Asymptotic Safety in the ADM formalism of gravity

Asymptotic Safety in the ADM formalism of gravity Frank Saueressig Research Institute for Mathematics, Astrophysics and Particle Physics Radboud University Nijmegen J. Biemans, A. Platania and F. Saueressig, arxiv:1609.04813 J. Biemans, A. Platania and F. Saueressig, arxiv:1702.06539 W. Houthoff, A. Kurov and F. Saueressig, in preparation FRG2017 Heidelberg, March 10th, 2017 p. 1/33

p. 2/33 What s the name of the game? Goal: a consistent quantum field theory describing gravity on all scales

p. 3/33 What s the name of the game? Goal: a consistent quantum field theory describing gravity Nature on all scales

p. 4/33 Asymptotic Safety as a successful quantum theoy (for nature) a) non-gaussian fixed point (NGFP) controls the UV-behavior of the RG-trajectory ensures the absence of UV-divergences

p. 4/33 Asymptotic Safety as a successful quantum theoy (for nature) a) non-gaussian fixed point (NGFP) controls the UV-behavior of the RG-trajectory ensures the absence of UV-divergences b) NGFP has finite-dimensional UV-critical surface S UV fixing the position of a RG-trajectory in S UV experimental determination of relevant parameters required for predictivity c) connection to observable low-energy physics: tests of general relativity (cosmological signatures,...) compatibility with standard model of particle physics at 1 TeV (light fermions, Yukawa couplings, Higgs potential,...)

p. 5/33 The Asymptotic Safety sloganizer Gravity rules!

p. 5/33 The Asymptotic Safety sloganizer Gravity rules! Matter matters!

p. 5/33 The Asymptotic Safety sloganizer Gravity rules! Matter matters! Time matters!

p. 6/33 Perturbative quantization of General Relativity Dynamics of General Relativity governed by Einstein-Hilbert action S EH 1 = d 4 x g [ R+2Λ ] 16πG N Newton s constant G N has negative mass-dimension Wilsonian picture of perturbative renormalization: dimensionless coupling constant attracted to GFP (free theory) in UV introduce dimensionless coupling constants g 0.04 g k = k 2 G N, λ k Λk 2 GFP: flow governed by mass-dimension: k k g k =2g +O(g 2 ) k k λ k = 2λ+O(g) 0.02 GFP 0.04 0.02 0.02 0.04 0.02 0.04 Λ

p. 7/33 Quantizing general relativity [M. H. Goroff and A. Sagnotti, Phys. Lett. B160 (1985) 81] [A. E. M. van de Ven, Nucl. Phys. B378 (1992) 309] quantizing the Einstein-Hilbert action S = 1 16πG d 4 x gr has a two-loop divergence = hallmark of perturbative non-renormalizability S div = 1 ǫ 209 2880 1 (16π 2 ) 2 d 4 x gc µν ρσ C ρσ αβ C αβ µν.

Quantizing general relativity [M. H. Goroff and A. Sagnotti, Phys. Lett. B160 (1985) 81] [A. E. M. van de Ven, Nucl. Phys. B378 (1992) 309] quantizing the Einstein-Hilbert action S = 1 16πG d 4 x gr has a two-loop divergence = hallmark of perturbative non-renormalizability S div = 1 ǫ 209 2880 1 (16π 2 ) 2 d 4 x gc µν ρσ C ρσ αβ C αβ µν. the Goroff-Sagnotti challenge mastered: Γ k = 1 16πG k [H. Gies, S. Lippholdt, B. Knorr, F.S., Phys. Rev. Lett. 116 (2016) 211302] d 4 x g ( R+2Λ k )+ σ k d 4 x gc µν ρσ C ρσ αβ C αβ µν +... possess a NGFP with two relevant directions: NGFP GS : λ = 0.193, g = 0.707, σ = 0.305 θ 1,2 = 1.475±3.043i, θ 3 = 79.39. p. 7/33

p. 8/33 renormalization group flows in the presence of a foliation

p. 9/33 Technical routes towards asymptotic safety starting point: path integral over metrics Z = Dĝe S[ĝ] FRGE t Γ k [h µν ;ḡ µν ] = 1 2 STr [ ( Γ (2) k +R k ) 1 t R k ] linear split: ĝ µν = ḡ µν +h µν exponential split: ĝ µν = ḡ µρ [ e h ] ρ ν

Technical routes towards asymptotic safety starting point: path integral over metrics Z = Dĝe S[ĝ] FRGE t Γ k [h µν ;ḡ µν ] = 1 2 STr [ ( Γ (2) k +R k ) 1 t R k ] linear split: ĝ µν = ḡ µν +h µν exponential split: ĝ µν = ḡ µρ [ e h ] ρ ν Euclidean Dynamical Triangulations (EDT) Z = triangulations (detĝ) β e SEH [ĝ] Causal Dynamical Triangulations (CDT) Z = causal triangulations e SEH [ĝ] p. 9/33

p. 10/33 A fundamental puzzle FRGE: existence of NGFP well-established extension to gravity-matter models possible Causal Dynamical Triangulations (CDT): phase diagram contains 2nd order phase transition = possible continuum limit Euclidean Dynamical Triangulations (EDT): phase transition is 1st order = existence of continuum limit questionable Conclusions: measure in partition function can manifestly change physics foliation structure underlying CDT is important

p. 11/33 renormalization group flows in the ADM-formalism

p. 12/33 Introducing time through the ADM formalism Preferred time -direction via foliation of space-time foliation structure M d+1 = R M d with y µ (t,x a ): ds 2 = N 2 dt 2 +σ ij ( dx i +N i dt )( dx j +N j dt ) fundamental fields: g µν (N,N i,σ ij ) g µν = N2 +N i N i N i N j σ ij

p. 13/33 FRGE for the ADM-formalism fundamental fields: {N(t,x),N i (t,x),σ ij (t,x)} N = N + ˆN, N i = N i + ˆN i, σ ij = σ ij + ˆσ ij symmetry: general coordinate invariance inherited from γ µν : δγ µν = L v (γ µν ), v α = (f(t,x), ζ a (t,x)) induces δn = f t N +ζ k k N +N t f NN i i f, δn i = N i t f +N k N k i f +σ ki t ζ k +N k i ζ k +f t Ñ i +ζ k k N i +N 2 i f δσ ij = f t σ ij +ζ k k σ ij +N j i f +N i j f +σ jk i ζ k +σ ik j ζ k Non-linearity of ADM-decomposition: symmetry realized non-linearly = background Diff(M)-symmetry broken to foliation preserving diffeos [ ( ) ] k k Γ k [h µν ;ḡ µν ] = 1 2 STr Γ (2) 1k k k +R k R k

p. 14/33 renormalization group flows on cosmological backgrounds

p. 15/33 Off-shell flows in the ADM formalism Einstein-Hilbert action in ADM variables Γ grav 1 k = dtd d xn σ [ K ij K ij K 2 ] R+2Λ k 16πG k background: flat Friedmann-Robertson-Walker spacetime N = 1, N i = 0, σ ij = a 2 (t)δ ij. fluctuations: adapted to cosmological perturbation theory ˆN i =u i + i 1 B, i u i = 0 ˆσ ij =h ij ( σ ) 1 ij + i j ψ 1 + i j E + 1 i v j + j 1 v i.

p. 16/33 Gauge-fixing in the ADM formalism (flat space) δ 2 Γ grav k Index hh vv EE ψψ ψe matrix element of 32πG k δ 2 Γ grav k t 2 + 2Λ k 2 [ t 2 2Λ ] k Λ k (d 1) [ (d 2)( t 2 + ) (d 3)Λ ] k 1 2 (d 1)[ t 2 2Λ ] k ˆN ˆN 0 BB 0 uu uv 2 2 t Bψ 2(d 1) t ˆN ψ 2(d 1) [ ] Λ k ˆN E 2Λ k

p. 17/33 Gauge-fixing in the ADM formalism (flat space) δ 2 Γ grav k +S gf Index hh vv EE ψψ ψe matrix element of 32πG k ( δ 2 Γ grav k t 2 + 2Λ k 2 [ t 2 + 2Λ ] k [ ] 2 t + Λ k 1 2 (d 1)(d 3) 4 [ 2 t + 2Λ k ] 1 2 (d 1)[ t 2 + 2Λ ] k +S gf) ˆN ˆN 2 t + BB 2 t + uu uv Bψ ˆN ψ ˆN E 2 [ 2 t + ] 2 t 2(d 1) t (d 1) [ t 2 + Λ ] k 2 t + 2Λ k

p. 18/33 RG flows in the ADM formalism Einstein-Hilbert action in ADM variables Γ grav 1 k = dtd d xn σ [ K ij K ij K 2 ] R+2Λ k 16πG k background: flat Friedmann-Robertson-Walker spacetime N = 1, N i = 0, σ ij = a 2 (t)δ ij. fluctuations: adapted to cosmological perturbation theory ˆN i =u i + i 1 B, i u i = 0 ˆσ ij =h ij ( σ ) 1 ij + i j ψ 1 + i j E + 1 i v j + j 1 v i. unique gauge-fixing of diffeomorphisms F = t ˆN + i ˆNi 1 2 tˆσ + 2(d 1) d K ˆN, F i = t ˆNi i ˆN 1 2 iˆσ + jˆσ ji +(d 2) K ij ˆNj, well-defined Hessian Γ (2) k with relativistic propagators

p. 19/33 beta functions for the gravity-matter system β g =(2+η)g, β λ =(η 2)λ+ g 4π anomalous dimension of Newton s constant: [ (3+ 4 1 2λ + 6 10λ )( B det (λ) 1 η) ] 6 8 η = 16πgB 1 (λ) (4π) 2 +16πgB 2 (λ) and B 1 (λ) 41 9 + 11 6(1 2λ) 5 6(1 2λ) 2 + 3 5λ 3B det (λ) + 7 124λ+196λ2 24B det (λ) 2 B 2 (λ) = 1 36 + 11 24(1 2λ) 5 36(1 2λ) 2 + 3 5λ 12B det (λ) + 7 124λ+196λ2 144B det (λ) 2

p. 20/33 Einstein-Hilbert-truncation on cosmological background J. Biemans, A. Platania and F. Saueressig, arxiv:1609.04813

adding minimally coupled matter fields p. 21/33

p. 22/33 Minimally coupled matter fields in ADM Supplement Einstein-Hilbert action by N S minimally coupled scalar fields S scalar = 1 2 N S i=1 dtd d xn σ [ φ i 0 φ i] N V abelian gauge fields (gauge-fixed) S vector = 1 4 N V i=1 + 1 2ξ N V i=1 dtd d xn σ [ g µν g αβ Fµα i ] Fi νβ dtd d x N σ [ N ḡ µν Dµ A i ] 2 V ν + i=1 dtd d x N σ [ Ci 0 C i] N D Dirac spinors S fermion =i N D i=1 dtd d xn σ [ ψi / ψ i].

p. 23/33 beta functions for the gravity-matter system β g =(2+η)g, β λ =(η 2)λ+ g 4π [ (3+ 4 1 2λ + 6 10λ )( B det (λ) 1 η) ] 8+dλ 6 anomalous dimension of Newton s constant: η = 16πgB 1 (λ) (4π) 2 +16πgB 2 (λ) and B 1 (λ) 41 9 + 11 6(1 2λ) 5 6(1 2λ) 2 + 3 5λ 3B det (λ) + 7 124λ+196λ2 24B det (λ) 2 + 1 6 d g B 2 (λ) = 1 36 + 11 24(1 2λ) 5 36(1 2λ) 2 + 3 5λ 12B det (λ) + 7 124λ+196λ2 144B det (λ) 2 Type I regulator: d g = N S N V N D, d λ = N S +2N V 4N D

Gravity-matter fixed points are quite common d g = N S N V N D, d λ = N S +2N V 4N D 150 100 50 0 50 100 150 100 50 0 50 100 d g d blue: 1 NGFP black: 0 NGFPs red: 3 NGFP green: 2 NGFPs p. 24/33

most of them are UV fixed points d λ 100 50 0-50 -100-150 -100-50 0 50 100 150 d g green: UV-FP: real exponents black: 0 NGFPs blue: UV-FP: complex exponents rest: saddle/ir fixed points p. 25/33

multi-fixed point systems exist 5 d λ 0-5 -10-15 -10 0 10 20 30 d g filled symbols: 3 fixed point systems open symbols: 2 fixed point systems p. 26/33

p. 27/33 most matter sectors are in a good region! model N S N D N V g λ θ 1 θ 2 pure gravity 0 0 0 0.78 + 0.32 0.50 ± 5.38 i Standard Model (SM) 4 45 2 12 0.75 0.93 3.871 2.057 SM, dark matter (dm) 5 45 2 12 0.76 0.94 3.869 2.058 SM, 3 ν 4 24 12 0.72 0.99 3.884 2.057 SM, 3 ν, dm, axion 6 24 12 0.75 1.00 3.882 2.059 MSSM 49 61 2 12 2.26 2.30 3.911 2.154 SU(5) GUT 124 24 24 0.17 + 0.41 25.26 6.008 SO(10) GUT 97 24 45 0.15 + 0.40 19.20 6.010

p. 28/33 renormalization group flows on CDT backgrounds S 1 S d

p. 29/33 Off-shell flows in the ADM formalism Einstein-Hilbert action in ADM variables Γ grav 1 k = dtd d xn σ [ K ij K ij K 2 ] R+2Λ k 16πG k background: S 1 S d N = 1, N i = 0, σ ij = σ ij (r) S d. fluctuations: adapted to cosmological perturbation theory ˆN i =u i + D i 1 B, Di u i = 0 ˆσ ij =h ij + D i 1 v j + D j 1 v i + ( D i D j + 1 d σ ij ) ψ + 1 d σ ijh. unique gauge-fixing of diffeomorphisms S gf 1 = dtd d y σ [ F 2 +F i σ ij ] F j 32πG k F = t ˆN + D i ˆN i 1 2 tˆσ, F i = t ˆN i D i ˆN 1 2 D iˆσ + D jˆσ ji

p. 30/33 Phase diagrams on S 1 S 3 Type I, linear g Type I, exponential g 1.4 1.4 1.2 1.2 1.0 0.8 A 1.0 0.8 A 0.6 0.6 0.4 0.4 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.2 O C Λ O 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.2 C Λ Type II, linear Type II, exponential g g 1.5 1.5 1.0 A 1.0 A 0.5 B 0.5 O 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 Λ O C 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 Λ

p. 31/33 IR completion of the flow: Type II - linear 10 6 G 10 4 10 4 100 10 5 1 0.01 10 6 4 2 0 2 4 6 8 ln k 4 2 0 2 4 6 8 ln k flow approaches quasi-fixed point C: dimensionful Newton s constant: lim k 0 G k = 0 dimensionful cosmological constant: lim k 0 Λ k = 0!!! transition to IR-phase at Hubble-horizon scales!!!

outlook p. 32/33

p. 33/33 CDT correlation functions [J. Ambjørn, et. al., Phys. Rev. D 94 (2016) 044010] Best measured correlation functions build from spatial volumes V 3 : expectation value of volume profiles: V 3 (t) correlators for volume fluctuations: (δv 3 (t))(δv 3 (t )) S 1 S 3 S 1 T 3 now within reach of the FRGE based on the ADM-formalism