The non-linear σ-model near two dimensions: Phase structure

2021 ◽  
pp. 458-488
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Phase Structure ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Two Dimensions ◽  
Leading Order ◽  
Non Linear ◽  
Two Parameters ◽  
Infrared Divergences ◽  
Goldstone Modes

This chapter is devoted to the study of the non-linear σ-model, a quantum field theory (QFT) where the (scalar) field is an N-component vector of fixed length, mostly in dimensions close to 2. The model possesses a global, non-linearly realized symmetry, O(N) symmetry: under a group transformation, the transformed field is a non-linear function of the field itself. The non-linear σ-model belongs to a class of models constructed on special homogeneous spaces, symmetric spaces that, as Riemannian manifolds, admit a unique metric. Unlike what happens in a (ϕ2)2 -like field theory with the same symmetry, in the non-linear σ-model, in the tree approximation, the O(N) symmetry is always spontaneously broken: the action describes the interactions of (N−1) massless fields, the Goldstone modes. Since the fields are massless, in two dimensions infrared divergences appear in the perturbative expansion and an infrared regulator is required. To understand the phase structure beyond leading order, a renormalization group (RG) analysis is necessary. This requires understanding how the model renormalizes. Power counting shows that the model is renormalizable in two dimensions. Since the field then is dimensionless, although the degree of divergence of Feynman diagrams is bounded, an infinite number of counterterms is generated, because all correlation functions are divergent. A quadratic master equation satisfied by the generating functional of vertex functions is derived, which makes it possible to prove that the coefficients of all counterterms are related, and that the renormalized theory depends only on two parameters.

Generalized non-linear σ-models in two dimensions

2021 ◽  
pp. 692-720
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Two Dimensions ◽  
Asymptotic Freedom ◽  
Quantum Field Theories ◽  
Field Equations ◽  
Local Conservation ◽  
Non Linear ◽  
Coset Spaces ◽  
Formal Properties

This chapter describes the formal properties, and discusses the renormalization, of quantum field theories (QFT) based on homogeneous spaces: coset spaces of the form G/H, where G is a compact Lie group and H a Lie subgroup. In physics, they appear naturally in the case of spontaneous symmetry breaking, and describe the interaction between Goldstone modes. Homogeneous spaces are associated with non-linear realizations of group representations. There exist natural ways to embed these manifolds in flat Euclidean spaces, spaces in which the symmetry group acts linearly. As in the example of the non-linear σ-model, this embedding is first used, because the renormalization properties are simpler, and the physical interpretation of the more direct correlation functions. Then, in a generic parametrization, the renormalization problem is solved by the introduction of a Becchi–Rouet–Stora–Tyutin (BRST)-like symmetry with anticommuting (Grassmann) parameters, which also plays an essential role in quantized gauge theories. The more specific properties of models corresponding to a special class of homogeneous spaces, symmetric spaces (like the non-linear σ-model), are studied. These models are characterized by the uniqueness of the metric and thus, of the classical action. In two dimensions, from the classical field equations an infinite number of non-local conservation laws can be derived. The field and the unique coupling renormalization group (RG) functions are calculated at one-loop order, in two dimensions, and shown to imply asymptotic freedom.

scholarly journals Kinetic field theory: Non-linear cosmic power spectra in the mean-field approximation

2021 ◽  
Vol 10 (6) ◽  
Author(s):  
Matthias Bartelmann ◽  
Johannes Dombrowski ◽  
Sara Konrad ◽  
Elena Kozlikin ◽  
Robert Lilow ◽  
...  
Keyword(s):  
Field Theory ◽  
Power Spectrum ◽  
Mean Field ◽  
Power Spectra ◽  
Field Approximation ◽  
Density Fluctuations ◽  
Mean Field Approximation ◽  
Non Linear ◽  
Two Parameters ◽  
Kinetic Field

We use the recently developed Kinetic Field Theory (KFT) for cosmic structure formation to show how non-linear power spectra for cosmic density fluctuations can be calculated in a mean-field approximation to the particle interactions. Our main result is a simple, closed and analytic, approximate expression for this power spectrum. This expression has two parameters characterising non-linear structure growth which can be calibrated within KFT itself. Using this self-calibration, the non-linear power spectrum agrees with results obtained from numerical simulations to within typically \lesssim10\,\%≲10% up to wave numbers k\lesssim10\,h\,\mathrm{Mpc}^{-1}k≲10hMpc−1 at redshift z = 0z=0. Adjusting the two parameters to optimise agreement with numerical simulations, the relative difference to numerical results shrinks to typically \lesssim 5\,\%≲5%. As part of the derivation of our mean-field approximation, we show that the effective interaction potential between dark-matter particles relative to Zel’dovich trajectories is sourced by non-linear cosmic density fluctuations only, and is approximately of Yukawa rather than Newtonian shape.

scholarly journals A master exceptional field theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Guillaume Bossard ◽  
Axel Kleinschmidt ◽  
Ergin Sezgin
Keyword(s):  
Field Theory ◽  
Gauge Invariance ◽  
Sigma Model ◽  
Linear Equations ◽  
Field Theories ◽  
Gauge Invariant ◽  
Non Linear ◽  
Non Linear Equations

Abstract We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.

scholarly journals On the three-particle analog of the Lellouch-Lüscher formula

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Fabian Müller ◽  
Akaki Rusetsky
Keyword(s):  
Field Theory ◽  
Finite Volume ◽  
Effective Field Theory ◽  
Effective Field ◽  
Infinite Volume ◽  
Leading Order ◽  
Particle Decay ◽  
Particle Decays

Abstract Using non-relativistic effective field theory, we derive a three-particle analog of the Lellouch-Lüscher formula at the leading order. This formula relates the three-particle decay amplitudes in a finite volume with their infinite-volume counterparts and, hence, can be used to study the three-particle decays on the lattice. The generalization of the approach to higher orders is briefly discussed.

scholarly journals Interface Roughening in Two Dimensions

2021 ◽  
Vol 182 (3) ◽  
Author(s):  
Gernot Münster ◽  
Manuel Cañizares Guerrero
Keyword(s):  
Field Theory ◽  
Monte Carlo ◽  
Capillary Wave ◽  
System Size ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Wave Approximation ◽  
Statistical Field ◽  
Statistical Field Theory ◽  
Interface Roughening

AbstractRoughening of interfaces implies the divergence of the interface width w with the system size L. For two-dimensional systems the divergence of $$w^2$$ w 2 is linear in L. In the framework of a detailed capillary wave approximation and of statistical field theory we derive an expression for the asymptotic behaviour of $$w^2$$ w 2 , which differs from results in the literature. It is confirmed by Monte Carlo simulations.

scholarly journals Dual subtractions

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Renato Maria Prisco ◽  
Francesco Tramontano
Keyword(s):  
Field Theory ◽  
Quantum Field ◽  
Matrix Elements ◽  
Leading Order ◽  
Initial State ◽  
Final State ◽  
Dual Representation ◽  
Subtraction Scheme ◽  
Theoretical Predictions ◽  
Perturbative Quantum

Abstract We propose a novel local subtraction scheme for the computation of Next-to-Leading Order contributions to theoretical predictions for scattering processes in perturbative Quantum Field Theory. With respect to well known schemes proposed since many years that build upon the analysis of the real radiation matrix elements, our construction starts from the loop diagrams and exploits their dual representation. Our scheme implements exact phase space factorization, handles final state as well as initial state singularities and is suitable for both massless and massive particles.

scholarly journals Solving puzzles in deformed JT gravity: phase transitions and non-perturbative effects

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Clifford V. Johnson ◽  
Felipe Rosso
Keyword(s):  
Phase Structure ◽  
Scalar Potential ◽  
String Equation ◽  
Two Dimensional ◽  
Leading Order ◽  
Classical Analysis ◽  
First Order ◽  
First Order Phase Transition ◽  
Definition Of ◽  
First Order Phase

Abstract Recent work has shown that certain deformations of the scalar potential in Jackiw-Teitelboim gravity can be written as double-scaled matrix models. However, some of the deformations exhibit an apparent breakdown of unitarity in the form of a negative spectral density at disc order. We show here that the source of the problem is the presence of a multi-valued solution of the leading order matrix model string equation. While for a class of deformations we fix the problem by identifying a first order phase transition, for others we show that the theory is both perturbatively and non-perturbatively inconsistent. Aspects of the phase structure of the deformations are mapped out, using methods known to supply a non-perturbative definition of undeformed JT gravity. Some features are in qualitative agreement with a semi-classical analysis of the phase structure of two-dimensional black holes in these deformed theories.

scholarly journals Steady Motion of Ice Sheets

1980 ◽  
Vol 25 (92) ◽  
pp. 229-246 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Power Law ◽  
Perturbation Expansion ◽  
Ice Sheet ◽  
Leading Order ◽  
Plane Flow ◽  
General Law ◽  
Non Linear ◽  
Linear Differential

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramter ν occurs, but the ν = 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factor ε(ν) is required, so that the aspect ratio ε of a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion in ε gives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.

Solvent Polarity Studies of the Water+t-Butyl Alcohol and Water+t-Butylamine Binary Systems With the Solvatochromic Dyes Nile Red and Pyridinium-N-phenoxide Betaine, Refractometry and Permittivity Measurements

1994 ◽  
Vol 47 (9) ◽  
pp. 1771 ◽  
Author(s):  
PK Kipkemboi ◽  
AJ Easteal
Keyword(s):  
Binary Systems ◽  
Hydrophobic Interactions ◽  
Nile Red ◽  
Solvent Polarity ◽  
Butyl Alcohol ◽  
Linear Functions ◽  
Non Linear ◽  
Organic Cosolvent ◽  
Two Parameters ◽  
Polarity Parameters

The empirical solvent polarity parameters ENR and ET for the solvatochromic compounds Nile Red (1) and pyridinium-N-phenoxide betaine (2), respectively, have been determined as a function of composition for water+t -butyl alcohol and water+t-butylamine binary mixtures, over the whole composition range at 298 K. For both systems the two parameters vary with composition in a strongly non-linear fashion, and the polarity of the mixture decreases with increasing proportion of the organic cosolvent. The non-linear variation of the polarity parameters is attributed to water-cosolvent hydrophobic interactions at low cosolvent contents, and hydrogen-bonding interactions at higher cosolvent contents. Permittivity and refractive index have also been measured at 298 K for both systems, and both properties are strongly non-linear functions of composition.

Non-linear dynamics, chaos, complexity and enclaves in granitoid magmas

1996 ◽  
Vol 87 (1-2) ◽  
pp. 217-223 ◽  
Author(s):  
James Flinders ◽  
John D. Clemens
Keyword(s):  
Deterministic Chaos ◽  
Interfacial Area ◽  
Two Dimensions ◽  
Mixing Processes ◽  
Nd Isotope ◽  
Natural Systems ◽  
Non Linear ◽  
The Difference ◽  
Magma System ◽  
Isotope Systematics

ABSTRACT:Most natural systems display non-linear dynamic behaviour. This should be true for magma mingling and mixing processes, which may be chaotic. The equations that most nearly represent how a chaotic natural system behaves are insoluble, so modelling involves linearisation. The difference between the solution of the linearised and ‘true’ equation is assumed to be small because the discarded terms are assumed to be unimportant. This may be very misleading because the importance of such terms is both unknown and unknowable. Linearised equations are generally poor descriptors of nature and are incapable of either predicting or retrodicting the evolution of most natural systems. Viewed in two dimensions, the mixing of two or more visually contrasting fluids produces patterns by folding and stretching. This increases the interfacial area and reduces striation thickness. This provides visual analogues of the deterministic chaos within a dynamic magma system, in which an enclave magma is mingling and mixing with a host magma. Here, two initially adjacent enclave blobs may be driven arbitrarily and exponentially far apart, while undergoing independent (and possibly dissimilar) changes in their composition. Examples are given of the wildly different morphologies, chemical characteristics and Nd isotope systematics of microgranitoid enclaves within individual felsic magmas, and it is concluded that these contrasts represent different stages in the temporal evolution of a complex magma system driven by nonlinear dynamics. If this is true, there are major implications for the interpretation of the parts played by enclaves in the genesis and evolution of granitoid magmas.

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