Recursive Graphical Construction of Feynman Diagrams in phi^4 Theory: Asymmetric Case and Effective Energy

TL;DR

This paper develops recursive methods to systematically construct and analyze Feynman diagrams in phi^4 theory, including asymmetric cases and effective energies, enabling non-perturbative investigations and resummations of graph classes.

Contribution

It introduces a recursive graphical construction approach for Feynman diagrams in phi^4 theory, extending to asymmetric cases and effective energy formulations, with non-perturbative capabilities.

Findings

Derived recursion relations for connected and 1PI Green's functions.

Proved generating functionals produce only connected and 1PI graphs.

Demonstrated resummation techniques for tadpole diagrams.

Abstract

The free energy of a multi-component scalar field theory is considered as a functional W[G,J] of the free correlation function G and an external current J. It obeys non-linear functional differential equations which are turned into recursion relations for the connected Greens functions in a loop expansion. These relations amount to a simple proof that W[G,J] generates only connected graphs and can be used to find all such graphs with their combinatoric weights. A Legendre transformation with respect to the external current converts the functional differential equations for the free energy into those for the effective energy Gamma[G,Phi], which is considered as a functional of the free correlation function G and the field expectation Phi. These equations are turned into recursion relations for the one-particle irreducible Greens functions. These relations amount to a simple proof that Gamma[G,J] generates only one-particle irreducible graphs and can be used to find all such graphs with their combinatoric weights. The techniques used also allow for a systematic investigation into resummations of classes of graphs. Examples are given for resumming one-loop and multi-loop tadpoles, both through all orders of perturbation theory. Since the functional differential equations derived are non-perturbative, they constitute also a convenient starting point for other expansions than those in numbers of loops or powers of coupling constants. We work with general interactions through four powers in the field.