Observations on the Wilson fermions in the ϵ regime

Abstract

We make several observations concerning the low quark mass region with Wilson fermions and how this is connected with the ϵ regime in the continuum. A transition from tiny cutoff effects to rather large discretization errors would take place in general with Wilson fermions if we lower the quark mass at finite lattice spacing. We argue that these two regions exhibit rather different behaviours concerning the coupling between cutoff effects and zero-modes. We interpolate between these two regimes adding to the continuum ϵ regime formulae, in the spirit of the Symanzik expansion, the relevant operators parametrising the leading cutoff effects. We compute the partition function, the chiral condensate, scalar and pseudo-scalar correlation functions. The final formulae can be used to fit lattice data to extract physical low energy constants, and to estimate systematic uncertainties coming from discretization errors. Moreover they suggest ways on how to remove these cutoff effects, the core of which are captured by the continuum zero modes.