Some exact properties of the gluon propagator

TL;DR

This paper proves exact bounds on the gluon propagator in various gauges and dimensions, clarifying its behavior at zero momentum and aligning with numerical and perturbative results.

Contribution

It establishes rigorous bounds on the gluon propagator in SU(N) gauge theories for Landau and Coulomb gauges across different dimensions, including lattice regularization.

Findings

D(0) = 0 in 2D space-time for Landau and Coulomb gauges

Bound on high-momentum behavior in 4D consistent with asymptotic freedom

Results align with numerical studies of the gluon propagator

Abstract

Recent numerical studies of the gluon propagator in the minimal Landau and Coulomb gauges in space-time dimension 2, 3, and 4 pose a challenge to the Gribov confinement scenario. We prove, without approximation, that for these gauges, the continuum gluon propagator $D(k)$ in SU(N) gauge theory satisfies the bound ${d-1 \over d} {1 \over (2 \pi)^d} \int d^dk {D(k) \over k^2} \leq N$. This holds for Landau gauge, in which case $d$ is the dimension of space-time, and for Coulomb gauge, in which case $d$ is the dimension of ordinary space and $D(k)$ is the instantaneous spatial gluon propagator. This bound implies that $\lim_{k \to 0}k^{d-2} D(k) = 0$, where $D(k)$ is the gluon propagator at momentum $k$, and consequently $D(0) = 0$ in Landau gauge in space-time $d = 2$, and in Coulomb gauge in space dimension $d = 2$, but D(0) may be finite in higher dimension. These results are compatible with numerical studies of the Landau-and Coulomb-gauge propagator. In 4-dimensional space-time a regularization is required, and we also prove an analogous bound on the lattice gluon propagator, ${1 \over d (2 \pi)^d} \int_{- \pi}^{\pi} d^dk {\sum_\mu \cos^2(k_\mu/2) D_{\mu \mu}(k) \over 4 \sum_\lambda \sin^2(k_\lambda/2)} \leq N$. Here we have taken the infinite-volume limit of lattice gauge theory at fixed lattice spacing, and the lattice momentum componant $k_\mu$ is a continuous angle $- \pi \leq k_\mu \leq \pi$. Unexpectedly, this implies a bound on the {\it high-momentum} behavior of the continuum propagator in minimum Landau and Coulomb gauge in 4 space-time dimensions which, moreover, is compatible with the perturbative renormalization group when the theory is asymptotically free.