Scaling in lattice QCD: Glueball masses and string tension

We study the scaling behaviour of lattice quantum chromodynamics by comparing the β dependence of the string tension and the 0⁺⁺ glueball mass. We use a source method at β=5.7, β=5.9 and β=6.1, on lattices from 9³· 24 to 16³· 32. Assuming a string tension of about (420 MeV)², the lattice spacing ranges from 0.16 to 0.08 fm. In order to separate finite volume from scaling violation effects we have compared data from lattices having approximately the same overall physical size at the different values of β. We find deviations from scaling to be very small.     

Plaquette expansion study of compact U(1) lattice gauge theory in (2 + 1) dimensions

The plaquette expansion is applied to compact U(1) lattice gauge theory in (2 + 1) dimensions to high order: 1/N ⁸ for the ground state energy density and 1/N ⁷ for the anti-symmetric or photon mass gap, where N is defined as the number of plaquettes. Evidence of scaling in the photon mass gap is observed for low order (≤ 1/N ⁴) for inverse coupling values β = 0. 7 to 1. 25 with scaling behaviour given by the weak-coupling formula: M ²/β = exp(-5. 01β + 5. 82) in good agreement with other studies. Higher order results appear to diverge from the scaling slope past the transition point portenting the prospect that the strong coupling trial state in this region gives vise to an asymtotic series in 1/N for the photon mass gap.     

Potentials in pure QCD on 324 lattices

A range of properties of pure SU (3) gauge theory are studied numerically on euclidean lattices of 32⁴ points at β values of 6.29, 6.585 and 6.88. Potentials between static pairs of colour charges in the triplet, octet and sextet representations are calculated. The lowest energy (the A_1g) gluonic modes are considered for each of these plus the E_u mode for the triplet case. In the calculations, the use of extended operators on the lattice is examined. The results are assessed for scaling but do not agree adequately with asymptotic behaviour based on the two-loop β-function.     

Determination of the renormalized heavy-quark mass in lattice QCD

We study on the lattice the correlator of heavy-quark currents in the vicinity of vanishing momentum. The renormalized charmed quark mass, the renormalized strong coupling constant and gluon condensate can be defined in terms of the derivatives of that correlator at zero momentum. We analyze quenched Monte Carlo data on a small 8³ × 16 lattice for β = 6. We generalize dispersion relations to the lattice theory in a simple way and use them successfully to fit both the short-distance part and long-distance part of the correlator in such a small volume.     

A study of the bulk phase transitions of the SU(2) lattice gauge theory with mixed action

Using the finite size scaling theory, we reexamine the nature of the bulk phase transition in the fundamental-adjoint coupling plane of the SU(2) lattice gauge theory at β_A = 1.25 where previous finite size scaling investigations of the deconfinement phase transition showed it to be of first order for temporal lattices with four sites. Our simulations on N⁴ lattices with N = 6, 8, 10, 12 and 16 show an absence of a first order bulk phase transition. We find the discontinuity in the average plaquette to decrease approximately linearly with N. Correspondingly, the plaquette susceptibility grows a lot slower with the 4-volume of the lattice than expected from a first order bulk phase transition.     

Spatial evolutionary game with bond dilution

In this paper, we study numerically the prisoner’s dilemma game (PDG) and snowdrift game (SG) on a two-dimensional square lattice with both quenched and annealed bond dilution. For quenched bond dilution, the system undergoes a dynamical transition at the critical occupation probability q^∗, which is higher than the bond percolation transition point for a square lattice. In the critical region, the defined order parameter has a scaling form as P_e∼(q−q^∗)^β for q<q^∗ with the critical exponents β=1.42 for PDG and β=1.52 for SG, which differ from those with quenched site dilution. For annealed bond dilution, the system exhibits a distinct cooperative behavior. We find that the cooperation is much enhanced in the range of small payoff parameters on a lattice with slightly annealed bond dilution.     

On the thermodynamics and scaling behaviour of SU(2) gauge theory with fermion feedback

We study SU(2) lattice gauge theory with four species (N_f = 4) of light dynamical fermions by microcanonical simulation methods. Relatively large lattices, long runs and small quark masses are considered. On a 8³ × 16 lattice $\text{〈}\text{ψ}\text{ψ〉}$ is measured and good evidence for asymptotic freedom with fermion feedback is found. The scaling window begins at β = 4/g² ≈ 1.85. On a 12³ × 6 lattice SU(2) thermodynamics is studied systematically. The chiral symmetry restoration transition is found at β = 1.925 ± .025. The crossover from hadronic matter to the quark-gluon plasma is abrupt.     

The transition from the lattice to the continuum: CPN−1 models at large N

We study in detail the large N solution for the CP^N?1 models on a euclidean lattice and their phase structure in D dimensions. In the two-dimensional case, we evaluate some physical quantities and discuss their behaviour in the continuum limit. In particular, for the internal energy, we found the analytic expression for the perturbative tail which, at large β, obscures the predicted renormalization group behaviour; such a tail, which, as we show, is reproduced by a mean field calculation at N=∞, must be subtracted in order to define a suitable scaling quantity. We also investigated the behaviour of the topological charge and we found no perturbative tail in agreement with previous Lüscher's results. Finally, some aspects of the supersymmetric case are discussed.     

How well do Monte Carlo simulations distinguish between U(1) and SU(2)?: A study of the string tension in U(1) lattice gauge theory

To investigate how a system with a known deconfining phase transition behaves when studied on finite lattices via Monte Carlo simulations, we have made such studies of compact U(1) lattice gauge theory for 8⁴, 10⁴, and 12⁴ lattices. We have concentrated on the mean plaquette energy and the string tension. The string tension does not vanish on a finite lattice, but using finite size scaling arguments the indications are that it does vanish on an infinite lattice, where we predict the critical coupling β_c = 1.008 and the correlation length exponent $\text{ν =}\text{1}\text{3}$. We compare our results to those for SU(2) and find that although there are differences, they are not yet definitive.     

A finite size scaling analysis of the O(4)-invariant lattice δ4 theory

The critical behaviour of the three- and four-dimensional N=4 vector model is investigated by means of a Monte Carlo simulation on lattices with size between 4³ and 16³, and between 4⁴ and 12⁴, respectively. For obtaining information about some critical properties of the model, we use a method due to Binder which is based on the theory of finite size scaling. For the three-dimensional model we get estimates of the critical exponents ν and η which are compatible with estimates obtained from the ϵ-expansion. In four dimensions we study for two different values of the bare self-coupling λ (λ=1 in our normalization, and λ=∞) the scaling behaviour of some Green function ratios at the phase boundary. In both cases we find compatibility with the “predicted” scaling behaviour at the gaussian fixed point. This is another independent numerical hint that the continuum limit of the four-dimensional O(4)-invariant lattice δ⁴-model is a free field theory.     

耦合常数小于1时SU(2)格点规范理论的质量隙及β函数计算

通过改变格点规范理论的作用量,我们把Monte Carlo计算的“窗口”移到了耦合常数小于1的区域。这就提供了在较弱耦合下作Monte Carlo计算的另一种途径。在3.9≤β≤4.7区域内,在10⁴格点上,我们计算了SU(2)规范的重整化群β函数及质量间隙,观察到类似于Wilson作用量的标度行为,估算出M≈(3.9±1.0)σ^1/2。 关键词：     

Glueball masses and string tension Smeared loop-loop correlation functions

By using results obtained by the Ape Collaboration [1] we study the glueball spectrum and the string tension in lattice QCD by analyzing correlation functions between operators. We use and discuss the smearing method. We consider a 5³ × 16 lattice at β = 5.6 as a test case, and a 12³ × 32 lattice at β = 5.9 in order to compare with the old wall results of ref. [3]. We study the structure of the excited states.     

Polymers and g|φ|4 theory in four dimensions

We investigate the approach to the critical point and the scaling limit of a variety of models on a four-dimensional lattice, including g|φ|₄⁴ theory and the self-avoiding random walk. Our results, both theoretical and numerical, provide strong evidence for the triviality of the scaling limit and for logarithmic corrections to mean field scaling laws, as predicted by the perturbative renormalization group. We relate logarithmic corrections to scaling to the triviality of the scaling limit. Our numerical analysis is based on a novel, high-precision Monte Carlo technique.     

Strong-Disorder Paramagnetic-Ferromagnetic Fixed Point in the Square-Lattice ±J Ising Model

We consider the random-bond ±J Ising model on a square lattice as a function of the temperature T and of the disorder parameter p (p=1 corresponds to the pure Ising model). We investigate the critical behavior along the paramagnetic-ferromagnetic transition line at low temperatures, below the temperature of the multicritical Nishimori point at T ^*=0.9527(1), p ^*=0.89083(3). We present finite-size scaling analyses of Monte Carlo results at two temperature values, T≈0.645 and T=0.5. The results show that the paramagnetic-ferromagnetic transition line is reentrant for T<T ^*, that the transitions are continuous and controlled by a strong-disorder fixed point with critical exponents ν=1.50(4), η=0.128(8), and β=0.095(5). This fixed point is definitely different from the Ising fixed point controlling the paramagnetic-ferromagnetic transitions for T>T ^*. Our results for the critical exponents are consistent with the hyperscaling relation 2β/ν?η=d?2=0.     

Charge polarization in compact lattice QED

We investigate the polarization cloud around static electric charges in the framework of compact lattice QED. We calculate the correlations of Polyakov loops with the electric charge densityψ ^? ψ normalized to the chiral condensate. It turns out that virtual anticharges screen static charges as expected from continuum QED. Charge polarization is more pronounced at strong couplings and extends to a longer range on the lattice in the weak coupling phase. In both phases the correlations decrease with larger fermion mass.     

The goldstone pion in SU(2) lattice QCD at low quark mass: Scaling and finite size effects

Results for the pion sector of an extensive numerical simulation of quenched SU(2) QCD with staggered fermions are presented. A wide range of quark masses, gauge couplings and lattice sizes is investigated. The systematic and statistical errors are carefully analyzed. Finite size effects occurring on small lattices and/or at low quark masses are studied in detail. When they are absent or weak, we find that the pion mass and coupling constant verify approximate scaling behaviour at low quark mass for β ≳ 2.4. We determine the K and π coupling constants to be in the ratio 1.25(5). At β = 2.3, we show that the square pion mass contains a substantial contribution which is non-linear in the quark mass and renders chiral limit extrapolations less reliable.     

The string tension and the scaling behavior of SU(2) gauge theory on a random lattice

The SU(2) gauge theory on an 8⁴ random lattice has been studied by the Monte Carlo method. The string tensions have been evaluated. They display the expected scaling behavior for β=1.2−1.3. The scale parameter Λ_RAN has been determined approximately.     

The nature of the transition in d = 4 U(1) lattice gauge theory

The phase transition for the d = 4 compact U(1) lattice gauge theory has been studied using the Monte Carlo renormalization-group method. A single relevant eigenvalue is observed on the Wilson axis. The MCRG estimate for the exponent v changes with the coupling β; an extrapolation towards β^∞_c provides an estimate v ∼ 0.42 rather than ∼ 0.33 as obtained by finite-size scaling. It is confirmed that loops of monopole current are the mechanism driving the transition. It is shown that at the transition the largest loop of monopole current undergoes a discontinuity in size and begins to span the lattice. On the basis of these findings, it is conjectured that on the Wilson axis the discontinuity is a finite-size effect.     

Hadron mass calculations with Susskind and Wilson fermions in the fundamental-adjoint plane

We report the results of hadron mass calculations in the valence (quenched) approximation, on an 8³ × 16 lattice. For Wilson fermions with the standard Wilson action we find good agreement with results form the hopping parameter expansion on a 16⁴ lattice at β = 5.7, with only a small finite size effect in the anticipated direction. The proton-to-rho mass ration is however too high by 60% and the delta-proton mass difference is too small. We have repeated these calculations at two points of the same string tension in the plane of the fundamental and adjoint couplings, in an attempt to avoid the unphysical critical point there; within statistical errors the meson masses remain the same, there is an improvement in the delta-proton splitting and the proton mass decreases slightly, but not enough to produce agreement with experiment. The estimates of lines of constant string tension obtained as a preliminary to the mass calculation are in good agreement with weak coupling expansions at the larger β values and cross over towards strong coupling predictions around β = 5.7. Also, crude estimates reveal the disappearance of the specific heat peak as one moves away from the unphysical singularity. For Susskind fermions we find some measure of agreement with other results form a 10³ × 16 lattice, but large, apparently finite size, effects in the rho mass at low quark mass. Meson masses in lattice units disagree with the Wilson fermion results by as much as a factor of 2. This disagreement persists in the fundamental-adjoint plane, suggesting the importance of studying improved fermion actions. At β = 6.0, Wilson fermion results show clear finite size effects on the 8³ × 16 lattice.