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An approach to calculating approximate solutions to the continuum Schwinger-Dyson equations is outlined, with examples for phi^4 in D=1. This approach is based on the source Galerkin methods developed by Garcia, Guralnik and Lawson. Numerical issues and opportunities for future calculations are also discussed briefly.
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The Source Galerkin method finds approximate solutions to the functional differential equations of field theories in the presence of external sources. While developing this process, it was recognized that approximations of the spectral representations of the Greens functions by Sinc function expansions are an extremely powerful calculative tool. Specifically, this understanding makes it not only possible to apply the Source Galerkin method to higher dimensional field theories, but also leads to a new approach to perturbation theory calculations in scalar and fermionic field theories. This report summarizes the methodologies for solving quantum field theories with the Source Galerkin method and for performing perturbation theory calculations using Sinc approximations.
The Source Galerkin Method is a new numerical technique that is being developed to solve Quantum Field Theories on the continuum. It is not based on Monte Carlo techniques and has a measure to evaluate relative errors. It promises to increase the accuracy and speed of calculations, and takes full advantage of symmetries of the theory. The application of this method to the non-linear sigma model is outlined.
We compute the Green function of the massless scalar field theory in the infrared till the next-to-leading order, providing a fully covariant strong coupling expansion. Applying Callan-Symanzik equation we obtain the exact running coupling for this case by computing the beta function. This result is applied using a recently proved mapping theorem between a massless scalar field theory and Yang-Mills theory. This beta function gives a running coupling going to zero as $p^4$ in agreement with lattice results presented in Boucaud et al. [JHEP 0304 (2003) 005] and showing that the right definition of the running coupling for a Yang-Mills theory in the infrared is given in a MOM scheme. The emerging scenario is supporting a quantum field theory based on instantons.
We extend our study of deriving the local gauge invariance with spontaneous symmetry breaking in the context of an effective field theory by considering self-interactions of the scalar field and inclusion of the electromagnetic interaction. By analyzing renormalizability and the scale separation conditions of three-, four- and five-point vertex functions of the scalar field, we fix the two couplings of the scalar field self-interactions of the leading order Lagrangian. Next we add the electromagnetic interaction and derive conditions relating the magnetic moment of the charged vector boson to its charge and the masses of the charged and neutral massive vector bosons to each other and the two independent couplings of the theory. We obtain the bosonic part of the Lagrangian of the electroweak Standard Model as a unique solution to the conditions imposed by the self-consistency conditions of the considered effective field theory.
We revisit the problem of deriving local gauge invariance with spontaneous symmetry breaking in the context of an effective field theory. Previous derivations were based on the condition of tree-order unitarity. However, the modern point of view considers the Standard Model as the leading order approximation to an effective field theory. As tree-order unitarity is in any case violated by higher-order terms in an effective field theory, it is instructive to investigate a formalism which can be also applied to analyze higher-order interactions. In the current work we consider an effective field theory of massive vector bosons interacting with a massive scalar field. We impose the conditions of generating the right number of constraints for systems with spin-one particles and perturbative renormalizability as well as the separation of scales at one-loop order. We find that the above conditions impose severe restrictions on the coupling constants of the interaction terms. Except for the strengths of the self-interactions of the scalar field, that can not be determined at this order from the analysis of three- and four-point functions, we recover the gauge-invariant Lagrangian with spontaneous symmetry breaking taken in the unitary gauge as the leading order approximation to an effective field theory. We also outline the additional work that is required to finish this program.
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