Background field method for nonlinear σ-model in stochastic quantization

We formulate the background field method for the nonlinear σ-model in stochastic quantization. We demonstrate a one-loop calculation for a two-dimensional non-linear σ-model on a general riemannian manifold based on our formulation. The formulation is consistent with the known results in ordinary quantization. As a simple application, we also analyse the multiplicative renormalization of the O(N) nonlinear σ-model.     

Finite temperature glueball-contributions in the framework of an extended nonlinear σ-model

We use an effective Lagrangian that implements aspects of chiral and conformal symmetry to compute the finite-temperature quark-antiquark condensate. The calculation is performed by means of the path integral approach, where we will take into account finite-temperature corrections up to the two loop-level by exploiting the real time formalism (RTF). To prove the validity of the RTF within the framework of the nonlinear σ-model we will first confirm the result for the finite-temperature quark condensate without glueball contributions and then will extend the computation by including a conformally extended Lagrangian density.     

Keldysh technique and non-linear σ-model: basic principles and applications

The purpose of this review is to provide a comprehensive pedagogical introduction into Keldysh technique for interacting out-of-equilibrium fermionic and bosonic systems. The emphasis is placed on a functional integral representation of the underlying microscopic models. A large part of the review is devoted to derivation and applications of the non-linear σ-model for disordered metals and superconductors. We discuss topics such as transport properties, mesoscopic effects, counting statistics, interaction corrections, kinetic equations, etc. The section devoted to disordered superconductors includes the Usadel equation, fluctuation corrections, time-dependent Ginzburg–Landau theory, proximity and Josephson effects, etc.     

Massive inter-particle symmetry and Ward identity in fermionic string theory

We show that there exist enlarged stringy (α′→∞) symmetries for (evenG-parity) massive modes in the 10D fermionic string theory. These symmetries are derived from on-shell Ward identities corresponding to the decoupling of massive gauge states in the spectrum. In the generalized massive supersymmetric σ-model formalism, some symmetry transformations relate particles with different spins in the first order weak field approximation.     

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.     

Non-Fermi liquid theory of quantum Hall effects

Within the Grassmannian U(2N)/U(N) × U(N) nonlinear σ-model representation of localization, one can study the low-energy dynamics of both a free and interacting electron gas. We study the crossover between these two fundamentally different physical problems. We show how the topological arguments for the exact quantization of the Hall conductance are extended to include the Coulomb interaction problem. We discuss dynamical scaling and make contact with the theory of variable range hopping.     

Bosonization for disordered and chaotic systems

Using a supersymmetry formalism, we reduce exactly the problem of electron motion in an external potential to a new supermatrix model valid at all distances. All approximate nonlinear sigma models obtained previously for disordered systems can be derived from our exact model using a coarse-graining procedure. As an example, we consider a model for a smooth disorder and demonstrate that using our approach does not lead to a "mode-locking" problem. As a new application, we consider scattering on strong impurities for which the Born approximation cannot be used. Our method provides a new calculational scheme for disordered and chaotic systems.     

Two-loop β-function for the generalized bosonic sigma model

We independently compute: (i) the order α′ terms in the effective action for the massless fields in the Bose string theory and (ii) the two-loop β-function in the corresponding two-dimensional σ-model. It is proved that the “massless string fields equations of motion equal the σ-model Weyl invariance conditions” conjecture is true to the given order     

Field theory description of continuous phase transitions

We present a formalism of a scalar, classical, and time-independent field theory of the type proposed by Ferrell for the treatment of continuous phase transitions. The formalism is developed along lines similar to those of many-body theory. All physical quantities, e.g., susceptibility, correlation length, and free energy, are expressed as functionals of the two-point time-independent correlation function and the order parameter. This is done both in the ordered and in the disordered phase. We obtain renormalized equations and diagram expansions of all quantities and self-consistent approximation schemes arc presented. It is shown that near the transition temperature, which is defined within the theory, no weak coupling limit exists. The generalization to more complicated field symmetries is straight-forward.     

Anomalous gauge theories as constrained Hamiltonian systems

Anomalous gauge theories considered as constrained systems are investigated. The effects of chiral anomaly on the canonical structure are examined first for nonlinear σ-model and later for fermionic theory. The breakdown of the Gauss law constraints and the anomalous commutators among them are studied in a systematic way. An intrinsic mass term for gauge fields makes it possible to solve the Gauss law relations as second class constraints. Dirac brackets between the time components of gauge fields are shown to involve anomalous terms. Based upon the Ward-Takahashi identities for gauge symmetry, we investigate anomalous fermionic theory within the framework of path integral approach.     

Dirac quantization of the supersymmetric nonlinearσ-model on a Riemannian manifold in superspace

We canonically quantize the supersymmetric nonlinear σ-model on a Riemannian manifold using the superfield language. The resulting quantization is consistent with those of the component fields.     

Weak currents in Fermi systems

In the framework of a relativistic field theory for the many-body system we use the hypothesis of a partially-conserved axial-vector current to relate weak-interaction coupling constants to strong-interaction parameters. The density-dependent renormalizations of weak-interaction coupling constants are investigated using the Ward-Takahashi identities of the σ-model. We compare the renormalization procedures for the vector current and for the axial-vector current and generally discuss differences between renormalizations at zero density and in the medium.     

The O(5) nonlinear σ-model as a SU(2) gauge theory

We show how the O(5) nonlinear σ-model, coupled with “gravity” in a generally covariant way, can be translated into the quaternionic model HP¹, invariant under a local SU (2) gauge, and we discuss the classical solutions.     

On the instanton effect in the finite temperature Yang-Mills theory and in the non-linear σ-model

We have considered the effect of instantons at finite temperature on the interquark force and the order parameter of the Yang-Mills theory, finding a sharp change of regime at a temperature close to the one obtained in the Monte Carlo study. We have also considered the problem of the generation of the mass gap in the non linear σ-model in 2 dimensions. This illustrates the role of the instantons, the intrinsic limitations of the approximations made and suggests a physical picture as in the case of Yang-Mills theory.     

Quantizing the N=2 super sigma-model in two dimensions

A manifestly covariant background field formalism for the N=2 supersymmetric non-linear σ-model is presented. The formalism allows the symmetries of the model to be exploited to the full in the discussion of the ultraviolet divergences in the quantum theory. This proves the cohomological triviality of the metric counterterms at the l⩾2 loop orders. The formalism confirms the finiteness of models with Ricci-flat metrics through the three-loop order. However, it seems unlikely that these cancellations will persist to higher orders. This general analysis is borne out by a study of the supercurrent structure. It is shown that while there is a component axial U(1) current which obeys an Adler-Bardeen theorem, this current is not in the supercurrent multiplet and its existence cannot therefore be used to prove conformal invariance at the quantum level.     

Current commutators for the non-linear σ-model with Wess-Zumino term

We consider equal-time commutation relations of chiralSU(N) _L ×SU(N) _R charge densities in the non-linear σ-model. These commutators are derived using the cocycle formalism and from the usual canonical theory. Both methods give the same result. The charge density commutator of the symmetry currents contains operator valued Schwinger terms arising from the Wess-Zumino term.     

Field theory for the global density of states distribution function of disordered conductors

A field-theoretical representation is suggested for the electron global density of states distribution function P(nu) in extended disordered conductors. This opens a way to study the complete statistics of fluctuations. The approach is based on a functional integration over bilocal functions Psir(1)(r(2)) instead of the integration over local functions in the usual functional representation for moments of physical quantities. The formalism allows one to perform the disorder averaging and to derive an analog of the usual nonlinear sigma model-a slow functional of a supermatrix field Qr(1)(r(2))(r) approximately Psi(rr(1)) composite functionPsi (r(2)r). As an application of the formalism, the long-tail asymptotics of P(nu) is derived.     

J/gy suppression as evidence for high densities in nuclear collisions

It is shown that the scattering amplitudes of some degenerate massive string states can be expressed in terms of that of higher spin states at the same mass level. This phenomenon is discussed from σ-model point of view and demonstrated explicitly by using massive Ward identities generated by zero-norm states atD=26. The subtlety of the scalar propagating state is briefly pointed out. We also compare our on-shell gauge symmetry derived from σ-model with off-shell gauge symmetry of covariant string field theory.     

Radiative pion decay and the SU(3) × SU(3) σ-model

We claim that there is a complete disagreement between the experimental values and those predicted by the SU(3) × SU(3) σ-model for the radiative pion decay.     

On localization in the theory of the quantized hall effect: A two-dimensional realization of the θ-vacuum

It is shown that the localization problem in the theory of the quantized Hall effect is governed by the zero-component grassmannian U(2m) non-linear σ-model with θ-term, a two-dimensional analogue of the θ-vacuum in Yang-Mills theory. In this case, θ is to be interpreted as the “bare” value for the Hall conductivity, determined by an underlying non-critical theory. A detailed derivation is presented starting from the replica method and a delta function distribution for the impurities.