KEK 6/27, 2011

Renormalizable 4D Quantum Gravity and

Cosmological Constant ProblemK. Hamada

http://research.kek.jp/people/hamada/

Reference

Renormalizable 4D Quantum Gravity as a Perturbed Theory from CFT, arXiv:0907.3969[hep-th].

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Recent developments for

Resolve the problem of space-time singularities

At , particle excitations become black holesgive up graviton picture

I proposed non-perturbative CFT approach

Understand space-time dynamics of the early universe

I proposed Inflation scenario without a scalar field:

New scale space-time transitionNumber of e-foldingsInitial condition of the universe CFT spectraAmplitude reduces during inflation

Physical quantities given by ratios of two gravitational scales

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Evolution of fluctuation K.H., Horata, Yukawa, arXiv:0908.0192

Planck phenomena (CFT) space-time transition （big bang） today (CMB)

From Planck length tocosmological distance

293059 101010 +=

inflation Friedmann

Spectrum at transition point= almost HZ spectrum

inflation 0.00

0.10

0.20

-2

-1

0

1 10-3

10-2

0.00

0.10

0.20

Bardeen Potential Φ(b1=10, m=0.0156)

proper time, log10(τ/τp)

k [Mpc-1]

58.0

58.5

59.0

59.5

60.0 10-3

10-2

1 × 10-4

3 × 10-4

5 × 10-4

proper time τ

k [Mpc-1]

0

2000

4000

6000

1 10 100 500 1500

wmap 5yrsacbar2008

Inflation era

Scale-inv. spectrum at Planck time

Amplitude decreases during inflationResolve the flatness problem

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The Aim of This Talk

Explain small cosmological constant in the contextof quantum gravity dynamics

Renomalizable 4D Quantum Gravity as a Perturbed Theory from CFT

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Renormalizable Quantum GravityThe Action (Weyl + Euler + Einstein) weight

conformally invariant (no R^2) Planck constant

“t” is a unique dimensionless gravitational coupling constant indicating asymptotic freedom

conformally flatAt high energies(Perturbation is defined

about this config.)“b” is not independent coupling, which is expanded by t

At the Einstein action dominates

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Conformal Symmetry is “Gauge” Symmetry in Quantum Gravity

Diffeomorphism includes conformal transformation at UV limit (t 0)No coupling const.

Metric field is expanded about

with

(# other gauge d.o.f. are gauge-fixed in usual way)

Field dep.transf.

(cf. Field indep.)

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Induced Riegert-Wess-Zumino Action

Jacobian to preserve diff. inv.= WZ action for conformal anomaly

Practical measure definedon the background

Lowest term of S (=Riegert action) is coupling-independent

cf. Liouville action

Dynamics of conformal mode is induced from the measure

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The Perturbation about CFTThis model :

CFT + perturbations (by single coupling “t”)Riegert + Weyl

Non-perturbative (conformal mode is treated exactly)

No graviton

cf. Early 4-derivative models in 1970’s

Free + perturbations (by two couplings)R^2 + Weyl

perturbative (all modes are treated in perturbation)graviton picture

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Dimensional Regularization Euclidean sign.

coupled with QEDBare action D-dimensional WZ integrability

Renormalization factors

( )

conformal mode is not renormalizedbecause this mode has no its own coupling constant

Ward-Takahashi identity

Ambiguity is fixed !

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Induced InteractionsResidues are functionsof renormalized coupling

beta functionBare action vertices and counterterms

ordinary countertermsfor gauge field

new vertices and new counterterms

Bare Weyl action Wess-Zumino actionfor conformal anomaly

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Laurent expansion of b

Euler term

counterterms

new WZ actions andnew counterterms

Dynamics of conformalmode is induced

Positive constant

Kinetic term (Riegert action) is induced

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Beta functions

Negative for n_F > 24 UV fixed point !?

Residues b_n [ ]

Hathrell, Ann.Phys.142(1982)34

( corrections from diagrams with internal gravitational lines)K.H., PTP 108(2002)399

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Non-renormalization of Conformal Mode（ ）

+ = UV finite

z: infinitesimal fictitious mass (IR regularization)Not gauge invariant cancel out !

propagator

[Remark : Einstein action cannot be considered as the mass termdue to the existence of exponential factor of conformal mode]

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The e^4 order correction

= UV finite ( )

Two-point functions at e^6 and e^6/b_1 have been checked.

It was also checked that vertex functions ( ) at e^6 and e^6/b_1 are renormalized by the condition .

See ref. K.H., PTP 108(2002)399

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Running Coupling Constant

where

: comoving momentum defined on the flat background

Asymptotic Freedom

Physical momentum: with

(# Conf. anomaly is necessary to preserve diff. inv.)

New IR scale breaking conformal invariance:

Renormalization of Cosmological Constants

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Renormalization of Cosmological Constant at Large b_1

(expand in and )

+ ・・・ ++ +

Lower derivative gravitational actions

+ ・・・

Anomalous dim.

(and also )See reference

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Effective Cosmological ConstantIntroduce constant background by the shift:Large b_1 approx. one-loop approximation becomes good

Effective potential

tree part(# IR divergences cancel out)

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Running Mass ConstantsRenormalization group equation (RGE)

Physical momentum

Solutions of RGE

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Renormalization GroupUV limit : IR limit :

Small cosmological constant

If

GUTs

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Renormalization Group ImprovementSolution to ’t Hooft-Weinberg equation

RG improved effective cosmological constant

Can analytically continue to >1 using

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Plot of Effective Potential

is real

Conclusion and Discussion

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I investigated the cosmological constant problem in the context of renormalizable 4D quantum gravity formulated as a perturbed theory from CFT.

Renormalization of the cosmological constant was carried out.

I studied the RG equation and found a solution that running cosmological constant becomes small in IR limit.

RG improved effective cosmological constant was calculated.

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Non-perturbative effects by gravitational instantons

study of self-dual Weyl configurations

Gravitational solitons

If unstable (=graviballs) dynamics of space-time transition, baryogenesis, …

If stable candidate for cold dark matter

Supersymmetrization of Riegert action