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 $SU(2{)}_L\times SU(2{)}_R$ and $SU(3{)}_L\times SU(3{)}_R$ chiral symmetries in QCD. The method starts with a reorganization of the ordinary perturbation theory with the addition of an arbitrary quark mass $m$. The new perturbative series can be summed to all orders thanks to renormalization group properties, with specific boundary conditions, and advocated analytic continuation in $m$ properties. In the approximation where the explicit breakdown of the chiral symmetries due to small current quark masses is neglected, we derive ansatzes for the dynamical contribution to the "constituent" masses $M_q$ of the $u,d,s$ quarks; the pion decay constant $F_{\pi }$; and the quark condensate $<\bar{q}q>$ in terms of the basic QCD scale ${\Lambda }_{\overline{MS}}$. Those ansatzes are then optimized, in a sense to be specified, and also explicit symmetry breaking mass terms can be consistently introduced in the framework. The obtained values of $F_{\pi }$ and $M_q$ are roughly in agreement with what is expected from other non-perturbative methods. In contrast we obtain quite a small value of $|<\bar{q}q>|$ within our approach. The possible interpretation of the latter results is briefly discussed.

40 pages, LaTex, 2 PS figures. Additions in section 2.2 to better explain the relation between the current mass and the dynamical mass ansatz. Minor misprints corrected. Version to appear in Phys. Rev. D

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