Abstract
The free energy of a multicomponent scalar field theory is considered as a functional 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 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 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 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