Plaquette expectation value and gluon condensate in three dimensions

TL;DR

In three-dimensional SU(3) gauge theory, the gluon condensate contains ultraviolet divergences up to 4-loop order. The authors extract the finite, lattice-regularized condensate by subtracting known perturbative UV terms from plaquette measurements and performing a continuum extrapolation, enabling a potential translation to the MSbar scheme. The master lattice–continuum relation connects the nonperturbative input B_G to the subtracted plaquette, with c1–c4 known and c4' unknown, providing a path to the first nonperturbative contribution to hot QCD pressure once MSbar matching is completed. The work delivers a precise lattice determination of the subtracted condensate and demonstrates a viable route to constrain finite-temperature QCD via dimensional reduction, pending the four-loop MSbar matching calculation.

Abstract

In three dimensions, the gluon condensate of pure SU(3) gauge theory has ultraviolet divergences up to 4-loop level only. By subtracting the corresponding terms from lattice measurements of the plaquette expectation value and extrapolating to the continuum limit, we extract the finite part of the gluon condensate in lattice regularization. Through a change of regularization scheme to MSbar and (inverse) dimensional reduction, this result would determine the first non-perturbative coefficient in the weak-coupling expansion of hot QCD pressure.

Figures (4)

• Figure 1: The plaquette expectation value, "plaq" $\equiv \langle 1 - \frac{1}{3} {\rm Tr\,}[P_{12}] \rangle_a$, as a function of $1/\beta$. Statistical errors are (much) smaller than the symbol sizes. The dotted curve contains the four known terms $c_1/\beta+c_2/\beta^2+c_3/\beta^3+c_4 \ln\beta/\beta^4$ from Eq. (\ref{['eq:betaG']}), together with terms of the type $1/\beta^4$, $1/\beta^5$ and $1/\beta^6$ with fitted coefficients.
• Figure 2: The significance loss due to the subtractions of the various ultraviolet divergent contributions in the gluon condensate. Here again "plaq" $\equiv \langle 1 - \frac{1}{3} {\rm Tr\,}[P_{12}] \rangle_a$, and the symbols $c_i$ in the curly brackets indicate which subtractions of Eq. (\ref{['eq:betaG']}) have been taken into account.
• Figure 3: Finite-volume values for $\beta^4\{\langle 1 - \frac{1}{3} {\rm Tr\,}[P_{12}] \rangle_a - [c_1/\beta+c_2/\beta^2+c_3/\beta^3+c_4 \ln\beta/\beta^4]\}$, as a function of the physical extent $\beta/N = 6/ g_3^2 L$ of the box. The solid symbols indicate the infinite-volume estimates, obtained by fitting a constant to data in the range $\beta/N < 1$.
• Figure 4: The infinite-volume extrapolated data, plotted as in Fig. \ref{['fig:orders']}. The effect of the 4-loop logarithmic divergence is to cause additional upwards "curvature" in the upper data set. The lower set includes all the subtractions, and should thus have a finite continuum limit. The continuum extrapolation (as described in the text) is indicated with the dashed line. The gray points have error bars so large that they are insignificant as far as the fit is concerned.