Ulli Wolff

Non-perturbative O(a) improvement of lattice QCD

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.

Monte Carlo errors with less errors

Abstract We explain in detail how to estimate mean values and assess statistical errors for arbitrary functions of elementary observables in Monte Carlo simulations. The method is to estimate and sum the relevant autocorrelation functions, which is argued to produce more certain error estimates than binning techniques and hence to help toward a better exploitation of expensive simulations. An effective integrated autocorrelation time is computed which is suitable to benchmark efficiencies of simulation algorithms with regard to specific observables of interest. A Matlab code is offered for download that implements the method. It can also combine independent runs (replica) allowing to judge their consistency.

Non-perturbative renormalization of lattice QCD at all scales

Abstract A general strategy to solve the non-perturbative renormalization problem in lattice QCD, using finite-size techniques and numerical simulations, is described. As an illustration we discuss the computation of the axial current normalization constant, the running coupling at zero quark masses and the scale evolution of the renormalized axial density. The non-perturbative calculation of O(a) correction terms (as they appear in Symanziks improvement programme) is another important field of application.

Computation of the strong coupling in QCD with two dynamical flavors

Abstract We present a non-perturbative computation of the running of the coupling α s in QCD with two flavors of dynamical fermions in the Schrodinger functional scheme. We improve our previous results by a reliable continuum extrapolation. The Λ-parameter characterizing the high-energy running is related to the value of the coupling at low energy in the continuum limit. An estimate of Λ r 0 is given using large-volume data with lattice spacings a from 0.07 fm to 0.1 fm . It translates into Λ MS ¯ ( 2 ) = 245 ( 16 ) ( 16 ) MeV [assuming r 0 = 0.5 fm ]. The last step still has to be improved to reduce the uncertainty.

Comparison between cluster Monte Carlo algorithms in the Ising model

We report autocorrelatron trmes for the Swendsen-Wang algortthm and for a recently proposed smgle cluster variant m the 2D and 3D Ismg models at crtttcahty The new algorithm decorrelates faster m all cases and gams about an order of magmtude on a 643 lattice Crtttcal slowmg down IS practically neghgrble and possibly completely absent m three dtmenstons Results on static properties of the 3D model are consrstent with pubhshed data Among the attempts to circumvent critical slowing-down m stmulations of field theories and critical systems a line of developments related to percolation and initiated by Swendsen and Wang (SW) [ 1 ] has been very successful recently. While the original SW proposal works for Potts spins only, our recent generalization [ 21 has been apphed to the x-y model [ 31 and to the 0 (3) nonlinear a-model [ 41 in two dtmensions with the result of no detectable slowmg down and further advantages related to variance reduction. Apart from the generalization to continuous spins our proposal [ 21 also modifies the SW algorithm in another way: the single cluster (1C) construction (see below). Consequently the new 1C algorithm does not comxde with SW even for Potts models. Here we study two-state Potts ( =Ising) models to evaluate the effect of the 1C variation in isolation. This complements recent studies [ 5,6] where the combination of continuous spms with the original SW cluster decomposition has also been found to drastically reduce or eliminate slowing down. The results of ref. [ 5 ] combined with ref. [ 1 ] or with the present study actually imply that the SW cluster constructton leads to a smaller dynamical exponent z for continuous spins than for Ising spms. The latter are thus “harder” to stmulate than O(n) o-models with IZ> 1. The SW and 1C algorithms for an Ising model with spins s, at the sites x of a d-dimensional hypertorus of Ld sites are most easily described m words: For given spins {sX} one activates bonds xp (links) with probability

Collective Monte Carlo Updating in a High Precision Study of the X-y Model

Abstract We employ a recently developed collective Monte Carlo algorithm to simulate the two-dimensional x−y model at correlation lengths up to 69. No critical slowing down manifests itself in the measured observables. In the high-temperature phase of the model we observe an approach to criticality of the type predicted by Kosterlitz and Thouless and exclude a conventional critical point. The exponent η exceeds 1 4 for the observed range of correlation lengths. In the spinwave phase we see a variable η and scaling behavior in agreement with the absence of ultraviolet coupling constant renormalization as expected for the abelian model.

Two-loop computation of the Schrödinger functional in lattice QCD

Abstract We compute the Schrodinger functional (SF) for the case of lattice QCD with Wilson fermions (with and without SW improvement) at two-loop order in lattice perturbation theory. This allows us to extract the three-loop β -function in the SF-scheme. These results are required to compute the running coupling, the Λ -parameter and quark masses by finite size techniques with negligible systematic errors. In addition our results enable the implementation of two-loop O( a ) improvement in SF-simulations.

Non-perturbative quark mass renormalization in two-flavor QCD

Abstract The running of renormalized quark masses is computed in lattice QCD with two flavors of massless O ( a ) improved Wilson quarks. The regularization and flavor independent factor that relates running quark masses to the renormalization group invariant ones is evaluated in the Schrodinger functional scheme. Using existing data for the scale r 0 and the pseudoscalar meson masses, we define a reference quark mass in QCD with two degenerate quark flavors. We then compute the renormalization group invariant reference quark mass at three different lattice spacings. Our estimate for the continuum value is converted to the strange quark mass with the help of chiral perturbation theory.

Asymptotic freedom and mass generation in the O(3) nonlinear σ-model

Abstract We complement a recently discovered collective Monte Carlo algorithm with a cluster variance reduction technique for the two-point function of σ-models. It is used to determine both the nonperturbative mass and the asymptotic freedom scale Λ MS from simulations of the O(3)-model with correlation lengths up to 121 on large lattices. We also estimate Λ MS in lattice units by studying the mass gap in physically small volumes. Asymptotic scaling with the bare coupling still does not occur. We propose an approximate lattice β-function that goes beyond finite-order perturbation theory and leads to better scaling behavior. The various methods give m/Λ MS = 2.5−3.0 for the O(3)-model.

Universality and the approach to the continuum limit in lattice gauge theory

Abstract The universality of the continuum limit and the applicability of renormalized perturbation theory are tested in the SU(2) lattice gauge theory by computing two different non-perturbatively defined running couplings over a large range of energies. The lattice data (which were generated on the powerful APE computers at Rome II and DESY) are extrapolated to the continuum limit by simulating sequences of lattices with decreasing spacings. Our results confirm the expected universality at all energies to a precision of a few percent. We find, however, that perturbation theory must be used with care when matching different renormalized couplings at high energies.