The Semileptonic decays of the B(c) meson

Semileptonic decays of the B

_c

meson

M. A. Ivanov

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia J. G. Ko¨rner

Johannes Gutenberg-Universita¨t, Institut fu¨r Physik, Staudinger Weg 7, D-55099, Mainz, Germany P. Santorelli

Dipartimento di Scienze Fisiche, Universita` di Napoli ‘‘Federico II,’’ Italy and INFN Sezione di Napoli, Via Cintia, I-80126 Napoli, Italy

共Received 17 July 2000; published 6 March 2001兲

We study the semileptonic transitions Bc→␩c, J/␺, D, D*, B, B*, Bs, Bs* in the framework of a

relativistic constituent quark model. We use experimental data on leptonic J/␺ decay, lattice and QCD sum rule results on leptonic Bcdecay, and experimental data on radiative␩ctransitions to adjust the quark model

parameters. We compute all form factors of the above semileptonic Bc transitions and give predictions for

various semileptonic Bc decay modes including their␶ modes when they are kinematically accessible. The

implications of heavy quark symmetry for the semileptonic decays are discussed and are shown to be manifest in our explicit relativistic quark model calculation. A comparison of our results with the results of other calculations is performed.

DOI: 10.1103/PhysRevD.63.074010 PACS number共s兲: 13.20.He, 12.39.Ki I. INTRODUCTION

Recently, the observation of the bottom-charm Bcmeson at the Fermilab Tevatron has been reported by the Collider Detector at Fermilab 共CDF兲 Collaboration 关1兴. The Bc me-sons were found in the analysis of their semileptonic decays,

B_c⫾→J/␺l⫾X. Values for the mass and the lifetime of the Bc meson were given as M (Bc)⫽6.40⫾0.39⫾0.13 GeV and

␶(Bc)⫽0.46_⫺0.16⫹0.18(stat)⫾0.03(syst) ps, respectively. The branching fraction for Bc→J/␺l␯ relative to that for Bc →J/␺K was found to be

␴共Bc兲⫻Br共Bc→J/␺l␯兲

␴共B兲⫻Br共Bc→J/␺K兲 ⫽0.132⫺0.037

⫹0.041_共stat兲

⫾0.031共syst兲⫺0.020⫹0.032.

The study of the Bcmeson is of great interest due to some of its outstanding features. It is the lowest bound state of two heavy quarks共charm and bottom兲 with open 共explicit兲 flavor that can be compared with the charmonium (cc¯ bound state兲 and the bottomium (bb¯ bound state兲 which have hidden 共im-plicit兲 flavor. The states with hidden flavor decay strongly and electromagnetically whereas the B_c meson decays weakly since it is below the BD¯ threshold. Naively it might appear that the weak decays of the Bc meson are similar to those of the B and D mesons. However, the situation is quite different. The new spin-flavor symmetry arises for the sys-tems containing one heavy quark when the mass of the heavy quark goes to infinity关2兴. It gives some relations between the form factors of the physical processes. The deviations from heavy quark symmetry are large for the D meson and negli-gibly small for the B meson. On the contrary, in the case of the Bc meson a consistent heavy quark effective theory共for

both constituent quarks兲 cannot include the heavy flavor symmetry关3兴. However, the residual heavy quark spin sym-metry can be used to reduce the number of independent semileptonic form factors at least near the zero recoil point 关4兴.

In the naive spectator model, one would expect that ⌫(Bc)⬇⌫(B)⫹⌫(D) which gives ␶(Bc)⬇0.3 ps, i.e. 1.5 times less than the central CDF value. The dominance of the

c→s transition will have to be investigated in future analysis

when more data become available. Thus a reliable evaluation of the long distance contributions is very important for studying the weak Bc decay properties.

The theoretical status of the Bc meson was reviewed in 关5兴. In this paper we focus on its exclusive leptonic and semileptonic decays which are sensitive to the description of long distance effects and are free of further assumptions, such as, for example, factorization of amplitudes in non-leptonic processes. Our results on the seminon-leptonic transition form factors can of course be used for a calculation of the nonleptonic decays of the Bc meson using the factorization approach.

The exclusive semileptonic and nonleptonic 共assuming factorization兲 decays of the Bcmeson were calculated before in a potential model approach 关6兴. The binding energy and the wave function of the B_cmeson were computed by using a flavor-independent potential with the parameters fixed by the cc¯ and bb¯ spectra and decays. The same processes were also studied in the framework of the Bethe-Salpeter equation in关7兴 and in the relativistic constituent quark model formu-lated on the light front in关8兴. Three-point sum rules of QCD and nonrelativistic QCD 共NRQCD兲 were analyzed in 关9,10兴 to obtain the form factors of the semileptonic decays of B_c⫹ →J/␺(␩c)l⫹␯ and B⫹c→Bs(Bs*)l⫹␯.

As shown by the authors of 关4兴, the form factors param-etrizing the B_c semileptonic matrix elements can be related

to a smaller set of form factors if the decoupling of the spin of the heavy quarks in Bcand in the mesons produced in the semileptonic decays is exploited. The reduced form factors can be evaluated as overlap integral of the meson wave func-tions obtained, for example, using a relativistic potential model. This was performed in 关11兴, where the Bc semilep-tonic form factors were computed and predictions for semi-leptonic and non-semi-leptonic decay modes were given.

In this paper we employ the relativistic constituent quark

model 共RCQM兲 关12兴 for the description of Bc semileptonic meson decays. The RCQM is based on an effective Lagrang-ian describing the coupling of hadrons H to their constituent quarks, the coupling strength of which is determined by the compositeness condition ZH⫽0 关13兴 where ZH is the wave function renormalization constant of the hadron H. Here Z_H1/2 is the matrix element between a physical particle state and the corresponding bare state. The compositeness condition

ZH⫽0 enables us to represent a bound state by introducing a

quasiparticle interacting with its constituents so that the renormalization factor is equal to zero. This does not mean that we can solve the QCD bound state equations but we are able to show that the condition ZH⫽0 provides an effective and self-consistent way to describe the coupling of the par-ticle to its constituents. One starts with an effective Lagrang-ian written down in terms of quark and hadron variables. Then, by using Feynman rules, the S-matrix elements de-scribing hadronic interactions are given in terms of a set of quark diagrams. In particular, the compositeness condition enables one to avoid a double counting of hadronic degrees of freedom. This approach is self-consistent and all calcula-tions of physical observables are straightforward. There is a small set of model parameters: the values of the constituent quark masses and the scale parameters that define the size of the distribution of the constituent quarks inside a given had-ron. This distribution can be related to the relevant Bethe-Salpeter amplitudes.

The shapes of the vertex functions and the quark propa-gators can in principle be found from an analysis of the Bethe-Salpeter and Dyson-Schwinger equations, respec-tively, as done e.g. in关15兴. The Dyson-Schwinger equation 共DSE兲 has been employed to entail a unified and uniformly accurate description of light- and heavy-meson observables 关16兴. In this paper we, however, choose a more phenomeno-logical approach where the vertex function is modeled by a Gaussian form, the size parameter of which is determined by a fit to the leptonic and radiative decays of the lowest lying charm and bottom mesons. For the quark propagators we use the local representation.

The leptonic and semileptonic decays of the lower-lying pseudoscalar mesons (␲, K, D, D_s, B, Bs兲 have been de-scribed in Ref. 关17兴 in which a Gaussian form was used for the vertex function and free propagators were adopted for the constituent quarks. The adjustable parameters, the widths of Bethe-Salpeter amplitudes in momentum space and the con-stituent quark masses were determined from a least squares fit to available experimental data and some lattice determi-nations. We found that our results are in good agreement with experimental data and other approaches. It was also

shown that the scaling relations resulting from the spin-flavor symmetries are reproduced by the model in the heavy quark limit.

Using this approach we have elaborated the so-called

relativistic three-quark model 共RTQM兲 to study the

proper-ties of heavy baryons containing a single heavy quark 共bot-tom or charm兲. For the heavy quarks we used propagators appropriate for the heavy quark limit. Physical observables for the semileptonic and nonleptonic decays as well as for the one-pion and one-photon transitions have been success-fully described in this approach 关18兴. Recently, the RTQM was extended to include the effects of finite quark masses 关19兴. We mention that the authors of 关20兴 have developed a relativistic quark model approach to the description of meson transitions which has similarities to our approach. They also use an effective heavy meson Lagrangian to describe the couplings of mesons to quarks. They use, however, point-like meson-quark interactions. Loop momenta are explicitly cut off at around 1 GeV in the approach 关20兴. In our ap-proach we use momentum dependent meson-quark interac-tions, which provides for an effective cutoff of the loop in-tegration. We would also like to mention a recent investigation 关21兴 where the same quark-meson Lagrangian employed in 关12兴 was used. The authors of 关21兴 employed dipole vertex to describe various leptonic and semileptonic decays of both the heavy-light mesons and the Bcmeson.

In this paper we follow the strategy adopted in Refs. 关16,17兴. The basic assumption on the choice of the vertex function in the hadronic matrix elements is made after tran-sition to momentum space. We employ the impulse approxi-mation in calculating these matrix elements which has been used widely in phenomenological DSE studies 共see, e.g., Ref. 关16兴兲. In the impulse approximation one assumes that the vertex functions depend only on the loop momentum flowing through the vertex. We present a general method which greatly facilitates the numerical evaluations that occur in the Feynman-type calculations involving quark loops 共see also 关12,17兴兲.

The basic emphasis of this work is to study leptonic and semileptonic decays of the Bc meson. We use Gaussian ver-tex functions with size parameters for heavy-light mesons as in Ref.关17兴. In this paper we limit our attention to the basic semileptonic decay modes of the B_cmeson. A new feature of our calculation is that we also discuss semileptonic decays involving the ␶ lepton. We discuss in some detail how our quark loop calculations reproduce the heavy quark limit re-lations between form factors at zero recoil. Explicit expres-sions for the reduced set of form factors in this limit are given.

II. MODEL

We employ an approach关12兴 based on the effective inter-action Lagrangian which describes the coupling between hadrons and their constituent quarks. For example, the cou-pling of the meson H into its constituents q1 and q2 is given

Lint共x兲⫽gHH共x兲

冕

dx1

冕

dx2⌽H共x,x1,x2兲

⫻q¯共x1兲⌫H␭Hq共x2兲. 共2.1兲

Here,␭Hand⌫_Hare Gell-Mann and Dirac matrices, respec-tively, which entail the flavor and spin quantum numbers of the meson H. The function⌽His related to the scalar part of Bethe-Salpeter amplitude and characterizes the finite size of the meson. ⌽H is invariant under the translation ⌽H(x ⫹a,x1⫹a,x2⫹a)⫽⌽H(x,x1,x2) which is necessary for the

Lorence invariance of the Lagrangian共2.1兲. For instance, the separable form

⌽H共x,x1,x2兲⫽␦

冉

x⫺ x1⫹x2

2

冊

f„共x1⫺x2兲

2_{… 共2.2兲}

has been used in关12兴 for pions with f (x2) being a Gaussian. The straightforward generalization of the vertex function 共2.2兲 to the case of an arbitrary pair of quarks with different masses is given by ⌽H共x,x1,x2兲⫽␦

冉

x⫺ m1x1⫹m2x2 m1⫹m2

冊

f„共x1⫺x2兲2…. 共2.3兲 The authors of关21兴 used a dipole form for the Fourier trans-form of the function f (x2). Here we follow the slightly dif-ferent strategy as proposed in Refs. 关16,17兴. The choice of the vertex function in the hadronic matrix elements is speci-fied after transition to momentum space. We employ the im-pulse approximation in calculating the one-loop transition amplitudes. In the impulse approximation one assumes that the vertex functions depend only on the loop momentum flowing through the vertex. The impulse approximation has been used widely in phenomenological DSE studies 共see, e.g., Ref.关16兴兲. The final results of calculating a quark loop diagram depend on the choice of loop momentum flow. In the heavy quark transitions discussed in this paper the loop momentum flow is, however, fixed if one wants to reproduce the heavy quark symmetry results.

To demonstrate our assumption, we consider the meson mass function defined by the diagram in Fig. 1. We have

⌸H共x⫺y兲⫽

冕

dx₁

冕

dx₂⌽H共x,x₁,x₂兲 ⫻

冕

d y1

冕

d y2⌽H共y,y1, y2兲

⫻tr兵S共y1⫺x1兲⌫HS共x2⫺y2兲⌫H其. 共2.4兲

Then we calculate the Fourier transform of the meson mass function共2.4兲: ⌸˜_H共p兲⫽

冕

_e⫺ipx_⌸H共x兲 ⫽

冕

dq 共2␲兲4

冕

dk1 共2␲兲4

冕

dk2 共2␲兲4⌽˜H ⫻共⫺p,k1,⫺k2兲⌽˜H共q,⫺k1,k2兲 ⫻tr兵S共k”₁兲⌫HS共k”₂兲⌫_H其. 共2.5兲 The Fourier transform of the function⌽(x1, . . . ,xn) which is invariant under the translation xi→xi⫹a can be written as ⌽˜_共q 1, . . . ,qn兲⫽

冕

dx1•••

冕

dxn ⫻exp

冉

i

兺

i_⫽1 n xiqi

冊

⌽共x1, . . . ,xn兲 ⫻共2␲兲4_␦

冉

兺

i⫽1 n qi

冊

n4

冕

dx1•••

冕

dxn ⫻␦

冉

兺

i⫽1 n xi

冊

exp

冉

i

兺

i⫽1 n xiqi

冊

⌽共x1, . . . ,xn兲 ⬅共2␲兲4_␦

冉

兺

i⫽1 n qi

冊

␾共q1, . . . ,qn_⫺1兲. 共2.6兲 Using this property one finds

⌸˜_H共p兲⫽

冕

dk

共2␲兲4␾H 2_共k,p兲tr

兵S共k”⫹p”兲⌫HS共k” 兲⌫H其. 共2.7兲 Here, we assume that the vertex function ␾_H depends only on the loop momentum k. Besides, we assume that ␾H is analytical function which decreases sufficiently fast in the Euclidean momentum space to render all loop diagrams UV finite.

The coupling constants gH are determined by the so-called compositeness condition proposed in 关13兴 and exten-sively used in关14兴. The compositeness condition means that the renormalization constant of the meson field is equal to zero, ZH⫽1⫺3gH 2 4␲2⌸ ˜ H

⬘

共mH 2_{兲⫽0, 共2.8兲}

FIG. 1. One-loop self-energy type diagram needed for the evalu-ation of the compositeness condition.

where⌸˜_H

⬘

is the derivative of the meson mass function de-fined by the diagram in Fig. 1:

⌸P共p2_兲⫽

冕

d 4_k 4␲2i␾P 2_共⫺k2_兲tr关␥5_S 3共k” 兲␥5S1共k”⫹p”兲兴, 共2.9兲 ⌸V共p2_兲⫽1 3

冋

g ␮␯_⫺p␮p␯ p2

册

冕

d4k 4␲2i␾V 2_共⫺k2_兲 ⫻tr关␥␮_S 3共k” 兲␥␯S1共k”⫹p”兲兴. 共2.10兲

For simplicity, we extract the factor 1/4␲2 from the defini-tion of the meson mass operator. We use the local quark propagators

Si共k” 兲⫽ 1

mi⫺k”, 共2.11兲

where miis the constituent quark mass. As discussed in关12兴, we assume that mH⬍mq

1⫹mq2 in order to avoid the

appear-ance of imaginary parts in the physical amplitudes. This is a reliable approximation for the heavy pseudoscalar mesons. The above condition is not always met for heavy vector me-sons. As discussed in Sec. VI we shall therefore employ equal masses for the heavy pseudoscalar and vector mesons in our matrix element calculations but use physical masses for the phase space.

III. METHOD FOR THE EVALUATION OF ONE-LOOP DIAGRAMS WITH ARBITRARY VERTEX

FUNCTIONS

For the present purposes one has to evaluate one-loop integrals of two- and three-point functions involving tensor integrands and product of vertex functions. In this section we describe a general method to efficiently enact these calcula-tions for the general case of n-point one-loop funccalcula-tions. We note two simplifying features of our integration technique. The arising tensor integrals are reduced to simple invariant integrations. The sequence of integrations is arranged such that the product of vertex functions is kept to the very end and allowing for a full flexibility in the choice of vertex functions.

We consider a rank s tensor integral in the Minkowski space as it appears in a general one fermion-loop calculation of an n-point function共see the diagram in Fig. 2兲. One has

I_[n,s]␮1, . . . ,␮s⫽

冕

d 4_k i␲2F共⫺k 2_兲 k ␮1•••k␮s

兿

i_⫽1 n 关mi 2 ⫺共k⫹li兲2_兴 . 共3.1兲

The outer momenta p_j ( j⫽1, . . . ,n) are all taken to be in-coming. The momenta of the inner lines are given by k⫹li with li⫽兺i_j⫽1pj such that ln⫽0. The maximum degree of

the momentum tensor in the numerator arising from the n fermion propagators is smax⫽n. We employ the impulse

ap-proximation dropping the dependence on the external mo-menta inside the vertex functions and denote the product of all vertex functions byF(⫺k2).

Using the␣ parametrization of Feynman one finds

I_[n,s]␮1, . . . ,␮s⫽⌫共n兲

冕

_dn_␣␦

冉

_1⫺

兺

i⫽1 n ␣i

冊

⫻

冕

d 4_k i␲2F共⫺k 2_兲 k ␮1•••k␮s 关Dn共␣兲⫺共k⫹P兲2_兴n, 共3.2兲 where P⫽兺in_⫽1␣ili, and Dn(␣)⫽兺i, j␣i␣jdi j with di, j ⫽(1/2) 关mi

2_⫹m

j

2_⫺(li⫺lj

)2兴.

Next we use the Cauchy integral representation for the functionF(⫺k2) leading to I_[n,s]␮1, . . . ,␮s⫽⌫共n兲

冕

_dn_␣␦

冉

_1⫺

兺

i⫽1 n ␣i

冊

⫻

冕

d 4_k i␲2

冖

d␨F共⫺␨兲 2␲i ⫻ k␮1•••k␮s 关␨⫺k2_{兴关Dn共␣兲⫺共k⫹P兲}2_兴n.

The new denominator factor is then included again via Feyn-man parametrization, giving

FIG. 2. One-loop diagram with n legs and arbitrary vertex func-tions.

I_[n,s]␮1, . . . ,␮s_{⫽⌫共n⫹1兲}

冕

0 1 d␤␤n⫺1

冕

dn␣␦

冉

1⫺

兺

i⫽1 n ␣i

冊

冕

d4k i␲2

冖

d␨F共⫺␨兲 2␲i ⫻ k␮1•••k␮s 关共1⫺␤兲␨⫺共k⫹␤P兲2⫹␤Dn共␣兲⫺␤共1⫺␤兲P2兴n⫹1.

One then factors out the (1⫺␤) in the denominator and shifts the integration variable k to k

⬘

⫽(k⫹␤P)/

冑

1⫺␤ to obtain I_[n,s]␮1, . . . ,␮s⫽⌫共n⫹1兲

冕

0 1 d␤

冉

␤ 1⫺␤

冊

n⫺1

冕

dn␣ ⫻␦

冉

1⫺

兺

i⫽1 n ␣i

冊

冕

d4k i␲2

冖

d␨F共⫺␨兲 2␲i ⫻共

冑

1⫺␤k⫺␤P兲 ␮1•••共

冑

₁⫺␤_k⫺␤_P兲␮s 关␨⫺k2_⫹z兴n⫹1 . The contour integral can be done again by Cauchy’s theo-rem. On substitution of␤⫽t/(1⫹t) one then has

I_[n,s]␮1, . . . ,␮s⫽共⫺兲n

冕

0 ⬁ dt t n⫺1 共1⫹t兲2

冕

d n_␣ ⫻␦

冉

1⫺

兺

i⫽1 n ␣i

冊

冕

d4_k i␲2F 共n兲_共⫺k2 ⫹z兲K␮1•••K␮s_,

whereF(n)denotes the nth derivative of the functionF and where K␮⫽ 1

冑

1⫹tk ␮_⫺ t 1⫹tP ␮_{, z⫽t Dn共␣兲⫺} t 1⫹tP 2_.

The momentum integration of the tensor integral can be trivi-ally done by invariant integration. Fintrivi-ally we go to the Eu-clidean space by rotating k0→ik4 which gives k2→⫺kE 2

⬅u; then one encounters the scalar integrals

I_[n,m]⫽共⫺兲n

冕

0 ⬁ dt t n⫺1 共1⫹t兲2_⫹m

冕

d n_␣ ⫻␦

冉

1⫺

兺

i⫽1 n ␣i

冊

冕

0 ⬁ duum⫹1F(n)共u⫹z兲. 共3.3兲

The u integration can be performed by partial integration and one finally obtains

I[n,m]⫽共⫺兲n⫹m⌫共m⫹2兲

冕

0 ⬁ dt t n_⫺1 共1⫹t兲2_⫹m

冕

d n_␣ ⫻␦

冉

1⫺

兺

i_⫽1 n ␣i

冊

F(n⫺m⫺2)共z兲. 共3.4兲

Since m_max⫽关n/2兴, Eq. 共3.4兲 holds true for all n and m except for the case n⫽2 and m⫽1. In this case we have

I_[2,1]⫽2

冕

0 ⬁ dt t 共1⫹t兲3

冕

d 2_␣␦

冉

_1⫺

兺

i⫽1 2 ␣i

冊

冕

z ⬁ duF共u兲 ⫽

冕

0 ⬁ dt

冉

t 1⫹t

冊

2

冕

d2␣␦

冉

1⫺

兺

i_⫽1 2 ␣i

冊

zt

⬘

F共z兲, 共3.5兲 where z_t

⬘

⫽dz(t)/dt.

One has to remark that the integration over the␣ param-eters in Eq.共3.4兲 can be done analytically up to a remaining one-fold integral. However, the ease with which the numeri-cal␣ integrations can be done does not warrant the effort of further analytical integrations. Using the integration tech-niques described in this section all necessary numerical inte-grations encountered in this investigation can be performed within minutes using a fast modern PC.

IV. HADRONIC MATRIX ELEMENTS A. Quark-meson coupling constants

As already discussed in Sec. II, the quark-meson coupling constants are determined by the compositeness condition, Eq. 共2.8兲. The derivatives of the meson-mass functions can be written as d d p2⌸P共p 2_兲⫽ 1 2 p2p ␣ d d p␣

冕

d4k 4␲2_i␾P 2_共⫺k2_兲 ⫻tr关␥5_S 3共k” 兲␥5S1共k”⫹p”兲兴 ⫽ 1 2 p2

冕

d4k 4␲2i␾P 2_共⫺k2_兲 ⫻tr关␥5_S 3共k” 兲␥5S1共k”⫹p”兲p” S1共k”⫹p”兲兴, 共4.1兲 d d p2⌸V共p 2_兲⫽1 3 1 2 p2p ␣ d d p␣

冋

g ␮␯_⫺p␮p␯ p2

册

⫻

冕

d 4_k 4␲2i␾V 2_共⫺k2_兲 ⫻tr关␥␮_S 3共k” 兲␥␯S1共k”⫹p”兲兴

⫽1₃

冋

g␮␯⫺p ␮_p␯ p2

册

1 2 p2

冕

d4k 4␲2i␾V 2_共⫺k2_兲 ⫻tr关␥␮_S 3共k” 兲␥␯S1共k”⫹p”兲p” S1共k”⫹p”兲兴. 共4.2兲 The evaluation of the integrals is done by using the method outlined in Sec. III. The compositeness condition reads 3g_H2 4␲2NH⫽1, where NH⫽ d d p2⌸H共p 2_兲

冏

p2_⫽m H 2 共4.3兲 NH⫽1 2

冕

0 ⬁ dt

冉

t 1⫹t

冊

2

冕

0 1 d␣ ␣兵•••其H 兵•••其P⫽FP共z兲_1⫹t1

冋

4⫺3 ␣t 1⫹t

册

⫺F

⬘

P共z兲 ⫻

再

2m1m3⫹ ␣t 1⫹t 关m1 2_⫺2m 1m3⫹p2兴 ⫺p2

冉

␣t 1⫹t

冊

2

冉

2⫺ ␣t 1⫹t

冊冎

, 兵•••其V⫽FV共z兲_1⫹t1

冋

2⫺ ␣t 1⫹t

册

⫺FV

⬘

共z兲 ⫻

再

2m1m3⫹ ␣t 1⫹t 关m1 2_⫺2m 1m3⫹p2兴 ⫺p2

冉

␣t 1⫹t

冊

2

冉

2⫺ ␣t 1⫹t

冊冎

.

Here, m₁stands for the heavy quark (b or c) and m₂for the light quarks (u, d, s) in the case of heavy-light systems, and for c in the case of double-heavy systems. The function FH(z) is the product of two vertex functions FH(z) ⫽␾H 2 (z) with z⫽t关␣m1 2 ⫹共1⫺␣兲m3 2_{⫺␣共1⫺␣兲p}2_兴⫺ ␣ 2_t 1⫹tp 2_.

B. Leptonic and radiative decays

The matrix elements of the leptonic and radiative decays are defined by the diagrams in Figs. 3–6 and given by

i M_P␮共p兲⫽3gP 4␲2

冕

d4_k 4␲2i␾P共⫺k 2_兲 ⫻tr关␥5_S 3共k” 兲O␮S1共k”⫹p”兲兴 ⫽ fPp␮, 共4.4兲 M_V␮共p兲⫽⫺CV•3gV 4␲2

冕

d4k 4␲2_i␾V共⫺k 2_兲 ⫻tr关⑀”*Sq共k” 兲␥␮Sq共k”⫹p”兲兴 ⫽⫺mVCVfV⑀*␮, 共4.5兲 i MP␥␥共q1,q2兲⫽CP␥␥ 3g_P 4␲2

冕

d4k 4␲2_i␾P共⫺k 2_兲 ⫻tr关␥5_{Sq共k”⫺q”} 2兲⑀”2*Sq共k” 兲⑀”1*Sq共k”⫹q”1兲兴, ⫽igP␥␥␧␮␯␣␤⑀1*␮⑀2*␯q1␣q2␤, 共4.6兲 i MV P␥共p,p

⬘

兲⫽CV P␥ 3gVgP 4␲2

冕

d4k 4␲2iFPV共⫺k 2_兲 ⫻tr关␥5_{Sq共k”⫹p”兲⑀”*Sq共k” 兲⑀”} ␥ *Sq共k”⫹q”兲兴 ⫽igV P␥␧␮␯␣␤⑀␥*␮⑀*␯p␣q␤. 共4.7兲

For ease of presentation, the expression for the V→P␥ de-cay is given for neutral-flavored mesons. Using the integra-tion techniques described in Sec. III one then arrives at the following analytical representation of the various one-loop matrix elements: fP⫽3gP 4␲2

冕

0 ⬁ dt t 共1⫹t兲2

冕

₀ 1 d␣␾P共zP兲 ⫻

冋

m3⫹共m1⫺m3兲 ␣t 1⫹t

册

,

FIG. 4. Quark model diagram for vector meson radiative decays. FIG. 3. Quark model diagram for leptonic meson decays.

zP⫽t关␣m₁2⫹共1⫺␣兲m₃2⫺␣共1⫺␣兲p2兴 ⫺t␣ 2 1⫹tp 2_{, 共4.8兲} fV⫽ 1 mV 3g_V 4␲2

冕

0 ⬁ dt t 共1⫹t兲2

冕

₀ 1 d␣␾V共zV兲 ⫻

冋

m_q2⫹1 2tzt

⬘

⫹ ␣t 1⫹t

冉

1⫺ ␣t 1⫹t

冊

p 2

册

_, zV⫽t关m_q2⫺␣共1⫺␣兲p2兴⫺ t␣ 2 1⫹tp 2_, ⌫共V→e⫹_e⫺_兲⫽4␲ 3 ␣2 mV fV 2 CV 2 , CV⫽eq 2_共V⫽␾ ,J/␺,⌼兲, 共4.9兲 gP␥␥⫽CP␥␥mq 3gP 4␲2

冕

0 ⬁ dt

冉

t 1⫹t

冊

2 ⫻

冕

d3_␣␦

冉

_1⫺

兺

i⫽1 3 ␣i

冊

关⫺␾

⬘

P共z0兲兴, z0⫽t共mq 2_⫺␣1␣2 p2兲⫺ t 1⫹t␣1␣2p 2_, ⌫共P→␥␥兲⫽␲₄ ␣2_m P 3 g_P2_␥␥, C_␩ c␥␥⫽2ec 2 , 共4.10兲 gV P␥⫽CV P␥mq 3gVgP 4␲2

冕

0 ⬁ dt

冉

t 1⫹t

冊

2 ⫻

冕

d3␣␦

冉

1⫺

兺

i⫽1 3 ␣i

冊

关⫺FV P

⬘

共zV P兲兴, zV P⫽t共m_q2⫺␣1␣3m_V2⫺␣1␣₂m_P2兲 ⫺_{1⫹t 关}t ␣1共␣1⫹␣2兲mV2_⫺␣1␣2 m_P2兴, ⌫共V→P␥兲⫽₂₄␣ m_V3

冉

1⫺mP 2 m_V2

冊

3 g_{V P}2 _␥ CJ/␺␩_c␥⫽2ec. 共4.11兲 The electric quark charges eq are given in units of e.

C. Semileptonic form factors

The semileptonic decays of the Bcmeson can be induced by either a beauty quark or a charm quark transition. In the relativistic quark model, the hadronic matrix element corre-sponding to b decay is defined by the diagram in Fig. 7 and is given by M_b␮„P共p兲→H共p

⬘

兲… ⫽3gPgH 4␲2

冕

d4k 4␲2iFPH共⫺k 2_兲 ⫻tr关␥5_S 3共k” 兲⌫HS2共k”⫹p”

⬘

兲O␮S1共k”⫹p”兲兴, 共4.12兲 where FPH⫽␾P•␾H, ⌫P⫽␥5, and ⌫V⫽⫺i⑀”* with ⑀*•p

⬘

⫽0. For the b-decay case one has the Cabibbo-Kobayaski-Maskawa- 共CKM兲-enhanced decays

b→c: Bc⫹→共␩c,J/␺兲l⫹␯, m1⫽mb, m2⫽m3⫽mc, and the CKM-suppressed decays

b→u: Bc⫹→共D0,D*0_兲l⫹_{␯, m} 1⫽mb,

m₂⫽mu, m₃⫽m_c.

FIG. 5. Quark model diagram for the decays of a neutral meson.

FIG. 6. Quark model diagram for the radiative decay of a vector meson into a pseudoscalar one.

FIG. 7. Quark model diagram for the semileptonic Bc decays

involving b→c,u transitions. The lower leg in the loop is the c quark.

The c-decay option of the Bcmeson is represented by the Feynman diagram in Fig. 8 which gives

M_c␮„P共p兲→H共p

⬘

兲… ⫽3gPgH 4␲2

冕

d4k 4␲2iFPH共⫺k 2_兲 ⫻tr关␥5_S 3共k” 兲O␮S2共k”⫹q”兲⌫HS1共k”⫹p”兲兴. 共4.13兲 Again one has the CKM-enhanced decays

c→s: B_c⫹→共B¯_s0,B¯_s*0_兲l⫹

␯, m1⫽mb,

m2⫽ms, m3⫽mc,

and the CKM-suppressed decays

c→d: Bc⫹→共B¯0,B¯*0_兲l⫹_{␯, m} 1⫽mb,

m₂⫽m_d, m₃⫽m_c.

It is convenient to present the results of the matrix element evaluations in terms of invariant form factors. A standard decomposition of the transition matrix elements into invari-ant form factors is given by

M␮„P共p兲→P

⬘

共p

⬘

兲…⫽ f_⫹共q2兲 共p⫹p

⬘

兲␮ ⫹ f_⫺共q2_{兲 共p⫺p}

_⬘

_兲␮ _共4.14兲 and i M␮„P共p兲→V共p

⬘

兲… ⫽⫺g␮␯_⑀*␯_共mP⫹m V兲A1共q2兲⫹共p⫹p

⬘

兲␮p•⑀* ⫻ A2共q2兲 mP⫹mV⫹共p⫺p

⬘

兲 ␮_p•⑀* ⫻ A3共q 2_兲 mP⫹mV⫺i␧ ␮␯␣␤_⑀*␯_p␣_p⬘␤ 2 V共q 2_兲 mP⫹mV . 共4.15兲 The various invariant form factors can be extracted from the one-loop expressions共4.12兲 and 共4.13兲 by using the tech-niques described in Sec. III. One finds that the form factor integrands factorize into a common piece times a piece spe-cific to the different form factors. One can thus write

F共q2兲⫽ 3 4␲2gPgH 1 2

冕

0 ⬁ dt

冉

t 1⫹t

冊

2

冕

d3␣␦

冉

1⫺

兺

i⫽1 3 ␣i

冊

⫻兵•••其F 共4.16兲

where F⫽ f_⫾,Ai,V. For the 0⫺→0⫺ b→c,u form factors

f_⫹one has 兵•••其f_⫹ b _{⫽FP P共zb兲} 1 1⫹t

冋

4⫺3共␣1⫹␣2兲 t 1⫹t

册

⫺FP P

⬘

共zb兲

再

共m1⫹m2兲m3 ⫹ t 1⫹t 关⫺共␣1⫹␣2兲共m1m3⫹m2m3⫺m1m2兲⫹␣1p 2_⫹␣2p

_⬘

2_兴 ⫺

冉

_1⫹tt

冊

2

冉

2⫺共␣1⫹␣2兲 t 1⫹t

冊

关共␣1⫹␣2兲共␣1p 2_⫹␣ 2p

⬘

2兲⫺␣1␣2q2兴

冎

, zb⫽t

冉

兺

i⫽1 3 ␣imi 2_⫺␣1␣ 3p2⫺␣2␣3p

⬘

2⫺␣1␣2q2

冊

⫺ t 1⫹tPb 2 , Pb⫽␣1p⫹␣2p

⬘

. For the corresponding c→s,d form factor one has

兵•••其f_⫹ c _{⫽⫺FP P共zc兲} 1 1⫹t

冋

1⫹3␣1 t 1⫹t

册

⫹FP P

⬘

共zc兲

再

m2m3 ⫹_{1⫹t 关}t ␣1共m1m2⫹m1m3⫺m2m3兲⫹␣2q2兴⫹

冉

t 1⫹t

冊

2 关␣1 2 p2⫺␣₂2q2兲] ⫺

冉

t 1⫹t

冊

3 ␣1关␣1共␣1⫹␣2兲p2⫺␣1␣2p

⬘

2⫹␣2共␣1⫹␣2兲q2兴

冎

, FIG. 8. Quark model diagram for the semileptonic Bc decays

involving c→s,d transitions. The upper leg in the loop is the b quark.

zc⫽t

冉

兺

i_⫽1 3 ␣imi 2_⫺␣1␣ 3p2⫺␣2␣3q2⫺␣1␣2p

⬘

2

冊

⫺ t 1⫹tPc 2 , Pc⫽␣1p⫹␣2q.

Expressions for the remaining 0⫺→0⫺ and 0⫺→1⫺ form factors f_⫺, Aiand V are given in the Appendix. The masses

m_i (i⫽1,2,3) appearing in the form factor expressions are constituent quark masses with a labeling according to Eqs. 共4.12兲 and 共4.13兲. The values of the constituent quark masses as well as the vertex functions entering the form factor ex-pressions will be specified in Sec. VI.

For the calculation of physical quantities it is more con-venient to use helicity amplitudes. They are linearly related to the invariant form factors 关22兴. For the 0⫺→0⫺ transi-tions one has

H0共q2兲⫽ 2mPP

冑

q2 f⫹共q 2_{兲, 共4.17兲} Ht共q2兲⫽ 1

冑

q2兵共mP 2_⫺m P⬘ 2 兲f⫹共q2兲⫹q2f⫺共q2兲其. 共4.18兲 For the 0⫺→1⫺ transitions one has

H_⫾共q2兲⫽⫺共mP⫹mV兲A1共q2兲⫿ 2mPP 共mP⫹mV兲 V共q2兲, 共4.19兲 H₀共q2_兲⫽ 1 2mV

冑

q2

再

⫺共mP2_⫺mV2_⫺q2_{兲共mP⫹mV兲} ⫻A1共q2兲⫹ 4m_P2P2 mP⫹mV A2共q2兲

冎

, 共4.20兲 Ht共q2_兲⫽ mPP mV

冑

q2

再

⫺共mP⫹mV兲A1共q2兲⫹共mP⫺mV兲 ⫻A2共q2兲⫹ q2 mP⫹mV A3共q2兲

冎

, 共4.21兲 where P⫽

冑

␭共mP 2 ,m_H2 ,q2兲 2m_P ⫽ 关共q⫹2⫺q2兲共q⫺2⫺q2兲兴1/2 2m_P with q_⫾2⫽(mP⫾mH)2.

Then the partial helicity rates are defined as

d⌫i dq2⫽ G_F2 共2␲兲3兩Vf f⬘兩 2共q 2_⫺ml2_兲2_P 12m_P2q2 兩Hi共q 2_兲兩2_{, i⫽⫾,0,t,} 共4.22兲 where Vf f⬘is the relevant element of the CKM matrix, and

ml is the mass of charged lepton.

Finally, the total partial rates including lepton mass ef-fects can be written as关22兴

d⌫PP⬘ dq2 ⫽

冉

1⫹ ml 2 2q2

冊

d⌫0 PP⬘ dq2 ⫹3 ml 2 2q2 d⌫tPP⬘ dq2 , 共4.23兲 d⌫PV dq2 ⫽

冉

1⫹ m_l2 2q2

冊

冋

d⌫_⫹PV dq2 ⫹ d⌫_⫺PV dq2 ⫹ d⌫₀PV dq2

册

⫹3ml 2 2q2 d⌫tPV dq2 . 共4.24兲

In the following we shall present numerical results of the total decay widths, polarization ratio and forward-backward asymmetry. The relevant expressions are given by

⌫⫽

冕

m_l2 (m_P⫺m_H)2 dq2 d⌫ dq2, ␣⫽2 ⌫0 ⌫_⫹⫹⌫_⫺⫺1, AFB⫽3 4 ⌫⫺⫺⌫⫹ ⌫ . 共4.25兲

V. HEAVY QUARK SPIN SYMMETRY

Our model allows us to evaluate form factors directly from Eq. 共4.16兲 without any approximation. However, it would be interesting to explore whether the heavy quark spin symmetry relations derived in Ref.关4兴 can be reproduced in our approach. As was shown 共see, for instance, 关17兴兲 our model exhibits all consequences of the spin-flavor symmetry for the heavy-light systems in the heavy quark limit. For example, the quark-meson coupling and leptonic decay con-stants behave as gH→

冑

2m1 2␲

冑

3N˜H , N˜H⫽

冕

0 ⬁ du␾_H2共u⫺2E

冑

u兲 m3⫹

冑

u m₃2⫹u⫺2E

冑

u, 共5.1兲 fH→ 1

冑

m₁

冑

3 2␲2_N˜ H

冕

0 ⬁ du关

冑

u⫺E兴␾H共u⫺2E

冑

u兲 ⫻ m3⫹

冑

u/2 m₃2⫹u⫺2E

冑

u, 共5.2兲

in the heavy quark limit: p2⫽m_H2⫽(m1⫹E)2 when m1

→⬁. Equations 共5.1兲 and 共5.2兲 make the heavy quark mass dependence of the coupling factors gHand fHexplicit since

we have factorized the coupling factor contributions into a heavy mass dependent piece and a remaining heavy mass independent piece. Moreover, Eqs.共5.1兲 and 共5.2兲 show gH and fHscale as m1

1/2_{and m} 1

⫺1/2_{, respectively.}

As is well known 共see Ref. 关4兴兲, heavy flavor symmetry cannot be used for hadrons containing two heavy quarks. But one can still derive relations near zero recoil by using heavy quark spin symmetry.

First, we consider the semileptonic decays Bc →B¯s(B¯0)e⫹␯ and Bc→B¯s*(B¯*

0_)e⫹_{␯ which correspond to c}

decay into light s and d quarks, respectively. Since the en-ergy released in such decays is much less than the mass of the b quark, the four-velocity of the Bc meson is almost unaffected. Then the initial and final meson momenta can be written as

p⫽m_B

cv, p

⬘

⫽mBv⫹r,

where r is a small residual momentum关v•r⫽⫺r2/(2mB)兴. The heavy quark spin symmetry can be realized in the fol-lowing way. We split the B-meson masses into the sum of

b-quark mass and binding energy:

mB_c⬅mP⫽m1⫹E1, mB⬅mH⫽m1⫹E2.

Then we go to the heavy quark mass limit mb⬅m₁→⬁ in which the b-quark propagator acquires the form

1

m1⫺p” ⫺k” ⇒

1⫹v” ⫺2共kv⫹E1兲

. 共5.3兲

The decoupling of the c-quark spin allows us to reliably neglect the k integration because k is small compare to the heavy c-quark mass. One has

1

m3⫺k” ⇒

1

m3

. 共5.4兲

As a consequence, the hadronic matrix element describing the weak c-quark decay simplifies:

M_c␮⫽

冑

2mP•2mH

冑

N˜P•N˜H 1 m3

冕

d4k 4␲2iFPH共⫺k 2_兲 ⫻tr关O ␮_共m 2⫹k”⫹q”兲⌫H共1⫹v” 兲␥5兴 关⫺2kv⫺2E1兴关m2 2 ⫺共k⫹q兲2_兴 , 共5.5兲

where q⫽p⫺p

⬘

⫽(mP⫺mH)v⫺r⫽(E1⫺E2)v⫺r⬅⌬Ev

⫺r and m2 stands for the light quark mass (m2⫽ms or md). One has to emphasize that all above approximations are valid only close to the zero-recoil point q_max2 ⫽⌬E2_{. Recalling the}

transversality of the final vector meson field p

⬘

•⑀*⫽m_Bv ⫹r•⑀*⫽0 and applying the integrations as described in Sec. III, one finds

M_c␮⫽

冑

2mP•2mH

冑

N˜P•N˜H 1 m3

冕

0 ⬁ dt t 共1⫹t兲2

冕

₀ ⬁ d␣FPH共zc兲 ⫻兵•••其PH, 兵•••其P P⫽⫺

冉

m2⫹ ␣t⫺⌬E 1⫹t

冊

v ␮_⫺ 1 1⫹tr ␮_, i兵•••其PV⫽⫺i␧␮␯␣␤⑀*␯v␣r␤ ⫻ 1 1⫹t ⫹

冉

m2⫹ ␣t⫺⌬E 1⫹t

冊

⑀* ␮ ⫺_1⫹t1 v␮⑀*•r. 共5.6兲 Here, zc⫽关␣t2/(1⫹t)兴(␣⫹2⌬E)⫹t(m₂2⫺2␣E₁)⫺关t/(1 ⫹t)兴⌬E2_{. It is readily seen that the amplitudes of c decay in}

the heavy quark limit are expressed through two independent functions 兵•••其1⫽

冉

m2⫹ ␣t⫺⌬E 1⫹t

冊

, 兵•••其2⫽ 1 1⫹t.

To complete the description of the heavy quark limit in the c-decay modes, we give the expressions for the form factors in this limit. One has

F共q_max2 兲→

冑

2m_P2m_H

冑

N˜PN˜H

冕

0 ⬁ dt t 共1⫹t兲2

冕

₀ ⬁ d␣FPH共zc兲兵•••其F 共5.7兲 where F⫽ f_⫾,Ai,V. The form factor specific pieces are given by 兵•••其f_⫹⫽⫺ 1 2m1m3

冉

m2⫹ ␣t 1⫹t

冊

, 兵•••其f⫺⫽ 1 m3 1 1⫹t, 兵•••其A₁⫽⫺ 1 mP⫹mV 1 m3

冉

m2⫹ ␣t⫺⌬E 1⫹t

冊

, 兵•••其A₂⫽兵•••其V⫽ mP⫹m_V 2 1 m1m3 1 1⫹t, 兵•••其A₃⫽ mP⫹mV 2 1 m1m3

冋

⫺ 3 1⫹t ⫹4

冉

m1 m3 ⫺1

冊

t 1⫹t

册

. Superficially it appears that the form factors f_⫹ and A1 are

suppressed by a factor of 1/m1. However, they must be kept

in the full amplitude to obtain the correct result in Eq.共5.6兲; for instance, one has

f_⫹共p⫹p

⬘

兲␮⫹ f_⫺共p⫺p

⬘

兲␮

⫽共2m1f⫹⫹⌬E f⫺兲v␮⫹共 f⫹⫺ f⫺兲r␮.

A similar analysis applies to the b→u decays Bc →(D0_,D*0_)e⫹_{␯. Again the heavy quark symmetry analysis}

is only reliable close to zero recoil where the u quark from the b→u decay has small momentum. One has

M_b␮⫽

冑

2mP•2mH

冑

N˜P•N˜H 1 m3

冕

d4k 4␲2iFPH共⫺k 2_兲 ⫻tr关␥ 5_⌫H共m 2⫹k”⫹p”

⬘

兲O␮共1⫹v” 兲兴 关⫺2kv⫺2E1兴关m2 2_⫺共k⫹p

_⬘

_兲2_兴 , 共5.8兲

where q⫽p⫺p

⬘

⫽(m₁⫹E₁⫺m_H)v⫺r. The light quark mass m2 in Eq.共5.8兲 is the u-quark mass. One finds

M_b␮⫽

冑

2mP2mH

冑

N˜PN˜H 1 m3

冕

0 ⬁ dt t 共1⫹t兲2

冕

₀ ⬁ d␣FPH共zb兲兵•••其PH 共5.9兲 with 兵•••其P P⫽

冉

m2⫹ mH⫺␣t 1⫹t

冊

v ␮_⫹ 1 1⫹tr ␮_, i兵•••其PV⫽⫺i␧␮␯␣␤⑀*␯v␣r␤

冉

m2⫹ mH⫺␣t 1⫹t

冊

⫺_1⫹t1 v␮⑀*•r. Here, zb⫽关␣t2_{/(1⫹t)兴 (␣⫹2m} H)⫹t (m2 2_⫺2␣E 1)⫺关t/(1 ⫹t)兴mH

2 _{. Again, the amplitudes for the b→u decays are}

expressed through two independent functions. The expres-sions for the form factors in the heavy quark limit close to zero recoil read

F共q_max2 兲→

冑

2mP2mH

冑

N˜PN˜H

冕

0 ⬁ dt t 共1⫹t兲2

冕

₀ ⬁ d␣FPH共zc兲兵•••其F, 共5.10兲 where F⫽ f_⫾,Ai,V and where

兵•••其f_⫹⫽⫺兵•••其f_⫺⫽ 1 2m3

1 1⫹t, TABLE I. Leptonic decay constants fH共MeV兲 used in the least-squares fit.

Meson This model Other Ref.

␲⫹ _{131 130.7⫾0.1⫾0.36 Expt.关35兴} K⫹ 160 159.8⫾1.4⫾0.44 Expt.关35兴 D⫹ 191 191_⫺28⫹19 Lattice关23,24兴 192⫾11_⫺8⫺0⫹16⫹15 Lattice关25兴 194_⫺10⫹14⫾10 Lattice关26兴 D_s⫹ 206 206_⫺28⫹18 Lattice关23,24兴 210⫾9_⫺9⫺1⫹25⫹17 Lattice关25兴 213_⫺11⫹14⫾11 Lattice关26兴 B⫹ 172 172_⫺31⫹27 Lattice关23,24兴 157⫾11_⫺9⫺0⫹25⫹23 Lattice关25兴 164_⫺11⫹14⫾8 Lattice关26兴 Bs⫹ 196 171⫾10⫺9⫺2⫹34⫹27 Lattice关25兴 185_⫺8⫹13⫾9 Lattice关26兴 Bc 479 Logarithmic potential关27兴 500 Buchmu¨ller-Tye potential关27兴 512 Power-law potential关27兴 687 Cornell potential关27兴 480 Potential model关6兴 432 QCD-inspired QM关28兴 400⫾20 QCD spectral SR关29兴 300 QCD SR关30兴 360⫾60 QCD SR关31兴 300⫾65 QCD SR关32兴 385⫾25 QCD SR关33兴 420共13兲 Lattice NRQCD关34兴 Bc 360 360 Our average of QCD SR J/␺ 404 405⫾17 Expt.关35兴 ⌼ 711 710⫾37 Expt.关35兴

兵•••其A₁⫽ 1 mP⫹mV 1 m3

冉

m2⫹ mH⫺␣t 1⫹t

冊

, 兵•••其A₂⫽⫺兵•••其A₃⫽兵•••其V⫽ mP⫹mV 2 1 m₁m₃ 1 1⫹t. Note that one needs to keep the next-to-leading term in the sum ( f_⫹⫹ f_⫺) to obtain the above amplitudes.

FIG. 9. q2_{dependence of the B}

c→␩cform factors. Note that we

plot the negative of the f_⫺(q2) form factor.

FIG. 10. q2_{dependence of the B}

c→J/␺ form factors. Note that

we plot the negative of the A3(q2) form factor.

FIG. 11. q2_{dependence of the B}

c→Bsform factors. Note that

we plot the negative of the f_⫹(q2) form factor.

The hadronic matrix elements of the b→c decays Bc →(␩c,J/␺)e⫹␯ simplify significantly in the heavy quark limit. In this case both b and c propagators may be replaced by their heavy quark limit forms in Eq. 共5.3兲 with the same velocity v. Again, the results will be valid only near zero recoil. One has

Mcc␮⫽

冑

2mP2mH

冑

N˜PN˜H 1 m₃

冕

d4k 4␲2iFPH共⫺k 2_兲 ⫻ tr关␥ 5_{⌫H共1⫹v” 兲O}␮_{共1⫹v” 兲兴} 关⫺2kv⫺2E1兴关⫺2kv⫺2E2兴 共5.11兲 where p⫽(m₁⫹E₁)v, p

⬘

⫽(m3⫹E2)v⫹r. One finds

M_cc␮⫽

冑

2mP2mH

冑

N˜PN˜H 1 2m3

冕

0 ⬁ du

冕

0 1 d␣ ⫻FPH共u⫺2

冑

u„␣E1⫹共1⫺␣兲E2…兲其兵•••其PH, 共5.12兲 兵•••其P P⫽⫹2v␮_{, i兵•••其PV⫽⫺2⑀*}␮_.

The form factors are written down

F共qmax 2 _兲→

冑

2mP2mH

冑

N˜PN˜H

冕

0 ⬁ du

冕

0 1 d␣ ⫻FPH共u⫺2

冑

u„␣E₁⫹共1⫺␣兲E₂兲…兵•••其_F 共5.13兲 where F⫽ f_⫾,A_i,V. We have 兵•••其f_⫹⫽ m₁⫹m₃ 4m1m3 2 , 兵•••其f⫺⫽⫺ m₁⫺m₃ 4m1m3 2 , 兵•••其A₁⫽ 1 mP⫹mV 1 m3 , 兵•••其A₂⫽⫺兵•••其A₃⫽兵•••其V⫽ mP⫹mV 4m1m3 2 .

Thus, our quark loop calculations reproduce the heavy quark limit relations between form factors obtained in关4兴 near zero recoil. Moreover, we give explicit expressions for the re-duced set of form factors in this limit.

VI. RESULTS AND DISCUSSION

Before presenting our numerical results we need to specify our values for the constituent quark masses and shapes of the vertex functions. As concerns the vertex func-tions, we found a good description of various physical quan-tities关17兴 adopting a Gaussian form for them. Here we apply the same procedure using ␾H(k2)⫽exp兵⫺k2/⌳H

2_其

in the Eu-clidean region. The magnitude of ⌳H characterizes the size of the vertex function and is an adjustable parameter in our model. We reiterate that all the analytical results presented in Sec. V are valid for any choice of form factor ␾H(k2). For example, we have reproduced the results of 关21兴 where di-pole form factor was adopted by using our general formula. In关17兴 we have studied various decay modes of the␲, K,

D, Ds, B and Bs mesons. The ⌳ parameters and the con-stituent quark masses were determined by a least-squares fit TABLE II. Predictions for the form factors at q2⫽0 and q2

⫽qmax 2 for Bc→P decays. P f_⫹(0) f_⫹(qmax 2 ) f_⫺(0) f_⫺(qmax 2 ) ␩c 0.76 1.07 ⫺0.38 ⫺0.55 D 0.69 2.20 ⫺0.64 ⫺2.14 B ⫺0.58 ⫺0.96 2.14 2.98 Bs ⫺0.61 ⫺0.92 1.83 2.35

TABLE III. Predictions for the form factors at q2⫽0 for Bc →V decays. V A1(0) A2(0) A3(0) V(0) J/␺ 0.68 0.66 ⫺1.13 0.96 D* 0.56 0.64 ⫺1.17 0.98 B* ⫺0.27 0.60 10.8 3.27 Bs* ⫺0.33 0.40 10.4 3.25

TABLE IV. Predictions for the form factors at q2_⫽q max 2 for Bc→V decays. V A1(qmax 2 ) A2(qmax 2 ) A3(qmax 2 ) V(qmax 2 ) J/␺ 0.86 0.97 ⫺1.71 1.45 D* 0.85 1.76 ⫺3.69 3.26 B* ⫺0.42 0.49 18.0 5.32 B_s* ⫺0.49 0.21 15.9 4.91

TABLE V. Comparison of the form factors at the zero recoil point q2⫽qmax

2

calculated in the heavy quark limit with exact re-sults. H f_⫹ f_⫺ A1 A2 A3 V ␩c,J/␺ 1.07 ⫺0.55 0.86 0.97 ⫺1.71 1.45 ␩c,J/␺ 共HQL兲 0.70 ⫺0.35 0.37 0.69 ⫺0.69 0.69 D, D* 2.20 ⫺2.14 0.85 1.76 ⫺3.69 3.26 D, D*共HQL兲 0.59 ⫺0.59 0.18 1.50 ⫺1.50 1.50 B, B* ⫺0.96 2.98 ⫺0.42 0.49 18.0 5.32 B, B*共HQL兲 ⫺0.47 1.67 ⫺0.25 1.91 21.47 1.91

to experimental data and lattice determinations. The obtained values for the charm and bottom quarks关see Eq. 共6.1兲兴 allow us to consider the low-lying charmonium (␩cand J/␺) and bottonium (⌼) states, and also the newly observed Bc meson: m_u 0.235 m_s 0.333 m_c 1.67 m_b 5.06. 共6.1兲

Basically we use either the available experimental values or the values of lattice simulations for the leptonic decay constants to adjust the size parameters ⌳H. The value of fB_cis unknown and theoretical predictions for it lie within the 300–600 MeV range. We choose the value of fB_c⫽360 MeV, being the average QCD sum rule predictions, for fitting ⌳B

c. The obtained values of

⌳H are listed in Eq. 共6.2兲 as well as the values of fH in Table I: ⌳␲ 1.16 ⌳K 1.82 ⌳D 1.87 ⌳D_s 1.95 ⌳J/␺ 2.12 ⌳B 2.16 ⌳B_s 2.27 ⌳B_c 2.43 ⌳⌼ 4.425. 共6.2兲

The values of ⌳H are such that ⌳m

i⬍⌳mj if mi⬍mj.

This corresponds to the ordering law for sizes of bound heavy-light states.

The situation with the determination of ⌳_␩

c is quite

un-usual. Naively one expects that ⌳_␩

c should be the same as

⌳J/␺. However, in this case the value of the␩c→␥␥ decay

width comes out to be 2.5 less than the experimental aver-age. The experimental average can be reached only for a relatively large value of⌳_␩

c⫽4.51 GeV. Note that the

val-ues of the other observables (J/␺→␩c␥ and Bc→␩cl␯ de-cay rates兲 are not so sensitive to the choice of ⌳_␩

c:

Br共␩c→␥␥兲⫽0.031共0.012兲 %, expt⫽共0.031⫾0.012兲 %,

Br共J/␺→␩c␥兲⫽0.90共1.00兲 %, expt⫽共1.3⫾0.4兲 %, Br共Bc→␩cl␯兲⫽0.98共1.02兲 %.

The values in parentheses correspond to the case of equal sizes for the charmonium states.

We concentrate our study on the semileptonic decays of the Bcmeson. To extend the number of modes, we consider TABLE VI. The numerical values of mfit

2

(GeV2_{) and␦in the form factor parametrization, Eq.共6.3兲.}

f_⫹ f_⫺ A1 A2 A3 V Bc→␩c,J/␺ mfit 2 (6.37)2 (6.22)2 (8.20)2 (5.91)2 (5.67)2 (5.65)2 ␦ 0.087 0.060 1.40 0.052 ⫺0.004 0.0013 Bc→Bs,Bs* mfit 2 (1.73)2 (2.21)2 (1.86)2 (3.44)2 (1.73)2 (1.76)2 ␦ ⫺0.09 0.07 0.13 ⫺107 ⫺0.09 ⫺0.052

TABLE VII. Branching ratios BR(%) for the semileptonic decays Bc⫹→Hl⫹␯, calculated with the CDF

central value␶(Bc)⫽0.46 ps 关1兴. H This model 关9,10兴 关7兴 关6兴 关11兴 关8兴 关21兴 ␩ce␯ 0.98 0.8⫾0.1 0.78 1.0 0.15共0.5兲 0.6 0.52 ␩c␶ ␯ 0.27 J/␺ e ␯ 2.30 2.1⫾0.4 2.11 2.4 1.5共3.3兲 1.2 1.47 J/␺ ␶ ␯ 0.59 D0e␯ 0.018 0.003 0.006 0.0003共0.002兲 D0_{␶ ␯ 0.0094} D*0_{e␯ 0.034 0.013 0.019 0.008共0.03兲} D*0_{␶ ␯ 0.019} B0_{e␯ 0.15 0.08 0.16 0.06共0.07兲} B*0_{e␯ 0.16 0.25 0.23 0.19共0.22兲} Bs 0 e␯ 2.00 4.0 1.0 1.86 0.8共0.9兲 1.0 0.94 Bs* 0 e␯ 2.6 5.0 3.52 3.07 2.3共2.5兲 1.44

also the decays into the vector mesons D*, B*and B_s*. We will use the masses and sizes of their pseudoscalar partners for the numerical evaluation of the form factors to avoid the appearance of imaginary parts in the amplitudes. Such an assumption is justified by the small differences of their physical masses.

In Figs. 9–12 we show the calculated q2 _dependence

in the full physical regions of the semileptonic form factors of the CKM-enhanced transitions Bc→␩c, Bc →J/␺ and Bc→Bs, Bc→Bs*. The values of form factors at maximum and zero recoil are listed in Tables II–IV. The comparison of the exact values of form factors at zero recoil and those obtained in the heavy quark limit is given in Table V. Our results indicate that the corrections to the heavy quark limit at the zero recoil point q2⫽q_max2 can be as large as a factor 2 in b-c transitions and a factor of almost 5 in b-u and c-d transitions. This is not so surprising considering the semileptonic decays of the D meson where similar corrections can amount to a factor of two 关16兴.

The form factors can be approximated by the form

f共q2兲⫽ f共0兲

1⫺q2/m_fit2⫺␦共q2/m_fit2兲2 共6.3兲 with the dimensionless values of f (0) given in Tables II and III. Note that the form factor f_⫹(q2) for the B_c →␩c transition rises with q2 as is appropriate in the time-like region. When plotted against ␻⫽p•p

⬘

/

(m_in•m_out)⫽(m_in2⫹m2_out⫺q2)/(2 m_in•m_out), they would fall with ␻ as one is familiar with heavy quark effective theory.

It is interesting that the obtained values of m_fit2 for the CKM-enhanced transitions 共see Table VI兲 are very close to the values of the appropriate lower-lying (q¯ q

⬘

) vector mesons (m_B

c

*⬇mB_c⫽6.4 GeV for b→c, mD

s

* ⫽2.11 GeV for c→s.兲. The parameter ␦ characterizes the admixture of a q4 term in the denominator. Its magnitude is relatively small for all form factors of the CKM-enhanced transitions except A1 for the Bc→J/␺ transition and A2 for Bc→B_s* which have a rather flat behavior. This means that those form factors can be reliably approximated by a vector dominance form. However, one cannot approximate the form factors for the CKM-suppressed transitions by a pole-like function only.

We use the calculated form factors in Eq. 共4.25兲 to evaluate the branching ratios for various semileptonic Bc decay modes including their ␶ modes when they are kinematically accessible. We report the calculated values of a wide range of branching ratios in Table VII. The results of other approaches are also given for comparison. The values of branching ratios of the CKM-enhanced modes with an electron in the final state are of order 1 –2 %. The values of branching ratios of the CKM-suppressed modes are considerably less. The modes with a ␶ lepton in the final state are suppressed due to the reduced phase space in these modes. To complete our predictions for the physical observables we give in Table VIII the values of the polarization ratio and forward-backward asymmetry for the prominent decay modes.

ACKNOWLEDGMENTS

We would like to thank F. Buccella for many interesting discussions. M.A.I. gratefully acknowledges the hospitality of the Theory Groups at Mainz and Naples Universities where this work was completed. His visit at Mainz University was supported by the DFG 共Germany兲 and the Heisenberg-Landau fund. J.G.K. acknowledges partial support by the BMBF 共Germany兲 under contract 06MZ865.

APPENDIX

In this appendix we list the remaining form factor expressions appearing in the curly brackets in Eq.共4.16兲 which have not been listed in the main text.

b decay: 兵•••其f_⫺ b _{⫽⫺FP P共zb兲} 1 1⫹t

冋

3共␣1⫺␣2兲 t 1⫹t

册

⫹FP P

⬘

共zb兲

再

共m1⫺m2兲m3⫹ t 1⫹t 关共␣1⫺␣2兲共m1m3⫹m2m3⫺m1m2兲 ⫹␣1p2⫺␣2p

⬘

2兴⫺共␣1⫺␣2兲关共␣1⫹␣2兲共␣1p2⫹␣2p

⬘

2兲⫺␣1␣2q2兴

冉

t 1⫹t

冊

3

冎

, TABLE VIII. The polarization ratio ␣ and forward-backward

asymmetry AFB. H ␣ AFB J/␺ 1.15 ⫺0.21 D*0 0.10 ⫺0.46 B*0 0.94 0.35 Bs* 0 1.09 0.29

兵•••其A₁ b _⫽ 2 mP⫹mV

再

FPV共zb兲_{1⫹t 共}1 m1⫹2m2⫺m3兲⫺FPV

⬘

共zb兲

冋

m1m2m3⫹ 1 2共p 2_⫹p

_⬘

2_⫺q2_兲m 3 ⫹1_{2 1⫹t}t 兵关␣1m1⫹共2␣1⫹␣2兲 m2⫺共3␣1⫹␣2兲m3兴p2⫹关共␣1⫹2␣2兲m1⫹␣2m2⫺共␣1⫹3␣2兲m3兴p

⬘

2 ⫺关␣1m1⫹␣2m2⫺共␣1⫹␣2兲m3兴q2其⫺共m1⫹m2⫺m3兲关共␣1⫹␣2兲共␣1p2⫹␣2p

⬘

2兲⫺␣1␣2q2兴

冉

t 1⫹t

冊

2

册冎

, 兵•••其A₂ b ⫽共mP⫹mV兲关FPV

⬘

共zb兲兴

再

⫺m3⫺ t 1⫹t 关␣1m1⫹␣2m2⫺共3␣1⫹␣2兲m3兴⫹2共m1⫺m3兲␣1共␣1⫹␣2兲

冉

t 1⫹t

冊

2

冎

, 兵•••其A₃ b ⫽共mP⫹mV兲关FPV

⬘

共zb兲兴

再

m3⫹ t 1⫹t 关␣1m1⫹␣2m2⫹共␣1⫺␣2兲m3兴⫹2共m1⫺m3兲␣1共␣1⫺␣2兲

冉

t 1⫹t

冊

2

冎

, 兵•••其V b ⫽共mP⫹mV兲关⫺FPV

⬘

共zb兲兴

再

m3⫹ t 1⫹t 关␣1共m1⫺m3兲⫹␣2共m2⫺m3兲兴

冎

. We use the abbreviations

Pb⫽␣1p⫹␣2p

⬘

, zb⫽t

冉

兺

i⫽1 3 ␣imi 2_⫺␣1␣3 p2⫺␣2␣3p

⬘

2⫺␣1␣2q2

冊

⫺ t 1⫹tPb 2 . c decay: 兵•••其f_⫺ c _{⫽3 FP P共zc兲} 1 1⫹t

冋

1⫺共␣1⫹2␣2兲 t 1⫹t

册

⫹FP P

⬘

共zc兲

再

⫺2m1m3⫹m2m3⫹ t 1⫹t 关共␣1⫹2␣2兲 ⫻共m1m2⫹m1m3⫺m2m3兲⫺2共␣1⫹␣2兲p2⫹2␣2p

⬘

2⫺␣2q2兴⫹

冉

t 1⫹t

冊

2 ⫻关共3␣12_{⫹6␣1␣2⫹2␣2}2_兲p2_{⫺2␣2共␣} 1⫹␣2兲p

⬘

2_{⫹␣2共2␣} 1⫹3␣2兲q 2_兴 ⫺

冉

_1⫹tt

冊

3 共␣1⫹2␣2兲关␣1共␣1⫹␣2兲p2⫺␣1␣2p

⬘

2⫹␣2共␣1⫹␣2兲q2兴

冎

, 兵•••其A₁ c _⫽ 2 mP⫹mV

再

FPV共zc兲 1 1⫹t 共m1⫺2m2⫺m3兲⫺FPV

⬘

共zc兲

冋

⫺m1m2m3⫹ 1 2共p 2_⫹q2_⫺p

_⬘

2_兲m 3

册

⫹1_{2 1⫹t}t 兵关␣1m1⫺共2␣1⫹␣2兲m2⫺共3␣1⫹␣2兲m3兴p2⫹关⫺␣1m1⫹␣2m2⫹共␣1⫹␣2兲m3兴p

⬘

2 ⫹关共␣1⫹2␣2兲m1⫺␣2m2⫺共␣1⫹3␣2兲m3兴q 2_其⫺

冉

t 1⫹t

冊

2 共m1⫺m2⫺m3兲关␣1共␣1⫹␣2兲p 2 ⫹␣2共␣1⫹␣2兲q2⫺␣1␣2p

⬘

2兴

冎

, 兵•••其A₂ c _{⫽共mP⫹mV兲关⫺FPV}

_⬘

_共zc兲兴

再

_m 3⫹ t 1⫹t 关␣1m1⫺␣2m2⫺共3␣1⫹␣2兲m3兴 ⫺2共m1⫺m3兲␣1共␣1⫹␣2兲

冉

t 1⫹t

冊

2

冎

,