Propagation properties and condensate formation of the confined Yang-Mills field

Abstract

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 ${b}^{4}$ of dimension (mass${)}^{4}$---on the one hand provides for a nonvanishing value of 〈0\ensuremath{\Vert}(${\ensuremath{\partial}}_{\ensuremath{\mu}}$${A}_{\ensuremath{\nu}}$-${\ensuremath{\partial}}_{\ensuremath{\nu}}$${A}_{\ensuremath{\mu}}$${)}^{2}$ \ensuremath{\Vert}0〉, a contribution to the ``gluon condensate.'' On the other hand, it implies a dominant nonperturbative form of the propagator that has no particle singularity on the real ${k}^{2}$ axis; instead, it describes a quantized field whose elementary excitations are short lived. The dispersion law for these excitations is given and shows that they grow more particlelike (are asymptotically free) at large momenta, thus providing a qualitative description of the short-lived excitation at the origin of a gluon jet. At large ${k}^{2}$, the nonperturbative propagator reproduces nonperturbative corrections derived from the operator-product expansion. Moreover, it is a solution to the Euclidean Dyson-Schwinger equation for the Yang-Mills field in the following sense: there exist nonperturbative three-vector vertices ${\ensuremath{\Gamma}}_{3}$ and auxiliary ghost-ghost-vector vertices ${G}_{3}$, satisfying all symmetry and invariance requirements, and in conjunction with which this propagator solves both the Euclidean Dyson-Schwinger equation through one-dressed-loop terms and the ${\ensuremath{\Gamma}}_{3}$ Slavnov-Taylor identity up to perturbative corrections of order ${g}^{2}$. The consistency conditions for this solution give ${b}^{2}$=${\ensuremath{\mu}}_{0}$${}^{2}$exp[-(4\ensuremath{\pi}${)}^{2}$ /11${g}^{2}$] to this order, confirming the nonperturbative nature of the residue parameter, and providing a paradigm for the dynamical determination of condensates.