Some exact properties of the gluon propagator

Recent numerical studies of the gluon propagator in the minimal Landau and Coulomb gauges in space-time dimension two, three, and four pose a challenge to the Gribov confinement scenario. In these gauges all configurations are transverse, ∂·A=0, and lie inside the Gribov region Ω, where the Faddeev-Popov operator, M(A)=-∂_μD_μ(A), is positive, that is, (ψ,M(A)ψ)≥0 for all ψ. We prove, without approximation, that for these gauges the continuum gluon propagator D(k) in SU(N) gauge theory satisfies the bound d-1d1(2π)d∫ddkD(k)k2≤N. This holds for the Landau gauge, in which case d is the dimension of space-time, and for the 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 kd^-2D(k)=0, where D(k) is the gluon propagator at momentum k, and consequently D(0)=0 in the Landau gauge in space-time d=2 and in the Coulomb gauge in space dimension d=2, but D(0) may be finite in higher dimensions. These results are compatible with numerical studies of the Landau-and Coulomb-gauge propagator. In four-dimensional space-time a regularization is required, and we also prove an analogous bound on the lattice gluon propagator, 1d(2π)d∫-ππddkΣμ^cos-2(k _μ/2)D_μμ(k)4Σλ^{sin -}2(k_λ/2)≤N. Here we have taken the infinite-volume limit of lattice gauge theory at fixed lattice spacing, and the lattice momentum componant k_μ is a continuous angle, -π≤k_μ≤π. Unexpectedly, this implies a bound on a renormalization-group invariant that governs the overall normalization of the continuum gluon propagator in the minimum Landau and Coulomb gauges in four space-time dimensions, which, moreover, is compatible with the perturbative renormalization group when the theory is asymptotically free.