Constraints on matter from asymptotic safety

Transcription

1 Constraints on matter from asymptotic safety Roberto Percacci SISSA, Trieste Padova, February 19, 2014

2 Fixed points Γ k (φ) = i λ i (k)o i (φ) β i (λ j,k) = dλ i dt, t = log(k/k 0) λ i = k d iλ i β i = d λ j dt = d i λi +k d iβ i Fixed point: βi ( λ j ) = 0 Asymptotically safe RG trajectory: λ i λ i UV attractive=ir repulsive=relevant UV-repulsive=IR attractive=irrelevant predictivity demands finitely many relevant directions

3 General picture

4 Examples Gaußian Fixed Point at λ i = 0. M ij = β i λ j = d i δ ij relevant=renormalizable - QCD - non-af examples: 4-Fermi interactions in d = 3 - maybe gravity?

5 Outline 1 Asymptotic safety 2 1-loop 3 FRGE 4 Matter 5 Conclusions

6 One Loop Corrections in Einstein s Theory k d dk 1 16πG(k) = ck2 k dg dk = 16πcG2 k 2 G = Gk 2 k d G dk = 2 G 16πc G 2 if c > 0 fixed point at G = 1/8πc one calculation: c = 23/48π 2

7 Perturbative beta functions β G = 2 G 46 G 2 6π, β Λ = 2 Λ+ 2 G 4π 16 G Λ 6π Λ = 3 62 G = 12π 46

8 Perturbative flow G

9 Higher derivative gravity Γ k = d 4 x g [ 2ZΛ ZR + 1 ( C 2 2ω )] 2λ 3 R2 + 2θE Z = 1 16πG K.S. Stelle, Phys. Rev. D16, 953 (1977). J. Julve, M. Tonin, Nuovo Cim. 46B, 137 (1978). E.S. Fradkin, A.A. Tseytlin,Phys. Lett. 104 B, 377 (1981). I.G. Avramidi, A.O. Barvinski,Phys. Lett. 159 B, 269 (1985). G. de Berredo Peixoto and I. Shapiro, Phys.Rev. D (2005). A. Codello and R. P., Phys.Rev.Lett (2006) N. Ohta and R.P. Class. Quant. Grav (2014); arxiv:

10 Beta functions I β λ = 1 (4π) λ2 β ω = ω + 200ω 2 (4π) 2 λ 60 β θ = 1 7(56 171θ) (4π) 2 λ 90 λ(k) = λ 0 1+λ 0 1 (4π) log ( k k 0 ) ω(k) ω θ(k) θ 0.327

11 Beta functions II β Λ = 2 Λ+ 1 [ 1+20ω 2 (4π) 2 256π Gω + 2λ2 ] 1+86ω + 40ω2 λ Λ 12ω 1+10ω2 64π 2 ω λ+ 2 G q(ω) G Λ π β G = 2 G ω 40ω 2 (4π) 2 λ G q(ω) G 2 12ω where q(ω) = (83+70ω + 8ω 2 )/18π

12 Flow in Λ G plane I where q = q(ω ) β Λ = 2 Λ+ 2 G π q G Λ β G = 2 G q G2 Λ = 1 πq 0.221, G = 2 q

13 Flow in Λ G plane II 2.0 G

14 Topologically massive gravity Action S(g)= 1 16πG d 3 x g Dimensionless combinations of couplings ( 2Λ R + 1 2µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτγ τ νρ) ) ν = µg ; τ = ΛG 2 ; φ = µ/ Λ R.P., E. Sezgin, Class.Quant.Grav. 27 (2010) , arxiv: [hep-th] Recently extended to TM SUGRA: R.P., M. Perry, C. Pope, E. Sezgin, arxiv

15 Beta functions of β ν = 0, β G = G +B( µ) G 2, β Λ = 2 Λ+ 1 2 G ( ) A( µ, Λ)+2B( µ) Λ Since ν = µg = µ G is constant can replace µ by ν/ G

16 G G Figure : The flow in the Λ- G plane for α = 0, ν = 5. Right: enlargement of the region around the origin, showing the Gaussian FP. The beta functions become singular at G =

17 G G Figure : The flow in the Λ- G plane for α = 0, ν = 0.1. Right: enlargement of the region around the origin, showing that there is no Gaussian FP. The beta functions diverge on the Λ axis.

18 Functional renormalization Define Γ k (φ). lim k 0 Γ k (φ) = Γ(φ). It satisfies a FRGE k dγ k(φ) dk = β(φ) The quantity β is UV and IR finite. Use FRGE to calculate the effective action.

19 Application to gravity Background field method: q µν = g µν +h µν FRGE can only be defined for Γ k (g,h). Expand Γ k (g,h) = Γ k (g)+γ (1) k (g,h)+γ(2) k (g,h)+... Γ k (g) = Γ k (g,0) Single-field truncations: neglect Γ (n) k (g,h), n > 0. [M. Reuter, Phys. Rev. D (1998)] [D. Dou and R. Percacci, Class. and Quantum Grav (1998)]

20 Single-field truncation 1 In the FRGE identify Z h with 16πG. Consequently identify with η h = k d logz h dk η h = k d logg dk

21 Einstein Hilbert truncation I Γ k (ḡ,h) = S EH (ḡ µν +h)+s GF (ḡ,h)+s ghost (ḡ, C µ,c ν ) S EH (g µν ) = dx gz(2λ R) ; Z = 1 16πG β Λ = 2(1 2 Λ) 2 Λ Λ+42 Λ Λ 3 72π G Λ 288π 2 (1 2 Λ) Λ 72π G β =2(1 2 Λ) 2 G Λ+600 Λ 2 72π G 2 G (1 2 Λ) Λ 72π G G2

22 Einstein Hilbert truncation II G Cutoff type Ia G Cutoff type II

23 Fourth order gravity R 2 O.Lauscher, M. Reuter, Phys. Rev. D 66, (2002) arxiv:hep-th/ R 2 +C 2 D. Benedetti, P.F. Machado, F. Saueressig, Mod. Phys. Lett. A24, (2009) arxiv: [hep-th] Nucl. Phys. B824, (2010), arxiv: [hep-th] M. Niedermaier, Nucl. Phys. B833, (2010)

24 f(r) gravity Γ k (g µν ) = d 4 x gf(r) f(r) = n g i (k)r i i=0 n=6 A. Codello, R.P. and C. Rahmede Int.J.Mod.Phys.A23: arxiv: [hep-th]; n=8 A. Codello, R.P. and C. Rahmede Annals Phys (2009) arxiv: arxiv: ; P.F. Machado, F. Saueressig, Phys. Rev. D arxiv: arxiv: [hep-th] n=35 K. Falls, D.F. Litim, K. Nikolakopoulos, C. Rahmede, arxiv: [hep-th] n= Dario Benedetti, Francesco Caravelli, JHEP 1206 (2012) 017, Erratum-ibid (2012) 157 arxiv: [hep-th] Juergen A. Dietz, Tim R. Morris, JHEP 1301 (2013) 108 arxiv: [hep-th] Dario Benedetti, arxiv: [hep-th]

25 Enter matter because it s there because it may help (large N limit) because pure gravity has no local observables because experimental constraints more likely e.g. M. Shaposhnikov and C. Wetterich, Phys.Lett. B683, 196 (2010) [L. Griguolo, R.P. Phys. Rev. D 52, 5787 (1995)] [R.P., D.Perini, Phys.Rev. D (2003)] [R.P., D.Perini, Phys. Rev. D68, (2004)] [G. Narain. R.P. Class. Quant. Grav (2010)] [P. Donà, A. Eichhorn, R.P. arxiv: [hep-th](2013)]

26 Perturbative beta functions with matter β G = 2 G + G 2 6π (N S + 2N D 4N V 46), β Λ = 2 Λ+ G 4π (N S 4N D + 2N V + 2) + G Λ 6π (N S + 2N D 4N V 16) Λ = 3 N S 4N D + 2N V + 2 4N S + 2N D 4N V 31, 12π G = N S + 2N D 4N V 46.

27 Exclusion plots N V = 0,6,12,24,

28 Position of FP for N V =

29 Truncated FRGE, bimetric formalism Γ k (ḡ,h) = + Z h 2 2Z c 1 16πG d d x ḡ ( R + 2Λ ) d d x ḡ h µν K µναβ (( D 2 2Λ)1 ρσ αβ +Wρσ αβ )h ρσ d d x ḡ c µ ( Dρḡ µκ g κν D ρ + D ρ ḡ µκ g ρν D κ D µ ḡ ρσ g ρν D σ ) c ν S S = Z S 2 S D = iz D S V = Z V 4 d d x N S g g µν µ φ i ν φ i d d x g N D i=1 N V i=1 ψ i / ψ i, d d x g g µν g κλ Fµκ i Fi νλ +... i=1

30 Gravitational beta functions d Λ dt [ 8π G d(d + 1)(d + 2 ηh ) = 2 Λ + (4π) d/2 4d(d + 2 η c) d(d + 2)Γ[d/2] 1 2 Λ ] +2N S (2 + d η S ) 2N D 2 [d/2] (2 + d η D ) + 2N V (d 2 4 d η V ) [ 4π G Λ d(5d 7)(d ηh ) 3d(4π) d/2 + 4(d + 6)(d η c) Γ[d/2] 1 2 Λ ] 2N S (d η S ) N D 2 [d/2] (d η D ) + 2N V (d (8 d) (6 d)η V ) d G dt 4π G = (d 2) G 2 [ d(5d 7)(d ηh ) 3d(4π) d/2 + 4(d + 6)(d η c) Γ(d/2) 1 2 Λ ] 2N S (d η S ) N D 2 [d/2] (d η D ) + 2N V (d(8 d) (6 d)η V )

31 Graviton+ghost contributions to graviton η t γ (2,0,0;0) k =

32 Graviton contributions to η h η h = η h gravity + η h matter [ ] gravity η h = a( Λ k ) + c( Λ k )η h + e( Λ k )η c G k a( Λ) = a 0 + a 1 Λ + a 2 Λ 2 + a 3 Λ 3 + a 4 Λ 4 (4π) d/2 Γ(d/2)d 2 (d 2 4)(3d 2)(1 2 Λ) 4, a 0 = 4π (d 2)( d d 2 434d d 4 + d 5 ), a 1 = 16π (d 2)( d 318d 2 125d 3 + 2d 4 + d 5 ), a 2 = 16π( d d d d 4 17d 5 + d 6 ), a 3 = 4096π(d 2)( d 19d 2 + 2d 3 ), a 4 = 2048π(d 2)( d 19d 2 + 2d 3 ) ; [ ] 8π(d 1) d 350d d (d 2)(d + 4) Λ c( Λ) = (4π) d/2 Γ(d/2)d 2 (d + 2)(d + 4)(3d 2)(1 2 Λ) 3 128π (32 50d + 23d 2) e( Λ) = (4π) d/2 Γ(d/2)d 2 (d + 2)(d + 4)(3d 2)

33 Matter contribution to graviton η

34 Matter contributions to η h η h matter = N S 32π G (4π) d/2 Γ(d/2) [ ] 1 d 2 (d 2) d + d2 + 2 η S (d + 2)(3d 2) d + 4 [ 2 + d 2 ] d + 1 η D +N D 2 [d/2] 16π G (d 1)(d 2) (4π) d/2 Γ(d/2) d 3 (3d 2) 32π G (d 1)(d 2) N V (4π) d/2 Γ(d/2) d 2 (d + 2)(3d 2) [ d 2 12d d d + 4 η V ]

35 Graviton+ghost contributions to ghost η t γ (0,1,1;0) k = +

36 Ghost η c η c = [ b( Λ k )+d( Λ k )η h +f( Λ k )η c ] Gk with b( Λ) = d( Λ) = f( Λ) = [ ] 64π 8+4d + 18d 2 7d 3 + 2(4 9d 2 + 3d 3 ) Λ (4π) d/2 Γ(d/2)d 2 (d 2 4)(d + 4)(1 2 Λ) 2 64π(4 4d 9d 2 + 4d 3 ) (4π) d/2 Γ(d/2)d 2 (d 2 4)(d + 4)(1 2 Λ) 2 64π(4 9d 2 + 3d 3 ) (4π) d/2 Γ(d/2)d 2 (d 2 4)(d + 4)(1 2 Λ) 2

37 Graviton contribution to matter η

38 Matter anomalous dimensions [ 32π G 2 1 η S = (4π) d/2 Γ(d/2) d + 2 (1 2 Λ) 2 η D = [ (d + 1)(d 4) + 2d(1 2 Λ) 2 ( 1 η ) h + 2 d + 4 ( 1 η h d d Λ ) ], ( 1 η ) S d π G (d 1)(d 2 ( + 9d 8) (4π) d/2 Γ(d/2) 8d (d 2)(d + 1)(1 2 Λ) 2 1 η ) h d + 3 (d 1) 2 ( η ) D (d 1)(2d2 ( 3d 4) 2d(d + 1)(d 2) 1 2 Λ d + 2 4d(d 2)(1 2 Λ) 2 1 η ) ] h d + 2 [ 32π G (d 1)( d 9d 2 + d 3 ) η V = (4π) d/2 Γ(d/2) 2d 2 (d 2)(1 2 Λ) 2 ( 4(d 1)(2d 5) + d(d 2 1 η ) V + 4)(1 2 Λ) d + 4 ( 1 η ) h d + 2 4(d 1)(2d 5) d(d 2 4)(1 2 Λ) 2 ( 1 η ) ] h d + 4

39 General structure of anomalous dimensions For Ψ = h,c,s,d,v, η Ψ = 1 k dz Ψ Z Ψ dk η = (η h,η c,η S,η D,η V ) η = η 1 ( G, Λ)+A( G, Λ) η one loop anomalous dimensions η = η 1 RG improved anomalous dimensions η = (1 A) 1 η 1

40 Gravity+scalars G NS NS ReΘ NS Ηh Ηc ΗS NS NS NS

41 Gravity+fermions G, Θ1,Θ ND ND 2.5 Ηh Ηc ΗD ND ND ND

42 Gravity+vectors G NV NV Θ1,Θ NV Ηh NV Ηc NV ΗV NV

43 Exclusion plot N V = 0 N D N S

44 Exclusion plot N V = 12 N D N S

45 Exclusion plot N D = 0 N V 50 N V N S N S

46 Exclusion plot N S = 0 N V N D

47 Standard model matter 1L-II full-ii 1L-Ia full-ia Λ G θ θ η h η c η S η D η V

48 Specific models model N S N D N V G Λ θ 1 θ 2 η h no matter SM 4 45/ SM +dm scalar 5 45/ SM+ 3 ν s SM+3ν s + axion+dm MSSM 49 61/ SU(5) GUT SO(10) GUT

49 Exclusion plot N V = 12, d = 5 N D N S

50 Exclusion plot N V = 12, d = 6 10 N D N S

51 Gravity+scalar Γ k [g,φ] = d d x g ( V(φ 2 ) F(φ 2 )R + 1 ) 2 gµν µ φ ν φ [R.P., D.Perini, Phys. Rev. D68, (2004)] [G. Narain, R.P., Class. and Quantum Grav. 27, (2010)]

52 Functional flow of F, V { tv = k4 192π V Ψ + 6(k ( 2 Ψ + 24φ 2 k 2 F Ψ + k 2 FΣ1 ) 4 + F + 5k 2 Ψ + k ) 2 Σ1 tf + 24φ2 k 2 Ψ } tf, k 2 { tf = 2304π k2 F (3k 2 F V) Ψ 2 24 (24φ 2 k 2 F Ψ + k 2 Ψ + k 2 ) FΣ 1 36 [ 2 4φ 2 (6k 4 F 2 + Ψ 2 ) + 4φ 2 ΨΨ (7k 2 F V )(Σ 1 k 2 ) + 4φ 2 Σ 1 (7k 2 F V )(2ΨV V Ψ ) ] +2k 4 Ψ 2 Σ k 4 F φ 2 ΨΨ Σ 2 24k 4 F φ 2 Ψ 2 Σ 2 tf [ 30 10k2 F (7Ψ + 4V) F Ψ ( 2 k 2 F Σ 1 + 4φ 2 V Ψ 24k 4 F φ 2 Ψ 2 Σ 2 )] 4φ 2 k 2 F Ψ Σ 1 (7k 2 F V ) + tf 24k2 φ 2 ]} [(k 2 2 F + 5V ) 12k 2 ΨΨ Σ 2 2(7k 2 F V )ΨΣ 1 where we have defined the shorthands: Ψ = k 2 F V ; Σ 1 = k 2 + 2V + 4φ 2 V ; Σ 2 = 2F + 4φ 2 F ; = (12φ 2 Ψ 2 ) + ΨΣ 1.

53 Polynomial truncations Ṽ( φ 2 ) = λ 0 + λ 2 φ2 + λ 4 φ F( φ 2 ) = ξ 0 + ξ 2 φ t λ4 = 9λ2 4 2π 2 + Gλ 4 π +...

54 SM+gravity dh dt = βsm h + a h 8π G(µ)h β SM 1 = β SM y = β SM λ = π 2 6 g3 1 ; β2 SM = π 2 6 g3 2 ; β3 SM = 1 16π 27g3 3 ; [ π 2 2 y3 8g3 2 y 9 4 g2 2 y 17 ] 21 g2 1 y [ ( 1 16π 2 24λ λy 2 9λ g ) 3 g2 1 6y g g ] 4 g2 2g1 2 If a 1 = a 2 = a 3 < 0, a y < 0, a λ > 0, predict m h =126GeV [M. Shaposhnikov and C. Wetterich, Phys.Lett. B683, 196 (2010) ]

55 AS: strengths and weaknesses but use continuum covariant QFT bottom up approach guaranteed to give correct low energy limit highly predictive strong coupling problem scheme dependence off shell: gauge dependence no observables computed yet

56 Summary/outlook Pure gravity: truly functional truncations other approximations, e.g. two loops describe as CFT Gravity+matter: BSM physics strongly constrained running gravitational mass from 2 point function matter interactions: LPA for scalar+gravity turn on gauge and Yukawa couplings