Analytic and numerical study of the free energy in gauge theory

Abstract

at h = 0 and k = 0, and the value of D(k, h) at this point depends on the order of limits. We also present numerical evaluations of the free energy W (k, h) and the gluon propagator D(k, h) for the case of SU(2) Yang-Mills theory in various dimensions which support all of these findings.PACS numbers: 11.10. Kk; 11.15.Ha; 12.38.Aw; 14.70.Dj * Supported by the DFG under grant number MA 3935/5-1; axelmaas@web.de † dz2@nyu.edu 2 I. INTRODUCTIONThe free energy, and its Legendre transform, the quantum effective action, play a central role in quantum field theories. As the generating functionals of correlation functions, their knowledge, in principle, grants access to all there is to know about a theory. It is often assured that these functionals are analytic in their external sources, at least away from phase transitions, and thus their derivatives yield, in a well-defined manner, the correlation functions.However, recently this assumption yielded colliding results: Assuming this analyticity, the minimal Landau gauge gluon propagator has necessarily to vanish at zero (Euclidean) momentum [1,2]. At the same time, lattice calculations, which do not need external sources, found this propagator to be finite, at least in three and four dimensions 1 [3-8] for a review.At the same time, continuum calculations using functional methods, where the functional equations were derived under this assumption, found both solutions [14][15][16][17][18]. See [9] for a review of the situation.The logical starting point to resolve this discrepancy is therefore to check the analyticity of the free energy. This is the aim in this work.To this end, we shall be concerned with the Euclidean correlators of gluons in QCD with an arbitrary gauge group, here chosen to be SU(N), for the local gauge symmetry that are fixed to the minimal Landau gauge. These are the fundamental quantities in quantum field theory.The minimal Landau gauge [9] is obtained by minimizing the Hilbert square normto some local minimum (in general not an absolute minimum) with respect to local gauge transformations g(x). These act according tog A|| 2 is stationary and its second variation is positive. It is well known that these two properties imply respectively that the Landau gauge (transversality) condition is satisfied, ∂ · A = 0, and that the Faddeev-Popov operator is positive i. e.(ω, M(A)ω) ≥ 0 for all ω. Here the Faddeev-Popov operator acts according to [9] that in general there are more than one local minimum of F A (g), and we do not specify which local minimum is achieved. This gauge is realized numerically by minimizing a lattice analog of F A (g) by some algorithm, and the local minimum achieved is in general algorithm dependent. However, for all commonly employed algorithms this does not yield different expectation values, as they all are equivalent to an averaging over the first Gribov region [9,20].The analytic bounds which we shall obtain in section II follow from the restriction of the gauge-fixed configurations to the Gribov region Ω, and are the same wheth...

Bounds on free energy in QCD

“…In the following some preliminary results from such an evaluation will be presented. More results will be presented elsewhere [15]. These will be given for two, three, and four dimensions, as it has been found that gauge-dependent quantities depend significantly on the dimensionality [14].…”

“…The three-dimensional lattice has volume of 72 3 lattice sites at β = 3.73, i. e. (14.4 fm) 3 and a = 0.2 fm. To investigate whether this is a lattice artifact and/or depends on the dimensionality, the remaining constant of proportionality for a range of lattice sizes and discretizations has been determined [15], and preliminary results are shown in figure 3. At first sight, no qualitative difference is found between different dimensions.…”

Gluon propagator in an external field; what happens when the field is removed?

“…Because Ω is bounded in every direction [6], this bound is finite. It is not hard to show that the maximum occurs when A lies on the boundary ∂ Ω of Ω, and we have the more precise bound on the free energy [7],…”

“…The expectation-value is calculated by Monte Carlo simulation, with specifics provided in [7]. Gauge fixing is done by minimization to minimal Landau gauge, so only the inside of the first Gribov region is sampled, A ∈ Ω;…”

Constraining the gauge-fixed Lagrangian in minimal Landau gauge

“…It should be possible to do this also for FRGs, where the regulator can be included by reweighting, in a similar manner as sources[68] 4. Note that slightly higher statistics than listed in table 1 have been used to create the figures in the appendices.…”