Spontaneous breaking of Weyl quadratic gravity to Einstein action and Higgs potential

We consider the (gauged) Weyl gravity action, quadratic in the scalar curvature ($\tilde R$) and in the Weyl tensor ($\tilde C_{\mu\nu\rho\sigma}$) of the Weyl conformal geometry. In the absence of matter fields, this action has spontaneous breaking in which the Weyl gauge field $\omega_\mu$ becomes massive (mass $m_\omega\sim$ Planck scale) after"eating"the dilaton in the $\tilde R^2$ term, in a Stueckelberg mechanism. As a result, one recovers the Einstein-Hilbert action with a positive cosmological constant and the Proca action for the massive Weyl gauge field $\omega_\mu$. Below $m_\omega$ this field decouples and Weyl geometry becomes Riemannian. The Einstein-Hilbert action is then just a"low-energy"limit of Weyl quadratic gravity which thus avoids its previous, long-held criticisms. In the presence of matter scalar field $\phi_1$ (Higgs-like), with couplings allowed by Weyl gauge symmetry, after its spontaneous breaking one obtains in addition, at low scales, a Higgs potential with spontaneous electroweak symmetry breaking. This is induced by the non-minimal coupling $\xi_1\phi_1^2 \tilde R$ to Weyl geometry, with Higgs mass $\propto\xi_1/\xi_0$ ($\xi_0$ is the coefficient of the $\tilde R^2$ term). In realistic models $\xi_1$ must be classically tuned $\xi_1\ll \xi_0$. We comment on the quantum stability of this value.

In the absence of matter and up to topological terms [18], the Weyl gravity action is the sum of two quadratic terms: the square of the Weyl-geometry scalar curvature term,R 2 , and the square of the Weyl-tensor:C 2 ≡C µνρσC µνρσ . These generalise their counterparts R 2 and C 2 of the Riemannian geometry, to include effects due to the Weyl gauge field ω µ . Indeed, in Weyl geometry ∇ µ g νσ ∝ ω µ g ρσ , so the Weyl connection coefficientsΓ ρ µν are not determined by the metric alone (as in the Riemannian case), but also depend on ω µ . ThenR 2 (orC 2 ) can be expressed in terms of their Riemannian counterparts R 2 (C 2 ) plus a function of ω µ , and become equal if ω µ = 0 (decoupled); in this limit, Weyl geometry becomes Riemannian. We consider only a torsion-free Weyl geometry.
In Section 2 we show that in the absence of matter fields, Weyl quadratic gravity has spontaneous breaking via a Stueckelberg mechanism [19] in which the Weyl gauge field ω µ becomes massive after "eating" the Goldstone (dilaton) field (φ 0 ); φ 0 is "extracted" from theR 2 term which propagates an extra scalar field. Below the mass m ω of the Weyl gauge field, m ω ∼ Planck scale, this field decouples, Weyl geometry becomes Riemannian and we recover the Einstein-Hilbert action and a positive cosmological constant. So Einstein-Hilbert action is a "low-energy" limit of Weyl quadratic gravity which thus avoids previous long-held criticisms against it, see e.g. [2]. No additional matter field (scalar, etc) is required.
The Stueckelberg mechanism of breaking is known in local Weyl-invariant models but these contain, however, additional (matter) scalars and are linear (rather than quadratic) in the scalar curvatureR of Weyl geometry [5,6] (see also recent [11]). In these models the Einstein action follows from the presence of a scalar matter field φ with coupling φ 2R , absent in our quadratic Weyl action without matter. To our knowledge, the breaking we study in the "pure" geometric case ofR 2 andC 2 -terms only, was not yet discussed. Our goal is to show that this symmetry breaking mechanism is still at work in this case.
In Section 3 we also consider the presence of matter fields in addition to Weyl quadratic action. We study the case of the SM Higgs field (φ 1 ) with all dimension-four couplings allowed by Weyl gauge symmetry. We show that the Stueckelberg mechanism is still present, with an additional benefit: the Higgs potential has spontaneous electroweak symmetry breaking. This follows from a Weyl-invariant non-minimal coupling ξ 1 φ 2 1R ; the higgs mass becomes ∝ ξ 1 /ξ 0 (Planck units), where ξ 0 is the coefficient of theR 2 term.

Spontaneous breaking from Weyl gauge symmetry to Einstein gravity
In the absence of matter, the Weyl action has two quadratic terms, with Here Under transformations (1), (2), L 1 is invariant. Another invariant under (1), (2) is where η is the coupling in Weyl geometry and in the second step we used eq.(6). L 2 is decomposed into the sum of two terms of Riemannian geometry, each of them Weyl invariant. Note the presence of a kinetic term for ω µ , relevant below. For this term to be canonically normalized in the presence of L 2 , we must set q 2 = −η/6 (so η < 0). In the absence of matter fields, L 1,2 are the only independent terms allowed by Weyl gauge symmetry, up to a topological term that is counterpart to the Riemannian Gauss-Bonnet term and that we do not include here (see e.g. Appendix C in [18]).
Returning to L 1 ,R 2 has higher derivatives, so it propagates an additional scalar state, see e.g. [41,42]. We extract from theR 2 term this (dynamical) degree of freedom via a Lagrangian constraint; this brings a term linear inR and an additional scalar φ 0 . We have The equation of motion for φ 0 gives φ 2 0 = −R. Using this back in (10), one recovers L 1 of (8), so Lagrangians (8) and (10) are classically equivalent. Given its definition, φ 0 transforms just like any matter scalar under eq.(1). Then a (local) shift symmetry exists, as for a Goldstone (dilaton) field: ln . This results from the Weyl invariance (1), (2) of the term φ 2 0R ; note that its Riemannian counterpart (φ 2 0 R) does not have this symmetry, hence the importance of this step (obviously (10) is invariant under (1), (2)).
Using eq.(5) to replaceR by its Riemannian version R and an integration by parts, then where we introduced Let us assume first that L 1 would be the total Lagrangian of our model; its dependence on ω µ is algebraic and this field can be integrated out. Its equation of motion is Inserting this solution back in (11) gives L eff is however uninteresting: although derived from (8) and invariant under (1), it has a remaining "fake" conformal symmetry [44]: its associated current is vanishing. This follows the absence of a kinetic term for the Weyl field which allowed its integration 4 ,5 . This situation is avoided if ω µ is dynamical, since then the current does not vanish anymore. A Weyl-invariant kinetic term δL 2 for ω µ can be added 'by hand'; but there is no need to do so since L 2 of (9) already contains δL 2 ! To conclude, hereafter we shall consider that the defining action of our model is If one insists to also include the term C 2 ≡ C µνρσ C µνρσ , then the action of our model is actually L 1 + L 2 . In both cases, the equation of motion for ω µ gives By applying ∂ µ with F αµ antisymmetric in (α, µ), we now find a non-vanishing current soK µ is conserved. Further, on the ground state, assuming φ 0 (x)=constant, it follows that ∂ µ ( √ g ω µ ) = 0, which is a condition similar to that for a Proca (massive) gauge field, 4 A second issue is that with ξ > 0, φ0 becomes ghost-like while if ξ0 < 0 and with φ0 = 0, Newton constant would be negative (with our conventions). The Einstein term in our convention is (−1/2) √ gM 2 p R. 5 A third issue: a conformal transformation (1), α = − ln(6Mp/ξ0φ), (Mp: Planck scale) on (14) removes φ from spectrum and we recover the Einstein action from L eff , but then the number of degrees of freedom in Jordan vs Einstein frame does not match, so "something" is missing; see text after eq.(25) for our solution.
leaving three degrees of freedom for ω µ . In fact, in the case of a Friedmann-Robertson-Walker (FRW) metric, g µν = (1, −a 2 (t), −a 2 (t), −a 2 (t)), with φ only t−dependent, the current conservation (in covariant form D µK µ = 0) leads naturally to φ 0 =constant [32]. The other equations of motion, of g µν (after trace) and of φ 0 , derived from L 1 +δL 2 , are and (20) leads to φ 2 0 = − R that we already know; thus the ground state hasR=constant (this is called Weyl gauge [17]). Further, when adding eqs. (19), (20), with the last one multiplied by φ 0 , one finds that on the ground state The potential is thus a homogeneous function of fields, as expected (given the symmetry); with our V , it is automatically respected; φ 0 is thus a parameter, not fixed by theory; in a Weyl-invariant theory only (dimensionless) ratios of vev's can be fixed. To see how ω µ becomes massive consider a conformal transformation to Einstein framê M is a mass scale present for dimensional reasons 6 ; its role is discussed shortly. Then Thus the field φ 0 is indeed dynamical, it has a kinetic term. Also note that δL 2 of (15) is invariant under (22) since the metric part and F µν are invariant. Further, introduce (24) where also ∂ µ ln K = ∂ µ ln Ω. Using (24), replace ω µ in L 1 and denote by F ′ µν the field strength of ω ′ µ . Then the total Lagrangian is It is important to note that there is no kinetic term left for φ 0 , since it was cancelled by that generated when expressing ω µ -dependence in eq.(23) in terms of ω ′ µ . The massless Weyl field has become massive after "eating" the Goldstone mode (φ 0 ), via a Stueckelberg mechanism, without a corresponding Higgs mode in the spectrum or a potential. This mechanism essentially re-distributes the degrees of freedom in the action: the initial massless ω µ and the real scalar φ 0 are converted into a single massive Weyl field ω µ with three degrees of freedom (recall ∂ µ ( √ g ω µ ) = 0); so the number of degrees of freedom is indeed conserved when going from the Jordan to the Einstein frame (as it should).
In eq.(25) we obtained the Einstein-Hilbert action, a positive cosmological constant and the Proca action for a massive Weyl field 7 ω ′ µ ; its mass is related to the Planck scale (M p ) The value of m ω depends on the gauge coupling q in L 1 +δL 2 , see L 1 of (11) and canonically normalised δL 2 of (15). Below the scale m ω the Weyl field decouples and Einstein gravity is obtained as a "low-energy" effective theory limit. The scale M introduced on dimensional grounds in (22), remains undetermined by the theory. From (11), M 2 = ξ 0 φ 0 2 /6 which is equally undetermined in a theory invariant under (1), (2), as discussed 8 .
In the decoupling limit of the massive Weyl gauge field, Weyl geometry "flows" into a Riemannian geometry. This may also be seen dynamically from the conserved current in (18) which for a FRW metric is driving φ 0 to a constant value [32] (in this case ∝ M ). Below m ω the Weyl gauge field is absent, so Weyl connection becomes that of the Riemannian geometry,Γ ρ µν = Γ ρ µν . As a result, long-held criticisms of Weyl quadratic gravity without matter [2,17] are avoided: the change of the norm of a vector under parallel transport on a closed curve or the change of the atomic spectral lines spacing under Weyl transformation are effects strongly suppressed by a very high mass scale of Weyl gauge field, m ω ∝ M p .
So far our analysis was based on the Lagrangian L 1 + δL 2 . Considering instead the Lagrangian L = L 1 + L 2 , one has to include the remaining term in the rhs of (9), i.e. C 2 ≡ C µνρσ C µνρσ ; this is immediate, since this is invariant under (22), (24). This term provides the kinetic term for the metric [23] and is needed at the quantum level to renormalize divergences like k 4 . However, in this case the coupling q cannot be adjusted at will anymore, being proportional to η (see text after (9)). Lowering q too much brings a too light mass m 2 g ∼ η M 2 of the spin-two ghost of the C 2 term, together with its instability. It is interesting that Lagrangian (8), (9) dictated by Weyl geometry (no matter) is so rich in structure, encoding Stueckelberg mechanism, dilaton φ 0 , Einstein action, Proca action for massive ω µ , a positive cosmological constant and fields kinetic terms and interactions. 7 Our above result is consistent with those in [41][42][43] where it was shown that 'pure' R 2 gravity in the Riemannian geometry, describes Einstein gravity plus a cosmological constant and a scalar (Goldstone) field. In our case the Goldstone mode is eaten by the Weyl gauge field present in theR 2 -term in Weyl geometry. 8 In conformal theory only ratios of scales can be predicted in terms of dimensionless couplings. If this symmetry is broken explicitly (by quantum corrections) dimensional transmutation can determine a field vev.

Adding matter fields 3.1 Spontaneous breaking of Weyl gauge symmetry
Let us now consider the SM scalar sector in addition to Weyl quadratic gravity action, with all dimension-four couplings allowed by the Weyl gauge symmetry. Note that the SM fermions do not have couplings to the Weyl gauge field (in the absence of torsion) [9,11]. We consider the SM Higgs field and denote by φ 1 its neutral component. Then the Lagrangian we study, invariant under (1), (2), and written in a Weyl geometry language, is with Weyl-covariant derivativeD µ φ 1 = (∂ µ − q/2 ω µ )φ 1 and L 1 and δL 2 of eqs. (8), (15) The Weyl tensor squared term C µνρσ C µνρσ in L 2 of (9) is not included in L, but since it is not affecting the transformations below, it can easily be added (replace δL 2 → L 2 ). Also F µν is the same in both Riemann and Weyl geometry (in the absence of torsion, as here).
The only possible form of the Higgs potential V 1 consistent with the symmetry, is Using (10) and (5), one finds, following steps similar to the previous section 9 up to a total derivative, with In (30) a sum over the repeated index "a" is understood, with a = 0, 1. The generalisation of this action to more matter (scalar) fields is immediate. We perform a conformal transformation to the Einstein frame, to a new metricĝ µν In V1 there is no classical coupling of φ1 to the dilaton (φ0) hidden in Weyl's quadratic action (28) since φ0 is an intrinsic part of ourR 2 term from which is extracted by "linearisation" ofR 2 , eq.(10). Adding to V1 a term (λm/12) φ 2 0 φ 2 1 is also redundant, since together with (10), integrating out φ0 simply restores the originalR 2 term after a redefinition of initial couplings of φ1: λ1 → λ1+λ 2 m /ξ0, ξ1 → ξ1+λm. Also, adding a termξφ 2 0R to (28) would introduce an extra (dynamical) degree of freedom beyond the dilaton inR 2 term! One has Further, introduce The kinetic terms in L for φ 0 and φ 1 (hereafter L k.t. ) become, after using eqs.(33), (35) We see there is only one kinetic term left in the action, for the new variable Z which is a combination of initial φ 0 , φ 1 . One could introduce polar coordinates fields (ρ, θ), such as In such basis Z is an "angular" variable field while K entering in (35) becomes K = ρ 2 and is the "radial" direction in field space. After transformation (35) the terms 10 in (34) other than L k.t. also depend only on the ratio φ 0 /φ 1 ∼ Z and not on the radial direction field! To anticipate, this is explained by ω ′ µ that must have "eaten" the radial direction in (35) (Stueckelberg mechanism), see later.
From L of (34), using (37) and notations (35), (38), we obtain our final Lagrangian where F ′ µν is the field strength of ω ′ µ and 10 These are the terms in (34) proportional to V/Ω 2 and K/Ω and clearly depend only of the ratio φ0/φ1.

For small higgs field values h ≪ M one has
The first term on the rhs of (42) is the mass of the Weyl gauge field ω ′ µ , up to additional corrections of order O( h 2 /M 2 ) due to the Higgs mechanism itself, if h = 0 (see below). Therefore, following eq.(35) the "radial" degree of freedom in the field space (dilaton) was "eaten" by the gauge field ω ′ µ which has become massive, via the Stueckelberg mechanism. The number of degrees of freedom is conserved: initially we had 2 real scalars (φ 0 , φ 1 ) and massless ω µ which were re-arranged into one real scalar h and a massive gauge field ω ′ µ . L in eq.(39) includes the Einstein action with a positive cosmological constant, a massive Weyl gauge field and a potential for the Higgs field h. For a vanishing non-minimal coupling ξ 1 = 0, see our starting L in (27), one recovers the initial potential for φ 1 plus a cosmological constant similar to that in Weyl quadratic action without matter, eq.(25).
Eqs. (39) to (41) give the Higgs sector for the SM enlarged with Weyl gauge symmetry and can be used for further investigations of this symmetry. These equations bring no restrictions at the classical level for the value of h relative to (otherwise arbitrary) M .

Electroweak symmetry breaking
From (41), (43), we see that a non-minimal coupling ξ 1R 2 φ 2 1 , (ξ 1 > 0) induced a negative quadratic term in the potential and spontaneous electroweak symmetry breaking, with A hierarchy m 2 h ≪ M 2 can be arranged by a classical tuning ξ 1 ≪ ξ 0 to an ultraweak value of ξ 1 . This is a gravitational higgs mechanism (which forbids the presence of TeV-scale squarks!) 11 .
Regarding the scale M , in the limit h ≪ M , can be identified with a constant value of the dilaton φ 0 (in this limit φ 0 ≫ φ 1 ); like M , φ 0 is a parameter not fixed by the theory and must be tuned to the actual Planck scale value (as mentioned, only ratios of scales can be determined in a Weyl invariant theory). Finally, the vacuum energy is still positive, dominated by the dilaton contributionV . A study of the quantum corrections to the Higgs mass is beyond the goal of this paper. However, we stress here the role Weyl gauge symmetry may play at the quantum level. Note that classically only the Higgs sector of the SM couples to the Weyl field ω µ [9,11]. Using a Weyl-invariant regularization [39] one could answer whether Weyl gauge symmetry can protect ξ 1 and thus the Higgs mass m h ∝ ξ 1 against large quantum corrections 12 . Note that each of the terms in our L of (27), (28) is separately Weyl gauge invariant, see eqs.(1), (2). We expect that this symmetry bring some "protection" for the ultraviolet (UV) behaviour of this theory. In particular we expect a better UV behaviour than in Riemannian gravity e.g. [37] where no similar local symmetry (Weyl, conformal) exists. This motivates a quantum analysis of Weyl quadratic gravity.
In the light of the results in [38] for renormalizability, one would expect a Weyl quadratic theory given by L 1 + L 2 of (8), (9) be renormalizable. One must however pay attention to the analytical continuation from Minkowski to the Euclidean space which is non-trivial in the presence of higher derivative terms [45].

Conclusions
We considered the general action of Weyl gravity in the absence of matter, which is the sum of two terms quadratic in the curvature scalar (R) and in the Weyl tensor (C µνρσ ) of the Weyl conformal geometry, then studied its spontaneous breaking. We also studied the effect of coupling Weyl (quadratic) gravity to a Higgs-like matter sector.
In the absence of matter fields, the Weyl gauge field ω µ in the action becomes massive, with a mass m ω ∼ q M p where q is the coupling and M p the Planck scale. This happens via a Stueckelberg mechanism which is essentially a re-arrangement of the degrees of freedom (without a higgs vev or potential needed): the field ω µ "eats" a scalar degree of freedom (dilaton) "extracted" from theR 2 term. The necessary presence of a kinetic term for the Weyl gauge field originates from the termC µνρσC µνρσ of Weyl geometry. However, if this is not included in the initial Weyl quadratic action, the gauge kinetic term may be added on its own in Weyl geometry, on symmetry arguments only.
After the Stueckelberg mechanism one obtains the Einstein-Hilbert action, a positive cosmological constant and Proca action for the massive Weyl gauge field (and a Riemannian Weyl-tensor-squared term, ifC µνρσC µνρσ was included initially). No additional matter scalar field (Higgs, etc) is needed to this purpose. Below the mass m ω the Weyl field ω µ decouples and Weyl geometry (connection) becomes Riemannian. Therefore, the Einstein-Hilbert action is a "low-energy" effective theory limit of Weyl quadratic gravity (without matter). In this way Weyl quadratic gravity avoids previous, long-held criticisms against it.
This result has consequences for physics at high scale where Weyl gauge symmetry may be present (inflation, black hole physics, conformal supergravity). During the spontaneous breaking of this symmetry the number of degrees of freedom is indeed conserved, also when going to the unitary gaugeφ 0 =constant (unlike in models invariant under "usual" Weyl symmetry, eq.(1)). It is remarkable that the simple Weyl quadratic action dictated by Weyl geometry alone is so rich in structure, encoding a Stueckelberg mechanism, the Einstein-Hilbert action, Proca action, positive cosmological constant, the dilaton, the metric and their interactions. Further study of this symmetry should use the Weyl geometry formulation which is easier (than the Riemannian one) since then the scalar curvature transforms covariantly, so each operator respects this symmetry. This places on equal footing, in the Lagrangian, Weyl gauge symmetry and (internal) gauge symmetries.
In the presence of a scalar matter field φ 1 (Higgs), the Weyl gauge symmetry allows a non-minimal coupling ξ 1 φ 2 1R , in addition to the mentioned Weyl quadratic action and to the matter action; the latter is that of the SM Higgs sector with a potential λ 1 φ 4 1 /4! (as the only one allowed by this symmetry). The Stueckelberg breaking mechanism is still at work and the Weyl gauge field is "eating" the dilaton (the radial direction in the field space of φ 0 , φ 1 ) which subsequently disappears from the action. At the same time, in the Riemannian limit (ω µ decoupled), the scalar potential of the remaining Higgs degree of freedom acquires at low energy (h ≪ M ) a negative quadratic term ∝ ξ 1 /ξ 0 . This is gravitationally-induced spontaneous electroweak symmetry breaking. It is worth investigating further if the Higgs mass value (ξ 1 /ξ 0 in Planck units) is stable at the quantum level in SM with spontaneously broken Weyl gauge symmetry.
Note added: After completing this work we became aware of [46] where the connection in a general theory of gravity is shown to acquire mass via a Higgs mechanism, while at low scales is "frozen" to the Levi-Civita connection. This is consistent with our result.