Functional differential equations for the free energy and the effective energy in the broken-symmetry phase of φ4-theory and their recursive graphical solution

Extending recent work on QED and the symmetric phase of the euclidean multicomponent scalar φ4-theory, we construct the vacuum diagrams of the free energy and the effective energy in the ordered phase of φ4-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.