Radiative Leptonic Bc Decays

We analyze the radiative leptonic Bc decay mode: Bc → `νγ (` = e, μ) using a QCD-inspired constituent quark model. The prediction: B(Bc → `νγ) ' 3× 10−5 confirms that this channel is experimentally promising in view of the large number of Bc mesons which are expected to be produced at the future hadron facilities. E-mail address: [email protected] E-mail address: [email protected] Recently, the CDF Collaboration at the Fermilab Tevatron has reported the observation of the Bc meson, the lowest mass b̄c (bc̄) bound state, through the semileptonic decay mode B± c → J/ψ`±X [1]. The measured mass and lifetime of the meson are MBc = 6.40± 0.39 (stat) ± 0.13 (syst) GeV (1) τ(Bc) = 0.46 +0.18 −0.16 (stat)± 0.03 (syst) ps . (2) The particular interest of this observation is related to the fact that the meson groundstate with open beauty and charm can decay only weakly, thus providing the rather unique opportunity of investigating weak decays in a heavy quarkonium-like system. Moreover, studying this meson, important information can be obtained, not only concerning fundamental parameters, such as, for example, the element Vcb of the Cabibbo-KobayashiMaskawa mixing matrix, but also the strong dynamics responsible of the binding of the quarks inside the hadron. Understanding such dynamics is one of the most important issues in the analysis of heavy hadrons [2]. The observation of the Bc meson at the Tevatron confirms that Bc physics will gain an important role at the future hadron facilities, where a large production rate of such particles is expected; in particular, at the Large Hadron Collider (LHC), which will be operating at CERN, it is estimated that 4.5× 10 B c mesons will be produced per year for a machine luminosity of L = 10 cm−2 sec−1 at √ s = 14 TeV [3]. Bc meson decays can be classified according to the mechanism inducing the processes at quark level. Neglecting Cabibbo-suppressed and penguin-induced transitions, such mechanisms are: • the b-quark transition b→ c W−, with the c̄ quark having the role of spectator; the corresponding final states are of the kind J/ψ π, J/ψ `ν; • the charm quark transition c̄→ s̄ W−, with b as spectator and possible final states Bs π, Bs `ν, etc.; • the annihilation modes c̄b→W−. The first two mechanisms are responsible of the largest part of the Bc decay width [5, 6, 7]. In particular, the measurement (2) provides us with an indication that the dominant Bc decay mechanism is the c-quark decay, which implies a Bc lifetime in the range τBc = (0.4−0.7) ps [7], whereas dominance of the b−quark decay mechanism would 1 produce a longer lifetime: τBc = (1.1 − 1.2) ps [6]. Various analyses of Bc transitions induced by the two mechanisms are available in the literature [2]; for example, a QCD sum rule calculation of Bc semileptonic decays suggested the dominance of the charm transition [8]. As far as the annihilation processes are concerned, the leptonic radiative decay mode Bc → `νγ and the leptonic decay without photon in the final state represent a minor fraction of the Bc full width. Nevertheless, their analysis is of particular interest, both from the phenomenological and the theoretical point of view. From the phenomenological side, Bc annihilation modes are governed by Vcb; therefore they are Cabibbo-enhanced with respect to the analogous Bu decays, and represent new channels to access this matrix element. From the theoretical viewpoint, the purely leptonic and the radiative leptonic Bc transitions are interesting since, in the nonrelativistic limit of the quark dynamics, both their rates can be expressed in terms of a single hadronic parameter, the Bc leptonic decay constant fBc [9, 10]. In this limit, a relation between the widths of the processes Bc → `νγ and Bc → `ν can be worked out [9, 10]: R` = Γ(Bc → `νγ) Γ(Bc → `ν) ' 0.40 r` (3) with r` = α 4π m2Bc m` . Eq.(3) implies that the width of the radiative leptonic Bc decay into muons is nearly equal to the purely leptonic width Γ(Bc → μν), whereas, in the case of electrons in the final state, the radiative leptonic mode is largely dominant. Eq.(3) presents uncertainties coming from the used values of the charm and beauty quark masses. Moreover, there could be corrections if the quark dynamics in the Bc meson deviates from the nonrelativistic regime, and the size of the corrections is useful for understanding the theoretical uncertainty affecting the ratio (3). Corrections to the ratio (3) can be estimated by considering a model for the Bc meson where relativistic effects in the constituent quark dynamics are, at least partially, taken into account; the present letter is devoted to such a study. In order to analyze the decay mode Bc → `νγ (` = e, μ), we follow the method adopted in ref.[11] to investigate the analogous Bu transition. Let us consider the process B− c (p) → `−(p1) ν̄(p2) γ(k, ) (4)