Lattice QCD on PCs?

Current PC processors are equipped with vector processing units and have other advanced features that can be used to accelerate lattice QCD programs. Clusters of PCs with a high-bandwidth network thus become powerful and cost-effective machines for numerical simulations.Comment: Lattice2001(plenary), LaTeX source, 8 pages, figures include

Computational Strategies in Lattice QCD

Lectures given at the Summer School on "Modern perspectives in lattice QCD", Les Houches, August 3-28, 2009Comment: Latex source, 72 pages, 23 figures; v2: misprints corrected, minor text change

Lattice QCD -- from quark confinement to asymptotic freedom

According to the present understanding, the observed diversity of the strong interaction phenomena is described by Quantum Chromodynamics, a gauge field theory with only very few parameters. One of the fundamental questions in this context is how precisely the world of mesons and nucleons is related to the properties of the theory at high energies, where quarks and gluons are the important degrees of freedom. The lattice formulation of QCD combined with numerical simulations and standard perturbation theory are the tools that allow one to address this issue at a quantitative level.Comment: Plenary talk, International Conference on Theoretical Physics, Paris, UNESCO, 22--27 July 2002; TeX source, 15 pages, figures include

Step scaling and the Yang-Mills gradient flow

The use of the Yang-Mills gradient flow in step-scaling studies of lattice QCD is expected to lead to results of unprecedented precision. Step scaling is usually based on the Schr\"odinger functional, where time ranges over an interval [0,T] and all fields satisfy Dirichlet boundary conditions at time 0 and T. In these calculations, potentially important sources of systematic errors are boundary lattice effects and the infamous topology-freezing problem. The latter is here shown to be absent if Neumann instead of Dirichlet boundary conditions are imposed on the gauge field at time 0. Moreover, the expectation values of gauge-invariant local fields at positive flow time (and of other well localized observables) that reside in the center of the space-time volume are found to be largely insensitive to the boundary lattice effects.Comment: Plain TeX source, 26 pages, 7 figures; v2: corrected typos and updated reference

Lattice regularization of chiral gauge theories to all orders of perturbation theory

In the framework of perturbation theory, it is possible to put chiral gauge theories on the lattice without violating the gauge symmetry or other fundamental principles, provided the fermion representation of the gauge group is anomaly-free. The basic elements of this construction (which starts from the Ginsparg-Wilson relation) are briefly recalled and the exact cancellation of the gauge anomaly, at any fixed value of the lattice spacing and for any compact gauge group, is then proved rigorously through a recursive procedure.Comment: Plain TeX source, 26 pages, figures include

The Schr\"odinger functional in lattice QCD with exact chiral symmetry

Similarly to the interaction lagrangian, the possible boundary conditions in quantum field theories on space-time manifolds with boundaries are strongly constrained by the symmetry and scaling properties of the theory. Based on this general insight, a lattice formulation of the QCD Schr\"odinger functional is proposed for the case where the lattice Dirac operator in the bulk of the lattice coincides with the Neuberger--Dirac operator. The construction satisfies all basic requirements (locality, symmetries, hermiticity) and is suitable for numerical simulations.Comment: Plain TeX source, 20 pages, no figure

Chiral gauge theories revisited

Contents: 1. Introduction, 2. Chiral gauge theories & the gauge anomaly, 3. The regularization problem, 4. Weyl fermions from 4+1 dimensions, 5. The Ginsparg-Wilson relation, 6. Gauge-invariant lattice regularization of anomaly-free theories.Comment: Lectures given at the International School of Subnuclear Physics, Erice, 27 August - 5 September 2000, Plain TeX source, 40 pages, figures included; reference added, minor text correction

Universality of the topological susceptibility in the SU(3) gauge theory

The definition and computation of the topological susceptibility in non-abelian gauge theories is complicated by the presence of non-integrable short-distance singularities. Recently, alternative representations of the susceptibility were discovered, which are singularity-free and do not require renormalization. Such an expression is here studied quantitatively, using the lattice formulation of the SU(3) gauge theory and numerical simulations. The results confirm the expected scaling of the susceptibility with respect to the lattice spacing and they also agree, within errors, with computations of the susceptibility based on the use of a chiral lattice Dirac operator.Comment: Plain TeX source, 14 pages, 1 figure; v3: further typos corrected, version published in JHE

Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation

It is shown that the Ginsparg-Wilson relation implies an exact symmetry of the fermion action, which may be regarded as a lattice form of an infinitesimal chiral rotation. Using this result it is straightforward to construct lattice Yukawa models with unbroken flavour and chiral symmetries and no doubling of the fermion spectrum. A contradiction with the Nielsen-Ninomiya theorem is avoided, because the chiral symmetry is realized in a different way than has been assumed when proving the theorem.Comment: plain tex source, 8 pages, no figure