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Scale Invariance and the Standard Model.Pdf 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.
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