Scale Invariance, Stueckelberg Breaking of Weyl Gravity and Inflation

Home , Proca action

Scale invariance, Stueckelberg breaking of Weyl gravity and inﬂation

D. Ghilencea

“Beyond General Relativity, Beyond Cosmological Standard Model”

Warsaw 1-5 July 2019”

———————– Based on: arXiv:1904.06596, 1812.08613, 1906.11572 [1]

“New physics” beyond SM: new symmetry? •

- SUSY @ TeV: - nice [in] theory.... protected a hierarchy of scales. Not found @ LHC.

- scale invariance (SI); - discrete

- global SI. e.g.: SM with higgs mφ =0 has scale invariance. [Bardeen 1995] - local SI

- gauged local SI = Weyl gauge symmetry Weyl gravity [Weyl 1919] → - quantum scale invariance (QSI): global, local. [Englert, Gastmans, Truﬃn 1976] [2]

(1). Global scale invariance x = ρx ; φ (ρx)= ρ 1 φ(x), forbids d4xm2φ2 µ′ µ ′ − Z

- no dim-ful coupling; all scales (MPlanck,...) from vev’s; e.g. SM with mh =0 I show: QSI in ﬂat spacetime protects a hierarchy of ﬁelds vev’s φ σ. • ≪

2 1 (2). Local scale invariance: g′ = Ω(x) gµν, φ′ = φ µν Ω(x)

2 1 (3). Gauged local scale invariance g′ = Ω(x) gµν, φ′ = φ, ω′ = ωµ ∂µ lnΩ(x) µν Ω(x) µ −

- also called Weyl gauge symmetry. Weyl gravity/conformal geometry (1918). - Non-metricity g = ω g . Einstein critique due to ω (massless) ∇µ αβ µ αβ µ - Not a problem if ωµ massive/decouples (ωµ =0) at very high scale.

I show: Weyl gravity spontaneously broken to Einstein-Proca+cc; m M • ω ∼ Planck MPlanck=phase transition emergent scale. Weyl inﬂation: 0.00257

(1). Global Scale Invariance – at Quantum level: [Englert 1976, Itzykson/Zuber, Shaposhnikov 2008]

1 2 2ǫ 4 DR: d=4 2ǫ : L = (∂µφ) λ µ φ , explicit SI by UV regulators, to avoid. − 2 − . replace µ ﬁeld σ, spontaneous σ =0 Goldstone (dilaton): σ = σ eτ , τ τ ln ρ → h i6 ⇒ h i → − [nothing new, recall Mstring moduli dep]

4 2 4 Action: d=4, S = d x (∂µφj) V (φj) + d y Lh(σ,∂σ) spectrum extended by σ! • Z h − i Z visible hidden - each sector SI (shift symmetry) enhanced shift symmetry: S S λ φ2σ2: λ naturally small → h × v ⇒ m m [Fubini 1976; Volkas, Kobakhidze, Foot 2013]

2 2ǫ 4 2ǫ 4 2ǫ σ˜ σ˜ 2 4 µ φ V˜ (z σ) φ = z σ 1+2 ǫ + ... + (ǫ ) φ , σ = σ +˜σ, (z : dim-less) → ∼ h i h σ − 2 σ 2 O i h i h i h i scale inv reg=DR+dilaton with -many ǫ-couplings. σ :’new physics’. φ σ quantum stable? ⇒ ∞ h i h i≪h i

4 [σσ σσ at 3 loops! (∂µ ln σ) ] → [4]

One-loop SI potential: =V (φ) • 1 1 1 L = (∂ φ)2 + (∂ σ)2 λφ4 , V V˜ = z2ǫ σ2ǫ V (φ). 2 µ 2 µ − z4!}| { → d ˜ 4 ˜ 2 i d p 2 ˜ Ms 1 Ms 2ǫ V1 = d tr ln p Vαβ + iε = − + ln µ(σ) , −2Z (2π) − 4κ ǫ c0 sX=φ,σ h i

∂2 V˜ ∂2 V... + ǫ (..); M˜ 4 = M 4 + ǫ..., M˜ 4 ǫ2, κ=(4π)2 σ... ∼ σ... φ φ σ ∼

λ2φ4 λφ2 1 λ2φ4 λφ2 1 λ2φ4 σ˜ σ˜2 U = V + ln = V + ln + − + + ... 16κ h (z σ)2 − 2 i 16κ h (z σ )2 − 2i 16κ h σ 2 σ 2 i h i h i h i CW 0; small yet maintains SI → | {z } | {z } σ = σ +˜σ, σ˜: ﬂuctuations h i U scale invariant due to dilaton (ln σ). ⇒ (1) B 2 B (1) 2 No new poles, same beta function βλ ; λ =λZλZφ− , dλ /d(ln z)=0, βλ = dλ/(d ln z)=3λ /κ. QSI with β =0, dilation current ∂ Dµ =0. β =0 is suﬃcient, not necessary for QSI. CS: dU/d(ln z)=0 λ 6 µ λ [Englert et al 1979, Shaposhnikov et al; Tamarit 2014] [5]

2ǫ 2ǫ Two-loop SI potential: using: V˜ (φ, σ) z σ V (φ) [background ﬁeld method] • ∼ 1 1 1 V˜ (φ+δ ,σ+δ )= V˜ (φ,σ)+V˜ δ + V˜ δ δ + V˜ δ δ δ + V˜ δ δ δ δ + α, β =φ,σ. ⇒ φ σ α α 2 αβ α β 3! αβγ α β γ 4! αβγρ α β γ ρ ··· V˜αβ... = ∂α∂β..V˜

i i i i ddp ddq V = + + = V˜ V˜ (D˜ ) (D˜ ) (D˜ ) + 2 12 8 2 12 αβγ α′β′γ′ Z (2π)d (2π)d p αα′ q ββ′ p+q γγ′ ···

3 4 2ǫ λ φ 3 2 0 2 = (zσ) + + (ǫ ) ; (D˜ p)αβ = (Dp)αβ + ǫ (...)αβ + ǫ (...)αβ 32κ2 n − ǫ2 ǫ O o 2 same poles, ǫ-shifts to propagators, vertices: V˜αβγ... = Vαβγ...+ ǫ (...)αβγ... + ǫ (...)αβγ...

[Z. Lalak, P. Olszewski,DG] [1712.06024]

Two-loop corrected U

2 2 2 2 4 λ 4 3λ Vφφ 1 3λ Vφφ 2 Vφφ 5λ φ 7λ φ U = φ 1+ ln + 4+ A0 4 ln +3 ln + + , 4! n 2κ (zσ)2 − 2 4κ2 − (zσ)2 (zσ)2 κ2 σ2 24κ2 σ4 o V V (1) V (2) new V (2,n) ﬁnite z-independent [6]

Two-loop: Taylor expand about σ = σ +˜σ : • h i

2 λ 4 3λ Vφφ 1 3λ Vφφ 2 Vφφ 1 U = φ 1+ ln + 4+ A0 4 ln +3 ln + 4! n 2κ (z σ )2 − 2 4κ2 − (z σ )2 (z σ )2 O σ h i h i h i h i

- This is the “usual” CW result with µ = σ , broken SI, no dilaton present. [Cheng, I. Jack, T. Jones, S. Martin] h i - new terms comparable/larger than standard two-loop terms

φn φ φ σ˜ σ˜2 1., n =1, 2; = 1 + + , 0, if φ σ σn ∼ σ σ − σ σ 2 ··· → ≪ h i h i h i

Non-polynomial terms: - SI, vanish if φ σ, only log σ left; ⇒ ≪ - no λnφ2σ2 =λn σ 2φ2+... no tuning of higgs self-coupling λ. h i ⇒ - ﬁnite c-terms, cannot be seen in a scheme that breaks this symmetry. - not forbidden by symmetry ops mandatory, quantum generated; non-renorm → - similar operators at three-loop SI potential (computed), as counterterms. [7]

SM+dilaton: one-loop SI potential [Shaposhnikov et al 2008; DG, Z. Lalak, P. Olszewski] • 3 λφ 2 λm 2 λσ 4 4 λ6 (H†H) 2 V = (H†H) + (H†H) σ + σ + + 1) 9λ = λ λ + loops 3! 2 4! 3 σ2 ··· m φ σ min σ =0: λ λ λ λ φ6 h i6 φ 2 3λ = φ φ4 + m φ2 σ2 + σ λ σ4 + 6 ; (λ <0), 2) h i = − m (1 + loops) 4! 4 4! σ 6 σ2 ··· m σ 2 λ h i φ λ 3λ 2 φ φ2 + m σ2 + loops. → 4 λφ

2 2 2 SI : tuning (i) or Vmin =0 EWSB; mφ˜ λφ φ =( 3)λm σ σ if λm λφ one classical . ⇒ ≈ h i − h i ≪h i | |≪ tuning

V (1) = old CW with µ zσ → (1,n) 1 2 4 2 2 2 4 V ( 16λmλφ 18λm + λφλσ) φ λm(48λm + 25λσ)φ σ 7λσσ ≡ 48κh − − − − 6 8 10 12 φ φ φ 2 φ + (λφλm +6λ6λσ) +8(4λφ 2λm) λ6 + (192λ6 +2λφ)λ6 + 40λ6 , large σ : Sv Sh σ2 − σ4 σ6 σ8 i ×

2 2 2 2 2 0 if φ σ and λm 0. no λ φ σ λ φ σ → ≪ → φ ∼ φ h i [8]

One-loop corrected potential U = V + V (1) + V (1,n) gives:

2 2 2 2 2 λm σ 4 g 1 4 g2 1 4 ht 1 ∆mφ˜ = − h i 27 g ln + +2 g2 ln + 16 ht ln λφ 16κ n h 4 3 4 3 − 2 − 3i 2 2 λφ 2 2 2 2 + 4 λφ 5ln 8+ln27 = λmλφ σ + σ , but no λφ σ . h 12 − io h i ··· ≪h i h i

- β λ , so λ stays ultraweak, natural once chosen classically small. λm ∝ m m - result is not a DR artefact, since all scales from ﬁelds’ vev’s. More scalars?

(classical) hierarchy φ σ protected by quantum scale inv Sv Sh, (all orders, spont SI) ⇒ h i≪h i × .

2 2 Curved spacetime - Einstein gravity breaks Sv Sh by ξ1φ +ξ2σ R βλm = ξ1(...)+ξ2(...)+λm(...) • × ⇒ - hierarchy of vev’s (Higgs vs dilaton)? maybe preserved, ﬁxed point for ξ1,ξ2. - what is the fate of the dilaton? can it decouple? [9]

(2). Local scale invariance: [t’Hooft 1104.4543; 1410.6675, IJMP 2016]

σ invariance under gˆ = Ω(x)2 g , σˆ = (a) µν µν Ω(x)

1 2 1 1 2 µν want to generate spontaneously: LEH = √gMp R L1 = √g σ R + g ∂µσ∂νσ (inv.) −2 ⇒ − 2 h 6 i a) - has a ghost (σ)! b) - Fake conformal symmetry? [Jackiw, Pi 2015], c) - Going to Einstein frame: d.o.f. number not conserved! something is missing.... if σ ⇒ h i somehow ﬁxed, is this really spontaneous breaking? does not look no.

To avoid!

Next: solution to (a),(b),(c): gauged local scale invariance: Weyl gravity: dilaton σ eaten by Weyl “photon”; ⇒ hence, no ghost. [ 10 ]

(3). Gauged local scale invariance - Weyl quadratic gravity:

σ(x) 1 invariance under gˆ (x)=Ω(x)2 g (x), σˆ(x)= , ωˆ (x)= ω (x) ∂ lnΩ(x)2 (b) µν µν Ω(x) µ µ − q µ

˜ ˜ρ ρ ρ ρ ρ Weyl geometry µ gαβ = q ωµ gαβ; with Γµν = Γµν + (q/2) δµ ων + δν ωµ gµν ω inv of (b) ∇ − h − i ρ Γµν = Levi-Civita. . 3 R˜ D˜ σ R˜ = R 3 q D ωµ q2 ωµω . R˜ˆ = , D˜ σ = (∂ q/2 ω ) σ D˜ˆ σˆ = µ − µ − 2 µ ⇒ Ω2 µ µ − µ ⇒ µ Ω

if ω 0: Γ˜ρ Γρ , Weyl geometry Riemann geometry; R˜ R, Weyl tensor C˜ C ⇒ µ → µν → µν → → µνρσ → µνρσ

2 2 2 2 2 invariants of (b): √g R˜ , √gσ R,˜ √g (D˜ µσ) , √gF , √g C˜ where Fµν = ∂µων ∂νωµ ⇒ µν µναβ −

no other invariants like higher dimensional ops (no scale to suppress them!) X˜ denotes a quantity in Weyl geometry [ 11 ]

Weyl gravity = Einstein gravity + c.c. + massive ω [ D.G. arXiv:1812.08613, 1904.06596 ] • µ

ξ0 ˜2 1 2 L0 = √g R Fµν , ξ0 > 0, original Weyl action (1918) n 4! − 4 o ξ0 2 ˜ 4 1 2 2 ˜ = √g 2 σ R σ Fµν eom: σ = R; dilaton: ln σ ln σ lnΩ n4! − − − 4 o − → − + Gauss-Bonnet + Weyl-tensor-squared.

Weyl gauge transf of “gauge ﬁxing”: • 2 ξ0 σ 2 2 6 Mp 1 2 ˜ µ 3 2 µ Ω= 2 gˆµν =Ω gµν, σˆ = , ωˆµ =ωµ ∂µ ln σ ; use R = R 3qDµω q ωµω 6Mp ⇒ ξ0 −q − − 2

3 M 4 1 2 ˆ p 3 2 2 µ 1 ˆ2 L0 = gˆ Mp R + q Mp ωˆµ ωˆ Fµν . p n − 2 − 2 ξ0 4 − 4 o

Einstein-Proca action for massive ω which “absorbed” the dilaton ln σ ﬁeld! mass of ω qM . ⇒ µ µ ∝ p Stueckelberg mechanism, massless σ + ω massive ω . no ghost! # dof=3 conserved! ⇒ µ → µ massive ω decouples; M : emergent scale of Weyl gauge symmetry breaking. Weyl gravity viable ⇒ µ p ⇒ M σ dynamically in FRW universe [arxiv:1801.07676 by G. Ross, C. Hill, P. Ferreira] see talk by Graham Ross ⇒ p ∼h i [ 12 ]

Geometric Stueckelberg∗ mechanism: A Weyl gauge invariant action: •

1 2 1 ˜ 2 L1 = √g Fµν + (Dµσ) + , Fµν = ∂µων ∂νωµ h − 4 2 ··· i −

2 D˜ µσ = (∂µ q/2 ωµ) σ = ( q/2) σ ωµ (1/q) ∂µ ln σ . − − −

σ2 1 1 “gauge ﬁxing” transformation: Ω2 = , gˆ =Ω2g σˆ = σ= M, ωˆ = ω ∂ lnΩ2 M 2 µν µν Ω µ µ − q µ

2 1 ˆ2 q 2 µ Proca action L1 = gˆ Fµν + M ωˆµωˆ + . → p h − 4 8 ··· i

- spontaneous breaking; ω has eaten the scalar ln σ (note ln σ ln σ lnΩ) µ → − - number of d.o.f. is conserved (=3).

————– ∗ Baron Ernst Carl Gerlach Stueckelberg von Breidenbach zu Breidenstein und Melsbach [ 13 ]

Weyl gravity with matter φ- scalar ﬁeld, higgs-like, etc • g ξ0 ˜2 1 2 √ 2 ˜ 1 ˜ 2 λ 4 ˜2 2 ˜ 4 L2 = √g R Fµν ξφ R + √g (Dµφ) φ , R 2σ R σ n 4! − 4 o − 12 n 2 − 12 o →− − 1 2 2 ˜ 1 2 1 ˜ 2 1 4 4 2 1 2 2 = √g (ξ0 σ + ξφ ) R Fµν + (Dµφ) (λφ + ξ0 σ ) , ρ = (ξ0σ + ξφ ) n − 4! − 4 2 − 4! o 6

ρ2 1 Weyl gauge transformation (b): “gauge ﬁxing”: 2 Ω= 2 , ρˆ = Mp, ωˆµ = ωµ ln ρ • Mp − q

1 2 ˆ 3 2 2 µ 1 ˆ2 1 ˜ˆ ˆ 2 Einstein-Proca action for ωµ : L2 = gˆ Mp R + Mp q ωˆµωˆ Fµν + (Dµφ) V , p n − 2 4 − 4 2 − io

3M 4 ˆ2 2 2 p ξφ λ ˆ4 φ V = 1 2 + φ + 2 , φ Mp. 2 ξ0 h − 6 Mp i 4! OMp ≪

No dilaton ln σ left (eaten by ω ). Gravitational higgs mechanism. Also m2 = ( ξ/ξ ) M 2. ⇒ µ φ − 0 p Fixed point for ξ? The limit ξ 0 S S β ξ? not (ξ +1/6)! Quantum corrections? ⇒ → ⇒ v × h ⇒ ξ ∝ [ 14 ]

2 4 2 φ 4 φ 3 Mp Weyl inﬂation: V = V0 1 ξ1 sinh + λ1ξ0 sinh , V0 = ; φ>Mp : OK! • nh − Mp√6i Mp√6o 2 ξ0

2 2 ξ1 2 2φ 3 2ξ1 2φ 2 λ1ξ0 ξ1 1: ǫ = sinh + (ξ1); η = cosh + (ξ1) ≪ ≪ 3 Mp√6 O − 3 Mp√6 O

4 2φ 2 2 2 ns = 1+2η 6ǫ =1 ξ1 cosh ∗ + (ξ1); r = 3(1 ns) + (ξ1) − − 3 Mp√6 O ⇒ − O

2 0.002567 r 0.00303 if n =0.9670 0.0037; n 1 2/N, r = 12/N , N N +4.1; N efolds ⇒ ≤ ≤ s ± s ≈ − ≈ r midly smaller than Starobinsky, where: N N; r 0.0031, n 0.968 (N =60) → s ≈ s ≈ [D.G. arxiv:1906.11572; G. Ross, C. Hill, P, Ferreira, J. Noller 1906.03415] see next talk by Graham Ross. [ 15 ]

Conclusions: •

I. Flat space:

Quantum SI can protect a classical hierarchy of: higgs vev dilaton vev ⇒ ≪ - All scales from ﬁelds vev’s: hierarchy stability not a DR artefact

II. Curved space:

- Weyl’s original gauge symmetry/gravity, viable theory; no ghost dilaton; # dof conserved. Weyl gravity = Einstein-Proca for ω + cc, after Stueckelberg, dilaton eaten by Weyl “photon” ⇒ µ m qM . ω ∼ p Weyl inﬂation predicts: 0.00257 r 0.00303 for N = 60, n = measured 68%CL ⇒ ≤ ≤ s (such r values will soon be tested, LiteBIRD, CMB-S4). [ 16 ] ———— BACKUP SLIDES ————

µ ∂ ν µ Dilatation current Dµ: D = L (x ∂νφj + dφ) x , dφ = (d 2)/2 (scalars), • ∂(∂µφj) − L −

µ ∂ ∂ - in d =4 2ǫ, potential V˜ : ∂µD = (dφ +1)(∂µφj) L + dφ φj L d − ∂(∂µφj) ∂φj − L d 2 ∂V˜ = d V˜ − φj , φj = φ,σ; (onshell, canonical k.t.) − 2 ∂φj

2d/(d 2) µ in SI theory: V˜ homogeneous in d dim’s: V˜ (ρφ )= ρ − V˜ (φ ) ∂ D =0, conserved. • j j ⇒ µ in “usual” reg: µ =const, no σ: V˜ = µ2ǫ V (φ), and V is scale inv in d =4: •

µ 2ǫ 2ǫ βλ ∂V ∂V ∂µD =2 ǫ µ V 2 ǫµ λ + + βλ , [=0 only if βλ = 0]. (scale anomaly) ∼ h ǫ ··· i ∂λ ∝ ∂λ - diﬀerent ﬁeld content! [Shaposhnikov et al, Tamarit 2013]

“For scale invariance, though, the situation is hopeless; any cutoﬀ procedure necessarily involves a large mass and a large mass necessarily breaks scale invariance in a large way. This argument does not show that the occurence of anomalies is inevitable [.....]” (S. Coleman: Aspects of Symmetry, p.82, 1985). [ 17 ]

Three-loop SI potential: Counterterms! [Monin 2015] • σ˜ 4 ǫ σ φ σ 6 8 h i 1 (3) 2 2ǫ 1 (3) 4 1 (3) φ 1 (3) φ δL3 = δ (∂µφ) µ δ λφ + δ λ6 + δ λ8 2 φ − n 4! λ 6 λ6 σ2 8 λ8 σ4 o

4 4 (3) 3 λ (3) 275 λ (3) δλ6 = 3 , δλ8 = 3 γφ , βλ6, βλ8 = .... 2 λ6 κ ǫ 864 λ8 κ ǫ. ⇒

······

Integrate CS: Three-loop term: U = ∆V + V (3) + V (3,n). V (3)=’old’[ µ zσ ]; new: V (3,n) and ∆V 3 →

6 8 4 4 λ6 φ λ8 φ (3) λ φ 97 9 ζ[3] Vφφ 31 2 Vφφ 9 3 Vφφ ∆V = + , V = + + A0 + ln ln + ln σ2 σ4 κ3 nQ 128 64 4 (zσ)2 − 96 (zσ)2 64 (zσ)2 o

3 4 2 4 (3,n) λ φ λ6 φ 401 λ φ Vφφ λ 2 V = 27λ + λ8 ln . Vφφ φ . 2 κ3 n − 2 8 σ2 72 − 16 σ4 o (zσ)2 ≡ 2 [D.G. arXiv:1712.06024]

SI eﬀective operators always suppressed by σ . At large σ enhanced symmetry S S restored ⇒ ⇒ v × h ⇒ forbids c-terms λ2 φ2σ2 = λ2 σ 2 φ2 + No tunning of Higgs selfcoupling λ for large σ. ⇒ h i ··· [ 18 ] σ˜ 4 ǫ σ φ h i Three-loop SI potential • 6 8 1 (3) 2 2ǫ 1 (3) 4 1 (3) φ 1 (3) φ Counterterm: δL3 = δ (∂µφ) µ δ λφ + δ λ6 + δ λ8 [Monin 2015] 2 φ − 4! λ 6 λ6 σ2 8 λ8 σ4

3 4 4 (3) λ 1 1 (3) 3 λ (3) 275 λ δφ = 3 2 , δλ6 = 3 , δλ8 = 3 −4κ 6ǫ − 12ǫ 2 λ6 κ ǫ 864 λ8 κ ǫ.

(3) 3 3 B 2ǫ 3 B So γφ = λ /(16κ ). With ZX =1+ δX and λ6 = µ (σ)λ6Zλ6Zφ− Zσ and (d/d ln z) λ6 =0,

2 3 λ λ6 λ 3 βλ6 = + 9λ λ6 , (similar βλ8). 2κ2 κ3 − 8

Callan-Symanzik eq: V (3) [“usual” with µ σ]+V (3,n) [new]: gives solution shown on previous page: →

∂V (3) ∂V (2) ∂V (1) ∂V ∂V (1) ∂V + β(1) + β(2) + β(3) + γ(2) + γ(3) = (λ5). ∂ ln z λ ∂λ λ ∂λ λ ∂λ φ ∂ ln φ φ ∂ ln φ O j

∂V (3,n) ∂V (2,n) ∂V + β(1) + β(3,n) = (λ5), λ = λ,λ ,λ . λj λj j j 6 8 ∂ ln z ∂λj ∂λj O [ 19 ]

SM+dilaton: one-loop beta functions (similar to when µ = constant, same ﬁeld content): •

1 9 4 3 4 3 2 2 4 3 2 9 2 2 2 2 βλ = 3 g2 + g1 + g1g2 12ht 4λφ g1 + g2 3ht +4λφ +3λm + 96λmλ6 φ κ h 4 4 2 − − 4 4 − i

2λm 1 3 2 9 2 2 βλm = λφ +2λm + λσ g1 + g2 3ht λm, λm stays small (f.p.). κ h 2 − 4 4 − i ∝ ⇒

for new couplings 3λ6 3 2 9 2 2 βλ6 = 6λφ 8λm + λσ 2 g1 + g2 3ht • κ h − − 4 4 − i 2 3 2 9 2 2 βλ8 = 2λ6 (28λ6 + λm) 4λ8 g1 + g2 3ht κh − 4 4 − i 2 3 2 9 2 2 βλ10 = 10 4λ6 λ10 g1 + g2 3ht h − 4 4 − i 2 3 2 9 2 2 βλ12 = 2 3λ6 6λ12 g1 + g2 3ht h − 4 4 − i

if one sets λ =0 at tree level, then β =0 at one-loop, but emerge at 2-loops. ⇒ 6,8,10,.. λ6,8,10,12 [ 20 ]

Weyl quadratic gravity with matter Einstein gravity + massive ω [DG arXiv:1812.08613, 1904.06596] • → µ

ξ0 ˜2 1 2 L1 = √g R Fµν , ξ0 > 0, original Weyl action (1918) h4! − 4 i ξ0 2 ˜ 4 1 2 2 ˜ = √g 2 σ R σ Fµν eom: σ = R; dilaton: ln σ ln σ lnΩ h4! − − − 4 i − → −

Use R˜ = R 3q D ωµ 3/2 q2 ω ωµ then, in Riemann language: − µ − µ

2 ξ0 1 2 2 ξ0 4 q 2 2 2 1 2 L1 = √g σ R + (∂µσ) σ + ξ0 σ ωµ 1/q∂µ ln σ Fµν . n − 2 h6 i − 4! 8 − − 4 o

ξ σ2 6 M 1 gauge transf (b) “gauge ﬁxing!” 0 2 2 p 2 Ω= 2 gˆµν =Ω gµν, σˆ = , ωˆµ =ωµ ∂µ ln σ • 6Mp ⇒ ξ0 −q

3 M 4 1 2 ˆ p 3 2 2 µ 1 ˆ2 L1 = gˆ Mp R + q M ωˆµ ωˆ Fµν . p n − 2 − 2ξ0 4 − 4 o

Einstein-Proca action for (massive) ω which has “eaten” ln σ ﬁeld! no ghost! # dof conserved! ⇒ µ α 1 q cons. current ∂ (Fαµ √g)+ √gξ0 φ0 ∂µ ωµ φ0 =0 2 − 2