Abstract. An outline is given of an extended perturbative solution of Euclidean QCD which systematically accounts for a class of nonperturbative effects, while still allowing renormalization by the perturbative counterterms. Euclidean proper vertices F are approximated by a double sequence F [r'pJ, where r denotes the degree of rational approximation with respect to the spontaneous mass scale AQCD, nonanalytic in the coupling g 2, while p represents the order of perturbative corrections in {]2 calculated from F rr'~ -rather than from the perturbative Feynman rules F (~ -as a starting point. The mechanism allowing the nonperturbative terms to reproduce themselves in the Dyson-Schwinger equations preserves perturbative renormalizability and is intimately tied to the divergence structure of the theory. As a result, it restricts the self-consistency problem for the F [r'~ rigorously -i.e. without decoupling approximations -to the seven superficially divergent vertices. An interesting aspect of the solution is that rational-function sequences for the QCD propagators contain subsequences describing short-lived elementary excitations. The method is calculational, in that it allows the known techniques of loop computation to be used while dealing with integrands of truly nonperturbative content.

The dynamical generation of a pole in the self-energy of a Yang-Mills field-an extension of the Schwinger mechanism-establishes a link between the tendency of the field to form nonperturbative vacuum condensates and its "noninterpolating" property in the confining phase-the fact that it has no particles associated with it. The nonvanishing residue of such a pole-a parameter b4 of dimension (mas~)~--on the one hand provides for a nonvanishing value of ( 0 (a,A, -&A, )2 / 0 ), a contribution to the "gluon condensate." On the other hand, it implies a dominant nonperturbative form of the propagator that has no particle singularity on the real k2 axis; instead, it describes a quantized field whose elementary excitations are short lived. The dispersion law for these excitations is given and shows that they grow more particlelike (are asymptotically free) at large momenta, thus providing a qualitative description of the short-lived excitation at the origin of a gluon jet. At large k2, the nonperturbative propagator reproduces nonperturbative corrections derived from the operator-product expansion. Moreover, it is a solution to the Euclidean Dyson-Schwinger equation for the Yang-Mills field in the following sense: there exist nonperturbative three-vector vertices T 3 and auxiliary ghost-ghost-vector vertices G 3 , satisfying all symmetry and invariance requirements, and in conjunction with which this propagator solves both the Euclidean Dyson-Schwinger equation through one-dressed-loop terms and the I'3 Slavnov-Taylor identity up to perturbative corrections of order g2. The consistency conditions for this solution give b2=po2exp[ --( 4~)~/ 1 lg2] to this order, confirming the nonperturbative nature of the residue parameter, and providing a paradigm for the dynamical determination of condensates. I. THE EXTENDED SCHWINGER MECHANISMThe spontaneous generation of mass for a gauge vector boson, whether by mechanisms visible already at the tree level or by pure quantum effects,' can be represented formally as the development of a pole at lightlike fourmomentum in the polarization function-the general Schwinger mechanisms2 Schematically, where the polarization function KI is defined in the usual way through the transverse invariant D T ( k 2 ) = [-k2[1+11(k2)]j-I (1.2) in the tensor decomposition ~p~( k ) = t~~i k )~~( k~) + l p~( k ) D~( k~) , (1.3) of the momentum-space gauge-field propagatorIn general, the residue m 2, representing the squared mass of the gauge boson grown massive, will depend on the renormalized gauge coupling g and the associated renormalization-mass scale po. The essential, nonperturbative feature of the process indicated in (1.1) is that the function 1 + KI(k2), equal to unity in zeroth-order perturbation theory, develops a term with a lower power of the squared four-momentum, whereas perturbation theory, to any finite order, can at best produce logarithmic corrections, indicated by 0 (g2) in Eq. (1.1).The present paper discusses what happens when this process of spontaneous generation of lower powers of...

We continue the study, initiated in the preceding paper, of a mechanism for generating nonperturbatively modified, zeroth-order QCD vertices F (~ in the sense of a generalized perturbation expansion. In a pure gluon theory, we discuss the one-loop self-consistency conditions for the 2-gluon and 3-gluon zeroth-order vertices, using a version of the 3-point Dyson-Schwinger equation which at the one-loop level has no coupling to the 4gluon function. We find one of several solutions which fulfills the self-consistency requirement of the nonperturbative gluon propagator remarkably well, so that the latter may be taken as well-confirmed. The solution also has unphysical features which can be traced to a partial violation of BRS invariance in the specific one-loop approximation adopted. We then discuss the following general properties of the nonperturbative vertex set: (1) Smatrix elements for production of free gluons and quarks vanish.(2) Vacuum condensates exist and can be calculated in terms of the spontaneous mass scale in the F(~ (3) Propagators of the elementary fields have no physical singularities on the real axis and describe short-lived elementary excitations, whose lifetime grows with energy. (4) F(~ fulfill the Slavnov-Taylor identities among themselves. (5) Perturbative renormalization counterterms remain applicable. (6) Bound qc~ systems exist and display an unusual spectrum, with a small finite number of bound states and no q ~ scattering continuum.PACS" ll.15Tk; 12.38.Lg

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VOLUME 36, NUMBER 2 15 JULY 1987 Erratum Erratum: Propagation properties and condensate formation of the confined Yang-Mills field [Phys. Rev. D 34, 3863 (1986)l Equation (4.13) contains a misprint; the right-hand side should read