Decoupling of heavy-quark loops in light-light and heavy-light quark currents

Matching of light-light quark currents in QCD with a heavy flavour and in the lowenergy effective QCD is calculated in MS at two loops. Heavy-light HQET currents are similarly considered. Consider QCD with nl light flavours and a single heavy one having mass M . Processes involving only light fields with momenta pi ≪ M can be described by an effective field theory — QCD with nl flavours plus 1/M n suppressed local operators in the Lagrangian, which are the remnants of heavy quark loops shrinked to a point. This low-energy effective theory is constructed to reproduce S-matrix elements of the full theory expanded to some order in pi/M . This condition fixes MS coupling α ′ s(μ) of the effective theory, its gauge parameter a(μ) and running light quark masses m′i(μ) in terms of the parameters of the full theory. Two-loop matching was considered in [1, 2], and three-loop one — in [3]. Operators of the full QCD can be expanded in 1/M in operators of the effective theory. Coefficients of these expansions are fixed by equating on-shell matrix elements. In this Letter, I consider bilinear quark currents jn(μ) = Z −1 n (μ)jn0, where the bare currents are jn0 = q̄0γ [μ1 . . . γq0, and Zn(μ) are their MS renormalization constants. In the low-energy theory, they are equal to Cn(μ)j ′ n(μ) plus power-suppressed terms (which will be omitted throughout this paper). It is natural to match the currents at μ = M , because Cn(M) contains no large logarithms. Currents at arbitrary normalization scales can be related by