Two-Loop Relations for Heavy-Quark Parameters in the Shape-Function Scheme

Moments of the renormalized B-meson shape function provide a natural way to define short-distance, running heavy-quark parameters such as the b-quark mass and kinetic energy. These parameters are particularly well suited for studies of inclusive decay distributions. The definitions of mb and μ 2 π in this “shape-function scheme” are derived to two-loop order. Using previous determinations of heavy-quark parameters in other schemes, we find mb(μf ) = (4.64 ± 0.10) GeV and μ 2 π(μf ) = (0.12 ± 0.09) GeV 2 at a reference scale μf = 1.5GeV. Introduction. The past decade has seen a revolution in the precision with which fundamental physics can be probed using measurements of the decay properties of b quarks. For instance, the element |Vcb| of the quark mixing matrix is now known with a precision of 2%, which is close to the accuracy of our knowledge of the Cabbibo angle [1]. Measurements of rare decay processes such as B → Xsγ impose stringent constraints on model building. The b-quark mass has been determined with an uncertainty of about 60MeV [1], which is much less than the QCD scale. To achieve such precision requires that heavy-quark parameters can be defined unambiguously using short-distance techniques. In particular, these definitions should not refer to the notion of on-shell quark states, which would introduce uncontrollable uncertainties (“renormalon ambiguities”) that far exceed the experimental precision achieved at the B factories. The question of a short-distance definition of the b-quark mass has received much attention. While an ad hoc subtraction scheme such as MS in principle defines the quark mass in an unambiguous way, such a definition is not appropriate for the discussion of B decays, where the typical scales are often significantly below mb. A more fruitful concept is that of a low-scale subtracted quark mass [2], which is based on the idea that non-perturbative contributions to the heavy-quark pole mass can be subtracted by making contact to some physical observable. The result is an expression mb(μf), which differs from the pole mass by an amount proportional to a subtraction scale μf = few× ΛQCD. Several examples of such low-scale subtracted quark masses have been discussed in the literature. In the potential-subtraction scheme, long-distance contributions to the pole mass are subtracted by relating it to the static potential between two heavy quarks [3]. Similarly, in the Υ(1S) scheme the quark mass is related to the mass of the lowest-lying bottomonium resonance [4]. Both schemes are well suited to study heavy-quark systems in the non-relativistic regime, such as bb̄ spectroscopy or heavy-quark production near threshold. A quark-mass definition more closely related to the non-perturbative physics probed in B-meson decays is provided by the kinetic scheme [2, 5, 6], in which the non-perturbative subtraction is accomplished with the help of heavy-quark sum rules [7]. Because these sum rules constrain the properties of B-meson form factors in the “small-velocity limit”, the kinetic scheme is well suited for heavy-quark expansions applied to B decays into charm particles. Inclusive B decays into final states consisting of only light hadrons, such as B → Xsγ or B → Xu l ν, probe yet different aspects of non-perturbative physics. The bound-state effects relevant in these processes are encoded in B-meson shape functions defined in terms of the forward matrix elements of non-local string operators on the light cone [8, 9, 10]. A sensible definition of heavy-quark parameters for such processes should incorporate the bulk properties of the leading-order shape function S(ω). Schematically, inclusive decay spectra are given in terms of convolution integrals of the form