Scaling Symmetry Research Articles

A formal notion of genericity and term-by-term vanishing superpotentials at supersymmetric vacua from R-symmetric Wess-Zumino models

It is known in previous literature that if a Wess-Zumino model with an R-symmetry gives a supersymmetric vacuum, the superpotential vanishes at the vacuum. In this work, we establish a formal notion of genericity, and show that if the R-symmetric superpotential has generic coefficients, the superpotential vanishes term-by-term at a supersymmetric vacuum. This result constrains the form of the superpotential which leads to a supersymmetric vacuum. It may contribute to a refined classification of R-symmetric Wess-Zumino models, and find applications in string constructions of vacua with small superpotentials. A similar result for a scalar potential system with a scaling symmetry is discussed.

Detecting scale anomaly in chiral phase transition of QCD: new critical endpoint pinned down

Violation of scale symmetry, scale anomaly, being a radical concept in quantum field theory, is of importance to comprehend the vacuum structure of QCD, and should potentially contribute to the chiral phase transition in thermal QCD, as well as the chiral and U(1) axial symmetry. Though it should be essential, direct evidence of scale anomalies has never been observed in the chiral phase transition. We propose a methodology to detect a scale anomaly in the chiral phase transition, which is an electromagnetically induced scale anomaly: apply a weak magnetic field background onto two-flavor massless QCD with an extremely heavy strange quark, first observe the chiral crossover; second, adjusting the strange quark mass to be smaller and smaller, observe the second-order chiral phase transition, and then the first-order one in the massless-three flavor limit. Thus, the second-order chiral phase transition, observed as the evidence of the quantum scale anomaly, is a new critical endpoint. It turns out that this electromagnetic scale anomaly gets most operative in the weak magnetic field regime, rather than a strong field region. We also briefly address accessibility of lattice QCD, a prospected application to dense matter system, and implications to astrophysical observations, such as gravitational wave productions provided from thermomagnetic QCD-like theories.

Complexity of scalar collapse in anti-de Sitter spacetime

We calculate the volume and action forms of holographic complexity for the gravitational collapse of scalar field matter in asymptotically anti-de Sitter spacetime, using numerical methods to reproduce the geometry responding to the oscillating field over multiple crossing times. Like the scalar field pulse, the volume complexity oscillates quasiperiodically before horizon formation. It also shows a scaling symmetry with the amplitude of the scalar field. The action complexity is also quasiperiodic with spikes of increasing amplitude.

Non-relativistic ten-dimensional minimal supergravity

We construct a non-relativistic limit of ten-dimensional mathcal{N} = 1 supergravity from the point of view of the symmetries, the action, and the equations of motion. This limit can only be realized in a supersymmetric way provided we impose by hand a set of geometric constraints, invariant under all the symmetries of the non-relativistic theory, that define a so-called ‘self-dual’ Dilatation-invariant String Newton-Cartan geometry. The non-relativistic action exhibits three emerging symmetries: one local scale symmetry and two local conformal supersymmetries. Due to these emerging symmetries the Poisson equation for the Newton potential and two partner fermionic equations do not follow from a variation of the non-relativistic action but, instead, are obtained by a supersymmetry variation of the other equations of motion that do follow from a variation of the non-relativistic action. We shortly discuss the inclusion of the Yang-Mills sector that would lead to a non-relativistic heterotic supergravity action.

Carroll limit of four-dimensional gravity theories in the first order formalism

We explore the ultra-relativistic (UR) limit of a class of four dimensional gravity theories, known as Lovelock–Cartan (LC) gravities, in the first order formalism. First, we review the well known limit of the Einstein–Hilbert (EH) action. A very useful scale symmetry involving the vierbeins and the boost connection is presented. Moreover, we explore the field equations in order to find formal solutions. Some remarkable results are obtained: Riemann and Weitzenböck like manifolds are discussed; Birkhoff’s theorem is verified for the torsionless case; an explicit solution with non-trivial geometry is discussed; a quite general solution in the presence of matter is obtained. Latter, we consider the UR limit of the more general LC gravity. The previously scale symmetry is also discussed. The field equations are studied in vacuum and in the presence of matter. In comparison with the EH case, a few relevant results are found: Birkhoff’s theorem is also verified for the torsionless case; a quite general solution in the presence of matter is obtained. This solution generalizes the previous case; Riemann and Weitzenböck like manifolds are derived in the same lines of the EH case.

Cosmology from pregeometry

We discuss cosmological solutions for a diffeomorphism invariant gauge theory of the non-compact Lorentz group $SO(1,3)$. Besides the gauge bosons our model of pregeometry contains a vector field in the vector representation of $SO(1,3)$ and a scalar singlet. General relativity and variable gravity emerge as effective theories for large distances and times in Planck units. We propose an approximation to the effective action with up to two derivatives. For a suitable range of parameters the universe approaches for large times stable Minkowski space. For late cosmology the model predicts dynamical dark energy and provides for a candidate for dark matter. Early cosmology is characterized by an inflationary epoch. The beginning of the universe in the infinite past is great emptiness, corresponding to an ultraviolet fixed point with the associated quantum scale symmetry. The beginning universe is a vacuum state with vanishing expectation values and finite non-vanishing correlation functions for the fluctuations of all fields. There is no physical big bang singularity.

Mirror map for Fermat polynomials with a nonabelian group of symmetries

We study Landau-Ginzburg orbifolds $(f,G)$ with $f=x_1^n+\ldots+x_N^n$ and $G=S\ltimes G^d$, where $S\subseteq S_N$ and $G^d$ is either the maximal group of scalar symmetries of $f$ or the intersection of the maximal diagonal symmetries of $f$ with $\mathrm{SL}_N(\mathbb{C})$. We construct a mirror map between the corresponding phase spaces and prove that it is an isomorphism restricted to a certain subspace of the phase space when $n=N$ is a prime number. When $S$ satisfies the condition PC of Ebeling and Gusein-Zade this subspace coincides with the full space. We also show that two phase spaces are isomorphic for $n=N=5$.

When scale is surplus

We study a long-recognised but under-appreciated symmetry called dynamical similarity and illustrate its relevance to many important conceptual problems in fundamental physics. Dynamical similarities are general transformations of a system where the unit of Hamilton’s principal function is rescaled, and therefore represent a kind of dynamical scaling symmetry with formal properties that differ from many standard symmetries. To study this symmetry, we develop a general framework for symmetries that distinguishes the observable and surplus structures of a theory by using the minimal freely specifiable initial data for the theory that is necessary to achieve empirical adequacy. This framework is then applied to well-studied examples including Galilean invariance and the symmetries of the Kepler problem. We find that our framework gives a precise dynamical criterion for identifying the observables of those systems, and that those observables agree with epistemic expectations. We then apply our framework to dynamical similarity. First we give a general definition of dynamical similarity. Then we show, with the help of some previous results, how the dynamics of our observables leads to singularity resolution and the emergence of an arrow of time in cosmology.

On testing structural identifiability by a simple scaling method: Relying on scaling symmetries can be misleading.

A recent paper published in PLOS Computational Biology [1] introduces the Scaling Invariance Method (SIM) for analysing structural local identifiability and observability. These two properties define mathematically the possibility of determining the values of the parameters (identifiability) and states (observability) of a dynamic model by observing its output. In this note we warn that SIM considers scaling symmetries as the only possible cause of non-identifiability and non-observability. We show that other types of symmetries can cause the same problems without being detected by SIM, and that in those cases the method may lead one to conclude that the model is identifiable and observable when it is actually not.

The limitations, dangers, and benefits of simple methods for testing identifiability.

In their Commentary paper, Villaverde and Massonis (On testing structural identifiability by a simple scaling method: relying on scaling symmetries can be misleading) have commented on our paper in which we proposed a simple scaling method to test structural identifiability. Our scaling invariance method (SIM) tests for scaling symmetries only, and Villaverde and Massonis correctly show the SIM may fail to detect identifiability problems when a model has other types of symmetries. We agree with the limitations raised by these authors but, also, we emphasize that the method is still valuable for its applicability to a wide variety of models, its simplicity, and even as a tool to introduce the problem of identifiability to investigators with little training in mathematics.

Primordial flat frame: A new view on inflation

Models of inflationary cosmology admit a choice of the metric for which the geometry of homogeneous isotropic solutions becomes flat Minkowski space in the infinite past. In this primordial flat frame all mass scales vanish in the infinite past and quantum scale symmetry is realized. The cosmological evolution is dominantly described by the slow increase of a scalar field which sets the scale of all masses. We construct the primordial flat frame for standard models of inflation as Starobinsky inflation or chaotic inflation. In particular, we discuss the evolution of inhomogeneous solutions in the neighborhood of the homogeneous isotropic background solution and their relation to the observable primordial fluctuation spectrum. If the propagators for the graviton and scalar field remain regular, our observed inhomogeneous Universe can be extrapolated back to the infinite past in physical time. In this case there is no physical big-bang singularity -- the latter reflects only a singular choice of ``field coordinates". Independently of the issue of singularity the primordial flat frame offers a new view on the physical properties of the inflationary universe, which can be characterized as a very slowly evolving almost empty vacuum state with approximate scale symmetry.

The foundation of the hyperunified field theory I — Fundamental building block and symmetry

Starting from the motional property of functional field based on the action principle of path integral formulation while proposing maximum coherence motion principle and maximum locally entangled-qubits motion principle as guiding principles, we show that such a functional field as fundamental building block appears naturally as an entangled qubit-spinor field expressed by a locally entangled state of qubits. Its motion brings about the appearance of Minkowski space–time with dimension determined by the motion-correlation [Formula: see text]-spin charge and the emergence of [Formula: see text]-spin/hyperspin symmetry as fundamental symmetry. Intrinsic [Formula: see text]-spin charge displays a periodic feature as the mod 4 qubit number, which enables us to classify all entangled qubit-spinor fields and space–time dimensions into four categories with respect to four [Formula: see text]-spin charges [Formula: see text]. An entangled decaqubit-spinor field in 19-dimensional hyper-space–time is found to be a hyperunified qubit-spinor field which unifies all discovered leptons and quarks and brings on the existence of mirror lepton–quark states. The inhomogeneous hyperspin symmetry [Formula: see text] as hyperunified symmetry in association with inhomogeneous Lorentz-type symmetry [Formula: see text] and global scaling symmetry provides a unified fundamental symmetry. The maximum locally entangled-qubits motion principle is shown to lay the foundation of hyperunified field theory, which enables us to comprehend long-standing questions raised in particle physics and quantum field theory.

Exact four-point function and OPE for an interacting quantum field theory with space/time anisotropic scale invariance

We identify a nontrivial yet tractable quantum field theory model with space/time anisotropic scale invariance, for which one can exactly compute certain four-point correlation functions and their decompositions via the operator-product expansion(OPE). The model is the Calogero model, non-relativistic particles interacting with a pair potential frac{g}{{left|x-yright|}^2} in one dimension, considered as a quantum field theory in one space and one time dimension via the second quantisation. This model has the anisotropic scale symmetry with the anisotropy exponent z = 2. The symmetry is also enhanced to the Schrödinger symmetry. The model has one coupling constant g and thus provides an example of a fixed line in the renormalisation group flow of anisotropic theories.We exactly compute a nontrivial four-point function of the fundamental fields of the theory. We decompose the four-point function via OPE in two different ways, thereby explicitly verifying the associativity of OPE for the first time for an interacting quantum field theory with anisotropic scale invariance. From the decompositions, one can read off the OPE coefficients and the scaling dimensions of the operators appearing in the intermediate channels. One of the decompositions is given by a convergent series, and only one primary operator and its descendants appear in the OPE. The scaling dimension of the primary operator we computed depends on the coupling constant. The dimension correctly reproduces the value expected from the well-known spectrum of the Calogero model combined with the so-called state-operator map which is valid for theories with the Schrödinger symmetry. The other decomposition is given by an asymptotic series. The asymptotic series comes with exponentially small correction terms, which also have a natural interpretation in terms of OPE.

Scale-invariant supergravity theory in component formulation

We formulate N=1 supergravity theory in four-dimensions with local scale invariance in semi-on-shell component formulation in four dimensions. The algebra we adopt has the generators (Pm,Mmn,Qα,S) consisting of the generators in super-Poincaré algebra: (Pm,Mmn,Qα) with 4+6+4 components, and that of local scale-transformation S with one additional component. Our field content consists of the on-shell multiplet of supergravity (eμm,ψμα), the tensor multiplet (Bμν,χα,φ) and the ‘Weylon multiplet’ (Sμ,ρα,Cμνρ), where Sμ is the Weylon that gauges local scale symmetry, ψμ,χ and ρ are Majorana spinors, while Cμνρ is a tensor auxiliary field. Similar to the non-supersymmetric case that has no minimal coupling between the Sμ-field and spin 1/2 field, the Weylon does not minimally couple to spin 3/2 gravitino field in our locally supersymmetric generalization. By going to the Einstein frame from the original Jordan frame, we show that our system is equivalent to a system with the gauged dilaton-shift symmetry. This implies that our system bypasses persistent dilaton-related problems, avoiding a long-distance force for a massless dilaton, a minimum value of its potential, or a run-away vacuum.

Restricted Weyl symmetry and spontaneous symmetry breakdown of conformal symmetry

In this paper, we elucidate the relation between the restricted Weyl symmetry and spontaneous symmetry breakdown of conformal symmetry. Using a scalar–tensor gravity, we show that the restricted Weyl symmetry leads to spontaneous symmetry breakdown of a global scale symmetry when the vacuum expectation value of a scalar field takes a nonzero value. It is then shown that this spontaneous symmetry breakdown induces spontaneous symmetry breakdown of special conformal symmetry in a flat Minkowski spacetime, but the resultant Nambu–Goldstone boson is not an independent physical mode but expressed in terms of the derivative of the dilaton which is the Nambu–Goldstone boson of the global scale symmetry. In other words, the theories which are invariant under the general coordinate transformation and the restricted Weyl transformation exhibit a Nambu–Goldstone phase where both special conformal transformation and dilatation are spontaneously broken while preserving the Poincaré symmetry.

Scale symmetry breaking, quantum anomalous energy and proton mass decomposition

We study the anomalous scale symmetry breaking effects on the proton mass in QCD due to quantum fluctuations at ultraviolet scales. We confirm that a novel contribution naturally arises as a part of the proton mass, which we call the quantum anomalous energy (QAE). We discuss the QAE origins in both lattice and dimensional regularizations and demonstrate its role as a scheme-and-scale independent component in the mass decomposition. We further argue that QAE role in the proton mass resembles a dynamical Higgs mechanism, in which the anomalous scale symmetry breaking field generates mass scales through its vacuum condensate, as well as its static and dynamical responses to the valence quarks. We demonstrate some of our points in two simpler but closely related quantum field theories, namely the 1+1 dimensional non-linear sigma model in which QAE is non-perturbative and scheme-independent, and QED where the anomalous energy effect is perturbative calculable.

Scale Symmetry and Friction

Dynamical similarities are non-standard symmetries found in a wide range of physical systems that identify solutions related by a change of scale. In this paper, we will show through a series of examples how this symmetry extends to the space of couplings, as measured through observations of a system. This can be exploited to focus on observations that can be used to distinguish between different theories and identify those which give rise to identical physical evolutions. These can be reduced into a description that makes no reference to scale. The resultant systems can be derived from Herglotz’s principle and generally exhibit friction. Here, we will demonstrate this through three example systems: the Kepler problem, the N-body system and Friedmann–Lemaître–Robertson–Walker cosmology.

Entropic destruction of heavy quarkonium with hyperscaling violation * *Supported by the Innovation Fund of Key Laboratory of Quark and Lepton Physics (QLPL2020P01)

We study the entropic destruction of heavy quarkonium in strongly coupled theories with an anisotropic scaling symmetry in time and a spatial direction. We consider Lifshitz and hyperscaling violation theories, which are covariant under a generalized Lifshitz scaling symmetry with the dynamical exponent z and hyperscaling violation exponent . It is shown that the entropic force depends on the parameters of these theories. In particular, increasing z decreases the entropic force, thus reducing the quarkonium dissociation, while increasing has the opposite effect.

Entanglement entropy with Lifshitz fermions

We investigate fermions with Lifshitz scaling symmetry and study their entanglement entropy in 1+1 dimensions as a function of the scaling exponent z. Remarkably, in the ground state the entanglement entropy vanishes for even values of z, whereas for odd values it is independent of z and equal to the relativistic case with z=1. We show this using the correlation method on the lattice, and also using a holographic cMERA approach. The entanglement entropy in a thermal state is a more detailed function of z and T which we plot using the lattice correlation method. The dependence on the even- or oddness of z still shows for small temperatures, but is washed out for large temperatures or large values of z.

Towards Possible Laminar Channels through Turbulent Atmospheres in a Multifractal Paradigm

In this paper, developments are made towards simulating complex atmospheric behavior using turbulent energy cascade staging models developed through scale relativity theories. Such theoretical considerations imply gauges that describe atmospheric parameters as multifractal functions undertaking scale symmetry breaking at each stage of the turbulent energy cascade. It is found that gauges of higher complexity (in this case, a Riccati-type gauge) can exhibit more complex behavior accordingly, such as both dilation and contraction, but properly parameterizing the solutions formed by these gauges in terms of turbulent staging can be challenging given the multiple constants and parameters. However, it is found that a logistic-type approximation of the multifractal equations of motion that describe turbulent atmospheric entities can be coupled with a model produced by a simpler gauge, and this combination can reveal instances of laminar, or otherwise non-chaotic, behavior in a given turbulent flow at certain scales. Employing the theory with elastic lidar data, quasi-laminar behavior is found in the vicinity of the planetary boundary layer height, and laminar channels are revealed throughout an atmospheric column—these might be used to reveal complex vertical transport behavior in the atmospheric column.