Recently, it has been pointed out that dimensionless actions in four dimensional curved spacetime possess a symmetry which goes beyond scale invariance but is smaller than full Weyl invariance. This symmetry was dubbed {\it restricted Weyl invariance}. We show that starting with a restricted Weyl invariant action that includes a Higgs sector with no explicit mass, one can generate the Einstein-Hilbert action with cosmological constant and a Higgs mass. The model also contains an extra massless scalar field which couples to the Higgs field (and gravity). If the coupling of this extra scalar field to the Higgs field is negligibly small, this fixes the coefficient of the nonminimal coupling $R Φ^2$ between the Higgs field and gravity. Besides the Higgs sector, all the other fields of the standard model can be incorporated into the original restricted Weyl invariant action.
We discuss the physics of {\it restricted Weyl invariance}, a symmetry of dimensionless actions in four dimensional curved space time. When we study a scalar field nonminimally coupled to gravity with Weyl(conformal) weight of $-1$ (i.e. scalar field with the usual two-derivative kinetic term), we find that dimensionless terms are either fully Weyl invariant or are Weyl invariant if the conformal factor $Ω(x)$ obeys the condition $g^{μν}\nabla_μ\nabla_νΩ=0$. We refer to the latter as {\it restricted Weyl invariance}. We show that all the dimensionless geometric terms such as $R^2$, $R_{μν}R^{μ…
We consider matter fields conformally coupled to a background metric and dilaton and describe in detail a quantization procedure and related renormalization group flow that preserve Weyl invariance. Even though the resulting effective action is Weyl-invariant, the trace anomaly is still present, with all its physical consequences. We discuss first the case of free matter and then extend the result to interacting matter. We also consider the case when the metric and dilaton are dynamical and gravitons enter in the loops.
We obtain scale-invariant scalar and tensor power spectra from a Weyl-invariant scalar-tensor theory in de Sitter spacetime. This implies that the Weyl-invariance guarantees to implement the scale-invariance of power spectrum in de Sitter spacetime. We establish a deep connection between the Weyl-invariance of the action and scale-invariance of power spectrum in de Sitter spacetime.
We consider a non-relativistic free scalar field theory with a type of anisotropic scale invariance in which the number of coordinates "scaling like time" is generically greater than one. We propose the Cartesian product of two curved spaces, the metric of each space being parameterized by the other space, as a notion of curved background to which the theory can be extended. We study this type of geometries, and find a family of extensions of the theory to curved backgrounds in which the anisotropic scale invariance is promoted to a local, Weyl-type symmetry.
The generalization of scale invariance when gravitational effects are considered is Weyl invariance, namely, invariance under (global or local) rescalings of the metric. In this work, we discuss in some details the implications of the fact that the value of the anomaly for the global Weyl invariant coupling of scalar fields to gravity is sensitive to the dynamics (or absence thereof) of the gravitational field.
The most general lagrangian describing spin 2 particles in flat spacetime and containing operators up to (mass) dimension 6 is carefully analyzed, determining the precise conditions for it to be invariant under linearized (transverse) diffeomorphisms, linearized Weyl rescalings, and conformal transformations.
We examine some Weyl invariant cosmological models in the framework of generalized dilaton gravity, in which the action is made of a set of $N$ conformally coupled scalar fields. It will be shown that when the FRW ansatz for the spacetime metric is assumed, the Ward identity for conformal invariance guarantees that the gravitational equations hold whenever the scalar fields EM do so. It follows that any scale factor can solve the theory provided a non-trivial profile for a dilaton field. In particular, accelerated expansion is a natural solution to the full set of equations.
A theory with a mass scale and yet Weyl invariant is presented. The theory is not invariant under all diffeomorphisms but only under transverse ones. This is the reason why Weyl invariance does not imply scale invariance in a free falling frame. Physical implications of this framework are discussed.
We discuss the physics of a restricted Weyl symmetry in a curved space-time where a gauge parameter $Ω(x)$ of Weyl transformation satisfies a constraint $\Box Ω= 0$. First, we present a model of QED where we have a restricted gauge symmetry in the sense that a $U(1)$ gauge parameter $θ(x)$ obeys a similar constraint $\Box θ= 0$ 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…
It was argued recently that conformal invariance in flat spacetime implies Weyl invariance in a general curved background for unitary theories and possible anomalies in the Weyl variation of scalar operators are identified. We argue that generically unitarity alone is not sufficient for a conformal field theory to be Weyl invariant. Furthermore, we show explicitly that when a unitary conformal field theory couples to gravity in a Weyl invariant way, each primary scalar operator that is either relevant or marginal in the unitary conformal field theory corresponds to a Weyl-covariant operator in…
We show how Einstein-Cartan gravity can accommodate both global scale and local scale (Weyl) invariance. To this end, we construct a wide class of models with nonpropagaing torsion and a nonminimally coupled scalar field. In phenomenological applications the scalar field is associated with the Higgs boson. For global scale invariance, an additional field -- dilaton -- is needed to make the theory phenomenologically viable. In the case of the Weyl symmetry, the dilaton is spurious and the theory reduces to a sub-class of one-field models. In both scenarios of scale invariance, we derive an equi…
Equations for cosmological evolution are formulated in a Weyl invariant formalism to take into account possible Weyl anomalies. Near two dimensions, the renormalized cosmological term leads to a nonlocal energy-momentum tensor and a slowly decaying vacuum energy. A natural generalization to four dimensions implies a quantum modification of Einstein field equations at long distances. It offers a new perspective on time-dependence of couplings and naturalness with potentially far-reaching consequences for the cosmological constant problem, inflation, and dark energy.
We investigate the relationship between quadratic gravity and a restricted Weyl symmetry where a gauge parameter $Ω(x)$ of Weyl transformation satisfies a constraint $\Box Ω= 0$ in a curved space-time. First, we briefly review a model with a restricted gauge symmetry on the basis of QED where a $U(1)$ gauge parameter $θ(x)$ obeys a similar constraint $\Box θ= 0$ in a flat Minkowski space-time, and explain that the restricted gauge symmetry removes one on-shell mode of gauge field, which together with the Feynman gauge leaves only two transverse polarizations as physical states. Next, it is sho…
The present work deals with two different but subtilely related kinds of conformal mappings: Weyl rescaling in $d>2$ dimensional spaces and SO(2,d) transformations. We express how the difference between the two can be compensated by diffeomorphic transformations. This is well known in the framework of String Theory but in the particular case of $d=2$ spaces. Indeed, the Polyakov formalism describes world-sheets in terms of two-dimensional conformal field theory. On the other hand, B. Zumino had shown that a classical four-dimensional Weyl-invariant field theory restricted to live in Minkowski …
We examine the local conformal invariance (Weyl invariance) in tensor-scalar theories used in recently proposed conformal cosmological models. We show that the Noether currents associated with Weyl invariance in these theories vanish. We assert that the corresponding Weyl symmetry does not have any dynamical role.
Pure $R^2$ gravity was considered originally to possess only global scale symmetry. It was later shown to have the larger restricted Weyl symmetry where it is invariant under the Weyl transformation $g_{μν} \to Ω^2(x)\, g_{μν}$ when the conformal factor $Ω(x)$ obeys the harmonic condition $\Box Ω(x)=0$. Restricted Weyl symmetry has an analog in gauge theory. Under a gauge transformation $A_μ\to A_μ + \frac{1}{e}\partial_μ f(x)$, the gauge-fixing term $(\partial_μA^μ)^2$ has a residual gauge symmetry when $\Box f=0$. In this paper, we consider scenarios where the symmetry of pure $R^2$ gravity …
We argue that conformal invariance in flat spacetime implies Weyl invariance in a general curved background metric for all unitary theories in spacetime dimensions $d \leq 10$. We also study possible curvature corrections to the Weyl transformations of operators, and show that these are absent for operators of sufficiently low dimensionality and spin. We find possible `anomalous' Weyl transformations proportional to the Weyl (Cotton) tensor for $d > 3$ ($d = 3$). The arguments are based on algebraic consistency conditions similar to the Wess-Zumino consistency conditions that classify possible…
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 non-zero value. It is then shown that this spontaneous symmetry breakdown induces spontaneous symmetry breakdown of special conformal symmetry in a flat Minkowski space-time, but the resultant Nambu-Goldstone boson is not an independent physical mode but expressed in terms of th…
It is shown that in the quadratic gravity based on Weyl's conformal geometry, the Planck mass scale can be generated from quantum effects of the gravitational field and the Weyl gauge field via the Coleman-Weinberg mechanism where a local scale symmetry, that is, conformal symmetry, is broken. At the same time, the Weyl gauge field acquires mass less than the Planck mass by absorbing the dilaton. The shape of the effective potential is almost flat owing to a gravitational character and high symmetries, so our model would provide an attractive model for the inflationary universe. We also presen…
We revisit Weyl invariance of string theories in generalized supergravity backgrounds. In the previous work arXiv:1703.09213, a possible counterterm was constructed, but it seems to be a point of controversy in some literatures whether it is non-local or not. To settle down this issue, we show that the counterterm may be local and exactly cancels out the one-loop trace anomaly in generalized supergravity backgrounds.
Weyl invariant theories of scalars and gravity can generate all mass scales spontaneously, initiated by a dynamical process of "inertial spontaneous symmetry breaking" that does not involve a potential. This is dictated by the structure of the Weyl current, $K_μ$, and a cosmological phase during which the universe expands and the Einstein-Hilbert effective action is formed. Maintaining exact Weyl invariance in the renormalised quantum theory is straightforward when renormalisation conditions are referred back to the VEV's of fields in the action of the theory, which implies a conserved Weyl cu…
It is shown that the explicit calculation of the Wess-Zumino functional pertaining to the breaking term of the Weyl symmetry for the Einstein-Hilbert action allows to restore the Weyl symmetry by introducing the extra dilaton field as Goldstone field. Adding the Wess-Zumino counter-term to the Einstein-Hilbert action reproduces the usual Weyl invariant action used in standard literature. Further consideration might confer to the Einstein-Hilbert action a new status.
We consider a globally neutral Lorentzian plasma as a possible remnant of a preinflationary stage of expansion and pose the problem of the suitable initial conditions for the evolution of the large-scale electromagnetic inhomogeneities. During the protoinflationary regime the Weyl invariance of the Ohmic current guarantees that the comoving conductivity is approximately constant. The subsequent breaking of Weyl invariance by the masses of the charge carriers drives the conductivity to zero. The newly derived conducting initial conditions for the amplification of large-scale magnetic fields are…
We consider conformally invariant form of the actions in Einstein, Weyl, Einstein-Cartan and Einstein-Cartan-Weyl space in general dimensions($>2$) and investigate the relations among them. In Weyl space, the observational consistency condition for the vector field determining non-metricity of the connection can be obtained from the equation of motion. In Einstein-Cartan space a similar role is played by the vector part of the torsion tensor. We consider the case where the trace part of the torsion is the Kalb-Ramond type of field. In this case, we express conformally invariant action in terms…
In the first part, we discuss the interplay between local scale invariance and metric-affine degrees of freedom from few distinct points of view. We argue, rather generally, that the gauging of Weyl symmetry is a natural byproduct of requiring that scale invariance is a symmetry of a gravitational theory that is based on a metric and on an independent affine structure degrees of freedom. In the second part, we compute the Nöther identities associated with all the gauge symmetries, including Weyl, Lorentz and diffeomorphisms invariances, for general actions with matter degrees of freedom, explo…
In this work, we investigate a theory of linear Weyl gravity coupled to a scalar field and study the scenario in which Lorentz symmetry is broken by a non-vanishing vacuum expectation value of the Weyl field in the flat space limit after Weyl symmetry breaking. We show that a $CPT$-odd Lorentz-violating interaction is generated after symmetry breaking. Features of different symmetry-broken phases and their dependence on the spacetime character of the generated Lorentz-violating background are discussed. Also, we analyze the naturalness of the theory by showing that the light mass scale is prot…
We show that in a quadratic gravity based on Weyl's conformal geometry, the Planck mass scale can be generated from quantum effects of the gravitational field and the Weyl gauge field via the Coleman-Weinberg mechanism where a local scale symmetry is broken. At the same time, the Weyl gauge field acquires a mass less than the Planck mass by absorbing the scalar graviton. The shape of the effective potential is almost flat owing to a gravitational character and high symmetries, so our model would provide for an attractive model for the inflationary universe. We also present a toy model showing …
In the theories of gravity with non-minimally coupled scalar fields there are "mavericks" -- unexpected solutions with odd properties, e.g., black holes with scalar hair in theories with scalar potential bounded from below. Probably the most famous example is Bocharova-Bronnikov-Melnikov-Bekenstein (BBMB) black hole solution in a theory with a scalar field conformally coupled to the gravity and with vanishing potential. Its existence naively violates no-hair conjecture without violating no-hair theorems because of the singular behavior of the scalar field at the horizon. Despite being discover…
We consider a simple but generic model of gravity where Weyl--invariance is realized thanks to the presence of a gauge field for dilatations. We quantize the theory by suitably defining renormalization group flows that describe the integration of successive momentum shells, in such a way that Weyl--invariance is maintained in the flow. When the gauge fields are massless the theory has, in addition to Weyl invariance, an abelian gauge symmetry. According to the definition of the cutoff, the flow can break or preserve this extended symmetry. We discuss the fixed points of these flows.
We analyze gravitational theories with quadratic curvature terms, including the case of conformally invariant Weyl gravity, motivated by the intention to find a renormalizable theory of gravity in the ultraviolet region, yet yielding general relativity at long distances. In the Hamiltonian formulation of Weyl gravity, the number of local constraints is equal to the number of unstable directions in phase space, which in principle could be sufficient for eliminating the unstable degrees of freedom in the full nonlinear theory. All the other theories of quadratic type are unstable -- a problem ap…
In a new Weyl 2-spinor approach to Non abelian Gauge Theories, starting with the local U(1) Gauge group of a previous work, we study now the SU(3) case corresponding to quarks (antiquarks) interacting with color fields. The principal difference with the conventional approach is that particle-field interactions are not described by means of potentials but by the field strength magnitudes. Some analytical expressions showing similarities with electrodynamics are obtained. Classical equations that describe the behavior of quarks under gluon fields might be in principle applied to the quark-gluon …
It has recently been shown that a set of the generalized type IIB supergravity equations follows from the requirement of kappa symmetry of the type IIB Green-Schwarz superstring theory defined on an arbitrary background. In this paper, we show that the whole bosonic part of the generalized type II supergravity equations can be reproduced from the T-duality covariant equations of motion of the double field theory by choosing a non-standard solution of the strong constraint. Then, by using the doubled formalism, we show the Weyl invariance of the bosonic string sigma model on a generalized gravi…
The pure $R^2$ gravity is equivalent to Einstein gravity with cosmological constant and a massless scalar field and it further possesses the so-called restricted Weyl symmetry which is a symmetry larger than scale symmetry. To incorporate matter, we consider a restricted Weyl invariant action composed of pure $R^2$ gravity, SU(2) Yang-Mills fields and a non-minimally coupled massless Higgs field (a triplet of scalars). When the restricted Weyl symmetry is spontaneously broken, it is equivalent to an Einstein-Yang-Mills-Higgs (EYMH) action with a cosmological constant and a massive Higgs non-mi…
We revisit a particular ghost-free bimetric model which is related to both partial masslessness (PM) and conformal gravity. Linearly, the model propagates six instead of seven degrees of freedom not only around de Sitter but also around flat spacetime. Nonlinearly, the equations of motion can be recast in the form of expansions in powers of curvatures, and exhibit a remarkable amount of structure. In this form, the equations are shown to be invariant under scalar gauge transformations, at least up to six orders in derivatives, the lowest order term being a Weyl scaling of the metrics. The term…
The fundamental string length, which is an essential part of string theory, explicitly breaks scale invariance. However, in field theory we demonstrated recently that the gravitational constant, which is directly related to the string length, can be promoted to a dynamical field if the standard model coupled to gravity (SM+GR) is lifted to a locally scale (Weyl) invariant theory. The higher gauge symmetry reveals previously unknown field patches whose inclusion turn the classically conformally invariant SM+GR into a geodesically complete theory with new cosmological and possibly further physic…
It is a well-known property of holographic theories that diffeomorphism invariance in the bulk space-time implies Weyl invariance of the dual holographic field theory in the sense that the field theory couples to a conformal class of background metrics. The usual Fefferman-Graham formalism, which provides us with a holographic dictionary between the two theories, breaks explicitly this symmetry by choosing a specific boundary metric and a corresponding specific metric ansatz in the bulk. In this paper, we show that a simple extension of the Fefferman-Graham formalism allows us to sidestep this…
We comment on some peculiarities of matter with and without Weyl invariance coupled to classical $2d$ Einstein-Hilbert gravity for several models, in particular, related to the counting of degrees of freedom and on the dynamics. We find that theories where the matter action is Weyl invariant has generically more degrees of freedom than action without the invariance. This follows from the Weyl invariance of the metric equations of motion independently of the invariance of the action. Then, we study another set of models with scalar fields and show that solutions to the equations of motion are e…
Here we show that local scale invariance -- invariance under Weyl rescalings -- may safely coexist with broken electroweak symmetry if assume the Weyl geometric theory to govern the affine structure of spacetime. We find that within the resulting scale invariant theory of gravity cosmological inflation is enough to explain the large hierarchy between the Higgs and Planck masses.
We study a system of $n$ Abelian vector fields coupled to $\frac 12 n(n+1)$ complex scalars parametrising the Hermitian symmetric space $\mathsf{Sp}(2n, {\mathbb R})/ \mathsf{U}(n)$. This model is Weyl invariant and possesses the maximal non-compact duality group $\mathsf{Sp}(2n, {\mathbb R})$. Although both symmetries are anomalous in the quantum theory, they should be respected by the logarithmic divergent term (the ``induced action'') of the effective action obtained by integrating out the vector fields. We compute this induced action and demonstrate its Weyl and $\mathsf{Sp}(2n, {\mathbb R…
It is well-known that the beta functions on a string worldsheet correspond to the target space equations of motion, e.g. the Einstein equations. We show that the BPS equations, i.e. the conditions of vanishing supersymmetry variations of the space-time fermions, can be directly derived from the worldsheet. To this end we consider the RNS-formulation of the heterotic string with (2,0) supersymmetry, which describes a complex torsion target space that supports a holomorphic vector bundle. After a detailed account of its quantization and renormalization, we establish that the cancellation of the …
We revisit Weyl's metrication (geometrization) of electromagnetism. We show that by making Weyl's proposed geometric connection be pure imaginary, not only are we able to metricate electromagnetism, an underlying local conformal invariance makes the geometry be strictly Riemannian and prevents observational gravity from being complex. Via torsion we achieve an analogous metrication for axial-vector fields. We generalize our procedure to Yang-Mills theories, and achieve a metrication of all the fundamental forces. Only in the gravity sector does our approach differ from the standard picture of …
