B-meson decay constant on the lattice and renormalization

Abstract

We compute in perturbation theory the relation between the B-meson leptonic decay constant FB computed on a lattice by the 1 / m b expansion in the manner of Eichten and the continuum: i.e., the physical value of FB. To that aim we compare the QCD radiative corrections up to order a, of the axial-vector-current correlator for different quark masses with the radiative corrections of the effective operator which replaces the correlator in the I / 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 y = 2 already found by Shifman and Voloshin. Our final result is that F, --0. 8 F p 1 , only weakly dependent on lattice spacing and ApcD. I. 1/M EXPANSION AND EFFECTIVE OPERATORLattice calculations lead to rather successful estimates of hadron properties: masses, decay constants such as F,, F K , etc. When considering states which include one or two charmed quarks the situation becomes trickier since present lattices have am, --t and one has no control on corrections of the order am, where a is the lattice spacing and m , the charmed-quark mass. For the b quark the same techniques are out of question since am, >> 1 for any presently realistic lattice. A new approach is necessary. The basic idea was proposed by Eichten. ' It consists of performing a 1 / m b expansion of the heavy-quark propagator. Then any useful Green's function such as correlation functions can be calculated, to some order in l / m b , only from the knowledge of the light-quark propagator and the gluon fields, which in turn can be estimated on a lattice. In principle with this method one can get an answer from the lattice up to corrections of the order 1 / m b a to some power. In fact in what follows we shall stick to the lowest order in 1 / m b a . Numerical calculations on lattice have already been done'32 but they can only give rather loose bounds on FB by lack of statistics.We need also to know the exact relation between the lattice result for FB and its continuum value, and to that aim we must compute the radiative corrections both in the continuum and on the lattice. However, some care is needed since the l / m , expansion breaks down for the frequencies of gluon fields which are not small compared t o m , .T o state more precisely the problem, let us first recall what happens in the case of F , (Refs. 3 and 4). One computes the correlation function of two axial-vector currents. These have a vanishing anomalous dimension due to partial conservation of the axial-vector current. The local axial-vector current on the lattice has also no anomalous dimension but it differs from the corresponding operator in the continuum by some finite radiative corrections. These corrections come from the high frequencies which differ on a lattice and in the continuum.Let us now consider the current 6yPySq. As in the case of F , the correlation function between two such currents leads to F B . As we have argued before, the radiative corrections in the continuum a...

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