Weyl conformal geometry vs Weyl anomaly

Weyl conformal geometry is a gauge theory of scale invariance that naturally brings together the Standard Model (SM) and Einstein gravity. The SM embedding in this geometry is possible without new degrees of freedom beyond SM and Weyl geometry, while Einstein gravity is generated by the broken phase of this symmetry. This follows a Stueckelberg breaking mechanism in which the Weyl gauge boson becomes massive and decouples, as discussed in the past [1–3]. However, Weyl anomaly could break explicitly this gauge symmetry, hence we study it in Weyl geometry. We first note that in Weyl geometry metricity can be restored with respect to a new differential operator (∇̂\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hat{\nabla} $$\end{document}) that also enforces simultaneously a Weyl-covariant formulation. This leads to a metric-like Weyl gauge invariant formalism that enables one to do quantum calculations directly in Weyl geometry, rather than use a Riemannian (metric) geometry picture. The result is the Weyl-covariance in d dimensions of all geometric operators (R̂\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hat{R} $$\end{document}, etc) and of their derivatives (∇̂\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hat{\nabla} $$\end{document}μR̂\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hat{R} $$\end{document}, etc), including the Euler-Gauss-Bonnet term. A natural (geometric) Weyl-invariant dimensional regularisation of quantum corrections exists and Weyl gauge symmetry is then maintained and manifest at the quantum level. This is related to a non-trivial current of this symmetry, the divergence of which cancels the trace of the energy-momentum tensor. The “usual” Weyl anomaly and Riemannian geometry are recovered in the (spontaneously) broken phase. The relation to holographic Weyl anomaly is discussed.


Motivation
Scale symmetry in its various forms (global, local/gauged) may play a role in physics beyond the Standard Model (SM).For example at high energies e.g. in the early Universe, the SM states are essentially massless and the theory has such symmetry.Moreover, the SM with the Higgs mass parameter set to zero is scale invariant [1].In this work we consider a local, gauged scale symmetry, also known as Weyl gauge symmetry.While it does not have such symmetry, Einstein gravity can actually be a spontaneously broken phase of this symmetry in a more fundamental theory.One such theory is the original Weyl quadratic gravity which is a gauge theory of scale invariance -a symmetry inherited from its underlying Weyl conformal geometry [2][3][4].Using the gauge principle, Weyl geometry then brings together SM and Einstein gravity in a gauge theory of scale invariance, as outlined below.
In Weyl geometry, both the Weyl connection and the spin connection have a Weyl gauge symmetry.Hence, this geometry is a well-suited framework to study this symmetry.Then in theories with this symmetry both the action and its underlying geometry i.e. its associated gravity, naturally share this symmetry.This is important since ultimately geometry "is" physics and in curved space-time one cannot really "separate" the action from the geometry 1 .This geometry generates the Weyl quadratic gravity theory [2][3][4].
From this theory, Einstein gravity and a small positive cosmological constant are naturally obtained [5] in the broken phase of this symmetry after a Stueckelberg mechanism: the Weyl gauge boson (ω µ ) of scale invariance becomes a massive Proca field2 , after "eating" the dilaton propagated by the R2 term of Weyl quadratic gravity [5], and then decouples.Further, one can show that the SM (with higgs mass parameter set to zero) has a natural, truly minimal embedding in Weyl geometry [6,7] with no new states beyond SM and Weyl geometry.This gives an interesting UV completion of SM and Einstein gravity in a gauge theory of scale invariance.In such case only the Higgs field of SM acquires a tree-level coupling to ω µ and may be generated in the early Universe by Weyl boson fusion ω µ ω µ → hh [6].With ω µ of geometric origin, one can explain the origin of mass by Weyl geometry [9].Successful inflation is obtained [12][13][14] giving a gauged version of Starobinsky inflation [15].
What happens at quantum level?With scale symmetry a (quantum) gauge symmetry, an immediate question arises.It is well-known that Weyl symmetry is anomalous [16][17][18][19], so one should address how Weyl anomaly is reconciled with a gauged scale symmetry.This issue arises partly because quantum corrections (more correctly, their regularisation) do not respect this classical symmetry.For example, in dimensional regularisation (DR) the analytic continuation to d = 4 − 2ǫ dimensions breaks explicitly this symmetry.Weyl invariant regularisations could address this matter [20], but Weyl anomaly is more than a regularisation issue: it involves the Euler-Gauss-Bonnet term which is not Weyl-covariant in d dimensions.Also our Weyl gauge symmetry has a current -what is its role?these are the right questions to ask, to answer how Weyl gauge symmetry can avoid Weyl anomaly.
This brings us to another motivation for this study.In an interesting work [21,22] in a flat spacetime approach it was shown that the only anomaly-free Weyl symmetry in the presence of dynamical gravity is global scale invariance.This result uses an effective action of the dilaton in a flat space-time limit that arises from general quadratic curvature actions.So can we avoid this no-go theorem?The reason this result does not apply here is that we have a different symmetry, current and geometry -Weyl geometry (WG) and then: a) the dilaton is absent at low scales, having been "eaten" by the Weyl-Proca gauge field to all orders; b) there is a Weyl gauge symmetry which has a non-trivial current, the divergence of which cancels the energy-momentum tensor trace T µ µ ; c) Very important, in WG the Euler-Gauss-Bonnet operator is Weyl-covariant in d dimensions; e) Finally, models in WG have a de Sitter ground state, with a small cosmological constant Λ = 3H 2 0 > 0 [6] preventing an exact flat spacetime limit approach.We detail these results in this work.
Finally, another motivation for an interest in Weyl anomaly in Weyl geometry comes from a holographic perspective of AdS/CFT.In recent [23,24] it was shown that Weyl geometry (in d dimensions) is induced on the conformal boundary of a d + 1 asymptotically locally anti-de Sitter (AlAdS) spacetime; the local Weyl symmetry and Weyl geometry are induced by diffeomorphism invariance in the bulk spacetime when working in the Weyl-Fefferman-Graham (WFG) gauge [23,24].Unlike in a Fefferman-Graham (FG) gauge where a bulk Levi-Civita (LC) connection induces on the conformal boundary also a LC connection (of the boundary metric), in the WFG gauge the bulk LC connection induces on the boundary the Weyl-covariant geometry.The induced metric and Weyl connection act as non-dynamical backgrounds of the dual quantum field theory 3 .Nevertheless, the Weyl connection makes the geometric quantities on the boundary Weyl-covariant.As a result, Weyl anomaly in the 4D boundary in the WFG gauge (while of similar form to that in FG gauge) has now become Weyl-covariant.We recover this result, but we have in addition a dynamical connection and non-trivial current (as mentioned), relevant for the anomaly.
These motivations and the good results so far of Weyl gauge invariance as a symmetry and gauge principle beyond SM and Einstein gravity, justify this study in Weyl geometry.
The plan of the paper is as follows: section 2 reviews Weyl anomaly and regularisations that respect Weyl gauge symmetry.New results are shown in Section 3: we first note that Weyl geometry can be made metric with respect to a new differential operator ( ∇), in which case geometric operators (curvature tensors/scalar) and their derivatives in d dimensions are Weyl-covariant (this includes the Euler-Gauss-Bonnet operator).This enables a metric-like geometry similar to Riemannian geometry, that preserves Weyl gauge invariance of the action; it allows quantum calculations in Weyl geometry that would otherwise require one use a Riemannian picture, as usual.With these ideas and symmetry we show by construction, how Weyl gauge symmetry is maintained at the quantum level in Weyl geometry-based models and how Weyl anomaly is generated by the (spontaneously) broken phase.Conclusions are in Section 4 and the Appendix gives additional technical details.

Weyl anomaly and beyond
Let us first review the Weyl anomaly [16][17][18][19].By Weyl symmetry we mean the invariance of the action under a transformation of the metric g µν and, if present, of its scalar(s) φ and fermion(s) ψ, as shown below in d dimensions Here q is the Weyl charge of the metric.Its normalization is arbitrary and conventions for q vary4 , e.g.q = 1 in [5,25] or q = 2 in [26]; if q = 2 and d = 4 we have q φ = −1, q ψ = −3/2 i.e. the fields' dimension in length units.For simplicity, one can set below q = 2. Following [16], consider a Weyl invariant action of some (unspecified) massless matter fields, in interaction with an external gravitational field and an external spin-1 gauge field.Assuming the absence of a self-interaction of these matter fields, one obtains at one-loop a gravitational effective action induced by (divergent) quantum contributions from them.In dimensional regularisation (DR) this part of the action has a structure [16] 5 where g = | det g µν | and A(d) is a function of the metric and its derivatives.A(d) contains higher derivative operators such as C αβγδ , etc, in a Weyl invariant combination, so the one-loop divergent W d is Weyl invariant [28].Here we used = ∇ µ ∇ µ in Riemannian notation.
The counterterm action W c can be written in a basis of independent operators as below, depends on the DR subtraction scale µ and has the general structure [16] where b, b ′ , c are constants and E 4 is a total derivative but G is not -this is the Euler-Gauss-Bonnet term; C 2 µνρσ is the Weyl tensor-squared in d dimensions; F µν is a field strength of the Abelian (external) vector field.W c is chosen such as its pole 1/(d − 4) cancels that in W d so the total action is finite.
The trace of the energy-momentum tensor given by W d + W c is where we used that W d is Weyl invariant.Further [16] 2 For convenience let us outline here a derivation of (6) [28].With notation (4) we have . Next, for a functional A(g ′ µν ) under transformation (1) For an infinitesimal transformation (1) (Σ → 1, g ′ µν → g µν ) and with )K which was used in (6).This misses the R term due to an ambiguity in the local conformal case [16,29], but it is easily accounted for, see e.g.[28] (section 17.2.2).
Using ( 6), ( 7), (8) in the expression for T µ µ we obtain a finite correction: the pole in W c is cancelled by the factor (d − 4) in these equations; one then takes the limit d → 4, then Since T µ µ = 0, Weyl symmetry is anomalous and a Weyl invariant quantum theory does not seem possible: the symmetry was broken explicitly by the DR scheme in (3) leading to the (finite) correction generated in T µ µ (the coefficient of R can be altered if an R 2 term exists in initial action, whose variation with respect to g µν will induce R; here no such initial R 2 term was assumed, as it would have broken initial Weyl symmetry).The coefficients b, b ′ depend on the matter fields only.For N s scalars, N f fermions and An attempt to avoid the anomaly due to the explicit breaking of Weyl symmetry by the DR scheme and to realise a quantum conformal gravity (with a spontaneous-only breaking of this symmetry) was first made in [20].The authors of [20] considered (massless) QED corrections to conformal gravity.It was shown that one could avoid the anomaly (of type B [31]) associated with the Weyl term d 4 x √ g C 2 µνρσ .This was possible with an analytical continuation different from the DR scheme used in (3), in order to preserve Weyl symmetry in d = 4 − 2ǫ.This is done with the aid of the scalar field (dilaton) φ with a Weyl invariant action in d dimensions φ transforms as in (1), so ln φ → ln φ + q φ ln Σ i.e. ln φ has a shift symmetry.When φ has a (non-zero) constant vev, Einstein gravity is recovered.The field φ is used to replace the subtraction scale µ in the counterterm W c shown below, to maintain Weyl symmetry [20] Then the simple pole in W d of ( 2) is cancelled as before by W c , while µ is generated spontaneously when the dilaton acquires a vev, µ ∼ φ .In this way W c respects symmetry (1), so the variation of W c is now δW c /δ ln Σ = 0, hence there is no contribution from C 2 µνρσ to (10) and there is no anomaly from ( 12): essentially, a mixing of φ with the (external) graviton (see figs. 4 and 6 in [20]) leads to Weyl anomaly cancellation.This result led to various interpretations [16].However, the absence of its associated anomaly here is nothing magic: the theory has an additional dynamical degree of freedom (dilaton φ).Its decoupling restores the anomaly contribution to T µ µ 6 .Hence, Weyl anomaly as we know it merely signals the missing (decoupling) of an additional degree of freedom that would otherwise enable the symmetry at quantum level.
Let us explain in a different way this absence of the anomaly of the Weyl term alone in this Weyl-invariant regularisation, using non-local form factors (ln ) to which it is related [28,30].In the usual approach with the DR scale µ (instead of a dilaton) as a regulator, the renormalised action W r = W d + W c for the Weyl term alone [28] is shown below, where ln (ln µ 2 ) arise from W d (W c ), respectively, see eqs.( 2), (3)7,8 [28, 30] Under (1), ; here f (ln Σ) depends on the derivatives and is neglected for an infinitesimal variation in (1), when g µν → g ′ µν (Σ → 1), used below.Next, for a functional A(g ′ µν ) we have eq.(9), then This is exactly the anomaly due to C 2 µνρσ of (10), re-derived using non-local ln term [28].Alternatively, a Weyl invariant regularisation eq.( 12) generates in d = 4 a finite Then in eq.( 13) the factor ln( /µ 2 ) is replaced by ln( /φ 2 ) which is invariant under an infinitesimal transformation (1) in d = 4, then so is the new W r , hence there is no anomaly (in this symmetric phase).This conclusion is reached using the ln form factor, enforcing the similar conclusion in the text after eq.( 12).The anomaly re-appears only after spontaneous breaking of Weyl symmetry of quantum W r : indeed, with φ = φ + δφ, Taylor expanding ln φ about φ = µ and after neglecting (decoupling) the series of dilaton fluctuations δφ suppressed by φ , one recovers eq.( 13) and the anomaly.
More recently, the above Weyl-invariant regularisation [20] was rediscovered and implemented in (global) scale invariant and conformal theories in flat space-time [32,33] to obtain quantum theories with this symmetry broken only spontaneously.As said, the subtraction scale is replaced by the dilaton field and all the counterterms can then respect the classical symmetry, as shown in detail at one-loop [32,34], two-loop [35] and three loops [36,37].The absence of a scale anomaly in the quantum action is then due to the additional presence of a dynamical degree of freedom in the theory (the dilaton).When this field acquires a vev and decouples (negligible fluctuations relative to its vev) the anomaly is recovered in the broken phase only.Although the scale anomaly vanishes, that does not necessarily mean that beta functions of the couplings vanish9 ; beta functions are now defined with respect to the rescaling of the dilaton; with this definition one can check that Callan-Symanzik equations are respected at 2-loop [34,35] together with Ward identities [38].
Returning to the symmetry preserving regularisation [20] for C 2 µνρσ , the most general action can contain additional terms such as the Euler-Gauss-Bonnet term G in which case the Weyl anomaly (of type A [31]) cannot be avoided due to the explicit breaking of Weyl symmetry in d dimensions: in such case G is not a total derivative and its action not Weyl invariant -then its anomaly cannot be removed by local counterterms and some special regularisation (unlike type B, it is µ independent).This situation will change in Weyl geometry.

Weyl anomaly in Weyl geometry
Weyl geometry is non-metric i.e. ∇λ g µν = 0.Here we first give a brief review of Weyl geometry and make the important observation that this geometry can actually be treated as a metric geometry with respect to a new differential operator ( ∇), so ∇λ g µν = 0; ∇λ preserves Weyl-covariance when acting on geometric operators like curvature tensors/scalar (which are themselves Weyl-covariant in d dimensions), much like in the matter sector of a gauge theory.These aspects then help us to explain how Weyl gauge symmetry is reconciled with Weyl anomaly which is recovered in the (spontaneously) broken phase.

Weyl geometry with a metric description
Weyl geometry10 is defined by classes of equivalence (g αβ , ω µ ) of the metric (g αβ ) and the Weyl gauge field (ω µ ), related by the Weyl gauge transformation shown below in d = 4 − 2ǫ dimensions, in the absence (a) and presence (b) of scalars (φ) and fermions (ψ) This defines the (non-compact) gauged dilatation symmetry or Weyl gauge symmetry.This extends eq.( 1) which is recovered if ω µ is "pure gauge" or zero everywhere.Note that here ω µ is essentially of geometric origin.By definition, in Weyl geometry we have that: ( ∇λ + q αω λ )g µν = 0, where ∇λ Weyl connection Γλ µν is found by standard calculation or via Γλ It is easy to check that Weyl connection ( Γ) is invariant under (16); the same is true for the Weyl spin connection [6] (Appendix A).This has important consequences, as seen below.Let us first show the "standard" definition of curvature tensors in Weyl geometry.The Riemann tensor in Weyl geometry Rµ νρσ is found as usual from a commutator acting on a vector field (v λ ), [ ∇µ , ∇ν ]v λ = Rλ ρµν v ρ .This gives the usual expression, now in terms of Γ with an explicit form in terms of ω µ shown in Appendix A. Since Γ is invariant under ( 16) then Rµ νρσ is invariant, too and the same is true for the Ricci tensor of Weyl geometry Rνσ ≡ Rµ νµσ .This is different from Riemannian geometry where the Riemann and Ricci tensors transform in a complicated way.One finds that in d dimensions: with R µν the Ricci tensor in Riemannian geometry, ∇ is that of Riemannian geometry (with LC connection) and where R is that of Riemannian geometry.Note that R transforms covariantly under (16), like the inverse metric g µν that enters in its definition.Further, the field strength of ω µ , regarded as the length curvature tensor, is where we used that Weyl connection is symmetric Γρ µν = Γρ νµ and ∇µ ω ν = ∂ µ ω ν − Γρ νµ ω ρ .Finally, the Weyl tensor of Weyl geometry Cµνρσ ( Cµ νµσ = 0) defined by Rµνρσ , is To summarise, under (16) we have: and therefore as detailed in Appendix A, eq.(A-14).G is the Euler-Gauss-Bonnet term of Weyl geometry.
It can be shown that G does not change the equations of motion [39] and [40] (eq.C1).In the case ω µ = 0 (F µν = 0) the familiar Riemannian version of the Euler-Gauss-Bonnet term is recovered, G → R µνρσ R µνρσ − 4R µν R µν + R 2 .To conclude, under ( 16) All these terms are thus Weyl gauge covariant in d dimensions.This makes it obvious why Weyl geometry is the right framework for implementing Weyl gauge symmetry, making it easy to write a Lagrangian invariant under ( 16) using these terms integrated with a √ g d 4 x measure.This is unlike in Riemannian geometry where they transform in a complicated way.The reason for this difference is the invariance of the Weyl connection.Despite this advantage, the above "standard" definition of curvature tensors is not satisfactory for Weyl geometry as a gauge theory, because the partial derivative ∂ µ in ∇µ when acting on the tensors fields of geometric origin is not Weyl covariant, while when acting on matter fields, Weyl covariantisation is indeed implemented in the literature, see e.g.[6].This different treatment prevents a consistent Weyl-covariant approach to all operators, both geometric and matter fields, and of their derivatives.One consequence is the presence of F 2 µν in both C2 and G terms above, showing that in this basis of operators, F 2 µν , C2 µνρσ , G are not independent.Another effect is that the theory is not metric ( ∇µ g αβ = 0) making calculations difficult and forcing one to go to a Riemannian (metric) picture to do them.
Since ( ∇λ + qαω λ )g µν = 0, where q is the charge of the metric g αβ , this suggests that for any given tensor T , including g µν , of Weyl charge q T (T ′ = Σ q T T ) one should introduce a new differential operator (we suppress the tensor indices) which transforms covariantly under (16), using that Γ is invariant: ∇′ µ T ′ = Σ q T ∇µ T .Regarding q T , a given tensor of type T (m) (n) has q T = (q/2)(n − m) e.g.n = 2, m = 0 for g µν .Applying this observation at the more fundamental level of tetrads, one defines a more suitable Riemann tensor (with a "hat") from which the length curvature tensor (F µν ) effect is removed, see Appendix A. The new Riemannian tensor Rτ νρσ of Weyl geometry is (A-19) with Fµν = The quantities with a "hat" have the same transformation as in (24).Using eqs.( 19), (20), (21) one immediately expresses these curvatures in terms of their Riemannian geometry counterparts (Appendix A).One also shows that the Weyl tensor associated to Rµνρσ is equal to that in Riemannian geometry (C µνρσ ), as shown in eq.(A-22) With these we express G of (25) in the new "basis" (note the position of summation indices): Ĝ is a natural generalisation to Weyl geometry of the usual Euler-Gauss-Bonnet term and, what is important here, it is Weyl-covariant under (16) as seen using ( 24), (29).Notice that now there is no contribution of F 2 µν to Ĉ2 µνρσ or Ĝ, which are now independent -this is welcome for identifying an action without redundant operators.
To conclude, objects with a "hat" transform under ( 16) just like those in ( 26) but the advantage of this more natural definition of operators is that differential operators ( ∇) acting on curvatures are now Weyl gauge covariant, too.In particular which can be seen using that Γ is Weyl invariant; also note that we now have that i.e. the theory is metric (with respect to ∇) in this natural Weyl "basis" (with a "hat").
To conclude, the formulation using geometric operators with a "hat", largely overlooked in the literature (except [24,40]) in favour of that in (24), is important: it enables a metric-like formulation giving at the same time a manifestly Weyl-covariant description of geometric operators and of their derivatives (acting on curvature tensors/scalar of the theory), as in any gauge theory!11 .This is important, since it enables us to do metric-like calculations in Weyl geometry (e.g.quantum corrections, see next), which would otherwise require one to go to a Riemannian picture, as usual.Relation (34) gives a close analogy to metric Riemannian geometry (via ∇ ↔ ∇) with the advantage of Weyl covariance/Weyl gauge symmetry manifest (in d dimensions), as in any gauge theory.This means that, to implement the symmetry, one could in principle take Riemannian results and Weylcovariantise them using the "hat" notation.This formalism will be used below.

Weyl anomaly in Weyl geometry
The most general Lagrangian of Weyl geometry/gravity in the absence of matter, in the original non-metric formulation (with a tilde), is [2][3][4] Using the relations of these operators to those in the new basis (with a hat), such as (A-13), (A-14), (A-22), and up to a redefinition of the couplings such as b 0 , this action is invariant Each term in ( 36) is separately invariant under the Weyl gauge transformation eq.( 16) for d = 4, see eq. (32).In a quantum theory, even if one of these terms is not included classically it will eventually be generated by quantum corrections, hence we included all possible terms 12 for a vacuum action and they are all independent in the natural Weyl basis 13 .Given the symmetry, any higher dimensional operators are not allowed since there is no fundamental scale to suppress them (except non-polynomial terms such as Ĉ4 µνρσ / R2 etc, not considered here).For the couplings we take First, if F 2 µν were absent (b 0 = 0), we would have an integrable Weyl geometry i.e. locally ω µ is "pure gauge" or zero, and then it could be integrated out via its equations of motion; the theory would become metric in the Riemannian sense ∇ µ g αβ = 0 [9] (instead of current ∇µ g αβ = 0).Since the symmetry allows it, we keep F 2 µν to have a general Weyl geometry, with a dynamical ω µ 14 hence we set b 0 = −1/4 for a canonical kinetic term.Regarding a 0 we take a 0 ∝ 1/ξ 2 where ξ ≪ 1 is the perturbative coupling of Weyl quadratic gravity.We also take c 0 = −1/η 2 , η < 1 but make no assumption about d 0 .
Action (36) gives a Weyl gauge invariant theory, in a metric formulation ( ∇µ g αβ = 0) that is spontaneously broken (via Stueckelberg mechanism) to an Einstein -Proca action for the dilatation gauge field ω µ and a small positive cosmological constant [5,6].
Let us then explore the corrections to W 0 from some massless matter states with an action with symmetry (16).One can consider the SM action which can be endowed with such Weyl gauge symmetry -this is obtained by a minimal embedding of the SM in Weyl geometry (with higgs mass parameter set to zero); this is immediate and natural, with no additional degrees of freedom beyond SM and Weyl geometry [6].The SM fermions and gauge bosons actions are simply those of flat space-time upgraded to curved space-time by multiplying them by √ g and they are invariant under ( 16) for d = 4 [6].The Higgs action is easily made Weyl gauge invariant and for a Higgs singlet state h has the form 15 [6] (eq.( 25)) Note that now each term is invariant under (16) for d = 4.
12 C 2 µνρσ , generated in all theories of gravity, may lead to non-unitarity due to its higher derivatives acting on gµν.In Weyl geometry gµν and Γ are independent, in which case this issue seems to be avoided [42]. 13Other terms like ∇µV µ , ˆ R, etc give a boundary term. 14The term F 2 µν also breaks the special conformal symmetry [43]. 15We ignore the self-coupling, not relevant here.
With these remarks, we can then consider the quantum corrections to vacuum action (36) due to the Weyl gauge invariant action of the Higgs or of SM action, or of its massless QED part only as done in conformal gravity [20] -the discussion below is independent of this choice; g µν and also ω µ that are part of "geometry" are regarded as external fields.
At the quantum level, the divergent vacuum action (denoted W d ) which is Weyl gauge invariant, will have a structure similar to that in eq.( 2) but now each individual operator is actually Weyl-covariant in the new "hat" basis.On dimensional and symmetry grounds A(d) of ( 2) will now include Weyl-covariant operators (see (33)) When integrated over d dimensions these operators give a Weyl gauge invariant action 16 .
The associated simple poles 1/(d − 4) in W d can be cancelled by a counterterm W c that is Weyl gauge invariant in d dimensions and has a general structure similar to ( 12) where the one-loop coefficients a 1 , b 1 , c 1 , d 1 are fixed (beta functions) by the one-loop divergences and depend on the matter field content considered, see e.g.[30].
In the light of the previous discussion for the Weyl term eq.( 12), we implemented a Weyl-invariant regularisation of W c : we replaced the usual DR subtraction scale µ −2ǫ of (3) by φ 2(d−4)/(d−2) where φ is the dilaton field.This is the field that linearises the quadratic term R2 in the action (as shown later, Section 3.4) 17 .A side-remark is in order here: if the higgs field contributes to the vacuum action, the "true" dilaton is actually the radial combination of φ and the higgs, while the "angular" combination of these fields becomes the physical (neutral) Higgs field at low scales [6].To a first approximation we neglect the higgs field contribution, in which case the factor in ( 40) is φ, justifying our notation there.Therefore, we did not add "by hand" any extra field: the dilaton field is itself part of the spectrum and has a geometric origin in R2 (as mentioned).
Similar to the previous section where Weyl-invariance was restored for Ĉ2 µνρσ √ g in d dimensions (and avoided the anomaly), each term in W c is here separately Weyl gauge invariant in d dimensions.This can be verified with eqs.( 16), (32).This is true in particular for Ĝ φ 2(d−4)/(d−2) √ g.While in Riemannian geometry (topological) G is a total derivative in d = 4, in Weyl geometry Ĝ is Weyl covariant in d dimensions and its contribution to the action is Weyl gauge invariant for this analytical continuation.Had Ĝ not been Weyl covariant then its contribution in (40) could not have been made invariant by φ (..) regulator.This shows the important role played by Weyl geometry.
There is a more natural analytical continuation of geometric origin for W c , due to Weyl covariance, that does not use the dilaton as regulator; one replaces it in (40) by This only apparently leads to a new regularisation or subtraction scale, since actually the dilaton has an equation of motion φ 2 = | R|, as shown in Section 3.4 (see also recent [46,47]).Action ( 36), ( 40) is now Weyl gauge invariant in d dimensions.As shown in Appendix B this has the consequence that the energy-momentum tensor is now cancelled by the divergence of the Weyl current, so the Ward identity is now: The current is conserved onshell Here ∇ µ is that of Riemannian geometry (with Levi-Civita connection Γ).This dilaton current is trivial (vanishes) if ω µ is "pure gauge" (ω µ = (1/α)∂ µ ln φ 2 ) or zero everywhere [9] 18 .The current has a vanishing divergence if we use the equation of motion of ω µ .The form of the current is shown in eq.(B-10) J µ = κ(∂ µ − α q ω µ )φ 2 where κ = −α/(4ξ 2 ), where φ is the dilaton field, used in Appendix B to linearise the R2 term in the action.
In the notation of Weyl geometry we have This current generalises that present in the global scale invariant case The vacuum part of the renormalised gravitational action is W r = W 0 + W d + W c and has the form below, using ( 36), ( 39), (40): with a notation W r includes UV non-local terms ln where = ∇µ ∇µ .All terms in (45) remain Weyl invariant, given that W 0 , W d and W c are invariant.The second line in ( 45) is due to the Euler-Gauss-Bonnet term.If we use regulator (41), replace φ 2 → | R| under log terms.Let us detail the origin of the log terms; terms in the divergent (16), see (39).If one expands it in powers of ǫ and retains only the leading −1/(2ǫ) + (1/2) ln + O(ǫ), the symmetry is violated at O(ǫ 0 ) by this truncation of the expansion, since ln transforms.The same applies to W c when expanded in ǫ that involves 1/(2ǫ) − (1/2) ln φ 2 + O(ǫ) terms.The two (finite) log terms combine into an invariant ln( /φ 2 ), thus keeping the Weyl invariance of W r in d = 4 ( and φ 2 have the same Weyl charge, equal to −q in d = 4).
Let us note that the form of W r can also be "guessed" using only symmetry arguments, Weyl-covariance of each operator and metricity, by writing an "upgraded" Weyl-invariant version of the usual result in DR such as eq.( 395) in [30] (see also [28,53]).Note however, that there is an additional term beyond Ĉ2 µνρσ , F 2 µν and Ĝ: this is the R2 term that plays a crucial role in the symmetry breaking, as we discuss in Section 3.4.
One can replace the Weyl curvature terms in (45) in terms of their Riemannian expressions, but then the result is not very illuminating.The presence of ln φ 2 as a coefficient of the various terms, including the kinetic term F 2 µν , does not allow a flat space time limit 19 ; this is expected from the spontaneous breaking of the Weyl symmetry which assumes a non-zero vev of the dilaton and this leads to a non-vanishing value of | R| = φ 2 .
To conclude, we showed that Weyl-covariance of individual operators in the action (such as R2 , Ĝ, etc) and of their derivatives in the "hat" basis (e.g.R, etc), together with a Weyl gauge invariant regularisation enabled by R, ensure that Weyl gauge symmetry is manifestly present at the quantum level (not broken explicitly by the anomaly).

Relation to holographic Weyl anomaly and Riemannian limit
Let us consider here using the "standard" regularisation (with a scale µ) which is the same as having a constant vev for φ in W c and W r .Then in the last line of eq.( 45) ln φ ∼ ln µ=constant simply cancels out; further, under ( 16), ln → ln + ln Σ −q ; then the last line in (45) generates an anomaly This is the holographic anomaly from the Euler-Gauss-Bonnet term as found in [23,24] in WFG gauge, where Weyl geometry is generated on the boundary.A good consistency check of our result is that, in agreement with the holographic picture [23,24], the term Ĝ is now Weyl-covariant.This is due to Weyl geometry and differs from the original anomaly in Riemannian space-time.The main difference between [23,24] and this work is that here ω µ is dynamical in which case there is a non-trivial current [5] as seen in ( 42), also mentioned in [23].The presence of this current is crucial in Weyl anomaly absence in the symmetric phase since its divergence cancels the trace T µ µ .It would be interesting to have the holographic picture for a dynamical ω µ to compare to our result.
The Riemannian limit of eq.( 45) can formally be recovered for ω µ = 0: then Γα µν → Γ α µν i.e.Weyl connection becomes Levi-Civita, then R → R, Rµν → R µν , → .If one also formally replaces ln φ → ln µ (corresponding to explicitly broken Weyl gauge symmetry by the DR scheme) then ln µ cancels out in the Euler-Gauss-Bonnet term of the second line of (45) and this is restoring the usual (type A) Weyl anomaly, see [30,53].The same applies for the Weyl-tensor-squared part of the anomaly (of type B).This is exactly the situation found in the broken phase of Weyl gauge symmetry where ω µ becomes massive and thus decouples (as we clarify shortly); then below the (large) mass of ω µ formally ω µ = 0, Weyl connection (geometry) becomes Levi-Civita (Riemannian), respectively, Einstein gravity is recovered [5,6], and the "usual" Weyl anomaly is generated from (45) in this broken phase.

Stueckelberg breaking of the symmetry
The results above rely on the spontaneous breaking of Weyl gauge symmetry and in particular on the role of the dilaton field.This was discussed extensively [5,6,9] but we review it here for a self-contained analysis.The breaking is closely related to the R2 term in the action.First let us ignore the presence of higgs/matter action, eq.(38).Then, at the tree-level one linearises the Weyl-covariant term R2 via a replacement R2 → −2φ 2 R − φ 4 in the action, where φ is a scalar field.One obtains in this way an equivalent form of the action.Here we display only the relevant terms involving R2 and F 2 µν (we ignore quantum corrections), to obtain an equivalent action The equation of motion for φ has a solution φ 2 = − R ( R < 0) 20 which when replaced back in W r recovers the initial action.This assumes a non-vanishing vev of φ, something already used in the regularisation.Since ln φ transforms with a shift under (16), it plays the role of the dilaton, as anticipated.Next, using (A-9), one can express R in a Riemannian notation; the action becomes [5] (see also Section 2.1 in [6]) When φ acquires a constant vev, W r becomes21 and (using (37)) Hence we obtained the Einstein-Proca action for the Weyl gauge field which became massive via Stueckelberg mechanism: ω µ has absorbed the ln φ field which is the wouldbe-Goldstone field (dilaton) of gauged dilatations (16) [5] 22 .Note that the cosmological constant is much smaller than Planck scale because gravity is weak (ξ ≪ 1). 23fter the massive Weyl gauge field decouples (together with the dilaton), the usual Weyl anomaly emerges in the broken phase of the quantum theory, as discussed (Section 3.3).Note that the equation of motion of φ, φ 2 = − R, gives after the symmetry breaking R = −4Λ = 0 [6,7].Then, ln φ 2 in (45) or (41) generates ln R terms which prevent one from taking an exactly flat metric.This is expected since we implicitly assumed φ 2 = 0 when we linearised the R2 term.
Finally, if the higgs is included, this discussion remains valid with the only change that now φ 2 R of (49) combines with h 2 R term of eq.( 38) to generate M 2 p R. Hence, the dilaton in the absence of higgs is now replaced by the radial direction in field space φ 2 → φ 2 + ξ 2 ξ h h 2 (ξ ≪ 1).It is then this combination that is used as regulator in (40) as already explained, or one is using directly eq.( 41).

Conclusions
We studied at the quantum level the gauged scale symmetry (also called Weyl gauge symmetry) that is built in Weyl conformal geometry and discussed Weyl anomaly in this geometry.
One motivation for this study was that Weyl geometry is interesting since it naturally brings together the SM and Einstein gravity in a gauge theory: as shown in the past, the SM (with vanishing higgs mass parameter) and Einstein gravity admit a truly minimal embedding in Weyl geometry without any new degrees of freedom beyond the SM and this geometry.This leads to a UV completion in a fundamental, gauge theory of scale invariance that recovers both Einstein gravity and SM in the spontaneously broken phase.This phase follows a Stueckelberg mechanism in which the Weyl gauge boson ω µ becomes massive after "eating" the dilaton ln φ 2 propagated by R2 term in the action.Another motivation of this study was that, as a (quantum) gauge theory, consistency requires it be anomaly free.A third motivation was to understand the connection with the holographic Weyl anomaly.
Our result was constructed on the important observation that there exists a natural Weyl "basis" for the geometric operators (curvature tensors and scalar), little used in the literature, that has an important advantage: in this "basis" one can restore metricity with respect to a new, Weyl-covariant differential operator ( ∇) acting on these operators.This gives: 1) a metric-like geometry formalism that enables one to do quantum calculations directly in Weyl geometry without going to a Riemannian picture, as usually done and: 2) all individual (geometric) operators and their derivatives are now Weyl gauge covariant (similar to matter fields operators), as in ordinary gauge theories.Weyl covariance of the operators is important also because it allows a Weyl invariant regularisation of geometric origin (with DR scale µ replaced by scalar curvature).Weyl gauge symmetry is then maintained and manifest at the quantum level, anomaly-free.
This result is possible because in Weyl geometry the vacuum action in the natural Weyl basis then contains only operators (and counterterms, etc) that are individually Weyl gauge covariant in d dimensions and this includes the Euler-Gauss-Bonnet term, and their derivatives.From a symmetry viewpoint, this brings on equal footing the Ĝ and Ĉ2 µνρσ terms.After the Stueckelberg symmetry breaking mechanism, ω µ decouples and Riemannian geometry is recovered together with "usual" Weyl anomaly, in the broken phase.
Our result remains consistent with that of a Weyl anomaly derived from the holographic perspective of AdS/CFT in the WFG gauge, where Weyl geometry is generated on the conformal boundary but without dynamical ω µ .The Euler-Gauss-Bonnet term Ĝ is Weyl-covariant (in d dimensions), in agreement to our result.Our case however, having a dynamical ω µ , has in addition a non-trivial current as well as a Weyl-invariant regularisation, both relevant for anomaly absence.Further study is needed of the role of this current from the holographic view and of the Ward identities in Weyl geometry.The results so far suggest that Weyl conformal geometry can be the right framework for a fundamental gauge theory and symmetry beyond both the SM and Einstein gravity.
T of charge q T , with T ′ = Σ q T T (the indices of T are not shown) we have ∇µ T ≡ ( ∇µ + α q T ω µ )T ⇒ ∇′ µ T ′ = Σ q T ∇µ T. (A-17) ∇ has the usual geometric action (with Γ).Similar Weyl covariant transformation applies to ∇µ ∇µ T , etc. Also note that Fµν = ∇µ At a more fundamental level, in the basis e a = e µ a ∂ µ where e µ a e ν b η ab = g µν , with η ab the Minkowski metric, one has that Γc

B Weyl gauge symmetry current
• For an arbitrary Weyl gauge invariant action we show there is a non-trivial, conserved current J µ in d = 4, information used in Section 3.2.Consider a Weyl gauge transformation in d = 4 dimensions: where φ is here some scalar field.For an infinitesimal transformation δΣ δg ′ µν = δ(ln Σ q ) g ′ µν , δ φ′ = − Consider a Weyl gauge invariant total action given by the sum W g + W , where W g is the Weyl gauge field kinetic term while W is the remaining action that can depend on ω µ but not on Fµν , hence W g and W are each Weyl gauge invariant.Under (B-2) where T µν and J µ are the energy-momentum tensor associated with W and the Weyl gauge symmetry current, respectively.The last term in δW vanishes by the equation of motion for φ.Since W is Weyl gauge invariant (δW = 0) and using (B-2), then (after removing the "prime" notation): where we used that in Riemannian geometry √ g∇ µ J µ = ∂ µ (J µ √ g).Therefore, for a Weyl gauge invariant action used in the text, eq.( 42).Finally, from the total action W + W g one can easily write the equation of motion for ω µ : with Riemannian ∇ σ .Multiply this equation by √ g and apply ∂ µ and use the antisymmetry of F σµ to find ∇ µ J µ = 0, i.e. there is a conserved current onshell.
• Let us now take a particular case for the Weyl action in d = 4 (no matter): where we linearised R2 as explained in Section 3.4 with a scalar (dilaton) φ of equation of motion φ 2 = − R. The Euler-Gauss-Bonnet Ĝ term was not added to the above action since it does not change the equations of motion here.In the second line we used a Riemannian notation (with ∇ µ given by the Levi-Civita connection) and the relation between R, Ĉµνρσ and their Riemannian counterparts (without a hat), see Appendix A, eqs.(A-9), (A-20), (A-22).We find a current (B-10) The total action W + W g gives the following equation of motion for ω µ √ g α 2 q 2 4 ξ 2 φ 2 ω ρ − α q 4ξ 2 ∇ ρ φ 2 + ∇ σ F σρ = 0, (B-11) This equation is Weyl gauge invariant (expected, since the action is invariant).Apply ∂ ρ on the last equation, use where we used that φ 2 = − R.Here = ∇ µ ∇ µ is in Riemannian notation.This result is actually valid for the total action W + W g since the contribution to the trace by the (conformal) gauge kinetic term F 2 µν √ g is vanishing.Therefore We thus have in agreement with general result (B-6).
ab e c = ∇a e b = ( ∇a − αq/2 ω a )e b because e b = e µ b ∂ µ has Weyl charge −q/2 i.e. half of that of g µν = e µ a e ν b η ab (∂ µ and dx µ have zero charge).Here we denoted ω a = ω µ e µ a .Then the Riemann tensor in Weyl geometry in the "basis" with a "hat" (called the natural Weyl "basis") is (with notation α ′ ≡ αq/2):Ra bcd e a = [ ∇c , ∇d ] e b = [ ∇c , ∇d ] e b − α ′ F cd e b = Ra bcd e a − α ′ δ a b F cd e a (A-18)Further, by direct calculation of the rhs of the equation below (in terms of their Riemannian counterparts, see (A-20) with (A-7), (A-8), (A-9), (A-12)) one can show a generalisation to Weyl geometry of the similar relation in Riemannian case, see eq.(A-12).Finally, using (A-23) in the last equation,23), (A-24) from which it was derived, eq.(A-25) also extends to Weyl geometry a similar relation of Riemannian geometry.