Derivative expansions of real-time thermal effective actions
  We use a generalised real-time path formalism with properly regularised propagators based on Le Bellac and Mabilat \cite{belmab} and calculate the effective potential and the higher order derivative terms of the effective action in the case of real scalar fields at finite temperature. We consider time-dependent fields in thermal equilibrium and concentrate on the quadratic part of the expanded effective action which has been associated with problems of non-analyticity at the zero limits of the four external momenta at finite temperature. We derive the effective potential and we explicitly show its independence of the initial time of the system when we include both paths of our time contour. We also derive the second derivative in the field term and recover the Real Time (RTF) and the Imaginary Time Formalism (ITF) and show that the divergences associated with the former are cancelled as long as we set the regulators zero in the end. Using an alternative method we write the field in its Taylor series form and we first derive RTF and ITF in the appropriate limits, check the analyticity properties in each case and do the actual time derivative expansion of the field up to second order in the end. We agree with our previous results and discuss an interesting term which arises in this expansion. Finally we discuss the initial time-dependence of the quadratic part of the effective action before the expansion of the field as well as of the individual terms after the expansion. 
Maria Asprouli, Victor Galan-Gonzalez