CONSTRUCTION OF Y-M(4) WITH AN INFRARED CUTOFF

We provide the basis for a rigorous construction of the Schwinger func tions of the pure SU(2) Yang-Mills field theory in four dimensions (in the trivial topological sector) with a fixed infrared cutoff but no u ltraviolet cutoff, in a regularized axial gauge. The construction expl oits the positivity of the axial gauge at large field. For small field s, a different gauge, more suited to perturbative computations is used ; this gauge and the corresponding propagator depends on large backgro und fields of lower momenta. The crucial point is to control (in a non -perturbative way) the combined effect of the functional integrals ove r small field regions associated to a large background field and of th e counterterms which restore the gauge invariance broken by the cutoff . We prove that this combined effect is stabilizing if we use cutoffs of a certain type in momentum space. We check the validity of the cons truction by showing that Slavnov identities (which express infinitesim al gauge invariance) do hold non-perturbatively.