Renormalization of the effective theory for heavy quarks at small velocity

TL;DR

This paper studies the lattice renormalization of the effective theory for slow heavy quarks, showing that the key parameter $\xi^{(1)}(1)$ can be renormalized without ultraviolet divergences, facilitating its computation for $V_{cb}$ extraction.

Contribution

It provides a one-loop renormalization analysis of the slow heavy quark effective theory, demonstrating the absence of ultraviolet divergences for $\xi^{(1)}(1)$ and addressing higher derivatives.

Findings

Renormalization of $\xi^{(1)}(1)$ is free of ultraviolet power divergences.

One-loop renormalization constants are computed to order $v^2$.

Higher derivatives face ultraviolet divergences requiring non-perturbative subtractions.

Abstract

The slope of the Isgur-Wise function at the normalization point, $\xi^{(1)}(1)$,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 $\xi^{(1)}(1)$ is not affected by ultraviolet power divergences, implying no need of difficult non-perturbative subtractions. A lattice computation of $\xi^{(1)}(1)$ 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 $\xi^{(1)}(1)$ . 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 $\xi$ 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.