Scale-invariant alternatives to general relativity

Scale-invariant alternatives to general relativity Mikhail Shaposhnikov Zurich, 21 June 2011 Zurich, 21 June 2011 p. 1

Based on: M.S., Daniel Zenhäusern, Phys. Lett. B 671 (2009) 162 M.S., Daniel Zenhäusern, Phys. Lett. B 671 (2009) 187 Diego Blas, M.S., Daniel Zenhäusern, arxiv:1104.1392 [hep-th] (2011) Zurich, 21 June 2011 p. 2

GR Einstein gravity is a theory which is invariant under all diffeomorphisms, Diff: x µ f µ (x ν ). Pros - consistence with all tests of GR. One of the main predictions - existence of massless graviton. Problems of GR Large dimensionfull coupling constant G 1 N = M2 P, leading to hierarchy problem m H M P. Extra arbitrary fundamental parameter - cosmological constant - which is known to be very small. Quantum gravity? Zurich, 21 June 2011 p. 3

Our proposal Diff TDiff Dilatations TDiff - volume conserving coordinate transformations, [ f µ ] det = 1. x ν Dilatations - global scale transformations (σ = const) Ψ(x) σ n Ψ(σx), n = 0 for the metric, n = 1 for scalars and vectors and n = 3/2 for fermions. Zurich, 21 June 2011 p. 4

Similarity with GR: consistency with all tests Differences with GR: Dynamical origin of all mass scales Hierarchy problem gets a different meaning - an alternative (to SUSY, techicolor, little Higgs or large extra dimensions) solution of it may be possible. Cosmological constant problem acquires another formulation. Natural chaotic cosmological inflation Low energy sector contains a massless dilaton There is Dark Energy even without cosmological constant Quantum gravity? Zurich, 21 June 2011 p. 5

Outline Field theory: classical scale invariance and its spontaneous breakdown Unimodular gravity Scale invariance, unimodular gravity, cosmological constant, inflation and dark energy Quantum scale invariance Dilaton as a part of the metric in TDiff gravity Conclusions Zurich, 21 June 2011 p. 6

Scale invariance Trivial statement: multiply all mass parameters in the theory M W, Λ QCD, M H,M Pl,... by one and the same number : M σm. Physics is not changed! Indeed, this change, supplemented by a dilatation of space-time coordinates x µ σx µ and an appropriate redefinition of the fields does not change the complete quantum effective action of the theory. This transformation, however, is not a symmetry of the theory (symmetry = transformation of dynamical variables which does not change the action) Zurich, 21 June 2011 p. 7

Dilatation symmetry Dilatations: φ(x) σ n φ(σx) n-canonical dimension of the field: n = 1 for scalars and vectors, n = 3/2 for fermions, while the metric transforms as g µν (x) g µν (σx). Dilatation symmetry forbids all dimensionfull couplings: Higgs mass, Newton gravity constant, cosmological constant, etc Ruled out by observations? No, if it is spontaneously broken! Zurich, 21 June 2011 p. 8

Classical scale invariant theory Unique regular scale-invariant Lagrangian L SM = L SM[M 0] + L G + 1 2 ( µχ) 2 V (ϕ,χ) Potential ( χ - dilaton, ϕ - Higgs, ϕ ϕ = 2h 2 ): V (ϕ,χ) = λ ( ϕ ϕ α 2λ χ2 ) 2 + βχ 4, Gravity part L G = ( ξ χ χ 2 + 2ξ h ϕ ϕ ) R 2, Zurich, 21 June 2011 p. 9

Spontaneous breaking of scale invariance Forget first about gravity. Consider scalar potential V (ϕ,χ) = λ ( ϕ ϕ α 2λ χ2 ) 2 + βχ 4, Requirements: vacuum state exists if λ 0, β 0 For λ > 0, β > 0 the vacuum state is unique: χ = 0, ϕ = 0 and scale invariance is exact. Field propagators: scalar 1/p 2, fermion /p/p 2. Greenberg, 1961: free quantum field theory!! If not - theory does not describe particles!! Zurich, 21 June 2011 p. 10

Gravity included - argument for β = 0 gets weaker: for β > 0 there is a de Sitter solution for β < 0 there is an AdS solution However, ds solution is not stable in the presence of massless scalar - no ds invariant ground sate exists AdS solution has different pathologies Zurich, 21 June 2011 p. 11

For λ > 0, β = 0 the scale invariance can be spontaneously broken. The vacuum manifold: h 2 0 = α λ χ2 0 Particles are massive, Planck constant is non-zero: M 2 H M W M t M N χ 0, M Pl χ 0 Phenomenological requirement: α v2 M 2 Pl 10 38 1 Zurich, 21 June 2011 p. 12

Good news: cosmological constant may be zero due to scale invariance and requirement of presence of particles Zurich, 21 June 2011 p. 13

Good news: cosmological constant may be zero due to scale invariance and requirement of presence of particles Bado news: cosmological constant may be zero due to scale invariance and requirement of presence of particles Zurich, 21 June 2011 p. 13

Unimodular gravity Ordinary gravity: the metric g µν is an arbitrary function of space-time coordinates. Invariant under general coordinate transformations Unimodular gravity: the metric g µν is an arbitrary function of space-time coordinates with set[g] = 1. Invariant under general coordinate transformations which conserve the 4-volume. van red Bi, van Dam, N Origin of UG: Field theory describing spin 2 massless particles is either GR or UG Number of physical degrees of freedom is the same. Zurich, 21 June 2011 p. 14

Unimodular gravity and cosmological constant Theories are equivalent everywhere except the way the cosmological constant appears GR. Λ is the fundamental constant: S = 1 d 4 x g[r + Λ] MP 2 UG. Λ does not appear in the action: S = 1 M 2 P d 4 xr Zurich, 21 June 2011 p. 15

UG is equivalent to S = 1 M 2 P d 4 x g [ ( R + Λ(x) 1 1 )] g Equations of motion (G µν - Einstein tensor): G µν = Λ(x)g µν, g = 1 Bianchi identity: Λ(x) ; = 0 Λ(x) = const. Solutions of UG are the same as solutions of GR with an arbitrary cosmological constant. Conclusion: in UG cosmological constant reappears, but as an integral of motion, related to initial conditions Zurich, 21 June 2011 p. 16

Scale invariance + unimodular gravity Solutions of scale-invariant UG are the same as the solutions of scale-invariant GR with the action S = d 4 x [ (ξχ g χ 2 + 2ξ h ϕ ϕ ) ] R 2 + Λ +... Physical interpretation: Einstein frame, g µν = Ω(x) 2 g µν, (ξ χ χ 2 + ξ h h 2 )Ω 2 = M 2 P Λ is not a cosmological constant, it is the strength of a peculiar potential! Zurich, 21 June 2011 p. 17

Relevant part of the Lagrangian (scalars + gravity) in Einstein frame: L E = g ( M 2 P R 2 + K U E(h,χ) ), K - complicated non-linear kinetic term for the scalar fields, ( 1 K = Ω 2 2 ( µχ) 2 + 1 ) 2 ( µh) 2 ) 3M 2 P ( µω) 2. The Einstein-frame potential U E (h,χ): U E (h,χ) = M 4 P [ ( λ h 2 α λ χ2) 2 ] 4(ξ χ χ 2 + ξ h h 2 ) + Λ 2 (ξ χ χ 2 + ξ h h 2 ) 2, Zurich, 21 June 2011 p. 18

U E U E Higgs Dilaton Higgs Dilaton Potential for the Higgs field and dilaton in the Einstein frame. Left: Λ > 0, right Λ < 0. 50% chance (Λ < 0): inflation + late collapse 50% chance (Λ > 0): inflation + late acceleration Zurich, 21 June 2011 p. 19

Inflation Chaotic initial condition: fields χ and h are away from their equilibrium values. Choice of parameters: ξ h 1, ξ χ 1 (will be justified later) Then - dynamics of the Higgs field is more essential, χ const and is frozen. Denote ξ χ χ 2 = M 2 P. Zurich, 21 June 2011 p. 20

Redefinition of the Higgs field to make canonical kinetic term d h dh = Ω 2 + 6ξ 2 h h2 /M 2 P Ω 4 = h h for h < M P /ξ ( ) h M P h ξ exp 6MP for h > M P / ξ Resulting action (Einstein frame action) S E = d 4 x ĝ { M2 P 2 ˆR + µ h µ h 2 1 } λ Ω( h) 4 4 h( h) 4 Potential: U( h) = λ 4 h 4 for h < M P /ξ ( ) 2 λm 4 P 1 e 2 h 6MP for h > M 4ξ 2 P /ξ. Zurich, 21 June 2011 p. 21

Potential in Einstein frame U(χ) λm 4 /ξ 2 /4 Reheating Standard Model λm 4 /ξ 2 λ v 4 /4 /16 0 0 v 0 0 χ end χ COBE χ Zurich, 21 June 2011 p. 22

ǫ = M2 P 2 η = M 2 P ( du/dχ U d 2 U/dχ 2 U ) 2 4 ( 3 exp 4χ ) 6MP 4 ( 3 exp 2χ ) 6MP Slow roll stage Slow roll ends at χ end M P Number of e-folds of inflation at the moment h N is N 6 8 h 2 N h2 end M 2 P /ξ χ 60 5M P COBE normalization U/ǫ = (0.027M P ) 4 gives ξ λ 3 N COBE 0.027 2 49000 λ = 49000 m H 2v Connection of ξ and the Higgs mass! Zurich, 21 June 2011 p. 23

CMB parameters spectrum and tensor modes 0.4 0.3 WMAP5 50 60 0.2 2 SM+ ξh R 0.1 0.0 0.94 0.96 0.98 1.00 1.02 Zurich, 21 June 2011 p. 24

Dark energy Late time evolution of dilaton ρ along the valley, related to χ as ( ) γρ χ = M P exp, γ = 4M P 4. 6 + 1 ξ χ Potential: Wetterich; Ratra, Peebles U ρ = Λ ξ 2 χ exp ( γρ M P ). From observed equation of state: 0 < ξ χ < 0.09 Result: equation of state parameter ω = P/E for dark energy must be different from that of the cosmological constant, but ω < 1 is not allowed. Zurich, 21 June 2011 p. 25

Intermediate summary Spontaneously broken scale invariance : All mass scales originate from one and the same source - vev of the massless dilaton Zero cosmological constant β = 0 existence of particles Scale invariance naturally leads to flat potentials and thus to cosmological inflation TDiff or Unimodular gravity: New parameter strength of a particular potential for the dilaton Dynamical Dark Energy Zurich, 21 June 2011 p. 27

Quantum scale invariance Zurich, 21 June 2011 p. 28

Quantum scale invariance Common lore: quantum scale invariance does not exist, divergence of dilatation current is not-zero due to quantum corrections: µ J µ β(g)g a αβ Gαβ a, Zurich, 21 June 2011 p. 28

Quantum scale invariance Common lore: quantum scale invariance does not exist, divergence of dilatation current is not-zero due to quantum corrections: µ J µ β(g)g a αβ Gαβ a, Sidney Coleman: For scale invariance,..., the situation is hopeless; any cutoff procedure necessarily involves a large mass, and a large mass necessarily breaks scale invariance in a large way. Zurich, 21 June 2011 p. 28

Standard reasoning Dimensional regularisation d = 4 2ǫ, M S subtraction scheme: mass dimension of the scalar fields: 1 ǫ, mass dimension of the coupling constant: 2ǫ Counter-terms: λ = µ 2ǫ [λ R + k=1 a n ǫ n ], µ is a dimensionfull parameter!! One-loop effective potential along the flat direction: V 1 (χ) = m4 H (χ) 64π 2 [ log m2 H (χ) 3 ] µ 2 2, Zurich, 21 June 2011 p. 29

Result: explicit breaking of the dilatation symmetry. Dilaton acquires a nonzero mass due to radiative corrections. Zurich, 21 June 2011 p. 30

Result: explicit breaking of the dilatation symmetry. Dilaton acquires a nonzero mass due to radiative corrections. Reason: mismatch in mass dimensions of bare (λ) and renormalized couplings (λ R ) Idea: Replace µ 2ǫ by combinations of fields χ and h, which have the correct mass dimension: µ 2ǫ χ 2ǫ 1 ǫ Fǫ (x), where x = h/χ. F ǫ (x) is a function depending on the parameter ǫ with the property F 0 (x) = 1. Zenhäusern, M.S Englert, Truffin, Gastmans, 1976 Zurich, 21 June 2011 p. 30

Example of computation Natural choice: µ 2ǫ [ ω 2] ǫ 1 ǫ, ( ξ χ χ 2 + ξ h h 2) ω 2 Potential: Counter-terms U cc = [ ω 2] ǫ 1 ǫ U = λ R 4 [ ω 2 ] ǫ 1 ǫ [ h 2 ζ 2 R χ2] 2, [Ah 2 χ 2 ( 1 ǫ + a )+Bχ 4 ( 1 ǫ + b ) +Ch 4 ( 1 ǫ + c ) ], To be fixed from conditions of absence of divergences and presence of spontaneous breaking of scale-invariance Zurich, 21 June 2011 p. 31

U 1 = + m 4 (h) 64π 2 [ log m2 (h) v 2 + O ( ζ 2 R ) ] λ 2 R 64π 2 [ C0 v 4 + C 2 v 2 h 2 + C 4 h 4] + O ( h 6), χ 2 where m 2 (h) = λ R (3h 2 v 2 ) and C 0 = 3 2 C 2 = 3 C 4 = 3 2 [ ( ζ 2 2a 1 + 2log R [ 2a 3 + 2log [ 2a 5 + 2log ξ χ ) + 43 log2λ R + O(ζ 2R ) ] ( ζ 2 ) ] R + O(ζ 2R ) ξ χ ( ζ 2 ) ] R 4log2λ R + O(ζ 2R ) ξ χ,., Zurich, 21 June 2011 p. 32

Origin of Λ QCD Consider the high energy ( s v but s χ 0 ) behaviour of scattering amplitudes on the example of Higgs-Higgs scattering (assuming, that ζ R 1). In one-loop approximation Γ 4 = λ R + 9λ2 R 64π 2 [ ( ) s log ξ χ χ 2 0 ] + const + O ( ζ 2 R). This implies that at v s χ 0 the effective Higgs self-coupling runs in a way prescribed by the ordinary renormalization group! For QCD: Λ QCD = χ 0 e 1 2b 0 αs, β(αs ) = b 0 α 2 s Zurich, 21 June 2011 p. 33

Quantum effective action is scale invariant in all orders of perturbation theory!!! Zurich, 21 June 2011 p. 34

Quantum effective action is scale invariant in all orders of perturbation theory!!! Problems Zurich, 21 June 2011 p. 34

Quantum effective action is scale invariant in all orders of perturbation theory!!! Problems Renormalizability: Can we remove all divergences with the similar structure counter-terms? The answer is no" (Tkachov, MS). However, this is not essential for the issue of scale invariance. We get scale-invariant effective theory Unitarity and high-energy behaviour: What is the high-energy behaviour (E > M Pl ) of the scattering amplitudes? Is the theory Unitary? Can it have a scale-invariant UV completion? Zurich, 21 June 2011 p. 34

Consequences The dilaton is massless in all orders of perturbation theory Zurich, 21 June 2011 p. 35

Consequences The dilaton is massless in all orders of perturbation theory Since it is a Goldstone boson of spontaneously broken symmetry it has only derivative couplings to matter (inclusion of gravity is essential!) Fifth force or Brans-Dicke constraints are not applicable to it Higgs mass is stable against radiative corrections (in dimensional regularisation) Zurich, 21 June 2011 p. 35

Dilaton as a part of the metric Previous discussion - ad hoc introduction of scalar field χ. It is massless, as is the graviton. Can it come from gravity? Yes - it automatically appears in scale-invariant TDiff gravity as a part of the metric! Consider arbitrary metric g µν (no constraints). Determinant g of g µν is TDiff invariant. Generic scale-invariant action for scalar field and gravity: S = d 4 x [ g 1 2 φ2 f( g)r 1 2 φ2 G gg ( g)( g) 2 1 ] 2 G φφ( g)( φ) 2 + G gφ ( g)φ g φ φ 4 v( g). Zurich, 21 June 2011 p. 36

Equivalence theorem This TDiff theory is equivalent (at the classical level) to the following Diff scalar tensor theory: L e g = 1 2 φ2 f(σ)r 1 2 φ2 G gg (σ)( σ) 2 1 2 G φφ(σ)( φ) 2 G gφ (σ)φ σ φ φ 4 v(σ) Λ 0 σ. Zurich, 21 June 2011 p. 37

Transformation to Einstein frame: L e g = 1 2 M2 R 1 2 M2 K σσ (σ)( σ) 2 1 2 M2 K φφ (σ)( ln(φ/m)) 2 M 2 K σφ (σ) σ ln(φ/m) M 4 V (σ) M4 Λ 0 φ 4 f(σ) 2 σ, As expected, φ is a Goldstone boson with derivative couplings only (except the term containing Λ 0 ). So, TDiff scale invariant theory automatically contains a massless dilaton. All previous results can be reproduced in it. Zurich, 21 June 2011 p. 38

Conclusions Scale-invariance and TDiff gravity lead to: Zurich, 21 June 2011 p. 39

Conclusions Scale-invariance and TDiff gravity lead to: Unique source for all mass scales. Zurich, 21 June 2011 p. 39

Conclusions Scale-invariance and TDiff gravity lead to: Unique source for all mass scales. Natural inflation Zurich, 21 June 2011 p. 39

Conclusions Scale-invariance and TDiff gravity lead to: Unique source for all mass scales. Natural inflation Higgs mass is stable against radiative corrections (at least if dimensional regularization is used) - no SUSY, or technicolor, or little Higgs, or large extra dimensions are needed Cosmological constant may be zero due to quantum scale-invariance and the requirement of existence of particles Zurich, 21 June 2011 p. 39

Problems to solve Non-perturbative regularisation? Lattice proposal: Tkachev, M.S. Zurich, 21 June 2011 p. 40

Problems to solve Non-perturbative regularisation? Lattice proposal: Tkachev, M.S. Though the stability of the electroweak scale against quantum corrections may be achieved, it is unclear why the electroweak scale is so much smaller than the Planck scale (or why ζ 1). Zurich, 21 June 2011 p. 40