Uniqueness of the Thermal Effective Potential

We discuss the use of derivative expansion techniques for the construction of thermal effective potentials. We present a theory for which the thermal bubble is analytic at the origin of the momentum-frequency space, although the internal propagators in the loop have the same mass. This means that, for this theory, the thermal effective potential is uniquely defined. We then examine a slightly different theory for which the thermal bubble displays the usual non-analyticity at the origin and the thermal effective potential is not uniquely defined. For this latter theory we compare our results with those of other works in the literature which employ the derivative expansion but find a uniquely defined thermal effective potential. We raise several questions concerning the interchange of the order of the perturbative and the derivative expansions, the thermal generalization of some non-perturbative zero temperature methods and the use of the periodicity of the external bosonic field. Finally, we re-examine the physical interpretation given to the imaginary part of the thermal bubble in the literature.