Scaling Symmetry Research Articles

Restricted Weyl symmetry

We discuss the physics of a restricted Weyl symmetry in a curved space-time where a gauge parameter $\Omega(x)$ of Weyl transformation satisfies a constraint $\Box \Omega = 0$. First, we present a model of QED where we have a restricted gauge symmetry in the sense that a $U(1)$ gauge parameter $\theta(x)$ obeys a similar constraint $\Box \theta = 0$ in a flat Minkowski space-time. Next, it is precisely shown that a global scale symmetry must be spontaneously broken at the quantum level. Finally, we discuss the origin of the restricted Weyl symmetry and show that its symmetry can be derived from a full Weyl symmetry by taking a gauge condition $R = 0$ in the BRST formalism.

Convective Effect on Magnetohydrodynamic (MHD) Stagnation Point Flow of Casson Fluid over a Vertical Exponentially Stretching/Shrinking Surface: Triple Solutions

In the current study, the characteristics of heat transfer of a steady, two-dimensional, stagnation point, and magnetohydrodynamic (MHD) flow of shear thickening Casson fluid on an exponentially vertical shrinking/stretching surface are examined in attendance of convective boundary conditions. The impact of the suction parameter is also considered. The system of governing partial differential equations (PDEs) and boundary conditions is converted into ordinary differential equations (ODEs) with the suitable exponential similarity variables of transformations and then solved using the shooting method with the fourth order Runge–Kutta method. Similarity transformation is an important class of phenomena in which scale symmetry allows one to reduce the number of independent variables of the problem. It should be noted that solutions of the ODEs show the symmetrical behavior of the PDES for the profiles of velocity and temperature. Similarity solutions are obtained for the case of stretching/shrinking and suction parameters. It is revealed that there exist two ranges of the solutions in the specific ranges of the physical parameters, three solutions depend on the opposing flow case where stagnation point (A) should be equal to 0.1, two solutions exist when λ1 = 0 where λ1 is a mixed convection parameter and A > 0.1, and a single solution exists when λ1 > 0. Moreover, the effects of numerous applied parameters on velocity, temperature distributions, skin friction, and local Nusselt number are examined and given through tables and graphs for both shrinking and stretching surfaces.

New scaling laws of passive scalar with a constant mean gradient in decaying isotropic turbulence

We use the Lie symmetry theory to derive new scaling laws for passive scalar dynamics under the influence of a constant mean gradient of the scalar in decaying homogeneous isotropic turbulence. For this purpose, we apply symmetry analysis to the equations for two-point correlation of the scalar and velocity fluctuations. It is shown that, in contrast to the classical self-similarity approach, the general invariant solutions, respectively scaling laws, of the two-point functions are constructed using the symmetry approach, without requiring an a priori set of similarity scales to carry on the analysis. In the context of the current analysis also, scaling laws for one-point quantities of the scalar variance , the transverse scalar flux and the variance of the turbulent velocity fluctuations are established, which are essentially related to the scaling symmetries. A key step to derive the scaling laws is the symmetry breaking induced by the constant mean scalar gradient. We use the results of a highly resolved direct numerical simulation of Gauding et al. (Comput. Fluids, vol. 180, 2019, pp. 206–217) to verify the scaling laws and the self-similarity of the two-point correlation functions. It is shown that the general symmetry solutions obtained from symmetry results provide a very good similarity to these functions.

Thin-jet scaling in large-scale shallow water quasigeostrophic flow

The evolution of thin frontal jets in large-scale shallow water quasigeostrophic flow is studied, with a focus on jet curvature and arclength. The flow is large-scale in the sense that mixed regions of potential vorticity (PV) are much larger than the deformation length . However the presence of sharp PV fronts with widths drives the ongoing growth of the flow's length scale; in particular the PV fronts and collocated jets support long undulations that facilitate jet interactions and the merger of mixed regions. The flow develops large dynamically active multilevel vortices containing two main mixed levels of PV, as well as small dynamically inactive vortices that persist for long times; these regions and their frontal jets display markedly different scaling properties. The frontal jets bounding the large dynamically active mixed regions follow power laws consistent with the scaling symmetries of the modified Korteweg-de Vries (mKdV) equation, which governs the motion of the jet axis in the thin-jet limit. These jets have population total arc length decaying approximately as , average arc length growing like , rms curvature as and typical curvature fluctuation as .

Brane Mechanism of Spontaneously Generated Gravity

The braneworld scenario is studied and the effective action of 3-brane living in 5-dim. Minkowski space is constructed. This action is proved to be invariant under spontaneously broken scale and Weyl symmetries and to encode a model of quadratic gravity generalizing the Starobinsky model. The symmetry breaking generates the Hilbert-Einstein term with the Newton constant $G_{N}\sim\frac{1}{\mu^{2}}$, where $\mu$ is a mass scale equal to the mean curvature of the vacuum hyper-worldsheet of 3-brane. This result proposes a brane modification of the mechanism of spontaneously generated gravity.

Conformal theory of gravitation and cosmology

The postulate of universal local Weyl scaling (conformal) symmetry modifies both general relativity and the Higgs scalar field model. Conformal gravity (CG) has recently been fitted to rotation data for 138 galaxies. The conformal Higgs model (CHM) acquires a gravitational effect that fits observed Hubble expansion for redshifts (7.33 Gyr) accurately with only one free constant parameter. Conformal theory explains dark energy and does not require dark matter, providing a viable alternative to the ΛCDM standard paradigm. Vanishing of centripetal acceleration outside a galactic halo boundary is a unique implication of the theory. The fit to observed astrophysical data by conformal models CG and CHM is shown here to account for both parameters w2 and λ of postulated Higgs potential , responsible for symmetry-breaking finite in particle theory. Recent criticism of CG is resolved here by showing that CG and CHM are interdependent but compatible. Nonclassical CG radial acceleration parameter γ is determined by the CHM. A recently established empirical relationship between classical and nonclassical galactic radial acceleration requires parameter γ to be independent of galactic mass. Conformal theory is shown here to be consistent with this and with the v4 baryonic Tully-Fisher relation for galactic rotation velocities.

Baryon-Interacting Dark Matter: heating dark matter and the emergence of galaxy scaling relations

The empirical scaling relations observed in disk galaxies remain challenging for models of galaxy formation. The most striking among these is the Mass Discrepancy-Acceleration Relation (MDAR), which encodes both a tight baryonic Tully-Fisher relation (BTFR) and the observed diversity of galaxy rotation curves through the central surface density relation (CSDR) . Building on our earlier work [1], we propose here that the MDAR is the result of interactions between baryons and `Baryon-Interacting Dark Matter' (BIDM), which heat up the dark matter. Following a bottom-up, hydrodynamical approach, we find that the MDAR follows if: i) the BIDM equation of state approximates that of an ideal gas; ii) the BIDM relaxation time is order the Jeans time; iii) the heating rate is inversely proportional to the BIDM density. Remarkably, under these assumptions the set of hydrodynamical equations together with Poisson's equation enjoy an anisotropic scaling symmetry. In the BIDM-dominated regime, this gives rise to an enhanced symmetry which fully captures the low-acceleration limit of the MDAR . We then show that, assuming a cored pseudo-isothermal profile at equilibrium, this set of equations gives rise to parameters reproducing the MDAR . Specifically, in the flat part of the rotation curve the asymptotic rotational velocity matches the parametric dependence of the BTFR . Moreover, in the central region of high-surface brightness galaxies, the profile reproduces the CSDR . Finally, by studying the time-dependent approach to equilibrium, we derive a global combination of the BTFR and CSDR, which matches the expectations in low surface-brightness galaxies. The form of the heating rate also makes model-independent predictions for various cosmological observables. We argue that our scenario satisfies existing observational constraints, and, intriguingly, offers a possible explanation to the EDGES anomaly.

Minimal neutral naturalness model

We build a minimal neutral naturalness model in which the top partners are not charged under QCD, with a pseudo Goldstone Higgs arising from $SO(5)/SO(4)$ breaking. The color-neutral top partners generate the Higgs potential radiatively without quadratic divergence. The misalignment between the electroweak scale and global symmetry breaking scale is naturally obtained from suppression of the Higgs quadratic term, due to cancellation between singlet and doublet top partner contributions. This model can be embedded into ultraviolet holographic setup in composite Higgs framework, which even realizes finite Higgs potential.

Collective modes of ultracold fermionic alkaline-earth-metal gases with SU( N ) symmetry

We calculate the collective modes of ultracold trapped alkaline-earth fermionic atoms, which possess an SU($N$) symmetry of the nuclear spin degree of freedom, and a controllable $N$, with $N$ as large as $10$. We calculate the breathing and quadrupole modes of two-dimensional and three-dimensional harmonically trapped gases in the normal phase. We particularly concentrate on two-dimensional gases, where the shift is more accessible experimentally, and the physics has special features. We present results as a function of temperature, interaction strength, density, and $N$. We include calculations across the collisionless to hydrodynamic crossover. We assume the gas is interacting weakly, such that it can be described by a Boltzmann-Vlasov equation that includes both mean-field terms and the collision integral. We solve this with an approximate scaling ansatz, taking care in two-dimensions to preserve the scaling symmetry of the system. We predict the collective mode frequency shifts and damping, showing that these are measurable in experimentally relevant regimes. We expect these results to furnish powerful tools to characterize interactions and the state of alkaline-earth gases, as well as to lay the foundation for future work, for example on strongly interacting gases and SU($N$) spin modes.

Conservation laws by virtue of scale symmetries in neural systems.

In contrast to the symmetries of translation in space, rotation in space, and translation in time, the known laws of physics are not universally invariant under transformation of scale. However, a special case exists in which the action is scale invariant if it satisfies the following two constraints: 1) it must depend upon a scale-free Lagrangian, and 2) the Lagrangian must change under scale in the same way as the inverse time, . Our contribution lies in the derivation of a generalised Lagrangian, in the form of a power series expansion, that satisfies these constraints. This generalised Lagrangian furnishes a normal form for dynamic causal models–state space models based upon differential equations–that can be used to distinguish scale symmetry from scale freeness in empirical data. We establish face validity with an analysis of simulated data, in which we show how scale symmetry can be identified and how the associated conserved quantities can be estimated in neuronal time series.

Symmetric scalars

We provide a complete classification of Poincar'e-invariant scalar field theories with an enlarged set of classical symmetries to leading order in derivatives, namely for the so-called P(X,ϕ) theories, in two or more spacetime dimensions. We find only three possibilities: Dirac-Born-Infeld, Cuscuton and Scaling theories. The latter two classes of actions involve an arbitrary function of the scalar field. As an application, we use the scaling symmetry to derive an infinite set of constraints on the Wilsonian coefficients of the low-energy Effective Field Theory. Furthermore, we study the extension of these results to cosmological (FLRW) and (Anti-)de Sitter spacetimes. We find in particular that the Cuscuton action has a generic set of symmetries around any background spacetime that possesses Killing vector fields, while the DBI actions have well-known analogues that we summarize explicitly.

SIMILARITY SOLUTIONS AND CONSERVATION LAWS FOR THE BEAM EQUATIONS: A COMPLETE STUDY

We study the similarity solutions and we determine the conservation laws of various forms of beam equations, such as Euler-Bernoulli, Rayleigh and Timoshenko-Prescott. The travelling-wave reduction leads to solvable fourth-order odes for all the forms. In addition, the reduction based on the scaling symmetry for the Euler-Bernoulli form leads to certain odes for which there exists zero symmetries. Therefore, we conduct the singularity analysis to ascertain the integrability. We study two reduced odes of second and third orders. The reduced second-order ode is a perturbed form of Painlevé-Ince equation, which is integrable and the third-order ode falls into the category of equations studied by Chazy, Bureau and Cosgrove. Moreover, we derived the symmetries and its corresponding reductions and conservation laws for the forced form of the abovementioned beam forms. The Lie Algebra is mentioned explicitly for all the cases.

Scale Symmetry in the Universe

Scale symmetry is a fundamental symmetry of physics that seems however not to be fully realized in the universe. Here, we focus on the astronomical scales ruled by gravity, where scale symmetry holds and gives rise to a truly scale invariant distribution of matter, namely it gives rise to a fractal geometry. A suitable explanation of the features of the fractal cosmic mass distribution is provided by the nonlinear Poisson–Boltzmann–Emden equation. An alternative interpretation of this equation is connected with theories of quantum gravity. We study the fractal solutions of the equation and connect them with the statistical theory of random multiplicative cascades, which originated in the theory of fluid turbulence. The type of multifractal mass distributions so obtained agrees with results from the analysis of cosmological simulations and of observations of the galaxy distribution.

Symmetries of the [formula omitted] approximation to the Boltzmann neutron transport equation

This work explores conditional translation and scaling symmetries of a P3 approximation to the one-dimensional (1D) Boltzmann neutron transport equation with arbitrary material properties. The principal outcomes of this analysis are twofold: 1) the identification of symmetries using Lie group theory is exhaustive, and thus provides definitive insight into scenarios where the differential equation(s) under consideration are expected to hold across phase space position or dimensional scale; and 2) invariance under transformations may be leveraged to construct changes of variables under which reduced-order structures or exact solutions with special properties may be derived. In support of these goals, this work includes the application of Lie group theory to the aforementioned neutron transport model. A variety of conditional translation and scaling symmetries are obtained (e.g., depending on the functional form of various macroscopic cross section functions), and an example scaleinvariant numerical solution of the 1D spherical P3 equations is provided.

New Interpretations of Normalization Methods in Deep Learning

In recent years, a variety of normalization methods have been proposed to help training neural networks, such as batch normalization (BN), layer normalization (LN), weight normalization (WN), group normalization (GN), etc. However, some necessary tools to analyze all these normalization methods are lacking. In this paper, we first propose a lemma to define some necessary tools. Then, we use these tools to make a deep analysis on popular normalization methods and obtain the following conclusions: 1) Most of the normalization methods can be interpreted in a unified framework, namely normalizing pre-activations or weights onto a sphere; 2) Since most of the existing normalization methods are scaling invariant, we can conduct optimization on a sphere with scaling symmetry removed, which can help to stabilize the training of network; 3) We prove that training with these normalization methods can make the norm of weights increase, which could cause adversarial vulnerability as it amplifies the attack. Finally, a series of experiments are conducted to verify these claims.

Holographic spontaneous anisotropy

We construct a family of holographic duals to anisotropic states in a strongly coupled gauge theory. On the field theory side the anisotropy is generated by giving a vacuum expectation value to a dimension three operator. We obtain our gravity duals by considering the geometry corresponding to the intersection of D3- and D5- branes along 2+1 dimensions. Our backgrounds are supersymmetric and solve the fully backreacted equations of motion of ten-dimensional supergravity with smeared D5-brane sources. In all cases the geometry flows to AdS5× \U0001d54a5 in the UV, signaling an isotropic UV fixed point of the dual field theory. In the IR, depending on the parameters of the solution, we find two possible behaviors: an isotropic fixed point or a geometry with anisotropic Lifshitz-like scaling symmetry. We study several properties of the solutions, including the entanglement entropy of strips. We show that any natural extension of existing c-functions will display non-monotonic behavior, conforming with the presence of new degrees of freedom only at intermediate regions between the boundary and the origin of the holographic dual.

2D Galilean field theories with anisotropic scaling

In this work, we study two-dimensional Galilean field theories with global translations and anisotropic scaling symmetries. We show that such theories have enhanced local symmetries, generated by the infinite dimensional spin-l Galilean algebra with possible central extensions, under the assumption that the dilation operator is diagonalizable and has a discrete and non-negative spectrum. We study the Newton-Cartan geometry with anisotropic scaling, on which the field theories could be defined in a covariant way. With the well-defined Newton-Cartan geometry we establish the state-operator correspondence in anisotropic GCFT, determine the two-point functions of primary operators, and discuss the modular properties of the torus partition function which allows us to derive Cardy-like formulae.

Phenomenology of the new light Higgs bosons in Gildener-Weinberg models

Gildener-Weinberg (GW) models of electroweak symmetry breaking are especially interesting because the low mass and nearly Standard Model couplings of the 125 GeV Higgs boson $H$ are protected by approximate scale symmetry. Another important but so far underappreciated feature of these models is that a sum rule bounds the masses of the new charged and neutral Higgs bosons appearing in all these models to be below about 500 GeV. Therefore, they are within reach of LHC data currently or soon to be in hand. Also so far unnoticed of these models, certain cubic and quartic Higgs scalar couplings vanish at the classical level. This is due to spontaneous breaking of the scale symmetry. These couplings become nonzero from explicit scale breaking in the Coleman-Weinberg loop expansion of the effective potential. In a two-Higgs doublet GW model, we calculate ${\ensuremath{\lambda}}_{HHH}\ensuremath{\simeq}2({\ensuremath{\lambda}}_{HHH}{)}_{\mathrm{SM}}=64\text{ }\text{ }\mathrm{GeV}$. This corresponds to $\ensuremath{\sigma}(pp\ensuremath{\rightarrow}HH)\ensuremath{\cong}15--20\text{ }\text{ }\mathrm{fb}$, its minimum value for $\sqrt{s}=13--14\text{ }\text{ }\mathrm{TeV}$ at the LHC. It will require at least the 27 TeV HE-LHC to observe this cross section. We also find ${\ensuremath{\lambda}}_{HHHH}\ensuremath{\simeq}4({\ensuremath{\lambda}}_{HHHH}{)}_{\mathrm{SM}}=0.129$, whose observation in $pp\ensuremath{\rightarrow}HHH$ requires a 100 TeV collider. Because of the above-mentioned sum rule, these results apply to all GW models. In view of this unpromising forecast, we stress that LHC searches for the new relatively light Higgs bosons of GW models are by far the surest way to test them in this decade.

Holographic Abelian Higgs model and the linear confinement

We consider the holographic abelian Higgs model and show that, in the absence of the scale symmetry breaking effect, chiral symmetry breaking gives linear confinement where the slope is given by the value of the chiral condensation. The model can be considered either as the theory of superconductivity or as the axial sector of QCD depending on the interpretation of the charge. We also provided a few models with linear confinement.

Supersymmetry shielding the scaling symmetry of conformal quantum mechanics

Renormalization of the inverse square potential usually breaks its classical conformal invariance. In a strongly attractive potential, the scaling symmetry is broken to a discrete subgroup while, in a strongly repulsive potential, it is preserved at quantum level. In the intermediate, weak-medium range of the coupling, an anomalous length scale appears due to a flow of the renormalization group away from a critical point. We show that potentials with couplings in the strongly-repulsive and in the weak-medium ranges can be related by a dynamical supersymmetry. Imposing SUSY invariance unifies these two ranges, and fixes the anomalous scale to zero, thus restoring the continuous scaling symmetry.