Principal Chiral Field at Large N

TL;DR

The paper provides an exact large-N solution for the two-dimensional principal chiral field with external Noether charges, yielding a closed-form ground-state energy density in terms of a single parameter B and revealing an emergent extra dimension via a semicircular rapidity distribution. It establishes asymptotic freedom through an exact running coupling and beta-function, and derives a perturbative expansion with factorially growing coefficients due to renormalons, indicating non-Borel-summability. At the threshold h→m, the energy exhibits an inverse-logarithmic singularity, signaling extended object excitations in the spectrum as N→∞. The work connects exact large-N results to perturbative QCD-like structure, discusses implications for string interpretations, and identifies finite-N corrections and potential master-field formulations as future directions.

Abstract

We present the exact and explicit solution of the principal chiral field in two dimensions for an infinitely large rank group manifold. The energy of the ground state is explicitly found for the external Noether's fields of an arbitrary magnitude. The exact Gell-Mann - Low function exhibits the asymptotic freedom behaviour at large value of the field in agreement with perturbative calculations. Coefficients of the perturbative expansion in the renormalized charge are calculated. They grow factorially with the order showing the presence of renormalons. At small field we found an inverse logarithmic singularity in the ground state energy at the mass gap which indicates that at $N=\infty$ the spectrum of the theory contains extended objects rather then pointlike particles.