ ) , (70) δ ˆ h − ,α s = 1 + w z cb ( w − w cb ) (cid:2) − (cid:0) − w − w ) z cb + z cb (cid:1) (1 − z cb ) r w ( w ) − ( w − w cb )(1 − z cb ) z cb − (cid:0) z cb + z cb ) − w (1 + 4 z cb + z cb ) (cid:1) z cb log z cb (cid:3) , (71) δ ˆ h V,α s = 16 z cb ( w − w cb ) (cid:2) z cb ( w − w cb )Ω w ( w ) + 2( w + 1)((3 w − z cb − z cb − r w ( w ) − z cb ( w − w cb ) − ( z cb −
1) log z cb (cid:3) + V ( µ ) , (72) δ ˆ h A ,α s = 16 z cb ( w − w cb ) (cid:2) z cb ( w − w cb )Ω w ( w ) + 2( w − w + 1) z cb − z cb − r w ( w )18 z cb ( w − w cb ) − ( z cb −
1) log z cb (cid:3) + V ( µ ) , (73) δ ˆ h A ,α s = 16 z cb ( w − w cb ) (cid:2)(cid:0) w − w − z cb + 2 w (2 w − z cb + (1 − w ) z cb (cid:1) r w ( w ) − z cb ( z cb + 1)( w − w cb ) + ( z cb − (4 w + 2) z cb + 3 + 2 w ) z cb log z cb (cid:3) , (74) δ ˆ h A ,α s = 16 z cb ( w − w cb ) (cid:2) w − w cb ) z cb Ω w ( w ) + (cid:0) w − w + 6 w z cb + z cb ( − z cb ) − wz cb (4 + 3 z cb ) − − w )( − z cb + 3 wz cb − z cb ) w cb (cid:1) r w ( w ) − wz cb (1 + 6 w + z cb ) + ( −
10 + 24 w + 2 z cb ) z cb w cb + ( − w + (2 + 4 w ) z cb − (2 + 3 w ) z cb ) log z cb (cid:3) + V ( µ ) , (75) δ ˆ h S,α s = 13 z cb ( w − w cb ) (cid:2) z cb ( w − w cb )Ω w ( w ) − ( w − z cb + 1) r w ( w )+( z cb −
1) log z cb (cid:3) + S ( µ ) , (76) δ ˆ h P,α s = 13 z cb ( w − w cb ) (cid:2) z cb ( w − w cb )Ω w ( w ) − ( w + 1)( z cb − r w ( w )+( z cb −
1) log z cb (cid:3) + S ( µ ) , (77) δ ˆ h T,α s = 13 z cb ( w − w cb ) (cid:2) z cb ( w − w cb )Ω w ( w ) + (cid:0) z cb w − (1 − z cb ) w − (1 + z cb ) (cid:1) r w ( w ) − z cb ( w − w cb ) + (1 − z cb ) log z cb (cid:3) + T ( µ ) , (78) δ ˆ h T ,α s = 13 z cb ( w − w cb ) (cid:2) z cb ( w − w cb )Ω w ( w ) + 2 z cb ( w − r w ( w ) − z cb ( w − w cb ) + (1 − z cb ) log z cb (cid:3) + T ( µ ) , (79) δ ˆ h T ,α s = w + 13 z cb ( w − w cb ) (cid:2) (1 − z cb ) r w ( w ) + 2 z cb log z cb (cid:3) , (80) δ ˆ h T ,α s = 13 z cb ( w − w cb ) [( z cb w − r w ( w ) − z cb log z cb ] , (81)with z cb = m c m b , w cb = 12 (cid:0) z cb + z − cb (cid:1) , w ± ( w ) = w ± √ w − , (82) r w ( w ) = log w + ( w ) √ w − , (83)Ω w ( w ) = w √ w − (cid:104) (1 − w − ( w ) z cb ) − (1 − w + ( w ) z cb )+ Li (1 − w ( w )) − Li (1 − w − ( w )) (cid:105) − wr w ( w ) log z cb + 1 , (84)19here Li ( x ) = (cid:82) x dt log(1 − t ) /t is dilogarithmical function. The above results are obtainedat the scale µ √ bc = √ m b m c , namely V ( µ √ bc ) = S ( µ √ bc ) = T ( µ √ bc ) = 0. Otherwise, the scalefactors are given as V ( µ ) = − (cid:0) wr w ( w ) − (cid:1) log m b m c µ , (85) S ( µ ) = − (cid:0) wr w ( w ) + 1 (cid:1) log m b m c µ , (86) T ( µ ) = − (cid:0) wr w ( w ) − (cid:1) log m b m c µ . (87)Note that we set the scale as µ b = 4 . /m b,c corrections involve four sub-leading IW functions, χ ( w ) and ξ ( w ), one ofwhich (usually χ ) can be absorbed into the definition of ξ ( w ). For the form of δ ˆ h X,m b,c , thesub-leading IW functions divided by ξ ( w ) are defined as in Eq. (11). Following Ref. [7], we canwrite δ ˆ h X,m b,c as δ ˆ h + ,m b = δ ˆ h + ,m c = δ ˆ h T ,m b = − w −
1) ˆ χ ( w ) + 12 ˆ χ ( w ) , (88) δ ˆ h − ,m b = − δ ˆ h − ,m c = δ ˆ h T ,m b = 1 − η ( w ) , (89) δ ˆ h V,m b = δ ˆ h A ,m b = δ ˆ h P,m b = δ ˆ h T,m b = δ ˆ h T,m c = 1 − η ( w ) − w −
1) ˆ χ ( w ) + 12 ˆ χ ( w ) , (90) δ ˆ h V,m c =1 − χ ( w ) , (91) δ ˆ h A ,m b = δ ˆ h S,m b = δ ˆ h S,m c = ( w − (cid:104) ( w + 1) − (cid:0) − η ( w ) (cid:1) − χ ( w ) (cid:105) + 12 ˆ χ ( w ) , (92) δ ˆ h A ,m c = ( w − w + 1) − − χ ( w ) , (93) δ ˆ h A ,m b = δ ˆ h T ,m b = 0 , (94) δ ˆ h A ,m c = − w + 1) − (cid:0) η ( w ) (cid:1) + 4 ˆ χ ( w ) , (95) δ ˆ h A ,m c = 1 − w + 1) − (cid:0) η ( w ) (cid:1) − χ ( w ) − χ ( w ) , (96) δ ˆ h P,m c = − (cid:0) η ( w ) (cid:1) + 4( w −
1) ˆ χ ( w ) − χ ( w ) , (97) δ ˆ h T ,m c = − χ ( w ) , (98) δ ˆ h T ,m c = − , (99)20 ˆ h T ,m c = ( w + 1) − (cid:0) η ( w ) (cid:1) + 2 ˆ χ ( w ) , (100)where χ ( w ) is absorbed.The 1 /m c corrections consist of six subsub-leading IW functions (cid:96) ( w ) in the absence ofthe 1 /m b and 1 / ( m b m c ) corrections [23]. The expressions for δ ˆ h X,m c can be obtained fromRef. [23] as δ ˆ h + ,m c = ˆ (cid:96) ( w ) , (101) δ ˆ h − ,m c = ˆ (cid:96) ( w ) , (102) δ ˆ h V,m c = ˆ (cid:96) ( w ) − ˆ (cid:96) ( w ) , (103) δ ˆ h A ,m c = ˆ (cid:96) ( w ) − w − w + 1 ˆ (cid:96) ( w ) , (104) δ ˆ h A ,m c = ˆ (cid:96) ( w ) + ˆ (cid:96) ( w ) , (105) δ ˆ h A ,m c = ˆ (cid:96) ( w ) − ˆ (cid:96) ( w ) − ˆ (cid:96) ( w ) + ˆ (cid:96) ( w ) , (106) δ ˆ h S,m c = ˆ (cid:96) ( w ) − w − w + 1 ˆ (cid:96) ( w ) , (107) δ ˆ h P,m c = ˆ (cid:96) ( w ) + ( w − (cid:96) ( w ) + ˆ (cid:96) ( w ) − ˆ (cid:96) ( w ) , (108) δ ˆ h T,m c = ˆ (cid:96) ( w ) − ˆ (cid:96) ( w ) , (109) δ ˆ h T ,m c = ˆ (cid:96) ( w ) , (110) δ ˆ h T ,m c = ˆ (cid:96) ( w ) , (111) δ ˆ h T ,m c = 12 (cid:0) ˆ (cid:96) ( w ) − ˆ (cid:96) ( w ) (cid:1) , (112)for ˆ (cid:96) ( w ) = (cid:96) ( w ) /ξ ( w ). B Angular dependence
Here we derive the full angular distribution of Eqs. (24) and (25). In the SM, the squared decayamplitude of M for B → D ∗− (cid:96) ¯ ν , followed by D ∗− → ¯ D π − , can be represented as |M ( q , θ (cid:96) , θ V , χ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) λ D ∗ ,λ (cid:48) D ∗ S λ D ∗ ( θ (cid:96) ) D λ D ∗ , λ (cid:48) D ∗ ( χ ) T λ (cid:48) D ∗ ( θ V ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (113)where S λ D ∗ ( q , θ (cid:96) ) = G F √ V cb (cid:88) λ W H λ D ∗ λ W L λ (cid:96) = − / λ W , (114)shows the usual helicity amplitude for B → D ∗− (cid:96) ¯ ν , which has been described in Ref. [26],whereas T λ (cid:48) D ∗ ( θ V ) indicates the amplitude for D ∗− → ¯ D π − and D λ D ∗ λ (cid:48) D ∗ ( χ ) is the Wignerrotation that connects two decay planes defined for θ (cid:96) ( (cid:96) - ν plane at W rest frame) and θ V ( D - π plane at D ∗ rest frame). Then the latter two can be obtained as T = N π cos θ V , T ± = ∓ N π sin θ V , (115)21nd D , = 1 , D ± , ± = e ± iχ , others = 0 , (116)where the normalization factor N is determined so that (cid:90) − d cos θ V (cid:90) π − π dχ d Γ full dw d cos θ (cid:96) d cos θ V dχ = d Γ( B → D ∗− (cid:96) ¯ ν ) dw d cos θ (cid:96) B ( D ∗− → ¯ D π − ) , (117)is satisfied. Following L λ (cid:96) = − / λ W by substituting θ τ = π − θ (cid:96) (due to difference in definition)given in Ref. [26] along with the above description, we can derive the SM contribution inEqs. (24) and (25). Note that we have defined H ±± ≡ H ± ( w ) and H ≡ H ( w ) in the maintext. The angular dependence for the case of the V type operator is given simply by replacing H ± ( w ) → − C V H ∓ ( w ) and H ( w ) → − C V H ( w ).As for the tensor NP operator, a similar procedure is applicable to obtain the angulardistribution by taking S λ D ∗ ( q , θ (cid:96) ) = 2 G F √ V cb C T (cid:88) λ,λ (cid:48) H λ D ∗ λ,λ (cid:48) L λ (cid:96) =+1 / λ,λ (cid:48) , (118)where L is again described in Ref. [26], H ±± , = ± H ±± ,s ≡ H T ± ( w ), and H , − = H ,s ≡ H T ( w ).Since the lepton helicity of the tensor current is flipped compared with the SM current, onefinds that the SM and tensor operators have no interference. C Constraints from QCDSR
The sub-leading IW functions, χ , ( w ), η ( w ), have been investigated by introducing QCDSRanalysis up to two-loop perturbative corrections in the literature [33, 34, 35]. In this approach,they are described as χ i ( w ) = [ α s (1GeV)] / ¯ χ i ( w ) , η ( w ) = 13 + ∆( w ) , (119)with ¯ χ ( w ) (cid:104) F ¯Λ e − /T (cid:105) = − α s T π (cid:18) w + 1 (cid:19) (cid:18) − r ( w ) w − (cid:19) δ ( ω T ) (120)+ α s T (cid:104) ¯ qq (cid:105) π (cid:18) − r ( w ) w − w + 1 (cid:19) δ ( ω T ) − (cid:104) α s GG (cid:105) π w + 1 , ¯ χ ( w ) (cid:104) F ¯Λ e − /T (cid:105) = α s T π (cid:18) w + 1 (cid:19) (cid:18) wr ( w ) − w + 12 (cid:19) δ ( ω T ) (121)+ 3 δω π ω e − ω /T (cid:34)(cid:18) w + 1 (cid:19) − ξ ( w ) (cid:35) + α s T (cid:104) ¯ qq (cid:105) π (cid:2) − r ( w ) − ξ ( w ) (cid:3) δ ( ω T )+ (cid:104) α s GG (cid:105) π (cid:20) w + 1 − ξ ( w ) (cid:21) − (cid:104) ¯ qg s σ µν G µν q (cid:105) T (cid:2) − ξ ( w ) (cid:3) , w ) (cid:104) ξ ( w ) F ¯Λ e − /T (cid:105) = α s T π (cid:18) w + 1 (cid:19) (11 + 6 w + (3 + w ) r ( w )) δ ( ω T ) (122) − α s T (cid:104) ¯ qq (cid:105) π (7 + (3 − w ) r ( w )) δ ( ω T )+ (cid:104) α s GG (cid:105) π w − w + 1 + (cid:104) ¯ qg s σ µν G µν q (cid:105) T ( w − , where r ( w ) = 1 √ w − w + √ w − , δ n ( x ) = 1Γ( n + 1) (cid:90) x dzz n e − z . (123)The continuum threshold ω and Borel parameter T control stability of the sum rule, as will beexplained below. The renormalized factor [ α s (1GeV)] / connects the sub-leading IW functionsin QCD ¯ χ i ( w ) to our basis χ i ( w ).The prefactors, presented with (cid:2) · · · (cid:3) in Eqs. (121)–(123), contain the leading IW function ξ ( w ), heavy meson decay constant F (HQET basis), and heavy quark-meson mass difference¯Λ. From two-current correlator, one finds F ¯Λ e − /T = 9 T π δ (cid:16) ω T (cid:17) − (cid:104) ¯ qg s σ µν G µν q (cid:105) T , (see Ref. [33]) (124) ξ ( w ) F ¯Λ e − /T = 9 T π (cid:18)
21 + w (cid:19) δ (cid:16) ω T (cid:17) − w + 13 (cid:104) ¯ qg s σ µν G µν q (cid:105) T , (see Ref. [35]) (125)while ξ ( w ) can be independently obtained as [34] ξ ( w ) = K ( T, ω , w ) K ( T, ω , , (cid:16) equivalently ξ ( w ) F e − /T = K ( T, ω , w ) , (cid:17) K ( T, ω , w ) = 3 T π (cid:18)
21 + w (cid:19) δ (cid:16) ω T (cid:17) − (cid:104) ¯ qq (cid:105) + 2 w + 13 (cid:104) ¯ qg s σ µν G µν q (cid:105) T . (126)Note that − ∂∂T − K ( T, ω , w ) equals to r.h.s. of Eq. (125) and hence these expressions areconsistent with each other.Input parameters for the QCDSR predictions consist of the decay constant F = (0 . ± . / [33], the mass difference ¯Λ = (0 . ± . δω = ( − . ± . χ ( w ) [34], and the following vacuum condensates: (cid:104) ¯ qq (cid:105) = − (0 . ± .
01 GeV) , (from Refs. [62, 63, 64]) (127) (cid:104) α s GG (cid:105) = (6 . ± . × − GeV , (from Ref. [65]) (128) (cid:104) ¯ qg s σ µν G µν q (cid:105) = m (cid:104) ¯ qq (cid:105) with m = (0 . ± . . (from Refs. [66, 67, 68]) (129)The continuum threshold ω and the Borel parameter T have been determined in the literatureso that ¯ χ , (1) and ∆(1) are stabilized. In our case, concerning higher derivatives such as ˆ χ (2)2 , and η (2) , we take0 . < T < , . < ω < . . (130)Substituting QCDSR for F and ¯Λ as in Eqs. (124) and (126), and then taking numerical inputwithin 1 σ uncertainties, we obtain the constraints as in Eqs. (50)–(52) of the main text. Notethat ˆ χ i ( w ) = χ ( w ) /ξ ( w ) and we take the conservative ranges for the uncertainties, which is inagreement with Ref. [7]. 23 Fit results with some details
Here we write down useful output data obtained by our fit analyses. First, we show correlationamong our fit results of the HQET parameters, for SM (3 / /
1) in Tables 5–8, and for SM(2 / /
0) in Tables 9–12. Note that the distributions for Belle17 (Belle18) are those of the decayrates (folded signal events) as explained in the main text.We then provide our fit results of G ( w ) and F ( w ) for SM (3 / /
1) with the z expansionforms as defined in Eqs. (63) and (66):corr.( G ) = . − . . − . − . . − . . . − . . − . − . . − . . , (131)for ( G (1) , g , g , g ) andcorr.( F ) = . − . . − . − . . − . . . − . . − . − . . − . . , (132)for ( F (1) , f , f , f ). In turn, R D – R D ∗ correlations are obtained as − . − . − .
81, and − .
57 for SM (2 / / / / V (2 / / V (3 / / w, cos θ (cid:96) , cos θ V , χ )with the comparisons between data and the fit results in the SM (2 / /
0) [red] and SM (3 / / -)[gray] scenarios, in order to visualize the improvement on the fits. corr. | V cb | ξ (1) ξ (2) ξ (3) ˆ χ (0)2 ˆ χ (1)2 ˆ χ (2)2 ˆ χ (1)3 ˆ χ (2)3 η (0) η (1) η (2) | V cb | . . − . . . − . − . . . . − . − . ξ (1) . . − . . . − . . − . . . . . ξ (2) − . − . . − . − . . − . . − . − . − . − . ξ (3) . . − . . . − . . . . . . . χ (0)2 . . − . . . − . − . . . . . − . χ (1)2 − . − . . − . − . . − . . . − . − . − . χ (2)2 − . . − . . − . − . . − . . . − . − . χ (1)3 . − . . . . . − . . . − . . . χ (2)3 . . − . . . . . . . . . . η (0) . . − . . . − . . − . . . − . − . η (1) − . . − . . . − . − . . . − . . − . η (2) − . . − . . − . − . − . . . − . − . . Table 5: Correlation among (cid:110) ξ ( n ) , ˆ χ ( n )2 , , η ( n ) (cid:111) – (cid:110) ξ ( n ) , ˆ χ ( n )2 , , η ( n ) (cid:111) in SM (3 / / orr. ˆ (cid:96) (0)1 ˆ (cid:96) (1)1 ˆ (cid:96) (0)2 ˆ (cid:96) (1)2 ˆ (cid:96) (0)3 ˆ (cid:96) (1)3 ˆ (cid:96) (0)4 ˆ (cid:96) (1)4 ˆ (cid:96) (0)5 ˆ (cid:96) (1)5 ˆ (cid:96) (0)6 ˆ (cid:96) (1)6 | V cb | . − . − . − . − . − . . . . − . . − . ξ (1) − . − . − . − . − . − . − . . . − . . − . ξ (2) . . . . . . . − . − . . − . . ξ (3) − . − . − . − . − . − . − . . . − . . − . χ (0)2 − . . − . − . − . − . − . . . − . . − . χ (1)2 . . . . − . − . − . − . − . . − . . χ (2)2 . − . . . . − . − . − . − . . . − . χ (1)3 − . − . . . − . . . . . − . − . . χ (2)3 . − . − . − . . − . − . . . − . . − . η (0) − . − . − . − . . . − . − . − . . . . η (1) . − . . − . − . − . . − . − . . − . − . η (2) . − . . − . − . − . . − . . − . − . − . Table 6: Correlation among (cid:110) ξ ( n ) , ˆ χ ( n )2 , , η ( n ) (cid:111) – (cid:110) ˆ (cid:96) ( n )1-6 (cid:111) in SM (3 / / corr. | V cb | ξ (1) ξ (2) ξ (3) ˆ χ (0)2 ˆ χ (1)2 ˆ χ (2)2 ˆ χ (1)3 ˆ χ (2)3 η (0) η (1) η (2) ˆ (cid:96) (0)1 . − . . − . − . . . − . . − . . . (cid:96) (1)1 − . − . . − . . . − . − . − . − . − . − . (cid:96) (0)2 − . − . . − . − . . . . − . − . . . (cid:96) (1)2 − . − . . − . − . . . . − . − . − . − . (cid:96) (0)3 − . − . . − . − . − . . − . . . − . − . (cid:96) (1)3 − . − . . − . − . − . − . . − . . − . − . (cid:96) (0)4 . − . . − . − . − . − . . − . − . . . (cid:96) (1)4 . . − . . . − . − . . . − . − . − . (cid:96) (0)5 . . − . . . − . − . . . − . − . . (cid:96) (1)5 − . − . . − . − . . . − . − . . . − . (cid:96) (0)6 . . − . . . − . . − . . . − . − . (cid:96) (1)6 − . − . . − . − . . − . . − . . − . − . Table 7: Correlation among (cid:110) ˆ (cid:96) ( n )1-6 (cid:111) – (cid:110) ξ ( n ) , ˆ χ ( n )2 , , η ( n ) (cid:111) in SM (3 / / corr. ˆ (cid:96) (0)1 ˆ (cid:96) (1)1 ˆ (cid:96) (0)2 ˆ (cid:96) (1)2 ˆ (cid:96) (0)3 ˆ (cid:96) (1)3 ˆ (cid:96) (0)4 ˆ (cid:96) (1)4 ˆ (cid:96) (0)5 ˆ (cid:96) (1)5 ˆ (cid:96) (0)6 ˆ (cid:96) (1)6 ˆ (cid:96) (0)1 . . . . . . . − . . . − . . (cid:96) (1)1 . . . . . . . . − . . − . . (cid:96) (0)2 . . . . . − . . − . − . . − . . (cid:96) (1)2 . . . . . . . − . − . . − . . (cid:96) (0)3 . . . . . − . − . − . . . . − . (cid:96) (1)3 . . − . . − . . − . − . − . . − . . (cid:96) (0)4 . . . . − . − . . . . − . − . − . (cid:96) (1)4 − . . − . − . − . − . . . . − . − . − . (cid:96) (0)5 . − . − . − . . − . . . . − . . − . (cid:96) (1)5 . . . . . . − . − . − . . − . . (cid:96) (0)6 − . − . − . − . . − . − . − . . − . . − . (cid:96) (1)6 . . . . − . . − . − . − . . − . . Table 8: Correlation among (cid:110) ˆ (cid:96) ( n )1-6 (cid:111) – (cid:110) ˆ (cid:96) ( n )1-6 (cid:111) in SM (3 / / orr. | V cb | ξ (1) ξ (2) ˆ χ (0)2 ˆ χ (1)2 ˆ χ (1)3 η (0) η (1) | V cb | . . . − . − . − . − . − . ξ (1) . . − . . . − . − . . ξ (2) . − . . − . . − . . − . χ (0)2 − . . − . . . . − . . χ (1)2 − . . . . . . . − . χ (1)3 − . − . − . . . . − . − . η (0) − . − . . − . . − . . . η (1) − . . − . . − . − . . . Table 9: Correlation among (cid:110) ξ ( n ) , ˆ χ ( n )2 , , η ( n ) (cid:111) – (cid:110) ξ ( n ) , ˆ χ ( n )2 , , η ( n ) (cid:111) in SM (2 / / corr. ˆ (cid:96) (0)1 ˆ (cid:96) (0)2 ˆ (cid:96) (0)3 ˆ (cid:96) (0)4 ˆ (cid:96) (0)5 ˆ (cid:96) (0)6 | V cb | − . − . . . . . ξ (1) − . − . . . . . ξ (2) . . − . − . − . − . χ (0)2 . − . − . . . . χ (1)2 − . . − . . . − . χ (1)3 − . . − . . − . − . η (0) . . − . − . − . . η (1) − . . − . − . − . . Table 10: Correlation among (cid:110) ξ ( n ) , ˆ χ ( n )2 , , η ( n ) (cid:111) – (cid:110) ˆ (cid:96) ( n )1-6 (cid:111) in SM (2 / / corr. | V cb | ξ (1) ξ (2) ˆ χ (0)2 ˆ χ (1)2 ˆ χ (1)3 η (0) η (1) ˆ (cid:96) (0)1 − . − . . . − . − . . − . (cid:96) (0)2 − . − . . − . . . . . (cid:96) (0)3 . . − . − . − . − . − . − . (cid:96) (0)4 . . − . . . . − . − . (cid:96) (0)5 . . − . . . − . − . − . (cid:96) (0)6 . . − . . − . − . . . Table 11: Correlation among (cid:110) ξ ( n ) , ˆ χ ( n )2 , , η ( n ) (cid:111) – (cid:110) ˆ (cid:96) ( n )1-6 (cid:111) in SM (2 / / corr. ˆ (cid:96) (0)1 ˆ (cid:96) (0)2 ˆ (cid:96) (0)3 ˆ (cid:96) (0)4 ˆ (cid:96) (0)5 ˆ (cid:96) (0)6 ˆ (cid:96) (0)1 . . . . − . . (cid:96) (0)2 . . . − . − . . (cid:96) (0)3 . . . . . . (cid:96) (0)4 . − . . . . − . (cid:96) (0)5 − . − . . . . . (cid:96) (0)6 . . . − . . . Table 12: Correlation among (cid:110) ˆ (cid:96) ( n )1-6 (cid:111) – (cid:110) ˆ (cid:96) ( n )1-6 (cid:111) in SM (2 / / .0 1.1 1.2 1.3 1.4 1.5w E ve n t s E ve n t s E ve n t s E ve n t s Belle18- e
Belle15
Figure 3: Binned decay distributions with respect to w with the comparisons between dataand the fit results from the SM (2 / /
0) [red] and SM (3 / / -) [gray] scenarios. Belle18- e