The Large-N _c Renormalization Group

Abstract

In this chapter¹, we review how effective theories of mesons and baryons become exactly soluble in the large-N _c limit. We start with a generic hadron Lagrangian constrained only by certain well-known large-N _c selection rules. The bare vertices of the theory are dressed by an infinite class of UV divergent Feynman diagrams at leading order in l/N _c . We show how all these leading-order diagrams can be summed exactly using semiclassical techniques. The saddle-point field configuration is reminiscent of the chiral bag: hedgehog pions outside a sphere of radius Λ⁻¹ (Λ being the UV cutoff of the effective theory) matched onto nucleon degrees of freedom for r ≤ Λ⁻¹. The effect of this pion cloud is to renormalize the bare nucleon mass, nucleon-Δ hyperfine mass splitting, and Yukawa couplings of the theory. The corresponding large-N _c renormalization group equations for these parameters are presented, and solved explicitly in a series of simple models. We explain under what conditions the Skyrmion emerges as a UV fixed-point of the RG flow as Λ → ∞.