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

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.

arXiv comment34 pages; abstract expanded; section IV.E about absorption of tadpoles and one related reference added; eqs. (20) and (23) corrected; further references added; some minor beautifications; to be published by Phys.Rev.E

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Cited by in corpus (13)

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• Non-universal Critical Quantities from Variational Perturbation Theory and Their Application to the BEC Temperature Shift
• A simple way to generate high order vacuum graphs
• Fluctuation pressure of a fluid membrane between walls through six loops
• Functional Closure of Schwinger-Dyson Equations in Quantum Electrodynamics, Part 1: Generation of Connected and One-Particle Irreducible Feynman Diagrams
• Recursive Graphical Solution of Closed Schwinger-Dyson Equations in phi^4-Theory -- Part1: Generation of Connected and One-Particle Irreducible Feynman Diagrams
• Recursive Graphical Construction for Feynman Diagrams and Their Weights in Ginzburg-Landau Theory
• Many-Body Vacuum Diagrams and Their Recursive Graphical Construction
• 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
• Fluctuation Pressure of a Membrane Between Walls Through Five Loops
• Effective Action for the Scalar Field Theory with Higher Vertices
• Two-Loop Effective Potential of O(N)-Symmetric Scalar QED in 4-epsilon Dimensions