Radiative corrections to b → cτ ν̄ τ

Analytical calculation is presented of the QCD radiative corrections to the rate of the process b → cτ ν̄τ and to the τ lepton longitudinal polarization in τ ν̄τ rest frame. The results are given in the form of one dimensional infrared finite integrals over the invariant mass of the leptons. We argue that this form may be optimal for phenomelogical applications due to a possible breakdown of semilocal hadron-parton duality in decays of heavy flavours. The semileptonic decay rate of B mesons is one of the key ingredients in the determination of weak mixing angles. Transitions of the b quark to the charmed as well as to the up quark have been analysed in great detail, exploiting either the inclusive decay rate or exclusive channels. The analysis of the inclusive rate is, however, affected by the uncertainty in the b mass and by bound state corrections. This problem is only partly circumvented by fixing mb −mc through the difference of bottom and charmed meson masses and by relating the bound state effects to phenomenological constants that can be determined from other observables in the context of the heavy quark effective theory (HQET). The decay rate is, furthermore, affected by perturbative QCD corrections, which have been calculated analytically up to order αs for arbitrary b and c masses and massless leptons [1, 2, 3]. Numerical results for perturbative QCD corrections to the partial decay rate b → τ ν̄X have been recently obtained in Ref.[4]. The bound state corrections to this decay chanel up to order 1/mb are also known [4, 5, 6] in the context of the HQET. The comparison between decays into light (μ, e) and heavy (τ) leptons may furthermore help to test the theoretical approach and allow to fix some of the free parameters. Including b → u transitions, four kinematically different leptonic decay modes are thus available for the comparison. In this paper analytical results for the decay rate b → cτ ν̄τ are presented in the form of an one dimensional integral over the invariant mass squared w of the leptonic system. This formulation allows, at least in principle, the separation of the region of relatively small w, where the inclusive parton model description based on local parton-hadron duality should work, from the region of large w where only one or few resonances are produced and the duality between the parton and hadron description may be doubtful. In the region of large w i.e. close to the Shifman-Voloshin limit[8] w = w max the theoretical description of lepton spectra based on summation over exclusive channels is particularly simple and reliable. The HQET approach for exclusive decays[9, 10] on the other hand becomes quite involved if not impractical in the region of small w which is dominated by multiparticle final states. The calculation of the lowest order rate as well as corrections can be related in a straightforward way to the corresponding calculations for the decay into a virtual W boson with the subsequent integration over the mass of the lν̄ system. The differential decay rate is proportional to H Lαβ dPS(b → cτ ν̄) Hαβ depends on quark and gluon fields and Lαβ(τ ; ν) ∼ ∑ s [ ūτγα(1− γ5)vν ] [ ūτγβ(1− γ5)vν ] ∼ νατβ + τανβ − ν · τgαβ − iεαβγδντ δ (1) In a recent preprint [7] the breakdown of the local parton-hadron duality has been invoked as the origin of problems with the semileptonic decay rate of D mesons in the framework of HQET. Let us remark that for large w the kinetic energy of the hadronic system in B decays can be similar to that in D decays. Thus the semileptonic branching ratios for b decays may be also affected for the effective mass of the hadronic system close to the resonance region.