Principal Chiral Field at Large N

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.