Seeing asymptotic freedom in an exact correlator of a large-$N$ matrix field theory

Understand

Exact expressions for correlation functions are known for the large-$N$ (planar) limit of the $(1+1)$-dimensional ${\rm SU}(N)\times {\rm SU}(N)$ principal chiral sigma model.

• These were obtained with the form-factor bootstrap, an entirely nonperturbative method.
• The large-$N$ solution of this asymptotically-free model is far less trivial than that of O($N$) sigma model (or other isovector models).
• Here we study the Euclidean two-point correlation function $N^{-1}< {\rm Tr}\,\Phi(0)^{\dagger} \Phi(x)>$, where $\Phi(x)\sim Z^{-1/2}U(x)$ is the scaling field and $U(x)\in SU(N)$ is the bare field.