Analyzing b→u transitions in semileptonic B ¯ s → K ∗+ (→Kπ) ℓ − ν ¯ ℓ decays
SSI-HEP-2015-11,QFET-2015-12,EOS-2015-01
Analyzing b → u transitions in semileptonic ¯ B s → K ∗ + ( → Kπ ) (cid:96) − ¯ ν (cid:96) decays Thorsten Feldmann, ∗ Bastian M¨uller, † and Danny van Dyk ‡ Theoretische Physik 1, Naturwissenschaftlich-Technische Fakult¨at,Universit¨at Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany (Dated: v1)We study the semileptonic decay ¯ B s → K ∗ + (cid:96) − ¯ ν (cid:96) , which is induced by b → u(cid:96) − ¯ ν (cid:96) transitions atthe quark level. We take into account the standard model (SM) operator from W -boson exchangeas well as possible extensions from physics beyond the SM. The secondary decay K ∗ + → Kπ canbe used to study a number of angular observables, which are worked out in terms of short-distanceWilson coefficients and hadronic form factors. Our analysis allows for an independent extraction ofthe Cabibbo-Kobayashi-Maskawa matrix element | V ub | and for the determination of certain ratiosof ¯ B s → K ∗ form factors. Moreover, a future precision measurement of the forward-backwardasymmetry in the ¯ B s → K ∗ + (cid:96) − ¯ ν (cid:96) decay can be used to unambiguously verify the left-handednature of the transition operator as predicted by the SM. We provide numerical estimates for therelevant angular observables and the resulting decay distributions on the basis of available form-factor information from lattice and sum-rule estimates. In addition, we pay particular attentionto suitable combinations of angular observables in the decays ¯ B s → K ∗ + ( → Kπ ) (cid:96) − ¯ ν (cid:96) and ¯ B → K ∗ ( → Kπ ) (cid:96) + (cid:96) − , and find that they provide complementary constraints on the relevant b → s short-distance coefficients. As a by-product, we perform a SM fit on the basis of selected experimentaldecay rates and hadronic input functions, which results in | V ub | = (4 . ± . · − . I. INTRODUCTION
The value of | V ub | represents one of the leastwell-measured parameters in the Cabibbo-Kobayashi-Maskawa (CKM) matrix of the standard model (SM).Moreover, at present, its inclusive determination from B → X u (cid:96)ν (cid:96) decays and the extraction from exclusivesemileptonic or leptonic decay modes lead to somewhatdifferent results (see e.g. the review in [1]). Indepen-dent phenomenological information on b → u transitionswill clearly help to better understand the origin of thesediscrepancies and the underlying theoretical uncertain-ties. As the solution to this | V ub | puzzle might also berelated to physics beyond the SM, one should also takeinto account possible new physics (NP) effects; see [2–4]for recent work in that direction.The proliferation of unknown parameters, whicharises in a model-independent approach with genericdimension-6 operators in the effective Hamiltonian, canbe handled with a sufficient number of independentexperimental observables in b → u transitions. Anexample is B → ρ ( → ππ ) (cid:96)ν (cid:96) where the analysis of thesecondary ρ → ππ decay introduces a large number ofangular observables with different sensitivities to theindividual short-distance coefficients [4]. This is similarto what has been extensively used in the analysis of rareexclusive b → s(cid:96) + (cid:96) − transitions [5–10]. Because of thelarge hadronic width of the ρ resonance and the questionof the S- and P-wave composition of the experimentallymeasured dipion final state, a precision determination of | V ub | from this decay also requires a better theoretical ∗ [email protected] † [email protected] ‡ [email protected] understanding of the B → ππ(cid:96)ν (cid:96) decay spectrum [11, 12].In this article, we focus on the decay ¯ B s → K ∗ + ( → Kπ ) (cid:96) − ¯ ν (cid:96) , which provides similar insight into the short-distance couplings as the decay B → ρ ( → ππ ) (cid:96) − ¯ ν (cid:96) . How-ever, the width of the K ∗ -meson is sufficiently smallerthan that of the ρ resonance, Γ K ∗ (cid:39) Γ ρ / (cid:39)
50 MeV.Moreover, from studies of the decay ¯ B → ¯ K ∗ J/ψ theS-wave background below the K ∗ resonance in B decaysis constrained to small values, with the S-wave fraction F s (cid:46)
7% on-resonance [13]. The decay ¯ B s → K ∗ + ( → Kπ ) (cid:96) − ¯ ν (cid:96) thus provides a promising alternative channelfor a precise determination of | V ub | in the SM, as hasalready been advocated for in [14]. For the same rea-son, it can also be used to constrain NP contributions in b → u transitions, in particular, as we will show below,to exclude possible effects from right-handed currents.Another benefit of the decay ¯ B s → K ∗ + (cid:96) − ¯ ν (cid:96) is theopportunity to combine it with the rare ¯ B → ¯ K ∗ (cid:96) + (cid:96) − decay. The secondary decay K ∗ → Kπ is identical inboth decays, which leads to a one-to-one correspondencebetween angular observables. Hadronic form factors inboth decays are related by the SU (3) f symmetry of thestrong interaction, and therefore hadronic uncertaintiesin ratios of angular observables from the two decays areexpected to be under control. The B → K ∗ (cid:96) + (cid:96) − decay amplitude also receives corrections fromnonfactorizable (i.e. not form-factor like) contributions involvinghadronic operators in the b → s effective Hamiltonian. Semilep-tonic b → u transitions are free of such effects. A comparison ofthe two decays can thus also shed light on the size of nonfactoriz-able hadronic matrix elements and the validity of the underlyingtheoretical framework. A detailed study along these lines is be-yond the scope of the present work. a r X i v : . [ h e p - ph ] M a r Furthermore, these ratios of angular observables aresensitive not only to | V ub | , but also to bilinear combina-tions of the Wilson coefficients describing semileptonic b → u and radiative b → s transitions in the SM and be-yond. In light of the present deviations between LHCbmeasurements and the respective SM predictions for afew angular observables in the ¯ B → ¯ K ∗ channel (see e.g.[15, 16], and also [17]), we will show how this can beexploited to obtain complementary information on the b → s Wilson coefficients.The outline of the article is as follows. In section IIwe introduce the effective Hamiltonian for semileptonic b → u(cid:96) ¯ ν (cid:96) transitions, including NP operators, and de-fine the angular observables for ¯ B s → K ∗ ( → Kπ ) (cid:96) ¯ ν (cid:96) transitions. In the following phenomenological sectionIII we identify SM null tests among the angular observ-ables, and derive expressions in a simplified scenario withonly right-handed NP contributions. We also define op-timized observables and highlight the synergies betweenthe angular observables in ¯ B s → K ∗ + ( → Kπ ) (cid:96) ¯ ν (cid:96) and¯ B → K ∗ ( → Kπ ) (cid:96) + (cid:96) − . In the numerical section IV wefirst perform a fit of the Wilson coefficients for (V-A) and(V+A) currents to experimental data for ¯ B → π + (cid:96) − ¯ ν (cid:96) , B − → τ − ¯ ν τ and ¯ B → X u (cid:96) − ¯ ν (cid:96) decays. On the basis ofthis fit and theoretical estimates for the relevant formfactors, we then provide numerical predictions for theangular observables and partially integrated branchingratios for ¯ B s → K ∗ + ( → Kπ ) (cid:96) ¯ ν (cid:96) decays, before we con-clude in section V. The helicity basis for the ¯ B s → K ∗ form factors is defined in appendix A, where we also inferthe form factor parameters from light-cone sum rule andlattice QCD results. The appendices B and C are dedi-cated to details on the determination of the hadronic am-plitudes and the angular observables of ¯ B s → K ∗ + (cid:96) − ¯ ν (cid:96) decays within and beyond the SM, respectively. II. DEFINITIONSA. Effective Hamiltonian for b → u(cid:96) ¯ ν (cid:96) We parametrize possible new physics contributions to b → u(cid:96) ¯ ν (cid:96) transitions in a model-independent fashion interms of a low-energy effective Hamiltonian, which canbe written in the form H eff b → u = − G F V eff ub √ (cid:88) X C X O X + h.c. . (1)Here the most general set of dimension-6 operators {O X } is given by O V,i = (cid:2) ¯ uγ µ P i b (cid:3)(cid:2) ¯ (cid:96)γ µ P L ν (cid:96) (cid:3) , O S,i = (cid:2) ¯ uP i b (cid:3)(cid:2) ¯ (cid:96)P L ν (cid:96) (cid:3) , O T = (cid:2) ¯ uσ µν b (cid:3)(cid:2) ¯ (cid:96)σ µν P L ν (cid:96) (cid:3) , (2)where P i ∈ { P L , P R } are chiral projectors, and we haverestricted ourselves to (massless) left-handed neutrinos and ignored the possibility of lepton-flavor violating cou-plings. (The generalization to more exotic scenarios withlight right-handed invisible neutral fermions is straight-forward, see e.g. [3].) Since in the presence of NP thenotion of V ub becomes ambiguous, we normalize the op-erators in eq. (1) to an effective parameter V eff ub , whichcan be taken, for instance, as the value of V ub that oneobtains from a global CKM fit within the SM. If NP ef-fects are small, one would then have C V,L (cid:39) C V,L ≡ V ub ≡ V eff ub , with all other Wil-son coefficients vanishing). Comparing with reference [2],where the modifications of left- and right-handed quarkcurrents has been parametrized in terms of ε L,R togetherwith a new mixing-matrix ˜ V for right-handed currents,our conventions are related via (cid:18) V ub V eff ub (cid:19) ε L = C V,L − , (cid:32) ˜ V ub V eff ub (cid:33) ε R = C V,R . (3) B. Angular distribution in ¯ B s → K ∗ (cid:96) ¯ ν (cid:96) The four-fold differential decay rate for ¯ B s → K ∗ + (cid:96) − ¯ ν (cid:96) is defined in terms of the dilepton invariant mass q , thepolar angles θ (cid:96) and θ K ∗ in the (cid:96)ν and K ∗ rest frames,respectively, and the azimuthal angle φ between the pri-mary and secondary decay planes,8 π Γ[ ¯ B s → K ∗ (cid:96) + ¯ ν (cid:96) ]d q d cos θ (cid:96) d cos θ K ∗ d φ = ˆ J ( q , θ (cid:96) , θ K ∗ , φ ) . (4)It can be expanded in a basis of trigonometric functionsof the decay angles. We defineˆ J ( q , θ (cid:96) , θ K ∗ , φ ) = ˆ J s sin θ K ∗ + ˆ J c cos θ K ∗ + ( ˆ J s sin θ K ∗ + ˆ J c cos θ K ∗ ) cos 2 θ (cid:96) + ˆ J sin θ K ∗ sin θ (cid:96) cos 2 φ + ˆ J sin 2 θ K ∗ sin 2 θ (cid:96) cos φ + ˆ J sin 2 θ K ∗ sin θ (cid:96) cos φ + ( ˆ J s sin θ K ∗ + ˆ J c cos θ K ∗ ) cos θ (cid:96) + ˆ J sin 2 θ K ∗ sin θ (cid:96) sin φ + ˆ J sin 2 θ K ∗ sin 2 θ (cid:96) sin φ + ˆ J sin θ K ∗ sin θ (cid:96) sin 2 φ , (5)with angular observables ˆ J i ( a ) ≡ ˆ J i ( a ) ( q ) for i = 1 , . . . , a = s, c . By construction, the functional dependenceof the angular distribution eq. (5) on the angular observ-ables is identical to the one for B → V ( → P P ) (cid:96) + (cid:96) − decays in [9], to which we refer for further details.Explicit expressions for the angular observables interms of hadronic form factors and Wilson coefficientsfor b → u(cid:96) ¯ ν (cid:96) in the general operator basis (1) are derivedin the appendices. III. PHENOMENOLOGY
For the remainder of this article we restrict our anal-ysis to vector-like couplings; i.e., we assume C S,i = C T = C T = 0 for simplicity. This leaves us with only two op-erators for left- and right-handed b → u currents, whichwe refer to as SM+SM’. We emphasize that with futureexperimental data one can also test for scalar and ten-sor currents on the basis of the formulae provided in ap-pendix C. A. Null tests of the SM
The twelve angular observables ˆ J i as introduced in eq.(5) are not independent. Within the SM, they can beexpressed in terms of four real-valued quantities: | N | ,and the three form factors F ⊥ , (cid:107) , . This fact can be usedto define a series of eight null tests that hold within theSM: 4 ˆ J c ˆ J + ˆ J − J = 0 , J s ˆ J c − J −
12 ˆ J = 0 , ˆ J c ˆ J s − J ˆ J = 0 ,
16 ˆ J s −
36 ˆ J − J s = 0 , ˆ J c = ˆ J = ˆ J = ˆ J = 0 . (6)Deviations from these relations are immediate signs ofphysics beyond the SM. This is in contrast to exclusive b → s(cid:96) + (cid:96) − decays, where such relations are broken bynonfactorizing long-distance contributions. B. Angular observables for SM+SM’
In the SM+SM’ scenario, we obtain a very simplestructure of the angular observables, which can be ex-pressed in terms of hadronic form factors (defined in thetransversity basis, see appendix A), and three indepen-dent combinations of Wilson coefficients, σ ± ≡ |C V,L ± C
V,R | , − σ ≡ ( C V,L − C
V,R )( C V,L + C V,R ) ∗ , (7)which depend on the absolute values | C V,L | and | C V,R | and the relative phase of the two Wilson coefficients(the absolute phase is irrelevant in the angular observ-ables). Notice that σ ± is even under parity transforma-tions ( L ↔ R ), while σ is odd. Neglecting the charged-lepton mass (which is valid as long as m (cid:96) / (cid:112) q (cid:28) J s = 3 ˆ J s = 9 | N | M B s (cid:104) σ +1 | F ⊥ | + σ − | F (cid:107) | (cid:105) , ˆ J c = − ˆ J c = 12 | N | M B s q σ − | F | , ˆ J = 6 | N | M B s (cid:104) σ +1 | F ⊥ | − σ − | F (cid:107) | (cid:105) , ˆ J = 6 √ | N | M B s (cid:112) q σ − F (cid:107) F , (8)and ˆ J = 24 √ | N | M B s (cid:112) q Re { σ } F ⊥ F , ˆ J s = 48 | N | M B s Re { σ } F ⊥ F (cid:107) , ˆ J = 12 √ | N | M B s (cid:112) q Im { σ } F ⊥ F , ˆ J = 24 | N | M B s Im { σ } F ⊥ F (cid:107) , (9)together with ˆ J c = ˆ J = 0 (all relations valid in theSM+SM’ scenario). Here, we introduce a normalizationfactor, | N | ≡ G | V eff ub | q √ λ · π M B s , (10)and λ ≡ λ ( M B , M K ∗ , q ) denotes the usual kinematicK¨all´en function. The normalization | N | is chosen suchthatdΓd q = (cid:88) λ =0 , ⊥ , (cid:107) | A Lλ | (11)= | N | M B s (cid:20) σ +1 | F ⊥ | + σ − (cid:18) | F (cid:107) | + M B s q | F | (cid:19)(cid:21) , where the transversity amplitudes A Lλ are defined inappendix B.Beside the decay rate, one can also define the leptonicforward-backward asymmetry A FB via the weighted in-tegral A FB ≡ / d q (cid:90) +1 − d cos θ (cid:96) sgn(cos θ (cid:96) ) d Γd q d cos θ (cid:96) . (12)In the SM+SM’ scenario, one finds that A FB takes therather simple form A FB = 2Re { σ } F ⊥ F (cid:107) σ +1 | F ⊥ | + σ − (cid:16) | F (cid:107) | + M Bs q | F | (cid:17) . (13)Note, that the bilinear σ is unconstrained by presentexperimental measurements of semileptonic b → u tran-sitions. Therefore a measurement of A FB would providecomplementary information on the Wilson coefficients.In particular, the sign of the forward-backward asymme-try resolves the present ambiguity between C V,L versus C V,R , see section IV.Similarly, the fraction of longitudinal K ∗ mesons isdefined as F L ≡ / d q (cid:90) +1 − dcos θ K ∗ ω F L (cos θ K ∗ ) d Γd q dcos θ K ∗ , (14)where ω F L ( z ) = (5 z − /
2. In the SM+SM’ scenariothis yields F L = σ − | F | σ +1 | F ⊥ | + σ − (cid:16) | F (cid:107) | + M Bs q | F | (cid:17) . (15) C. Optimized observables in SM+SM’
It is now possible to construct particular combinationsof angular observables where the hadronic form-factordependencies cancel (at least partially), and, as a con-sequence, these observables are sensitive to the short-distance Wilson coefficients, only; or vice-versa.We begin with observables where the form-factor de-pendencies cancel. These can be defined in completeanalogy to what has been discussed in [9],ˆ H (1) T = √ J (cid:113) − ˆ J c (2 ˆ J s − ˆ J ) , ˆ H (2) T = ˆ J (cid:113) − J c (2 ˆ J s + ˆ J ) , ˆ H (3) T = ˆ J s (cid:113) (2 ˆ J c ) − ( ˆ J ) , ˆ H (4) T = 2 ˆ J (cid:113) − J c (2 ˆ J s + ˆ J ) , ˆ H (5) T = − ˆ J (cid:113) (2 ˆ J c ) − ( ˆ J ) . (16)Within the SM+SM’ scenario, the form factors depen-dencies cancel exactly at every point in the q spectrum.However, for integrated angular observables one has totake into account the different kinematical prefactors,and a residual form-factor dependence will remain. Inthe SM+SM’ scenario these optimized observables readˆ H (1) T = 1 , ˆ H (2) T = ˆ H (3) T = 2 Re { σ } (cid:113) σ +1 σ − , ˆ H (4) T = ˆ H (5) T = 2 Im { σ } (cid:113) σ +1 σ − . (17)We continue with the construction of observables thatare only sensitive to form-factor ratios. Just as in ¯ B → We emphasize again that the cancellation of form-factor de-pendencies holds for the whole q spectrum, in contrast to¯ B → ¯ K ∗ (cid:96) + (cid:96) − where it can be spoiled by contributions withintermediate photons dissociating into (cid:96) + (cid:96) − . ¯ K ∗ (cid:96) + (cid:96) − , we find that the SM+SM’ scenario solely allowsus to extract one form factor ratio, namely F /F (cid:107) , in fivedifferent ratios of angular observables, M B s (cid:112) q F ( q ) F (cid:107) ( q ) = √ J ˆ J s = − ˆ J c √ J = ˆ J J s − ˆ J = (cid:115) − ˆ J c J s − ˆ J = √ J − ˆ J . (18)Inconsistencies among these relations would indicate NPbeyond the SM+SM’ scenario. D. Synergies with ¯ B → ¯ K ∗ (cid:96) + (cid:96) − The decay ¯ B → ¯ K ∗ ( → ¯ Kπ ) (cid:96) + (cid:96) − is induced by theflavor-changing neutral current (FCNC) transition b → s(cid:96) + (cid:96) − . At low hadronic recoil, q (cid:38)
15 GeV , it is againdominated by four-fermion operators which can be ex-tended to a SM+SM’ scenario. The structure of angu-lar observables J n ( q ) in those decays is similar as for¯ B s → ¯ K ∗ + (cid:96) ¯ ν (cid:96) . The analogous combinations of Wilsoncoefficients which enter the J n ( q ) now read ρ ± and ρ .(For the explicit definition and a detailed phenomenolog-ical discussion, we refer to [9].)With this we can define a number of useful ratios ofangular observables J n ( q ) in ¯ B → ¯ K ∗ (cid:96) + (cid:96) − and ˆ J n ( q )in ¯ B s → ¯ K ∗ + (cid:96) ¯ ν (cid:96) , R n ( q ) ≡ J n ( q )ˆ J n ( q ) , (19)for n = 1 c, c, , , s, ,
9, as well as R ± ( q ) ≡ J s ( q ) ± J ( q )2 ˆ J s ( q ) ± J ( q ) ,R ± ( q ) ≡ J s ( q ) ± J ( q )2 ˆ J s ( q ) ± ˆ J ( q ) . (20)Within these ratios, the dependence on the hadronic formfactors can be expected to cancel up to corrections fromthe violation of the SU (3) f symmetry of strong interac-tions, and from nonfactorizing hadronic matrix elementsin exclusive b → s(cid:96) + (cid:96) − transitions. In the limit wherethese corrections are neglected, we find R n (cid:39) α e π | V tb V ∗ ts | | V ub | ρ +1 σ +1 for n = 1+ , ρ − σ − for n = 1 − , c, − , c Re { ρ } Re { σ } for n = 4 , , s Im { ρ } Im { σ } for n = 8 , . (21)Of particular interest are ratios that are proportionalto the combination ρ ∝ Re (cid:8) C ( q ) C ∗ (cid:9) , where in the Decay q [GeV ] Measurement Reference B − → τ − ¯ ν τ – (1 . ± . ± . · − [18]– (1 . ± . ± . · − [19]– (1 . +0 . − . ± . · − [20]– (0 . +0 . − . ± . · − [21]¯ B → π + µ − ¯ ν τ [0 ,
2] (1 . ± . · − [22][2 ,
4] (1 . ± . · − [4 ,
6] (1 . ± . · − [6 ,
8] (1 . ± . · − [8 ,
10] (1 . ± . · − [10 ,
12] (1 . ± . · − [0 ,
2] (1 . ± . · − [23][2 ,
4] (1 . ± . · − [4 ,
6] (1 . ± . · − [6 ,
8] (1 . ± . · − [8 ,
10] (1 . ± . · − [10 ,
12] (1 . ± . · − [0 ,
2] (1 . ± . · − [24][2 ,
4] (1 . ± . · − [4 ,
6] (1 . ± . · − [6 ,
8] (1 . ± . · − [8 ,
10] (1 . ± . · − [10 ,
12] (1 . ± . · − [0 ,
2] (1 . ± . · − [25][2 ,
4] (1 . ± . · − [4 ,
6] (1 . ± . · − [6 ,
8] (0 . ± . · − [8 ,
10] (0 . ± . · − [10 ,
12] (1 . ± . · − TABLE I. Summary of the experimental likelihoods forbranching fractions of the exclusive b → u transitions. Weassume no correlation among the B − → τ − ¯ ν τ data, and usethe correlation matrices as given in [22, tables XI and XII],[23, table III and IV], [24, tables XXIX and XXXII] and [25,table XVII] for the respective data on ¯ B → π + µ − ¯ ν µ decays. SM C ( q ) is a linear combination of the Wilson coef-ficients C eff7 and C eff9 ( q ) in b → s transitions (see [9]for the explicit definitions). Optimized observables in¯ B → ¯ K ∗ (cid:96) + (cid:96) − only allow to access the ratio |C eff9 / C | ,whereas the ratios R n are sensitive to C eff9 · C . Measur-ing the corresponding ratios J n / ˆ J n thus allows to directlyaccess the q dependence of C eff9 and to test the theoret-ical predictions which are based on an operator productexpansion in the heavy b -quark limit. IV. NUMERICAL RESULTS
In this section we derive numerical results for theangular observables ˆ J n as introduced in section II B.Our analysis is carried out within a Bayesian frame-work, for which we use and extend EOS [26] for all numerical evaluations. As prerequisites to our numericalstudy of the angular observables, information on the¯ B s → K ∗ form factors, and constraints on the b → u Wilson coefficients are needed. These will be expressedthrough a-posteriori probability density functions(PDFs) labelled P ( (cid:126)θ FF | theory) and P ( (cid:126)θ ∆ B | exp. data),respectively. We refer to appendix A for the precisedefinition of P ( (cid:126)θ FF | theory). A. Determination of C V,L and C V,R
For the following numerical analysis we consider ex-perimental data on the branching ratios for leptonic B − → τ − ¯ ν τ and semileptonic ¯ B → π + µ − ¯ ν µ decays assummarized in table I, together with the averaged valuefor | V ub | from the inclusive determination, | V incl. ub | = (4 . ± . · − [1] . (22)Within the SM+SM’ scenario, the additional right-handed operator contributes differently to the individualdecay rates, corresponding to (see e.g. [2]) | V B → τνub | → | V eff ub | (cid:12)(cid:12) C V,L − C
V,R (cid:12)(cid:12) , | V B → π(cid:96)νub | → | V eff ub | (cid:12)(cid:12) C V,L + C V,R (cid:12)(cid:12) , | V incl. ub | → | V eff ub | (cid:0) |C V,L | + |C V,R | (cid:1) . (23)In order to illustrate the NP reach of our analysis, we fixthe auxiliary parameter V eff ub to a value that lies betweenthe exclusive and inclusive determinations of | V ub | withinthe SM, | V eff ub | ≡ . · − . With this we can constrain the absolute values and therelative phases of the Wilson coefficients C V,L and C V,R ,where the SM-like solution would correspond to |C V,L | ∼ C V,R ∼ P (data | (cid:126)θ ∆ B , M ) from(multi)normal distributions as indicated in table I andeq. (22). Note that we assume that the results for the B − → τ − ¯ ν τ branching ratios [18] and [20] are uncor-related, since the underlying sets of events use differenttagging methods for the selection process. The same as-sumption applies to the results of [21] and [27]. At thistime, we only use theoretical input from light-cone sumrules for the B → π transition form factors, and thereforerestrict ourselves to the kinematic range q ≤
12 GeV .For a consistent inclusion of lattice results on the B → π form factor in the high- q region (see e.g. [28–30], butalso note added below), we presently do not have accessto the necessary correlation information required for ourstatistical procedure.Within our analysis, we address the theoretical uncer-tainties using nuisance parameters for the hadronic ma-trix elements. These are the B -meson decay constant f B − , and the parameters of the the B → π vector formfactor f Bπ + ( q ): its normalization f Bπ + (0), as well as twoshape parameters b Bπ , ; see [31] for their definition. Forthe B -meson decay constant we use a Gaussian priorwith central value and minimal 68% probability inter-val f B − = (210 ±
11) MeV, as obtained from a recent2-point QCD sum rule at NNLO accuracy [32]. As priorfor the form factor parameters we use the a-posterioridistribution obtained from a recent Bayesian analysis ofthe LCSR prediction at NLO accuracy [31].In order to assess the physical implications of possibledeviations from the SM expectations, we compare thefit results for three different scenarios. In all cases weassume C V,L to be real-valued (i.e. a possible NP phasein the left-handed b → u transition should be associatedto V eff ub ). As already mentioned, the fit to the considereddata is only sensitive to the relative phase between theWilson coefficients C V,L and C V,R , and consequently wewill always encounter an irreducible degeneracy relatedto C V,L/R → − C V,L/R .1. First, we consider the scenario “left” that fea-tures only the left-handed current. In thiscase the number of parameters is five, (cid:126)θ left∆ B = (cid:0) C V,L , f Bπ + (0) , b Bπ , b Bπ , f B − (cid:1) .2. Next, we consider the scenario “real”, inwhich C V,R is present and real-valued. Theset of ∆ B parameters then reads (cid:126)θ real∆ B = (cid:0) C V,L , Re C V,R , f Bπ + (0) , b Bπ , b Bπ , f B − (cid:1) .3. Last but not least, we also consider the sce-nario “comp”, which includes a complex-valued C V,R , with the full seven parameters, (cid:126)θ comp∆ B = (cid:0) C V,L , Re C V,R , Im C V,R , f Bπ + (0) , b Bπ , b Bπ , f B − (cid:1) .For all scenarios ( M = left , real , comp), we obtain the a-posteriori PDF as usual via Bayes’ theorem, P ( (cid:126)θ ∆ B | data , M ) = P (data | (cid:126)θ ∆ B , M ) P ( (cid:126)θ ∆ B , M ) P (data , M ) , (24)where P (data , M ) ≡ (cid:90) d (cid:126)θ ∆ B P (data | (cid:126)θ ∆ B , M ) P ( (cid:126)θ ∆ B , M )(25)is the evidence for the scenario M . The likelihood P (data | (cid:126)θ ∆ B , M ) has already been introduced earlier. Inall three scenarios, we use for the priors of the Wilsoncoefficients uncorrelated, uniform distributions with thesupport − ≤ C i ≤ +2. For model comparisons, we nor-malize the model priors for the various fits scenarios. Thecorresponding relations read P (comp) : P (real) : P (left) = 1 : 4 : 16 (26) Significance [ σ ]Quantity “left” “real” “comp” d.o.f. Reference f + ( † ) 3 .
11 2 .
36 2 .
36 3 [31] B − → τ − ¯ ν τ +0 .
57 +0 .
39 +0 .
39 1 [18]+0 .
64 +0 .
34 +0 .
34 1 [19]+0 .
99 +0 .
75 +0 .
75 1 [20] − . − . − .
35 1 [21]¯ B → π + µ − ¯ ν τ .
85 1 .
08 1 .
08 6 [22]0 .
87 0 .
98 0 .
98 6 [23]1 .
70 1 .
97 1 .
97 6 [24]2 .
53 2 .
46 2 .
46 6 [25]¯ B → X u (cid:96) − ¯ ν (cid:96) +1 .
67 +1 .
45 +1 .
45 1 [1]TABLE II. Significances of the measurements at the best-fitpoint closest to the SM point for all three fit scenarios. Noticethat the pull for the LCSR calculation of the B → π vectorform factor f + , marked by a † , does not enter the goodness-of-fit calculation.
1. Scenario “Left”
Our findings for the scenario “left” can be summarizedas follows. We find two degenerate best-fit points corre-sponding to |C V,L | (cid:39)
1. The best-fit point (with positive C V,L ) reads (cid:126)θ left, ∗ ∆ B = (1 . , . , − . , +0 . , . . (27)We find at this point χ = 18 .
54, for 28 degrees offreedom (from 29 measurements reduced by 1 fit param-eter). As a consequence, this represents an excellent fitwith a p-value of 91%. The significances of the indi-vidual experimental inputs are collected in table II. Theone-dimensional marginalized posterior is approximatelyGaussian, and yields |C V,L | = 1 . ± .
05 at 68% probability . (28)Equivalently, this result can be expressed as | V ub | =(4 . ± . · − at 68% probability.
2. Scenario “Real”
For the scenario “real”, we find a four-fold ambiguityin the data; see figure 1 for an illustration. All localmodes are degenerate. We calculate the goodness of fitin the local mode closest to the SM, (cid:126)θ real, ∗ ∆ B =(1 . , − . , . , − . , +0 . , . , (29)and obtain χ = 20 .
47. This fit’s p-value of 81% is verygood. However, note that the χ value has increased incomparison to the previous scenario. This result war-rants a comment. The additional degree of freedom in − . − . − . . . . . C V,L − . − . − . . . . . C V , R − . − . − . . . . . C V,L − . − . − . . . . . C V , R FIG. 1. (left) Contours of the 68% (dark orange area) and 95% (orange area) probability regions for the Wilson coefficients C V,L and C V,R as obtained from our fit. See the text for details. Overlaid are the 68% and 95% contour lines for ¯ B → π + (cid:96) − ¯ ν (cid:96) (blue solid lines, negative slope), B − → (cid:96) − ¯ ν (cid:96) (blue solid lines, positive slope) and inclusive ¯ B → X u (cid:96) − ¯ ν (cid:96) (green solid rings).The black diamond marks the SM point. (right) Contours of the 68% and 95% probability regions for the Wilson coefficients(solid orange lines) overlaying the 68% (dark gray area) and 95% (light gray area) probability regions as obtained from ahypothetical measurement of A FB = A SMFB ± − . − . − . . . . . C V,L − . − . − . . . . . R e C V , R − . − . − . . . . . C V,R − . − . − . . . . . I m C V , R FIG. 2. Contours of the 68% (dark orange area) and 95% (orange area) probability regions for the Wilson coefficients C V,L and C V,R as obtained from our fit in scenario “comp”. See the text for details. The black diamond marks the SM point. form of C V,R allows the fit to move the form factor pa-rameters f + Bπ , b and b closer to the central values ofthe prior. This shift occurs at the expense of increas-ing the significances of the experimental data, while si-multaneously reducing the significance of the nuisanceparameters. For completeness, we also list these signifi-cances for all scenarios in table II. The one-dimensionalmarginalized posterior distributions for this scenario areapproximately Gaussian and symmetric under exchange C V,L ↔ Re C V,R . We find (at 68% propabality) |C V,L | = 1 . ± .
05 and | Re C V,R | ≤ . , (30)or | Re C V,R | = 1 . ± .
05 and |C V,L | ≤ . . (31)
3. Scenario “Comp”
We repeat the fit in scenario “comp”. As a consequenceof the additional degree of freedom, the four solutionsfrom the previous scenario now become connected. Thisis illustrated in figure 2. We calculate the goodness of fitin the local mode closest to the SM, which now reads: (cid:126)θ comp, ∗ ∆ B =(1 . , − . , . , . , − . , +0 . , . . (32)The individual significances are listed in table II, andamount to a total χ = 20 .
48. For the increase of χ withrespect to the “left” scenario, see our earlier comment.With 26 degrees of freedom the p-value is 77%, whichis still very good. It is not sensible to provide the 68%probability interval of the one-dimensional marginalizedposterior, since the solutions are strongly connected. Weshow the contours of the probability regions at 68% and95% probability in figure 2.
4. Comparison
We proceed with a comparison of the various fit sce-narios by means of the posterior odds. The latter can becalculated as P ( M | data) P ( M | data) = P (data | M ) P (data | M ) P ( M ) P ( M ) (33)We find P (“left” | data) P (“real” | data) = 27 . , (34)and P (“real” | data) P (“comp” | data) = 3 .
62 : 1 . (35)Using Jeffrey’s scale for the interpretation of the posteriorodds [33], we find that the data favour the interpretation with purely left-handed b → u currents over the otherscenarios very strongly . Moreover, the scenario “real” is substantially favored over the scenario “comp”.This means that – despite the observed tensions be-tween the different SM determinations of | V ub | – a NPscenario with right-handed currents does not lead to amore efficient description of the experimental data. Weemphasize again that the statistical treatment of the the-oretical uncertainties on the hadronic input parameters,which are still relatively large at present, has been crucialfor this argument. On the other hand, the experimentaldata on the inclusive and exclusive decay rates alone alsocannot exclude large right-handed currents. B. Predictions for Angular Observables ˆ J n We can now proceed to produce predictive distribu-tions for the angular observables ˆ J n in ¯ B s → K ∗ + ( → Kπ ) (cid:96) ¯ ν (cid:96) , for which we have two main applications inmind.
1. SM Scenario
First, we assume the SM case; i.e., we go back to V eff ub → V ub with C V,L ≡ C i ≡
0. In this case,only the a-posteriori
PDF on the ¯ B s → K ∗ form fac-tors is needed. We obtain the joint posterior-predictivedistribution for the angular observables by means of P ( (cid:126) ˆ J ) = (cid:90) d (cid:126)θ FF P ( (cid:126)θ FF | theory) δ ( (cid:126) ˆ J − ˆ J ( (cid:126)θ FF )) . (36)In practice, the above is carried out by calculating the ˆ J n for a set of samples drawn from the a-posteriori PDF.In our analysis 10 samples are used. Our results forthe angular observables, normalized to the decay width,are compiled in table III. We single out the branchingratio, which appears to be the most immediate candidatefor upcoming measurement. We present our results inunits of | V ub | − , which is convenient to extract | V ub | fromfuture data. Our results read (cid:90) d q d B d q = (5 . +0 . − . ) | V ub | − , (cid:90) q .
18 GeV d q d B d q = (8 . +0 . − . ) | V ub | − , (cid:90) q q d q d B d q = (27 . +2 . − . ) | V ub | − . (37)In the above, q = 0 .
02, and q = ( M B s − M K ∗ ) .
2. SM+SM’ Scenario
Second, we consider the interesting prospect of NP ef-fects entering the b → u transitions, which, according tothe discussion in the previous subsection, cannot yet beruled out. Based upon our model comparison, we chooseto give predictions for the scenario “real” only. In orderto investigate the NP effects on the angular observablesin ¯ B s → K ∗ (cid:96) ¯ ν (cid:96) , we compute the joint predictive distri-bution that arises from both posteriors P ( (cid:126)θ ∆ B | data) and P ( (cid:126)θ FF | theory). Our findings are listed in table IV for ourthree nominal choices of q bins. In addition, we find forthe partially integrated branching ratios in the scenario“real” (cid:90) d q d B d q = (9 . ± . · − , (cid:90) q .
18 GeV d q d B d q = (1 . ± . · − , (cid:90) q .
02 GeV d q d B d q = (4 . ± . · − . (38)We also consider suitable ratios of partial decay widthsin ¯ B s → K ∗ + µ − ¯ ν µ over either the ¯ B → π + µ − ¯ ν µ or B − → τ − ¯ ν τ widths. We define three such ratios,˜ R ≡ (cid:82) q q d q | A L | Γ( B − → τ − ¯ ν τ ) = 3 ˆ J c − ˆ J c B − → τ − ¯ ν τ ) , ˜ R (cid:107) ≡ (cid:82) q q d q | A L (cid:107) | Γ( B − → τ − ¯ ν τ ) = 8 ˆ J s −
12 ˆ J B − → τ − ¯ ν τ ) , ˜ R ⊥ ≡ (cid:82) q q d q | A L ⊥ | (cid:104) Γ( ¯ B → π + µ − ¯ ν µ ) (cid:105) = 8 ˆ J s + 12 ˆ J (cid:104) Γ( ¯ B → π + µ − ¯ ν µ ) (cid:105) , (39)where, as already explained above, we only use theLCSR-accessible part of the ¯ B → π + µ − ¯ ν µ phase space, (cid:104) Γ( ¯ B → π + µ − ¯ ν µ ) (cid:105) = (cid:90)
12 GeV q d q dΓ( ¯ B → π + µ − ¯ ν µ )d q . (40)The ratios ˜ R , (cid:107) , ⊥ are independent of NP effects in thisscenario. We find numerically,˜ R = 2 . +0 . − . , ˜ R (cid:107) = 1 . +0 . − . , ˜ R ⊥ = 0 . +0 . − . , (41)where the uncertainties are purely due to the imprecisetheoretical knowledge of the ¯ B s → K ∗ form factors, the¯ B → π form factors, and the B -meson decay constant.Here, correlation information among the various hadronicmatrix elements would help in reducing these uncertain-ties. (cid:104) ˆ J n (cid:105) / (cid:104) Γ (cid:105) n (a) (b) (c)1 s . + − . . . + − . . . + − . . c . + − . . . + − . . . + − . . s . + − . . . + − . . . + − . . c − . + − . . − . + − . . − . + − . . − . + − . . − . + − . . − . + − . . . + − . . . + − . . . + − . . − . + − . . − . + − . . − . + − . . s − . + − . . − . + − . . − . + − . . TABLE III. Estimates for the normalized nonvanishing an-gular observables in the SM. The integration ranges are (a)1 GeV ≤ q ≤ , (b) 14 .
18 GeV ≤ q ≤ .
71 GeV ,and 0 .
02 GeV ≤ q ≤ .
71 GeV . We normalize the in-tegrated angular observables (cid:104) ˆ J n (cid:105) to the partially integrateddecay width (cid:104) Γ (cid:105) for the same integration range. (cid:104) ˆ J n (cid:105) / (cid:104) Γ (cid:105) n (a) (b) (c)1 s . + − . . . + − . . . + − . . c . + − . . . + − . . . + − . . s . + − . . . + − . . . + − . . c − . + − . . − . + − . . − . + − . . − . + − . . − . + − . . − . + − . . . + − . . . + − . . . + − . . − . + − . . − . + − . . − . + − . . s − . + − . . − . + − . . − . + − . . TABLE IV. Estimates for the nonvanishing angular observ-ables ˆ J n in the SM+SM’ basis for real-valued Wilson co-efficients. Constraints on the Wilson coefficient are takenfrom data on exclusive semileptonic b → u transitions, seetext. The integration ranges are (a) 1 GeV ≤ q ≤ ,(b) 14 .
18 GeV ≤ q ≤ .
71 GeV , and 0 .
02 GeV ≤ q ≤ .
71 GeV . We normalize the angular observables to the par-tially integrated decays width (cid:104) Γ (cid:105) . Note that the quoted signfor the angular observables ˆ J and ˆ J s corresponds to the SM-like solution (30) with dominating left-handed current. Forthe solution (31), one simply has to flip the sign of ˆ J andˆ J s . V. CONCLUSIONS
The angular analysis of exclusive ¯ B s → K ∗ + ( → Kπ ) (cid:96) ¯ ν (cid:96) decays provides a powerful tool to measure theCabibbo-Kobayashi-Maskawa (CKM) element | V ub | inthe Standard Model (SM) and to constrain new physics(NP) contributions to the underlying semileptonic b → u(cid:96) ¯ ν (cid:96) transition. In this article, we have identified rela-tions among the angular observables that serve as nulltests of the SM. Furthermore, we have constructed op-timized observables where, also in the presence of NP,the dependence on either the hadronic form-factor or the0short-distance coefficients drops out. The fact that thesame secondary decay, K ∗ → Kπ , is used for the an-gular analysis of the rare B → K ∗ (cid:96) + (cid:96) − decay can bephenomenologically exploited by measuring certain ra-tios R n of angular observables from both decays. Inthe limit where nonfactorizable effects in B → K ∗ (cid:96) + (cid:96) − as well as SU (3) f symmetry corrections to form-factorratios can be neglected, the ratios R n are only sensi-tive to short-distance coefficients. In particular, we haveshown that in this way one can directly access the q -dependence of the effective Wilson coefficient function C eff9 ( q ) in B → K ∗ (cid:96) + (cid:96) − transitions.We have combined presently available experimentaldata on inclusive and exclusive, leptonic and semilep-tonic b → u transitions with theoretical information onhadronic form factors and decay constants, thereby ob-taining detailed numerical estimates for angular observ-ables and partially integrated decay widths in ¯ B s → K ∗ + ( → Kπ ) (cid:96) ¯ ν (cid:96) . Here, we also allowed for the presenceof right-handed currents that could arise from physicsbeyond the SM. Using a Bayesian approach for the sta-tistical treatment of theoretical uncertainties, we havefound that – despite the present tensions between dif-ferent | V ub | determinations – the SM is still more effi-cient in describing the experimental data than its right-handed extension. In a simultaneous SM fit to ¯ B → π + µ − ¯ ν µ (using light-cone sum rule results for low dilep-ton mass), B − → τ − ¯ ν τ and B → X u (cid:96)ν (cid:96) data, we find | V ub | = (4 . ± . · − with a p-value of 91%.On the other hand, right-handed contributions cannotbe excluded, either. In a SM-like scenario with domi-nating left-handed currents, we found that the ratio ofright-handed over left-handed currets is constrained to (cid:46) Again, some of the angular observ-ables in ¯ B s → K ∗ + ( → Kπ ) (cid:96) ¯ ν (cid:96) , e.g. the leptonic forward-backward asymmetry, are “parity”-odd and can thus un-ambigously test the (dominating) left-handed nature ofsemileptonic b → u currents. In this case, one wouldobtain strong constraints on the flavour sector of NPmodels with generic right-handed currents. (For a recentattempt to construct a left-right symmetric NP modelbased on the Pati-Salam gauge group, which can acco-modate naturally small right-handed b → u currents, see[34].)A crucial ingredient of our analysis has been the im-plementation of hadronic uncertainties. Improvements ofour theoretical understanding of nonperturbative QCDeffects (see also notes added below) would lead to morestringent constraints on the value of | V ub | and the possiblesize of right-handed b → u currents. In particular, predic-tions from lattice or light-cone sum rules for form-factor Notice that the lepton current with a light SM-like neutrino isalways considered to be left-handed, only. ratios with ¯ B and ¯ B s initial states (including correla-tions between input parameters), and similarly between B → π form factors and the B -meson decay constant,would be helpful in this respect. Notes added:
In the final phase of this work, the LHCb collaborationmeasured the ratio of the exclusive semileptonic branch-ing fractions of Λ b → pµ − ¯ ν µ and Λ b → Λ c µ − ¯ ν µ [35, 36].Assuming SM-like b → cµ − ¯ ν µ transitions, with knowl-edge of the magnitude of | V cb | and using information onthe relevant form factors [37], this ratio can be used toextract the branching fraction B (Λ b → pµ − ¯ ν µ ). As such,the branching fraction is a very powerful new constraint.However, in light of the present tension in the determi-nation of V cb from both inclusive and exclusive b → c(cid:96) ¯ ν (cid:96) decays, and in order to follow the logical line of this arti-cle, the new LHCb measurement should only be used ina setup that accounts for NP in both b → u and b → c transitions.Another article [38] that was published in the finalphase of this work provides updated LCSR results forthe hadronic form factors for ¯ B s → K ∗ transitions, whichinclude correlation information among the form factors.This development will help to further reduce theory un-certainties for this decay.In recent lattice studies of the B → π form factors[39, 40], also the correlation matrix between the relevanthadronic fit parameters has been provided. This will alsoallow to include the high- q data for the ¯ B → π(cid:96) ¯ ν (cid:96) decayin our statistical procedure, which could and should beused in future updates of our results. ACKNOWLEDGMENT
D.v.D. acknowledges time and effort spent by MartinJung on checking the angular distribution in the earlyphases of this work, and also for the initial idea to in-vestigate the full angular distribution of B → V (cid:96)ν (cid:96) de-cays for the complete basis of dimension-six operators.This work is supported in parts by the Bundesminis-terium f¨ur Bildung und Forschung (BMBF), and theDeutsche Forschungsgemeinschaft (DFG) within researchunit FOR 1873 (“QFET”).
Appendix A: Form Factors
There are in general 7 independent hadronic form fac-tors for B s → K ∗ transitions. Commonly, these are de-noted as V , A , , , T , , , see e.g. the definition in [41].For our purpose, it is more convenient to start with a1definition of form factors in a helicity basis, F ± ≡ iM B s (cid:104) K ∗ ( k, η ) | ¯ u/ε ∗± (1 − γ ) b | ¯ B s ( p ) (cid:105) ,F ≡ − i (cid:112) q M B s (cid:104) K ∗ ( k, η ) | ¯ u/ε ∗ (1 − γ ) b | ¯ B s ( p ) (cid:105) ,F t ≡ i (cid:112) q M B s (cid:104) K ∗ ( k, η ) | ¯ u/ε ∗ t (1 − γ ) b | ¯ B s ( p ) (cid:105) , (A1)and F T ± ≡ M B s (cid:104) K ∗ ( k, η ) | ¯ uσ µν (cid:15) µ ∗± q ν (1 + γ ) b | ¯ B s ( p ) (cid:105) ,F T ≡ M B s (cid:112) q (cid:104) K ∗ ( k, η ) | ¯ uσ µν (cid:15) µ ∗ q ν (1 + γ ) b | ¯ B s ( p ) (cid:105) , (A2)which is related to the one proposed in [42]. However,compared to [42], we have chosen a normalization con-vention such that all form factors are finite in the limit q → t − ≡ ( M B s − M K ∗ ) , and nonzero in the limit q →
0. In the above definition, η denotes the physi-cal polarization of the K ∗ meson, and (cid:15) stands for anauxiliary polarization vector of the dilepton system withpolarization states t, ± ,
0. Notice, that the form factorfor the pseudoscalar current is not independent, but fromthe equations of motion can be related to F t , (cid:104) K ∗ ( k, η ) | ¯ uγ b | ¯ B s (cid:105) = − i M B s m b + m u F t . (A3)Instead of the helicity form factors F ± , we will use thelinear combinations F (cid:107) ( ⊥ ) ≡ √ F − ± F + ) , F T (cid:107) ( ⊥ ) ≡ √ F T − ± F T + ) , (A4)which simplify the analytical expressions for the angularobservables. The explicit relations between our and thetraditional form factor basis read F ⊥ = √ λM B s ( M B s + M K ∗ ) V , (A5)for the vector form factor, F (cid:107) = √ M B s + M K ∗ M B s A ,F = ( M B s + M K ∗ ) ( M B s − M K ∗ − q ) A − λ A M K ∗ M B s ( M B s + M K ∗ )= 8 M K ∗ A M B s ,F t = √ λM B s A , (A6) q [GeV ] 0 15 .
00 19 . V ( q ) 0 . ± .
026 0 . ± .
066 1 . ± . A ( q ) 0 . ± .
023 0 . ± .
015 0 . ± . A ( q ) 0 . ± .
025 – – A ( q ) – 0 . ± .
016 0 . ± . V A A q [GeV ] 15 .
00 19 .
21 15 .
00 19 .
21 15 .
00 19 . .
00 1 .
000 0 .
271 1 .
000 0 .
305 1 .
000 0 . .
21 – 1 .
000 – 1 .
000 – 1 . B s → K ∗ form factor fits.Top: Form factor values at q = 0 are taken from LCSR cal-culations in [41]; values at q = 15 GeV and q = 19 .
21 GeV are taken from lattice QCD simulations [43]. Bottom: Cor-relation information for the lattice QCD inputs. The latticeQCD values and correlations are produced from the joint PDFgiven in [43, Table XXIX] using 5 · samples. for the axialvector currents, and F T ⊥ = √ λM B s T ,F T (cid:107) = √ M B s − M K ∗ ) M B s T ,F T = ( M B s − M K ∗ ) ( M B s + 3 M K ∗ − q ) T − λ T M K ∗ M B s ( M B s − M K ∗ )= 4 M K ∗ T M B s + M K ∗ , (A7)for the tensor current. In the above equations, the formfactors A and T are defined as in [43].The form factors fulfill endpoint relations [42, 44] which in our convention readlim q → t − F ⊥ = lim q → t − F t = 0 , lim q → t − F (cid:107) F = √ M B s M B s − M K ∗ , (A8)with t ± ≡ ( M B s ± M K ∗ ) . We will use these relationsfor our form-factor parametrization in the numerical fit.To this end, we consider a modified “ z -expansion” andwrite F ⊥ ( q ) = √ λM B s − M K ∗ P ( q , M B ∗ ) F ⊥ (0) × (cid:2) b ⊥ (cid:0) z ( q , t ) − z (0 , t ) (cid:1)(cid:3) ,F (cid:107) , ( q ) = P ( q , M B ) F (cid:107) , (0) Note that the endpoint relation for the ⊥ form factor in [42,appendix B] should read lim q → t − B V, /B V, = 0. × (cid:2) b (cid:107) , (cid:0) z ( q , t ) − z (0 , t ) (cid:1)(cid:3) ,F t ( q ) = √ λM B s − M K ∗ P ( q , M B ) F t (0) × (cid:2) b t (cid:0) z ( q , t ) − z (0 , t ) (cid:1)(cid:3) . (A9)Here, the prefactors contain global kinematic factors, theform-factor normalization at q = 0, together with theleading pole behaviour from the lowest resonances abovethe semileptonic decay region, P ( q , M ) − ≡ − q /M .The remaining q -dependence for each form factors isparametrized by a shape parameter b i . The variable z ( q , t ) is obtained from the conformal mapping, (see e.g. [45–47]) z ( a, b ) ≡ √ t + − a − (cid:112) t + − b √ t + − a + (cid:112) t + − b . (A10)Here we choose t = t + − (cid:112) t + ( t + − t − ) which minimizes | z | in the decay region. For the resonance masses weuse M B = 5279 MeV, M B ∗ = 5325 MeV and M B =5724 MeV [48]. The above parametrization eq. (A9) au-tomatically fulfills the end-point relation eq. (A8) for F ⊥ .The end-point relation for F (cid:107) /F is fulfilled by imposing b ≡ z (0 , t ) − z ( t − , t ) (cid:32) − F (cid:107) (0) F (0) (cid:115) t − M B s (cid:2) b (cid:107) ( z ( t − , t ) − z (0 , t )) (cid:3)(cid:33) . (A11)We fit the B s → K ∗ helicity form factors F ⊥ , (cid:107) , tothe nine constraints listed in table V. Our fit uses fiveparameters, (cid:126)θ FF = (cid:0) F ⊥ (0) , F (cid:107) (0) , F (0) , b ⊥ , b (cid:107) (cid:1) (A12)which represent the three normalizations F ⊥ , (cid:107) , ( q = 0),as well as two independent shape parameters b ⊥ , (cid:107) . As a-priori probability P ( (cid:126)θ F F ) we choose uncorrelated uni-form distributions with a generous support (to be com-pared with (A15) below),0 ≤ F ⊥ , (cid:107) , (0) ≤ , − ≤ b ⊥ ≤ , − ≤ b (cid:107) ≤ +5 . (A13)The likelihood P (theory | (cid:126)θ FF ) is constructed as theproduct of uncorrelated Gaussian likelihoods for each ofthe LCSR results for the form factors V , A and A ,as well as the joint multivariate Gaussian likelihood forthe lattice QCD results. All of these are listed in table V.The a-posteriori PDF is obtained as usual via Bayes’theorem, P ( (cid:126)θ FF | theory) = P (theory | (cid:126)θ FF ) P ( (cid:126)θ FF ) (cid:82) d (cid:126)θ FF P (theory | (cid:126)θ FF ) P ( (cid:126)θ FF ) . (A14)For all applications here and in section IV, we draw 10 samples from the a-posteriori distribution.The best-fit point, and the 1D-marginalized minimalintervals at 68% probability are found to be F ⊥ (0) = 0 . ± . , b ⊥ = − . +1 . − . ,F (cid:107) (0) = 0 . ± . , b (cid:107) = +0 . ± . .F (0) = 0 . ± . , (A15)Although the 1D-marginalized distributions are symmet-ric and resemble Gaussian distributions, we find that the distribution in eq. (A14) is distinctly non-Gaussian. Wetherefore use the posterior samples to carry out the un-certainty propagation. Appendix B: ¯ B s → K ∗ ( → Kπ ) (cid:96) − ¯ ν (cid:96) Decay Amplitude
In this appendix we give details on the parametrizationof the matrix element for the decay ¯ B s → K ∗ + (cid:96) − ¯ ν (cid:96) , withthe subsequent decay K ∗ + → ( Kπ ) + . We decompose thematrix element as in [9] M = F (cid:8) X S (cid:2) ¯ (cid:96)ν (cid:3) + X P (cid:2) ¯ (cid:96)γ ν (cid:3) + X µV (cid:2) ¯ (cid:96)γ µ ν (cid:3) + X µA (cid:2) ¯ (cid:96)γ µ γ ν (cid:3) + X µνT (cid:2) ¯ (cid:96)σ µν ν (cid:3)(cid:9) (B1)with the prefactor F = i √ G F V ub g K ∗ Kπ D K ∗ | (cid:126)k RF | , (B2)and | (cid:126)k RF | ≡ (cid:112) λ ( M K ∗ , M K , M π ) / M K ∗ . In the small-width approximation we replace the K ∗ resonance by | D K ∗ ( k ) | (cid:39) k − M K ∗ ) + M K ∗ Γ K ∗ → πM K ∗ Γ K ∗ δ ( k − M K ∗ ) (B3)where Γ K ∗ denotes the total decay width of the K ∗ me-son. Since Γ K ∗ (cid:39) Γ[ K ∗ → Kπ ] to very good approxima-tion, we use Γ K ∗ = | g K ∗ → Kπ | | (cid:126)k RF | πM K ∗ . (B4)Our parametrization of the hadronic matrix elementof B → V ( → P P ) (cid:96) − ¯ ν (cid:96) decays differes from the one in3[9] due to different conventions for the Levi-Civita tensor,the phase convention for the polarization vectors, and thefact that in this decay only left-handed lepton currentscontribute. We use N X S = i θ V A Lt = − N X P , (B5)and N X µV = − N X µA = + i θ V ε µ (0) A L + i θ V ε µ (+) e + iφ (cid:2) A L ⊥ + A L (cid:107) (cid:3) + i θ V ε µ ( − ) e − iφ (cid:2) A L ⊥ − A L (cid:107) (cid:3) , (B6)(B7)and N X µνT = cos θ V ε µ (+) ε ν ( − ) A (cid:107)⊥ + sin θ V √ ε µ ( t ) ε ν (+) e + iφ A t ⊥ + sin θ V √ ε µ ( t ) ε ν ( − ) e − iφ A t ⊥ − sin θ V √ ε µ (0) ε ν (+) e + iφ A (cid:107) − sin θ V √ ε µ (0) ε ν ( − ) e − iφ A (cid:107) . (B8)Using the normalisation constant N as given in eq. (10)and the general operator basis (2), we obtain for the in-dividual amplitude contributions A L = − N M B s (cid:112) q ( C V,L − C
V,R ) F ( q ) ,A L ⊥ = +4 N M B s ( C V,L + C V,R ) F ⊥ ( q ) ,A L (cid:107) = − N M B s ( C V,L − C
V,R ) F (cid:107) ( q ) ,A Lt = − N (cid:104) m (cid:96) M B s q (cid:0) C V,L − C
V,R (cid:1) + M B s m b (cid:0) C S,L − C
S,R (cid:1)(cid:105) F t ( q ) , (B9) and A (cid:107)⊥ = +8 N M B s C T F T ( q ) ,A t ⊥ = 4 √ N M B s (cid:112) q C T F T ⊥ ( q ) ,A (cid:107) = 4 √ N M B s (cid:112) q C T F T (cid:107) ( q ) . (B10) Appendix C: Angular Observables for B → V (cid:96)ν (cid:96)
In the limit m (cid:96) →
0, the angular observables ˆ J n readˆ J s = 316 (cid:2) | A L ⊥ | + 3 | A L (cid:107) | + 16 | A (cid:107) | + 16 | A t ⊥ | (cid:3) , ˆ J c = 34 (cid:2) | A L | + 2 | A Lt | + 8 | A (cid:107)⊥ | (cid:3) , ˆ J s = 316 (cid:2) | A L ⊥ | + | A L (cid:107) | − | A (cid:107) | − | A t ⊥ | (cid:3) , ˆ J c = − (cid:2) | A L | − | A (cid:107)⊥ | (cid:3) , ˆ J = 38 (cid:2) | A L ⊥ | − | A L (cid:107) | + 16 | A (cid:107) | − | A t ⊥ | (cid:3) , ˆ J = 34 √ (cid:110) A L A L ∗(cid:107) − √ A (cid:107)⊥ A ∗ (cid:107) (cid:111) , (C1)and ˆ J = 32 √ (cid:110) A L A L ⊥ + 2 √ A (cid:107) A L ∗ t (cid:111) , ˆ J s = 32 Re (cid:110) A L (cid:107) A L ∗⊥ (cid:111) , ˆ J c = − (cid:8) A (cid:107)⊥ A L ∗ t (cid:9) , ˆ J = 32 √ (cid:110) A L A L ∗(cid:107) − √ A t ⊥ A L ∗ t (cid:111) , ˆ J = 34 √ (cid:8) A L A L ∗⊥ (cid:9) , ˆ J = 34 Im (cid:110) A L ⊥ A L ∗(cid:107) (cid:111) . (C2) [1] R. Kowalewski and T. Mannel, in Review of ParticlePhysics , edited by K. Olive et al. (Chin.Phys. C38, 2014).[2] A. J. Buras, K. Gemmler, and G. Isidori, Nucl.Phys.
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