Scale invariance and the Standard Model

Theory Institute in Particle Physics and Cosmology, Warsaw, October 2019

G. G. Ross (work done with P.Ferreira, C.Hill and J.Noller) Motivation: Hierarchies

Λ ≡ ρ ∼ 10−122 M 4 • CC vac P

m ∼ 10−16 M • Higgs p

m ≪ H (slow roll) • inflaton I Scale Invariance & m Higgs t,W , Renormalisable Field theory: h Z,h h 3G δm2 = F 4m2 −2m2 − m2 − m2 Λ2 h 2 ( t W Z h ) 4 2π

but….

2 δm not measureable

2 2 2 …only m = m0 +δ m “physical”

d m2 3m2 ⎛ 3g 2 3g 2 ⎞ 2 h = h 2λ + y2 − 2 − 1 + ... Only m = 0 special d ln µ 8π 2 ⎜ t 4 20 ⎟ h ⎝ ⎠

2 Wetterich Symmetry enhanced Small m h natural Bardeen -scale invariant Scale Invariance & mHiggs Heavy

Scale invariance at high scale, Λ M X h h

2 2 ⎛ Q + M ⎞ d m2 2 2 X h M 2 m M ln ∝ X δ h ∝ X ⎜ 2 ⎟ d ln µ Λ ⎝ ⎠

2 2 2 2 2 2 m (Q = Λ )= m +δm = 0 ⇒ m (0)≈ 0 (MX large) 0 /

The “real” hierarchy problem Scale Invariance & m Higgs

Scale invariance at high scale, Λ M X h h

2 2 ⎛ Q + M ⎞ d m2 2 2 X h M 2 m M ln ∝ X δ h ∝ X ⎜ 2 ⎟ d ln µ Λ ⎝ ⎠

2 2 2 2 2 2 m (Q = Λ )= m +δm = 0 ⇒ m (0)≈ 0 (MX large) 0 /

The “real” hierarchy problem

✔ 2 Small m natural (no heavy states) Standard Model: h Scale Invariance & m Higgs

Scale invariance at high scale, Λ M X h h

2 2 ⎛ Q + M ⎞ d m2 2 2 X h M 2 m M ln ∝ X δ h ∝ X ⎜ 2 ⎟ d ln µ Λ ⎝ ⎠

2 2 2 2 2 2 m (Q = Λ )= m +δm = 0 ⇒ m (0)≈ 0 (MX large) 0 /

The “real” hierarchy problem

✔ 2 Small m natural (no heavy states) Standard Model: h

…but what about gravity ? Scale Invariance & m Higgs

Scale invariance at high scale, Λ M X h h

2 2 ⎛ Q + M ⎞ d m2 2 2 X h M 2 m M ln ∝ X δ h ∝ X ⎜ 2 ⎟ d ln µ Λ ⎝ ⎠

2 2 2 2 2 2 m (Q = Λ )= m +δm = 0 ⇒ m (0)≈ 0 (MX large) 0 /

The “real” hierarchy problem

2 Small m natural (heavy states?) ? Standard Model + Gravity: h

Λ ? Small CC unnatural (IR RG running? ) Tsamis, Woodard Wetterich….. Scale invariance : Jordan-Brans-Dicke gravity

⎡⎛ 1 1 ⎞ ⎤ S = −g g µν ∂ φ ∂ φ − α φ 2 R − λφ 4 ∫ ⎢⎜ 2 µ ν 12 ⎟ ⎥ ⎣⎝ ⎠ ⎦

ε φ(x) → e φ(x) Global Weyl (scale) invariance : g (x) → e−2ε g (x), det(−g(x)) → e−4ε det(−g(x)) { µν µν Jordan-Brans-Dicke gravity

⎡⎛ 1 1 ⎞ ⎤ S = −g g µν ∂ φ ∂ φ − α φ 2 R − λφ 4 ∫ ⎢⎜ 2 µ ν 12 ⎟ ⎥ ⎣⎝ ⎠ ⎦

ε φ(x) → e φ(x) Global Weyl (scale) invariance : g (x) → e−2ε g (x), det(−g(x)) → e−4ε det(−g(x)) { µν µν

…how is scale symmetry broken? Jordan-Brans-Dicke gravity …spontaneous breaking

⎡⎛ 1 1 ⎞ ⎤ S = −g g µν ∂ φ ∂ φ − α φ 2 R − λφ 4 ∫ ⎢⎜ 2 µ ν 12 ⎟ ⎥ ⎣⎝ ⎠ ⎦

ε φ(x) → e φ(x) Global Weyl (scale) invariance : g (x) → e−2ε g (x), det(−g(x)) → e−4ε det(−g(x)) { µν µν

Noether current : 1 δ S K = = (1−α ) φ ∂ φ µ det( g) δ ∂ ε ( µ ) − µ

FRW: Dµ K = K!! + 3HK! = 0 1 2 µ K K, K (1 ) t µ = ∂µ = −α φ ⎛ dt' ⎞ 2 { K(t) = c + c → constant 1 2 ∫ ⎜ 3 ⎟ t ⎝ a(t ') ⎠ 0

“Inertial” symmetry breaking : 1 K = (1−α )φ 2 → constant Scale breaking order parameter 2 independent of potential ! - set by initial (chaotic) conditions 2 1 2 M Planck = − α φ - only ratios of masses physical 6 Ferreira, Hill, GGR

∂V (φ) ( KG : (1−α )⎡φD2φ + ∂µφ ∂ φ⎤ = φ − 4V (φ) = 0, independent of potential) ⎣ µ ⎦ ∂φ Inflation?

1 ⎛ 1 ⎞ ns ~ 0.96 Starobinsky 1980 S = d 4x −g M 2 R + R2 2 ∫ ⎜ P 6 f 2 ⎟ { r ~ 0.004 ⎝ 0 ⎠

Scale invariant version :

⎛ 1 λ 1 1 ⎞ S = d 4x −g g µν ∂ φ ∂ φ − φ 4 − α φ 2 R + R2 . ∫ ⎜ 2 µ ν 4 12 1 6 f 2 ⎟ ⎝ 0 ⎠

4th order derivatives

⇒ new scalar degree of freedom

Herrera, Contreras, del Campo Maeda Tambalo, Rinaldi Rinaldi, Vanzo Bamba,Odintsov,Tretyakov Ghilencea Karam, Pappas, Tamvakis Ferreira, Hill, Noller, GGR Inflation?

1 ⎛ 1 ⎞ ns ~ 0.96 Starobinsky 1980 S = d 4x −g M 2 R + R2 2 ∫ ⎜ P 6 f 2 ⎟ { r ~ 0.004 ⎝ 0 ⎠

Scale invariant version :

⎛ 1 λ 1 1 ⎞ S d 4x g g µν 4 2 R R2 . = − ⎜ ∂µφ ∂ν φ − φ − α1φ + 2 ⎟ α ∫ 2 4 12 6 f 2 2 2 ⎝ 0 ⎠ ξ = 6 f0 ( ) 12

⎛ 1 λ 1 1 ξ ⎞ 2 α 2 S ≡ d 4x −g g µν ∂ φ ∂ φ − φ 4 − α φ 2 R − α η2 R − η4 ) η = − R ∫ ⎜ µ ν 1 2 ⎟ 6ξ ⎝ 2 4 12 12 4 ⎠ Inflation?

1 ⎛ 1 ⎞ ns ~ 0.96 Starobinsky 1980 S = d 4x −g M 2 R + R2 2 ∫ ⎜ P 6 f 2 ⎟ { r ~ 0.004 ⎝ 0 ⎠

Scale invariant version :

⎛ 1 λ 1 1 ⎞ S d 4x g g µν 4 2 R R2 . = − ⎜ ∂µφ ∂ν φ − φ − α1φ + 2 ⎟ α ∫ 2 4 12 6 f 2 2 2 ⎝ 0 ⎠ ξ = 6 f0 ( ) 12

⎛ 1 λ 1 1 ξ ⎞ 2 α 2 S ≡ d 4x −g g µν ∂ φ ∂ φ − φ 4 − α φ 2 R − α η2 R − η4 ) η = − R ∫ ⎜ µ ν 1 2 ⎟ 6ξ ⎝ 2 4 12 12 4 ⎠

Inertial symmetry breaking

δ S 1 2 2 K ≡ = (1−α )φ ∂ φ −α η ∂ η, K = (1−α )φ −α η → constant µ µ 1 µ 2 µ ( 1 2 ) δ ∂ ε 2 i.e. Single field inflation Dilaton decoupling

⎛ 1 λ 1 1 ξ ⎞ S ≡ d 4x −g g µν ∂ φ ∂ φ − φ 4 − α φ 2 R − α η2 R − η4 ) ∫ ⎜ µ ν 1 2 ⎟ ⎝ 2 4 12 12 4 ⎠

Change variables : σ (x) → σ (x) − ε, φ!(x) → φ!(x) ( ) φ = e−σ ( x)/ f φˆ, η = e−σ ( x)/ f ηˆ, g = e2σ ( x)/ f gˆ , R(g) = e2σ / f R(gˆ) + 6Dˆ 2σ / f + 6(Dˆσ / f )2 µν µν ( )

⎛ 1 1 1 1 λ ξ ⎞ L gˆ ( ˆ2 ˆ 2 )Rˆ ˆ µ ˆ Kˆ µ µ Kˆ ˆ4 ˆ 4 = − ⎜ − α1φ +α 2η + ∂µφ ∂ φ + 2 ∂µ σ ∂ σ + ∂µ σ ∂ − φ − η ⎟ ⎝ 12 2 2 f f 4 4 ⎠ Dilaton decoupling

⎛ 1 λ 1 1 ξ ⎞ S ≡ d 4x −g g µν ∂ φ ∂ φ − φ 4 − α φ 2 R − α η2 R − η4 ) ∫ ⎜ µ ν 1 2 ⎟ ⎝ 2 4 12 12 4 ⎠

Change variables :

φ = e−σ ( x)/ f φˆ, η = e−σ ( x)/ f ηˆ, g = e2σ ( x)/ f gˆ , R(g) = e2σ / f R(gˆ) + 6Dˆ 2σ / f + 6(Dˆσ / f )2 µν µν ( )

⎛ 1 1 1 1 λ ξ ⎞ L gˆ ( ˆ2 ˆ 2 )Rˆ ˆ µ ˆ Kˆ µ µ Kˆ ˆ4 ˆ 4 = − ⎜ − α1φ +α 2η + ∂µφ ∂ φ + 2 ∂µ σ ∂ σ + ∂µ σ ∂ − φ − η ⎟ ⎝ 12 2 2 f f 4 4 ⎠

1 Kˆ (1 ) ˆ2 ˆ 2 For constant = ( −α1 φ −α 2η ) 2 Dilaton

⎛ 1 1 λ ξ 1 ⎞ L gˆ ( ˆ2 ˆ 2 )Rˆ ˆ µ ˆ ˆ4 ˆ 4 Kˆ µ = − ⎜ − α1φ +α 2η + ∂µφ ∂ φ − φ − η + 2 ∂µ σ ∂ σ ⎟ ⎝ 12 2 4 4 2 f ⎠

Decoupled dilaton…

…avoids astrophysical constraints on 5th force in Brans Dicke Ferreira, Hill, GGR Brax, Davis Local Weyl symmetry Smolin Nishino, Rajpoot φ(x),η(x) → eε ( x)φ(x),η(x); g (x) → e−2ε ( x)g (x) µν µν

Need Weyl “photon” A → A + ∂ ε(x) • µ µ µ D! φ ≡ (∂ − A )φ, D! φ → e−ε( x) D! φ µ µ µ µ µ

R → R! • Riemannian Ricci scalar Weyl geometric version

R! = R − 6D Aµ − 6A A , R! → e2ε( x) R! µ µ µ

⎛ 1 λ 1 1 1 ⎞ S d 4x g g µν D! D! 4 2 R! R! 2 F µν F L = − ⎜ µφ νφ − φ − α1φ + 2 − 2 µν ⎟ ∫ ⎝ 2 4 12 6 f 4e ⎠ 0

Ghilencea ⎛ 1 λ 1 1 1 ⎞ S d 4x g g µν D! D! 4 2 R! R! 2 F µν F L = − ⎜ µφ νφ − φ − α1φ + 2 − 2 µν ⎟ ∫ ⎝ 2 4 12 6 f 4e ⎠ 0 ⎛ 1 λ 1 ξ 1 ⎞ S ≡ d 4x −g g µν D! φD! φ − φ 4 − (α φ 2 +α η2 )R! − η4 − F µν F L ∫ ⎝⎜ 2 µ ν 4 12 1 2 4 4e2 µν ⎠⎟

R! Substituting for and integrating by parts

⎛ 1 1 1 1 1 λ ξ ⎞ S = d 4x −g ∂ φ ∂µφ − F µν F − A K µ + K A Aµ − (α φ 2 +α η2 )R − φ 4 − η4 L ∫ ⎜ µ 2 µν µ µ 1 2 ⎟ ⎝ 2 4e 2 2 12 4 4 ⎠ where K = ∂ K, K = (1+α )φ 2 +α η2 µ µ 1 2 Weyl photon eq of motion:

1 δ L − = Dµ F − K − 2A K = 0 ⇒ K → constant 1 µ µν µ µ B = A − ∂ σ −g δ A µ µ µ f decoupled Changing variables (here a gauge choice)

⎛ 1 1 λ ξ 1 1 ⎞ L g ( ˆ2 ˆ 2 )Rˆ ˆ µ ˆ ˆ4 ˆ 4 F µν F K B Bµ = ⎜ − α1φ +α 2η + ∂µφ ∂ φ − φ − η − 2 µν + µ ⎟ ⎝ 12 2 4 4 4e 2 ⎠ Ferreira, Hill, Noller, GGR c.f. Higgs/Dilaton (Weyl photon) model

~Brans-Dicke scalar 2nd singlet models “Higgs” sector

4 2 4 2 2 2 ⎛ 1 λ 1 1 µν ξ η 1 ⎞ ! µν ! ! ! ! " + g ∂ χ! ∂ χ! − χ! + φ" χ! − α χ! R! S = −g g ∂µφ ∂ν φ − φ − α1φ R µ ν 2 ⎟ ∫ ⎝⎜ 2 4 12 2 4 2 12 ⎠

Absent in R2 case

Shaposhnikov, Zenhausern Garcia Bellida, Rubio, Shaposhnikov, Zenhausern Hill, Ferrera, GGR Ghilencea, Lee Inflation (global / local) (Jordan frame) Initial flow to ellipse ⎛ 1 1 λ ξ ⎞ S ≡ d 4x −g − (α φˆ2 +α ηˆ 2 )Rˆ + ∂ φˆ∂µφˆ − φˆ4 − ηˆ 4 ) ∫ ⎜ 1 2 µ ⎟ ⎝ 12 2 4 4 ⎠ 1 Kˆ (1 ) ˆ2 ˆ 2 constant = ( −α1 φ −α 2η ) = 2

(dropping decoupled dilaton/Weyl photon inflation sector identical) Inflation (global / local) (Jordan frame) Initial flow to ellipse ⎛ 1 1 λ ξ ⎞ S ≡ d 4x −g − (α φˆ2 +α ηˆ 2 )Rˆ + ∂ φˆ∂µφˆ − φˆ4 − ηˆ 4 ) ∫ ⎜ 1 2 µ ⎟ ⎝ 12 2 4 4 ⎠ 1 Kˆ (1 ) ˆ2 ˆ 2 constant = ( −α1 φ −α 2η ) = 2

Klein-Gordon equations : 1 1 2 ˆ ˆ3 ˆ ˆ 3 ˆ D φ + λφ + α1φR = 0, ξη + α 2ηR = 0 6 6 D2φˆ α = −α λφˆ2 +α ξηˆ 2 2 ˆ 2 1 φ Inflation (global / local) (Jordan frame) Initial flow to ellipse ⎛ 1 1 λ ξ ⎞ S ≡ d 4x −g − (α φˆ2 +α ηˆ 2 )Rˆ + ∂ φˆ∂µφˆ − φˆ4 − ηˆ 4 ) ∫ ⎜ 1 2 µ ⎟ ⎝ 12 2 4 4 ⎠ 1 Kˆ (1 ) ˆ2 ˆ 2 constant = ( −α1 φ −α 2η ) = 2

Klein-Gordon equations : 1 1 2 ˆ ˆ3 ˆ ˆ 3 ˆ D φ + λφ + α1φR = 0, ξη + α 2ηR = 0 6 6 D2φˆ α = −α λφˆ2 +α ξηˆ 2 2 ˆ 2 1 φ

, 1 Slow roll : λ << ξ α1 <<

2 4 2 2 ˆ ˆ! 2 ˆ ∂N φ = − α1φ D φ → 3Hφ = 3H ∂N φ, N = ln a(t) 3 ( )

4 4 − α N − α1 N J 1 1 J φ 2 = φ 2 e 3 , η2 = C − φ 2 e 3 E | | E η α 2 Inflation

Karamitses,Pilaftsis Frame independent analysis Ferreira, Hill, Noller, GGR

δ 3 f f λ θ 4 + ξ θ 4 1 2 G 11 ,A ,B , U 1 2 , f M 2 2 , ˆ, ˆ AB = + 2 = 2 ≡ P ≡ − ∑α Aθ A θ1 = φ θ2 = η f 2 f 4 f 6 A=1

X = (lnU ) ... A ,A

1 A A ε U = X X A, ηU = − X lnε U ( ), 2 φ A

n 1 2 U s = − ε −ηU ... Inflation

Frame independent analysis δ 3 f f λ θ 4 + ξ θ 4 1 2 G 11 ,A ,B , U 1 2 , f M 2 2 , ˆ, ˆ AB = + 2 = 2 ≡ P ≡ − ∑α Aθ A θ1 = φ θ2 = η f 2 f 4 f 6 A=1

scalar spectral index

tensor to scalar ratio

scalar running

tensor spectral index

tensor running

1 ⎛ 3 ⎞ N = N + ln + O(α ) N 60, N 58 E J ⎜ ⎟ 1 ( J = E ∼ ) 2 ⎝ 2N J ⎠ Ferreira, Hill, Noller, GGR

Inflation

Frame independent analysis

δ 3 f f λ θ 4 + ξ θ 4 1 2 G 11 ,A ,B , U 1 2 , f M 2 2 , ˆ, ˆ AB = + 2 = 2 ≡ P ≡ − ∑α Aθ A θ1 = φ θ2 = η f 2 f 4 f 6 A=1

N 4α (e−ν J +1) n = 1+ 1 + O(α 2 ), S −ν N J 1 3(1− e )

N 64α 2e−ν J r = 1 + O(α 3) −ν N J 2 1 3(e −1) α1

n = 0.9652 ± 0.0047 ⇒ α < 0.019, n < 0.967 s 1 s r 0.0026 c. f . r 0.07 | > < Planck

/ 2 O(10−10 ) ξ α 2 = α1 Comparison with Higgs/dilaton model

β ≡ α c. f .Starobinsky n ~ 0.96, r ~ 0.004 0 ……..coloured lines ( 2 ) s ≡ α1 → Ferreira, Hill, Noller, GGR

Hierarchy generation

⎛ 1 1 λ ξ 2 2 ⎞ S ≡ d 4x −g − (α φˆ2 +α ηˆ 2 )Rˆ + ∂ φˆ∂µφˆ − φˆ4 − ηˆ 4 −δφ! η! ) ∫ ⎝⎜ 12 1 2 2 µ 4 4 ⎠⎟

Infrared fixed point: η2 η2 λα +δα IR 2 1 (η χ dilaton-Higgs model) 2 → 2 = ∼ φ φ ξα +δα IR 1 2 Hierarchy of couplings ⇒ hierarchy of scales

λ ≪ δ ≪ ξ

Radiative corrections? Hierarchy generation

⎛ 1 1 λ ξ 2 2 ⎞ S ≡ d 4x −g − (α φˆ2 +α ηˆ 2 )Rˆ + ∂ φˆ∂µφˆ − φˆ4 − ηˆ 4 −δφ! η! ) ∫ ⎝⎜ 12 1 2 2 µ 4 4 ⎠⎟

Infrared fixed point: η2 χ 2 λα +δα → IR = 2 1 χ 2 φ 2 ξα +δα IR 1 2 Hierarchy of couplings ⇒ hierarchy of scales

λ ≪ δ ≪ ξ

Hierarchy stability? Scale invariant regularisation

Scale invariance doesn’t constrain couplings…radiative corrections?

Radiative corrections? Scale invariance - SM radiative corrections

⎡1 2 λ 4 ⎤ S = d 4x −g! ∂ φ! ∂µφ! − φ! ∫ ⎢2 ∑ µ i i 4 1 ⎥ Scale invariant ⎣ i=1 ⎦ f f ⎛ 1 2 2 ⎞ ! sin , ! cos (1 )! f 2 φ1 = θ φ 2 = θ ⎜ ∑ −α i φ i = ⎟ ⎝ 2 i 1 ⎠ 1−α 1−α −

⎛ fθ ⎞ ⎜ Φ = , θ small⎟ ⎜ 1−α ⎟ ⎝ 1 ⎠

Renormalisation conditions:

λ λ 2Φ4 ⎛ Φ2 25⎞ W 4 ln = Φ + 2 ⎜ 2 − ⎟ 4! 256 M 6 Coleman-Weinberg π ⎝ ⎠

Weyl invariant f 2 1 C = 2 4 2 M 2 1−α ⎛ ⎞ 2 λ 4 λ φ1 ⎛ Cφ1 ⎞ 25 W ln 2 ≈ φ1 + 2 ⎜ ⎜ 2 ⎟ − ⎟ dλ 3λ Usual 4! 256 6 renormalisation π ⎝ ⎝ φ2 ⎠ ⎠ = 2 d logC 32π group equations Hierarchy generation

⎛ 1 1 λ ξ 2 2 ⎞ S ≡ d 4x −g − (α φˆ2 +α ηˆ 2 )Rˆ + ∂ φˆ∂µφˆ − φˆ4 − ηˆ 4 −δφ! η! ) ∫ ⎝⎜ 12 1 2 2 µ 4 4 ⎠⎟

Infrared fixed point: η2 χ 2 λα +δα → IR = 2 1 λ ≪ δ ≪ ξ χ 2 φ 2 ξα +δα IR 1 2

Hierarchy stability?

Non gravitational corrections:

λ = δ = 0 enhances symmetry: φ → φ + c, η → η

✔ 2 dλ dδ η , ∝ δ ,η (C = 2 ) d logC d logC φ Hierarchy generation

⎛ 1 1 λ ξ 2 2 ⎞ S ≡ d 4x −g − (α φˆ2 +α ηˆ 2 )Rˆ + ∂ φˆ∂µφˆ − φˆ4 − ηˆ 4 −δφ! η! ) ∫ ⎝⎜ 12 1 2 2 µ 4 4 ⎠⎟

Infrared fixed point: η2 χ 2 λα +δα → IR = 2 1 λ ≪ δ ≪ ξ χ 2 φ 2 ξα +δα IR 1 2

Hierarchy stability? …including gravitational corrections

0 enhances symmetry: ! ! c, ! ! λ = δ = = α1 φ → φ + η → η

, , , Δλ Δδ ∝ δ λ α1

ΔM 2 2 ✗ (δ term also breaks η Δδ φ α1α 2 2 ∼ 2 ∝ ∼ α 2 M α φ α ! !, ! ! c) P 1 1 φ → φ η → η + Gravitational corrections can destabilise hierarchy Gravitational corrections α ≠ 0 ⇒ gravitational corrections to λ,δ 1,2 n ⎛ Λ ⎞ ✔ Pure Einstein gravity : Δδ ∝α 1 ⎜ M ⎟ ⎝ P ⎠ Gravitational corrections α ≠ 0 ⇒ gravitational corrections to λ,δ 1,2 n ⎛ Λ ⎞ ✔ Pure Einstein gravity : Δδ ∝α 1 ⎜ M ⎟ ⎝ P ⎠

What happens if divergences are softened/eliminated? e.g.I A-gravity :

⎡ 1 2 2 ⎤ 2 R R ⎢ − µν ⎥ 4 R 3 α1 2 α 2 2 S = d x −g ⎢ + − φ R − η R...⎥ ∫ 6 f 2 f 2 12 12 ⎢ 0 2 ⎥ ⎣ ⎦

1 1 ⎡ 1 1 ⎤ 2 1 2 2 = ⎢ − ⎥ ghost M = f M Graviton propagator: M 2 p2 − p4 M 2 p2 p2 − M 2 2 2 P 2 2 ⎣ 2 ⎦ 2

dδ ⎛ δ 2 2 ⎞ (4 )2 5 f 4 f 4 (6 1 (6 1) ⎜ φ η ⎟ π = −α1α 2 ( 2 + 0 α1 + ) α 2 + ⎝ 2 ⎠ d logC Salvio, Strumia

Real hierarchy problem associated with new massive (normal+ghost) states

2 δ M α f 4 ⎛ Λ2 ⎞ η 2 i ln , i 0,2 2 ∼ 2 ⎜ 2 2 ⎟ = M 16π ⎝ M + Q ⎠ P 2 A-gravity :

⎡ 1 2 2 ⎤ 2 R R ⎢ − µν ⎥ 4 R 3 α1 2 α 2 2 S = d x −g ⎢ + − φ R − η R...⎥ ∫ 6 f 2 f 2 12 12 ⎢ 0 2 ⎥ ⎣ ⎦

1 1 ⎡ 1 1 ⎤ 2 1 2 2 = ⎢ − ⎥ ghost M = f M Graviton propagator: M 2 p2 − p4 M 2 p2 p2 − M 2 2 2 P 2 2 ⎣ 2 ⎦ 2

dδ ⎛ δ 2 2 ⎞ (4 )2 5 f 4 f 4 (6 1 (6 1) ⎜ φ η ⎟ π = −α1α 2 ( 2 + 0 α1 + ) α 2 + ⎝ 2 ⎠ d logC Salvio, Strumia

Real hierarchy problem associated with new massive (normal+ghost) states

2 δ M α f 4 ⎛ Λ2 ⎞ η 2 i ln , i 0,2 2 ∼ 2 ⎜ 2 2 ⎟ = M 16π ⎝ M + Q ⎠ P 2

… possibilities ? A-gravity :

⎡ 1 2 2 ⎤ 2 R R ⎢ − µν ⎥ 4 R 3 α1 2 α 2 2 S = d x −g ⎢ + − φ R − η R...⎥ ∫ 6 f 2 f 2 12 12 ⎢ 0 2 ⎥ ⎣ ⎦

1 1 ⎡ 1 1 ⎤ 2 1 2 2 = ⎢ − ⎥ ghost M = f M Graviton propagator: M 2 p2 − p4 M 2 p2 p2 − M 2 2 2 P 2 2 ⎣ 2 ⎦ 2

dδ ⎛ δ 2 2 ⎞ (4 )2 5 f 4 f 4 (6 1 (6 1) ⎜ φ η ⎟ π = −α1α 2 ( 2 + 0 α1 + ) α 2 + ⎝ 2 ⎠ d logC Salvio, Strumia

Real hierarchy problem associated with new massive (normal+ghost) states

2 δ M α f 4 ⎛ Λ2 ⎞ η 2 i ln , i 0,2 2 ∼ 2 ⎜ 2 2 ⎟ = M 16π ⎝ M + Q ⎠ P 2 α → 0? • 2 ✗ 2 dα ⎛ 3 3 ⎞ 4π Higgs ∼ 1+ 6α 2y2 − g 2 − g 2 + 2ξ Rajantie ( ) d lnC ( Higgs )⎝⎜ t 4 2 20 1 ⎠⎟

A-gravity :

⎡ 1 2 2 ⎤ 2 R R ⎢ − µν ⎥ 4 R 3 α1 2 α 2 2 S = d x −g ⎢ + − φ R − η R...⎥ ∫ 6 f 2 f 2 12 12 ⎢ 0 2 ⎥ ⎣ ⎦

1 1 ⎡ 1 1 ⎤ 2 1 2 2 = ⎢ − ⎥ ghost M = f M Graviton propagator: M 2 p2 − p4 M 2 p2 p2 − M 2 2 2 P 2 2 ⎣ 2 ⎦ 2

dδ ⎛ δ 2 2 ⎞ (4 )2 5 f 4 f 4 (6 1 (6 1) ⎜ φ η ⎟ π = −α1α 2 ( 2 + 0 α1 + ) α 2 + ⎝ 2 ⎠ d logC Salvio, Strumia

Real hierarchy problem associated with new massive (normal+ghost) states

2 δ M α f 4 ⎛ Λ2 ⎞ χ 2 i ln , i 0,2 2 ∼ 2 ⎜ 2 2 ⎟ = M 16π ⎝ M + Q ⎠ P 2

f ∼ 10−8 , M ∼ 1010 GeV • 0,2 0,2 New (normal+ghost) states well below Planck scale?? Salvio e.g.II Asymptotic safety: Weinberg Reuter

Gravitational radiative corrections contribute to RG equations above Planck scale

2 M Planck Nonperturbative UV fixed point: 2 → constant k

2 2 2 2 1 ⇒ G k | 2 2 ∼ M k = M + 2ξ k N ( ) k ≫M 2 P ( ) P 0 P 16πξ k 0 Regular behaviour of high energy amplitudes

2 2 Robinson, Wilczek +… f j x j , k > M P β grav ∼ x j { 0, k 2 < M 2 x = g ,g g ,h ,... P j 1 2, 3 t Ghosts in Asymptotic Safety

Functional RG analysis shows Ostrogradski instabilities may be avoided within Asymptotic Safety

(RG flow can drive decoupling of ghosts) Becker, Ripken, Saueressig

S = S EinsteinHilbert + S Matter g G k 2 k = k

y = Y k 2 k −2 k k λk = Λk

2 2 1 δ M Y = 0, µ = ghost decouples η = 0 ∀k k Y M 2 k P Asymptotic safety - prediction

• Higgs mass V(H ) = −m2 |φ |2 +λ |φ |4

∂ µ lnλ = fλ ∂µ λ f > 0, λ k → 0 λ ( tr ) “Infra red” fixed point

m (173.34 0.27 0.71)GeV t = ± ±

Buttazzo et al Wetterich, Shaposhnikov Jan Kwapisz M ∼ 1019 GeV P Summary “Inertial” symmetry breaking : Weyl (scale) invariance is always • spontaneously broken independently of the potential. • Massless dilaton – decouples … avoids BD bounds on 5th force • Slow roll inflation with acceptable properties possible in scale invariant R 2 model r 0.003, n 0.967 > s ≤

Spontaneously broken scale-invariant “SM”+gravity • - only dimensionless ratios meaningful

up to IR corrections H 2 ξ m2 δ H 2 1 ⎛ λ ξµ4 δµ2 ⎞ I ∝ , "h" ∝ , 0 ∝ + + M 2 α 2 M 2 α M 2 α 2 ⎜ 4 4 2 ⎟ P 2 P 1 P 2 ⎝ ⎠ Hierarchy related to hierarchy of couplings

Hierarchy stability : • ✔ Non-gravitational corrections (Landau pole ...above Λ ?) ✔ ? Gravitational corrections (Ghosts...above Λ ?)

Asymptotic safety - postdictions

• U(1)Y : taming the Landau pole 2 g* 2 2 41 g Y ,2 k >ktr Y βg = − fg gY + 2 + ... y 6 16π * gY ,1 2 * * 6.16π fg gY ,1 |UVSFP = 0, gY ,2 |IRSFP = fg > 0 41

* * g ≤ g ( M ) ≤ g Test of asymptotic safety? Y ,1 Y p Y ,2

Asymptotic safety - postdictions

• U(1)Y : taming the Landau pole 2 g* 2 2 41 g Y ,2 k >ktr Y βg = − fg gY + 2 + ... y 6 16π * gY ,1 2 * * 6.16π fg gY ,1 |UVSFP = 0, gY ,2 |IRSFP = fg > 0 41

* * g ≤ g ( M ) ≤ g Test of asymptotic safety? Y ,1 Y p Y ,2

k 2 M 2 yt(b) ⎛ 9 ⎞ • β > P = 3y2 + 9y2 − g 2 − 8g 2 − 6(Y 2 + Y 2 )g 2 − f y yt ( b) 2 ⎜ b(t) t(b) 2 3 Q t(b) Y ⎟ y t(b) 32π ⎝ 2 ⎠

2* 2* 1 * yt − yb = gY ,2 3 RG evolution of couplings from far UV

UV Initial conditions

Scale invariant g* = 0 2,3 2* 2* 1 * yt − yb = gY ,2 3

Eichhorn, Held m 178GeV t =