We construct an effective field theory valid for processes in which highly energetic light-like particles interact with collinear and soft degrees of freedom, using the decay B → X s γ near the endpoint of the photon spectrum, x = 2E γ /m b → 1, as an example. Below the scale µ = m b both soft and collinear degrees of freedom are included in the effective theory, while below the scale µ = m b √ x − y, where 1 − y is the lightcone momentum fraction of the b quark in the B meson, we match onto a theory of bilocal operators. We show that at one loop large logarithms cancel in the matching conditions, and that we recover the well known renormalization group equations that sum leading Sudakov logarithms. *

Since fields in the heavy quark effective theory are described by both a velocity and a residual momentum, there is redundancy in the theory: small shifts in velocity may be absorbed into a redefinition of the residual momentum. We demonstrate that this trivial reparameterisation invariance has non-trivial consequences: it relates coefficients of terms of different orders in the 1/m expansion and requires linear combinations of these operators to be multiplicatively renormalised. For example, the operator −D 2 /2m in the effective lagrangian has zero anomalous dimension, coefficient one, and does not receive any non-perturbative contributions from matching conditions. We also demonstrate that this invariance severely restricts the forms of operators which may appear in chiral lagrangians for heavy particles.

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In light of the large amount of new experimental data, we revisit the determination of |V cb | and m b from inclusive semileptonic and radiative B decays. We study shape variables to order Λ 3 QCD /m 3 b and α 2 s β0, and include the order αs ΛQCD/m b correction to the hadron mass spectrum in semileptonic decay, which improves the agreement with the data. We focus on the 1S and kinetic mass schemes for the b quark, with and without expanding m b − mc in HQET. We perform fits to all available data from BABAR, BELLE, CDF, CLEO, and DELPHI, discuss the theoretical uncertainties, and compare with earlier results. We find |V cb | = (41.9 ± 0.6 ± 0.1τ B )×10 −3 and m 1S b = 4.68±0.04 GeV, including our estimate of the theoretical uncertainty in the fit.

We discuss the matching conditions and renormalization group evolution of non-relativistic QCD. A variant of the conventional MS-bar scheme is proposed in which a subtraction velocity nu is used rather than a subtraction scale mu. We derive a novel renormalization group equation in velocity space which can be used to sum logarithms of v in the effective theory. We apply our method to several examples. In particular we show that our formulation correctly reproduces the two-loop anomalous dimension of the heavy quark production current near threshold.Comment: (27 pages, revtex