Some exact properties 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 $\frac{d-1}{d}\frac{1}{(2\pi {)}^d}\int d^{d}k\frac{D(k)}{k^2}\le N$. 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, $\frac{1}{d(2\pi {)}^d}{\int }_{-\pi }^{\pi }{d}^{d}k\frac{{\sum }_{\mu }{\cos }^2(k_{\mu }/2)D_{\mu \mu }(k)}{4{\sum }_{\lambda }{\sin }^2(k_{\lambda }/2)}\le N$. 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 $-\pi \le k_{\mu }\le \pi$. 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.

13 pages

References in corpus (11)

• A refinement of the Gribov-Zwanziger approach in the Landau gauge: infrared propagators in harmony with the lattice results
• Lattice gluodynamics computation of Landau-gauge Green's functions in the deep infrared
• Constraints on the IR behavior of the gluon propagator in Yang-Mills theories
• New features of the gluon and ghost propagator in the infrared region from the Gribov-Zwanziger approach
• Two- and three-point Green's functions in two-dimensional Landau-gauge Yang-Mills theory
• Coulomb gauge gluon propagator and the Gribov formula
• The Landau gauge gluon and ghost propagator in the refined Gribov-Zwanziger framework in 3 dimensions
• The effects of Gribov copies in 2D gauge theories
• The Saga of Landau-Gauge Propagators: Gathering New Ammo
• Scaling study of the gluon propagator in Coulomb gauge QCD on isotropic and anisotropic lattices
• Infrared-suppressed gluon propagator in 4d Yang-Mills theory in a Landau-like gauge

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