The background field method and the non-linear σ-model

We prove the renormalizability in the background field method of the two-dimensional bosonic non-linear σ-model with an arbitrary riemannian manifold as target space. Particular attention is paid to the question of non-linear renormalization of the quantum fields and its effect on subdivergences. We give an algorithm that allows one to compute to arbitrary loop order.     

A one-loop finite contorted non-linear σ-model without Wess-Zumino term

An explicit breaking interaction term of (1, 0) supersymmetry, governed by a dimensionless coupling parameter, is added to the σ-model action in such a way as to describe the torsion to which the fermions couple. Using (1, 0) supergraph methods, it is shown that, under certain assumptions on the torsion-like explicit breaking term, one-loop finiteness is achieved without spoiling the Ricci-flatness of the target manifold.     

Nonlinear σ-model and nucleon structure

The nonlinear σ-model with the Wess-Zumino action describes the nucleon as a soliton and incorporates the non-abelian chiral anomalies. Several studies have shown that the model works well except for the nucleon mass, which comes out consistently too large. We investigate this question beginning with the more general framework of the linear σ-model, which has besides a pseudoscalar meson sector, a fermion or quark sector, a scalar field and an interaction between the fermions via the scalar field. Using a path integral formulation, we express the fermion measure of the model as the product of a Jacobian and an invariant measure. Identifying this Jacobian as exp[iΓ _wz] , we find that the model breaks up into two parts, when in the pseudoscalar meson sector the scalar field is replaced by its vacuum value. The pseudoscalar part of the model becomes the nonlinear σ-model with the Wess-Zumino actionΓ _wz. The other part involves chiral fermions, the scalar field and their interaction. We continue this part back to the Minkowski space to determine its ground state and energy levels. We find that for a scalar field that vanishes at smallr, but rises sharply to its vacuum value at someR, the ground state energy of the interacting quark-scalar-field system can be lower than the ground state energy of the non-interacting quark system. This means the interaction between quarks and the scalar field can lead to a condensed ground state or vacuum and can reduce the overall energy of the system (a phase transition as in superconductivity). It is, therfore, not surprising that the nonlinear σ-model predicts too large a nucleon mass, since it implicitly assumes a normal non-interacting vacuum in the quark sector. Quarks are now quasiparticles that appear as excitations of the condensed vacuum. The nucleon structure that emerges from this investigation agrees fully with the phenomenological nucleon structure found from analysis of high energy elasticpp and \(\bar p\) p scattering at CERN ISR and SPS Collider.     

Universality of the Bethe-ansatz solution for the chiral non-linear σ-model

A euclidean lattice action is constructed to test the idea of universality of the Bethe-ansatz solution for the two-dimensional principal chiral model. The response of the system to uniform external fields is studied using the Monte Carlo simulation. A transition to the gapless phase is observed at the critical external field predicted by the Bethe-ansatz method.     

Continuum limit of the hierarchicalO(N) non-linear σ-model

We construct rigorously the continuum limit of theO(N) non-linear model in two euclidean dimensions with a hierarchical kinetic term. Asymptotic freedom and weak coupling Wilson renormalization group flow are established.     

A classical solution of the non-linear complex Grassmann σ-model with higher derivatives

We construct a soliton solution of the non-linear complex Grassmann-model with higher derivatives, and show that this solution, as a continuous map, represents a generator of theK-group of a sphere.     

Bosonization and the O(3) σ-model

The rules of vector bosonization with respect to the O(4) σ-model with a Wess—Zumino term, and the O(3) σ-model with an instanton term are derived. The critical points at which the theories become conformally invariant are found.     

String theory effective actions and σ-model β-functions

String theory effective actions compatible with two recent computations of the two-loop β-functions in σ-models (including torsion) are constructed. The equivalence of these actions is explicitly demonstrated by means of field redefinitions.     

Geometrical structure and ultraviolet finiteness in the supersymmetric σ-model

A complete geometrical classification of supersymmetric -models is given. Extended supersymmetry requires covariantly constant complex structures, and Kahler and hyperkahler manifolds play a special role. As an application of the classification, it is shown that a particular class of these models is on-shell ultraviolet finite to all orders in perturbation theory.     

The two-dimensionalO(N) nonlinear σ-model: Renormalisation and effective actions

We establish the existence of the Wilson Renormalised trajectory of theO(N) sigma model in perturbation theory in the effective charge. This yields a proof of perturbative renormalisability, and is also relevant in the small-field analysis of the rigorous renormalisation group construction of the continuum theory.Laboratoire associé au CNRS UA 280On leave from the Tata Institute of Fundamental Research, Bombay, 400 005, IndiaResearch funded in part by NSF grant DMS 8601978     

Pion,sigma and nucleon propagators in the linear σ-model

We study the pion, sigma and nucleon propagators of the linear -model in the one-loop approximation. We show that in the renormalized model the meson propagators exhibit imaginary poles. Such nonphysical poles disappear when a cutoff is introduced. The maximum of this cutoff value is shown to be of the order _c1.2 Gev. Once this value is fixed it is possible to find a set of parameters which allow for a selfconsistent interpretation of physical properties associated with the propagators. A 400 MeV wide resonance appears in the sigma propagator, the nucleon is a bound state and in its continuum two Roper resonances can be identified.Work supported in part by CNPq and FAPEMIG (Brasil) and CERN project no. PCERN/C/FAE/74/91     

Effective action and vacuum instability in the chiral σ-model and the standard electroweak model

We develop a partial expansion of the effective action from fermion integration relevant for spatially small configurations of the meson fields in the chiral σ-model. It is shown that this leads (in this semiclassical approximation) to a vacuum instability. For weak Yukawa coupling, when the semiclassical calculation is valid, this would indicate an instability of the Weinberg-Salam model, implying that the latter is only an effective theory. We give an estimate of the approximate length scale at which this occurs.     

The equivalence theorem for the heavy-Higgs Standard Model and the gauged nonlinear σ-model

The equivalence theorem states that the leading part of the amplitude for a process with external longitudinally polarized vector bosons is given by the amplitude in which the longitudinal vector bosons are replaced by the corresponding pseudo-Goldstone bosons. The validity of this theorem within the standard model with a heavy Higgs boson and within the gauged nonlinear -model (in which the Higgs boson is absent) is shown. Furthermore it is examined to what extent also internal lines other than scalar lines can be neglected. A simple power-counting method is developed which determines the leading diagrams for a given process at an arbitrary loop order. This method is also applied to effective Lagrangians with additional nonstandard interaction terms of higher dimension (chiral Lagrangians).     

Renormalizability and infrared finiteness of non-linear σ-models: A regularization-independent analysis for compact coset spaces

The non-linear models in two space-time dimensions corresponding to compact homogeneous coset spacesG/H are studied with particular attention to three problems: first, independence of coordinate choice and regularization, second, the physical content of the theory, and finally the regularity of the physics in the infrared limit. Concerning in particular the physical content of the theory, we construct a set of local observables whose correlation functions depend on a finite number of parameters identified among those defining the metric tensor of the coset space. For these models, we give a general proof of renormalizability based on the introduction of a nilpotent BRS operator which describes the non-linear isometries and a classical action which contains a mass term for all quantized fields. The mass term belongs to a finite dimensional representation of the groupG, which allows us to prove the conjecture that the correlation functions of local observables, i.e., the local operators invariant underG, are regular in the infrared limit.     

Diffractive-optical processing of temporal signals, part I: basic principles

The topic of this paper is the utilization of time for optical information processing. As clock rates in computing and communication systems increase and reach the THz border, optical techniques for signal filtering, shaping and clock distribution become attractive. We discuss the use of optics in temporal processing and consider in particular diffractive solutions. In part one of this paper, we discuss the basic concepts of temporal optics.     

Analytical results for the four-loop RG functions in the 2D non-linear O(n) σ-model on the lattice

We recalculate four-loop renormalization group functions in 2-dimensional nonlinear O(n) σ-model using coordinate-space method. The high accuracy of calculation allow us to find the analytical form of β- and γ-function (anomaluos dimension).     

Locally supersymmetric σ-model with Wess-Zumino term in two dimensions and critical dimensions for strings

We construct an N = 1 locally supersymmetric σ-model with a Wess-Zumino term coupled to supergravity in two dimensions. If one takes the σ-model manifold to be the product of d-dimensional Minkowski space M^d and a group manifold G, and if the radius of G is quantized in appropriate units of the string tension, then the model describes a Neveu-Schwarz-Ramond (NSR)-type string moving on M_d × G. (Our model generalizes earlier work of refs. [1,2] which do not contain a Wess-Zumino term and that of refs. [5,6] which is not locally supersymmetric.) The zweibein and the gravitino field equations yield constraints which generalize those of the NSR model to the case of a non-abelian group manifold. In particular, the fermionic constraint contains a new term trilinear in the fermionic fields. We quantize the theory in the light-cone gauge and derive the critical dimensions. We compute the mass spectrum of a closed string moving on M_d × G and show that massless fermions do not arise for non-abelian G for the spinning string, in agreement with the result of Friedan and Shenker [22].     

Critical behaviour of SU(n) quantum chains and topological non-linear σ-models

The critical behaviour of SU(n) quantum “spin” chains, Wess-Zumino-Witten σ-models and grassmanian σ-models at topological angle θ = π (of possible relevance to the quantum Hall effect) is reexamined. It is argued that an additional Z_n symmetry is generally necessary to stabilize the massless phase. This symmetry is not present for the σ-models for n > 2 and is only present for certain representations of “spin” chains.