Spectral decomposition of the ghost propagator and a necessary condition for confinement (original) (raw)
In this article we present exact calculations that substantiate a clear picture relating the confining force of QCD to the zero-modes of the Faddeev-Popov (FP) operator M (gA) = −∂ • D(gA). This is done in two steps. First we calculate the spectral decomposition of the FP operator and show that the ghost propagator G(k;gA) = k|M −1 (gA)| k in an external gauge potential A is enhanced at low k in Fourier space for configurations A on the Gribov horizon. This results from the new formula in the low-k regime G ab (k, gA) = δ ab λ −1 | k| (gA), where λ | k| (gA) is the eigenvalue of the FP operator that emerges from λ | k| (0) = k 2 at A = 0. Next we derive a strict inequality signaling the divergence of the color-Coulomb potential at low momentum k namely, V (k) ≥ k 2 G 2 (k) for k → 0, where V (k) is the Fourier transform of the color-Coulomb potential V (r) and G(k) is the ghost propagator in momentum space. Although the color-Coulomb potential is a gauge-dependent quantity, we recall that it is bounded below by the gauge-invariant Wilson potential, and thus its long range provides a necessary condition for confinement. The first result holds in the Landau and Coulomb gauges, whereas the second holds in the Coulomb gauge only.
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