Polyakov loop correlator in perturbation theory

We study the Polyakov loop correlator in the weak coupling expansion and show how the perturbative series reexponentiates into singlet and adjoint contributions. We calculate the order $g^7$ correction to the Polyakov loop correlator in the short distance limit. We show how the singlet and adjoint free energies arising from the reexponentiation formula of the Polyakov loop correlator are related to the gauge invariant singlet and octet free energies that can be defined in pNRQCD, namely we find that the two definitions agree at leading order in the multipole expansion, but differ at first order in the quark-antiquark distance.

Figure 1

All unconnected two-gluon diagrams. Some diagrams can be flipped to give in total four other diagrams that are not explicitly displayed. The line with the right arrow (omitted in the last two diagrams) is the Polyakov loop contour for the quark, and the line with the left arrow corresponds to the antiquark.

Figure 2

Additional diagrams in the Wilson loop that contribute with terms of order $z^2$.