Higher-order perturbation theory for highly-improved actions

Lattice QCD simulations are routinely done nowadays using highly-improved actions, which are designed to remove the leading errors arising from the discretization of the continuum theory. Improved actions can be designed from both perturbative and nonperturbative considerations. In this review I describe techniques for doing the higher-order perturbation theory (PT) calculations that are necessary in the design of highlyimproved lattice discretizations for gluons, staggered quarks, and heavy quarks. Recent results obtained with these methods are also reviewed. Much of the work described here is part of the program of the HPQCD collaboration, a major goal of which is to make precision calculations of hadronic matrix elements relevant to b-physics experiments. In order to fully realize the potential impact of these experiments on the parameters of flavor mixing requires calculations of the relevant hadronic matrix elements to a few percent accuracy. The CLEO-c program also presents an enormous opportunity to validate lattice QCD methods for b physics, by testing predictions for analogous quantities in the charm system. These stringent requirements for accuracy and timeliness are unlikely to be met without significant algorithmic developments, in both the efficiency of unquenched simulations, and in the technical challenges posed by lattice PT. The development of an improved action for staggered quarks [1,2] has at last made accurate unquenched simulations feasible [3] at dynamical quark masses that are small enough to allow for reliable chiral extrapolations to the physical region [4]. This is one particularly striking success of the perturbative analysis of lattice discretizations. More generally one must do perturbative matching calculations for a wide array of coupling constants, action parameters and hadronic matrix elements. The scope of the charge to lattice PT is set by two expansion parameters: aΛQCD, where a is the lattice spacing and ΛQCD is a typical lowenergy scale; and the strong coupling αs(q ), evaluated near the ultraviolet cutoff (q ∼ 1/a) at which the lattice theories are to be matched onto continuum QCD. Affordable unquenched simulations can only be done for lattice spacings around 0.1 fm, where these two expansion parameters have about the same value: