Principal Chiral Sigma Model Research Articles

Universal coefficient of the exact correlator of a large-Nmatrix field theory

Exact expressions have been proposed for correlation functions of 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. The short-distance form of the two-point function of the scaling field $\Phi(x)$, was found to be $N^{-1}\langle {\rm Tr}\,\Phi(0)^{\dagger} \Phi(x)\rangle=C_{2}\ln^{2}mx$, where $m$ is the mass gap, in agreement with the perturbative renormalization group. Here we point out that the universal coefficient $C_{2}$, is proportional to the mean first-passage time of a L\'{e}vy flight in one dimension. This observation enables us to calculate $C_{2}=1/16\pi$.

Confinement-Higgs phase crossover as a lattice artifact in 1 + 1 dimensions

We examine the phase structure of massive Yang-Mills theory in 1+1 dimensions. This theory is equivalent to a gauged principal chiral sigma model. It has been previously shown that the gauged theory has only a confined phase, and no Higgs phase in the continuum, and at infinite volume. There are no massive gluons, but only hadron-like bound states of sigma-model particles. The reason is that the gluon mass diverges, being proportional to the two-point correlation function of the renormalized field of the sigma model at $x=0$. We use exact large-$N$ results to show that after introducing a lattice regularization and typical values of the coupling constants used in Monte Carlo simulations, the gluon mass becomes finite, and even sometimes small. A smooth crossover into a Higgs phase can then appear. For small volumes and large $N$, we find an analytic expression for the gluon mass, which depends on the coupling constants and the volume. We argue that this Higgs phase is qualitatively similar to the one observed in lattice computations at $N=2$.

Nontrivial thermodynamics in ’t Hooft’s large-Nlimit

We study the finite volume/temperature correlation functions of the (1+1)-dimensional ${\rm SU}(N)$ principal chiral sigma model in the planar limit. The exact S-matrix of the sigma model is known to simplify drastically at large $N$, and this leads to trivial thermodynamic Bethe ansatz (TBA) equations. The partition function, if derived using the TBA, can be shown to be that of free particles. We show that the correlation functions and expectation values of operators at finite volume/temperature are not those of the free theory, and that the TBA does not give enough information to calculate them. Our analysis is done using the Leclair-Mussardo formula for finite-volume correlators, and knowledge of the exact infinite-volume form factors. We present analytical results for the one-point function of the energy-momentum tensor, and the two-point function of the renormalized field operator. The results for the energy-momentum tensor can be used to define a nontrivial partition function.

Seeing asymptotic freedom in an exact correlator of a large-Nmatrix field theory

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. We express the two-point function in terms of the spectrum of the operator $\sqrt{-d^{2}/du^{2}}$, where $u\in (-1,1)$. At short distances, this expression perfectly matches the result from the perturbative renormalization group.

(2+1)-dimensional Yang-Mills theory and form factor perturbation theory

We study Yang Mills theory in $2+1$ dimensions, as an array of coupled ($1+1$)-dimensional principal chiral sigma models. This can be understood as an anisotropic limit where one of the space-time dimensions is discrete and the others are continuous. The $SU(N)\ifmmode\times\else\texttimes\fi{}SU(N)$ principal chiral sigma model in $1+1$ dimensions is integrable, asymptotically free and has massive excitations. New exact form factors and correlation functions of the sigma model have recently been found by the author and P. Orland. In this paper, we use these new results to calculate physical quantities in ($2+1$)-dimensional Yang-Mills theory, generalizing previous $SU(2)$ results by Orland, which include the string tensions and the low-lying glueball spectrum. We also present a new approach to calculate two-point correlation functions of operators using the light glueball states. The anisotropy of the theory yields different correlation functions for operators separated in the ${x}^{1}$ and ${x}^{2}$ directions.

Dynamical mass reduction in the massive Yang-Mills spectrum in1+1dimensions

The (1+1)-dimensional SU}(N) Yang-Mills Lagrangian, with bare mass M, and gauge coupling e, naively describes gluons of mass M. In fact, renormalization forces M to infinity. The system is in a confined phase, instead of a Higgs phase. The spectrum of this diverging-bare-mass theory contains particles of finite mass. There are an infinite number of physical particles, which are confined hadron-like bound states of fundamental colored excitations. These particles transform under irreducible representations of the global subgroup of the explicitly-broken gauge symmetry. The fundamental excitations are those of the SU(N) X SU(N) principal chiral sigma model, with coupling e/M. We find the masses of meson-like bound states of two elementary excitations. This is done using the exact S matrix of the sigma model. We point out that the color-singlet spectrum coincides with that of the weakly-coupled anisotropic SU(N) gauge theory in 2+1 dimensions. We also briefly comment on how the spectrum behaves in the 't Hooft limit, with N approaching infinity.

Summing planar diagrams by an integrable bootstrap. II

We continue our investigation of correlation functions of the large-N (planar) limit of the (1+1)-dimensional principal chiral sigma model, whose bare field U(x) lies in the fundamental matrix representation of SU(N). We find all the form factors of the renormalized field. An exact formula for Wightman and time-ordered correlation functions is found.

Kac–Moody and Virasoro symmetries of principal chiral sigma models

It is commonly asserted that there is a G ˆ × G centreless Kac–Moody extension of the manifest G × G global symmetry of the two-dimensional principal chiral model (PCM) for the group manifold G. Here, we show that the symmetry is in fact larger, namely G ˆ × G ˆ , the full centreless Kac–Moody extension of the entire manifest G × G global symmetry. Extending previous results in the literature, we also obtain an explicit realisation of the Virasoro-like symmetry of the PCM, generated by K n = L n + 1 − L n − 1 for both positive and negative n. We show that these generators obey Sugawara-type commutation relations with the two commuting copies of the Kac–Moody algebra G ˆ .

Glueball masses in(2+1)-dimensional anistropic weakly-coupled Yang-Mills theory

The confinement problem has been solved in the anisotropic (2+1)-dimensional SU(N) Yang-Mills theory at weak coupling. In this paper, we find the low-lying spectrum for N=2. The lightest excitations are pairs of fundamental particles of the (1+1)-dimensional SU(2)xSU(2) principal chiral sigma model bound in a linear potential, with a specified matching condition where the particles overlap. This matching condition can be determined from the exactly-known S-matrix for the sigma model.

Integrable models and confinement in(2+1)-dimensional weakly-coupled Yang-Mills theory

We generalize the $(2+1)$-dimensional Yang-Mills theory to an anisotropic form with two gauge coupling constants $e$ and ${e}^{\ensuremath{'}}$. In axial gauge, a regularized version of the Hamiltonian of this gauge theory is ${H}_{0}+{e}^{\ensuremath{'}2}{H}_{1}$, where ${H}_{0}$ is the Hamiltonian of a set of $(1+1)$-dimensional principal chiral sigma nonlinear models and ${H}_{1}$ couples charge densities of these sigma models. We treat ${H}_{1}$ as the interaction Hamiltonian. For gauge group SU(2), we use form factors of the currents of the principal chiral sigma models to compute the string tension for small ${e}^{\ensuremath{'}}$, after reviewing exact S-matrix and form-factor methods. In the anisotropic regime, the dependence of the string tension on the coupling constants is not in accord with generally-accepted dimensional arguments.