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

Abstract

Exact expressions for correlation functions are known for the large-$N$ (planar) limit of the $(1+1)$-dimensional $\mathrm{SU}(N)\times \mathrm{SU}(N)$ 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 $N^{-1}<\mathrm{Tr}\Phi (0{)}^{†}\Phi (x)>$, where $\Phi (x)\sim Z^{-1/2}U(x)$ is the scaling field and $U(x)\in SU(N)$ is the bare field. We express the two-point function in terms of the spectrum of the operator $\sqrt{-d^2/d{u}^2}$, where $u\in (-1,1)$. At short distances, this expression perfectly matches the result from the perturbative renormalization group.

arXiv commentSmall corrections in text and references. Published in PRD. Still 4 pages, revtex

References in corpus (14)

• Fractional Laplacian in Bounded Domains
• Resurgence in QFT: Unitons, Fractons and Renormalons in the Principal Chiral Model
• Spectral properties of the Cauchy process
• Principal Chiral Field at Large N
• Exact form factors of the O(N) $\sigma$-model
• Two dimensional SU(N)xSU(N) Chiral Models on the Lattice (II): the Green's Function
• Multiparticle Form Factors of the Principal Chiral Model At Large N
• Summing Planar Diagrams by an Integrable Bootstrap II
• Summing Planar Diagrams by an Integrable Bootstrap
• The Spectrum of the Fractional Laplacian and First Passage Time Statistics
• Correlation Functions of the SU(infinity) Principal Chiral Model
• A large N phase transition in the continuum two dimensional SU(N) X SU(N) principal chiral model
• (2+1)-Dimensional Yang-Mills Theory and Form Factor Perturbation Theory
• The spectrum of large powers of the Laplacian in bounded domains

Cited by in corpus (5)

• Lectures on S-matrices and Integrability
• Generating a non-perturbative mass gap using Feynman diagrams in an asymptotically free theory
• Nontrivial Thermodynamics in 't Hooft's Large-$N$ Limit
• The universal coefficient of the exact correlator of a large-$N$ matrix field theory
• A diagrammatic approach towards the thermodynamics of integrable systems