Martin Lüscher

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

Abstract 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.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.

Two particle states on a torus and their relation to the scattering matrix

Abstract The energy spectrum of a system of two particles enclosed in a box with periodic boundary conditions is characteristic for the forces between the particles. For box sizes greater than the interaction range, and for energies below the inelastic threshold, the spectrum is shown to be determined by the scattering phases at these energies. Simple exact formulae are derived which can be used to compute the energy levels given the scattering phases or, conversely, to calculate the scattering phases if the energy spectrum is known.

The Schrodinger Functional — a Renormalizable Probe for Non-Abelian Gauge Theories

Following Symanzik we argue that the Schrodinger functional in lattice gauge theories without matter fields has a well-defined continuum limit. Due to gauge invariance no extra counter terms are required. The Schrodinger functional is, moreover, accessible to numerical simulations. It may hence be used to study the scaling properties of the theory and in particular the evolution of the renormalized gauge coupling from low to high energies. A concrete proposition along this line is made and the necessary perturbative analysis of the Schrodinger functional is carried through to 1-loop order.

Chiral symmetry and O(a) improvement in lattice QCD

Abstract The dominant cutoff effects in lattice QCD with Wilson quarks are proportional to the lattice spacing a . In particular, the isovector axial current satisfies the PCAC relation only up to such effects. Following a suggestion of Symanzik, they can be cancelled by adding local O( a ) correction terms to the action and the axial current. We here address a number of theoretical issues in connection with the O( a ) improvement of lattice QCD and then show that chiral symmetry can be used to fix the coefficients multiplying the correction terms.

How to Calculate the Elastic Scattering Matrix in Two-dimensional Quantum Field Theories by Numerical Simulation

Abstract A method to calculate the elastic scattering amplitude at low energies in two-dimensional quantum field theories is proposed and tested in a numerical simulation of the O(3) non-linear σ-model on a simple square lattice. We also compute the isospin current form factor in this model and compare our results with the known exact expressions for the S -matrix and the form factor in the continuum limit. As a technical improvement, we introduce a two-cluster simulation algorithm which leads to significantly reduced statistical errors in the calculation of four-point correlation functions.

Properties and uses of the Wilson flow in lattice QCD

Theoretical and numerical studies of the Wilson flow in lattice QCD suggest that the gauge field obtained at flow time t > 0 is a smooth renormalized field. The expectation values of local gauge-invariant expressions in this field are thus well-defined physical quantities that probe the theory at length scales on the order of √ t. Moreover, by transforming the QCD functional integral to an integral over the gauge field at a specified flow time, the emergence of the topological (instanton) sectors in the continuum limit becomes transparent and is seen to be caused by a dynamical effect that rapidly separates the sectors when the lattice spacing is reduced from 0.1 fm to smaller values.Theoretical and numerical studies of the Wilson flow in lattice QCD suggest that the gauge field obtained at flow time t > 0 is a smooth renormalized field. The expectation values of local gauge-invariant expressions in this field are thus well-defined physical quantities that probe the theory at length scales on the order of

Non-perturbative O(a) improvement of lattice QCD

\sqrt {t}

Non-perturbative quark mass renormalization in quenched lattice QCD

. Moreover, by transforming the QCD functional integral to an integral over the gauge field at a specified flow time, the emergence of the topological (instanton) sectors in the continuum limit becomes transparent and is seen to be caused by a dynamical effect that rapidly separates the sectors when the lattice spacing is reduced from 0.1fm to smaller values.

Locality properties of Neuberger's lattice Dirac operator

The coefficients multiplying the counterterms required for O(a) improvement of the action and the isovector axial current in lattice QCD are computed non-perturbatively, in the quenched approximation and for bare gauge couplings g0 i range 0 ⩽ g0 ⩽ 1. A finite-size method based on the Schrodinger functional is employed, which enables us to perform all calculations at z nearly zero quark mass. As a by-product the critical hopping parameter κc is obtained at all couplings considered.

A precise determination of the running coupling in the SU(3) Yang-Mills theory

Abstract The renormalization factor relating the bare to the renormalization group invariant quark masses is accurately calculated in quenched lattice QCD using a recursive finite-size technique. The result is presented in the form of a product of a universal factor times another factor, which depends on the details of the lattice theory but is easy to compute, since it does not involve any large scale differences. As a byproduct the A-parameter of the theory is obtained with a total error of 8%.