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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
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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:
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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
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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