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 $Γ$ for the $b$-quark, using mass-independent renormalization. To finite order of perturbation theory the series for $Γ$ 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 $Γ$, 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 $Γ$ (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 $Γ$ in terms of $a(μ)$ and $m(μ)$ both evaluated at a renormalization scheme independent mass scale $I\!\!M$ 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 $β(a)$ and $γ(a)$, associated with $a(μ)$ and $m(μ)$ respectively, to characterize the remaining renormalization scheme arbitrariness of $Γ$. We further show that all terms in the expansion of $Γ$ can be written in terms of the $c_i$ and $g_i$ coefficients and a set of renormalization scheme independent parameters $τ_i$.