Scale-invariant Limit Research Articles

Downscale energy fluxes in scale-invariant oceanic internal wave turbulence

Abstract

Non-local non-linear sigma models

We study non-local non-linear sigma models in arbitrary dimension, focusing on the scale invariant limit in which the scalar fields naturally have scaling dimension zero, so that the free propagator is logarithmic. The classical action is a bi-local integral of the square of the arc length between points on the target manifold. One-loop divergences can be canceled by introducing an additional bi-local term in the action, proportional to the target space laplacian of the square of the arc length. The metric renormalization that one encounters in the two-derivative non-linear sigma model is absent in the non-local case. In our analysis, the target space manifold is assumed to be smooth and Archimedean; however, the base space may be either Archimedean or ultrametric. We comment on the relation to higher derivative non-linear sigma models and speculate on a possible application to the dynamics of M2-branes.

A Hamilton-Jacobi formulation of anisotropic inflation

Classifying inflationary scenarios according to their scaling properties is a powerful way to connect theory with observations. A useful tool to make such a classification is the β-function formalism. By describing inflation in terms of renormalization group equations, within this framework, it is possible to define universality classes, which can be considered as sets of theories that share a common scale invariant limit. In this paper we apply the formalism to define such classes of universality for models of inflation where the inflaton is coupled to gauge fields. We show that the formalism may consistently be extended to capture the peculiar features of these models such as statistical anisotropy. We also obtain some consistency conditions which serve as useful guidelines for model building.

Temporal Scale Selection in Time-Causal Scale Space

When designing and developing scale selection mechanisms for generating hypotheses about characteristic scales in signals, it is essential that the selected scale levels reflect the extent of the underlying structures in the signal. This paper presents a theory and in-depth theoretical analysis about the scale selection properties of methods for automatically selecting local temporal scales in time-dependent signals based on local extrema over temporal scales of scale-normalized temporal derivative responses. Specifically, this paper develops a novel theoretical framework for performing such temporal scale selection over a time-causal and time-recursive temporal domain as is necessary when processing continuous video or audio streams in real time or when modelling biological perception. For a recently developed time-causal and time-recursive scale-space concept defined by convolution with a scale-invariant limit kernel, we show that it is possible to transfer a large number of the desirable scale selection properties that hold for the Gaussian scale-space concept over a non-causal temporal domain to this temporal scale-space concept over a truly time-causal domain. Specifically, we show that for this temporal scale-space concept, it is possible to achieve true temporal scale invariance although the temporal scale levels have to be discrete, which is a novel theoretical construction. The analysis starts from a detailed comparison of different temporal scale-space concepts and their relative advantages and disadvantages, leading the focus to a class of recently extended time-causal and time-recursive temporal scale-space concepts based on first-order integrators or equivalently truncated exponential kernels coupled in cascade. Specifically, by the discrete nature of the temporal scale levels in this class of time-causal scale-space concepts, we study two special cases of distributing the intermediate temporal scale levels, by using either a uniform distribution in terms of the variance of the composed temporal scale-space kernel or a logarithmic distribution. In the case of a uniform distribution of the temporal scale levels, we show that scale selection based on local extrema of scale-normalized derivatives over temporal scales makes it possible to estimate the temporal duration of sparse local features defined in terms of temporal extrema of first- or second-order temporal derivative responses. For dense features modelled as a sine wave, the lack of temporal scale invariance does, however, constitute a major limitation for handling dense temporal structures of different temporal duration in a uniform manner. In the case of a logarithmic distribution of the temporal scale levels, specifically taken to the limit of a time-causal limit kernel with an infinitely dense distribution of the temporal scale levels towards zero temporal scale, we show that it is possible to achieve true temporal scale invariance to handle dense features modelled as a sine wave in a uniform manner over different temporal durations of the temporal structures as well to achieve more general temporal scale invariance for any signal over any temporal scaling transformation with a scaling factor that is an integer power of the distribution parameter of the time-causal limit kernel. It is shown how these temporal scale selection properties developed for a pure temporal domain carry over to feature detectors defined over time-causal spatio-temporal and spectro-temporal domains.

Ab initio Lattice Results for Fermi Polarons in Two Dimensions.

We investigate the attractive Fermi polaron problem in two dimensions using nonperturbative Monte Carlo simulations. We introduce a new Monte Carlo algorithm called the impurity lattice Monte Carlo method. This algorithm samples the path integral in a computationally efficient manner and has only small sign oscillations for systems with a single impurity. As a benchmark of the method, we calculate the universal polaron energy in three dimensions in the scale-invariant unitarity limit and find agreement with published results. We then present the first fully nonperturbative calculations of the polaron energy in two dimensions and density correlations between the impurity and majority particles in the limit of zero-range interactions. We find evidence for a smooth crossover transition from fermionic quasiparticle to molecular state as a function of the interaction strength.

Statistical Variability and Tokunaga Branching of Aftershock Sequences Utilizing BASS Model Simulations

Aftershock statistics provide a wealth of data that can be used to better understand earthquake physics. Aftershocks satisfy scale-invariant Gutenberg–Richter (GR) frequency–magnitude statistics. They also satisfy Omori’s law for power-law seismicity rate decay and Bath’s law for maximum-magnitude scaling. The branching aftershock sequence (BASS) model, which is the scale-invariant limit of the epidemic-type aftershock sequence model (ETAS), uses these scaling laws to generate synthetic aftershock sequences. One objective of this paper is to show that the branching process in these models satisfies Tokunaga branching statistics. Tokunaga branching statistics were originally developed for drainage networks and have been subsequently shown to be valid in many other applications associated with complex phenomena. Specifically, these are characteristic of a universality class in statistical physics associated with diffusion-limited aggregation. We first present a deterministic version of the BASS model and show that it satisfies the Tokunaga side-branching statistics. We then show that a fully stochastic BASS simulation gives similar results. We also study foreshock statistics using our BASS simulations. We show that the frequency–magnitude statistics in BASS simulations scale as the exponential of the magnitude difference between the mainshock and the foreshock, inverse GR scaling. We also show that the rate of foreshock occurrence in BASS simulations decays inversely with the time difference between foreshock and mainshock, an inverse Omori scaling. Both inverse scaling laws have been previously introduced empirically to explain observed foreshock statistics. Observations have demonstrated both of these scaling relations to be valid, consistent with our simulations. ETAS simulations, in general, do not generate Bath’s law and do not generate inverse GR scaling.

Oceanic Internal-Wave Field: Theory of Scale-Invariant Spectra

Abstract Steady scale-invariant solutions of a kinetic equation describing the statistics of oceanic internal gravity waves based on wave turbulence theory are investigated. It is shown in the nonrotating scale-invariant limit that the collision integral in the kinetic equation diverges for almost all spectral power-law exponents. These divergences come from resonant interactions with the smallest horizontal wavenumbers and/or the largest horizontal wavenumbers with extreme scale separations. A small domain is identified in which the scale-invariant collision integral converges and numerically find a convergent power-law solution. This numerical solution is close to the Garrett–Munk spectrum. Power-law exponents that potentially permit a balance between the infrared and ultraviolet divergences are investigated. The balanced exponents are generalizations of an exact solution of the scale-invariant kinetic equation, the Pelinovsky–Raevsky spectrum. A small but finite Coriolis parameter representing the effects of rotation is introduced into the kinetic equation to determine solutions over the divergent part of the domain using rigorous asymptotic arguments. This gives rise to the induced diffusion regime. The derivation of the kinetic equation is based on an assumption of weak nonlinearity. Dominance of the nonlocal interactions puts the self-consistency of the kinetic equation at risk. However, these weakly nonlinear stationary states are consistent with much of the observational evidence.

Sachs-Wolfe at second order: the CMB bispectrum on large angular scales

We calculate the Cosmic Microwave Background anisotropy bispectrum on large angular scales in the absence of primordial non-Gaussianities,assuming exact matter dominance and extendingat second order the classic Sachs-Wolfe result δT/T = Φ/3. The calculation is done in Poisson gauge.Besides intrinsic contributions calculated at last scattering, one must consider integrated effects. These are associated tolensing, and to the time dependence of the potentials (Rees-Sciama) and ofthe vector and tensor components ofthe metric generated at second order. The bispectrum is explicitly computed in the flat-sky approximation. It scales as l−4 inthe scale invariant limit and the shape dependence of its variouscontributions is represented in 3d plots.Although all the contributions to the bispectrum are parametrically of the same order, the full bispectrum is dominated by lensing. In the squeezed limit it corresponds tofNLlocal = −1/6−cos(2θ), where θ is theangle between the short and the long modes; the angle dependentcontribution comes from lensing. In the equilateral limit itcorresponds to fNLequil ≃ 3.13.

Efimov effect from functional renormalization

We apply a field-theoretic functional renormalization-group technique to the few-body (vacuum) physics of nonrelativistic atoms near a Feshbach resonance. Three systems are considered: one-component bosons with a U(1) symmetry, two-component fermions with a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{SU}(2)$ symmetry, and three-component fermions with a $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{SU}(3)$ symmetry. We focus on the scale-invariant unitarity limit of an infinite scattering length. The exact solution for the two-body sector is consistent with the unitary fixed-point behavior of the considered systems. Nevertheless, the numerical three-body solution in the $s$-wave sector develops a limit cycle scaling in case of U(1) bosons and SU(3) fermions. The Efimov parameter for the one-component bosons and the three-component fermions is found to be $s\ensuremath{\approx}1.006$, consistent with the result of Efimov.

The 't Hooft model as a hologram

We consider the 3d dual of 1+1 dimensional large-N_c QCD with quarks in the fundamental representation, also known as the 't Hooft model. 't Hooft solved this model by deriving a Schroedinger equation for the wavefunction of a parton inside the meson. In the scale-invariant limit, we show how this equation is related by a transform to the equation of motion for a scalar field in AdS_3. We thus find an explicit map between the `parton-x' variable and the radial coordinate of AdS_3. This direct map allows us to check the AdS/CFT prescription from the 2d side. We describe various features of the dual in the conformal limit and to the leading order in conformal symmetry breaking, and make some comments on the 3d theory in the fully non-conformal regime.

One-Parameter Scaling at the Dirac Point in Graphene

We numerically calculate the conductivity sigma of an undoped graphene sheet (size L) in the limit of a vanishingly small lattice constant. We demonstrate one-parameter scaling for random impurity scattering and determine the scaling function beta(sigma)=dlnsigma/dlnL. Contrary to a recent prediction, the scaling flow has no fixed point (beta>0) for conductivities up to and beyond the symplectic metal-insulator transition. Instead, the data support an alternative scaling flow for which the conductivity at the Dirac point increases logarithmically with sample size in the absence of intervalley scattering--without reaching a scale-invariant limit.

Aspects of scalar field dynamics in Gauss-Bonnet brane worlds

The Einstein-Gauss-Bonnet equations projected from the bulk to brane lead to a complicated Friedmann equation which simplifies to $H^2 \sim \rho^q$ in the asymptotic regimes. The Randall-Sundrum (RS) scenario corresponds to $q=2$ whereas $q=2/3$ $&$ $q=1$ give rise to high energy Gauss-Bonnet (GB) regime and the standard GR respectively. Amazingly, while evolving from RS regime to high energy GB limit, one passes through a GR like region which has important implications for brane world inflation. For tachyon GB inflation with potentials $V(\phi) \sim \phi^p$ investigated in this paper, the scalar to tensor ratio of perturbations $R$ is maximum around the RS region and is generally suppressed in the high energy regime for the positive values of $p$. The ratio is very low for $p>0$ at all energy scales relative to GB inflation with ordinary scalar field. The models based upon tachyon inflation with polynomial type of potentials with generic positive values of $p$ turn out to be in the $1 \sigma$ observational contour bound at all energy scales varying from GR to high energy GB limit. The spectral index $n_S$ improves for the lower values of $p$ and approaches its scale invariant limit for $p=-2$ in the high energy GB regime. The ratio $R$ also remains small for large negative values of $p$, however, difference arises for models close to scale invariance limit. In this case, the tensor to scale ratio is large in the GB regime whereas it is suppressed in the intermediate region between RS and GB. Within the frame work of patch cosmologies governed by $H^2 \sim \rho^q$, the behavior of ordinary scalar field near cosmological singularity and the nature of scaling solutions are distinguished for the values of $q < 1$ and $q > 1$.

Conformal black hole solutions of axidilaton gravity inDdimensions

Static, spherically symmetric solutions of axi-dilaton gravity in $D$ dimensions is given in the Brans-Dicke frame for arbitrary values of the Brans-Dicke constant $\omega$ and an axion-dilaton coupling parameter $k$. The mass and the dilaton and axion charges are determined and a BPS bound is derived. There exists a one parameter family of black hole solutions in the scale invariant limit.

Particle versus field structure in conformal quantum field theories

I show that a particle structure in conformal field theory is incompatible with interactions. As a substitute one has particle-like excitations whose interpolating fields have in addition to their canonical dimension an anomalous contribution. The spectra of anomalous dimension is given in terms of the Lorentz invariant quadratic invariant (compact mass operator) of a conformal generator R μ with pure discrete spectrum. The perturbative reading of R 0 as a Hamiltonian in its own right, i.e., associated with an action in a functional integral setting naturally leads to the AdS formulation. The formal service role of AdS in order to access CQFT by a standard perturbative formalism (without being forced to understand first massive theories and then taking their scale-invariant limit) vastly increases the realm of conventionally accessible 4-dimensional CQFT beyond those for which one had to use Lagrangians with supersymmetry in order to have a vanishing Beta-function.

Second virial coefficient of anyons without hard core

We calculate the second virial coefficient of anyons whose wave function does not vanish at coincidence points. This kind of anyons appear naturally when one generalizes the hard-core boundary condition according to self-adjoint extension method in quantum mechanics, and also when anyons are treated field theoretically by applying renormalization procedure to nonrelativistic Chern-Simons field theory. For the anyons which do not satisfy hard-core boundary condition, it is argued that the other scale-invariant limit is more relevant in high-temperature limit where virial expansion is useful. Furthermore, the cusp existing at the bosonic point for hard-core anyons disappears in all the other cases; instead it is shown that a new cusp is generated at the fermionic point. A physical explanation is given.

Walking technicolor beyond the ladder approximation.

We prove that in the scale-invariant limit of walking technicolor the fermion dynamical mass function falls off exactly as 1/p when the gauge coupling \ensuremath{\alpha} just reaches the critical value required to trigger spontaneous chiral-symmetry breaking. The proof is given to all orders in \ensuremath{\alpha}.

Weyl scalings and spinor matter interactions in scalar-tensor theories of gravitation

An SL(2, C) gauge formulation of a scalar-tensor theory of gravitation is related to the Brans-Dicke theory in the absence of spinorial matter couplings although the natural connection of space-time has a torsion determined by the scalar field. Simplified scaling properties of gravitational variables are elucidated in this geometry. In the presence of matter with intrinsic spin a natural scalar-spinor coupling is exposed. A locally scale invariant limit exists without a Weyl gauge field.

The order/disorder quantum field operators associated with the two-dimensional Ising model in the continuum limit

We show that the sin/cos functions of the axial potential of the free massive Dirac field appropriately describe the correlation functions of the ( T > T c order/disorder variable of two independent superimposed Ising systems taken in the continuum limit. The well known Kramers-Wannier transformation has a simple implementation in this description. It is also demonstrated that the square relationship at the level of correlation functions between the doubled and the single versions of the Ising model originates from an operator relationship between Dirac and Majorana spinor fields in two-dimensional space-time. Finally a connection is made between the Wightman functions of these local field operators and the statistical correlation functions obtained by McCoy, Tracy and Wu. The use of the doubled version of the Ising model based on the Dirac field allows a simple description of the scale invariant limit and therefore bridges the gap between the rigorous work and that of Kadanoff and Ceva as well as that of Luther and Peschel.

Differential dispersive analysis of e+e− annihilation into hadrons

The previously reported differential dispersive interpretation of the narrow resonances in e+e− annihilation is described in more detail. In addition, the final asymptotic value of the ratioR=σ(e+e−→hadrons)/σ(e+e−→μ+μ−) is found to be related to, but not simply the same as, the sum over the squares of the quark charges. Both this final asymptotic value ofR and the narrow resonances are argued to be consistent with the renormalizability of electrodynamics together with the scale invariance of the hadronic electromagnetic current at large momentum transfer, where only the conventional colored, fractionally charged p, n and λ quarks are involved in this scale-invariant limit. No new quark or new quantum number is necessary.

Differential dispersive analysis of e+-e- annihilation into hadrons

The previously reported differential dispersive interpretation of the narrow resonances in e+e− annihilation is described in more detail. In addition, the final asymptotic value of the ratioR=σ(e+e−→hadrons)/σ(e+e−→μ+μ−) is found to be related to, but not simply the same as, the sum over the squares of the quark charges. Both this final asymptotic value ofR and the narrow resonances are argued to be consistent with the renormalizability of electrodynamics together with the scale invariance of the hadronic electromagnetic current at large momentum transfer, where only the conventional colored, fractionally charged p, n and λ quarks are involved in this scale-invariant limit. No new quark or new quantum number is necessary.