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Using Heavy Quark Spin Symmetry in Semileptonic $B


BARI-TH/99-351 UGVA-DPT 1999/09-1051 August 1999

Using Heavy Quark Spin Symmetry in Semileptonic Bc Decays
Pietro Colangeloa and Fulvia De Fazio?b

arXiv:hep-ph/9909423v1 17 Sep 1999

b

Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Italy D?partement de Physique Th?orique, Universit? de Gen`ve, Switzerland e e e e

a

Abstract
The form factors parameterizing the Bc semileptonic matrix elements can be related to a few invariant functions if the decoupling of the spin of the heavy quarks in Bc and in the mesons produced in the semileptonic decays is exploited. We compute the form factors as overlap integral of the meson wave-functions obtained using a QCD relativistic potential model, and give predictions for semileptonic and non-leptonic Bc decay modes. We also discuss possible experimental tests of the heavy quark spin symmetry in Bc decays.

PACS numbers: 13.25.Hw,12.39.Pn

? “Fondazione

Angelo Della Riccia” Fellow

1

I. INTRODUCTION
+ The discovery of the Bc meson by the CDF Collaboration at the Fermilab Tevatron [1] opens for interesting investigations concerning the structure of strong and weak interactions in the quarkonium-like ? hadronic system. The studies will be further developed at the bc hadronic machines currently under construction, such as the LHC accelerator at CERN, where a copious production of Bc meson and of its radial and orbital excitations is expected [2,3]; at these experimental facilities, together with the measurement of the mass of the particles belonging to the ? (b?) family, it will be possible to observe the decay chains bc c reaching the 1 S0 ground state, the Bc , which decays weakly. A peculiarity of the Bc decays, with respect to the decays of the Bu,d and Bs mesons, is that both the quarks are involved in the weak decay process with analogous probability. The weak decays of the charm quark, whose mass is lighter than the b quark mass, are mainly governed by the CKM matrix element Vcs which is larger than Vcb mainly controlling the b quark transitions; the result is that both the quark decay processes contribute on a comparable footing to the Bc decay width. Another peculiar aspect is that the ? annihibc lation amplitude, proportional to Vcb , is enhanced with respect to the analogous amplitude describing the B + annihilation mode. The above considerations have inspired several theoretical analyses [4–8] aimed at predicting the Bc lifetime. Namely, a QCD analysis [7], based on the OPE expansion in the inverse mass of the heavy quarks and on the assumption of quark-hadron duality, provides for τBc a prediction in agreement (at least within the current experimental accuracy) with the CDF measurement: τ (Bc ) = 0.46+0.18 (stat) ± 0.03 (syst)10?12 s [1]. The agreement ?0.16 supports the overall picture of the inclusive Bc decays. The calculation of the Bc exclusive decay modes can be carried out either using QCDbased methods, such as lattice QCD or QCD sum rules, or adopting some constituent quark model. So far, lattice QCD has only been employed to calculate the Bc purely leptonic width [9]. As for QCD sum rules [10], the Bc leptonic constant, as well as the matrix elements relevant for the semileptonic decays, were computed in refs. [11,6,12]. These analyses identi?ed a di?culty in correctly considering the Coulomb pole contribution in the three-point functions needed for the calculation of the semileptonic matrix elements. Attempts aimed at taking this correction into account are described in [13]; however, the problem of including the contribution of the Coulomb pole for all the values of the squared momentum transfer t to the lepton pair has not been solved, yet. Extending to all values of t the expression of the Coulomb contribution valid at tmax only allows to conclude that it represents a large correction to the lowest order quark spectral functions. It is worth looking at the outcome of constituent quark models which, although less established on the QCD theoretical ground, can nevertheless provide us with signi?cant information to be compared to the experimental results. The models in refs. [14,15] have been used in the past [4,16] to estimate the semileptonic Bc decay rates. More recently, di?erent versions of the constituent quark model have been used to analyze the decays induced both by the b → c(u) and c → s(d) transitions [17,18]. It is noticeable that the calculations can be put on a ?rmer theoretical ground if some dynamical features of the Bc decays are taken into account. Such features are mainly related to the decoupling of the spin of the heavy quarks of the Bc meson, as well as of the

2

meson produced in the semileptonic decays, i.e. mesons belonging to the cc family (ηc , J/ψ, ? (?) (?) (?) etc.) and mesons containing a single heavy quark (Bs , Bd , D ). The decoupling occurs in the heavy quark limit (mb , mc ? ΛQCD ), and produces a symmetry, the heavy quark spin symmetry, allowing to relate the form factors governing the Bc decays into a 0? and 1? ?nal meson to a few invariant functions [19]. The main consequence is that the number of form factors parameterizing the matrix elements is reduced, and the description of the semileptonic transitions is greatly simpli?ed. However, at odds of the heavy quark ?avour symmetry, holding for heavy-light mesons, spin symmetry does not ?x the normalization of the form factors at any point of the phase space. The normalization, as well as the functional dependence near the zero-recoil point, must be computed by some nonperturbative approach. So far, the “universal” form factors of semileptonic Bc decays have been estimated using nonrelativistic meson wave-functions [19] and employing the ISGW model at the zero-recoil point [20]. An analysis in the framework of a di?erent quark model is described in [17]. In this paper we present a calculation based on a constituent quark model which has been used to describe several aspects of the heavy meson phenomenology [21]. The peculiar features of the model are related to the interquark potential, which follows general QCD properties, such as scalar ?avour-independent con?nement at large distances, and asymptotically free QCD coulombic behaviour at short distances. Moreover, the use of the relativistic form of the quark kinematics allows to describe heavy-light as well as heavy-heavy mesons, and to account for deviations from the nonrelativistic limit. As a result, the Bc form factors can be written as overlap integrals of meson wave-functions, obtained by solving the wave equation de?ning the model. As discussed in the following, the representation as overlap integral of meson wave-functions allows to predict, in the heavy quark limit, the normalization of the invariant functions at the zero-recoil point and to obtain, for example, the suppression factor between the form factors of the Bc transitions into heavy-light mesons with respect to the corresponding functions governing the decays Bc → ηc ?ν and Bc → J/ψ?ν. The calculation of the overlap integrals and of the Bc semileptonic form factors is presented in Sec. III, after having reviewed in Sec. II the consequences of the heavy quark spin symmetry in Bc decays. In Sec. IV, using the obtained invariant functions, we analyze the semileptonic decay modes, and in Sec. V, assuming the factorization ansatz, we estimate several non-leptonic Bc decay rates. Sec. VI is devoted to the conclusions.
II. HEAVY QUARK SPIN SYMMETRY

Heavy quark spin symmetry amounts to assume the decoupling between the spin of the heavy quarks in the Bc meson, since the ? spin-spin interaction vanishes in the in?nite bc heavy quark mass limit, as well as the vanishing of the heavy quark-gluon vertex. This symmetry has been invoked in [19] to work out relations among the semileptonic matrix elements between Bc and other heavy mesons (both heavy-heavy and heavy-light). The main di?erence with respect to the most well known case of the heavy-light systems is that in the latter case one can exploit heavy quark ?avour symmetry, which also holds in the heavy quark limit and allows to relate B to D form factors. In order to apply spin symmetry to Bc decays one should distinguish decays due to charm transitions from b quark transitions. To the ?rst category belong processes such as 3

? ? Bc → (Bs , Bs )?ν and Bc → (Bd , Bd )?ν, induced at the quark level by the transitions c → s and d, respectively. Since mc ? mb , the energy released in such decays to the ?nal hadronic system is much less than mb , and therefore the b quark remains almost una?ected. As a consequence, the ?nal Ba meson (a is a light SU(3)F index) keeps the same Bc four-velocity v, apart from a small residual momentum q. The initial and ?nal meson momenta can then 1 be written as: pBc = MBc v and pBa = MBa v + q, with v · q = O( mQ ). The relation between the residual momentum q and the momentum k transferred to the lepton pair is

k ? = p? c ? p? a = (MBc ? MBa )v ? ? q ? . B B

(2.1)

In this kinematic situation, exploiting the decoupling of the spin of the heavy quarks in the mesons, several relations can be worked out among the semileptonic Bc form factors. A straightforward way to derive such relations is to use the trace formalism [22,23] 1 . This has been done in ref. [19], and we repeat here the derivation for the sake of completeness. ? ? One introduces a 4 × 4 matrix H cb describing the doublet (Bc , Bc ) of c? mesons of fourb velocity v [19]: H cb =
?

(1+ v ) ?? (1? v ) [Bc γ? ? Bc γ5 ] , 2 2

(2.2)

?? ? where Bc and Bc annihilate a vector Bc and a pseudoscalar Bc meson of four-velocity v. ? ? ? ? Under spin rotations of the heavy quarks, H cb transforms as H cb → Sc H cb S? . b ? On the other hand, for heavy-light Ba and Ba mesons, the analogous 4 × 4 matrix ? describing the (Ba , Ba ) spin multiplet reads:

Ha =

(1+ v ) ?? [Ba γ? ? Ba γ5 ] ; 2

(2.3)

all the ?elds in (2.2),(2.3) contain a factor MBc,a and have therefore dimension 3/2. Applying the trace formalism, one gets that the hadronic matrix elements relative to (?) the decays Bc → Ba ?ν have the following general form, compatible with heavy quark spin symmetry:
? (?) ? < Ba , v, q|?a Γc|Bc , v >= ? MBc MBa T r[Ha ?ΓH cb ] q

(2.4)

where ? is the most general Dirac matrix proportional to the four-velocity v and to the residual momentum q. The calculation using (2.2),(2.3) shows that the various matrix elements reduce to: < Ba , v, q|V?|Bc , v > = 2MBc 2MBa [?a v? + a0 ?a q? ] , 1 2
? 2MBc 2MBa a0 ?a ??ναβ ??ν q α v β , 2

? < Ba , v, q|V?|Bc , v > = ?i ? < Ba , v, q|A?|Bc , v > =

(2.5)

? 2MBc 2MBa [?a ?? + a0 ?a ?? · q v? ] , 1 ? 2

a discussion of the heavy quark formalism applied to the quarkonium system see ref. [24] and references therein.

1 For

4

where V? and A? represent the weak ?avour-changing (c → s, d) vector and axial current, ? respectively, and ? is the Ba polarization vector. Therefore, as shown by eq.(2.5), the six ? form factors parameterizing the Bc into Ba and Ba matrix elements can be expressed in terms of two invariant functions, ?a and ?a . The main di?erence with respect to the spin-?avour 1 2 symmetry, holding in heavy-light mesons, is that the normalization of the form factors is not predicted at any point of the kinematic range and, in particular, it is not ?xed at the non-recoil point q = 0. Actually, the form factors ?a give rise to terms proportional to the lepton mass in the 2 calculation of the semileptonic rates. Moreover, ?a do not contribute at zero-recoil. The 2 scale parameter a0 is related to the size of the Bc meson, it can be assumed as proportional to the Bc Bohr radius and represents the typical range of variation of the form factors [19]. The relations (2.5) are valid near the zero-recoil point, where both Bc and the meson (?) (?) produced in the decay are nearly at rest. In the case of the transitions Bc → Bs , Bd the physical phase space is quite narrow (the maximum momentum transfer t to the lepton pair is tmax ? 1 GeV2 ) and therefore one can assume that eqs.(2.5) completely determine the semileptonic matrix elements (modulo a set of corrections mentioned below). The situation is di?erent for processes induced, at the quark level, by the b?quark transitions. Let us consider the decays Bc → (D, D ?)?ν, induced by the b → u transition. In this case, the energy released to the ?nal meson is small only near the zero-recoil point, where q 2 ? m2 . c At such kinematic point one can repeat the considerations for the transition Bc → Bs ?ν, obtaining the relations: < D, v, q|V?|Bc , v > = 2MBc 2MD [Σ1 v? + a0 Σ2 q? ] , 2MBc 2MD? a0 Σ2 ??ναβ ??ν q α v β , (2.6)

< D? , v, q|V?|Bc , v > = ?i < D? , v, q|A?|Bc , v > =

2MBc 2MD? [Σ1 ?? + a0 Σ2 ?? · q v? ] . ?

Far from the non-recoil point, the light recoiling quark keeps a large momentum, and thereq fore terms of the order of mc cannot be neglected in the e?ective theory leading to (2.6). Finally, we consider Bc decays into quarkonium states, such as ηc and J/ψ. The spin decoupling of both the beauty and charm quark allows now to relate the six form factors to a single one: < ηc , v, q|V? |Bc , v > = < J/ψ, v, q|A?|Bc , v > = 2MBc 2Mηc ? v? 2MBc 2MJ/ψ ? ?? . ? (2.7)

Also in this case eqs.(2.7) are only valid near the zero-recoil point. Nevertheless, in the following we use them, as well as eqs. (2.6), for all physical values of the momentum transfer t, in order to compute semileptonic and non-leptonic Bc decay rates. This is admittedly a strong assumption, and the related uncertainty must be added to the uncertainties coming from ?nite mass and QCD corrections that in principle relate the invariant functions to the physical semileptonic matrix elements [19]. However, assuming eqs.(2.7) and (2.6) in the whole kinematic range, a number of predictions can be collected; the experimental results will then provide us with indications on the numerical importance of the corrections. 5

III. BC FORM FACTORS FROM A CONSTITUENT QUARK MODEL

In this section we compute the form factors ?, ?a and Σ1 by using a relativistic potential 1 model which allows to account for two QCD e?ects. The ?rst one is con?nement, which produces a suppression, at large distances, of the meson wave-functions, due to the linearly increasing interquark potential. The second e?ect is represented by the deviation of the quark dynamics from the nonrelativistic limit. By taking such two e?ects into account, we are able to compute the form factor ? in (2.7) as an overlap integral of Bc and J/ψ wave(?) (?) functions. Moreover, we can apply the formalism to the transitions Bc → Bs , Bd and (?) Dd at the non-recoil point, and then extrapolate the result to the whole kinematic region spanned by the various semileptonic transitions. Let us consider ? in (2.7). In order to compute it, we consider the costituent quark model studied in [21], whose essential features can be easily summarized. First, we write + + down an expression for the Bc meson state, in the Bc rest frame, in terms of quark and antiquark creation operators, and of a meson wave-function: δαβ δrs + |Bc >= i √ √ 3 2 dk ψBc (k) b? (?k, r, α) c? (k, s, β)|0 > (3.1)

where α and β are colour indices, r and s spin indices. The operator b? creates an anti-b quark with momentum ?k, while c? creates a charm quark with momentum k. A similar expression holds for the ηc (?c) state, as well as for vector 1? states, as described in [21]. c In the meson state, as written in (3.1), the contribution of other Fock states such as, e.g., states containing one or more gluons, is neglected. The wave-function ψBc (k) describes the momentum distribution of the quarks in the meson. It is obtained by solving the wave equation k 2 + m2 + b k 2 + m2 ? MBc ψBc (k) + c dk ′ V (k, k ′ ) ψBc (k ′ ) = 0 (3.2)

stemming from the quark-antiquark Bethe-Salpeter equation, in the approximation of an istantaneous interaction represented by the potential V . Eq.(3.2) partially takes into account the relativistic behaviour of the quarks in the kinetic term; mc and mb represent the mass of the constituent charm and beauty quark, and MBc the mass of the bound state. The QCD interaction is described assuming a static interquark potential having the form, in the coordinate space [25]: V (r) = 8π f (Λr) Λ Λr ? 33 ? 2nf Λr
∞ 0

,

(3.3)

with Λ a scale parameter, nf the number of active ?avours, and the function f (t) given by f (t) = 4 π dq sin(qt) 1 1 ? 2 . 2) q ln(1 + q q (3.4)

The interest for this form of the potential is that it continuously interpolates the linearly con?ning behaviour at large distances with the QCD coulombic behaviour at short distances, where the logarithmic reduction of the strong coupling constant, due to the asymptotic 6

freedom property of QCD, is implemented. A further smoothing of the potential at short distances is adopted, according to quark-hadron duality arguments [21]. The wave equation (3.2), together with the form (3.3) of the potential and (3.1) of the meson state, completely determines the model, which has been extensively studied to describe static as well as dynamic properties of mesons containing heavy quarks [26–28]. Notice that the spin interaction e?ects are neglected since, in the case of heavy mesons, the chromomagnetic coupling is of the order of the inverse heavy quark masses. Therefore, both the pseudoscalar and the vector mesons, being degenerate in mass, are described by the same wave-function. An equation for the form factor ?(q = 0) in (2.7) can be obtained expressing the b → c ?avour-changing weak currents in terms of quark and antiquark operators; for the vector current, the expression is V? = δαβ (2π)3 dqdq ′ mb mc Eb (q)Ec (q ′ )
1 2

: [?b (q, r)b? (q, r, α) + vb (q, r)db (q, r, α)]γ ? u ? b [uc (q ′ , s)bc (q ′ , s, β) + vc (q ′ , s)d? (q ′ , s, β)] : ? c (3.5)

(Eq (k) = k 2 + m2 , k = |k|); an analogous expression describes the axial current. Then, q writing down the matrix elements (2.7) and applying canonical anticommutation relations [21,26], we obtain: ?(q = 0) = 1 2 2MBc 2Mηc
0 ∞

uBc (k)uηc (k) (Eb + mb )(Ec + mc ) ? k 2 , dk √ [(Eb + mb )(Ec + mc )]1/2 Eb Ec

(3.6)

where the reduced wave-functions uM (k) are related to the L = 0 wave-functions ψM according to uM (k) = k ψM (|k|) √ 2π . (3.7)

The covariant normalization is adopted: 0∞ dk|uM (k)|2 = 2MM . The wave-functions uBc and uηc can be obtained by solving eq.(3.2) by numerical methods, choosing the values of the masses mc and mb of the constituent quarks, together with the scale parameter Λ, in such a way that the charmonium and bottomonium spectra are reproduced: mb = 4.89 GeV and mc = 1.452 GeV, with Λ = 397 MeV [21]. A ?t of the heavy-light meson masses also ?xes the values of the constituent light-quark masses: mu = md = 38 MeV and ms = 115 MeV [21]. It is worth observing that, for the ? system, bc all the input parameters needed in (3.2) are ?xed from the analysis of other channels, and the predictions do not depend on new external quantities. The numerical solution of (3.2) produces the spectrum of the ? bound states; the prebc dicted mass and the leptonic constant of the ?rst S?wave resonance are [28]: MBc = 6.28 GeV (the value we use in our analysis) and fBc = 432 MeV, in agreement with other theoretical determinations based on constituent quark models [29], QCD sum rules (MBc = 6.35 GeV [6]) and lattice QCD (MBc = 6.388 ± 9 ± 98 ± 15 GeV [30]). Within the errors, the Bc mass agrees with the CDF result: MBc = 6.40 ± 0.39 (stat) ± 0.13 (syst) GeV [1]. 7

The obtained Bc wave-function uBc (k) is depicted in ?g.1. In the same ?gure we plot the wave-functions of the other mesons involved in Bc semileptonic decays: Bs and Bd , the cc ? ′ states ηc and J/ψ together with the ?rst radial excitation ηc and ψ(2S), and the D meson. Let us come back to eq.(3.6) which provides the form factor ?. For quark masses larger than the typical relative quark-antiquark momentum k, eq.(3.6) becomes: ?(q = 0) = 1 (2π)3 1 2MBc 2Mηc 1 2MBc 2Mηc
? dk ψBc (k) ψηc (k)

= where ΨM (x) is de?ned as

dx ΨBc (x) Ψ?c (x) η

,

(3.8)

ΨM (x) =

1 (2π)3

dk eik·x ψM (k) .

(3.9)

Eq. (3.8) shows that the form factor ?, at the zero-recoil point, is simply given by the overlap integral of the Bc and ηc wave-functions in the coordinate space. This result has already been obtained in [19], as it is typical of the calculation of form factors by quark models [26,31]. The interest in eq.(3.8) is that no factors appear in the integral other than the wave functions; this implies that, in the limit where the Bc and ηc wave-functions are equal (modulo the normalization condition), the form factor ? is 1. Although such an overlap is not constrained by symmetry arguments, as in the case of the ?avour symmetry in heavy-light mesons, from eq.(3.8) it turns out that the deviation from unity of the invariant function at the zero-recoil point is due to the actual shapes of the meson wave-functions. In our speci?c case, as reported in Table I, the deviation from unity is a 5% e?ect. The calculation of ? near the zero-recoil point, for a small momentum q, can be performed by modifying eq.(3.8), as discussed in [19]: ?(q) = 1 2MBc 2Mηc dx eiq·x/2 ΨBc (x) Ψ?c (x) , η
p p

(3.10)

B η 2 and using the relation (valid near the zero-recoil point) y = MBc Mc = 1 + q 2 /Mηc . We ηc c choose to perform an extrapolation of the result in the whole kinematic region, obtaining the form factor depicted in ?g.2. The extrapolation provides a form factor having a nearly linear (with a small curvature term) y?dependence in the kinematic range of the decays Bc → ηc ?ν and Bc → J/ψ?ν. The same method and the same formulae can be used to calculate the form factor ?′ of ′ Bc → ηc and Bc → ψ(2S); the only new ingredient is the wave-function of the ψ(2S) radial excitation. Due to the oscillating behaviour of uψ(2S) , the function ?′ is suppressed with respect to ?; interestingly enough, it has a negligible y?dependence, as one can observe in ?g.2. Before discussing the phenomenology of the decays Bc → ηc (J/ψ)?ν and Bc → ′ ? ηc (ψ(2S))?ν, let us consider the matrix elements relevant for the transitions Bc → Bs (Bs ). A feature of the model we are considering is that both heavy-heavy and heavy-light mesons are

8

described by the same formalism. Therefore, eq.(3.6) can be applied to calculate ?s (q = 0), 1 substituting mb with ms and the wave-function uηc with uBs . In the limit ms → 0 and for a large value of the b?quark mass, eq.(3.6) becomes: 1 1 ?s (q = 0) = √ √ 1 2 2MBc 2MBs dx ΨBc (x) Ψ? s (x) , B (3.11)

1 which di?ers by a factor √2 with respect to the analogous relation for ?. This factor is a consequence of considering a heavy-light meson in the ?nal state instead of a heavy-heavy meson, and produces a suppression of the corresponding form factor. Eq.(3.11) suggests that, for similar (modulo the normalization condition) Bc and Bs wave-functions, the form √ factor ?s (q = 0) is close to the value ?s (q = 0) = 1/ 2. The actual value, reported in 1 1 Table I, di?ers from this value by a 7% e?ect. √ The two results ?(q = 0) ? 1 and ?s (q = 0) ? 1/ 2 are the main predictions of our 1 analysis. They would deserve independent checks by di?erent theoretical methods, namely by QCD sum rules in the heavy quark limit. From eq.(3.11) it is also possible to derive a relation, proposed in [19], between the form factor ?s and the leptonic constant of the Bs meson. As a matter of fact, in the framework 1 of the constituent quark model, the Bs leptonic constant, de?ned by the matrix element: < 0|A? |Bs (p) >= ifBs p? , is given by [21]: √ ∞ 3 (Eb + mb )(Es + ms ) 1/2 k2 fBs = dk k uBs (k) ] . (3.12) [1 ? 2πMBs 0 Eb Es (Eb + mb )(Es + ms )

For vanishing ms and large mb , fBs is simply related to the Bs wave-function at the origin: √ 3 (3.13) ΨB (0) , fBs = MBs s a relation analogous to the van Royen-Weisskopf formula for the quarkonium state. Expanding ΨBs (x) near the origin in (3.11), we obtain: √ 1 1 ?s (q = 0) ? √ fBs M Bs √ 1 2MBc 2 3 dx ΨBc (x) + corrections . (3.14)

The numerical comparison of (3.14) with (3.11), however, suggests that the next-to-leading corrections in (3.14) are sizeable, and therefore the expansion (truncated at the ?rst term) leading to eq.(3.14) appears to be of limited usefulness. The value of ?s at zero-recoil is reported in Table I, and the plot of the form factor, 1 extrapolated in the whole kinematic region, is depicted in ?g.2; the form factor presents a ? soft y-dependence in the narrow kinematic range spanned by the semileptonic Bc → Bs , Bs transitions. The same procedure can be applied to compute ?d and Σ1 , and the results are also 1 depicted in ?g.2. The only new information is that, keeping ?nite values of the light quark masses, a SU(3)F breaking e?ect between ?d and ?s of less than 3% is predicted. 1 1 All the invariant functions can be represented by the three-parameter formula F (y) = F (0) 1 ? ρ2 (y ? 1) + c (y ? 1)2 9 (3.15)

in terms of the value at zero-recoil, the slope ρ2 and the curvature c; the corresponding values are collected in Table I. A remark concerns the invariant functions ?s,d and Σ2 . As mentioned in Sect.II, such 2 form factors do not contribute at the zero-recoil point, since they appear in the term proportional to the small momentum q. In our approach, based on considering overlap integrals of wave-functions of mesons at rest, we cannot provide an independent calculation of ?s,d 2 and Σ2 , which therefore will be neglected in our analysis. Such an approximation, however, could have relevant consequences only in the case of the transitions Bc → D (?) ?ν; as already (?) underlined, for the decays Bc → Bs and Bc → B (?) the contribution from ?2 is always proportional to the momentum q, which remains small in these processes. Let us conclude the section comparing our form factors ?, ?a and Σ1 with the outcome 1 of the ISGW model [15], which has been widely applied to describe the heavy meson decays. In the ISGW approach, the form factors exponentially depend on the squared momentum transfer to the lepton pair, and at zero-recoil they are given by products of parameters relative to the mesons involved in the decays. We depict in ?g.2 the various invariant functions obtained in this approach, observing some agreement with our results in the case √ of ?; as for ?s , the result based on [15] deviates considerably from the value 1/ 2 suggested 1 by our model.
IV. BC SEMILEPTONIC DECAYS

The form factors ?s and ?d , ?, ?′ and Σ1 can be used to predict the semileptonic 1 1 Bc decay rates, as well as various decay distributions. Before doing the calculation let us stress again that an extrapolation is performed for the relevant matrix elements far from the symmetry point (zero-recoil) where the form factors are originally computed. Such a procedure would require the calculation of the corrections, which could be sizable far from the symmetry point, an analysis beyond the aim of the present work. Considering the small range of momentum transfer t involved in c → (s, d) transitions, it is plausible (?) (?) that the extrapolation is quite under control for the decays Bc → Bs ??, Bd ??. As for ν ν Bc → ηc , J/ψ??, the extrapolation is done on a wider range of momentum transfer to the ν lepton pair. However, also in this case it is interesting to make predictions and to compare them with the experimental results. Notice that we only consider massless charged leptons in the ?nal state. Concerning the parameters needed in the analysis, we use the experimental values of ′ ′ the masses of ηc , J/ψ, ψ(2S), D (?) , B (?) and Bs mesons; for the ηc we use Mηc = 3.66 ? ? ? GeV, and for MBs we put: MBs = MBs + (MBd ? MBd ). For the CKM matrix elements we use Vcb = 0.039 and Vub = 0.0032; the values of Vcs and Vcd are ?xed to Vcs = 0.975 and Vcd = 0.22. The results for the decay widths are reported in Table II where we also report the corresponding branching fractions, obtained assuming for τBc the CDF central value: τBc = 0.46 ps. In order to understand the e?ect of the t?dependence of the form factors, we also report in Table II the results obtained assuming t?independent invariant functions, with the values ?xed at the zero-recoil point. The results provide us with an upper bound for the various decay widths. As expected, the momentum transfer dependence is mild in the case of the (?) (?) Bc → Bs , Bd decays, where it only provides an e?ect of less than 10% in the decay rates. 10

This is mainly due to the narrow t? range spanned in such decay modes. In the case of Bc → ηc and J/ψ, there is a sizeable e?ect due to the t? dependence of the form factors. On ′ the contrary, in the case of decays into radial excited states, ηc and ψ(2S), the t dependence is negligible. The t?dependence is important for the Cabibbo suppressed Bc decays into D and D ? . From Table II we conclude that the semileptonic modes are dominated by two channels, ? Bc → Bs ?ν and Bc → Bs ?ν, in spite of the small phase space available for both the transitions; the two modes nearly represent the 60% of the semileptonic width, a result in agreement with the predictions available in the literature. As for the b → c induced semileptonic Bc transitions, a peculiar role is played by the Bc decay into J/ψ, due to the clear signature represented by three charged leptons from the same decay vertex, two of them coming from J/ψ. This signature has been exploited to identify the Bc meson at Tevatron [1], and will be mainly employed at the future colliders [34]. Our prediction for the width of the decay Bc → J/ψ?ν is: Γ(Bc → J/ψ?ν) ? 21.6 × 10?15 GeV, with an upper bound of 48 × 10?15 GeV obtained using a t?independent form factor ?. The agreement of this result with other calculations in the literature suggests that the ?nite mass corrections, responsible of subleading form factors in the matrix elements, should not be large. Tests on the size of such corrections can be performed by measuring the Bc decay rates into longitudinally and transversely polarized J/ψ: ΓL,T = Γ(Bc → J/ψL,T ?ν), together with the corresponding decay distributions. Using the parameterization in (2.7) the decay widths are given by:
2 5 G2 Vcb MJ/ψ F ΓL = 12π 3 2 5 G2 Vcb MJ/ψ F 12π 3 1+δ 1 1+δ 1 (M

dy [?(y)]2 y 2 ? 1[r y ? 1]2 dy [?(y)]2 y 2 ? 1[r 2 + 1 ? 2 r y]
?M )2

ΓT =

(4.1)

B where r = MBc /MJ/ψ and δ = 2McB MJ/ψ . The measurement of dΓi /dy provides inforJ/ψ c mation on ? and Vcb; in particular, if the curvature term in ?(y) is neglected, the ratio ΓT /ΓL gives access to the slope ρ2 . The combination Vcb ?(1) can be obtained from the measurement of ΓL and from the total width, and therefore a measurement of Vcb is possible using this decay channel [34,32]. Such new determinations of the CKM element Vcb , even though not accurate as from Bd and Bu decays, would represent an important consistency check of the Standard Model. Tests of the spin symmetry are provided by the measurement of the decay distributions in the y variable, whose deviations from the distributions related to a unique form factor ? would imply the presence of spin symmetry-breaking terms. Let us ?nally observe that our prediction for the rates of the decays into 0? (?c) states, c ′ Bc → ηc ?ν and Bc → ηc ?ν, is smaller than the value reported by other analyses.

V. NON-LEPTONIC BC DECAYS

Estimates of the decay rates of several two-body non-leptonic Bc transitions can be obtained adopting the factorization approximation. Such an approximation ?nds theoretical 11

support in few cases (large Nc limit; mb → ∞ limit in b → u transitions involving heavy-light meson systems [35]); nevertheless, it is widely used to estimate non-leptonic decay rates of mesons containing heavy quarks. Let us ?rst consider non-leptonic Bc decay modes induced, at the quark level, by the b → c and u transitions. The e?ective Hamiltonian governing the processes reads: GF Hef f = √ { Vcb [c1 (?)Qcb + c2 (?)Qcb] + Vub [c1 (?)Qub + c2 (?)Qub ] + h.c.} 1 2 1 2 2 + penguin operators ; (5.1)

GF is the Fermi constant, Vij are CKM matrix elements and ci (?) scale-dependent Wilson coe?cients. The four-quark operators Qcb and Qcb are given by 1 2
? ? ? ? ? ? Qcb = [Vud (du)V ?A + Vus (?u)V ?A + Vcd (dc)V ?A + Vcs (?c)V ?A ] (?b)V ?A s s c 1 ? ? ? ? ? ? Qcb = [Vud (?u)V ?A (db)V ?A + Vus (?u)V ?A (?b)V ?A + Vcd (?c)V ?A (db) + Vcs (?c)V ?A (?b)] c c s c c s 2

(5.2) with (?1 q2 )V ?A = q1 γ? (1 ? γ5 )q2 ; analogous relations hold for Qub and Qub . q ? 1 2 As well known, the factorization approximation amounts to evaluate the matrix elements of the four-quark operators in (5.2) between the initial Bc state and the ?nal two-body hadronic states as the product of quark-current matrix elements. We adopt this approximation in the calculation of the rates, neglecting the contribution of penguin operators, since their Wilson coe?cients are small with respect to c1 and c2 (interference e?ects of penguin diagrams are of prime importance in producing CP violating asymmetries in Bc decays). Moreover, we do not take into account the weak annihilation contribution represented by a Bc meson annihilating into a charged W ; in this amplitude, the ?nal hadronic state is entirely produced out of the vacuum, and therefore the contribution should be characterized by a sizeable form factor suppression. Annihilation processes are presumably relevant mainly for rare or suppressed Bc decays; in these cases they deserve a dedicated analysis. A further remark concerns the Wilson coe?cients c1 (?) and c2 (?). Writing the factorized amplitudes and taking into account the contribution of the Fierz reordered currents, it turns out that the relevant coe?cients are the combinations: a1 = c1 + ξc2 and a2 = c2 + ξc1, with the QCD parameter ξ given by ξ = 1/Nc . Several discussions concerning this parameter are available in the literature. We choose a1 = c1 and a2 = c2 , i.e. ξ = 0, in the spirit of the large Nc limit, and use c1 and c2 computed at an energy scale of the order of mb . A detailed analysis of 1/Nc corrections to the coe?cients a1 , a2 as well as of the role of color-octet current operators in B decays can be found in [36]. Analogous considerations hold for the decays induced by the c → s(d) transitions; in this case we choose the coe?cients c1 and c2 at the scale of the charm mass. The factorized amplitudes can be expressed in terms of the form factors in eqs.(2.5), (2.6) and (2.7), and of leptonic decay constants de?ned by the matrix elements: < 0|A? |M(p) >= ifM p? and < 0|V? |V (p, ?) >= fV MV ?? . We use the values: fπ+ = 0.131 GeV, fρ+ = 0.208 GeV and fa1 = 0.229 GeV; fK + = 0.159 GeV, fK ?+ = 0.214 GeV and fK1 = 0.229 GeV; ′ fηc = 0.31 GeV, fηc = 0.23 GeV, fψ = 0.38 GeV, fψ′ = 0.28 GeV, and ?nally fD = 0.2 12

? GeV, fDs = 0.24 GeV and fD? = 0.23 GeV, fDs = 0.275 GeV. Such values correspond to experimental results or to average values from lattice QCD and QCD sum rules 2 . The decay rates of several non-leptonic Bc transitions, obtained using c1 (mb ) = 1.132, c2 (mb ) = ?0.286 and c1 (mc ) = 1.351, c2 (mc ) = ?0.631, are collected in Tables III, IV. Also in this case we use the physical phase space together with the expression of the matrix elements in (2.5)-(2.7). Few comments are in order. We observe the dominance of the decay modes induced by the + ? charm transition, and in particular of the channel Bc → Bs ρ+ , which represents more than 10% of the total Bc width. It would be interesting to experimentally con?rm this prediction, even though the ?nal state presents severe reconstruction di?culties. From the experimental point of view, more promising are the decay modes having a J/ψ meson in the ?nal state; + + among such modes, the decay channels Bc → J/ψπ + and Bc → J/ψρ+ are particularly useful for the precise measurement of the Bc mass, by the complete reconstruction of the ?nal state. Also the decay into a1 is of particular interest, due to the large decay rate. Several tests of factorization can be carried out, mainly using the decay channels having a J/ψ in the ?nal state. For example, the assumption of the factorization approximation, together with the heavy quark spin symmetry, implies that the relation

+ dΓ(Bc →J/ψ?+ ν)

+ Γ(Bc → J/ψπ + ) dy

|y=yπ

=

2 2 3π 2 Vud a2 fπ 1 MBc MJ/ψ

(5.3)

Bc J/ψ holds in the limit Mπ → 0 (yπ = 2MB MJ/ψ ). An analogous relation holds for the Bc c transition into the radial excited state ψ(2S):

M 2 +M 2

+ dΓ(Bc →ψ(2S)?+ ν) |y=yπ dy

+ Γ(Bc → ψ(2S)π + )

=

2 2 3π 2 Vud a2 fπ 1 . MBc Mψ(2S)

(5.4)

In the case of a ρ meson in the ?nal state one has:
+ dΓ(Bc →J/ψ?+ ν) |y=yρ dy

+ Γ(Bc → J/ψρ+ )

=

2 2 2 2 2 2 2 3π 2 Vud a2 fρ [8MJ/Ψ Mρ + (MBc ? MJ/ψ ? Mρ )2 ] 1 5 2 8MBc MJ/ψ 2 2 2 λ 2 (MBc , MJ/ψ , Mρ ) 2 y 2 ? 1[r 2 yρ ? 6ryρ + 2r 2 + 3] M M 2 +M 2
2 ?Mρ 1

×√

,

(5.5)

Bc . λ being the triangular function, r = MJ/ψ and yρ = Bc B J/ψ 2M c MJ/ψ + To test eqs.(5.3)-(5.5) two-body decay rates and the di?erential Bc → J/ψ?+ ν decay width are required; the measurement of such quantities, possible at the hadronic facilities, would provide us with important information on the heavy quark spin symmetry as well as on the factorization approximation in Bc decays.

description of the current theoretical situation concerning the heavy meson leptonic decay constants is reported in the Appendices C and D of ref. [33].

2A

13

VI. CONCLUSIONS

We have presented a determination of the invariant functions parameterizing the semileptonic Bc matrix elements in the in?nite heavy quark mass limit. The form factors are obtained as overlap integrals of meson wave-functions, obtained in the framework of a QCD relativistic potential model. An interesting result is that, although not constrained by symmetry arguments, the normalization of the form factor ? describing the transition Bc → J/ψ?ν is close to 1 at the zero-recoil point, as being the overlap of similar wave-functions. On the contrary, the form factors relative to the transitions into heavy-light mesons, at zero-recoil √ point, are suppressed by a factor ? 1/ 2 with respect to ?. These results have several phenomenological consequences, in semileptonic and non-leptonic Bc decay processes, which can be experimentally tested. Moreover, they a?ect other important processes, such as radiative ?avour-changing Bc decays [37] and CP violating Bc transitions [38,18]. In particular, the invariant functions computed in this paper can be useful to identify the Bc decay channels characterized by a clean experimental signature, a large branching fraction and a visible CP asymmetry; the identi?cation of this kind of decay modes is of paramount importance for the physics program of the experiments at the future accelerators.

Acknowledgments (F.D.F.) thanks Prof. R. Gatto for hospitality at D?partement de Physique Th?orique, Unie e versit? de Gen`ve, and for interesting discussions and encouragement. She also acknowledges e e ”Fondazione Angelo Della Riccia” for a fellowship.

14

REFERENCES
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16

TABLES
TABLE I. Parameters of the form factors (ψ ′ = ψ(2S)). The functional dependence is in (3.15). Channel ? Bc → Bs (Bs ) ? Bc → Bd (Bd ) Bc → ηc (J/ψ) ′ Bc → ηc (ψ ′ ) Bc → D(D ? ) Form factor ?s 1 ?d 1 ? ?′ Σ1 F (1) 0.66 0.66 0.94 0.23 0.59 ρ2 8 8 2.9 0 1.3 c 0 0 3 0 0.4

+ TABLE II. Semileptonic Bc decay widths and branching fractions.

Channel + Bc → Bs e+ ν ? + Bc → Bs e+ ν + → B e+ ν Bc d ? + Bc → Bd e+ ν + Bc → ηc e+ ν + Bc → J/ψe+ ν ′ + Bc → ηc e+ ν + Bc → ψ ′ e+ ν + Bc → D 0 e+ ν + Bc → D ?0 e+ ν

Γ(10?15 GeV) 11.1(12.9) 33.5(37.0) 0.9(1.0) 2.8(3.2) 2.1(6.9) 21.6(48.3) 0.3(0.3) 1.7(1.7) 0.005(0.03) 0.12(0.5)

ΓL (10?15 GeV) 19.1(21.4) 1.6(1.8) 13.2(33.2) 1.1(1.1) 0.08(0.35)

ΓT (10?15 GeV) 7.2(7.8) 0.6(0.8) 4.2(7.6) 0.3(0.3) 0.02(0.05)

BR 0.8(0.9) × 10?2 2.3(2.5) × 10?2 0.06(0.07) × 10?2 0.19(0.22) × 10?2 0.15(0.5) × 10?2 1.5(3.3) × 10?2 0.02(0.02) × 10?2 0.12(0.12) × 10?2 0.0003(0.002) × 10?2 0.008(0.03) × 10?2

17

+ TABLE III. Non-leptonic (b → c, u) Bc decay widths and branching fractions.

Channel ηc π + ηc ρ+ ηc a+ 1
′ ηc π + ′ ηc ρ+ ′ ηc a+ 1

Γ(10?15 GeV) a2 0.28 1 a2 0.75 1 a2 0.96 1 a2 0.074 1 a2 0.16 1 a2 0.15 1 a2 1.48 1 a2 4.14 1 a2 5.78 1 a2 0.22 1 a2 0.54 1 a2 0.65 1 a2 0.15 2 a2 0.13 2 a2 1.46 2 a2 2.4 2

BR 2.6 × 10?4 6.7 × 10?4 8.6 × 10?4 6.6 × 10?5 1.5 × 10?4 1.4 × 10?4 1.3 × 10?3 3.7 × 10?3 5.2 × 10?3 1.9 × 10?4 4.86 × 10?4 5.8 × 10?4 8.4 × 10?6 7.5 × 10?6 8.4 × 10?5 51.4 × 10?4

Channel ηc K + ηc K ?+ + ηc K1
′ ηc K + ′ K ?+ ηc ′ + ηc K1

Γ(10?15 GeV) a2 0.023 1 a2 0.041 1 a2 0.05 1 a2 0.0055 1 a2 0.008 1 a2 0.0075 1 a2 0.076 1 a2 0.23 1 a2 0.3 1 a2 0.01 1 a2 0.03 1 a2 0.033 1 a2 0.01 2 a2 0.009 2 a2 0.087 2 a2 0.15 2 (a1 0.86 + a2 0.46)2 × 10?1 (a1 0.7 + a2 0.9)2 × 10?1 (a1 0.28 + a2 0.7)2 × 10?1 (a1 0.17 + a2 0.8)2 × 10?1 (a1 1.31 + a2 0.47)2 × 10?1 (a1 2.02 + a2 2.3)2 × 10?1 (a1 0.35 + a2 0.36)2 × 10?1 (a1 0.55 + a2 1.76)2 × 10?1

BR 2 × 10?5 3.6 × 10?5 4.4 × 10?5 5 × 10?6 7.4 × 10?6 6.7 × 10?6 6.8 × 10?5 2 × 10?4 2.7 × 10?4 9.3 × 10?6 2.6 × 10?5 3 × 10?5 6 × 10?7 5.3 × 10?7 5 × 10?6 8.4 × 10?6 5 × 10?5 2 × 10?5 1 × 10?6 6 × 10?8 1.3 × 10?4 1.9 × 10?4 5.8 × 10?6 8.7 × 10?7

J/ψπ + J/ψρ+ J/ψa+ 1 ψ′ π+ ψ ′ ρ+ ψ ′ a+ 1 ? D+ D0 ? D + D?0 ?+ D 0 D ? ? D?+ D ?0 ηc Ds ? ηc Ds ′D ηc s ′ ? ηc Ds J/ψDs ? J/ψDs ′D ψ s ? ψ ′ Ds

J/ψK + J/ψK ?+ + J/ψK1 ψ′ K + ψ ′ K ?+ + ψ ′ K1
+? Ds D 0 +? Ds D?0 ?+ ? Ds D0 ?+ ? Ds D ?0

(a1 7.8 + a2 1.6)2 × 10?1 5 × 10?3 (a1 3.6 + a2 6.05)2 × 10?1 3.8 × 10?4 (a1 1.5 + a2 3.2)2 × 10?1 3.7 × 10?5 (a1 0.79 + a2 1.8)2 × 10?1 1 × 10?5 (a1 6.7 + a2 2.3)2 × 10?1 3.4 × 10?3 (a1 11 + a2 10.4)2 × 10?1 5.9 × 10?3 (a1 1.4 + a2 1.33)2 × 10?1 1 × 10?4 (a1 2.75 + a2 7.8)2 × 10?1 5.7 × 10?5

ηc D+ ηc D ?+ ′ ηc D+ ′ ηc D ?+ J/ψD+ J/ψD?+ ψ′ D+ ψ ′ D?+

18

+ TABLE IV. Non-leptonic (c → s, d) Bc decay widths and branching fractions.

Channel Bs π + Bs ρ + ? Bs π + ? Bs ρ+ Bd π + Bd ρ+ ? Bd π + ? Bd ρ+

Γ(10?15 GeV) a2 30.6 1 a2 13.6 1 a2 35.6 1 a2 110.1 1 a2 1.97 1 a2 1.54 1 a2 2.4 1 a2 8.6 1

BR 4 × 10?2 1.7 × 10?2 4.5 × 10?2 1.4 × 10?1 2.5 × 10?3 2 × 10?3 3 × 10?3 1 × 10?2

Channel Bs K + Bs K ?+ ? Bs K +

Γ(10?15 GeV) a2 2.15 1 a2 0.043 1 a2 1.6 1

BR 2.7 × 10?3 5.4 × 10?5 2 × 10?3 1.8 × 10?4 4 × 10?5 1.6 × 10?4 4.4 × 10?4

Bd K + Bd K ?+ ? Bd K + ? Bd K ?+

a2 0.14 1 a2 0.032 1 a2 0.12 1 a2 0.34 1

19

FIGURE CAPTIONS

Fig. 1 Reduced L = 0 wave-functions uM (k) of heavy-heavy (Bc , J/Ψ, ψ(2S)) and heavy-light (Bs , Bd , D) mesons. The wave-functions are obtained by solving the wave equation (3.2); they describe both the pseudoscalar 0? and vector 1? mesons.

Fig. 2 Form factors of Bc semileptonic decays. The variable y is related to the squared momentum
M Bc t, transferred to the lepton pair, by the relation: y = 2MB MM . The solid lines correspond c to the form factors obtained by the model discussed in the paper; the dashed lines refer to the model in ref. [15].

M 2 +M 2 ?t

20

FIGURES

4 3 2 1 0 0 1 2 3

4 3 2 1 0 0 1 2 3

4 3 2

4 3 2

1 0 -1 0 1 2 3 1 0 0 1 2 3

FIG. 1.

21

1 0.8 0.6 0.4 0.2 0 1 1.005 1.01

1 0.8 0.6 0.4 0.2 0 1 1.005 1.01

1 0.8 0.6 0.4 0.2 0 1 1.1 1.2

1 0.8 0.6 0.4 0.2 0 1 1.2 1.4 1.6

1 0.8 0.6 0.4 0.2 0 1 1.05 1.1 1.15

FIG. 2.

22


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