Functional Differential Equations for the Free Energy and the Effective Energy in the Broken-Symmetry Phase of phi^4-Theory and Their Recursive Graphical Solution

TL;DR

This paper develops a graphical recursive method to compute vacuum diagrams for free and effective energies in the broken-symmetry phase of ^4-theory, extending previous work on symmetric phases.

Contribution

It introduces a novel recursive graphical approach to solve nonlinear functional differential equations for vacuum diagrams in the broken-symmetry phase of ^4-theory.

Findings

Successfully computes all connected vacuum diagrams with proper weights.

Extends the methodology from symmetric to broken-symmetry phases.

Provides a systematic loop-by-loop diagrammatic solution.

Abstract

Extending recent work on QED and the symmetric phase of the euclidean multicomponent scalar \phi^4-theory, we construct the vacuum diagrams of the free energy and the effective energy in the ordered phase of \phi^4-theory. By regarding them as functionals of the free correlation function and the interaction vertices, we graphically solve nonlinear functional differential equations, obtaining loop by loop all connected and one-particle irreducible vacuum diagrams with their proper weights.