The thermal heat kernel expansion and the one-loop effective action of QCD at finite temperature

The heat kernel expansion for field theory at finite temperature is constructed. It is based on the imaginary time formalism and applies to generic Klein-Gordon operators in flat space-time. Full gauge invariance is manifest at each order of the expansion and the Polyakov loop plays an important role at any temperature. The expansion is explicitly worked out up to operators of dimension six included. The method is then applied to compute the one loop effective action of QCD at finite temperature with massless quarks. The calculation is carried out within the background field method in the $\bar{\text{MS}}$ scheme up to dimension six operators. Further, the action of the dimensionally reduced effective theory at high temperature is also computed to the same order. Existing calculations are reproduced and new results are obtained in the quark sector for which only partial results existed up to dimension six.