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

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Abstract

Extending recent work on QED and the symmetric phase of the euclidean multicomponent scalar \phi^4-theory, we construct the vacuum diagrams of the free energy and the effective energy in the ordered phase of \phi^4-theory. By regarding them as functionals of the free correlation function and the interaction vertices, we graphically solve nonlinear functional differential equations, obtaining loop by loop all connected and one-particle irreducible vacuum diagrams with their proper weights.

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Recursive Graphical Construction of Feynman Diagrams and Their Multiplicities in phi^4- and in phi^2A-Theory

M. Bachmann, Boris Kastening, Axel Pelster … (1999)

The free energy of a field theory can be considered as a functional of the free correlation function. As such it obeys a nonlinear functional differential equation which can be turned into a recursion relation. This is solved order by order in the coupling constant to find all connected vacuum diagrams with their proper multiplicities. The procedure is applied to a multicomponent scalar field theory with a phi^4-self-interaction and then to a theory of two scalar fields phi and A with an interaction phi^2 A. All Feynman diagrams with external lines are obtained from functional derivatives of the connected vacuum diagrams with respect to the free correlation function. Finally, the recursive graphical construction is automatized by computer algebra with the help of a unique matrix notation for the Feynman diagrams.

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Four-Loop Vacuum Energy Beta Function in O(N) Symmetric Scalar Theory

Boris Kastening (1996)

The beta function of the vacuum energy density is computed at the four-loop level in massive O(N) symmetric phi^4 theory. Dimensional regularization is used in conjunction with the MSbar scheme and all calculations are in momentum space in the massive theory. The result is beta_v = g N/4+g^3 N(N+2)/96+g^4 N(N+2)(N+8)[12 zeta(3)-25]/1296+o(g^5).

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Five-Loop Vacuum Energy Beta Function in phi^4 Theory with O(N)-Symmetric and Cubic Interactions

Boris Kastening (1997)

The beta function of the vacuum energy density is analytically computed at the five-loop level in O(N)-symmetric phi^4 theory, using dimensional regularization in conjunction with the MSbar scheme. The result for the case of a cubic anisotropy is also given. It is pointed out how to also obtain the beta function of the coupling and the gamma function of the mass from vacuum graphs. This method may be easier than traditional approaches.

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Author and article information

Journal

Publication date Created: 20 June 2000

Article

DOI: 10.1016/S0378-4371(02)01932-5

ArXiV ID: hep-th/0006153

SO-VID: d5a84133-e1c8-4fdd-bf50-f56f99c94fe9

History

Custom metadata

Journal reference Physica A323 (2003) 370-400

Comments Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html Latest update of paper also at http://www.physik.fu-berlin.de/~kleinert/310

Categories hep-th

ScienceOpen disciplines: High energy & Particle physics

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ScienceOpen disciplines: High energy & Particle physics