Semileptonic B and B s decays into orbitally excited charmed mesons
J. Segovia, C. Albertus, D. R. Entem, F. Fernández, E. Hernández, M. A. Pérez-García
aa r X i v : . [ h e p - ph ] N ov Semileptonic B and B s decays into orbitally excited charmed mesons J. Segovia, C. Albertus, D.R. Entem, F. Fern´andez, E. Hern´andez, and M.A. P´erez-Garc´ıa
Departamento de F´ısica Fundamental e IUFFyM,Universidad de Salamanca, E-37008 Salamanca, Spain (Dated: November 13, 2018)The BaBar Collaboration has recently reported products of branching fractions that include B meson semileptonic decays into final states with charged and neutral D (2420) and D ∗ (2460), twonarrow orbitally excited charmed mesons. We evaluate these branching fractions, together withthose concerning D ∗ (2400) and D ′ (2430) mesons, within the framework of a constituent quarkmodel. The calculation is performed in two steps, one of which involves a semileptonic decay andthe other is mediated by a strong process. Our results are in agreement with the experimentaldata. We also extend the study to semileptonic decays of B s into orbitally excited charmed-strangemesons, providing predictions to the possible measurements to be carried out at LHC. PACS numbers: 12.39.Pn, 12.40.-y, 13.20.FcKeywords: potential models, models of strong interactions, leptonic and semileptonic decays
I. INTRODUCTION
Different collaborations have recently reportedsemileptonic B decays into orbitally excited charmedmesons providing detailed results of branching fractions.The theoretical analysis of these data, which includeboth weak and strong decays, offers the possibility for astringent test of meson models.Moreover, an accurate determination of the | V cb | and | V ub | Cabbibo - Kobayashi - Maskawa matrix elements de-mands a detailed knowledge of semileptonic decays of b -hadrons. Decays including orbitally excited charmed me-son in the final state provide a substantial contributionto the total semileptonic decay width. Furthermore, abetter understanding of these processes is also necessaryin the analysis of signals and backgrounds of inclusiveand exclusive measurements of b -hadron decays.The Belle Collaboration [1], using a full reconstruc-tion tagging method to suppress the large combinatorialbackground, reported data on the product of branchingfractions B ( B + → D ∗∗ l + ν l ) B ( D ∗∗ → D ( ∗ ) π ), where, inthe usual notation, l stands for a light e or µ lepton, the D ∗ , D ′ , D and D ∗ mesons are denoted generically as D ∗∗ , and the D ∗ and D mesons as D ( ∗ ) . D ∗∗ decays are reconstructed in the decay chains D ∗∗ → D ∗ π ± and D ∗∗ → Dπ ± . In particular, the D ∗ meson decays only through the Dπ channel, while the D ′ and D mesons decay only via D ∗ π . Both Dπ and D ∗ π channels are opened in the case of D ∗ .In the case of BaBar data [2, 3] the branching frac-tions B ( D ∗ → D ( ∗ ) π ) include both the D ∗ and D con-tributions. As they also provide the ratio B D/D ( ∗ ) weestimate the D ∗ and D contributions separately. Theexperimental results of both collaborations are given inTable I.A similar analysis can be done in the strange sec-tor for the B s meson semileptonic decays. Here theintermediate states are the orbitally charmed-strangemesons, D ∗∗ s , and the available final channels are DK and D ∗ K . The Particle Data Group (PDG) reports a value B ( B s → D s (2536) − µ + ν µ ) B ( D s (2536) − → D ∗− ¯ K ) =2 . ± . B (¯ b → B s )and the experimental data for B (¯ b → B s ) B ( B s → D s (2536) − µ + ν µ ) B ( D s (2536) − → D ∗− ¯ K ) measuredby the D0 Collaboration [5].All these magnitudes can be consistently calculatedin the framework of constituent quark models becausethey can simultaneously account for the hadronic part ofthe weak process and the strong meson decays. In thiscontext, meson strong decay has been described success-fully in phenomenological models, like the P model [6]or the flux-tube model [7], or in microscopic models(see Refs. [8, 9]). The difference between the two ap-proaches lies on the description of the q ¯ q creation vertex.While the P model assumes that the q ¯ q pair is createdfrom the vacuum with vacuum quantum numbers, in themicroscopic model the q ¯ q pair is created from the in-terquark interactions already acting in the model. Bothapproaches will be used here to evaluate the strong de-cays. As for the weak process the matrix elements factor-izes into a leptonic and a hadronic part. It is the hadronicpart that contains the nonperturbative strong interactioneffects and we shall evaluate it within a constituent quarkmodel (CQM). We will work within the CQM of Ref. [10]which successfully describes hadron phenomenology andhadronic reactions [11–13] and has recently been appliedto mesons containing heavy quarks in Refs. [14, 15].The paper is organized as follows: In Sec. II we in-troduce the model we have used to get the masses andwave functions of the mesons involved in the reactionsmentioned above. In Secs. III and IV we study thesemileptonic and strong decay mechanisms, which consti-tute the two steps of the processes under study. Finally,we present our results in Sec. V and give some conclusionsin Sec. VI. Belle [1] ( × − ) BaBar [2, 3] ( × − ) D ∗ (2400) B ( B + → ¯ D ∗ l + ν l ) B ( ¯ D ∗ → D − π + ) 2 . ± . ± . . ± . ± . B ( B → D ∗− l + ν l ) B ( D ∗− → ¯ D π − ) 2 . ± . ± . . ± . ± . D ′ (2430) B ( B + → ¯ D ′ l + ν l ) B ( ¯ D ′ → D ∗− π + ) < . . ± . ± . B ( B → D ′ − l + ν l ) B ( D ′ − → ¯ D ∗ π − ) < . ± . ± . D (2420) B ( B + → ¯ D l + ν l ) B ( ¯ D → D ∗− π + ) 4 . ± . ± . . ± . ± . B ( B → D − l + ν l ) B ( D − → ¯ D ∗ π − ) 5 . ± . ± . . ± . ± . D ∗ (2460) B ( B + → ¯ D ∗ l + ν l ) B ( ¯ D ∗ → D − π + ) 2 . ± . ± . . ± . ± . ( ∗ ) B ( B + → ¯ D ∗ l + ν l ) B ( ¯ D ∗ → D ∗− π + ) 1 . ± . ± . . ± . ± . ( ∗ ) B ( B + → ¯ D ∗ l + ν l ) B ( ¯ D ∗ → D ( ∗ ) − π + ) 4 . ± . ± . . ± . ± . B ( B → D ∗− l + ν l ) B ( D ∗− → ¯ D π − ) 2 . ± . ± . . ± . ± . ( ∗ ) B ( B → D ∗− l + ν l ) B ( D ∗− → ¯ D ∗ π − ) < . ± . ± . ( ∗ ) B ( B → D ∗− l + ν l ) B ( D ∗− → ¯ D ( ∗ )0 π − ) < . . ± . ± . B D/D ( ∗ ) . ± .
03 0 . ± . ± . l stands for a light e or µ lepton. The symbol ( ∗ ) indicates results estimated from the original data by using B D/D ( ∗ ) . II. CONSTITUENT QUARK MODEL
Spontaneous chiral symmetry breaking of the QCD La-grangian together with the perturbative one-gluon ex-change (OGE) and the nonperturbative confining inter-action are the main pieces of potential models. Usingthis idea, Vijande et al. [10] developed a model of thequark-quark interaction which is able to describe mesonphenomenology from the light to the heavy quark sector.We briefly explain the model below. Further details canbe found in Ref. [10].One consequence of the spontaneous chiral symmetrybreaking is that the nearly massless ’current’ light quarksacquire a dynamical, momentum-dependent mass M ( p )with M (0) ≈
300 MeV for the u and d quarks, namely,the constituent mass. To preserve chiral invariance ofthe QCD Lagrangian new interaction terms, given byGoldstone boson exchanges, should appear between con- stituent quarks.A simple Lagrangian invariant under chiral transfor-mations can be derived as [16] L = ¯ ψ ( iγ µ ∂ µ − M U γ ) ψ, (1)where U γ = exp( iπ a λ a γ /f π ), π a denotes the pseu-doscalar fields ( ~π, K, η ) and M is the constituent quarkmass. The momentum-dependent mass acts as a naturalcutoff of the theory. The chiral quark-quark interactioncan be written as V qq ( ~r ij ) = V C qq ( ~r ij ) + V T qq ( ~r ij ) + V SO qq ( ~r ij ) , (2)where C , T and SO stand for central, tensor and spin-orbit potentials. The central part presents four differentcontributions, V C qq ( ~r ij ) = V C π ( ~r ij ) + V C σ ( ~r ij )+ V C K ( ~r ij )+ V C η ( ~r ij ) , (3)given by V C π ( ~r ij ) = g ch π m π m i m j Λ π Λ π − m π m π (cid:20) Y ( m π r ij ) − Λ π m π Y (Λ π r ij ) (cid:21) ( ~σ i · ~σ j ) X a =1 ( λ ai · λ aj ) ,V C σ ( ~r ij ) = − g ch π Λ σ Λ σ − m σ m σ (cid:20) Y ( m σ r ij ) − Λ σ m σ Y (Λ σ r ij ) (cid:21) ,V C K ( ~r ij ) = g ch π m K m i m j Λ K Λ K − m K m K (cid:20) Y ( m K r ij ) − Λ K m K Y (Λ K r ij ) (cid:21) ( ~σ i · ~σ j ) X a =4 ( λ ai · λ aj ) ,V C η ( ~r ij ) = g ch π m η m i m j Λ η Λ η − m η m η " Y ( m η r ij ) − Λ η m η Y (Λ η r ij ) ( ~σ i · ~σ j ) (cid:2) cos θ p (cid:0) λ i · λ j (cid:1) − sin θ p (cid:3) , (4)where Y ( x ) is the standard Yukawa function defined by Y ( x ) = e − x /x . We consider the physical η meson insteadof the octet one and so we introduce the angle θ p . The λ a are the SU(3) flavor Gell-Mann matrices, m i is the quarkmass and m π , m K and m η are the masses of the SU(3)Goldstone bosons, taken from experimental values. m σ is determined through the PCAC relation m σ ≃ m π +4 m u,d [17]. Finally, the chiral coupling constant, g ch , isdetermined from the πN N coupling constant through g ch π = 925 g πNN π m u,d m N , (5) which assumes that flavor SU(3) is an exact symmetryonly broken by the different mass of the strange quark.There are three different contributions to the tensorpotential V T qq ( ~r ij ) = V T π ( ~r ij ) + V T K ( ~r ij ) + V T η ( ~r ij ) , (6)given by V T π ( ~r ij ) = g ch π m π m i m j Λ π Λ π − m π m π (cid:20) H ( m π r ij ) − Λ π m π H (Λ π r ij ) (cid:21) S ij X a =1 ( λ ai · λ aj ) ,V T K ( ~r ij ) = g ch π m K m i m j Λ K Λ K − m K m K (cid:20) H ( m K r ij ) − Λ K m K H (Λ K r ij ) (cid:21) S ij X a =4 ( λ ai · λ aj ) ,V T η ( ~r ij ) = g ch π m η m i m j Λ η Λ η − m η m η " H ( m η r ij ) − Λ η m η H (Λ η r ij ) S ij (cid:2) cos θ p (cid:0) λ i · λ j (cid:1) − sin θ p (cid:3) . (7) S ij = 3( ~σ i · ˆ r ij )( ~σ j · ˆ r ij ) − ~σ i · ~σ j is the quark tensoroperator and H ( x ) = (1 + 3 /x + 3 /x ) Y ( x ).Finally, the spin-orbit potential only presents a contri-bution coming from the scalar part of the interaction V SO qq ( ~r ij ) = V SO σ ( ~r ij ) = − g ch π m σ m i m j Λ σ Λ σ − m σ × (cid:20) G ( m σ r ij ) − Λ σ m σ G (Λ σ r ij ) (cid:21) ( ~L · ~S ) . (8)In the last equation G ( x ) is the function (1+1 /x ) Y ( x ) /x .Beyond the chiral symmetry breaking scale one expects the dynamics to be governed by QCD perturbative ef-fects. In this way one-gluon fluctuations around the in-stanton vacuum are taken into account through the qqg coupling L qqg = i √ πα s ¯ ψγ µ G µc λ c ψ, (9)with λ c being the SU (3) color matrices and G µc the gluonfield.The different terms of the potential derived from theLagrangian contain central, tensor, and spin-orbit con-tributions and are given by V COGE ( ~r ij ) = 14 α s ( ~λ ci · ~λ cj ) (cid:20) r ij − m i m j ( ~σ i · ~σ j ) e − r ij /r ( µ ) r ij r ( µ ) (cid:21) ,V TOGE ( ~r ij ) = − α s m i m j ( ~λ ci · ~λ cj ) " r ij − e − r ij /r g ( µ ) r ij r ij + 13 r g ( µ ) + 1 r ij r g ( µ ) ! S ij ,V SOOGE ( ~r ij ) = − α s m i m j ( ~λ ci · ~λ cj ) " r ij − e − r ij /r g ( µ ) r ij (cid:18) r ij r g ( µ ) (cid:19) ×× h (( m i + m j ) + 2 m i m j )( ~S + · ~L ) + ( m j − m i )( ~S − · ~L ) i , (10)where ~S ± = ( ~σ i ± ~σ j ). Besides, r ( µ ) = ˆ r µ nn µ ij and r g ( µ ) = ˆ r g µ nn µ ij are regulators which depend on µ ij , thereduced mass of the q ¯ q pair. The contact term of thecentral potential has been regularized as δ ( ~r ij ) ∼ πr e − r ij /r r ij (11)The wide energy range needed to provide a consistentdescription of light, strange and heavy mesons requires aneffective scale-dependent strong coupling constant. Weuse the frozen coupling constant of Ref. [10] α s ( µ ) = α ln (cid:16) µ + µ Λ (cid:17) , (12) in which µ is the reduced mass of the q ¯ q pair and α , µ and Λ are parameters of the model determined by aglobal fit to the meson spectra.Confinement is one of the crucial aspects of QCD.Color charges are confined inside hadrons. It is wellknown that multigluon exchanges produce an attractivelinearly rising potential proportional to the distance be-tween quarks. This idea has been confirmed, but not rig-orously proved, by quenched lattice gauge Wilson loopcalculations for heavy valence quark systems. However,sea quarks are also important ingredients of the stronginteraction dynamics. When included in the lattice cal-culations they contribute to the screening of the risingpotential at low momenta and eventually to the breakingof the quark-antiquark binding string. This fact, whichhas been observed in n f = 2 lattice QCD [18], has beentaken into account in our model by including the terms V CCON ( ~r ij ) = (cid:2) − a c (1 − e − µ c r ij ) + ∆ (cid:3) ( ~λ ci · ~λ cj ) ,V SOCON ( ~r ij ) = − (cid:16) ~λ ci · ~λ cj (cid:17) a c µ c e − µ c r ij m i m j r ij h (( m i + m j )(1 − a s ) + 4 m i m j (1 − a s ))( ~S + · ~L )+( m j − m i )(1 − a s )( ~S − · ~L ) i , (13)where a s controls the mixture between the scalar andvector Lorentz structures of the confinement. At shortdistances this potential presents a linear behavior withan effective confinement strength σ = − a c µ c ( ~λ ci · ~λ cj )and becomes constant at large distances with a thresholddefined by V thr = {− a c + ∆ } ( ~λ ci · ~λ cj ) . (14)No q ¯ q bound states can be found for energies higherthan this threshold. The system suffers a transitionfrom a color string configuration between two static colorsources into a pair of static mesons due to the breaking ofthe color string and the most favored decay into hadrons.Among the different methods to solve the Schr¨odinger equation in order to find the quark-antiquark boundstates, we use the Gaussian Expansion Method [19] be-cause it provides enough accuracy and it makes the subse-quent evaluation of the decay amplitude matrix elementseasier.This procedure provides the radial wave function solu-tion of the Schr¨odinger equation as an expansion in termsof basis functions R α ( r ) = n max X n =1 c αn φ Gnl ( r ) , (15)where α refers to the channel quantum numbers. Thecoefficients, c αn , and the eigenvalue, E , are determined Quark masses m n (MeV) 313 m s (MeV) 555 m c (MeV) 1763 m b (MeV) 5110Goldstone Bosons m π (fm − ) 0 . m σ (fm − ) 3 . m K (fm − ) 2 . m η (fm − ) 2 . π (fm − ) 4 . σ (fm − ) 4 . K (fm − ) 4 . η (fm − ) 5 . g ch / π . θ p ( ◦ ) − α . (fm − ) 0 . µ (MeV) 36 . r (fm) 0 . r g (fm) 0 . a c (MeV) 507 . µ c (fm − ) 0 . . a s . from the Rayleigh-Ritz variational principle n max X n =1 " ( T αn ′ n − EN αn ′ n ) c αn + X α ′ V αα ′ n ′ n c α ′ n = 0 , (16)where T αn ′ n , N αn ′ n and V αα ′ n ′ n are the matrix elements ofthe kinetic energy, the normalization and the potential,respectively. T αn ′ n and N αn ′ n are diagonal whereas themixing between different channels is given by V αα ′ n ′ n .Following Ref. [19], we employ Gaussian trial functionswith ranges in geometric progression. This enables theoptimization of ranges employing a small number of freeparameters. Moreover, the geometric progression is denseat short distances, so that it allows the description ofthe dynamics mediated by short range potentials. Thefast damping of the gaussian tail is not a problem, sincewe can choose the maximal range much longer than thehadronic size.Table II shows the model parameters fitted over allmeson spectra and taken from Refs. [10, 14]. III. WEAK DECAYS
In this section, we give an account of the semilep-tonic decays of the B ( B or B s ) meson into orbitally excited charmed mesons. In the nonstrange sector, thishas been studied before within heavy quark effectivetheory (HQET) in Refs. [20, 21]. There, only relativebranching ratios could be predicted and their resultsdepended on the approximation used and on two un-known functions, τ , τ , that describe corrections of or-der Λ QCD /m Q . Only the ratio Γ λ =0 D ∗∗ / Γ D ∗∗ , semileptonicdecay rate with a helicity 0 D ∗∗ final meson over totalsemileptonic decay rate to that meson, seemed to be sta-ble in the different approximations. We shall commenton this below.In the context of nonrelativistic constituent quarkmodels, the state of a meson is given by | M, λ ~P i NR = Z d p (2 π ) / X α ,α ( − / − s p E f ( ~p )2 E f ( ~p ) × ˆ φ ( M,λ ) α ,α ( ~p ) | ¯ q, α ~p i | q, α ~p i , (17)where ~P is the three-momentum of the meson and λ isthe spin projection in the meson center of mass. Thevector ~p is the relative momentum of the q ¯ q pair, ~p = m f m f + m f ~P − ~p and ~p = m f m f + m f ~P + ~p are the momenta ofthe antiquark and the quark, respectively, α and α arethe spin, flavor and color quantum numbers. ( E ( ~p i ) , ~p i )are the four-momenta and m i are the quark masses. Thefactor ( − / − s is included in order that the antiquarkspin states have the correct relative phase.The normalization of the quark-antiquark states is h α ′ ~p ′ | α ~p i = δ α ′ ,α (2 π ) E f ( ~p ) δ ( ~p ′ − ~p ) , (18)and the momentum space wave function ˆ φ ( M,λ ) α ,α ( ~p ) nor-malization is given by Z d p X α ,α ( ˆ φ ( M,λ ′ ) α ,α ( ~p )) ∗ ˆ φ ( M,λ ) α ,α ( ~p ) = δ λ ′ ,λ . (19)Finally, the normalization of our meson states is NR h M, λ ′ ~P ′ | M, λ ~P i NR = δ λ ′ ,λ (2 π ) δ ( ~P ′ − ~P ) . (20)In the decay we have a ¯ b → ¯ c transition at the quarklevel and we need to evaluate the hadronic matrix ele-ments of the weak current J bcµ (0) = ¯ ψ b (0) γ µ ( I − γ ) ψ c (0) . (21)The hadronic matrix elements can be parameterized interms of form factors as h D (0 + ) , λ ~P D | J bcµ (0) | B (0 − ) , ~P B i = P µ F + ( q ) + q µ F − ( q ) , h D (1 + ) , λ ~P D | J bcµ (0) | B (0 − ) , ~P B i = − m B + m D ǫ µναβ ǫ ν ∗ ( λ ) ( ~P D ) P α q β A ( q ) − i ( ( m B − m D ) ǫ ∗ ( λ ) µ ( ~P D ) V ( q ) − P · ǫ ∗ ( λ ) ( ~P D ) m B + m D (cid:2) P µ V + ( q ) + q µ V − ( q ) (cid:3)) , h D (2 + ) , λ ~P D (cid:12)(cid:12) J bcµ (0) (cid:12)(cid:12) B (0 − ) ~P B i = ǫ µναβ ǫ νδ ∗ ( λ ) ( ~P D ) P δ P α q β T ( q ) − i n ǫ ∗ ( λ ) µδ ( ~P D ) P δ T ( q ) + P ν P δ ǫ ∗ ( λ ) νδ ( ~P D ) (cid:2) P µ T ( q ) + q µ T ( q ) (cid:3)o . (22)In the expressions above, P = P B + P D and q = P B − P D , P B and P D being the meson four-momenta. m B and m D are the meson masses, ǫ µναβ is the fully antisymmetrictensor, for which the convention ǫ = +1 is taken, and ǫ ( λ ) µ ( ~P ) and ǫ ( λ ) µν ( ~P ) are the polarization vector andtensor of vector and tensor mesons, respectively. Themeson states in the Lorentz decompositions of Eq. (22)are normalized such that h M, λ ′ ~P ′ | M, λ ~P i = δ λ ′ ,λ (2 π ) E M ( ~P ) δ ( ~P ′ − ~P ) . (23)where E M ( ~P ) is the energy of the M meson with three-momentum ~P . Note the factor 2 E M difference with re-spect to Eq. (20).The form factors will be evaluated in the center of mass of the 0 − meson, taking ~q in the ˆ z direction, so that ~P B = ~ ~P D = − ~q = −| ~q | ~k , with ~k representing theunit vector in the ˆ z direction. We have taken the phasesof the states such that all form factors are real. F + , F − , A , V , V + , V − and T are dimensionless, whereas T , T and T have dimension of E − . Defining vector V µλ ( | ~q | )and axial A µλ ( | ~q | ) matrix elements such that V µλ ( | ~q | ) = h M F , λ − | ~q | ~k | J bcµV (0) | M I ,~ i ,A µλ ( | ~q | ) = h M F , λ − | ~q | ~k | J bcµA (0) | M I ,~ i , (24)we have for a 0 − → + decay, that the form factors aregiven in terms of vector and axial matrix elements as F + ( q ) = − m B (cid:20) A ( | ~q | ) + A ( | ~q | ) | ~q | ( E D ( − ~q ) − m B ) (cid:21) ,F − ( q ) = − m B (cid:20) A ( | ~q | ) + A ( | ~q | ) | ~q | ( E D ( − ~q ) + m B ) (cid:21) . (25)In the case of a 0 − → + transition, the corresponding expressions for the form factors are A ( q ) = − i √ m B + m D m B | ~q | A λ = − ( | ~q | ) ,V + ( q ) = + i m B + m D m B m D | ~q | m B (cid:26) V λ =0 ( | ~q | ) − m B − E D ( − ~q ) | ~q | V λ =0 ( | ~q | ) + √ m B E D ( − ~q ) − m D | ~q | m D V λ = − ( | ~q | ) (cid:27) ,V − ( q ) = − i m B + m D m B m D | ~q | m B (cid:26) − V λ =0 ( | ~q | ) − m B + E D ( − ~q ) | ~q | V λ =0 ( | ~q | ) + √ m B E D ( − ~q ) + m D | ~q | m D V λ = − ( | ~q | ) (cid:27) ,V ( q ) = + i √ m B − m D V λ = − ( | ~q | ) . (26)Finally, the form factors for a 0 − → + transition are given by the relations T ( q ) = − i m D m B | ~q | A T λ =+1 ( | ~q | ) ,T ( q ) = i m B ( − r m D | ~q | A T λ =0 ( | ~q | ) − r m D | ~q | ( E D ( − ~q ) − m B ) A T λ =0 ( | ~q | )+ 2 m D | ~q | (cid:18) − E D ( − ~q )( E D ( − ~q ) − m B ) | ~q | (cid:19) A T λ =+1 ( | ~q | ) ) ,T ( q ) = i m B ( − r m D | ~q | A T λ =0 ( | ~q | ) − r m D | ~q | ( E D ( − ~q ) + m B ) A T λ =0 ( | ~q | )+ 2 m D | ~q | (cid:18) − E D ( − ~q )( E D ( − ~q ) + m B ) | ~q | (cid:19) A T λ =+1 ( | ~q | ) ) ,T ( q ) = i m D m B | ~q | V T λ =+1 ( | ~q | ) . (27)The CQM evaluation of the vector and axial matrix ele-ments V µλ ( | ~q | ) and A µλ ( | ~q | ) can be found in the Appendix.For a B meson at rest and neglecting the neutrinomass, we have the double differential decay width d Γ dq dx l = G F m B | V bc | π λ / ( q , m B , m D )2 m B q − m l q × H αβ ( P B , P D ) L αβ ( p l , p ν ) , (28)where x l is the cosine of the angle between the final me-son momentum and the momentum of the final chargedlepton measured in the lepton-neutrino center of massframe. G F = 1 . × − GeV − is the Fermiconstant [4], m l is the charged lepton mass, λ ( a, b, c ) =( a + b − c ) − ab and V bc is the bc element of the Cab-bibo - Kobayashi - Maskawa matrix for which we shall use V bc = 0 . H αβ and L αβ represent the hadron andlepton tensors. P B , P D , p l and p ν are the meson andlepton momenta.Working in the helicity formalism of Ref. [22] and afterintegration on x l we have d Γ dq = G F π | V bc | ( q − m l ) m B q λ / ( q , m B , m D )2 m B × ( H U + H L + ˜ H U + ˜ H L + ˜ H S ) , (29)where the suffixes U, L, S stand for the unpolarized-transverse, longitudinal and scalar components of thehadronic tensor, and ˜ H = m l q H . Integrating over q we obtain the total decay width that can be written asΓ = Γ U + Γ L + ˜Γ U + ˜Γ L + ˜Γ S , (30)with Γ J and ˜Γ J partial helicity widths defined asΓ J = Z dq G F π | V bc | ( q − m l ) m B q λ / ( q , m B , m D )2 m B H J (31) and similarly for ˜Γ J in terms of ˜ H J . The evaluationof the different form factors, and thus of the differenthelicity amplitudes of the hadronic tensor, has been donefollowing Ref. [23]. IV. STRONG DECAYS
Meson strong decay is a complex nonperturbative pro-cess that has not yet been described from QCD first prin-ciples. Instead, several phenomenological models havebeen developed to deal with this topic, the P [6], theflux-tube [7], and the Cornell [8, 9] models being the mostpopular.Some models describe the decay process assuming thatthe extra quark-antiquark pair is created from the vac-uum. This is the case of the P model, which borrowsits name from the quantum numbers of the created pair,or the flux-tube model, which in addition to the creationvertex incorporates the overlaps between the color fluxtubes of the initial and final states.To address a more fundamental description of the de-cay mechanism, one has to describe hadron strong decaysin terms of quark and gluon degrees of freedom. However,there has been little previous work in this area. Two dif-ferent examples are the study of open-charm decays of c ¯ c resonances by Eichten et al. [8], who assumed that thedecays are due to pair production from the static part ofa Lorentz vector confining interaction, and the study of afew strong decays in the light sector by Ackleh et al. [9],where the q ¯ q pair production comes from the one-gluonexchange and a scalar confining interaction.As we mentioned in the introduction, we shall use boththe P model and a microscopic one, resembling those ofRefs. [8] and [9], that originates from the different inter-action pieces present in our interquark potential. Thesetwo approaches to meson production are introduced inthe following subsections. A. The P model It was first proposed by Micu [6] and further devel-oped by Le Yaouanc et al. [24]. To describe the mesondecay process A → B + C , the P model assumes that aquark-antiquark pair is created with vacuum J P C = 0 ++ quantum numbers. The created q ¯ q pair together with the q ¯ q pair present in the original meson regroup in the twooutgoing mesons via a quark rearrangement process.The interaction Hamiltonian which describes the pro-duction process is given by [9] H I = g Z d x ¯ ψ ( ~x ) ψ ( ~x ) (32)where g is related to the dimensionless constant givingthe strength of the q ¯ q pair creation from the vacuumas γ = g m q , m q being the mass of the created quark.Note that the operator g ¯ ψψ leads to the decay ( q ¯ q ) A → ( q ¯ q ) B + ( q ¯ q ) C through the a † b † term. B. The microscopic model
In microscopic decay models one attempts to describehadron strong decays in terms of quark and gluon degrees of freedom. The quark-gluon decay mechanism shouldgive similar predictions to the reasonably accurate P model and should determine the strength of the q ¯ q paircreation, γ , of the P model in terms of more fundamen-tal parameters.Following Ref. [9], the strong decays should be drivenby the same interquark Hamiltonian which determinesthe spectrum, the one-gluon exchange, and the confin-ing interaction appearing as the kernels. These inter-actions and their associated decay amplitudes are un-doubtedly all present and should be added coherently.We already mentioned that our constituent quark modelfor the heavy quark sector has a one-gluon exchangeterm and a mixture of Lorentz scalar and vector confin-ing interactions. This completely defines our microscopicmodel for strong decays. Unlike previous works we usea screening confinement interaction and also a mixturebetween scalar and vector Lorentz structures, which isalready fixed.The Hamiltonian of the interaction can be written as H I = 12 Z d xd y J a ( ~x ) K ( | ~x − ~y | ) J a ( ~y ) . (33)The current J a in Eq. (33) is assumed to be a color octet.The currents, J , with the color dependence λ a / K ( r ), for the interactions are • Currents J ( ~x ) = ¯ ψ ( ~x ) Γ ψ ( ~x ) = ¯ ψ ( ~x ) I ψ ( ~x ) Scalar Lorentz current,¯ ψ ( ~x ) γ ψ ( ~x ) Static part of vector Lorentz current,¯ ψ ( ~x ) ~γ ψ ( ~x ) Spatial part of vector Lorentz current, (34) • Kernels K ( r ) = − a s [ − a c (1 − e − µ c r ) + ∆] Confining interaction,+ α s r Color Coulomb OGE, − α s r Transverse OGE. (35)For the Lorentz vector structure of the confinementwe use K ( r ) = ± (1 − a s )4 [ − a c (1 − e − µ c r ) + ∆], where ± refers to static and transverse terms, respectively. Werefer, following Ref. [9], to this general type of interac-tion as a JKJ decay model, and to the specific casesconsidered here as sKs , j Kj and j T Kj T interactions.The wave functions for the mesons involved in the reac-tions are the solutions of the Schr¨odinger equation usingthe Gaussian Expansion Method mentioned above. De-tails of the resulting matrix elements for different casesare given in Ref. [25]. C. Strong decay width
The total width is the sum over the partial widthscharacterized by the quantum numbers J BC and l Γ A → BC = X J BC ,l Γ A → BC ( J BC , l ) (36)whereΓ A → BC ( J BC , l ) = 2 π Z dk δ ( E A − E BC ) |M A → BC ( k ) | (37)and M A → BC ( k ) is calculated according to Refs. [25, 26].Using relativistic phase space, we arrive atΓ A → BC ( J BC , l ) = 2 π E B E C m A k |M A → BC ( k ) | , (38)where k = λ / ( m A , m B , m C )2 m A (39)is the on shell relative momentum of mesons B and C . V. RESULTS
For the low-lying positive parity excitations, any quarkmodel predicts four states that in the S +1 L J basis cor-respond to P , P , P and P . As charge conjugationis not well defined in the heavy-light sector, P and P states can mix under the interaction.In the infinite heavy quark mass limit, heavy quarksymmetry (HQS) predicts two degenerated P -wave me-son doublets, labeled by j q = 1 / J P = 0 + , + ( | / , + i , | / , + i ) and j q = 3 / J P = 1 + , + ( | / , + i , | / , + i ). In this limit, the meson propertiesare governed by the dynamics of the light quark, which ischaracterized by its total angular momentum j q = s q + L ,where s q is the light quark spin and L the orbital angularmomentum. The total angular momentum of the meson J is obtained coupling j q to the heavy quark spin, s Q .Moreover, in the infinite heavy quark mass limit thestrong decays of the D J ( j q = 3 /
2) proceed only through D -waves, while the D J ( j q = 1 /
2) decays happen onlythrough S -waves [27]. The D -wave decay is suppressedby the barrier factor which behaves as q L +1 where q is the relative momentum of the two decaying mesons.Therefore, the states decaying through D -waves are ex-pected to be narrower than those decaying via S -waves.A change of basis allows to express the above statesin terms of the S +1 L J basis, by recoupling angular mo-menta, as | / , + i = + | P i| / , + i = + r | P i + r | P i| / , + i = − r | P i + r | P i| / , + i = + | P i (40)where in the S +1 L J wave functions we couple heavy andlight quark spins, in this order, to total spin S .In the actual calculation the ideal mixing in Eq. (40)between P and P states changes due to finite charmquark mass effects. Our CQM model predicts the mixedstates shown in Table III, which are very similar to theHQS states. This is expected since the c - quark is muchheavier ( m c = 1763 MeV) than the light ( m n = 313 MeV)or strange ( m s = 555 MeV) quarks. Note that now we D ∗ D D ′ D ∗ P + , . P - − , . − , . P - + , . − , . P - - - + , . / , + + , . / , + - + , . − , . / , + - + , . , . / , + - - - + , . D ∗ s D s D ′ s D ∗ s P + , . P - − , . − , . P - + , . − , . P - - - + , . / , + + , . / , + - − , . − , . / , + - + , . − , . / , + - - - + , . + strange sector the effects ofnon- q ¯ q components are included, see text for details. have mixing, even if small, between the P and F par-tial waves in 2 + mesons. This is due to the OGE tensorterm.In Ref. [15] we have studied the J P = 1 + charmed-strange mesons, finding that the J P = 1 + D s (2460)has an important non- q ¯ q contribution whereas the J P =1 + D s (2536) is almost a pure q ¯ q state. The presenceof non- q ¯ q degrees of freedom in the J P = 1 + charmed-strange meson sector enhances the j q = 3 / D s (2536). This wave function explains most ofthe experimental data, as shown in Ref. [15], and it isthe one we shall use here. For this sector only the q ¯ q probabilities are given in Table III. A. B semileptonic decays into D ∗∗ mesons
1. Semileptonic B → D ∗ (2400) lν l decay The measured branching fractions are B ( B + → ¯ D ∗ l + ν l ) B ( ¯ D ∗ → D − π + ) and B ( B → D ∗− l + ν l ) B ( D ∗− → ¯ D π − ). The meson D ∗ (2400) has J P = 0 + quantum numbers and, therefore,due to parity conservation, it decays only into Dπ , so thatwe have B ( ¯ D ∗ → D − π + ) = B ( D ∗− → ¯ D π − ) = 2 / L while the rest are negligible. Thedifference between the semileptonic width of the charged0 B + → ¯ D ∗ l + ν l B → D ∗− l + ν l Γ U .
00 0 . U .
00 0 . L .
30 1 . L . × − . × − ˜Γ S . × − . × − Γ 1 .
30 1 . − GeV, for the D ∗ meson. B + → ¯ D ′ l + ν l B → D ′ − l + ν l Γ U .
23 0 . U . × − . × − Γ L .
56 0 . L . × − . × − ˜Γ S . × − . × − Γ 0 .
79 0 . − GeV, for the D ′ meson. and neutral B meson is due to the large mass differencebetween the D ∗ and D ∗± mesons for which we take themasses reported in Ref. [4].Figure 1 shows the q dependence in the form factorsand in the differential decay width for B ( B + → ¯ D ∗ l + ν l ),panels (a) and (b), respectively. Similar results (notshown) are obtained for the B ( B → D ∗− l + ν l ) case.The final results for the product of branching fractionsare B ( B + → ¯ D ∗ l + ν l ) B ( ¯ D ∗ → D − π + ) = 2 . × − , B ( B → D ∗− l + ν l ) B ( D ∗− → ¯ D π − ) = 1 . × − , (41)which compare very well with Belle data [1].
2. Semileptonic B → D ′ (2430) lν l decay The only Okubo - Zweig - Iizuka (OZI)-allowed decaychannel for the D ′ meson is D ′ → D ∗ π so thatisospin symmetry predicts a branching fraction B ( D ′ → D ∗ π ± ) = 2 / B + → ¯ D ′ l + ν l and B → D ′ − l + ν l calculated in the framework of the CQM. In thiscase, Γ U and Γ L are of the same order of magnitude andgive the total semileptonic decay rate.Panels ( a ) and ( b ) of Fig. 2 show the q dependence ofthe form factors and the differential decay width for theneutral D ′ channel. A very similar result is obtained forthe D ∗ case. B + → D l + ν l B → D − l + ν l Γ U .
38 0 . U . × − . × − Γ L .
17 1 . L . × − . × − ˜Γ S . × − . × − Γ 1 .
55 1 . − GeV, for the D meson. We have in this case the product of branching fractions B ( B + → ¯ D ′ l + ν l ) B ( ¯ D ′ → D ∗− π + ) = 1 . × − , B ( B → D ′ − l + ν l ) B ( D ′ − → ¯ D ∗ π − ) = 1 . × − . (42)which are a rough factor of 2 smaller than the resultsfrom the BaBar Collaboration [2].
3. Semileptonic B → D (2420) lν l decay As in the previous case, the branching fraction B ( D → D ∗ π ± ) is again 2 / D → D ∗ π isthe only OZI-allowed decay channel.Table VI shows the different helicity contributions tothe semileptonic width of the reactions B + → ¯ D l + ν l and B → ¯ D − l + ν l . The most important contribution isgiven by Γ L . The ratio Γ L / Γ = 0 .
75 gives the probabilityfor the final D meson to have helicity 0. This result isin agreement with the values 0 . − .
81 obtained in theHQET calculation of Ref. [21].Figure 3 shows the q dependence of the form factorsand the differential decay width for neutral D channel,in panels ( a ) and ( b ), respectively. Again, a very similarresult is obtained for the charged case.The product of branching fractions are B ( B + → ¯ D l + ν l ) B ( ¯ D → D ∗− π + ) = 2 . × − , B ( B → D − l + ν l ) B ( D − → ¯ D ∗ π − ) = 2 . × − , (43)which in this case compare very well with the latestBaBar data [3].
4. Semileptonic B → D ∗ lν l decay The semileptonic decay is studied by reconstructingthe decay channel D ∗ → D ( ∗ ) π − , using the decaychain D ∗ → D π for D ∗ meson and D → K − π + or D + → K − π + π + for D meson. What is actu-ally measured is the product of branching fractions B ( B + → ¯ D ∗ l + ν l ) B ( ¯ D ∗ → D − π + ) and B ( B + → ¯ D ∗ l + ν l ) B ( ¯ D ∗ → D ∗− π + ).1 (GeV ) (a) F + -F - d Γ / dq ( - G e V - ) q (GeV ) (b) FIG. 1. Form factors and differential decay widths for the B + → ¯ D ∗ l + ν l decay as a function of q . Very similar results areobtained for the B → D ∗− l + ν l decay. (a) : Form factors predicted by CQM. (b) : Differential decay width predicted by CQM. -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 6 7 8q (GeV ) (a) -V-A + A - -A d Γ / dq ( - G e V - ) q (GeV ) (b) FIG. 2. Form factors and differential decay widths for the B + → ¯ D ′ l + ν l decay as a function of q . Very similar results areobtained for the B → D ′ − l + ν l decay. (a) : Form factors predicted by CQM. (b) : Differential decay width predicted by CQM. The first step of this decay involves a semileptonic pro-cess which can be calculated using Eq. (30). In Table VIIwe show the different helicity contributions to the totalwidth. The main contribution is Γ L in both neutral andcharged D ∗ channels, providing almost 2 / U , the rest of the contri-butions being negligible. Again our ratio Γ L / Γ = 0 . . − .
64 obtained inRef. [21] using HQET.Figure 4 shows the q dependence in the form factorsand in the differential decay width, panels (a) and (b)respectively, for the B + → ¯ D ∗ l + ν l decay. Very similarresults (not shown) are obtained for the B → D ∗− l + ν l B + → D ∗ l + ν l B → D ∗− l + ν l Γ U .
44 0 . U . × − . × − Γ L .
90 0 . L . × − . × − ˜Γ S . × − . × − Γ 1 .
34 1 . − GeV, for the D ∗ meson. -0.4-0.2 0 0.2 0.4 0.6 0 1 2 3 4 5 6 7 8q (GeV ) (a) -V-A + A - -A d Γ / dq ( - G e V - ) q (GeV ) (b) FIG. 3. Form factors and differential decay widths for the B + → D l + ν l decay as a function of q . The differences with respect B → D − l + ν l are negligible. (a) : Form factors predicted by CQM. (b) : Differential decay width predicted by CQM.Branching ratio Exp. P MicroscopicΓ( D π + ) / Γ( D ∗ π + ) 1 . ± . ± . .
80 1 . D + π − ) / Γ( D ∗ + π − ) 1 . ± .
16 1 .
82 1 . D + π − ) / Γ( D ( ∗ )+ π − ) 0 . ± . ± .
02 0 .
65 0 . D ∗ decays collected by thePDG [4] and our theoretical results calculated through thetwo strong decay models. case.The subsequent strong decays which appear are D ∗ → D ∗ π − and D ∗ → Dπ − . In Table VIII we show the strongdecay branching ratios obtained with the P and micro-scopic models. They are in good agreement with experi-mental data [4].Finally, we obtain the products of branching fractionsfor both decay chains considering that the total width ofthe D ∗ meson is the sum of the partial widths of D ∗ π and Dπ channels since these are the only OZI-allowedprocesses B ( B + → D ∗ l + ν l ) B ( D ∗ → D + π − ) = (cid:26) . × − . × − B ( B + → D ∗ l + ν l ) B ( D ∗ → D ∗ + π − ) = (cid:26) . × − . × − B ( B → D ∗− l + ν l ) B ( D ∗− → D π − ) = (cid:26) . × − . × − B ( B → D ∗− l + ν l ) B ( D ∗− → D ∗ π − ) = (cid:26) . × − . × − (44)where the first one refers to the calculation using the P model and the second one comes from the microscopicmodel. These results are in very good agreement withBaBar data [3].
5. Summary of the results
Final results and their comparisons with the exper-imental data are given in Table IX. Except for the D ′ (2430), the predictions are in very good agreementwith the latest experimental measurements, Belle for D (2400) and BaBar for D (2420) and D ∗ (2460). Forthe D ′ (2430) there is also a strong disagreement betweenexperimental data in the neutral case. B. B s semileptonic decays into D ∗∗ s mesons The semileptonic decays of B s meson into orbitally ex-cited P -wave charmed-strange mesons ( D ∗∗ s ) provides anextra opportunity to get more insight into this system.The j q = 1 / D ∗ s (2318) and D s (2460),shows surprisingly light masses which are below the DK and D ∗ K thresholds, respectively. These unexpectedproperties have triggered many theoretical interpreta-tions, including four quark states, molecules, and thecoupling of the q ¯ q components with different structures.As mentioned before, the D s (2460) meson has an im-portant non- q ¯ q contribution.We have calculated the semileptonic B s decays as-suming that the D ∗∗ s mesons are pure q ¯ q systems. Forthe D ∗ s (2318) and D s (2460), which are below the cor-responding D ( ∗ ) K thresholds, we only quote the weakdecay branching fractions. Concerning the D s (2460),and as shown in Ref. [15], the P and P probabilities3 (GeV ) (a) -T
100 x T (GeV -2 )-100 x T (GeV -2 ) 100 x T (GeV -2 ) d Γ / dq ( - G e V - ) q (GeV ) (b) FIG. 4. Form factors and differential decay widths for the B + → D ∗ l + ν l decay as a function of q . Very similar results areobtained for the B → D ∗− l + ν l decay. (a) : Form factors predicted by CQM. (b) : Differential decay width predicted by CQM.Belle [1] BaBar [2, 3] P Mic.( × − ) ( × − ) ( × − ) ( × − ) D ∗ (2400) B ( B + → ¯ D ∗ l + ν l ) B ( ¯ D ∗ → D − π + ) 2 . ± . ± . . ± . ± . .
15 2 . B ( B → D ∗− l + ν l ) B ( D ∗− → ¯ D π − ) 2 . ± . ± . . ± . ± . .
80 1 . D ′ (2430) B ( B + → ¯ D ′ l + ν l ) B ( ¯ D ′ → D ∗− π + ) < . . ± . ± . .
32 1 . B ( B → D ′ − l + ν l ) B ( D ′ − → ¯ D ∗ π − ) < . ± . ± . .
23 1 . D (2420) B ( B + → ¯ D l + ν l ) B ( ¯ D → D ∗− π + ) 4 . ± . ± . . ± . ± .
17 2 .
57 2 . B ( B → D − l + ν l ) B ( D − → ¯ D ∗ π − ) 5 . ± . ± . . ± . ± .
25 2 .
39 2 . D ∗ (2460) B ( B + → ¯ D ∗ l + ν l ) B ( ¯ D ∗ → D − π + ) 2 . ± . ± . . ± . ± . ( ∗ ) .
43 1 . B ( B + → ¯ D ∗ l + ν l ) B ( ¯ D ∗ → D ∗− π + ) 1 . ± . ± . . ± . ± . ( ∗ ) .
79 0 . B ( B + → ¯ D ∗ l + ν l ) B ( ¯ D ∗ → D ( ∗ ) − π + ) 4 . ± . ± . . ± . ± . .
22 2 . B ( B → D ∗− l + ν l ) B ( D ∗− → ¯ D π − ) 2 . ± . ± . . ± . ± . ( ∗ ) .
34 1 . B ( B → D ∗− l + ν l ) B ( D ∗− → ¯ D ∗ π − ) < . ± . ± . ( ∗ ) .
74 0 . B ( B → D ∗− l + ν l ) B ( D ∗− → ¯ D ( ∗ )0 π − ) < . . ± . ± . .
08 2 . B D/D ( ∗ ) . ± .
03 0 . ± . ± .
02 0 .
65 0 . ∗ ) indicates the estimated results from the original data using B D/D ( ∗ ) . change with the coupling to non- q ¯ q degrees of freedom.What we do here is to vary these probabilities (includ-ing the phase) in order to obtain the limits of the decaywidth in the case of the D s (2460) being a pure q ¯ q state,see Fig. 5. Assuming that non- q ¯ q components will givea small contribution to the weak decay, experimental re-sults lower than these limits will be an indication of a more complex structure for this meson.For the decay into D s (2536), our model pre-dicts the weak decay branching fraction B ( B s → D s (2536) µ + ν µ ) = 4 . × − and the strong branch-ing fractions B ( D s (2536) − → D ∗− ¯ K ) = 0 .
43 (0 .
47) forthe P (microscopic) models. The final result appearsin Table X. It is in good agreement with the existing4 Experiment Theory( × − ) ( × − ) D ∗ s (2318) B ( B s → D ∗ s (2318) − µ + ν µ ) - 4 . D s (2460) B ( B s → D s (2460) − µ + ν µ ) - 1 . − . D s (2536) P Mic. B ( B s → D s (2536) − µ + ν µ ) B ( D s (2536) − → D ∗− ¯ K ) 2 . ± . .
05 2 . D ∗ s (2573) P Mic. B ( B s → D ∗ s (2573) − µ + ν µ ) B ( D ∗ s (2573) − → D − ¯ K ) - 1 .
70 1 . B ( B s → D ∗ s (2573) − µ + ν µ ) B ( D ∗ s (2573) − → D ∗− ¯ K ) - 0 .
18 0 . B ( B s → D ∗ s (2573) − µ + ν µ ) B ( D ∗ s (2573) − → D ( ∗ ) − ¯ K ) - 1 .
88 1 . B s decays into orbitallyexcited charmed-strange mesons. Γ ( - G e V ) P( P ) FIG. 5. Decay width for the B s → D s (2460) − µ + ν µ decayas a function of the P component probability. The signreflects the relative phase between P and P components:-1 opposite phase and +1 same phase. experimental data [4], which to us is a confirmation ofour former result in Ref. [15] about the q ¯ q nature of thisstate.In the case of the D ∗ s (2573) the open strongdecays are DK and D ∗ K , so the experimen-tal measurements must be referred to B ( B s → D ∗ s (2573) − µ + ν µ ) B ( D ∗ s (2573) − → D − ¯ K ) and B ( B s → D ∗ s (2573) − µ + ν µ ) B ( D ∗ s (2573) − → D ∗− ¯ K ).For the weak branching fraction we get in this case B ( B s → D ∗ s (2573) − µ + ν µ ) = 3 . × − . For the strong decay part of the reaction, we obtain in our model B ( D ∗− s → D − ¯ K ) = ( . . , B ( D ∗− s → D ∗− ¯ K ) = ( . . , (45)where the first one refers to the calculation using the P model and the second one comes from the microscopicmodel. Our final results can be seen in Table. X.Besides we predict the ratioΓ( D ∗ s → DK )Γ( D ∗ s → DK ) + Γ( D ∗ s → D ∗ K ) = ( . P .
94 Mic . (46) VI. CONCLUSIONS
We have performed a calculation of the branching frac-tions for the semileptonic decays of B and B s mesonsinto final states containing orbitally excited charmed andcharmed-strange mesons, respectively.We worked in the framework of the constituent quarkmodel of Ref. [10]. The model parameters were fitted tothe meson spectra in Refs. [10, 14]. Our meson states areclose to the ones predicted by HQS as expected.We have calculated the semileptonic decay rates withinthe helicity formalism of Ref. [22] and following the workin Ref. [23]. The strong decay widths have been calcu-lated using two models, the P model and a microscopicmodel based on the quark-antiquark interactions presentin the CQM model of Ref. [10].From the experimental point of view, Belle and BaBarCollaborations provide their most recent measurementsfor the B meson in Refs. [1] and [2, 3] respectively. Forthe B s meson only the product of branching fractions5 B ( B s → D s (2536) − µ + ν µ ) B ( D s (2536) − → D ∗− ¯ K )has been determined [4] using the experimental data on B (¯ b → B s ) B ( B s → D s (2536) − µ + ν µ ) B ( D s (2536) − → D ∗− ¯ K ) measured by the D0 Collaboration [5] and thePDG’s best value for B (¯ b → B s ) [4].Our results for B semileptonic decays into D ∗ (2400), D (2420) and D (2460) are in good agreement withthe latest experimental measurements. In the case of D ′ (2430) the prediction lies a factor of 2 below BaBardata. Note however the disagreement between BaBarand Belle data for the neutral case.In the case of B s semileptonic decays, our predic-tion for the B ( B s → D s (2536) − µ + ν µ ) B ( D s (2536) − → D ∗− ¯ K ) product of branching fractions is in agreementwith the experimental data. This, together with theproperties calculated in Ref. [15], is to us evidence of a dominant q ¯ q structure for the D s (2536) meson. Wealso give predictions for decays into other D ∗∗ s mesonswhich can be useful to test the q ¯ q nature of these states. ACKNOWLEDGMENTS
This work has been partially funded by the SpanishMinisterio de Ciencia y Tecnolog´ıa under Contracts Nos.FIS2006-03438, FIS2009-07238 and FPA2010-21750-C02-02, by the Spanish Ingenio-Consolider 2010 Pro-grams CPAN CSD2007-00042 and MultiDark CSD2009-0064, and by the European Community-Research Infras-tructure Integrating Activity ’Study of Strongly Interact-ing Matter’ (HadronPhysics2 Grant No. 227431). C. A.thanks a Juan de la Cierva contract from the SpanishMinisterio de Educaci´on y Ciencia.
Appendix A: Form factor decomposition of hadronic matrix elements
Here we give general expressions valid for transitions between a pseudoscalar meson M I at rest with quark content¯ q f q f and a final M F meson with total angular momentum and parity J P = 0 + , + , + , three-momentum −| ~q | ~k ,and quark content ¯ q f ′ q f . The transition changes the antiquark flavor. Following Ref. [23] we evaluate V µλ ( | ~q | ) and A µλ ( | ~q | ) in the CQM through the relations V µλ ( | ~q | ) = p m I E F ( − ~q ) NR h M F , λ − | ~q | ~k | J bcµV (0) | M I ,~ i NR A µλ ( | ~q | ) = p m I E F ( − ~q ) NR h M F , λ − | ~q | ~k | J bcµA (0) | M I ,~ i NR (A1)For the different cases under study we will have the following.
1. Case − → + A ( | ~q | ) = p m I E F ( − ~q ) Z d p π | ~p | (cid:16) ˆ φ ( M (0 + )) f ′ f ( | ~p | ) (cid:17) ∗ ˆ φ ( M (0 − )) f f (cid:18) | ~p − m f m f ′ + m f q~k | (cid:19)s ˆ E f ′ ˆ E f E f ′ E f ~p · (cid:18) m f m f ′ + m f | ~q | ~k − ~p (cid:19) ˆ E f + ~p · (cid:18) − m f ′ m f ′ + m f | ~q | ~k − ~p (cid:19) ˆ E f ,A ( | ~q | ) = p m I E F ( − ~q ) × Z d p π | ~p | (cid:16) ˆ φ ( M (0 + )) f ′ f ( | ~p | ) (cid:17) ∗ ˆ φ ( M (0 − )) f f (cid:18) | ~p − m f m f ′ + m f q~k | (cid:19) s ˆ E f ′ ˆ E f E f ′ E f × ( p z − (cid:18) − m f ′ m f ′ + m f | ~q | ~k − ~p (cid:19) · (cid:18) m f m f ′ + m f | ~q | ~k − ~p (cid:19) ˆ E f ′ ˆ E f + 1ˆ E f ′ ˆ E f (cid:20)(cid:18) − m f ′ m f ′ + m f | ~q | − p z (cid:19) × ~p · (cid:18) m f m f ′ + m f | ~q | ~k − ~p (cid:19) + (cid:18) m f m f ′ + m f | ~q | − p z (cid:19) ~p · (cid:18) − m f ′ m f ′ + m f | ~q | ~k − ~p (cid:19)(cid:21) ) . (A2)6 E f ′ and E f are shorthand notations for E f ′ ( − m f ′ m f ′ + m f | ~q | ~k − ~p ) and E f ( m f m f ′ + m f | ~q | ~k − ~p ) respectively and ˆ E f = E f + m f .
2. Case − → + Here we have to distinguish two different cases that depend on the total spin S of the quark-antiquark system.i) Case S = 0 V (1 + ,S =0)0 λ =0 ( | ~q | ) = − i √ p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M F (1 + ,S =0)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M I (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) p z (cid:18) − m f ′ m f ′ + m f | ~q | ~k − ~p (cid:19) · (cid:18) m f m f ′ + m f | ~q | ~k − ~p (cid:19) ˆ E f ′ ˆ E f ,V (1 + ,S =0)1 λ = − ( | ~q | ) = i r p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M F (1 + ,S =0)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M I (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) p x E f + 1ˆ E f ′ ! ,V (1 + ,S =0)3 λ =0 ( | ~q | ) = − i √ p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M F (1 + ,S =0)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M I (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) p z m f m f ′ + m f | ~q | − p z ˆ E f − m f ′ m f ′ + m f | ~q | + p z ˆ E f ′ ,A (1 + ,S =0)1 λ = − ( | ~q | ) = − i r p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M F (1 + ,S =0)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M I (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) p y | ~q | ˆ E f ˆ E f ′ . (A3)ii) Case S = 17 V (1 + ,S =1)0 λ =0 ( | ~q | ) = i r p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M F (1 + ,S =1)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M I (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) | ~q | ( p z − p )ˆ E f ′ ˆ E f ,V (1 + ,S =1)1 λ = − ( | ~q | ) = − i √ p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M F (1 + ,S =1)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M I (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) p y + p z + p z | ~q | m f ′ m f ′ + m f ˆ E f ′ − p y + p z − p z | ~q | m f m f ′ + m f ˆ E f ,V (1 + ,S =1)3 λ =0 ( | ~q | ) = i r p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M F (1 + ,S =1)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M I (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) ( p x + p y ) E f − E f ′ ! ,A (1 + ,S =1)1 λ = − ( | ~q | ) = i √ p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M F (1 + ,S =1)) f ′ f ( p ) (cid:17) ∗ ˆ φ ( M I (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) × p z − (cid:18) − m f ′ m f ′ + m f | ~q | ~k − ~p (cid:19) · (cid:18) m f m f ′ + m f | ~q | ~k − ~p (cid:19) ˆ E f ′ ˆ E f + m f − m f ′ m f ′ + m f p x | ~q | ˆ E f ′ ˆ E f . (A4)8
3. Case − → + Here we have to distinguish between L = 1 and L = 3.i) Case L = 1 V (2 + ,L =1)1 λ =+1 ( | ~q | ) = − i √ p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M F (2 + ,L =1)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M I (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) p y − p z − p z | ~q | m f ′ m f ′ + m f ˆ E f ′ − p y − p z + p z | ~q | m f m f ′ + m f ˆ E f ,A (2 + ,L =1)0 λ =0 ( | ~q | ) = − i √ p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M F (2 + ,L =1)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M I (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) p x + p y − p z − p z | ~q | m f ′ m f ′ + m f ˆ E f ′ + p x + p y − p z + 2 p z | ~q | m f m f ′ + m f ˆ E f ,A (2 + ,L =1)1 λ =+1 ( | ~q | ) = i √ p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M F (2 + ,L =1)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M I (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) p z − (cid:18) − m f ′ m f ′ + m f | ~q | ~k − ~p (cid:19) · (cid:18) m f m f ′ + m f | ~q | ~k − ~p (cid:19) ˆ E f ′ ˆ E f + 4 p z p x − p x | ~q | m f − m f ′ m f ′ + m f ˆ E f ′ ˆ E f ,A (2 + ,L =1)3 λ =0 ( | ~q | ) = − i √ p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M F (2 + ,L =1)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M I (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) ( p z " − (cid:18) − m f ′ m f ′ + m f | ~q | ~k − ~p (cid:19) · (cid:18) m f m f ′ + m f | ~q | ~k − ~p (cid:19) ˆ E f ′ ˆ E f + 1ˆ E f ′ ˆ E f (cid:20) p z (cid:18) − m f ′ m f ′ + m f | ~q | − p z (cid:19) (cid:18) m f m f ′ + m f | ~q | − p z (cid:19) +( p x + p y ) (cid:18) − p z + m f − m f ′ m f ′ + m f ) | ~q | (cid:19)(cid:21) ) . (A5)ii) Case L = 39 V (2 + ,L =3)1 λ =+1 ( | ~q | ) = i √ p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M (2 + ,L =3)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) × " E f p (cid:16) p y − p z (cid:16) m f m f ′ + m f | ~q | − p z (cid:17)(cid:17) + 5 p z (cid:16) − p y p z + (cid:16) m f m f ′ + m f | ~q | − p z (cid:17) ( p x − p y + p z ) (cid:17)! + 1ˆ E f ′ p (cid:16) − p y + 3 p z (cid:16) − m f ′ m f ′ + m f | ~q | − p z (cid:17)(cid:17) − p z (cid:16) − p y p z + (cid:16) − m f ′ m f ′ + m f | ~q | − p z (cid:17) ( p x − p y + p z ) (cid:17)! ,A (2 + ,L =3)0 T λ =0 ( | ~q | ) = − i r p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M (2 + ,L =3) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) " (cid:18) p z p − (cid:19) p x + p y ˆ E f + p x + p y ˆ E f ′ ! − p z p (cid:18) p z p − (cid:19) m f m f ′ + m f | ~q | − p z ˆ E f − m f ′ m f ′ + m f | ~q | + p z ˆ E f ′ ! ,A (2 + ,L =3)3 λ =0 ( | ~q | ) = − i p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M (2 + ,L =3)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) ( p x + p y ) (cid:18) p z p − (cid:19) m f − m f ′ m f ′ + m f | ~q | − p z ˆ E f ˆ E f ′ − p z (cid:18) p z p − (cid:19) − p x + p y − (cid:18) − m f ′ m f ′ + m f | ~q | − p z (cid:19) (cid:18) m f m f ′ + m f | ~q | − p z (cid:19) ˆ E f ˆ E f ′ ,A (2 + ,L =3)1 λ =+1 ( | ~q | ) = − i √ p m I E F ( − ~q ) Z d p s ˆ E f ′ ˆ E f E f ′ E f πp (cid:16) ˆ φ ( M (2 + ,L =3)) f ′ f ( p ) (cid:17) ∗ × ˆ φ ( M (0 − )) f f ( | ~p − m f m f ′ + m f | ~q | ~k | ) " p z + 3 p z p x − p y − (cid:18) − m f ′ m f ′ + m f | ~q | − p z (cid:19) (cid:18) m f m f ′ + m f | ~q | − p z (cid:19) ˆ E f ˆ E f ′ + 5 p z p x p + p y p − p z p ! p x − p y − (cid:18) − m f ′ m f ′ + m f | ~q | − p z (cid:19) (cid:18) m f m f ′ + m f | ~q | − p z (cid:19) ˆ E f ˆ E f ′ − p x (cid:18) p z p − (cid:19) m f − m f ′ m f ′ + m f | ~q | − p z ˆ E f ˆ E f ′ + 20 p z p x p y ˆ E f ˆ E f ′ p . 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