Symanzik improvement and RG actions

Abstract

When introducing the QCD action on the lattice we had to discretize the derivative terms that show up in the continuum action. It was pointed out then that any discretization, e.g., symmetric differences for the first derivative in the fermion action, gives rise to discretization effects. Typically the discretization effects are of (a) for fermions and of (a 2) for the gauge fields. These disappear only in the continuum limit when the lattice spacing it a is sent to zero. Performing the continuum limit is, however, a nontrivial task. As one decreases a, the number of lattice points has to increase, such that the physical volume remains constant (ideally one would first send the number of lattice points to infinity before sending a to zero). Thus in a numerical simulation one always works with finite a and the discretization errors have to be dealt with, e.g., by including them in the extrapolation to vanishing a.