The Large-N Expansion

Abstract

Conventionally, one computes physical observables in power series of coupling strengths characterizing the interaction. In effective theories of strong interactions, this choice leads to unacceptably large higher-order corrections. One of the most successful propositions for an alternative expansion parameter is an expansion in the inverse powers of the number of degrees of freedom representing the dynamical variables. In connection with the chiral symmetry-breaking, the reciprocal of the number of light (nearly massless) quarks and correspondingly the reciprocal number of light (pseudo-Goldstone) mesons provide examples of such acceptably small (less than unity) parameters. In this chapter, we shall first give a technically detailed account of constructing the renormalized leading-order (LO) and next-to-leading-order (NLO) solutions of a scalar field with N components supplied with O(N)-symmetric dynamics. The chapter closes with a review of the quark–meson model, where the O(N)-symmetric meson theory is supplemented with quark degrees of freedom interacting with meson fields via Yukawa coupling.