A Rigourous Treatment of the Lattice Renormalization Problem of F_B

TL;DR

This paper rigorously analyzes the lattice renormalization of the $B$-meson decay constant, resolving previous discrepancies and clarifying the renormalization procedure for accurate lattice QCD calculations.

Contribution

It provides a detailed comparison of one-loop and two-loop renormalization schemes, resolving ambiguities and offering numerical values for different lattice approximations.

Findings

One-loop and two-loop schemes are consistent to order α_s.

Renormalization prescription ambiguity can significantly affect results.

Correct prescription for O(a) improved calculations is identified.

Abstract

The $B$-meson decay constant can be measured on the lattice using a $1/m_b$ expansion. To relate the physical quantity to Monte Carlo data one has to know the renormalization coefficient, $Z$, between the lattice operators and their continuum counterparts. We come back to this computation to resolve discrepancies found in previous calculations. We define and discuss in detail the renormalization procedure that allows the (perturbative) computation of $Z$. Comparing the one-loop calculations in the effective Lagrangian approach with the direct two-loop calculation of the two-point $B$-meson correlator in the limit of large $b$-quark mass, we prove that the two schemes give consistent results to order $\alpha_s$. We show that there is, however, a renormalization prescription ambiguity that can have sizeable numerical consequences. This ambiguity can be resolved in the framework of an $O(a)$ improved calculation, and we describe the correct prescription in that case. Finally we give the numerical values of $Z$ that correspond to the different types of lattice approximations discussed in the paper.