Renormalization of the effective theory for heavy quarks at small velocity

Abstract

The slope of the Isgur-Wise function at the normalization point, ξ (1) (1), is one of the basic parameters for the extraction of the CKM matrix element V cb from exclusive semileptonic decay data. A method for measuring this parameter on the lattice is the effective theory for heavy quarks at small velocity v. This theory is a variant of the heavy quark effective theory in which the motion of the quark is treated as a perturbation. In this work we study the lattice renormalization of the slow heavy quark effective theory. We show that the renormalization of ξ (1) (1) is not affected by ultraviolet power divergences, implying no need of difficult non-perturbative subtractions. A lattice computation of ξ (1) (1) with this method is therefore feasible in principle. The one-loop renormalization constants of the effective theory for slow heavy quarks are computed to order v 2 together with the lattice-continuum renormalization constant of ξ (1) (1). We demonstrate that the expansion in the heavy-quark velocity reproduces correctly the infrared structure of the original (non-expanded) theory to every order. We compute also the one-loop renormalization constants of the slow heavy quark effective theory to higher orders in v 2 and the lattice-continuum renormalization constants of the higher derivatives of the ξ function. Unfortunately, the renormalization constants of the higher derivatives are affected by ultraviolet power divergences, implying the necessity of numerical non-perturbative subtractions. The lattice computation of higher derivatives of the Isgur-Wise function seems therefore problematic.