Recursive graphical construction of Feynman diagrams in φ4 theory: Asymmetric case and effective energy

Boris Kastening
Phys. Rev. E 61, 3501 – Published 1 April 2000
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Abstract

The free energy of a multicomponent scalar field theory is considered as a functional W[G,J] of the free correlation function G and an external current J. It obeys nonlinear functional differential equations which are turned into recursion relations for the connected Green’s 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 Γ[G,Φ], which is considered as a functional of the free correlation function G and the field expectation Φ. These equations are turned into recursion relations for the one-particle irreducible Green’s functions. These relations amount to a simple proof that Γ[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 multiloop tadpoles, both through all orders of perturbation theory. Since the functional differential equations derived are nonperturbative, 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.

  • Received 27 August 1999

DOI:https://doi.org/10.1103/PhysRevE.61.3501

©2000 American Physical Society

Authors & Affiliations

Boris Kastening

  • Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany

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Vol. 61, Iss. 4 — April 2000

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