Variational quark mass expansion and the order parameters of chiral symmetry breaking

Abstract

We investigate in some detail a “variational mass” expansion approach, generalized from a similar construction developed in the Gross-Neveu model, to evaluate the basic order parameters of the dynamical breaking of the (Formula presented) and (Formula presented) chiral symmetries in QCD. The method starts with a reorganization of the ordinary perturbation theory with the addition of an arbitrary quark mass (Formula presented). The new perturbative series can be summed to all orders thanks to renormalization group properties, with specific boundary conditions, and advocated analytic continuation in (Formula presented) properties. In the approximation where the explicit breakdown of the chiral symmetries due to small current quark masses is neglected, we derive Ansätze for the dynamical contribution to the “constituent” masses (Formula presented) of the (Formula presented) quarks, the pion decay constant (Formula presented) and the quark condensate (Formula presented) in terms of the basic QCD scale (Formula presented) Those Ansätze are then optimized, in a sense to be specified, and also explicit symmetry breaking mass terms can be consistently introduced in the framework. The values of (Formula presented) and (Formula presented) obtained are roughly in agreement with what is expected from other nonperturbative methods. In contrast we obtain quite a small value of (Formula presented) within our approach. The possible interpretation of the latter results is briefly discussed. © 1998 The American Physical Society.