Second Order Perturbation Theory for Improved Gluon and Staggered Quark Actions

It has long been known that perturbative calculations are necessary for precision measurements of many quantities. Unfortunately perturbation theory using a lattice cutoff is difficult, due to extra diagrams and vastly more complicated Feynman rules. The complexity of these calculations is an impediment to doing the higher–order perturbation theory required for many important applications of improved actions. The present work attempts to automate as much of the perturbation theory as possible in order to make these types of computations more straightforward. For any given action some of the most basic things we would like to compute are various operators that can be built out of products of links. These include the static quark potential, small Wilson loops, the static quark self energy, and the mean link in Landau gauge. Perturbative determinations of these quantities are central to many other applications. For example, the static quark potential can be used to determine a physical, renormalized coupling, and the perturbative expansions of small Wilson loops can be used to extract the strong coupling from simulations (see [1]). We have the techniques in place to calculate all of these, for an arbitrary gluon action, along with the contributions from quark loops for highly improved quark actions. This paper reports preliminary results for the above quantities, for the one-loop Symanzik improved gluon action, and the improved staggered quark action. These calculations are similar to traditional continuum perturbative QCD. The major difference is that the ultraviolet regulator is the lattice cutoff, which leads to the computational difficulties mentioned above. A further problem is how to regulate infrared divergences beyond one– loop, in a gauge covariant manner. This can be done using twisted periodic boundary conditions [2]. Feynman rules and factors from the operators are generated using the Lüscher and Weisz vertex generation algorithm [2]. PYTHON or C++ scripts implement this program for a given set of links. An advantage to this method is the ease with which new actions and operators can be treated. We use FORTRAN programs to do the momentum sums. These are generally done on large but finite lattices (typically 100 volume). The action we consider here is the one–loop Symanzik improved action for isotropic lattices,