A quenched momentum prescription for large- N theories

Abstract

A general quenched momentum prescription is presented for treating theories with a global, or local gauge, U( N) symmetry, both on the lattice and in the continuum, in the limit of infinite N. It is shown that the quenched theory produces, to all orders in perturbation theory, the integrands of the standard Feynman diagrams for invariant Green functions. In the case of gauge theories the quenching of the momentum must be accompanied by a constraint on the eigenvalues of the covariant derivative. In ultraviolet finite theories integration over the values of the quenched momenta reproduces ordinary perturbation theory, and it is suggested that for theories with logarithmic ultraviolet divergences a quenched momentum cutoff can be used a gauge-invariant regularization. The quenched lattice gauge theory is shown to be equivalent to the quenched Eguchi-Kawai model, and is proven to be equivalent, to all orders in perturbation theory, to the standard model. The loop equations of this model are derived, shown to be true equations of the standard model, and are argued to have a smooth continuum limit. The coupling constant renormalization is calculated in the quenched continuum gauge theory to one-loop order, and the formal loop equations are derived. Application of the method to numerical calculations of continuum QCD are discussed, as well as the possibility of achieving a gauge-invariant continuum regularization of chiral gauge and supersymmetric theories.