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**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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