Critical exponents in quantum Einstein gravity Sándor Nagy Department of Theoretical physics, University of Debrecen MTA-DE Particle Physics Research Group, Debrecen Leibnitz, 28 June Critical exponents in quantum Einstein gravity p. 1
Outline Quantum field theory and gravity Functional renormalization group Wetterich equation regulators fixed points Asymptotic safety Quantum Einstein gravity UV criticality crossover criticality IR criticality Critical exponents in quantum Einstein gravity p. 2
Quantization and gravity In gravitational interaction the spacetime metrics becomes a dynamical object. The Einstein equation is R µν 1 2 g µνr = Λg µν +8πGT µν 11 m3 G Newton constant: G = 6.67 10 kgs 2 Λ cosmological constant: Λ 10 35 s 2 Questions inconsistency: R is classical, T µν is quantized singularities: black holes, big bang cosmological constant Quantum gravity may solve these problems. Critical exponents in quantum Einstein gravity p. 3
Quantum Einstein gravity quantum Einstein gravity (QEG) = gravitational interaction + quantum field theory We use the elements of quantum field theory the metrics g µν plays the role of the field variable in the path integral the action is a diffeomorphism invariant action couplings, the cosmological constant λ, Newton constant g The model cannot be extended beyond the ultraviolet (UV) limit, since around the Gaussian fixed point (GFP) λ = Λk 2 relevant, g = Gk d 2 irrelevant. The loop correction do not modify the tree level scaling behaviors at the GFP. The Newton coupling g diverges, the theory seems non-renormalizable. Critical exponents in quantum Einstein gravity p. 4
Quantum Einstein gravity The classical description of gravity is an effective theory which looses its validity around the Planck scale, therefore it is not complete. The gravity can be UV complete: the UV physics contains different degrees of freedom, e.g. string theory loop quantum gravity (new quantization scheme) Weinberg s conjecture (1976): the gravity has a UV non-gaussian fixed point (NGFP), where the newton coupling is relevant If the NGFP is UV attractive, the QEG can be extended to arbitrarily high energies, without divergences. This is the so called asymptotic safety. Critical exponents in quantum Einstein gravity p. 5
QEG UV NGFP First it was shown in2+ǫ dimensions that there exists a UV NGFP. Taking into account the 1-loop counterterms into the QEG action one can get the following perturbative RG equation k k g = ǫg bg 2, b > 0 by using expansion both ing and inǫ. The goal is to treat the model at d = 4, which cannot be reached perturbatively from d = 2. Similar problem: In the d-dimensional O(N)-model, we have a NGFP at d = 4 ǫ, the Wilson-Fisher fixed point. If we perform an expansion inǫthen we can reach thed = 3 case, and we can calculate the exponents for the NGFP. However there is a NGFP even in d = 1, which cannot be treated by perturbative considerations. We use functional RG instead of perturbative RG. Critical exponents in quantum Einstein gravity p. 6
QEG UV NGFP The QEG is perturbatively non-renormalizable, because every order in the perturbation requires new types of counterterms (new interactions) there is no physical predictivity. The scaling properties around the UV NGFP could modify the tree level scalings. One can find the fixed point perturbatively ing, i.e. k k g = 2g 16πcg 2, but its position in the phase space is not universal. The functional RG method can provide nonperturbative flow equations and universal (less sensitive) results. Critical exponents in quantum Einstein gravity p. 7
Renormalization The functional renormalization group method is a fundamental element of quantum field theory. We know the high energy (UV) action, which describes the small distance interaction between the elementary excitations. We look for the low energy IR (or large distance) behavior. The RG method gives a functional integro-differential equation for the effective action, which is called the Wetterich equation. The form of the blocked action is S k [φ,g i ] = i g i (k)f i (φ), where k is the (momentum or energy) scale, and g i (= g i (k)) are the dimensionful (scale dependent) couplings. The action takes into account the symmetries of the model. Critical exponents in quantum Einstein gravity p. 8
Renormalization The vacuum-vacuum transition amplitude (the generating functional) is Z = Dφe S k = dφ 0...dφ k k dφ k dφ k+ k...dφ e S k The path integral is performed by removing the degrees of freedom (quantum fluctuations, modes) one-by-one, systematically, whch gives scale dependent couplings. g k k g kgk+ k (IR) 0 k k k k k+ k k (UV) Critical exponents in quantum Einstein gravity p. 9
The Wetterich equation The form of the generating functional is Z = e Wk[J] = D[φ]e (S k+r k [φ] J φ). The integration of the modes we take the average of the field variable in a volume of k 1. The Wetterich equation is Γ k = 1 2 Tr Ṙ k R k +Γ k = 1 2, where = / ϕ, = / t, and the symbol Tr denotes the momentum integral and the summation over the internal indices. We assume that the functional form of the effective and the blocked action is similar, i.e. Γ k S k = i g i (k)f i (φ). Critical exponents in quantum Einstein gravity p. 10
Regulators The IR regulator has the form R k [φ] = 1 2 φ R k φ. It is a momentum dependent mass like term, which serves as an IR cutoff and has the following properties lim p 2 /k 2 0 R k > 0: it serves as an IR regulator lim k 2 /p 2 0 R k 0: in the limitk 0 we obtain back the form of Z lim R k : for the microscopic action S = lim k Λ Γ k (it k 2 serves as a UV regulator, too) Critical exponents in quantum Einstein gravity p. 11
Regulators The compactly supported smooth (css) regulator has the form r css = R p 2 = s 1 exp[s 1 y b /(1 s 2 y b )] 1 θ(1 s 2y b ), withy = p 2 /k 2. Its limiting cases provide us the following commonly used regulator functions lim r CSS = s 1 0,s 2 =1 ( ) 1 y b 1 θ(1 y), lim s 1 0,s 2 0 r CSS = 1 y b, lim r CSS = s 1 =1,s 2 0 1 exp[y b ] 1. where the first limit gives the Litim s optimized, the second gives the power law and the third gives the exponential regulator. The b = 1 case satisfies the normalization conditionslim y 0 yr = 1 and lim y yr = 0.. Critical exponents in quantum Einstein gravity p. 12
Gaussian fixed point (GFP) Free massless theory, i.e. g i = 0,s i = d i, where s i are the eigenvalues of the linearized RG equation around the fixed point andd i are the canonical dimensions. s i > 0 (d i < 0), when k. The fixed point repels the trajectories. when k 0 the coupling decreases irrelevant. 10-7 s i < 0 (d i > 0), when k. The fixed point attracts the trajectories. when k 0 the coupling diverges ~ g i ~ g i relevant. 10-7 10-1 10-2 10-3 10-4 10-5 10-6 10-1 10-2 10-3 10-4 10-5 10-6 IR 10-3 10-2 10-1 10 0 10 1 10 2 k/k Λ IR UV 10-3 10-2 10-1 10 0 10 1 10 2 k/k Λ UV Critical exponents in quantum Einstein gravity p. 13
UV behavior Asymptotic freedom: g i lim t g i = 0 Example models are QCD 3-dimensional φ 4 model... Asymptotic safety: there is an UV attractive NGFP in the phase space with a finite number of relevant directions. Example models are Gross-Neveu model non-linear σ model sine-gordon model quantum Einstein gravity Critical exponents in quantum Einstein gravity p. 14
QEG effective action The RG evolution equation is Γ k [ h,ξ, ξ,ḡ] = 1 2( 2 Tr Ṙ grav k ) h h ( 2R grav k +Γ [ ( 1 2 Tr κ 2 (Ṙgh k ) ξξ where the QEG effective action is Γ k = k ) h h 1 (κ 2 R gh k +Γ k ) ξξ )] 1 (κ 2 R gh k +Γ k ) ξ ξ d d x detg µν ( 1 16πG k (2Λ k R) ω k 3σ k R 2 +... f(r) + 1 C 2 + θ k E +... 2σ k σ k ) + gf. terms + gh. terms. The first two terms consitute the Einstein-Hilbert action. Critical exponents in quantum Einstein gravity p. 15
QEG evolution equation The flow equations with the Litim s regulator are (d = 4) λ = 2( 96g2 +λ+4λ 2 (11+λ)+g(8λ(1 3λ) 6)) 4g (1 2λ) 2, ġ = (2+η)g, 24g where η = 4g (1 2λ) is the gravitation anomalous dimension. 2 The fixed points are UV: (NGFP) repulsive focal point: λ UV = 1/4,g UV = 1/64 GFP: hyperbolic point: λ G = 0,g UV = 0 In order to find the IR fixed point we should rescale the equations as χ = 1 2λ, ω = 4g (1 2λ) 2 and τ = ωk k, which gives τ χ = 4ω +2χω(8+21χ)+24ω 2 +6χ 2 (3χ(χ+1) 1), τ ω = 8ω 2 (1 6χ) 2χ(42χ 2 +9χ 4) 6χ 3 (χ(6χ+5) 2). Critical exponents in quantum Einstein gravity p. 16
QEG evolution equation Symmetric phase: (strong coupling) g µν δ µν, Euclidean flat geometry, k 0: R = 4λ < 0, negative curvature Broken phase: (weak coupling) g µν = 0, degenerate geometry, at k 0λ > 0 The fixed points g 0.03 0.02 0.01 0 G -0.4-0.2 0 0.2 0.4 UV: (NGFP) repulsive focal point: λ UV = 1/4,g UV = 1/64, (ω UV = 3/16, χ UV = 1/2), eig.values: s UV1,2 = ( 5±i 167)/3, extension dependent s UV1,2 = θ ±iθ,θ 1.5 2 and θ 2.5 4.3,ν = 1/θ = 1/3. G: hyperbolic point: λ G = 0,g UV = 0, (ω G = 1 andχ G = 1), eig.values: s G1 = 2 ands G2 = d 2, extension dependent. IR: attracitve fixed point: λ IR = 1/2, g IR = 0. (ω IR = 0, χ IR = 0). λ UV Critical exponents in quantum Einstein gravity p. 17 IR
UV criticality In the case of UV NGFP the eigenvalues of the corresponding stability matrix can be written as 1/ν 1.8 1.75 1.7 1.65 1.6 1.55.P 0 0.2 L. 0.4 0.6 s 0.8 1 1 0 θ UV 1,2 = θ ±iθ, and ν = 1/θ.E 0.6 0.8 1 0.4 0.2 s 2 1.8 1.75 1.7 1.65 1.6 1.55 1/ν s 1 = s 2 = 0 Power law regulator s 1 = 0, s 2 = 1 Litim s regulator 1.52 1.5 1.48 1.46 1.44 1.42 1.4 1.38 s 1 = 1, s 2 = 0 Exponential regulator s 1 =0.001 s 1 =0.1 s 1 =1 1/ν 3.2 2.8 2.4 2.0 1.6 10 0 10 2 s 10 4 10 6 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 s 2 Critical exponents in quantum Einstein gravity p. 18
Crossover criticality 3-dimensional φ 4 model with the potential Ṽ = 1 2 g 1φ 2 + 1 24 g 2φ 4 +O(φ 6 ) The case of 2 and 4 couplings is considered, and both cases can possess extrema, with minimal sensitivity to the parameters s 1 ands 2. ν 0.7 0.68 0.66 0.64 0.62 0.6 0.58 0.56 0.54 0.52 0.5 b=1 b=2 b=3 0 1 2 3 4 5 6 s 2 Critical exponents in quantum Einstein gravity p. 19
IR criticality exponent UV G IR ν 3/5 1/2 1/2 η -2 k 2 k 3/2 η 10 2 10 0 10-2 10-4 10-6 10-8 IR k -3/2 k 2 G η=-2 UV 10-10 10-10 10-8 10-6 10-4 10-2 10 0 10 2 10 4 10 6 Taking into account the higher order terms in the curvature of the QEG action we get the exponents UV: RG: ν 1/2 2/3, lattice calculations: ν = 1/3 G: RG: ν = 1/2 exact IR: RG: ν = 1/2 exact k-k c Critical exponents in quantum Einstein gravity p. 20
Models and their fixed points model UV crossover IR 3d O(N) Gaussian Wilson-Fisher IR 3d non-linear σ non-gaussian non-gaussian Gaussian 3d Gross-Neveu Gaussian non-gaussian IR 2d sine-gordon Gaussian and non-gaussian Kosterlitz-Thouless Gaussian and IR 4d QEG non-gaussian Gaussian IR Critical exponents in quantum Einstein gravity p. 21
Thank You Critical exponents in quantum Einstein gravity p. 22