Proper heavy-quark potential from a spectral decomposition of the thermal Wilson loop

We propose a non-perturbative and gauge invariant derivation of the static potential between a heavy-quark ($Q$) and an anti-quark ($\bar{Q}$) at finite temperature. This proper potential is defined through the spectral function (SPF) of the thermal Wilson loop and can be shown to satisfy the Schr"{o}dinger equation for the heavy $Q\bar{Q}$ pair in the thermal medium. In general, the proper potential has a real and an imaginary part,corresponding to the peak position and width of the SPF. The validity of using a Schr"{o}dinger equation for heavy $Q\bar{Q}$ can also be checked from the structure of the SPF. To test this idea, quenched QCD simulations on anisotropic lattices ($a_{\sigma }=4a_{\tau }=0.039\mathrm{f}\mathrm{m}$, $N^3_\sigma \times N_{\tau} =20^2 \times (96-32)$) are performed. The real part of the proper potential below the deconfinement temperature ($T=0.78T_c$) exhibits the well known Coulombic and confining behavior. At ($T=2.33T_c$) we find that it coincides with the Debye screened potential obtained from Polyakov-line correlations in the color-singlet channel under Coulomb gauge fixing. The physical meaning of the spectral structure of the thermal Wilson loop and the use of the maximum entropy method (MEM) to extract the real and imaginary part of the proper potential are also discussed.