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 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.