Holography and the cosmological constant

Transcription

1 String Pheno Ioannina, 22 June 2016 Holography and the cosmological constant CCTP/IPP/QCN University of Crete APC, Paris 1-

2 Bibliography Ongoing work with Francesco Nitti (APC, Paris 7), Christos Charmousis, (U. d Orsay) based on earlier ideas in EPJ Web Conf. 71 (2014) arxiv: [hep-ph] Self-tuning 2.0, 2

3 Introduction The cosmological constant problem is arguably the most important shortcoming today of our understanding of the physical world. It signifies the violent clash between gravity and quantum field theory, more so than the black hole information paradox problem. In four-dimensional Einstein gravity a non-zero vacuum energy entails irrevocably the acceleration of the univers: G µν = 1 2 Λ g µν Fine-tuning the cosmological constant does not solve the problem as we can tune a single constant while the quantum corrections to the vacuum energy are scale dependent. reviews: Weinberg, Rubakov, Hebecker+Wetterich,Burgess Self-tuning 2.0, 3

4 The higher-dimensional hope It was argued that the existence of higher (than four) dimensions offers the possibility to alleviate the cosmological constant problem. Rubakov+Shaposhnikov The rough idea is that the SM-induced vacuum energy, instead of curving the 4-d world/brane, could be absorbed by bulk fields. For this idea to be effective, the mechanism must be quasi-generic: any cosmological constant must relax, absorbed by the bulk dynamics. Any such mechanism must be intertwined tightly with cosmology as we have good reasons to believe that a large cosmological constant played an important role in the early universe, with observable consequences today. D. Kazanas, Englert+Brout,Sato,Guth,Starobinsky,Muchanov+Chibisov Self-tuning 2.0, 4

5 Brane worlds and early attempts String Theory D-branes offered a concrete, calculable realization of a brane universe. Polchinski Branes in a cutoff-ads 5 space were used to argue that this offers a context in which brane-world scales run exponentially fast, putting the hierarchy problem in a a very advantageous framework. Randall+Sundrum It is in this context that the first attempts of self tuning of the brane cosmological constant were made. Arkani-Hamed+Dimopoulos+Kaloper+Sundrum,Kachru+Schulz+Silverstein, The models used a bulk scalar to absorb the brane cosmological constant, and provide solutions with a flat brane metric despite the non-zero brane vacuum energy. The attempts failed as such solutions had invariantly a bad/naked bulk singularity that rendered models incomplete. 5

6 More sophisticated setups were advanced and more general contexts have been explored but without success: the naked bulk singularity was always there. Csaki+Erlich+Grojean+Hollowood, The Randall-Sundrum Z 2 orbifold boundary conditions were relaxed to consider even more general setups, but this did not improve the situation. Padilla The RS setup and its siblings is related via holographic ideas to cutoff- CFTs and this provides independent intuition on the physics. Maldacena, Witten,Arkani-Hamed+Porrati+Randall In view of our current understanding of holography, these failures were to be expected. Our goal: provide a 2.0 version of the self-tuning mechanism that is in line with the dictums of holography. Self-tuning 2.0, 5-

7 Holographic gravity and the SM We can envisage the physics of the SM+gravity (plus maybe other ingredients) as emerging from 4d UV complete QFTs: Kiritsis a) A large N/strongly coupled stable (near-cft) b) The Standard Model c) A massive sector (of mass M) that couples the two theories. (a) has a holographic description in a 5d space-time. For E M we can integrate out the messenger sector and obtain directly the SM coupled to the bulk gravity. The holographic picture is that of a brane (the SM) embedded in the bulk at r 1 M. 6

8 Holography reproduces the brane world embedded in a higher dimensional bulk. This picture has a UV cutoff: the messenger mass M. The configuration resembles string theory orientifolds and possible SM embeddings have been classified in the past. Anastasopoulos+Dijkstra+Kiritsis+Schellekens The SM couples to all operators/fields of the bulk QFT. Most of them they will obtain large masses of O(M) due to SM quantum effects. The only protected fields are the metric, the universal axion T r[f F ] and possible vectors (aka graviphotos). Self-tuning 2.0, 6-

9 2nd order gravity equations and 1rst order RG flows Consider a bulk (5d) Einstein-scalar theory (dual to the QFT dynamics of a scalar operator O(x) and the stress tensor T µν (x)): S bulk = M 3 d 5 x [ g R 1 ] 2 ( ϕ)2 V (ϕ) and a Poincaré-invariant ansatz ds 2 = du 2 + e 2A(u) ( dt 2 + d x 2 ), ϕ(u) The independent bulk gravitational scalar-einstein equations are 12Ȧ(u) ϕ 2 + V (ϕ) = 0,, 6Ä(u) + ϕ 2 = 0, 7

10 They can be written as first order (holographic RG) flow equations Ȧ(u) := 1 6 W (ϕ), ϕ(u) = W (ϕ) in terms of the superpotential W (ϕ) that satisfies V (ϕ) = 1 2 W 2 (ϕ) 1 3 W 2 (ϕ) One of the integration constants is hidden in the non-linear superpotential equation but... It is fixed, by asking the gravitational solution is regular at the interior of the space-time. The solutions are dual to RG flows from the UV (AdS boundary) to the IR (typically AdS if IR CFT) with (holographic) β-function dϕ d = β(ϕ) = 6 da dϕ log W Conclusion: given a bulk action the solution is characterized by the unique superpotential function W (ϕ). Self-tuning 2.0, 7-

11 Brane+Bulk equations We are ready to consider bulk solutions with the SM brane inserted at some radial position u = u 0. S bulk = M 3 d 5 x [ g R 1 ] 2 ( Φ)2 V (Φ) S brane = M 2 δ(u u 0 ) d 4 x γ [ W B (Φ) 1 ] 2 Z(Φ)γµν µ Φ ν Φ + U(Φ)R B +, The localized action on the brane is due to quantum effects of the SM fields. W B (ϕ) is the cosmological term whose magnitude is of the order of the cutoff of the whole description namely M. The equations to solve are the bulk equations plus the Israel junction conditions at u = u 0. 8

12 We denote by gab UV, gir ab and Φ UV, Φ IR the solutions for the metric and scalar field on each side of the brane. [ ] IR X UV is the jump of a quantity X across the defect. The Israel matching conditions are: 1. Continuity of the metric and scalar field: [g ab ] UV IR = 0, [ ] IR Φ UV = 0 2. Discontinuity of the extrinsic curvature and normal derivative of Φ: [ ] IR K µν γ µν K UV = 1 γ δs brane δγ µν, [ ] IR n a a Φ UV = δs brane δφ, 8-

13 With the standard Poincaré invariant ansatz (with flat brane metric) we have Ȧ UV 1 (u) = 2(d 1) W UV (Φ(u)), Φ UV (u) = Ȧ IR 1 (u) = 2(d 1) W IR (Φ(u)), Φ IR (u) = dw UV dφ dw IR (Φ(u)) dφ (Φ(u)). The scalar functions W UV,IR are both solutions to the superpotential equation: 1 3 W 2 1 ( ) dw 2 = V. 2 dφ The continuity conditions A UV (u 0 ) = A IR (u 0 ) = A 0, Φ UV (u 0 ) = Φ IR (u 0 ) = Φ 0. Only one initial condition (A, Φ ) must be imposed in the UV, and this choice is trivial. The jump conditions are W IR W UV Φ0 = W B (Φ 0 ), dw IR dφ dw UV dφ = dw B Φ0 dϕ (Φ 0) Self-tuning 2.0, 8-

14 Old Self-Tuning W UV and W IR are determined from the superpotential equation up to one integration constant, C UV, C IR. For a generic brane potential W B (Φ), the two matching/jump equations will fix C UV, C IR for any generic value of Φ 0. The fixed value of C IR typically leads to a bad IR singularity. Self-tuning 2.0, 9

15 Self-Tuning 2.0 The IR constant C IR should be fixed by demanding that the IR singularity is a good one according to one of the typical criteria ie. Gubser. Typically there is only one such solution to the superpotential equation (or a discreet set). According to holography rules, the solution W IR should be fixed before we impose the matching conditions. Once W IR is fixed by regularity, the Israel conditions will determine: The integration constant C UV in the UV superpotential The brane position in field space, Φ 0. This is a desirable outcome as there would be no massless radion mode. It can be checked that generically such an equilibrium position exists. Self-tuning 2.0, 10

16 Tensor perturbations and induced gravity The tensor mode satisfies the Laplacian equation in the bulk 2 r ĥµν + (d 1)( r A) r ĥ µν + ρ ρ ĥ µν = 0 and the matching condition [ĥ IR ĥ ] UV = U(ϕ r 0 ) 0 e A 0 µ µ ĥ(r 0 ), This is the same condition as in DGP in flat space Dvali+Gabadadze+Porrati The main difference is e A factor that enhances the brane Planck scale U(ϕ 0 ) if the brane is in the IR part of the geometry. In other (string) realizations of the DGP setup, large unnatural values for moduli or number of particles are needed to obtain this. Kiritsis+Tetradis+Tomaras, Antoniadis+Minasian+Pioline+VanHove Self-tuning 2.0, 11

17 The gravitational interaction on the brane The brane-to-brane propagator in momentum space can be calculated: G 4 (p) = ea 0 M 3 D(p, r 0 ) 1 + [e A 0U 0 D(p, r 0 )]p 2 where D(p, r) solves the equation: [ r e 3A(r) r e 3A(r) p 2] D(p, r) = δ(r r 0 )e 3A. When e A 0U 0 D(p, r 0 ) p 2 1, G 4 (p) the propagator is 4-dimensional 1 1 e 2A 0M 3 U 0 p 2 The detailed behavior of the propagator is determined by the function D(p, r) evaluated at r 0. It is determined by the Laplacian in the UV and IR part of the geometry, with continuity and unit jump at the brane. The 4-d phase is always present at large enough p 2. Self-tuning 2.0, 12

18 The characteristic scales There are the following characteristic distance scales that play a role, besides r 0 (position of the SM brane). The transition scale r t around which D(r 0, p) changes from small to large momentum asymptotics: D(r 0, p) 1 2p p 1 r t, d 0 + O(p 2 ) p 1 r t, d 0 = e 3A 0 r0 0 dr e 3A UV (r ) The transition scale r t depends on r 0 and the bulk QFT dynamics. The crossover scale, or DGP scale, r c : r c e A 0U 0 ; 2 This scale determines the crossover between 5-dimensional and 4-dimensional behavior, and enters the 4D Planck scale and the graviton mass. 13

19 The gap scale d 0 d 0 D(r 0, 0) = e 3A 0 0 dr e 3A UV (r ), which governs the propagator at the largest distances (in particular it sets the graviton mass). r0 Typically, d 0 r 0 For example, in IR AdS we have. d 0 = r 0 4 In confining bulk backgrounds we have instead d 0 1 6Λ 2 r 0 In the far IR, Λr 0 1 and d 0 can be made arbitrarily small. Self-tuning 2.0, 13-

20 DGP and massive gravity When r t > r c we have three regimes for the gravitational interaction on the brane: G 4 (p) ea 0 2r c M d 1 1 p 2 ea 0 1 M d 1 2p ea 0 1 2r c M d 1 p 2 + m 2 0 p 1 r c 1 r c p 1 r t p r t 1, m r c d 0 The 4d gravitational coupling is the same in both IR and UV regimes. 14

21 Massive 4d gravity (r t < r c ) In this case, at all momenta above the transition scale, p 1/r t > 1/r c, we are in the 4-dimensional regime of the DGP-like propagator. The behavior is four-dimensional at all scales, and it interpolates between massless and massive four-dimensional gravity. Kiritsis+Tetradis+Tomaras Below the transition, p 1/r t, we have again a massive-graviton propagator. 14-

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26 Conclusions and Outlook A large-n QFT coupled holographically to the SM offers the possibility of tuning the SM vacuum energy. The graviton fluctuations have DGP behavior while the graviton is massive at large enough distances. No fine-tuning is needed to have a reasonable Planck scale. The graviton mass can be very small in a natural fashion (driven by a small confinement scale in the bulk). Most of the interesting physics is to be deciphered: The scalar perturbations must be analyzed in detail. The cosmological evolution must be elucidated. The setup offers the possibility to address both inflation and dark energy (late-time acceleration). Self-tuning 2.0, 15

27 . THANK YOU Self-tuning 2.0, 16

28 Linear perturbations around a flat brane Bulk Fluctuations equations ds 2 = a 2 (r) [ (1 + 2ϕ)dr 2 + 2A µ dx µ dr + (η µν + h µν )dx µ dx ν], φ(x) = φ 0 (r)+χ where the fields ϕ, A µ, h µν, χ depend on r, x µ and are small perturbations. We further decompose the bulk modes into tensor, vector and scalar perturbations as usual: A µ = µ W + A T µ, h µν = 2η µν ψ + µ ν E + 2 (µ V T ν) + ĥ µν with µ A T µ = µ V T µ = µ ĥ µν = ĥ µ µ = 0. The physical bulk scalar can be identified with the gauge-invariant combination: ζ = ψ A ϕ χ. o The non-trivial (propagating) fluctuations are ĥ µν and ζ. 17

29 In the presence of the brane there is also the embedding mode X A (σ α ) where X A = (r, x µ ) and σ α are world-volume coordinates. We choose the gauge σ α = x µ δ α µ, so the embedding is completely specified by the radial profile r(x µ ). We consider a small deviation from the equilibrium position r 0 : r(x µ ) = r 0 + ρ(x µ ) The brane scalar mode ρ represents brane bending. Self-tuning 2.0, 17-

30 The gravitational interaction on the brane The field equations together with the matching conditions can be obtained by extremizing S[h] = M d 1 d d xdr gg ab a ĥ b ĥ + M d 1 r=r 0 d d x γu B (ϕ)γ µν µ ĥ ν ĥ, where g ab = e A(r) η ab and γ µν = e A 0 η µν are the unperturbed bulk metric and induced metric on the brane, respectively. We introduce brane-localized matter, S m = d d x γ L m (γ µν, ψ i ) where ψ i denotes collectively the matter fields. The field equation for the tensor propagator in the presence of branelocalized sources is: ( [ r e 3A(r) ) [ ] r + e 3A(r) + U 0 e 2A 0δ(r r 0 ) ] µ µ G 4 (r, x; r, x ) = = e2a 0 M 3 T (x)δ(r r 0)δ (4) (x x ) 18

31 The static potential on the brane is given by V ( x) = m 1 m 2 dt G 4 ( x, t; 0, 0). The brane-to-brane propagator in momentum space is: G 4 (p) = ea 0 M 3 D(p, r 0 ) 1 + [e A 0U 0 D(p, r 0 )]p 2 where D(p, r) solves the equation: [ r e 3A(r) r e 3A(r) p 2] D(p, r) = δ(r r 0 )e 3A. When e A 0U 0 D(p, r 0 ) p 2 1, G 4 (p) the propagator is 4-dimensional 1 e 2A 0M 3 U 0 1 p 2 18-

32 A(p) G N T (p)t ( p) p 2, G N e2a 0 U 0 M 3, M 2 P = e 2A 0U 0 M 3 The detailed behavior of the propagator is determined by the function D(p, r) evaluated at r 0. It is determined by the Laplacian in the UV and IR part of the geometry, with continuity and unit jump at the brane. We will see that the 4-d phase is always present at large enough p 2. The crossover scale is determined by the equation: e A 0U 0 p 2 D(p, r 0 ) = 1 Self-tuning 2.0, 18-

33 The bulk propagator At large Euclidiean p 2, we can approximate the bulk equations as in flat space, 2 r Ψ (p) (r) = p 2 Ψ (p) (r) except for small r, where the effective Schrödinger potential is 1/r 2 and cannot be neglected. The solution satisfying appropriate boundary conditions (vanishing in the IR and for r 0) and jump condition is IR = sinh pr 0 e pr, p Ψ (p) For large p, it is like in flat space Ψ (p) UV = e pr 0 p sinh pr, D(p, r 0 ) = sinh pr 0 e pr 0 1 p 2p, pr 0 1 p p 2 At small momenta, things are simpler if the spectrum is gaped, i.e. if the IR is confining. 19

34 The bulk propagator has an analytic expansion in powers of p 2 and we can solve perturbatively in p 2. At p = 0 the solution can be found exactly: D(r, 0) = e 3A 0 e 3A 0 r 0 dr e 3A UV (r ) r0 0 dr e 3A UV (r ) r < r 0 r > r 0 For gapless geometries (that include an IR AdS) the result is valid but non-analytic terms appear higher up in the p-expansion Self-tuning 2.0, 19-

35 DGP and massive gravity When r t > r c we have three regimes for the gravitational interaction on the brane: G 4 (p) e(d 3)A 0 2M d 1 1 p + r c p 2 e(d 3)A 0 2r c M d 1 1 p 2 e(d 3)A 0 M d 1 1 2p p 1/r c 1/r c p 1/r t e(d 3)A 0 2r c M d 1 1 p 2 + m 2 0 p r t 1, m r c d 0 The 4d gravitational coupling is the same in both IR and UV regimes. 20

36 Massive 4d gravity (r t < r c ) In this case, at all momenta above the transition scale, p 1/r t > 1/r c, we are in the 4-dimensional regime of the DGP-like propagator. The behavior is four-dimensional at all scales, and it interpolates between massless and massive four-dimensional gravity. Kiritsis+Tetradis+Tomaras Below the transition, p 1/r t, we have again a massive-graviton propagator. 20-

37 An example: V (ϕ) = ( ) 2 ϕ , The flow is from ϕ = 0 (UV Fixed point) to ϕ = 1 (IR fixed point). W b (ϕ) = ω exp[γϕ]. ω = 0.01, γ = 5 ϕ 0 = This gives, in conformal coordinates, r 0 = Self-tuning 2.0, 20-

38 Detailed plan of the presentation Title page 0 minutes Bibliography 1 minutes Introduction 2 minutes Higher dimensions 3 minutes Brane worlds and early attempts 5 minutes Holographic gravity and the SM 7 minutes 2nd order gravity equations and 1rst order RG flows 10 minutes Brane and bulk equations 16 minutes Old Self-Tuning 17 minutes Self-Tuning minutes Linear Perturbations around a flat brane 21 minutes Induced Gravity 23 minutes The gravitational interaction on the brane 24 minutes The characteristic scales 27 minutes DGP and massive gravity 31 minutes Conclusions and Outlook 32 minutes 21

39 Linear Perturbations around a flat brane 35 minutes The gravitational interaction on the brane 36 minutes The bulk propagator 39 minutes DGP and massive gravity 42 minutes Self-tuning 2.0, 21-