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Non-perturbative Aspects of Schwinger-Dyson Equations

Schwinger-Dyson equations (SDEs) provide a natural staring point to study non-perturbative phenomena such as dynamical chiral symmetry breaking in gauge field theories. We briefly review this research in the context of quenched quantum electrodynamics (QED

a r X i v :h e p -p h /0301193v 1 22 J a n 2003Non-perturbative Aspects of Schwinger-Dyson Equations

A.Bashir

Instituto de Física y Matemáticas,Universidad Michoacana de San Nicolás de Hidalgo,Apartado

Postal 2-82,Morelia,Michoacán 58040,México.

Abstract.

Schwinger-Dyson equations (SDEs)provide a natural staring point to study non-perturbative phenomena such as dynamical chiral symmetry breaking in gauge field theories.We briefly review this research in the context of quenched quantum electrodynamics (QED)and discuss the advances made in the gradual improvement of the assumptions employed to solve these equations.We argue that these attempts render the corresponding studies more and more reliable and suitable for their future use in the more realistic cases of unquenched QED,quantum chromodynamics (QCD)and models alternative to the standard model of particle physics.

INTRODUCTION The standard model of particle physics is highly successful in collating experimental information on the basic forces.Yet,its key parameters,the masses of the quarks and leptons,are theoretically undetermined.In the simplest version of the model,these masses are specified by the couplings of the Higgs boson,couplings that are in turn undetermined.However,it could be that it is the dynamics of the fundamental gauge theories themselves that generate the masses of all the matter fields.To explore this possibility,the favorite starting point is to consider quenched QED as the simplest example of a gauge theory and study the behavior of the fermion propagator,using the corresponding SDE.Apart from the fermion propagator itself,the only unknown

ingredient in this equation is the fermion-boson vertex.As the SDE of the vertex is quite complicated,a common practice is to start from a suitable construction for it.One should ensure that every ansatz of a non-perturbative fermion-boson interaction must have the following characteristics :

•It should respect the Ward-Green-Takahashi identity (WGTI)which relates it to the fermion propagator.Moreover,in the limit when the fermion momenta are identical,it should also obey the limiting Ward identity (WI).

•In the weak coupling regime,it should match onto its perturbative loop expansion.•It should transform according to the Landau-Khalatnikov-Fradkin transformations (LKFT)under a variation of gauge.Moreover,it must guarantee that when used in the SDE for the fermion propagator,the resulting propagator also obeys its corresponding LKFT.

•It should not contain any kinematic singularities.

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