Renormalization Mass Scale and Scheme Dependence in the Perturbative Contribution to Inclusive Semileptonic b Decays

We examine the perturbative calculation of the inclusive semi-leptonic decay rate \Gamma for the b-quark, using mass-independent renormalization. To finite order of perturbation theory the series for \Gamma will depend on the unphysical renormalization scale parameter μ and on the particular choice of mass-independent renormalization scheme; these dependencies will only be removed after summing the series to all orders. In this paper we show that all explicit μ-dependence of \Gamma, through powers of ln(μ), can be summed by using the renormalization group equation. We then find that this explicit μ-dependence can be combined together with the implicit μ-dependence of \Gamma (through powers of both the running coupling a(μ) and the running b-quark mass m(μ)) to yield a μ-independent perturbative expansion for \Gamma in terms of a(μ) and m(μ) both evaluated at a renormalization scheme independent mass scale IM which is fixed in terms of either the ``\overline{MS} mass'' \overline{m}_b of the b quark or its pole mass m_{pole}. At finite order the resulting perturbative expansion retains a degree of arbitrariness associated with the particular choice of mass-independent renormalization scheme. We use the coefficients c_i and g_i of the perturbative expansions of the renormalization group functions \beta(a) and \gamma(a), associated with a(μ) and m(μ) respectively, to characterize the remaining renormalization scheme arbitrariness of \Gamma. We further show that all terms in the expansion of \Gamma can be written in terms of the c_i and g_i coefficients and a set of renormalization scheme independent parameters \tau_i. A second set of renormalization scheme independent parameters \sigma_i is shown to play a very similar role in the perturbative expansion of m_{pole} in terms of m(μ) and a(μ). We illustrate our approach by a perturbative computation of \Gamma using the \overline{MS} renormalization scheme. Two other particular mass independent renormalization schemes are briefly considered.