Bilocal Approximation Research Articles

Euclidean scalar field theory in the bilocal approximation

The blocking step of the renormalization group method is usually carried out by restricting it to fluctuations and to local blocked action. The tree-level, bi-local saddle point contribution to the blocking, defined by the infinitesimal decrease of the sharp cutoff in momentum space, is followed within the three dimensional Euclidean $\phi^6$ model in this work. The phase structure is changed, new phases and relevant operators are found and certain universality classes are restricted by the bi-local saddle point.

Confinement in the Abelian–Higgs-Type Theories: String Picture and Field Correlators

Field correlators and the string representation are used as two complementary approaches for the description of confinement in the SU(N)-inspired dual Abelian-Higgs-type model. In the London limit of the simplest, SU(2)-inspired, model, bilocal electric field-strength correlators have been derived accounting for the contributions to these averages produced by closed dual strings. The Debye screening in the plasma of such strings yields a novel long-range interaction between points lying on the contour of the Wilson loop. This interaction generates a Lüscher-type term, even when one restricts oneself to the minimal surface, as is usually done in the bilocal approximation to the stochastic vacuum model. Beyond the London limit, it has been shown that a modified interaction appears, which becomes reduced to the standard Yukawa one in the London limit. Finally, a string representation of the SU(N)-inspired model with the Θ term, in the London limit, can be constructed.

Instantons in the nonperturbative QCD vacuum

The influence of nonperturbative fields on instantons in quantum chromodynamics is studied. Nonperturbative vacuum is described in terms of nonlocal gauge invariant vacuum averages of gluon field strength.Effective action for instanton is derived in bilocal approximation and it is demonstrated that stochastic background gluon fields are responsible for infra-red (IR) stabilization of instantons. Dependence of characteristic instanton size on gluon condensate and correlation length in nonperturbative vacuum is found. Comparison of obtained instanton size distribution with lattice data is made.

EFFECTIVE PERMITTIVITY OF A STATISTICALLY INHOMOGENEOUS MEDIUM WITH STRONG PERMITTIVITY FLUCTUATIONS - Abstract

Most previous multiple-scattering theories for electromagnetic waves in strongly fluctuating media are limited by the assumption of statistical homogeneity of media. In the paper, a lossy electrically isotropic random medium is considered whose mean permittivity distribution, as well as the multipoint permittivity's moments are invariant under arbitrary rotations about and translations along a fixed symmetry axis, and are inhomogeneous in the radial direction. The goal of the paper is to calculate the effective permittivity operator (EPO) for such medium in the case of strong permittivity fluctuations. For this purpose, one has to eliminate the secular terms from the spectral representation of the-EPO in the basis set of waves suited to a statistically inhomogeneous medium. This is achieved via a renormalization approach which takes into proper account a delta function singularity of the spectral Green's function (rather than that of the spatial Green's function accounted for in the past) referring to a spatially inhomogeneous electrically anisotropic background medium. On this basis, the permittivity matrix of the background medium is explicitly found, a full perturbation series solution and a bilocal approximation for the EPO are derived, the macroscopic properties of the spatially dispersive effective medium are studied, and a perturbative solution for the propagation constants of guided modes of the mean field is obtained.

Effective Permittivity of a Statistically Inhomogeneous Medium with Strong Permittivity Fluctuations

Most previous multiple-scattering theories for electromagnetic waves in strongly fluctuating media are limited by the assumption of statistical homogeneity of media. In the paper, a lossy electrically isotropic random medium is considered whose mean permittivity distribution, as well as the multipoint permittivity's moments are invariant under arbitrary rotations about and translations along a fixed symmetry axis, and are inhomogeneous in the radial direction. The goal of the paper is to calculate the effective permittivity operator (EPO) for such medium in the case of strong permittivity fluctuations. For this purpose, one has to eliminate the secular terms from the spectral representation of the-EPO in the basis set of waves suited to a statistically inhomogeneous medium. This is achieved via a renormalization approach which takes into proper account a delta function singularity of the spectral Green's function (rather than that of the spatial Green's function accounted for in the past) referring to a spatially inhomogeneous electrically anisotropic background medium. On this basis, the permittivity matrix of the background medium is explicitly found, a full perturbation series solution and a bilocal approximation for the EPO are derived, the macroscopic properties of the spatially dispersive effective medium are studied, and a perturbative solution for the propagation constants of guided modes of the mean field is obtained.

Strong permittivity fluctuation theory of random media: Part I: Nonspherical scatterers

The strong permittivity fluctuation theory is applied to investigate the electromagnetic wave propagation in inhomogeneous media embedded with randomly positioned and randomly oriented nonspherical scatterers. Both bilocal approximation and low-frequency approximation are used to derive the random medium effective permittivities for the full space case. An isotropic model of random media is obtained due to the chosen geometry of scatterers as well as due to the chosen equal correlation lengths in two space directions. Using the given stochastic model, a first moment of random electromagnetic field can be directly computed. For the purposes of effective computation of permittivities, a FORTRAN 77 program is developed and approved. The developed random medium model was applied with success in sea ice investigation of its dielectric properties.

Instanton IR stabilization in the nonperturbative confining vacuum

The influence of nonperturbative fields on instantons in quantum chromodynamics is studied. Nonperturbative vacuum is described in terms of nonlocal gauge invariant vacuum averages of gluon field strength. Effective action for instanton is derived in bilocal approximation and it is demonstrated that stochastic background gluon fields are responsible for infra-red (IR) stabilization of instantons. Dependence of characteristic instanton size on gluon condensate and correlation length in nonperturbative vacuum is found. It is shown that instanton size in QCD is of order of 0.25 fm. Comparison of obtained instanton size distribution with lattice data is made.

Third-order implementation and convergence of the strong-property-fluctuation theory in electromagnetic homogenization.

The strong-property-fluctuation theory (SPFT) has been widely applied under the second-order approximation (also known as the bilocal approximation) to estimate the constitutive properties of effectively homogeneous composite mediums. A third-order mass operator approximation is developed here. The convergence of the long-wavelength, bilocally-approximated SPFT is demonstrated for isotropic chiral composite mediums, as well as for chiroferrite composite mediums.

Strong-property-fluctuation theory for homogenization of bianisotropic composites: formulation

The strong-property-fluctuation theory is developed for the homogenization of the linear dielectric, magnetic, and magnetoelectric properties of a two-constituent bianisotropic composite. The notion of a bianisotropic comparison medium (BCM) is introduced to serve as a springboard for the Dyson equation satisfied by the ensemble-averaged electromagnetic field. With the constitutive properties of the BCM serving as the zeroth-order solution of the Dyson equation, the first-order correction, known as the bilocal approximation, is obtained. Wave propagation in the composite can be described in this manner by a nonlocal effective medium containing information about the spatial correlations of the constitutive properties. For scales larger than the correlation length, the nonlocality vanishes and a local effective medium emerges. Analytical results for the local effective constitutive properties are presented after assuming a spherical particulate topology for the constituent mediums. Illustrative numerical results are provided.

Validity of the small perturbation method for volume scattering from marine sediments

For high-frequency volume scattering and propagation in marine sediments (at 10–500 kHz), the validity of single scattering methods (e.g., Born approximation) is not well established. Insight into whether single or multiple scattering is significant in a medium can be gained using effective medium theory (bilocal approximation to Dyson’s equation). An alternative analysis can be made using exact numerical Monte Carlo simulations. In this talk, both multiple scattering theory and numerical methods will be used to investigate the applicability of the small perturbation method for volume scattering from a sediment half-space. Parameters of the sediment are limited to marine sand and mud modeled as an acoustic fluid with Gaussian random fluctuations in density and compressibility. Sediment properties (from core sampling) are taken from published data on four experiment sites: Orcas Island; Eckernfoerde Bay; STRESS Midshelf; and Panama City. It is shown that strong scattering may be significant at the Panama City site (assuming the core estimates are accurate). However, even when strong scattering is predicted, the presence of a ‘‘transition layer’’ near the sediment–water interface reduces the effect of multiple scattering on the far-field scattering strength, and the first-order small perturbation approximation may still be applicable.

The covariance function, bilocal approximation and homogenization of chiral-in-chiral composite materials

In continuation of an earlier paper on the application of the bilocal approximation (BA) within the framework of the strong property fluctuation theory (SPFT), an important restriction on the covariance function is lifted and the consequences numerically examined.

Effective permittivity of a discrete random medium with anisotropic inclusions

We use the method of substitution of field variables in the bilocal approximation to find the effective permittivity of a two-phase composite random medium in the form of an ensemble of small, arbitrarily anisotropic spherical inclusions distributed inside an isotropic matrix. To illustrate the results we calculate the damping of the plane waves of the mean field in such a medium.

Gluodynamics string as a low-energy limit of the universal confining string theory

An effective string theory emerging from the bilocal approximation to the Method of Vacuum Correlators in gluodynamics is shown to be well described by the 4D theory of the massive Abelian Kalb-Ramond field interacting with the string, which is known to be the low-energy limit of the Universal Confining String Theory. The mass of the Kalb-Ramond field in this approach plays the role of the inverse correlation length of the vacuum, and it is shown that in the massless limit string picture disappears. The background field method, known in the theory of nonlinear sigma models, is applied to derivation of the effective action, quadratic in quantum fluctuations around a given (e.g. minimal) string world-sheet.

NONPERTURBATIVE FIELD CORRELATOR AND STRING REPRESENTATION FOR THE 't HOOFT LOOP AVERAGE IN THE ABELIAN HIGGS MODEL

Making use of the duality transformation, we derive in the Londons' limit of the Abelian Higgs model string representation for the 't Hooft loop average defined on the string worldsheet, which yields the values of two coefficient functions parametrizing the bilocal correlator of the dual field strength tensors. The asymptotic behaviors of these functions agree with the ones obtained within the method of vacuum correlators in QCD in the lowest order of perturbation theory. We demonstrate that the bilocal approximation to the method of vacuum correlators is an exact result in the Londons' limit, i.e. all the higher cumulants in this limit vanish. We also show that at large distances, apart from the integration over metrics, the obtained string effective theory (which in this case reduces to the nonlinear massive axionic sigma model) coincides with the low energy limit of the dual version of 4D compact QED, the so-called universal confining string theory. We derive string tension of the Nambu–Goto term and the coupling constant of the rigidity term for the obtained string effective theory and demonstrate that the latter is always negative, which means the stability of strings, while the positiveness of the former is confirmed by the present lattice data. These data enable us to find the Higgs boson charge and the vacuum expectation value of the Higgs field, which well-described QCD. We also study dynamics of the weight factor of the obtained string representation for the 't Hooft average in the loop space. In conclusion, we obtain string representation for the partition function of the correlators of an arbitrary number of Higgs currents, by virtue of which we rederive the structure of the bilocal correlator of the dual field strength tensors, which yields the surface term in the string effective action.

FLUCTUATING STRINGS IN THE UNIVERSAL CONFINING STRING THEORY AND GLUODYNAMICS

An effective string theory emerging from the bilocal approximation to the method of vacuum correlators in gluodynamics is shown to be well-described by the 4D theory of the massive Abelian Kalb–Ramond field interacting with the string, which is known to be the low energy limit of the universal confining string theory. This correspondence follows from the agreement of the behavior of the coefficient functions, which parametrize the gauge-invariant correlator of two gluonic field strength tensors, known from the lattice data, with their values obtained from the propagator of the Kalb–Ramond field. We discuss this correspondence in several aspects and demonstrate that the mass of the Kalb–Ramond field in this approach plays the role of the inverse correlation length of the vacuum, so that in the massless limit string picture disappears. Next, we apply the background field method, known in the theory of nonlinear sigma models, to obtain the action, which is quadratic in quantum fluctuations around a given (e.g. minimal) string worldsheet. Several nontrivial types of couplings of these fluctuations with the background worldsheet are obtained and discussed.

The random analogue of Epstein's transition layer

For a quasi-homogeneous, random medium with variance, varying along the same direction as tanh, the mean Green function is obtained as an exact solution of Dyson's equation in a bilocal approximation. The coherent part of the field of a plane wave, falling on a bounded, randomly fluctuating medium with a non-sharp boundary, is studied in detail. In the case of small-scale fluctuations, a medium of this kind is shown to be a random analogue of a transient Epstein layer.

NEW EQUATIONS FOR VACUUM CORRELATORS IN FIELD THEORY

Stochastic quantization is used to derive exact equations connecting an infinite set of multilocal field correlators in the φ3 theory and gluodynamics. In the latter case the obtained equations are explicitly gauge-invariant. In the bilocal approximation, corresponding to the Gaussian stochastic ensemble, a minimal finite system of equations is obtained and investigated in the lowest orders of perturbation theory. A gauge-invariant diagrammatic technique in gluodynamics is developed.

Extinction Properties of Dust Grains: A New Computational Technique

view Abstract Citations (23) References (19) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Extinction Properties of Dust Grains: A New Computational Technique Michel, B. ; Henning, Th. ; Stognienko, R. ; Rouleau, F. Abstract The Strong Permittivity Fluctuation Theory (SPFT) is a statistical approach to describe the optical properties of randomly oriented inhomogeneous dust grains. In contrast to most other methods that must calculate the extinction properties of individual particles and then sum the results to obtain ensemble averages, the SPFT starts with ensemble-averaged quantities from the very beginning. Therefore, the computation of the extinction has to be performed only once for a given ensemble. This makes this method especially suitable for astrophysical applications in which grain alignment is not required. We use the SPFT in bilocal approximation (BA) to calculate the orientationally and positionally averaged extinction cross section of dust agglomerates. In the BA, the average extinction cross section of a statistical ensemble of grains depends only on the complex refractive index of the constituent particles, and on the value and spatial correlation of the local density (filling factor) of the material. The BA is tested for the case of agglomerates of spherical grains by comparing its predictions to those of the discrete dipole approximation (DDA) and the generalized Mie theory (GMT), which embodies an exact solution of Maxwell's equations. We find that the BA works well for all size parameters tested (up to 0.6, taking the radius of the individual constituent spheres). We find a very good agreement between the BA, DDA, and GMT for a moderate refractive index (nref = 1.7 + 0.7i, typical of soot or "dirty" silicate cosmic dust analogs in the visual). For a somewhat larger refractive index (nref = 2.5 + 1.5i, typical of graphite in the UV), the BA is of comparable accuracy as the DDA. For the more extreme case of a quasimetallic refractive index (nref = 3.0 + 4.0i), the BA seems to be an inadequate approach. For this case, all methods are pushed to their limit. Publication: The Astrophysical Journal Pub Date: September 1996 DOI: 10.1086/177739 Bibcode: 1996ApJ...468..834M Keywords: ISM: DUST; EXTINCTION; METHODS: ANALYTICAL; METHODS: NUMERICAL full text sources ADS |

Acoustic wave scattering from a randomly rough surface

Acoustic wave scattering from a randomly rough surface is studied using a multiple scattering analysis. Assuming the surface asperities to be small, approximate boundary conditions are used to derive a pair of coupled integral equations for the velocity potentials. The Dyson equation is next derived on introducing symbolic operators. Using the bilocal approximation the Dyson equation is solved, thereby obtaining explicit expressions for the coherent reflection and transmission coefficients. The analysis is similarly carried on further to compute the incoherent intensity by using the ladder approximation, and hence an expression for the scattering coefficient is obtained.

On the application of the strong property fluctuation theory for homogenizing chiral-in-chiral composites

The bilocal approximation is used along with the strong property fluctuation theory (SPFT) to homogenize (i.e. estimate the effective constitutive properties of) a chiral-in-chiral composite. Important properties of the covariance function of the spatial distribution of the two materials are deduced and interpreted with respect to the homogenization results. Comparison is made with the Maxwell Garnett and the Bruggeman approaches. The correlation length is an additional parameter that distinguishes the SPFT in the bilocal approximation from the Bruggeman approach (as well as from the Maxwell Garnett approach). The SPFT in the bilocal approximation turns out to be a size-dependent extension of the Bruggeman approach, the two homogenization procedures yielding almost the same results at very long wavelengths but not at shorter wavelengths.