Perturbative QCD with Quark and Gluon Condensates

QCD perturbation theory using quark and gluon propagators that have an extra term at zero four-momentum is equivalent to standard PQCD in the presence of massless quark and gluon pairs in the perturbative vacuum. We verify at low orders in perturbation theory that 〈ψψ〉 6 = 0 and that quarks acquire a constituent mass. Thus chiral symmetry is spontaneously broken, and the conservation of the axial vector current ensures a massless pion. The gluon pairs generate an 〈FμνF μν〉 expectation value and a tachyonic gluon mass. Lorentz and gauge symmetry is preserved, and the short-distance structure of PQCD is unaffected. The modified perturbative expansion thus contains relevant elements of QCD at both short and long distances. This paper is published only in the hep-ph archive. Comments are welcome. On leave from the Department of Physics, University of Helsinki, Finland. Research supported in part by the European Commission under contract HPRN-CT-2000-00130. 1. Perturbing in the presence of parton pairs We wish to call attention to the interesting properties of a Perturbative QCD (PQCD) expansion where the free, massless quark and gluon propagators have an extra term at vanishing four-momentum. In Feynman gauge, S q (p) = δ AB [ ip/ p + iε + Cq(2π) δ(p) ] (1.1) D g (p) = −g δ [ i p + iε + Cg(2π) δ(p) ] (1.2) A corresponding modification of the ghost propagator is irrelevant since the ghost-gluon vertex is proportional to the ghost momentum. The gluon propagator modification (1.2) was proposed in [1], motivated by a Lorentz and gauge invariant formulation of QCD vacuum effects. In later work it was shown that such a modification would arise if gluon pairs are present in the asymptotic wave function [2], and some of its phenomenological consequences were studied [3]. Independently, it was noted in [4] that perturbative expansions using different iε prescriptions at the poles of the free gluon propagator are equivalent to standard expansions with gluons added to the inand out-states. This was inspired by earlier studies [5] of the relation between modifications of the on-shell propagators and boundary conditions. It should be emphasized that a perturbative expansion using propagators such as (1.1) and (1.2) is formally as justified as the standard expansion. Standard perturbation theory uses empty, perturbative vacua as inand out-states at asymptotic times (t = ±∞). In the infinite time separating the empty states from the scattering event they will relax to the true QCD ground state (assuming the overlap to be non-vanishing). At finite orders of PQCD one sees, however, only the beginning of this relaxation. Thus if the true ground state differs essentially from the perturbative vacuum, e.g., by containing quark and gluon condensates [6], the perturbative expansion will not capture this physics. This is a likely reason for the failure of standard PQCD at long distances, where soft condensate partons have important effects. It is thus natural to consider PQCD with asymptotic states that contain (free) quark and gluon pairs. Assuming again an overlap of those states with the true ground state the resulting expansion will be a priori as justified as standard PQCD. The addition of quark and gluon pairs to the asymptotic states implies no change in the QCD lagrangian nor in the short distance properties of the theory (including renormalizability). Such asymptotic states may on the other hand give rise to non-vanishing quark and gluon condensates, ‘constituent’ parton masses and a spontaneous breaking of chiral symmetry. There are phenomenological indications that perturbative methods are relevant for long distance QCD physics. Constituent quarks are effective degrees of freedom in hadrons. The transition from short distance (quark and gluon) to long-distance (hadron) physics is smooth [4, 7]. This suggests that the strong coupling freezes, αs(Q 2 = 0) ≃ 0.5, and that confinement and chiral symmetry breaking is due to quark and gluon ‘condensates’ in the QCD vacuum [6], rather than to a strong coupling regime.