Quantum black holes and effective quantum gravity approaches!

Transcription

1 Quantum black holes and effective quantum gravity approaches! Xavier Calmet Physics & Astronomy University of Sussex

2 Outline QBHs as a probe of the scale of quantum gravity Low energy effective action Higgs boson s nonminimal coupling to R How does the nonminimal coupling of the Higgs impact physics? Running of the Planck mass Higgs inflation Unitarity Frame dependence of gravitational theories

3 Why is gravity so much weaker than other forces?

4 TeV gravity extra-dimensions Arkani-Hamed, Dimopoulos, Dvali (ADD) brane world where M P is the effective Planck scale in 4-dim Randall Sundrum (RS) warped extra-dimension

5 Why is gravity so much weaker than other forces? or rather: At what energy do quantum gravity effects become important?

6 Typical problems of models with TeV Quantum Gravity: Light Kaluza-Klein gravitons in ADD: Graviton KKs lead to astrophysical constraints: supernovae cooling and neutron stars heating: limits on the scale/number of dimensions

7 Bounds (orders of magnitude) on ADD brane-world model n Gravity exp. LEP2/ Tevatron LHC (see Greg s talk) Astro. SN +NS Cosmic rays 10 7 km 0.2mm 0.1 fm 1 TeV 1 TeV 1 TeV 1 TeV 1 TeV ~5 TeV ~5 TeV ~5 TeV ~5 TeV ~5 TeV 10 3 TeV 10 2 TeV 5 TeV none none 1 TeV 1 TeV 1 TeV 1 TeV 1 TeV 1 TeV Note: mass gap grows with n. In RS bounds of the order of a few TeV due to mass gap.

8 BHs as probe of strong gravity Formation of small black holes in the collisions of particles would be a signal of strong gravity. LHC Cosmic rays Early universe Primordial black holes remnants? A brief review on the formation of black holes

9 When does a black hole form? This is well understood in general relativity with symmetrical distribution of matter: But, what happens in particle collisions at extremely high energies?

10 Small black hole formation (in collisions of particles) In trivial situations (spherical distribution of matter), one can solve explicitly Einstein s equations e.g. Schwarzschild metric. In more complicated cases one can t solve Einstein equations exactly and one needs some other criteria. Hoop conjecture (Kip Thorne): if an amount of energy E is confined to a ball of size R, where R < E, then that region will eventually evolve into a black hole.

11 Small black hole formation (in collisions of particles) In trivial situations (spherical distribution of matter), one can solve explicitly Einstein s equations e.g. Schwarzschild metric. In more complicated cases one can t solve Einstein equations exactly and one needs some other criteria. Hoop conjecture (Kip Thorne): if an amount of energy E is confined to a ball of size R, where R < E, then that region will eventually evolve into a black hole. Cross-section for semi-classical BHs (closed trapped surface constructed by Penrose; D Eath & Payne; Eardley & Giddings):

12 A CTS is a compact spacelike two-surface in space-time such that outgoing null rays perpendicular to the surface are not expanding. At some instant, the sphere S emits a flash of light. At a later time, the light from a point P forms a sphere F around P, and the envelopes S 1 and S 2 form the ingoing and outgoing wavefronts respectively. If the areas of both S 1 and S 2 are less than of S, then S is a closed trapped surface.

13 Small LHC (studied by Anchordoqui et al. and many other people) fb σ(pp->bh+x), M D =1 TeV For partons, σ increases with energy but note that PDFs go so fast to zero that they dominate. In other words quantum black holes dominate! This shows the significance of the inelasticity in BH production

14 Semi-classical (thermal) versus quantum black hole: calculate the entropy! m BH >M m BH ~M " N M BH M n+2 n+1 Keep in mind that E-G construction only works for m BH >>M

15 Assumptions on Quantum Black Holes decays Gauge invariance is preserved (conservation of U(1) and SU(3) C charges) Quantum Black Holes do not couple to long wavelength and highly off-shell perturbative modes. Global charges can be violated. Lepton flavor is not conserved. Lorentz invariance could be broken or not. Gravity is democratic. We can think of quantum black holes as gravitational bound states.

16 The Quark Model (1964) S=1 s Q=-1/3 Q=2/3 d u S=0 u d Q=-2/3 Q=-1/3 s S=-1

17 The Meson Octet S=1 K 0 K + S= 0 π - π 0 ; η π + S= 1 K - K 0 Q=1 Q=-1 Q=0

18 The Quantum Black Hole Octet S=1 QBH 01 QBH +1 S= 0 QBH -0 QBH 00 ; QBH 00 QBH +0 S= 1 QBH -1 QBH 01 Q=1 Q=-1 Q=0

19 QCD for Quantum Black Holes Quantum Black Holes are classified according to representations of SU(3) C. For LHC the following Quantum Black Holes are relevant: XC, W. Gong & S. Hsu They can have non-integer QED charges. They can carry a SU(3) C charge.

20 QBHs as particles We can think of QBHs as state in a path integral. LHC teaches us that the Planck scale is not in the few TeV region. Integrate them out. We can probe physics at at much higher scale. We know the symmetries: we can formulate an effective field theory.

21 Below the Planck scale Note: here we are conservative, we could have more exotics terms depending on which symmetries are conserved or broken. What do we know about the coefficient of this action?

22 This theory must contain the particles of the SM m H 125 GeV Is there something special about the Higgs boson from a gravitational point of view?

23 Is the Higgs boson the source of problems? Dominant point of view for the last 40 years: the Higgs boson s mass is not stable in the SM and should be protected by a symmetry (or there is no fundamental scalar) My point of view: it is not a question that we can address within our current theories of physics. We are only dealing with renormalizable theories: the Higgs mass is not calculable. Wilson called the hierarchy problem a blunder! Note that the lack of new physics at the LHC could be the second nail in the coffin for naturalness after the cosmological constant.

24 Or rather is the Higgs boson the solution to other fine tuning problems? It is difficult to imagine why our universe is so flat and homogenous. Why are there no monopols? What is at the origin of the cosmological perturbations? This is really an initial condition problem. Either we had very special initial conditions or something created them: inflation. Could the Higgs boson be the inflaton? Obviously inflation with scalar fields also has fine-tuning/ stability issues.

25 The Standard Model predicts precisely how the Higgs boson should be produced It is the Higgs boson

26 The Standard Model predicts precisely how the Higgs boson should decay It is the Higgs boson

27 It is the Higgs boson From arxiv:

28 It is the Higgs boson From arxiv:

29 Something special about the Higgs boson It can be coupled in a nonminimal way to gravity. This is a dimension 4 operator: it is a fundamental constant of nature. Is there any bound on its value?

30 The decoupling effect Let s consider the SM with a nonminimal coupling to R We can always go from the Jordan frame to the Einstein frame

31 The decoupling effect In the Einstein frame, the action reads One notices that the Higgs boson kinetic term is not canonically normalized. We need to diagonalize this term. Let me now use the unitary gauge The Planck mass is defined by

32 The decoupling effect To diagonalize the Higgs boson kinetic term: To leading order in

33 The decoupling effect The couplings of the Higgs boson to particles of the SM are rescaled! E.g. For a large nonminimal coupling, the Higgs boson decouples from the Standard Model:

34 The decoupling effect The decoupling can also be seen in the Jordan frame: same renormalization factor!

35 Bound on the nonminimal coupling from the LHC The LHC experiments produce fits to the data assuming that all Higgs boson couplings are modified by a single parameter (arxiv: [hep-ph]): In the narrow width approximation, one finds:

36 Bound on the nonminimal coupling from the LHC Current LHC data allows to bound Combining these two bounds one gets: ATLAS CMS which excludes Atkins & xc, PRL 110 (2013)

37 Bound on the nonminimal coupling from the LHC At a 14 TeV LHC with an integrated luminosity of 300 fb -1, could lead to an improved bound on the nonminimal coupling: while an ILC with a center of mass energy of 500 GeV and an integrated luminosity of 500 fb -1, could give It seems tough to push the bound below this limit within the foreseeable future.

38 How does the nonminimal coupling of the Higgs impact physics? Running of the Planck mass Higgs inflation Unitarity Much more but work in progress

39 Running of Newton s constant Consider GR with a massive scalar field Let me consider the renormalization of the Planck mass: Can be derived using the heat kernel method (regulator preserves symmetries!)

40 Running of Newton s constant Consider GR with a massive scalar field Let me consider the renormalization of the Planck mass: Gravity becomes strong if: To give you an idea ξ=10 15 implies μ GeV

41 Like any other coupling constant: Newton s constant runs! Theoretical physics can lead to anything even business ideas!

42 Higgs as the inflaton? Nice idea: try to unify two scalar fields Successful inflation requires This is suspiciously large! What can we learn from unitarity considerations?

43 Quick review of inflation Assuming a Friedmann, Robertson Walker (FRW) metric for the universe where a(t) is the scale factor, using Einstein's equations, one gets where ρ and p are the density and pressure appearing in the stress energy tensor of the vacuum of the universe. Inflation can be described as the condition occurs for

44 Quick review of inflation One can fix the potential of a scalar field such that with So one finds: The potential needs to be flat enough.

45 Quick review of inflation With a flat enough potential, this criteria will be met and inflation will occur as the scalar field slowly rolls down the slope. The potential also requires a minimum where inflation can eventually end. During the period of inflation the universe is supercooled. Following inflation, the inflaton oscillates around its final minimum transferring its potential energy into the standard model particles that fill the universe including electromagnetic radiation which starts the radiation dominated phase of the universe. This period after inflation ends and before the inflaton comes to rest is known as reheating.

46 Quick review of Higgs inflation Since we know of one scalar field in nature it is natural to try to describe inflation with it. The SM Higgs potential is not flat enough! But a nonminimal coupling will change the shape of the potential

47 Quick review of Higgs inflation In Einstein frame the action becomes with For small Higgs values the potential is the same as for the initial Higgs one, however for large field values i.e. the potential is exponentially flat

48 From (Bezrukov&Shaposhnikov) Standard analysis, slow role parameters: Number of e-foldings:

49 Unitarity in quantum field theory Follows from the conservation of probability in quantum mechanics. Implies that amplitudes do not grow too fast with energy. One of the few theoretical tools in quantum field theory to get information about the parameters of the model. Well known example is the bound on the Higgs boson s mass in the Standard Model (m<790 GeV).

50 Let us consider gravitational scattering of the particles included in the standard model (s-channel, we impose different in and out states)

51 Higgs as the inflaton? We obtained a bound on the non minimal gravitational coupling of scalar fields In the minimal model, we have the SM + gravity and no new physics. The cutoff should be the red. Planck mass! From the J=0 partial wave, we get: In today s background (small Higgs vev, flat spacetime) For ξ=10 4, unitarity breaks down at GeV New physics? Strong dynamics? Is the potential still flat enough?

52 However one needs to be careful, the bound depends on the background. In inflationary background, one finds This does not affect our conclusion though: the tightest bound on ξ is the one obtained in flat space-time and for a small Higgs vev Minimal model does not work, one needs new physics between the inflationary scale and the red. Planck mass. Way out: asymptotically safe gravity. If the Planck mass and ξ get weaker in the UV, their running can compensate the growth of the amplitude with energy. Or is there a self-healing mechanism at work?

53 Singlet scalar field What happens for N=1? In the high energy limit: Terms proportional to ξ (and not m) do not grow with energy: no bound.

54 Higgs+singlet inflation Singlets have been advocated as a possible solution Giudice & Lee but one finds by looking at 2 to 4 scattering. The self coupling must be finely tuned!

55 SM Higgs + Gravity alone cannot provide a full description of particle physics and inflation up to the Planck mass unless gravity is asymptotically safe. Singlets could work, but you need to fine-tune the selfcoupling. Is there a connection to dark matter? Interesting idea suggested by Donoghue: there is a selfhealing mechanism at work? But is the potential still flat enough?

56 More nonminmal couplings! We can describe any theory of quantum gravity below the Planck scale using effective field theory techniques: Electroweak symmetry breaking: Several energy scale: Λ C ~10-12 GeV cosmological constant M P or equivalently Newton s constant G= 1/(8π M P2 ) M energy scale up to which one trusts the effective theory Dimensionless coupling constants ξ, c 1, c 2 etc

57 What values to expect for the coefficients? It all depends whether they are truly new fundamental constants or whether the operators are induced by quantum gravitational effects. If fundamental constants, they are arbitrary If induced by quantum gravity we can estimate their magnitude. Usually induced dimension four operators are expected to be small λ is some low energy scale However, translates into in terms of the graviton h. -type operators lead to We thus expect the coefficients of these operators to be O(1). Naturalness arguments would imply M ~Λ C. However, there is not sign of new physics at this energy scale.

58 What do experiments tell us? In 1977, Stelle has shown that one obtains a modification of Newton s potential at short distances from R 2 terms c 1 and c 2 <10 61 xc, Hsu and Reeb (2008) NB: Bound has improved by 10 order of magnitude since Stelle s paper! Schematic drawing of the Eöt-Wash Short-range Experiment

59 Can better bounds be obtained in astrophysics? Bounds on Earth are obtained in weak curvature, binary pulsar systems are probing high curvature regime. Approximation: Ricci scalar in the binary system of pulsars by G M/(r^3c^2) where M is the mass of the pulsar and r is the distance to the center of the pulsar. But: if the distance is larger than the radius of the pulsar, then the Ricci scalar vanishes. This is a rather crude estimate.

60 Can better bounds be obtained in astrophysics? Let me be optimistic and assume one can probe gravity at the surface of the pulsar. I take r=13.1km and M=2 solar masses. I now request that the R 2 term should become comparable to the leading order Einstein-Hilbert term (1/2 M P 2 R) One could reach bounds of the order of only on c 1 or c 2 Such limits are obviously much weaker that those obtained on Earth.

61 Frame dependence in GR? Short and obvious answer: no you are just doing field redefinitions. BUT: semantic is important! The real question is what do you mean by the equivalence of two frames. Starting from the Jordan frame

62 Frame dependence in GR? Starting from the Jordan frame action using

63 Frame dependence in GR? Starting from the Jordan frame action one obtains

64 Frame dependence in GR? We thus find with the boundary term given by

65 Frame dependence in GR? At the quantum level, it is easier to work backwards Doing the same field transformation, we obtain with a Jacobian given by

66 Frame dependence in GR? One can easily show that the Jacobian is related to the expectation value of the energy momentum tensor with

67 Frame dependence in GR? Partitions functions are the same up to 2 terms To be compared with Physics is not affected as long as the transformations are done properly!

68 In the Higgs inflation case Frame dependence in GR? with This clearly does not affect the inflationary calculation.

69 Conclusions QBHs may be relevant at different energy scales. Instead of direct production, we can integrate them out. One parameter of the effective action is particularly interesting: The SM Higgs has been found: there is at least one more new fundamental constant in nature: the nonminimal coupling of the Higgs boson to gravity. This new parameter can have a dramatic impact on physics: It can make Newton s constant run It can lead to Higgs inflation within the SM It creates issues with unitarity (unless there is a self-healing mechanism at work) Much more to come Physics does not depend on the frame.

70 Conclusions QBHs may be relevant at different energy scales. Instead of direct production, we can integrate them out. One parameter of the effective action is particularly interesting: The SM Higgs has been found: there is at least one more new fundamental constant in nature: the nonminimal coupling of the Higgs boson to gravity. This new parameter can have a dramatic impact on physics: It can make Newton s constant run It can lead to Higgs inflation within the SM It creates issues with unitarity (unless there is a self-healing mechanism at work) Much more to come Physics does not depend on the frame. Thanks for your attention!

71 Back up slides

72 Derivation of the renormalization group equation (see e.g. Larsen & Wilczek (1995)) One loop effective action of a scalar field coupled to gravity: The heat kernel is defined as:

73 The integration over τ is divergent: introduce an ultra-violet cutoff ε 2 : One can define: Where the Green s function satisfies: In flat space one would have:

74 Expansion for small curvature yields: One thus finds: This was old-fashion perturbation theory. Wilsonian approach: let us integrate out modes with k >µ and consider physics at energies below µ: And thus:

75 Running of the Planck Mass With spin 0, spin 1/2 (Weyl) and spin 1 fields: Gravity becomes strong at Some definitions: In SUSY models: