Non-Perturbative Methods and Lattice QCD

The following sections are included:

• PREFACE

• CONTENTS

Lattice fermions obeying the Ginsparg-Wilson relation do correctly represent the physical properties related to chirality. This can be achieved by local fermions, which involve an infinite number of couplings, however. For practical purposes, it is useful to first construct approximate Ginsparg-Wilson fermions within a short range. We report on a successful construction in QCD at β = 6. The good quality of the approximation is observed from the spectrum, which is situated close to a Ginsparg-Wilson circle. These fermions also provide an excellent approximation to rotational symmetry and they are promising for a good scaling, since they arise from the perfect action framework. Their insertion into the overlap formula renders the Ginsparg-Wilson relation exact. It leads to an improved overlap fermion with a high level of locality. This insertion is statistically on safe grounds at β ≳ 5.6.

Chiral symmetry and locality property of low energy effective action of domain-wall fermion are discussed.

A block-spin transformation on the dual lattice leads us to an almost perfect lattice action for monopoles and strings in QCD. The perfect operator for a static quark potential is fixed when we compare the above action with the perfect action obtained analytically after infinite-step block-spin transformations in a simple case. The continuum rotational invariance is restored and the physical value of the string tension is reproduced fairly well. Gauge independence of the abelian and the monopole scenario is discussed. This talk is based on our recent works.^1,2,3,4,5

We present recent results of the light hadron spectrum and quark masses in QCD with two flavors of dynamical quarks from the CP-PACS computer. To compensate the increase in computer time, we use a renormalization-group improved gauge action and a tadpole-improved clover quark action. A comparison with quenched results with the same improved action and with the standard action shows clearly the existence of sea quark effects in meson and quark masses.

The mass spectra of the L = 1 orbitally excited heavy baryons with light quarks in both the spin-flavor symmetric and the mixed representations are studied by the 1/N_c expansion method in the framework of the heavy quark effective theory. The mixing effect between the baryons in the two representations is also considered. The general pattern of the spectrum is predicted which will be verified by the experiments in the near future.

Glueball spectrum is studied using an improved gluonic action on asymmetric lattices in the pure SU(3) gauge theory. The smallest spatial lattice spacing is about 0.08fm which makes the extrapolation to the continuum limit more reliable. In particular, attention is paid to the scalar glueball mass which is known to have problems in the extrapolation. Converting our lattice results to physical units using the scale set by the static quark potential, we obtain the following results for the glueball masses: M_G(0⁺⁺) = 1730(90)MeV for the scalar glueball mass and M_G(2⁺⁺) = 2400(95)MeV for the tensor glueball.

Preliminary results for light hadron masses are presented from lattice QCD calculations on asymmetric lattices with improved actions. The used lattices have a fixed spatial lattice spacing a_s at 0.4fm, and a temporal lattice spacing a_t from 0.14 to 0.08fm. The discretization errors of a_s is reduced through improving actions at certain level. We find that masses of mesons have a weak a_t dependence, while the a_t dependence for baryons is more significant. Extrapolating to a_t = 0 limit, we obtain the mass of the nucleon, which has a 9% difference compared with the experimental value.

We study lattice SU(2) configurations by fast cooling with Wilson action and an improved action. The classical lattice gauge configurations are obtained. Its some properties are presented. We also show its some properties in momentum space.

A 'forward walking' Green's Function Monte Carlo algorithm is used to obtain expectation values for SU(3) lattice Yang-Mills theory in (3+1) dimensions. The ground state energy and Wilson loops are calculated, and the finite-size scaling behaviour is explored. Crude estimates of the string tension are derived, which agree with previous results at intermediate couplings; but more accurate results for larger loops will be required to establish scaling behaviour at weak coupling.

We review several topics related to the diagonalization of quantum field Hamiltonians using the quasi-sparse eigenvector (QSE) method.

Monte Carlo techniques with importance sampling have been extensively applied to lattice gauge theory in the Lagrangian formulation. Unfortunately, it is extremely difficult to compute the excited states using the conventional Monte Carlo algorithm. Our recently developed approach: the Monte Carlo Hamiltonian method, has been designed to overcome the difficulties of the conventional approach. In this paper, we extend the method to many body systems and quantum field theory. The Klein-Gordon field theory is used as a testing ground.

The standard hybrid Monte Carlo algorithm is known to simulate even flavors QCD only. Simulations of odd flavors QCD, however, can be also performed in the framework of the hybrid Monte Carlo algorithm where the inverse of the fermion matrix is approximated by a polynomial. In this exploratory study we perform three flavors QCD simulations. We make a comparison of the hybrid Monte Carlo algorithm and the R-algorithm which also simulates odd flavors systems but has step-size errors. We find that results from our hybrid Monte Carlo algorithm are in agreement with those from the R-algorithm obtained at very small step-size.

Unquenched lattice SU(2) is studied at nonzero chemical potential in the strong coupling limit. The topic of diquark condensation is addressed analyzing the probability distribution function of the diquark condensate. We present results at zero external source without using any potentially dangerous extrapolation procedure. We find strong evidences for a (high density) second order phase transition where a diquark condensate appears, and show quantitative agreement of lattice calculations with low-energy effective Lagrangian calculations.

SU(2) lattice gauge theory with dynamical fermion at non-zero chemical potential and at finite temperature is studied. We focus on the influence of chemical potential for quark condensate and mass of pseudoscalar meson at finite temperature. Hybrid Monte Carlo simulations with N_f = 8 staggered fermions are carried out on 12 × 12 × 24 × 4 lattice. At β = 1.1 and m_q =0.05,0.07,0.1, we calculate the quark condensate and masses of pseudoscalar meson consisting of light and heavier quarks for chemical potential µ = 0.0,0.02,0.05,0.1,0.2.

At sufficiently high temperature and density, quantum chromodynamics (QCD) predicts phase transition from the hadronic phase to the quark-gluon plasma phase. Lattice QCD is the most useful tool to investigate this critical phenomenon, which status is briefly reviewed. The usual problem in the Lagrangian formulation at finite density is either an incorrect continuum limit or its complex action and a premature onset of the transition as the chemical potential is raised. We show how the difficulties are overcome in our Hamiltonian approach.

In this talk I will first give a short discussion of some lattice results for QCD at finite temperature. I will then describe in some detail the technique of dimensional reduction, which in principle is a powerful technique to obtain results on the long distance properties of the quark-gluon plasma. Finally I will describe some new results, which test the technique in a simpler model, namely three dimensional gauge theory.

We reanalyze in the first part of this paper the old question of P and CT realization in QCD. The second part is devoted to establish general results on the phase structure of this model in the presence of a θ-vacuum term.

The SU(2) gluonic correlation functions, glueball effective masses in the J^P = 0⁺, 2⁺ and 0^- channels were calculated from the lattice classical gauge configurations which were obtained by smoothing the thermal gauge configurations through the improved cooling method. The instanton-induced attractive force in the 0⁺ channel and the repulsive force in the 0^- channel are confirmed in the Monte Carlo simulation. There is evidence that the instanton vacuum contribution to the 0⁺ glueball mass is significant.

In the two-dimensional CP^N-l model one can parametrize exact many-instanton solutions via N 'constituents' (called 'zindons'). This parameterization allows, in principle, a complete 'melting' of individual instantons. The model is therefore well suited to study whether dynamics prefers a dilute or a strongly overlapping ensemble of instantons. We study the statistical mechanics of instantons both analytically and numerically. We find that at N = 2 the instanton system collapses into zero-size instantons. At N = 3,4 we find that well-isolated instantons are dynamically preferred though 15-25% of instantons have a considerable overlap with others.

We show how center vortices and Abelian monopoles both appear as local gauge ambiguities in the Laplacian Center gauge. Numerical results, for SU(2) and SU(3), support the view that the string tension obtained in the center-projected theory matches the full string tension when the continuum limit is taken.

Concerning the instanton sector, we study gauge invariant field strength correlators in pure SU(2) gauge theory by applying renormalization group based smoothing. The correlators are used to extract orientations of clusters of topological charge in configuration and color space. With respect to the fermion sector, we compute the quark condensate, the quark charge and the chiral density in full QCD. Classifying the lattice by elementary 3-cubes being associated to dual links occupied by (or free of) monopoles, the simultaneous occupation by chirality is investigated.

In this talk, I briefly review the reasons why quantum computation is an interesting subject and why sometimes quantum computation required in order to perform certain tasks efficiently. This talk is addressed to physicists who are new to this area.

We describe the construction and configuration of a parallel computer composed of a cluster of personal computers. Furthermore, we show that such a cluster is an extremely inexpensive way of building computational power.

We have calculated the large-q series of the energy cumulants, the magnetization cumulants and the correlation length at the first order phase transition point both in the ordered and disordered phases for the q-state Potts model in two dimensions. The series enables us to estimate the numerical values of the quantities more precisely by a factor of 10² - 10⁴ than the Monte Carlo simulations.From the large-q series of the eigenvalues of the transfer matrix, we also find that the excited states form a continuum spectrum and there is no particle state at the first order phase transition point.

We discuss recent high statistics Monte Carlo simulations of the Edwards-Anderson Ising spin-glass model in three and four dimensions based on a non-Boltzmann sampling technique. Particular emphasis is placed on those properties of the probability density of the Parisi overlap parameter q which are difficult to obtain with canonical simulations relying on the standard Boltzmann distribution. This comprises the free-energy barriers which are visible in and the behavior of the tails of the probability density.

An introductory review to short-time critical dynamics is given. From the scaling relation valid already in the early stage of the evolution of a system at or near the critical point, one derives power law behaviour for various quantities. By a numerical simulation of the system one can measure the critical exponents and, by searching for the best power law behaviour, one can determine the critical point. Critical slowing down as well as finite size corrections are nearly absent, since the correlation length is still small for times far before equilibrium is reached. By measuring the (pseudo) critical points it is also possible to distinguish (weak) first-order from second-order phase transitions.

Frustrated magnets are a notorious example where the usual perturbative methods are in conflict. We show that a nonperturbative approach, based on the concept of effective average action enables one to get a coherent picture of the physics of Heisenberg frustrated magnets everywhere between d = 2 and d = 4. We recover all known perturbative results in a single framework and find the transition to be weakly first order in d = 3. The effective critical exponents found by this method are in good agreement with numerical and experimental data.

The universal behavior of the short-time dynamics for spin models on a two-dimensional triangular lattice are investigated by using a dynamic Monte Carlo simulation. Our simulation results of the dynamic evolutions from fully ordered initial states show that the universal scaling exists already in the short-time regime by observing the power-law behavior of the magnetization and Binder cumulant. The values estimated for the dynamic and static critical exponents, θ, β and ν, confirm explicitly that the Potts models on the triangular lattices and square lattices are belong to the same universality class. Also our work strongly suggests that the simulation for the dynamic relaxations can be used to determine the universality.

We report on a simulation of the supersymmetric anharmonic oscillator computed using lattice path integral techniques¹. Our numerical work utilizes a Fourier accelerated hybrid Monte Carlo scheme to sample the path integral. Combining this with the one-dimensional nature of the problem we find that we can generate high statistics data on large lattices for very modest computational cost.

We first review the three known chiral anomalies in four dimensions and then use the anomaly free conditions to study the uniqueness of quark and lepton representations and charge quantizations in the standard model. We also extend our results to theory with an arbitrary number of color. Finally, we discuss the family problem.

Participants