Renormalization Group Scaling of the 1/m HQET Lagrangian

Heavy quark effective theory (HQET) [1,2] is a useful tool for studying the physics of hadrons containing a single heavy quark. The HQET Lagrangian has an expansion in powers of derivatives divided by the heavy quark mass m, which translates into an expansion of hadronic quantities in powers of ΛQCD/m, where ΛQCD is the non-perturbative scale parameter of the strong interactions. The HQET Lagrangian can be computed by matching with the full QCD Lagrangian at a scale μ ≈ m. This has been done at one loop to order 1/m for the two-Fermion terms in the Lagrangian [3–5], and at one loop to order 1/m for the two-Fermion terms [6]. The renormalization group running of the dimension five (1/m) operators in the HQET Lagrangian has been computed [3,4]. There are several computations of the running of the dimension six (1/m) operators [6–10] in the literature, but the various papers disagree with each other. In Refs. [7–10] the authors do not take into account the effect of a four fermion operator which is present in the dimension six operator basis and is related to the Darwin term by the equations of motion. A complete calculation including all dimension six operators was given in Ref. [6], but we disagree with this calculation in a few entries of the anomalous dimension matrix. In this paper we compute the running of the dimension six operators of the HQET Lagrangian. The coefficients are computed using one-loop running and tree-level matching, which makes the calculation particularly simple. We will use the notation of Ref. [6], to