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Recursive graphical construction of Feynman diagrams in φ^{4} theory: Asymmetric case and effective energy

Kastening, Boris (2000):
Recursive graphical construction of Feynman diagrams in φ^{4} theory: Asymmetric case and effective energy.
In: Physical Review E, American Physical Society, pp. 3501-3528, 61, (4), ISSN 1063-651X,
[Online-Edition: http://dx.doi.org/10.1103/PhysRevE.61.3501],
[Article]

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 G[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 G[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.

Item Type: Article
Erschienen: 2000
Creators: Kastening, Boris
Title: Recursive graphical construction of Feynman diagrams in φ^{4} theory: Asymmetric case and effective energy
Language: English
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 G[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 G[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.

Journal or Publication Title: Physical Review E
Volume: 61
Number: 4
Publisher: American Physical Society
Divisions: 11 Department of Materials and Earth Sciences > Material Science > Advanced Thin Film Technology
11 Department of Materials and Earth Sciences > Material Science
11 Department of Materials and Earth Sciences
Date Deposited: 04 Jan 2013 11:43
Official URL: http://dx.doi.org/10.1103/PhysRevE.61.3501
Identification Number: doi:10.1103/PhysRevE.61.3501
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