Schwinger-Dyson equations in QED and QCD the calculation of fermion-antifermion condensates

Abstract

We present non-perturbative solutions for the fermion and boson propagators of QED in both three- and four-dimensions, and QCD. In doing so, we solve the coupled system of Schwinger-Dyson equations numerically in Euclidean space, investigating criticality, gauge dependence and phenomenology of the solutions. We do so by exploiting a new and novel three-point ansatz, the Kizilersü-Pennington vertex, designed to satisfy multiplicative renormalisability in unquenched QED. The efficacy of this is demonstrated numerically for QED(_4), where we find a marked improvement in the gauge-invarance of the photon wave-function. The critical coupling associated with dynamical mass generation is investigated for a variety of gauges; remarkably a lessening of this dependence is seen, despite the ansatz’s origins from a massless theory, which is improved further by constructing a hybrid system. As with many studies in the past, we apply this ansatz to the three-dimensional non-compact formulation of QED, checking gauge covariance of the propagators through a momentum-space formulation of the Landau-Khalatnikov-Pradkin transformations. The critical dependence on the number of active fermions was investigated, with the gauge dependence of the condensate unresolved. As an aside, we found numerically that LKF transforming the propagators gave rise to a constant condensate; a fact supported analytically through an explicit proof. We turn our attention towards QCD, where we explore a variety of phenomeno-logical models, including the full ghost-gluon system, in which we make comparisons between traditional vertices and the new KP-Vertex. These models are used in a determination of the physical quark condensate for massive quarks, through the exploitation of a class of non-positive definite solutions accessible for small quark masses. Finally, we examine Generalised Ward-Takahashi identities, which hold promise to further constrain the tranvserse part of the vertex. The identity is shown to hold true at one-loop through an explicit calculation, and a constraint on one of the basis coefficients is given as an example of its use.