arXiv subjects

Peter Orland

Publications and source records attributed to Peter Orland.

At least 19 recordsLinked to original sources

arXiv2024

Operator Products in the SU($\infty$) Principal Chiral Model

The SU($N$) principal chiral model is asymptotically free and integrable in $1+1$ dimensions. In the large-$N$ limit, there is no scattering, but correlation functions are {\em not} those of a free field theory. We briefly review the derivation of form factors for local operators. Two-point functions for such operators are known exactly. The two-point function of scaling-field operators has the short-distance behavior expected from the renormalization group. We briefly discuss non-vacuum operator products. The ultimate goal is to derive the Lagrangian field theory from this axiomatic quantum-field-theory formalism.

hep-th↗

arXiv2021

A unitary renormalizable model of composite gravitons

A four-dimensional ${\rm SU}(4)$ confining Yang-Mills field, coupled to a fundamental fermion field and a bi-fundamental scalar field, has excitations with spin-2, but no other quantum numbers. These spin-2 excitations can be light or can condense, depending upon the scalar coupling. If condensation occurs, there is a massless spin-2 Goldstone boson with (possibly weakly) broken Lorentz invariance in the effective theory. The low-lying spectrum contains additional spin-0 and spin-1 particles. We discuss how to couple these new fields to other matter fields. To our knowledge, this is the only explicit proposal for a unitary and perturbatively-renormalizable local field theory of gravity.

hep-th↗

arXiv2020

Confinement for all couplings in a ${\mathbb Z}_{2}$ lattice gauge theory

For a particular lattice gauge theory with ${\mathbb Z}_2$ gauge invariance there is confinement for all couplings. The gauge fields, on lattice links, lie in the closed interval $[-1,1]$. It is proved that the expectation value of a gauge-invariant loop operator decays as the exponential of minus the area.

math-ph↗

arXiv2016

The universal coefficient of the exact correlator of a large-$N$ matrix 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 $Φ(x)$, was found to be $N^{-1}\langle {\rm Tr}\,Φ(0)^{\dagger} Φ(x)\rangle=C_{2}\ln^{2}mx$, 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 Lévy flight in one dimension. This observation enables us to calculate $C_{2}=1/16π$.

hep-th↗

arXiv2014

Seeing asymptotic freedom in an exact correlator of a large-$N$ matrix 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 $N^{-1}< {\rm Tr}\,Φ(0)^{\dagger} Φ(x)>$, where $Φ(x)\sim Z^{-1/2}U(x)$ 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.

hep-th↗

arXiv2014

Dynamical Mass Reduction in the Massive Yang-Mills Spectrum in 1+1 dimensions

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.

hep-th↗

arXiv2013

Subtleties of Non-Abelian Gauge Theories in Cold-Atomic Lattices

I point out two of the subtleties referred to in the title. The first is that gauge-invariant magnetic systems may realized under general circumstances, as suggested by a simple theorem. The second subtlety is that care is needed to identify the field theory simulated by a cold-atomic lattice gauge system. Though the simplest such model confines in 2+1 dimensions, it has non-relativistic ``gluon" excitations. Time-reversal invariance is spontaneously broken in this system. The confinement mechanism is related to an extra U(1) gauge invariance.There is a model, suggested long ago by D. Rohlich and me, which is known to have relativistic spin waves. One of the outstanding theoretical problems is a better determination of the energy-momentum relation of spin waves in different magnetic gauge systems.

hep-lat↗

arXiv2013

Correlation Functions of the SU(infinity) Principal Chiral Model

We obtain exact matrix elements of physical operators of the (1+1)-dimensional nonlinear sigma model of an SU(N)-valued bare field, in the 't Hooft limit N goes to infinity. Specifically, all the form factors of the Noether current and the stress-energy-momentum tensor are found with an integrable bootstrap method. These form factors are used to find vacuum expectation values of products of these operators.

hep-th↗

arXiv2012

Exact Correlators in the 't Hooft Limit of the Principal Chiral Model

The properties of (N X N)-matrix-valued-field theories, in the limit N goes to infinity, are harder to obtain than those for isovector-valued field theories. This is because we know less about the sum of planar diagrams than the sum of bubble/linear diagrams. Combining the 1/N-expansion with the axioms for form factors, exact form factors can be found for the integrable field theory of an SU(N)-valued field in 1+1 dimensions. These form factors can be used to find the vacuum expectation value of the product of two field operators. We briefly mention how the results can be applied to 2+1 dimensional gauge theories.

hep-lat↗

arXiv2012

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.

hep-th↗

arXiv2012

A reintroduction of dynamical SU(2) gauge fields in Hubbard models

This is a preprint from 1990, in which it is explained how non-Abelian gauge magnets originate as effective dynamics in models of hopping particles. In particular, an explicit model is discussed in which both link and plaquette terms appear. The motivation (as explained in a brief apology) to reintroduce the idea is some recent theoretical progress on the topic of optical lattices.

cond-mat.str-el↗

arXiv2012

Longitudinal Rescaling of Quantum Chromodynamics

We examine the effect of quantum longitudinal rescaling of coordinates, on the action of quantum chromodynamics (with quarks) to one loop. We use an aspherical Wilsonian integration (previously applied to the pure Yang-Mills theory and to quantum electrodynamics). Quantum fluctuations produce anomalous powers of the rescaling parameter in the coefficients of the rescaled action. Our results are valid for small rescalings only, because perturbation theory breaks down for large rescalings.

hep-th↗

arXiv2011

Longitudinal Rescaling of Quantum Electrodynamics

We investigate quantum longitudinal rescaling of electrodynamics, transforming coordinates as $x^{0,3}\toλx^{0,3}$ and $x^{1,2}\to x^{1,2}$, to one loop. We do this by an aspherical Wilsonian renormalization, which was applied earlier to pure Yang-Mills theory. We find the anomalous powers of $λ$ in the renormalized couplings. Our result is only valid for $λ\lesssim 1$, because perturbation theory breaks down for $λ\ll 1$.

hep-th↗

arXiv2011

Summing Planar Diagrams by an Integrable Bootstrap

Correlation functions of matrix-valued fields are not generally known for massive renormalized field theories. We find the large-N limit of form factors of the (1+1)-dimensional sigma model with SU(N) X SU(N) symmetry. These form factors give a correction to the free-field approximation for the N=infinity Wightman function. The method is a combination of the 1/N-expansion of the S-matrix and Smirnov's form-factor axioms. We expand the renormalized field in terms of a free massive Bosonic field as N goes to infinity.

hep-th↗

arXiv2010

Applying Integrability to Gauge Theories

Lattice Yang-Mills theories in any dimension may be regarded as coupled 1+1-dimensional integrable field theories. These integrable systems decouple at large center-of-mass energies, where the action becomes effectively anisotropic. This effective action is the high-energy center-of-mass limit of the gauge theory. In 2+1 dimensions, the quark-antiquark potential and the mass spectrum can be calculated, using the exact 1+1-dimensional S-matrix and form factors. The methods are quite similar to those applying integrability in statistical and condensed-matter physics. The high-energy anisotropic action at one loop in 3+1 dimensions has been found using a Wilsonian renormalization method. We briefly discuss the isotropic theory in 2+1 dimensions and the connection with soft scattering in 3+1 dimensions.

hep-lat↗

arXiv2009

Longitudinal Rescaling and High-Energy Effective Actions

Under a rescaling of longitudinal coordinates $x^{0,3}$ by a factor $λ$ which is taken to zero, the classical QCD action simplifies dramatically. This is the high-energy limit, as $λ$ is of order $s^{-1/2}$, where $s$ is the center-of-mass energy squared of a hadronic collision. We find the quantum corrections to the rescaled action at one loop, in particular finding the anomalous powers of $λ$ in this action, for $λ$ close to unity. The method is an integration over high-momentum components of the gauge field. This is a Wilsonian renormalization procedure, and counterterms are needed to make the sharp-momentum cut-off gauge invariant. Our result for the quantum action is found, assuming that the logarithm of $λ$ is small, which is essential for the validity of perturbation theory. If $λ$ is sufficiently small (so that its logarithm is large), then the perturbative renormalization group breaks down. This is due to uncontrollable fluctuations of the longitudinal chromomagnetic field.

hep-ph↗

arXiv2008

Composite Strings in (2+1)-Dimensional Anisotropic Weakly-Coupled Yang-Mills Theory

The small-scale structure of a string connecting a pair of static sources is explored for the weakly-coupled anisotropic SU(2) Yang-Mills theory in (2+1) dimensions. A crucial ingredient in the formulation of the string Hamiltonian is the phenomenon of color smearing of the string constituents. The quark-anti-quark potential is determined. We close with some discussion of the standard, fully Lorentz-invariant Yang-Mills theory.

hep-th↗

arXiv2008

Near-integrability and confinement for high-energy hadron-hadron collisions

We investigate an effective Hamiltonian for QCD at large s, in which longitudinal gauge degrees of freedom are suppressed, but not eliminated. In an axial gauge the effective field theory is a set of coupled (1+1)-dimensional principal-chiral models, which are completely integrable. The confinement problem is solvable in this context, and we find the longitudinal and transverse string tensions with techniques already used for a similar Hamiltonian in (2+1)-dimensions. We find some a posteriori justification for the effective Hamiltonian as an eikonal approximation. Hadrons in this approximation consist of partons, which are quarks and soliton-like excitations of the sigma models. Diffractive hadron-hadron scattering appears primarily due to exchange of longitudinal flux between partons.

hep-ph↗