Dyson-Schwinger Equations And Their Applications To QCD

Confinement and the dynamical chiral symmetry breaking are the two genuine effects of non-perturbative QCD (NPQCD). To study these two effects, non-perturbative methods are required. Recently the Dyson-Schwinger (DS) approach has been a useful tool to study these two effects [1, 2, 3]. It has been proven to be successful in developing a hadron phenomenology which interpolates smoothly between the infrared (non-perturbative) and the ultraviolet (perturbative) regime [1, 2, 3].In this paper, we firstly introduce briefly the DS equation (DSE) formalism and then use the DS approach in two aspects of non-perturbative studies. In the study of the infrared behavior of gluon and ghost propagators in the coupled gluon and ghost DSEs, where the three-gluon and ghost-gluon vertex functions are taken to be bare, we use two approaches to do renormalization: the analytic continuation approach and the subtraction approach. The results show that the two renormalization approaches give the same results (κ→1, α(0) = 4.19) in the infrared analysis of propagators. Then we do the infrared analysis using a special choice of three-gluon and ghost-gluon vertex functions which are derived by their Slavnov-Taylor identities (STIs) [3, 4]. Our calculations show that the vertices used in Refs. [3, 4] are not "full" ones, because if we do exact integrations of the functions, no solution of κ exists. In studying the linear response of the dressed quark propagator in the presence of the variable external field, we derive a general expression for the vacuum susceptibility calculation in the Quantum Chromodynamical (QCD) sum rule two-point external field formula, using the variable axial external field as an example. The expressions we have derived for the calculation of vacuum susceptibility are model independent. Then we use the expressions to the calculation of the axial vector, vector and tensor vacuum susceptibilities. The numerical values of the three vacuum susceptibilities are calculated within the framework of the rainbow-ladder approximation of the DS approach. A comparison with the results of the previous approach is given. We show that the differences between the results of our approach and the previous for the axial vector and vector vacuum susceptibilities are large, while that for the tensor vacuum susceptibility is small. The differences root in the dressed effects of the non-perturbative vertices.