Breaking and the Quark-quark Interaction 1 a Path from Qcd to Chiral Perturbation Theory

A global color symmetry model (GCM) that is based upon an effective quark-quark interaction arises from the QCD partition function by formally integrating over the gluon fields and truncating the expansion in gluon n point functions after n = 2. 1 This model truncation respects all global symmetries of QCD, in particular chiral symmetry. What is lost is invariance under local color gauge transformations. Fixing a model form for the gluon propagator D(s), or, what is equivalent, the running coupling α(s), specifies a quark-quark interaction. Furthermore the model allows a 1 Nc expansion. The lowest order (O(N c)) consists in solving the Dyson-Schwinger equation for the quark self energy including rainbow gluon dressings and the ladder Bethe-Salpeter equation for mesonic bound states. One then is able to change the degrees of freedom from quarks to hadrons by performing an appropriate variable transformation in the generating functional path integral. For the moment only the Goldstone bosons are taken into account. We obtain a non-local hadronic interaction Table 1: The chiral coefficients using a running coupling α(s). The parameter choices listed maintain fπ = 86MeV, which is the generic scale of dynamical chiral symmetry breaking. The numbers in parantheses are the " phenomenological " values.