Thermodynamics of SU(N) Yang-Mills theories in 2+1 dimensions I - The confining phase
Michele Caselle, Luca Castagnini, Alessandra Feo, Ferdinando Gliozzi, Marco Panero
备注：
1+22 pages, 8 eps figures; v2: added comments about the uncertainties on the temperature scale and the pressure, expanded the discussion in section 4, added references: version published in JHEP
We compute the equation of state in the confining phase of SU(N) Yang-Mills theories with N=2, 3, 4, 5 and 6 colors in 2+1 dimensions, via lattice simulations. At low enough temperatures, the results are accurately described by a gas of glueballs, including all known states below the two-particle threshold. Close to the deconfinement temperature, however, this prediction underestimates the numerical results, and the contribution from heavier glueballs has to be included. We show that the spectral density of the latter can be accurately described using a simple bosonic string model.
We study the structure of the confining string in Yang-Mills theory using the method of the field strength correlators. The method allows us to demonstrate that both the local fluctuations of the topological charge and the gluon condensate are suppressed in the vicinity of the string axis in agreement with results of lattice simulations.
We discuss thermodynamic properties of open confining strings introduced via static sources in the vacuum of Yang-Mills theory. We derive new sum rules for the chromoelectric and chromomagnetic condensates and use them to show that the presence of the confining string lowers the gluonic pressure in the bulk of the system. The pressure deficit of the gluon plasma is related to the potential energy in the system of heavy quarks and anti-quarks in the plasma.
We show that the QCD vacuum (without dynamical quarks) is a dual superconductor at least in the low-energy region in the sense that monopole condensation does really occur. In fact, we derive the dual Ginzburg-Landau theory (i.e., dual Abelian Higgs model) directly from the SU(2) Yang-Mills theory by adopting the maximal Abelian gauge. The dual superconductor can be on the border between type II and type I, excluding the London limit. The masses of the dual Abelian gauge field is expressed by the Yang-Mills gauge coupling constant and the mass of the off-diagonal gluon of the original Yang-Mil…
We investigate the transverse fluctuations of the confining string connecting two static quarks in (2+1)-d SU(2) Yang-Mills theory using Monte Carlo calculations. The exponentially suppressed signal is extracted from the large noise by a very efficient multi-level algorithm. The resulting width of the string increases logarithmically with the distance between the static quark charges. Corrections at intermediate distances due to universal higher order terms in the effective string action are calculated analytically. They accurately fit the numerical data.
The logarithmic broadening predicted by the systematic low-energy effective field theory for the confining string has recently been verified in numerical simulations of (2+1)-d SU(2) lattice Yang-Mills theory at zero temperature. The same effective theory predicts linear broadening of the string at low non-zero temperature. In this paper, we verify this prediction by comparison with very precise Monte Carlo data. The comparison involves no additional adjustable parameters, because the low-energy constants of the effective theory have already been fixed at zero temperature. It yields very good …
It is well known that confinings and asymptotic freedom are properties of quantum chromo-dynamics (QCD). But hints of these features can also be observed at purely classic levels. For this purpose we need to find solutions to the colorly-sourceful Yang-Mills equations with both confining and asymptotic freedom features. We provide such a solution in this paper which at the near-source region is of serial form, while at the far-away region is approximately expressed through simple elementary functions. From the solution, we derive out a classically non-perturbative beta function describing the …
We present a numerical method to compute path integrals in effective SU(2) Yang-Mills theories. The basic idea is to approximate the Yang-Mills path integral by summing over all gauge field configurations, which can be represented as a linear superposition of a small number of localized building blocks. With a suitable choice of building blocks many essential features of SU(2) Yang-Mills theory can be reproduced, particularly confinement. The analysis of our results leads to the conclusion that topological charge as well as extended structures are essential elements of confining gauge field co…
We continue our investigation of quark confinement using a particular variant of the Cho-Duan-Ge gauge independent Abelian decomposition. The decomposition splits the gauge field into a restricted Abelian part and a coloured part in a way that preserves gauge covariance. The restricted part of the gauge field can be divided into a Maxwell term and a topological term. Previously, we showed that by a particular choice of this decomposition we could fully describe the confining potential using only the restricted gauge field. We proposed that various topological objects (a form of magnetic monopo…
We discuss within the weak-field approximation and the derivative expansion how the area law of the Wilson loop follows directly from the vacuum condensate of mass dimension 2, i.e., simultaneous Bose-Einstein condensation of gluon pair and ghost-antighost pair. Such a novel vacuum condensate was recently claimed to exist as the non-vanishing vacuum expectation value of a BRST-invariant composite operator of mass dimension 2. First of all, we use a version of the non-Abelian Stokes theorem to rewrite the Wilson loop line integral to a surface integral. Then we convert the Yang-Mills theory wit…
We study the confining/deconfining phase transition in the mass deformed Yang-Mills matrix model which is obtained by the dimensional reduction of the bosonic sector of the four-dimensional maximally supersymmetric Yang-Mills theory compactified on the three sphere, i.e. the bosonic BMN model. The $1/D$ (with $D$ the number of matrices) expansion suggests that the model may have two closely separated transitions. However, using a second order lattice formulation of the model we find that for the small value of the mass parameter, $μ=2$, those two apparent critical temperatures merge at large $…
We study some features of the confining string connecting a quark-anti-quark pair in Yang-Mills theory. Monte Carlo investigations of the flux tube between two static quarks in the fundamental representation show that its thickness increases with the separation of the sources. The collected numerical data in (2+1)-d SU(2) Yang-Mills theory are in very good quantitative agreement with the next-to-leading order formula derived from the systematic low-energy effective theory of the confining string. At zero temperature, we observe the predicted logarithmic broadening. At finite and low temperatur…
We illustrate some physical application of a lattice formulation of the two-dimensional $\mathcal{N}=(2,2)$ supersymmetric SU(2) Yang-Mills theory with a (small) supersymmetry breaking scalar mass. Two aspects, power-like behavior of certain correlation functions (which implies the absence of the mass gap) and the static potential $V(R)$ between probe charges in the fundamental representation, are considered. For the latter, for $R\lesssim1/g$, we observe a linear confining potential with a finite string tension. This confining behavior appears distinct from a theoretical conjecture that a pro…
In this series of three papers, we generalize the derivation of dual photons and monopoles by Polyakov, and Banks, Myerson and Kogut, to obtain approximative models of SU(2) lattice gauge theory. The papers take three different representations as their starting points: the representation as a BF Yang-Mills theory, the spin foam representation and the plaquette representation. The derivations are based on stationary phase approximations. In this first article, we cast 3- and 4-dimensional SU(2) lattice gauge theory in the form of a lattice BF Yang-Mills theory. In several steps, the expectation…
We analyze the Wilson loop for a pure Yang-Mills theory, using a decoupling solution in close agreement with lattice computations. At one-gluon exchange level it is seen that the potential cannot yield a linear rising contribution as expected for a confining theory. Next-to-leading order correction gives rise to a quartic term for momenta in the gluon propagator that, in agreement with Gribov's view, yields a linear confining term. This correction is due to a two-loop or sunrise integral that we need to evaluate in the low-momenta limit. In the infrared regime, the physical consistency of the …
An effective theory based on ensembles of either regular gauge instantons or merons is shown to produce confinement in SU(2) Yang-Mills theory. When the scale is set by the string tension, the action density, topological susceptibility and low-lying glueball spectrum are similar to those arising in lattice QCD. The physical mechanism producing confinement is explained, and a number of analytical insights into the effective theory are presented.
For the four-dimensional SU(N) Euclidean Yang-Mills theory in the Landau gauge, we present two sets of gauge field configurations that satisfy the Euclidean equations of motion. These configurations generate four-dimensional chromo-field flux sheets whose spatial cross sections are three-dimensional chromo-field flux tubes. In lattice simulations, they may be detected as center vortices. The first set of gauge field configurations generates chromo-electric flux tubes that should contribute to a chromo-electric flux tube between two static color charges. The string tension for two static color …
In earlier papers we established quark confinement analytically in anisotropic $(2+1)$-dimensional Yang-Mills theory with two gauge coupling constants. Here we point out a few features of the confining phase. These are: 1) the string tension in the $x^{2}$-direction as a function of representation obeys a sine law, and 2) static adjoint sources are not confined.
The topological susceptibility is calculated within the Hamiltonian approach to Yang-Mills theory in Coulomb gauge, using the vacuum wave functional previously determined by a variational solution of the Yang-Mills Schroedinger equation. The numerical result agrees qualitatively with the predictions of lattice simulations.
An explicit canonical transformation is constructed to relate the physical subspace of Yang-Mills theory to the phase space of the ADM variables of general relativity. This maps 3+1 dimensional Yang-Mills theory to local evolution of metrics on 3 manifolds.
We analyze the leading Lyapunov exponents of SU(2) Yang-Mills field configurations on the lattice which are initialized by quantum Monte Carlo simulations. We find that configurations in the strong coupling phase at finite temperature are substantially more chaotic than in deconfinement.
The SU(2) Yang-Mills gluon and ghost Green-functions are studied in Landau gauge by means of lattice gauge simulations. A focal point is their low energy behavior since in particular the Kugo-Ojima confinement criterion relates a diverging ghost form factor at vanishing momentum transfer to the confining capabilities of the theory. This divergence is verified by numerical simulations. Removing the confining vortices from the lattice ensembles converts SU(2) Yang-Mills theory in a non-confining theory. It is shown that in this modified theory the divergence of the ghost propagator disappears.
The solution of symmetry equation of Yang-Mills self dual system is found in explicit form of its raising Hamiltonian operator. Thus explicit form of equations of self dual Yang Mills hierarchy is constructed.
We report about our ongoing computation of running coupling constants in asymptotically free theories using the recursive finite size scaling technique. The latest results for the SU(3) Yang-Mills theory are presented.
Monte Carlo simulations are performed in a five-dimensional lattice SU(2) Yang-Mills theory with a compactified extra dimension, and scaling laws are studied. Our simulations indicate that as the compactification radius $R$ decreases, the confining phase spreads more and more to the weak coupling regime, and the effective dimension of the theory changes gradually from five to four. Our simulations also indicate that the limit $a_4 to 0$ with $R/a_4$ kept fixed exists both in the confining and deconfining phases if $R/a_4$ is small enough, where $a_4$ is the lattice spacing in the four-dimensio…
