[Paper Review] On Daisy and Superdaisy Resummation of the Effective Potential at Finite Temperature

M. Quirós|ArXiv.org|Apr 21, 1993

Advanced Thermodynamics and Statistical Mechanics3 citations

TL;DR

This paper develops a consistent resummation scheme for daisy and superdaisy diagrams in the finite-temperature effective potential of a scalar φ⁴ theory. It shows that dressing only the zero-mode contribution via the infrared limit of the gap equation yields a consistent result, provided logarithmic corrections from self-energies and overlapping momenta are neglected; full consistency requires solving the gap equations exactly, which is feasible only in momentum-independent self-energy theories like the symmetric phase of φ⁴ theory.

ABSTRACT

We describe in detail, in the context of the simple scalar $ϕ^4$ 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 $ϕ=0$). However some partial dressing procedures are still possible in general.

Motivation & Objective

• To establish a consistent framework for resumming daisy and superdaisy diagrams in the finite-temperature effective potential.
• To clarify the conditions under which the superdaisy resummation procedure remains valid at finite temperature.
• To investigate the role of logarithmic corrections in self-energies and momentum integration in the consistency of the resummation.
• To determine whether partial dressing of non-zero modes is feasible in general theories beyond the symmetric phase.
• To explore the limitations of the infrared approximation in solving the gap equations for the effective potential.

Proposed method

• Applies the gap equation formalism to the scalar φ⁴ theory at finite temperature, focusing on the infrared limit of self-energies.
• Implements a resummation scheme that includes all daisy and superdaisy diagrams by dressing the zero-mode contribution to the effective potential.
• Neglects logarithmic terms arising from finite-temperature self-energies and overlapping momentum integrals to maintain consistency in the resummation.
• Considers the exact solution of the gap equations as the only fully consistent approach, but shows it is tractable only when self-energies are momentum-independent.
• Analyzes the symmetric phase (φ = 0) of the scalar theory as a case where exact gap equation solutions are possible.
• Proposes partial dressing procedures for non-zero modes in general theories, though full consistency remains unattainable without exact solutions.

Experimental results

Research questions

• RQ1 Under what conditions is the superdaisy resummation procedure consistent in the finite-temperature effective potential?
• RQ2 How do logarithmic corrections from self-energies and overlapping momenta affect the validity of the resummation scheme?
• RQ3 Can the zero-mode contribution be consistently dressed using the infrared limit of the gap equation?
• RQ4 What are the limitations of the infrared approximation when extending the resummation to non-zero modes?
• RQ5 In which theories can the full gap equations be solved exactly to ensure consistency in the resummation?

Key findings

• The superdaisy resummation is consistent only if logarithmic corrections from finite-temperature self-energies and overlapping momenta are neglected.
• Dressing only the zero-mode contribution via the infrared limit of the gap equation yields a consistent approximation to the effective potential.
• Exact solution of the gap equations is required for full consistency, but this is feasible only in theories with momentum-independent self-energies, such as the symmetric phase of the φ⁴ theory.
• Partial dressing of non-zero modes is still possible in general, but cannot guarantee full consistency without solving the gap equations exactly.
• The method breaks down when logarithmic terms are included, indicating the need for a more refined treatment in general cases.
• The framework provides a practical, though approximate, method for resumming self-energy corrections in finite-temperature scalar field theories.