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 −16 mHiggs ∼ 10 M 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 + ... m 0 2 ⎜ t ⎟ Only h = special d ln µ 8π ⎝ 4 20 ⎠ 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 & mHiggs 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 & mHiggs 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 & mHiggs 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 Small Λ unnatural (IR RG running? )? CC Tsamis, Woodard Wetterich….. Scale invariance : Jordan-Brans-Dicke gravity ⎡⎛ 1 µν 1 2 ⎞ 4 ⎤ S = −g ⎢ g ∂µφ ∂ν φ − α φ R − λφ ⎥ ∫ ⎝⎜ 2 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 µν 1 2 ⎞ 4 ⎤ S = −g ⎢ g ∂µφ ∂ν φ − α φ R − λφ ⎥ ∫ ⎝⎜ 2 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 µν 1 2 ⎞ 4 ⎤ S = −g ⎢ g ∂µφ ∂ν φ − α φ R − λφ ⎥ ∫ ⎝⎜ 2 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 ∫ ⎜ 2 µ ν 4 12 1 12 2 4 ⎟ 6ξ ⎝ ⎠ 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 ) ∫ ⎝⎜ 2 µ ν 4 12 1 12 2 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 ) ∫ ⎜ 2 µ ν 4 12 1 12 2 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 − (α φˆ2 +α ηˆ 2 )Rˆ + ∂ φˆ∂µφˆ + Kˆ ∂ σ ∂µ σ + ∂ σ ∂µ Kˆ − φˆ4 − ηˆ 4 ⎜ 12 1 2 2 µ 2 µ f µ 4 4 ⎟ ⎝ 2 f ⎠ 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 ! µ ! 2ε( x) ! R = R − 6Dµ A − 6Aµ Aµ , R → e R ⎛ 1 λ 1 1 1 ⎞ S = d 4x −g g µν D! φD! φ − φ 4 − α φ 2 R! + R! 2 − F µν F L ∫ ⎜ µ ν 1 2 2 µν ⎟ ⎝ 2 4 12 6 f0 4e ⎠ 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 ⎠⎟ ! Substituting for R 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 − (α φˆ2 +α ηˆ 2 )Rˆ + ∂ φˆ∂µφˆ − φˆ4 − ηˆ 4 − F µν F + K B Bµ ⎝⎜ 12 1 2 2 µ 4 4 4e2 µν 2 µ ⎠⎟ Ferreira, Hill, Noller, GGR c.f.