Optimised Dirac Operators on the Lattice: Construction, Properties and Applications

TL;DR

This paper reviews the development and application of perfect lattice Dirac operators, focusing on their construction, properties, and how they improve lattice fermion simulations, including chiral symmetry and low-energy QCD constants.

Contribution

It introduces optimized perfect lattice Dirac operators, combining exact chiral symmetry with improved properties, and demonstrates their effectiveness in QCD simulations and related models.

Findings

Successful implementation of overlap-hypercube fermions in QCD simulations

Estimation of chiral condensate and pion decay constant from lattice data

Linking lattice results to Chiral Perturbation Theory

Abstract

We review a number of topics related to block variable renormalisation group transformations of quantum fields on the lattice, and to the emerging perfect lattice actions. We first illustrate this procedure by considering scalar fields. Then we proceed to lattice fermions, where we discuss perfect actions for free fields, for the Gross-Neveu model and for a supersymmetric spin model. We also consider the extension to perfect lattice perturbation theory, in particular regarding the axial anomaly and the quark gluon vertex function. Next we deal with properties and applications of truncated perfect fermions, and their chiral correction by means of the overlap formula. This yields a formulation of lattice fermions, which combines exact chiral symmetry with an optimisation of further essential properties. We summarise simulation results for these so-called overlap-hypercube fermions in the two-flavour Schwinger model and in quenched QCD. In the latter framework we establish a link to Chiral Perturbation Theory, both, in the p-regime and in the epsilon-regime. In particular we present an evaluation of the leading Low Energy Constants of the chiral Lagrangian - the chiral condensate and the pion decay constant - from QCD simulations with extremely light quarks.