Perspectives in Lattice QCD

Preface

Contents

Unwanted symmetry breaking by the lattice regularization will produce cut-off effects and distort the results. Symmetries are realized on the lattice frequently in an unusual way. Fixed point actions preserve all the classical symmetries of the theory and help to abstract not only the symmetry conditions, but the form of the lattice symmetry transformations also.

This is the write-up of three lectures on algorithms for dynamical fermions that were given at the ILFTN workshop ‘Perspectives in Lattice QCD’ in Nara during November 2005. The first lecture is on the fundamentals of Markov Chain Monte Carlo methods and introduces the Hybrid Monte Carlo (HMC) algorithm and symplectic integrators; the second lecture covers topics in approximation theory and thereby introduces the Rational Hybrid Monte Carlo (RHMC) algorithm and ways of evading integrator instabilities by means of multiple pseudofermion fields; the third lecture introduces on-shell chiral (Ginsparg–Wilson) lattice fermions and discusses five-dimensional formulations for computing fermion propagators for such fermions.

These lectures describe the use of effective field theories to extrapolate results from the parameter region where numerical simulations of lattice QCD are possible to the physical parameters (physical quark masses, infinite volume, vanishing lattice spacing, etc.). After a brief introduction and overview, I discuss three topics: 1) Chiral perturbation theory (χPT) in the continuum; 2) The inclusion of discretization effects into χPT, focusing on the application to Wilson and twisted-mass lattice fermions; 3) Extending χPT to describe partially quenched QCD.

In these lectures I explain how chiral symmetry of continuum QCD naturally leads to a class of lattice regularisations known as twisted mass QCD (tmQCD). As compared to standard Wilson quarks, its advantages are the absence of unphysical zero modes, the possibility to circumvent lattice renormalisation problems and automatic O(a) improvement. On the other hand, the physical parity and flavour symmetries are explicitly broken. I discuss these aspects and then turn to the theory in a finite space-time volume with Schrödinger functional boundary conditions. Again, chiral transformations of the continuum theory may be used as a guide to formulate an alternative lattice regularisation of the Schrödinger functional, with interesting applications to renormalization problems in QCD.

We give an introduction to three topics in lattice gauge theory:

I. The Schrödinger functional and O(a) improvement.

O(a) improvement has been reviewed several times. Here we focus on explaining the basic ideas in detail and then proceed directly to an overview of the literature and our personal assessment of what has been achieved and what is missing.

II. The computation of the running coupling, running quark masses and the extraction of the renormalization group invariants.

We focus on the basic strategy and on the large effort that has been invested in understanding the continuum limit. We point out what remains to be done.

III. Non-perturbative Heavy Quark Effective Theory.

Since the literature on this subject is still rather sparse, we go beyond the basic ideas and discuss in some detail how the theory works in principle and in practice.