On Daisy and Superdaisy Resummation of the Effective Potential at Finite Temperature

TL;DR

This paper details a method for resumming daisy and superdaisy diagrams in the finite-temperature effective potential of scalar  theory, emphasizing the importance of zero-mode dressing and the limitations of the IR approximation.

Contribution

It provides a detailed prescription for resumming diagrams using gap equations in the infrared limit, clarifying the conditions for consistency and the necessity of approximations.

Findings

Resummation is consistent when neglecting logarithmic terms in self-energies.

Zero-mode dressing is essential for accurate finite-temperature effective potential.

Partial dressing of non-zero modes is possible but more complex.

Abstract

We describe in detail, in the context of the simple scalar $\phi^4$ theory, the prescription for resummation of daisy and superdaisy diagrams in the effective potential using the solution of the gap equations in the infrared limit. We find that the latter procedure is consistent provided we neglect logarithmic terms from the finite-temperature self energies and from the integration of overlapping momenta. This amounts to dressing only the zero-mode contribution to the finite-temperature effective potential. Improving also the non-zero modes, would require exactly solving (not in the IR limit) the gap equations. In general this can only be done in a theory where all self-energies are momentum independent ({\em e.g.} in the scalar theory at the symmetric phase $\phi=0$). However some partial dressing procedures are still possible in general.