Standard Model in Weyl conformal geometry

We study the Standard Model (SM) in Weyl conformal geometry. This embedding is natural and truly minimal {\it with no new fields} required beyond the SM spectrum and Weyl geometry. The action inherits a gauged scale symmetry $D(1)$ (known as Weyl gauge symmetry) from the underlying geometry. The associated Weyl quadratic gravity undergoes spontaneous breaking of $D(1)$ by a geometric Stueckelberg mechanism in which the Weyl gauge field ($\omega_\mu$) acquires mass by"absorbing"the spin-zero mode ($\phi_0$) of the $\tilde R^2$ term in the action. This mode also generates the Planck scale and the cosmological constant. The Einstein-Proca action of $\omega_\mu$ emerges in the broken phase. In the presence of the SM, this mechanism receives corrections (from the Higgs) and it can induce electroweak (EW) symmetry breaking. The EW scale is proportional to the vev of the Stueckelberg field ($\phi_0$). The Higgs field ($\sigma$) has direct couplings to the Weyl gauge field, and its mass may be protected at quantum level by the D(1) symmetry. The SM fermions can acquire couplings to $\omega_\mu$ only in the special case of a non-vanishing kinetic mixing of the gauge fields of $D(1)\times U(1)_Y$. If this mixing is indeed present, part of $Z$ boson mass is not due to the Higgs mechanism, but to its mixing with massive $\omega_\mu$. Precision measurements of $Z$ mass then set lower bounds on the mass of $\omega_\mu$ which can be light (few TeV). In the early Universe the Higgs field can have a {\it geometric} origin, by Weyl vector fusion, and the Stueckelberg-Higgs potential can drive inflation. The dependence of the tensor-to-scalar ratio $r$ on the spectral index $n_s$ is similar to that in Starobinsky inflation but shifted to lower $r$ by the Higgs non-minimal coupling to Weyl geometry.


Motivation
The Standard Model (SM) with the Higgs mass parameter set to zero has a scale symmetry. This may indicate that this symmetry plays a role in model building for physics beyond the SM [1]. Scale symmetry is natural in physics at higher scales or in the early Universe when all states are essentially massless. In such scenario, the mass terms and scales of the theory e.g. the Planck and electroweak (EW) scales must be generated by the vacuum expectations values (vev) of some scalar fields. In this work we consider the SM with a gauged scale symmetry (also called Weyl gauge symmetry) [2][3][4] which we prefer to the more popular global scale symmetry, since the latter is broken by black-hole physics [5]. A natural framework for this symmetry is Weyl geometry [2][3][4] where this symmetry is built in. We thus consider the SM embedded in the Weyl conformal geometry and study the implications.
The Weyl geometry 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, see (a) below. If matter is present, (a) must be extended by transformation (b) of the scalars (φ) and fermions (ψ) (a)ĝ µν = Σ d g µν ,ω µ = ω µ − 1 α ∂ µ ln Σ, ĝ = Σ 2d √ g, Here d is the Weyl charge of g µν , α is the Weyl gauge coupling 1 , g = | det g µν | and Σ > 0. This is a non-compact gauged dilatation symmetry, denoted D(1). Since it is Abelian, the normalization of the charge d is not fixed 2 . In this paper we take d = 1. The case of arbitrary d is recovered from our results by simply replacing α → d α. A discussion on symmetry (1) and a brief introduction to Weyl geometry are found in Appendix A.
Γ is invariant under (1), as it should be, since the parallel transport of a vector must be gauge independent. Taking the trace in (2), with a notationΓ µ =Γ ν µν and Γ µ = Γ ν µν , then The Weyl field is thus a measure of the (trace of the) deviation from a Levi-Civita connection. The general quadratic gravity action defined by Weyl geometry [2][3][4], invariant under (1), is written in terms of scalar and tensor curvatures of this geometry. UsingΓ of (2) and standard formulae one can express these curvatures in terms of their Riemannian counterparts and re-write the action in a more familiar Riemannian notation (as we shall do). In the limit ω µ = 0 i.e. if: i) ω µ is 'pure gauge' or if ii) ω µ becomes massive and decouples, thenΓ = Γ and then Weyl geometry becomes Riemannian! This is an interesting transition, relevant later. In i) invariance under (1) reduces to local scale invariance (no ω µ ).
In L 0 we replaceR 2 → −2φ 2R − φ 4 with φ a scalar field. Doing so gives a classically equivalent L 0 , since by using the solution φ 2 = −R of the equation of motion of φ in the modified L 0 , one recovers action (4). With eq.(5), L 0 becomes in a Riemannian notation or, making the symmetry manifest Every term of coefficient ∝ 1/ξ 2 and the entire L 0 are invariant under (1); we must then "fix the gauge" of this symmetry. This follows from the equations of motion of φ, ω µ (see later), while at the level of the Lagrangian this is done by applying to L 0 a specific form of transformation (1) that is scale-dependent Σ = φ 2 / φ 2 which is fixing φ to its vev (assumed to exist, generated e.g. at quantum level); naively, one simply sets φ → φ in (7). In terms of the transformed fields (with a "hat"), L 0 becomes In (8) a total divergence in the action, δS = α/(4ξ 2 ) φ 2 d 4 x √ĝ ∇ µω µ was ignored -it may be replaced by a local condition ∇ µω µ = 0. This constraint will be obtained shortly from the current conservation of the symmetric phase, eqs.(6), (7).
In (8) we identify M p with the Planck scale. Eq.(8) is the Einstein gauge (frame) and also the unitary gauge of action (7). By a Stueckelberg breaking mechanism [34][35][36], ω µ has become a massive Proca field, after "eating" in (7) the derivative ∂ µ ln φ of the Stueckelberg field ln φ [27], which transforms with a shift under (1). It is important to note here that the number of degrees of freedom (dof) is indeed conserved: in addition to the graviton, the real, massless φ (dof=1) and massless ω µ (dof=2) were replaced by massive ω µ (dof=3) of mass m 2 ω = (3/2)α 2 γ 2 M 2 p in eq. (8). We shall see shortly that φ is indeed a dynamical field. One may expect m ω ∼ M p but the Weyl gravity coupling may naturally be α ≪ 1, so m ω ≪ M p !
The Einstein-Proca action in (8) is a broken phase of L 0 of (7). After ω µ decouples from (8), below m ω the Einstein-Hilbert action is obtained as a 'low-energy' effective theory of Weyl gravity [27]. Hence, Einstein gravity appears to be the "Einstein gauge"-fixed version of the Weyl action. However, the breaking is more profound and is not the result of a mere 'gauge choice': it is accompanied by a Stueckelberg mechanism and by a transition from Weyl to Riemannian geometry: indeed, when massive ω µ decouples thenΓ of (2) is replaced by Γ.
In the other case, when ω µ is light (α ≪ 1), it may be present in the action at low energies, since the current non-metricity lower bound (set by the mass m ω of ω µ ) is actually very low, of few TeV only [37]! It can also be a dark matter candidate e.g. [38,39].
Note that the Stueckelberg term in (7) ( is simply a Weyl-covariant kinetic term of the Stueckelberg field that became the mass term of ω µ in (8). That is, a Weyl gauge-invariant kinetic term of a (Weyl-charged) scalar in Weyl geometry is a mass term for ω µ in the (pseudo)Riemannian geometry underlying (8). This gives an interesting geometric interpretation to the origin of mass, as a transition from Weyl to Riemannian geometry, without any scalar field present in the final spectrum. The field φ also generated the Planck mass and was "extracted" from theR 2 term i.e. is of geometric origin (like ω µ ), giving an elegant breaking mechanism. Further, from (6) one writes the equation of motion of ω µ and applying ∇ ρ to it, one finds a conserved current (see Appendix B and [27]) This current conservation equation confirms that φ is indeed a dynamical field, which is relevant for the above Stueckelberg mechanism to take place. This result also extends to the case of the gauged scale symmetry a conserved current K ρ = φ∂ ρ φ in global scale invariant theories, with ∇ ρ K ρ = 0. For a Friedmann-Robertson-Walker (FRW) metric, ∇ ρ K ρ = 0 had a solution φ(t) →constant for large enough time (t), so φ evolved to a vev [40][41][42][43][44]. In our case here, for ω µ (t) = (ω 0 (t), 0, 0, 0) consistent with a FRW metric, if ω 0 (t) 2 ∼ 1/φ(t) then a similar solution φ(t) →constant can exist. Assuming that φ acquires a vev by such mechanism or at the quantum level, etc, then equation (10) gives ∇ µ ω µ = 0. This is the "gauge fixing" condition, specific to a massive Proca field, that emerges from the conserved current of the Weyl gauge symmetry.
Finally, one may ask what Weyl geometry tells us about the cosmological constant (Λ). From Lagrangians (7) and (8) we find Both Λ and M p are generated by same φ and are thus related and Λ > 0. For M p fixed, Λ is small because gravity is weak (ξ ≪ 1). In the limit φ → 0 then Λ, M p → 0 and the Weyl gauge symmetry is restored 4 . This shows how Λ is protected by this symmetry.
In conclusion, Weyl action (4), (7) is more fundamental than Einstein-Proca action (8) which is its "low-energy", broken phase. When the massive Weyl gauge boson decouples, the geometry becomes Riemannian and the Einstein gravity is recovered. In a sense this picture is entirely geometrical [31] since we did not yet include any matter. Thus, ultimately the underlying geometry of our Universe may actually be Weyl conformal geometry. Its Weyl gauge symmetry could then explain a small (non-vanishing) positive cosmological constant.

Weyl quadratic gravity and "photon" -photon mixing
Consider now L 0 in the presence of the SM hypercharge gauge group U (1) Y . A kinetic mixing of ω µ (Weyl "photon") with the B µ gauge field of U(1) Y is allowed by the direct product symmetry U (1) Y × D(1). Such mixing was mentioned in the literature [20] but not investigated. Consider then where F y is the field strength of B µ . The source of B µ is the SM fermionic Lagrangian (not shown in (12)) which is invariant under (1) and does not depend on ω µ [8] (see next section). This mixing could eventually be forbidden by some discrete symmetry, not discussed here 5 . We repeat the steps in Section 2.1 and after transformation (1) under which B µ is invariant,B µ = B µ , we find L 1 in terms of the new fields (with a hat): The kinetic mixing is removed by a transformation [45] to new ('primed') fieldŝ where, for a simpler notation, we introduced sinχ (note that γ ≤ 1) 6 . The result is 4 This limit is formal, since the linearisation of (4) with φ 2 = −R implicitly assumes that φ is non-zero. 5 Note that ωµ is C-even [8] and the photon is C-odd and the mixing violates C and CP. Global or discrete symmetries (C, CP, Z2 etc) can be used to forbid the kinetic mixing; such symmetries can however be broken by black-hole physics/gravity [5]. Also, the CPT invariance theorem applies only if the theory is local, unitary and in flat space-time, so it cannot be used here: the Weyl-geometry actions are neither unitary (C 2 term in Weyl action has a ghost) nor in flat space-time. The consequences of χ = 0 are further studied in Section 2.7. 6 In the limit γ = 1 there is noC 2 µνρσ term in the initial action (formally η → ∞).
As in the previous section, we obtain again the Einstein-Proca action but with diagonal gauge kinetic terms for both gauge fields. However, the final, canonical hypercharge gauge field B ′ µ has acquired a dependence on the Weyl gauge field, see (14), due to the initial kinetic mixing. In the full model, upon the electroweak symmetry breaking the photon field (A µ ) is a mixing of the hypercharge (B ′ µ ) with SU(2) L neutral gauge field (A 3 µ ) where θ w is the Weinberg angle and in the second step we used eq. (14). Due to the gauge kinetic mixing the photon field includes a small component of the initial Weyl gauge field, suppressed by sin χ and by the mass (∼ M p ) of ω µ , but still present 7 ; however, it exists only in the presence of matter e.g. fermionic fields that act as the source of B µ . Such mixing in models with Abelian gauge fields beyond the hypercharge exists in string models, with similar massive and anomaly-free gauge fields (as ω µ , see later) and similar mass mechanism [46]. However, here ω µ is a gauge field of a space-time (dilatation) symmetry. The mixing is not forbidden by the Coleman-Mandula theorem -the overall symmetry is always a direct product U (1) Y × D(1) and both symmetries are subsequently broken spontaneously 8 .

Fermions
Consider now the SM fermions (ψ) in Weyl geometry and examine their action. To begin with, to avoid a complicated notation we do not display the SM gauge group dependence: Heres ab µ is the Weyl geometry spin connection. In (17), the Weyl charge of the fermions is (−3/4) according to our convention in (1) (d = 1). The relation of the Weyl spin connection to the spin connection s ab µ of (pseudo-)Riemannian geometry is (see Appendix A) where σ ab = 1 4 [γ a , γ b ] while Γ ν µλ is the Levi-Civita connection, g µν = e a µ e b ν η ab and e µ a e a ν = δ µ ν . It can be checked that, similar to the Weyl connection (Γ), the Weyl spin connections ab µ is 7 In some sense this says that Weyl's unfortunate attempt to identify ωµ to the photon was not entirely wrong, if the aforementioned mixing is present. 8 The theorem implies that D(1) cannot be part of an internal non-Abelian symmetry so d cannot be fixed.
invariant under (1). This is seen by using that s ab µ transforms under (1) aŝ Withs ab µ invariant, one checks that L f is Weyl gauge invariant. In fact one can easily show that (−3/4)αω µ in γ µ∇ µ ψ is cancelled by theω µ -presence in the Weyl spin connection. This cancellation also happens between fermions and anti-fermions [8] (eqs. 36, 37) 9 . This is so because both fermions and anti-fermions have the same real Weyl charge (no i factor in∇ µ ψ). As a result, we have Thus the SM fermions do not couple [8] to the Weyl field ω µ and there is no gauge anomaly.
We can now restore the SM gauge group dependence and the Lagrangian becomes with the usual quantum numbers of the fermions under the SM group (not shown), T = σ/2, and with g and g ′ the gauge couplings of SU (2) L and U (1) Y . But this is not the final result.
Since the fermions are U (1) Y charged and the initial fieldB µ in (21) is shifted by the gauge kinetic mixing, as seen in eq. (14), then ω ′ µ is still present in L f : We found a new coupling of the SM fermions to ω ′ µ , of strength Y g ′ tanχ. This coupling comes with the usual fermions hypercharge assignment (which is anomaly-free). After the electroweak symmetry breaking B ′ µ is replaced in terms of the mass eigenstates A µ , Z µ , Z ω µ and ω ′ µ is a combination of Z µ , Z ω µ (see later, eq. (47)). If χ ∼χ = 0, the fermions Lagrangian is identical to that in the (pseudo-)Riemannian case (with no Weyl gauge symmetry).
Regarding the Yukawa interactions notice that the SM Lagrangian is invariant under (1) where H is the Higgs SU (2) L doublet andH = iσ 2 H † , the sum is over leptons and quarks; Y, Y ′ are the SM Yukawa matrices. L Y is invariant under (1): indeed, since the Weyl charge is real, the sum of charges of the fields in each Yukawa term is vanishing: two fermions (charge 2 × (−3)/4), the Higgs (charge −1/2) and √ g (charge 2). Hence the Yukawa interactions have the same form as in SM in the (pseudo-)Riemannian space-time.

Gauge bosons
Regarding the SM gauge bosons, their SM action is invariant under transformation (1) [8]. A way to understand this is that a gauge boson of the SM enters under the corresponding covariant derivative acting on a field charged under it and should transform (have same weight) as ∂ µ acting on that field; since coordinates are kept fixed under (1), the gauge fields do not transform either. Their kinetic terms are then similar to those of the SM in flat space-time, since the Weyl connection is symmetric. Explicitly, this is seen from the equation below, where the sum is over the SM gauge group factors: where A is a generic notation for a SM gauge boson and since∇ µ A ν = ∂ µ A ν −Γ ρ µν A ρ , then for a symmetricΓ ρ µν =Γ ρ νµ one sees thatΓ and its ω µ -dependence cancel out in the field strength F µν . Hence, L g does not depend on ω µ and has the same form in Weyl and in (pseudo)Riemannian geometries.

Higgs sector
• The action: Let us now consider the SM Higgs doublet (H) in Weyl conformal geometry: The where The case of no gauge kinetic mixing in (25) (χ = 0) is obvious. We keep χ = 0 for generality. We consider the electroweak unitary gauge where H = (1/ √ 2) h ζ, with ζ T ≡ (0, 1). Then with As done earlier, in L H replaceR 2 → −2φ 2R − φ 4 to find a classically equivalent action; using the equation of motion of φ and its solution φ 2 = −R back in the action, one recovers (25). After this replacement, the non-minimal coupling term in (25) is modified It is interesting to notice that the initial term in the action, (1/ξ 2 )R 2 , (where ξ < 1) in (25) was replaced by a term above with a large non-minimal coupling 1/ξ 2 > 1 (plus an additional φ 4 ). For details, the full Lagrangian L H after step (29) is shown in the Appendix, eq.(C-1). Next, to fix the gauge, apply transformation (1) to L H with a special scale-dependent Σ which fixes the fields combination (φ 2 /ξ 2 + ξ h h 2 ) to a constant: In terms of the transformed fields and metric (with a 'hat'), L H becomes were we used (5), the notationẐ = Z(B µ →B µ , A µ →ˆ A µ ), with γ ≤ 1 defined in (7) and and finallyV We found again a massive ω µ in (31) by Stueckelberg mechanism after 'eating' the radial direction field (1/ξ 2 φ 2 + ξ h h 2 ), with constraint ∇ µ ω µ = 0. We identify M p with the Planck scale; M p and thus also m ω receive now contributions from both the Higgs and φ.
The term proportional to ξ 2 inV is ultimately due to the (1/ξ 2 )R 2 term in the action and is ultimately responsible for the EW symmetry breaking and for inflation, see later.
Eq.(31) contains a mixing termω µ ∂ µĥ from the Weyl-covariant derivative ofĥ. We choose the unitary gauge for the D(1) symmetry i.e. eliminate this term by replacinĝ Then L H becomes with the potentialV expressed now in terms of the actual Higgs field σ, using (33), (34 and L H becomes: where F ′ (F ′ y ) is the field strength of ω ′ (B ′ ) and Note the presence in L H of a coupling ∆L H = (1/8) σ 2 ω ′ µ ω ′µ (g ′2 tan 2χ +α 2 γ 2 sec 2χ ). This is due to 1) the gauge kinetic mixing χ and 2) to the Higgs coupling to ω µ , eq.(26). This coupling is non-zero even if there is no gauge kinetic mixing (χ = 0), when it becomes This is relevant for Higgs physics and can constrain α. If γ = 1 (i.e. if there is noC 2 µνρσ term in (25)) then this coupling is due entirely to the higgs kinetic term |D µ H| 2 in (25) and is the only coupling of the SM to the background Weyl geometry (apart from that to the graviton). In the symmetric phase ∆L H can generate higgs production via Weyl boson fusion. This coupling is further discussed in [31].
• Higgs potential: One may write L H in a more compact form The σ-dependent matrix M(σ) written in eq.(40) in the basis X ≡ (B ′ µ , A 3 µ , ω ′ µ ) is presented in the Appendix, eq.(C-3). Finally we have This is the Higgs potential in our SMW model in the unitary gauge for the EW and D(1) symmetries. The second line is valid for small field values σ ≪ M p when we recover a Higgs potential similar to that in the SM; the quadratic term has a negative coefficient (with ξ h > 0, as needed for inflation, see later). This follows when the Higgs field contributes positively to the Planck scale, eq.(32) and "to compensate" for its contribution to M p , a negative sign emerges in (33) and inV (σ). The EW symmetry is thus broken at tree level.

EW scale and Higgs mass
The small field regime σ ≪ M p in (44) gives realistic predictions in the limit ξ h ξ 2 ≪ 1; indeed, in this case the quartic Higgs coupling becomes λ and the EW scale σ and Higgs mass are To comply with the values of the Higgs mass and EW vev we must set ξ √ ξ h ∼ 3.5×10 −17 . This means one or both perturbative couplings ξ h and ξ take small values, while λ ∼ 0.12 as in the SM and the regime σ ≪ M p is respected. Recall that ξ is the coupling of the term (1/ξ 2 )R 2 in the action, hence we see the importance of this term for the hierarchy of scales! From (45), using the Planck scale expression eq.(32), then With ξ √ ξ h fixed earlier, one still has a freedom of either a hierarchy or comparable values of these two vev's, depending on the exact values of ξ h < 1. Eq.(46) relates the EW scale physics to the underlying Weyl geometry represented by theR 2 term in the action (from which φ is "extracted").
The SMW model with the Higgs action as in eqs. (25), (40) has similarities to Agravity [47,48] which is a global scale invariant model. Unlike in Agravity, we only have the Higgs scalar, while the role of the second scalar field (s) in [47], that generated the Planck scale and Higgs mass in Agravity is played in our model by the "geometric" Stueckelberg field (φ); φ was not added "ad-hoc" and cannot couple to the Higgs field, being extracted from theR 2 term itself (see eq.(25)). Hence, there is no classical coupling between the Higgs field and the field generating M p , while in [47] a coupling λ HS h 2 s 2 is present.
However, the SMW contains the field ω µ (part of Weyl geometry), not present in [47]. Our preference here for a local, gauged scale symmetry, that brought in the Weyl gauge field, is motivated by three aspects: firstly, we already have a "geometric" mass generation mechanism which does not need adding ad-hoc an extra scalar; secondly, global symmetries do not survive black-hole physics [5] and finally, the Weyl gauge symmetry of the action is also a symmetry of the underlying geometry (connectionΓ), as it should be the case.
At the quantum level, large loop corrections to m σ could in principle arise, as in the SM (plus those due to ω µ by the coupling ω µ w µ σ 2 ). But the Weyl gauge symmetry can change this. The mass m ω ∼ αM p may be light (if α ≪ 1), possibly not far above the lower bound (of few TeV) on the non-metricity scale (set by m ω ) [37]. This means the Weyl gauge symmetry breaking scale can be low. The mass of ω µ is then the highest physical scale ("cutoff") for the low-energy observer. Then all quantum corrections to m 2 σ are expected to be quadratic in the scale of "new physics" (m 2 ω ), so δm 2 σ ∝ m 2 ω . Above m ω the gauged scale symmetry is restored, together with its UV protection (for the Higgs mass) not affected by its spontaneous breaking. In this way the Weyl gauge symmetry (with ξ, α ≪ 1) could give a solution to the hierarchy problem. Intriguingly, since m ω also sets the non-metricity scale, this suggests the hierarchy problem and non-metricity scale are related!

Constraints from Z mass
Let us now compute the eigenvalues of the Higgs-dependent matrix M 2 (σ), eqs. (40), (C-3), and examine the constraints from the mass of Z on the model parameters α and χ. Since Z µ and ω µ mix, part of Z boson mass is not due the Higgs mechanism, but to this mixing and ultimately, to the Stueckelberg mechanism giving mass to ω µ . After the electroweak symmetry breaking, in the mass eigenstates basis of M 2 ( σ ), one has the photon field (A µ ) (it is massless, since det M 2 = 0), the neutral gauge boson (Z) and the Weyl field (Z ω ). M 2 (σ) is brought to diagonal form by two rotations (C-4), (C-5) giving Denote by U the matrix relating the gauge eigenstates ( The masses of Z boson (m Z ) and Weyl gauge field (m ω ) are then found 10 where P = 4 g ′2 (g 2 + g ′2 ) sin 2 2χ + g 2 (1 − 2 δ 2 ) + (g 2 + 2g ′2 ) cos 2χ Since σ ≪ M p (see conditions after eq. (45)) The factor in front is the mass of Z boson (hereafter m Z 0 ) in the SM; m Z has a negligible correction from Einstein gravity (∝ σ 2 /M 2 ). But there is also a correction (∝ sin 2 χ/α 2 ) from the Weyl field i.e. due to deviations from Einstein gravity induced by Weyl geometry. This can be significant and it reduces m Z by a relative amount: 10 If there is no mixing, χ = 0, then in eq.(48), also (47), ζ = 0, and with Mp of (32) and h of (34) then In the second step we replaced the mass of ω and the definition ofχ in eq. (36). The effect in (53) is significant if sin χ/α ≫ 1. From the mass of Z boson and with ∆m Z at 1 σ deviation, one has |ε| ≤ 2.3 × 10 −5 , then eq.(53) gives a lower bound on the Weyl gauge coupling α, for a given non-zero gauge kinetic mixing: Note that for an arbitrary charge d of the metric, the results depending on α are modified by replacing α → d × α. In terms of the mass of ω µ one finds This gives a lower bound on the mass of the Weyl field in terms of the mixing angle χ and γ. A larger m ω allows a larger amount of mixing. For a mixing angle of e.g.χ = π/4 then m w ≥ 6.35 TeV. Note that if there is no term (1/η)C 2 µνρσ in the original gravity action, then γ = 1 and then χ =χ. Alternatively, using the current lower bound on the non-metricity scale (represented by m ω ) which is of the order of the TeV scale [37], then tanχ ≤ 0.16 (56) This is consistent with the non-metricity constraint. These bounds are significant and affect other phenomenological studies. To give an example, consider the impact of ω µ on the g − 2 muon magnetic moment, due to the new coupling of ω µ in L f , eq.(22). Using [49,50] an estimate of the correction of ω µ to ∆a µ is where we used constraints (53), (55). These do not allow ∆a µ to account for the SM discrepancy with the experiment [51]; however, this discrepancy may be only apparent, according to lattice-based results [52]. One can also use these constraints when studying the role of ω µ for phenomenology in other examples, such as the dark matter problem [39], in which case it may even provide a solution (of geometric origin!) to this problem; other implications can be for example in the birefringence of the vacuum induced by ω µ . This can impact on the propagation of the observed polarization of the gamma-ray bursts [53] or of the CMB [54].

Inflation
The SMW model can have successful inflaton. For large σ ∼ M p , the Higgs potential in (43) can drive inflation [28,29,55]. But who "ordered" the Higgs in the early Universe? the Higgs could initially be produced by the Weyl gauge boson fusion via its coupling ω µ ω µ HH † dictated by the symmetry, eq.(25). This means, rather interestingly, that the Higgs can be regarded as having a geometric origin, just like ω µ which is part of the Weyl connection 11 . As seen from (37) this coupling becomes ω µ ω µ f (σ) with σ the neutral Higgs. But in a Friedmann-Robertson-Walker universe considered below, g µν = (1, −a(t) 2 , −a(t) 2 , −a(t) 2 ), the vector field background compatible with the metric is ω µ (t) = 0 [29]. The fluctuations of σ and of (longitudinal component of) ω µ do not mix since ω µ (t)δω µ δσ is then vanishing. As a result, the single-field inflation formalism in the Einstein gravity applies, with σ as the inflaton. Since M p is simply the scale of Weyl gauge symmetry breaking, σ > M p is natural. The predictions of the Higgs inflation are then [28,29] Here r is the tensor-to-scalar ratio and n s is the scalar spectral index. Up to small corrections from ξ h that can be neglected for ξ h < 10 −3 , the above dependence r = r(n s ) is similar to that in the Starobinsky model [56] of inflation where r = 3(1 − n s ) 2 . For mildly larger ξ h ∼ 10 −3 − 10 −2 , eq.(58) departs from the Starobinsky model prediction and r is mildly reduced relative to its value in the Starobinsky case, for given n s . These results require a hierarchy λ ≪ ξ 2 h ξ 2 which may be respected by a sufficiently small λ and 12 ξ h ∼ 10 −3 − 10 −2 . A relatively very small λ means that it is actually the squared term in (43) that is multiplied by ξ 2 (see also (33)) that is mostly responsible for inflation; this is the Stueckelberg field contribution, ultimately due to the initial term φ 4 arising from the initial quadratic curvature (1/ξ 2 )R 2 term in action (25). This then explains the close similarities to the Starobinsky R 2 -inflation. Thus, we actually have a Starobinsky-Higgs inflation. The initial Higgs field h (which has ξ h = 0) does play a role as it brings a minimum in 13V (σ) of (43). In conclusion, a negligible λ is required for successful inflation (as the numerical values of r below also show it). This is consistent with SM prediction for λ at the high scales, while a value of λ at the EW scale as in the SM can then be induced by the SM quantum corrections.
The numerical results give that for N = 60 efolds and with n s = 0.9670 ± 0.0037 at 68% CL (TT, TE, EE+low E + lensing + BK14 + BAO) [57] then [28, 29, 55] 14 0.00257 ≤ r ≤ 0.00303, (n s at 68% CL) (59) 0.00227 ≤ r ≤ 0.00303, (n s at 95% CL) The case of Starobinsky model for N = 60 corresponds to the upper limit of r above and is reached for the smallest ξ h , when this limit is saturated, according to relation (58). The small value of r found above may be reached by the next generation of CMB experiments CMB-S4 [58,59], LiteBIRD [60,61], PICO [62], PIXIE [63] that have sensitivity to r values as low as 0.0005. Such sensitivity will be able to test this inflation model and to distinguish it from other models. For example, similarly small but distinct values of r are found in other models with Weyl gauge symmetry [29,30] based on the Palatini approach to gravity action (4) used in this paper; however these models do not respect relation (58) and the slope of the curve r(n s ) is different, due to their different vectorial non-metricity. The above experiments also have the sensitivity to distinguish inflation in this model from the Starobinsky model for ξ h ∼ 10 −2 when the curve r(n s ) is shifted by ξ h below that of the Starobinsky model, towards smaller r (for fixed n s ).

SMW and its properties
In this section we discuss some features of our model and the differences from other SM-like models with local scale invariance. The main aspect of our model is that scale symmetry is gauged, eq.(1). The Weyl gauge symmetry is not only a symmetry of the action but also of the underlying Weyl geometry; indeed, the Weyl connectionΓ (A-5) is invariant under (1) and the same is true about the Weyl spin connection (A-18). This adds consistency to SMW and distinguishes it from models with an action that is Weyl or conformal invariant (with no ω µ ), built in a (pseudo-)Riemannian space; their Levi-Civita connection and thus their underlying geometry do not have the symmetry of the action -which may be a concern.
The breaking of the Weyl gauge symmetry is accompanied by a change of the underlying geometry. When massive ω µ decouples at some (high) scale, the Weyl connection becomes Levi-Civita, so Weyl geometry becomes Riemannian and the theory is then metric 15 . Thus, the breaking of the symmetry in Section 2.1 (see [27]) is not just a result of a "gauge fixing" to the Einstein frame, as it happens in Weyl or conformal theories with no ω µ ; it is accompanied by the Stueckelberg mechanism and by a change of the underlying geometry 16 .
The SMW avoids some situations present in interesting models with local scale invariance (without ω µ ), like a negative kinetic term of the scalar field [70] (also [71][72][73]), or an imaginary vev [74,75] of the scalar that generates 17 M p . Such situations may be a cause of concern according to [15,17]. Gauging the scale symmetry avoids such situations -in SMW this scalar field plays the role of a would-be Goldstone of the Weyl gauge symmetry (eaten by ω µ ). See also eq.(7) where the (negative) kinetic term in the first square bracket is cancelled by that in the second square bracket corresponding to a Stueckelberg mechanism 18 .
In local scale invariant models (without ω µ ) the associated current can be trivial, leading to so-called "fake conformal symmetry" [76,77]; in the SMW the current is non-trivial even in the absence of matter [27] due to dynamical ω µ . If ω µ were not dynamical (F µν = 0) it could be integrated out algebraically to leave a local scale-invariant action [27,31]; in this case Weyl 15 A similar Weyl gauge symmetry breaking and change of geometry exists in a Palatini version [29,30]. 16 An aspect of models with Weyl gauge symmetry relates to their geodesic completeness, see [15,17]. In conformal/Weyl invariant models (without ωµ) this aspect seems possible in the (metric) Riemannian spacetime where geodesic completeness or incompleteness is related to a gauge choice (and singularities due to an unphysical conformal frame) [65][66][67]. In models in Weyl geometry, the geodesics are determined by the affine structure. Differential geometry demands the existence of the Weyl gauge field [68] for the construction of the affine connection, because this ensures that geodesics are invariant (as necessary on physical grounds, the parallel transport of a vector should not depend on the gauge choice). Hence the Weyl gauge field/symmetry may actually be required! After the breaking of this symmetry, wµ decouples, we return to (pseudo)Riemannian geometry and geodesics are then given by extremal proper time condition. Since a dynamical ωµ also brings in non-metricity, geodesic completeness seems related to non-metricity. 17 It seems to us this means a negative Σ and therefore a metric signature change in transformation (1). 18 This Stueckelberg mechanism may apply to more general metric affine theories studied in [69].
geometry would be integrable and metric, see e.g. [16,17]. But since ω µ is dynamical, the theory is also non-metric. This non-metricity would indeed be a physical problem if ω µ were massless (assuming this, non-metricity of a theory was used as an argument against such theory by Einstein 19 [2]). However, non-metricity became here an advantage, since Weyl geometry with dynamical ω µ enabled the Stueckelberg breaking mechanism, ω µ acquired a mass (above current non-metricity bounds [37]), and the Einstein-Proca action was naturally obtained in the broken phase. The SMW differs from the SM with conformal symmetry of [79] or [74,75] and from conformal gravity models [80][81][82] formulated in the (pseudo)Riemannian space and based on C 2 µνρσ term; these models have metric geometry and do not have a gauged scale symmetry; in our case the C 2 µνρσ term is largely spectator and may be absent in a first instance; it was included because its Weyl geometry counterpart gave a correction to α and it is needed at a quantum level. And unlike the conformal gravity action [83] which is metric, the SMW has a gauge kinetic term for the Weyl field which 1) makes the geometry non-metric and 2) breaks the special conformal symmetry; this symmetry and non-metricity do not seem compatible.
Concerning the quantum calculations in the SMW, one could try to use the "traditional" dimensional regularization (DR), but that breaks explicitly the Weyl gauge symmetry by the presence of the subtraction scale (µ). One should use instead a regularisation similar to [84] that preserves Weyl gauge symmetry at the quantum level. This is possible by using our Stueckelberg field φ as a field-dependent regulator, to replace the subtraction scale µ generated later by µ ∼ φ (after symmetry breaking). This would allow the computation of the quantum corrections without explicitly breaking the Weyl gauge symmetry 20 .
It is interesting to study the renormalizability of the Weyl quadratic gravity and of the SMW. The usual (metric) quadratic gravity theory in the (pseudo-)Riemannian case is known to be renormalizable but not unitary due to the massive spin-2 ghost [89]. Considering now the Weyl quadratic gravity alone, note that for computing the quantum corrections eq.(8) is not appropriate since this is the (non-renormalizable) unitary gauge of Weyl gauge symmetry. Therefore, one should consider computing the necessary quantum corrections in the symmetric phase, for example in L 0 of eq.(6). Note that no higher order operators are allowed by the symmetry in eq.(4), (6), since there is no initial mass scale to suppress them, and this is an argument in favour of its renormalizability. Finally, regarding the SMW itself, in a Riemannian notation it simply has an additional (anomaly-free) Weyl gauge field which becomes massive by the Stueckelberg mechanism which cannot affect renormalizability; naively, one then expects the SMW be renormalizable.
The model has the special feature that both the action and its underlying geometry (connectionΓ and spin connectionw ab µ ) are Weyl gauge invariant. This adds consistency to the model and distinguishes it from previous SM-like models with local scale symmetry, built in a (pseudo-)Riemannian geometry whose connection is not local scale invariant.
The SMW model has another attractive feature. In Weyl geometry there exists a (geometric) Stueckelberg mechanism in which this symmetry is spontaneously broken. The Weyl quadratic gravity associated to this geometry is broken spontaneously to the Einstein-Proca action of ω µ . The Stueckelberg field φ has a geometric origin, being "extracted" fromR 2 in the Weyl action, and is subsequently eaten by ω µ . Once the Weyl gauge field decouples, the Weyl connection becomes Levi-Civita and Einstein gravity is recovered. The Planck scale and a positive cosmological constant are both generated by the Stueckelberg field vev. Also, the mass term of the Weyl field is on the Weyl geometry side just a Weyl-covariant kinetic term of the same Stueckelberg field. These aspects relate symmetry breaking and thus mass generation to a geometry change (from Weyl to Riemannian) which is itself related to the non-metricity induced by dynamical ω µ .
The SMW gauge group is a direct product of the SM gauge group and D(1) of the Weyl gauge symmetry, both broken spontaneously. Usually, of the SM spectrum only the Higgs field (σ) has a coupling to ω µ , the term α 2 ω µ ω µ σ 2 . The Weyl gauge symmetry can protect the Higgs mass at a quantum level, if this symmetry is broken at a low scale. The breaking scale is set by m ω ∼ αM p , and if the Weyl gauge coupling α ≪ 1, then m ω can be light, few TeV (which is the current lower bound on non-metricity). The mass of ω µ is then the highest physical scale for the low-energy observer and quantum corrections to m 2 σ will appear as ∝ m 2 ω . Above m ω the gauged scale symmetry is restored, together with its UV protection (for the Higgs) not affected by the spontaneous breaking. Hence Weyl gravity and its gauged scale symmetry could give a solution to the hierarchy problem. And since m ω also sets the non-metricity scale, then the hierarchy problem and non-metricity scale may be related!
The fermions can acquire a coupling (Y g ′ tanχ) to ω µ only in the case of a small kinetic mixing (χ) of the gauge fields of U (1) Y × D(1), if this mixing is not forbidden by a discrete symmetry. Due to such mixing part of Z boson mass is not due to the Higgs mechanism, but to the mixing of Z with the massive Weyl field which has a Stueckelberg mass; hence, part of Z mass has a geometric origin, due to a departure from the pseudo-Riemannian geometry and Einstein gravity. Since m Z is accurately measured, one finds bounds on the Weyl gauge coupling and the mass of ω µ , for a given amount of kinetic mixing. We showed how these bounds can be used in other phenomenological studies. If ω µ is light (few TeV, α ≪ 1) its effects may be amenable to experimental tests, with consequences for phenomenology e.g.: ω µ as a dark matter candidate, the vacuum birefringence, etc.
The SMW has successful inflation. Intriguingly, in the early Universe the Higgs may be produced via Weyl vector fusion, thus having a geometric origin. With M p a simple phase transition scale in Weyl gravity, Higgs field values larger than M p are natural. Note that while the inflationary potential is that of the Higgs, due to its mixing with φ, it is ultimately a contribution to this potential from the initial scalar mode (φ) in theR 2 term that is actually responsible for inflation. This explains the close similarities to the Starobinsky R 2 -inflation. With the scalar spectral index n s fixed to its measured value, the tensor-to-scalar ratio 0.00227 ≤ r ≤ 0.00303. Compared to the Starobinsky model, the curve r(n s ) is similar but shifted to smaller r (for same n s ) by the Higgs non-minimal coupling (ξ h ) to Weyl geometry. These results of the SMW model deserve further investigation.
Thus, the Weyl gauge field can be thought of as the trace of the departure of the Weyl connection from the Levi-Civita connection. UsingΓ one computes the scalar and tensor curvatures of Weyl geometry, using formulae similar to those in Riemannian case but with Γ instead of Γ. For examplẽ After some algebra one finds where the rhs is in a Riemannian notation, so ∇ µ is given by the Levi-Civita connection (Γ). An important property is thatR transforms covariantly under (A-1) which follows from the transformation of g µσ that enters its definition above and from the fact thatR µν is invariant (sinceΓ is so). Then the term √ gR 2 is Weyl gauge invariant.
In Weyl geometry one can also define a Weyl tensorC µνρσ that is related to that in Riemannian geometry C µνρσ as follows used in the text, eq.(4). √ gC 2 µνρσ and its above separation are invariant under (A-1). To introduce the Weyl spin connection, consider first the spin connection in the Riemannian geometry Under transformation (A-1) one checks thats ab µ is invariant, similar to Weyl connectionΓ. Let us now consider matter fields and find their charges in Weyl geometry by demanding that: a) their Weyl-covariant derivatives transform under (A-1) like the fields themselves and b) that their kinetic terms be invariant. More explicitly, take the kinetic term for a scalar of charge d φ : √ g(D µ φ) 2 whereD µ is the Weyl-covariant derivative which we demand it transform under (A-1) just like the scalar field itself, i.e. it has same charge d φ . From the invariance of this action under (A-1) one has that d φ = −d/2. The Weyl covariant derivative is then found according to (A-4) and the kinetic term is with L φ invariant, while φ transforms aŝ For a fermion ψ the Weyl charge is found in a similar way, by using (A-4) to write their Weyl covariant derivative, hence the action has the form L ψ = i 2 √ g ψ γ a e µ a∇µ ψ + h.c.,∇ µ ψ = ∂ µ + d ψ α ω µ + 1 2s ab µ σ ab ψ (A-21) where σ ab = 1/4[γ a , γ b ]. Since we saw earlier thats ab µ is Weyl gauge invariant then the above derivative∇ µ ψ transforms covariantly just like a fermion field itself of charge d ψ . From the structure of the kinetic term and its invariance it follows that d ψ = −3d/4 so, under (A-1) ψ = Σ −3d/4 ψ. (A-22) With this charge and using (A-21), (A-18) one shows that ω µ cancels out: γ a e µ a∇µ ψ = γ a e µ a ∂ µ + 1 2 s ab µ σ ab ψ. (A-23) Hence, the fermionic kinetic term has the same form as in the Riemannian geometry L ψ = i 2 √ g ψγ a e µ a ∇ µ ψ + h.c., ∇ µ ψ = ∂ µ + 1 2 s ab µ σ ab ψ, . For more information see also [8,19].
C Higgs sector: L H and the matrix M 2 (σ) For convenience, we write here in the Riemannian notation and in the symmetric phase the form of L H shown in the text in the Weyl geometry notation eq.(25) after step (29) L H = √ g −1 2 where θ 2 = (1/ξ 2 ) φ 2 + ξ h h 2 denotes the radial direction in the fields space with and θ 2 = 6M 2 p . The first line in L H is similar to that of a single field case, see eq.(7) for θ 2 ↔ (1/ξ 2 ) φ 2 . Note that L H is invariant under the Weyl gauge transformation eq.(1) (one checks that the first square bracket is invariant, while for the remaining terms this is easily verified). From this action eq.(31) then follows, via a Stueckelberg mechanism.
In the formal limit when the radial direction in field space θ → 0 (M p → 0) which restores the Weyl gauge symmetry, then from the definition of θ we see that φ → 0 and h → 0 (EW symmetry is also restored) and therefore the potential vanishes V → 0, as expected due to the Weyl gauge symmetry.
The Higgs-dependent matrix M 2 (σ) introduced in eq. (40) Notice that if there is no gauge kinetic mixing χ ∼χ = 0, then M 2 simplifies considerably. The mass matrix is diagonalised by two successive rotations of the fields; first: After this, Z 1 − Z 2 mass mixing usually exists, diagonalized by a final rotation of suitable ζ Combining these two rotations we find a matrix relating the mass eigenstates (A µ , Z µ , Z ω µ ) to the gauge eigenstates X µ = (B ′ µ , A 3 µ , ω ′ µ ). The inverse of this matrix is shown in eq.(47).