Origin of Confining Force

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 $\mathcal{M}(A)=-\mathrm{\partial }\cdot D(A)$. This is done in two steps. First we calculate the spectral decomposition of the FP operator and show that the ghost propagator $$\mathcal{G}$(k; A) = \langle \vec{k}| $\mathcal{M}$^{-1}(A) | \vec{k} \rangle$inanexternalgaugepotential$A$isenhancedatlow$k$inFourierspaceforconfigurations$A$ on the Gribov horizon. This results from the new formula in the low-$k$ regime ${\mathcal{G}}^{ab}(k,A)={\delta }^{ab}{\lambda }_{\mathrm{\mid }\vec{k}\mathrm{\mid }}^{-1}(gA)$, where ${\lambda }_{\mathrm{\mid }\vec{k}\mathrm{\mid }}(gA)$ is the eigenvalue of the FP operator that emerges from ${\lambda }_{\mathrm{\mid }\vec{k}\mathrm{\mid }}(0)={\vec{k}}^2$ at $A$ = 0. Next we derive a strict inequality signaling the divergence of the color-Coulomb potential at low momentum $k$ namely, $\tilde{\mathcal{V}}(k)\ge k^2{G}^2(k)$ for $k \to 0$,where$\widetilde{$\mathcal{V}$}(k)$ is the Fourier transform of the color-Coulomb potential $\mathcal{V}(r)$ and $G(k)$ is the ghost propagator in momentum space. The first result holds in the Landau and Coulomb gauges, whereas the second holds in the Coulomb gauge only. We propose a new numerical lattice gauge fixing that should be closer to the present analytic approach than other numerical gauges.