<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi></mml:math>-meson decay constant on the lattice and renormalization

Abstract

We compute in perturbation theory the relation between the $B$-meson leptonic decay constant ${F}_{B}$ computed on a lattice by the $\frac{1}{{m}_{b}}$ expansion in the manner of Eichten and the continuum: i.e., the physical value of ${F}_{B}$. To that aim we compare the QCD radiative corrections up to order ${\ensuremath{\alpha}}_{s}$ of the axial-vector-current correlator for different quark masses with the radiative corrections of the effective operator which replaces the correlator in the $\frac{1}{{m}_{b}}$ expansion. The latter radiative corrections are computed in the continuum and on a lattice. For this effective operator we recover the anomalous dimension $\ensuremath{\gamma}=2$ already found by Shifman and Voloshin. Our final result is that ${F}_{B}\ensuremath{\simeq}0.8{F}_{B}^{\mathrm{latt}}$, only weakly dependent on lattice spacing and ${\ensuremath{\Lambda}}_{\mathrm{QCD}}$.