Global Scale Symmetry Research Articles

Exact quantum conformal symmetry, its spontaneous breakdown, and gravitational Weyl anomaly

The classical Lagrangian of the Standard Model enjoys the symmetry of the full conformal group if the mass of the Higgs boson is put to zero. This is a hint that conformal symmetry may play a fundamental role in the ultimate theory describing Nature. The origin of scales, such as the Higgs vacuum expectation value (vev), may result from the spontaneous breakdown of the conformal symmetry by the dilaton field. In this work, we study whether this classical setup can be implemented in quantum theory and be phenomenologically viable by presenting an explicit construction where the exact conformal symmetry can be preserved and is anomaly free while being spontaneously broken. Not only the Higgs mass but also the genuine quantum scales like the QCD confinement radius are generated by the dilaton vev. We also discuss the extension of these ideas to the theories with dynamical gravity and show that the only finite subgroup of the local Weyl transformations which is anomaly free corresponds to the global scale symmetry. This means that the conformal invariance of the flat space theory is explicitly broken down to the scale symmetry by gravitational effects related to the Weyl anomaly.

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.

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.

Odd-dimensional self-duality for non-Abelian tensor-multiplet in D = 3 + 2 as master theory of integrable-systems

We present N=2 supersymmetric non-Abelian duality-symmetry between a tensor multiplet and an extra vector multiplet in D=3+2 dimensions. Our system has the Yang-Mills (YM) vector multiplet (AμI,λI), a tensor-multiplet (BμνI,χI,φI), and an extra vector multiplet (CμI,ρI,σI). The index I=1,2,⋯,dimG is for the adjoint representation of a non-Abelian group G. The AμI is the conventional YM gauge field, BμνI is a non-Abelian tensor field, while φI and σI are scalar fields. The λI,χI and ρI are Majorana fermions in the 2 of Sp(1). The BμνI and CμI-fields have their respective field-strengths defined by GμνρI≡+3D⌊⌈μBνρ⌋⌉I+3fIJKF⌊⌈μνJCρ⌋⌉K and HμνI≡+2D⌊⌈μCν⌋⌉I+mBμνI+fIJKϕJHμνK−fIJKσJFμνK. The duality relationship is HμνI=(1/6)ϵμνρστGρστI−(1/2)fIJK(λ‾JγμνχK), with its super-partner relationships: φI=−σI,χI=−ρI. Since HστI contains mBστI linearly, this is a ‘massive’ self-dual relationship. Interestingly, the closure of supersymmetries shows the intrinsic global scale symmetry: δζ(BμνI,χI,φI,CμI,ρI,σI)=+mζ(BμνI,χI,φI,CμI,ρI,σI). By certain dimensional-reduction scheme into D=2+2, we show that self-dual supersymmetric tensor multiplet is generated. We deduce that our present theory in D=3+2 can serve as the underlying ‘Master Theory’ of a similar system in D=2+2.

Weyl scaling invariant R2 gravity for inflation and dark matter

Inflation in the early universe can generate the nearly conformal invariant fluctuation that leads to the structures we observe at the present. The simple viable Starobinsky R2 inflation has an approximate global scale symmetry. We study the conformal symmetric Weyl Rˆ2 and general F(Rˆ) theories and demonstrate their equivalence to Einstein gravity coupled with a scalar and a Weyl gauge field. The scalar field in Weyl Rˆ2 gravity can be responsible for inflation with Starobinsky model as the attractor, potentially distinguishable from the latter by future experiments. The intrinsic Weyl gauge boson becomes massive once the Einstein frame is fixed, and constitutes as a dark matter candidate with mass up to ∼5×1016GeV.

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.

Spontaneous symmetry breakdown of scale symmetries

We discuss spontaneous symmetry breakdown (SSB) of both global and local scale symmetries in scalar-tensor gravity with two scalar fields, one of which couples nonminimally to scalar curvature while the other is a normal scalar field. In case of a global scale symmetry, by moving from the Jordan frame to the Einstein frame, a normal scalar field becomes massive while the dilaton remains massless after the SSB. In case of a local scale symmetry, we take a gauge fixing condition for the local scale invariance, $a R + b \phi^2 = k$, which was found in our previous study of a Weyl's quadratic gravity. Together with locally scale invariant potential terms in a classical action, this gauge condition generates a Higgs potential whose vacuum expectation value (VEV) produces the Einstein-Hilbert action with an $R^2$ term in the lowest level of approximation. One interesting aspect in this model is that a massless Nambu-Goldstone boson associated with a scale invariance is absorbed into the metric tensor field and consequently an $R^2$ term is induced in the action.

Vanishing trace anomaly in flat spacetime

Quantum scale invariant regularization is a variant of dimensional regularization where the renormalization scale is treated as a dynamical field. But, rather than be regarded as a novel regularization method on par with dimensional regularization, momentum cutoff, Pauli-Villars etc., it should be understood as a way to define a subset in the infinite space of nonrenormalizable models of certain type. The subset realizes the demand that renormalization scale, along with any other dimensionful parameters, should be interpreted as a dynamical field's homogeneous background. This restriction is most straightforwardly implemented using dimensional regularization but it can hypothetically be imposed with any regularization method. Theories that satisfy it offer a new perspective on the radiative violation of global scale symmetry associated with RGE functions. As a result of the quantum scale invariant regularization being implemented, the scale symmetry is preserved at the quantum level despite the RGE functions being non-zero, as can be inspected at the level of composite quantum operators that govern dilatation of Green functions. We analyze these statements in explicit detail using a specific but easily generalized toy model with scalar fields.

Small dark energy and stable vacuum from Dilaton\u2013Gauss\u2013Bonnet coupling in TMT

In two measures theories (TMT), in addition to the Riemannian measure of integration, being the square root of the determinant of the metric, we introduce a metric-independent density Phi in four dimensions defined in terms of scalars varphi _a by Phi =varepsilon ^{mu nu rho sigma } varepsilon _{abcd} (partial _{mu }varphi _a)(partial _{nu }varphi _b) (partial _{rho }varphi _c) (partial _{sigma }varphi _d). With the help of a dilaton field phi we construct theories that are globally scale invariant. In particular, by introducing couplings of the dilaton phi to the Gauss–Bonnet (GB) topological density , {sqrt{-g}} , phi left( R_{mu nu rho sigma }^2 - 4 R_{mu nu }^2 + R^2 right) , we obtain a theory that is scale invariant up to a total divergence. Integration of the varphi _a field equation leads to an integration constant that breaks the global scale symmetry. We discuss the stabilizing effects of the coupling of the dilaton to the GB-topological density on the vacua with a very small cosmological constant and the resolution of the ‘TMT Vacuum-Manifold Problem’ which exists in the zero cosmological-constant vacuum limit. This problem generically arises from an effective potential that is a perfect square, and it gives rise to a vacuum manifold instead of a unique vacuum solution in the presence of many different scalars, like the dilaton, the Higgs, etc. In the non-zero cosmological-constant case this problem disappears. Furthermore, the GB coupling to the dilaton eliminates flat directions in the effective potential, and it totally lifts the vacuum-manifold degeneracy.

Fake conformal symmetry in unimodular gravity

We study Weyl symmetry (local conformal symmetry) in unimodular gravity. It is shown that the Noether currents for both Weyl symmetry and global scale symmetry, identically vanish as in the conformally invariant scalar-tensor gravity. We clearly explain why in the class of conformally invariant gravitational theories, the Noether currents vanish by starting with the conformally invariant scalar-tensor gravity. Moreover, we comment on both classical and quantum-mechanical equivalences among Einstein's general relativity, the conformally invariant scalar-tensor gravity and the Weyl-transverse (WTDiff) gravity. Finally, we discuss the Weyl current in the conformally invariant scalar action and see that it is also vanishing.

Global scaling symmetry, Noether charge, and universality of shear viscosity

Recently, it was established in Einstein-Maxwell-Dilaton gravity that the Kovtun-Son-Starinets viscosity/entropy ratio associated with anti-de Sitter planar black holes can be viewed as the boundary dual to the generalized Smarr relation of the black holes in the bulk. In this paper, we establish this relation in Einstein gravity with general minimally coupled matter and also in theories with an additional nonminimally coupled scalar field. We consider two examples for explicit demonstrations.

Chiral Scale and Conformal Invariance in 2D Quantum Field Theory

It is well known that a local, unitary Poincaré-invariant 2D quantum field theory with a global scaling symmetry and a discrete non-negative spectrum of scaling dimensions necessarily has both a left and a right local conformal symmetry. In this Letter, we consider a chiral situation beginning with only a left global scaling symmetry and do not assume Lorentz invariance. We find that a left conformal symmetry is still implied, while right translations are enhanced either to a right conformal symmetry or a left U(1) Kac-Moody symmetry.

The very-extended trombone

Starting from the very-extended Kac-Moody algebra $E_{11}$, we consider the algebra $E_{11,D}^{local}$, obtained by adding to the non-negative level $E_{11}$ generators the $D$-dimensional momentum operator and an infinite set of additional generators that promote the global $E_{11}$ symmetries to gauge ones. We determine all the possible trombone deformations of this algebra, that is the deformations that involve the $D$-dimensional scaling operator. The Jacobi identities imply that such deformations are uniquely determined by a single tensor belonging to the same representation of the internal symmetry group as the vector generators and satisfying additional quadratic constraints. The non-linear realisation of the deformed algebra gives the field strengths of the theory which are those of any possible maximal supergravity theory in which the global scaling symmetry is gauged in any dimension. All the possible deformed algebras are in one to one correspondence with all such maximal supergravity theories. The tensor that parametrises the deformation is identified with the embedding tensor that is used to parametrise all maximal supergravity theories with gauged scaling symmetry, and the quadratic constraints that we determine exactly coincide with the field theory results.

Supergravities without an action: Gauging the trombone

We present a systematic account of supergravity theories in which the global scaling symmetry is gauged. This generalizes the standard gaugings of non-abelian off-shell symmetries. A particular feature of these theories is an additional positive contribution to the effective cosmological constant. As the scaling symmetry is an on-shell symmetry, the resulting gaugings do no longer possess an action. We develop the algebraic framework for the maximal theories in various dimensions and construct explicit solutions to the algebraic consistency constraints — related to “pure-spinor-like” structures for the exceptional groups. As an example, we explicitly work out the modified supersymmetry transformation rules and equations of motion in three dimensions. Finally, we speculate about the role of these theories from the perspective of very extended Kac–Moody algebras.

Scale symmetry spontaneously broken by asymptotic behaviour

Conserved quantities are obtained and analysed using new recently proposed models with global scale invariance. Such models allow for non-trivial scalar field potentials and masses for particles, so that the scale symmetry must somehow be broken. We obtain this conclusion by showing that the infrared behaviour of the conserved currents is singular so that there are no conserved charges associated with the global scale symmetry. Nevertheless, the scale symmetry plays a crucial role in determining the structure of the theory and it implies that in some high-field regions the potentials become flat.

Scale invariance and inflation

Solving a scale invariant effective theory, we obtain an inflationary solution which is equivalent to an asymptotic boundary condition that has been imposed previously by hand on the scalar φ, namely, φ( r →∞)→ v. This inflationary solution is unique for a slow roller φ. Possible implications are also discussed. This result signifies the importance of the global scale symmetry.