critical exponents in quantum einstein gravity€¦ · the model cannot be extended beyond the...

Critical exponents in quantumEinstein gravity

Sandor Nagy

Department of Theoretical physics, University of Debrecen

MTA-DE Particle Physics Research Group, Debrecen

Leibnitz, 28 June

Critical exponents in quantum Einstein gravity – p. 1

Outline• Quantum field theory and gravity

• Functional renormalization group

• Wetterich equation

• regulators

• fixed points

• Asymptotic safety

• Quantum Einstein gravity

• UV criticality

• crossover criticality

• IR criticality


Quantization and gravity

In gravitational interaction the spacetime metrics becomes a dynamical object.

The Einstein equation is

Rµν −1

2gµνR = −Λgµν + 8πGTµν

• G Newton constant: G = 6.67× 10−11 m3

kgs2

• Λ cosmological constant: Λ ≈ 10−35s−2

Questions

• inconsistency: R is classical, Tµν is quantized

• singularities: black holes, big bang

• cosmological constant

Quantum gravity may solve these problems.


Quantum Einstein gravity

quantum Einstein gravity (QEG) = gravitational interaction + quantum field

theory

We use the elements of quantum field theory

• the metrics gµν plays the role of the field variable in the path integral

• the action is a diffeomorphism invariant action

• couplings, the cosmological constant λ, Newton constant g

The model cannot be extended beyond the ultraviolet (UV) limit, since around

the Gaussian fixed point (GFP)

λ = Λk−2 relevant,

g = Gkd−2 irrelevant.

The loop correction do not modify the tree level scaling behaviors at the GFP.

The Newton coupling g diverges, the theory seems non-renormalizable.


Quantum Einstein gravity

The classical description of gravity is an effective theory which looses its

validity around the Planck scale, therefore it is not complete.

The gravity can be UV complete:

• the UV physics contains different degrees of freedom, e.g. string theory

• loop quantum gravity (new quantization scheme)

• Weinberg’s conjecture (1976): the gravity has a UV non-Gaussian fixed

point (NGFP), where the newton coupling is relevant

If the NGFP is UV attractive, the QEG can be extended to arbitrarily high

energies, without divergences. This is the so called asymptotic safety.


QEG UV NGFP

First it was shown in 2 + ǫ dimensions that there exists a UV NGFP.

Taking into account the 1-loop counterterms into the QEG action one can get

the following perturbative RG equation

k∂kg = ǫg − bg2, b > 0

by using expansion both in g and in ǫ.

The goal is to treat the model at d = 4, which cannot be reached

perturbatively from d = 2.

Similar problem:

In the d-dimensional O(N)-model, we have a NGFP at d = 4− ǫ, the

Wilson-Fisher fixed point.

If we perform an expansion in ǫ then we can reach the d = 3 case, and we can

calculate the exponents for the NGFP. However there is a NGFP even in

d = 1, which cannot be treated by perturbative considerations.

We use functional RG instead of perturbative RG.Critical exponents in quantum Einstein gravity – p. 6

QEG UV NGFP

The QEG is perturbatively non-renormalizable, because every order in the

perturbation requires new types of counterterms (new interactions)→ there is

no physical predictivity.

The scaling properties around the UV NGFP could modify the tree level

scalings.

One can find the fixed point perturbatively in g, i.e.

k∂kg = 2g − 16πcg2,

but its position in the phase space is not universal.

The functional RG method can provide nonperturbative flow equations and

universal (less sensitive) results.


Renormalization• The functional renormalization group method is a fundamental element

of quantum field theory.

• We know the high energy (UV) action, which describes the small

distance interaction between the elementary excitations. We look for

the low energy IR (or large distance) behavior.

• The RG method gives a functional integro-differential equation for the

effective action, which is called the Wetterich equation.

The form of the blocked action is

Sk[φ, gi] =∑

i

gi(k)Fi(φ),

where k is the (momentum or energy) scale, and gi(= gi(k)) are the

dimensionful (scale dependent) couplings. The action takes into account the

symmetries of the model.


Renormalization

The vacuum-vacuum transition amplitude (the generating functional) is

Z =

∫

Dφe−Sk =

∫

dφ0 . . . dφk−∆kdφkdφk+∆k . . . dφ∞e−Sk

The path integral is performed by removing the degrees of freedom (quantum

fluctuations, modes) one-by-one, systematically, whch gives scale dependent

couplings.

(IR) 0← k k →∞ (UV)

gk−∆kgk

gk+∆k

k−∆k k k+∆k


The Wetterich equation

The form of the generating functional is

Z = eWk[J] =

∫

D[φ]e−(Sk+Rk[φ]−J·φ).

The integration of the modes we take the average of the field variable in a

volume of k−1. The Wetterich equation is

Γk =1

2Tr

Rk

Rk + Γ′′

k

=1

2,

where ′ = ∂/∂ϕ, ˙ = ∂/∂t, and the symbol Tr denotes the momentum integral

and the summation over the internal indices. We assume that the functional

form of the effective and the blocked action is similar, i.e.

Γk ∼ Sk =∑

i

gi(k)Fi(φ).


Regulators

The IR regulator has the form Rk[φ] =12φ · Rk · φ. It is a momentum

dependent mass like term, which serves as an IR cutoff and has the following

properties• lim

p2/k2→0Rk > 0: it serves as an IR

regulator

• limk2/p2→0

Rk → 0: in the limit k → 0

we obtain back the form of Z

• limk2→∞

Rk → ∞: for the micro-

scopic action S = limk→Λ Γk (it

serves as a UV regulator, too)


Regulators

The compactly supported smooth (css) regulator has the form

rcss =R

p2=

s1exp[s1yb/(1− s2yb)]− 1

θ(1− s2yb),

with y = p2/k2. Its limiting cases provide us the following commonly used

regulator functions

lims1→0,s2=1

rCSS =

(

1

yb− 1

)

θ(1− y),

lims1→0,s2→0

rCSS =1

yb,

lims1=1,s2→0

rCSS =1

exp[yb]− 1.

where the first limit gives the Litim’s optimized, the second gives the power

law and the third gives the exponential regulator. The b = 1 case satisfies the

normalization conditions limy→0 yr = 1 and limy→∞ yr = 0..Critical exponents in quantum Einstein gravity – p. 12

Gaussian fixed point (GFP)

Free massless theory, i.e. g∗i = 0, si = −di, where si are the eigenvalues of

the linearized RG equation around the fixed point and di are the canonical

dimensions.

• si > 0 (di < 0), when k →∞.

• The fixed point repels the

trajectories.

• when k → 0 the coupling decreases

irrelevant. 10-7

10-6

10-5

10-4

10-3

10-2

10-1

10-3 10-2 10-1 100 101 102

g~ i

k/kΛ

UVIR

• si < 0 (di > 0), when k →∞.

• The fixed point attracts the

trajectories.

• when k → 0 the coupling diverges

relevant. 10-7

10-6

10-5

10-4

10-3

10-2

10-1

10-3 10-2 10-1 100 101 102

g~ i

k/kΛ

UVIR


UV behavior

Asymptotic freedom: ∀ gi limt→∞

gi = 0 Example models are

• QCD

• 3-dimensional φ4 model...

Asymptotic safety: there is an UV attractive NGFP in the phase space with a

finite number of relevant directions. Example models are

• Gross-Neveu model

• non-linear σ model

• sine-Gordon model

• quantum Einstein gravity


QEG effective action

The RG evolution equation is

Γk[h, ξ, ξ, g] =1

2Tr

√2(Rgrav

k )hh(√2Rgrav

k + Γ′′

k)hh

−1

2Tr

[

κ2(Rghk )ξξ

(

1

(κ2Rghk + Γ′′

k)ξξ− 1

(κ2Rghk + Γ′′

k)ξξ

)]

.

where the QEG effective action is

Γk =

∫

ddx√

detgµν

(

1

16πGk(2Λk −R)− ωk

3σkR2 + . . . → f(R)

+1

2σkC2 +

θkσk

E + . . .

+ gf. terms + gh. terms

)

.

The first two terms consitute the Einstein-Hilbert action.


QEG evolution equation

The flow equations with the Litim’s regulator are (d = 4)

λ =2(−96g2 + λ+ 4λ2(11 + λ) + g(8λ(1− 3λ)− 6))

4g − (1− 2λ)2,

g = (2 + η)g,

where η = 24g4g−(1−2λ)2 is the gravitation anomalous dimension.

The fixed points are

UV: (NGFP) repulsive focal point: λ∗

UV = 1/4, g∗UV = 1/64

GFP: hyperbolic point: λ∗

G = 0, g∗UV = 0

In order to find the IR fixed point we should rescale the equations as

χ = 1− 2λ, ω = 4g − (1− 2λ)2 and ∂τ = ωk∂k, which gives

∂τχ = −4ω + 2χω(8 + 21χ) + 24ω2 + 6χ2(3χ(χ+ 1)− 1),

∂τω = 8ω2(1− 6χ)− 2χ(42χ2 + 9χ− 4)− 6χ3(χ(6χ+ 5)− 2).


QEG evolution equation

• Symmetric phase: (strong coupling)

〈gµν〉 ∼ δµν , Euclidean flat geometry,

k → 0: R = 4λ < 0, negative

curvature

• Broken phase: (weak coupling)

〈gµν〉 = 0, degenerate geometry, at

k → 0 λ > 00

0.01

0.02

0.03

-0.4 -0.2 0 0.2 0.4

g

λ

G

UV

IR

The fixed points

UV: (NGFP) repulsive focal point: λ∗

UV = 1/4, g∗UV = 1/64,

(ω∗

UV = −3/16, χ∗

UV = 1/2), eig.values: sUV 1,2 = (−5± i√167)/3,

extension dependent sUV 1,2 = θ′ ± iθ′′, θ′ ≈ 1.5− 2 and

θ′′ ≈ 2.5− 4.3, ν = 1/θ′ = 1/3.

G: hyperbolic point: λ∗

G = 0, g∗UV = 0, (ω∗

G = −1 and χ∗

G = 1),

eig.values: sG1 = −2 and sG2 = d− 2, extension dependent.

IR: attracitve fixed point: λ∗

IR = 1/2, g∗IR = 0. (ω∗

IR = 0, χ∗

IR = 0).Critical exponents in quantum Einstein gravity – p. 17

UV criticality

In the case of UV NGFP the eigenvalues of the corresponding stability matrix

can be written as

θUV1,2 = θ′ ± iθ′′, and ν = 1/θ′

0 0.2 0.4 0.6 0.8 1s1 0 0.2

0.4 0.6

0.8 1

s2

1.55 1.6

1.65 1.7

1.75 1.8

1/ν 1.55 1.6 1.65 1.7 1.75 1.8

P.

L.

E.

1.38

1.4

1.42

1.44

1.46

1.48

1.5

1.52

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1/ν

s2

s1=0.001

s1=0.1

s1=1

1.6

2.0

2.4

2.8

3.2

100 102 104 106

1/ν

s1

• s1 = s2 = 0→ Power law regulator

• s1 = 0, s2 = 1→ Litim’s regulator

• s1 = 1, s2 = 0→ Exponential regulator


Crossover criticality

3-dimensional φ4 model with the potential

V =1

2g1φ

2 +1

24g2φ

4 +O(φ6)

The case of 2 and 4 couplings is

considered, and both cases can pos-

sess extrema, with minimal sensi-

tivity to the parameters s1 and s2.

0.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

0 1 2 3 4 5 6

ν

s2

b=1

b=2

b=3


IR criticality

exponent UV G IR

ν 3/5 1/2 1/2

η -2 k2 k−3/2

10-10

10-8

10-6

10-4

10-2

100

102

10-10 10-8 10-6 10-4 10-2 100 102 104 106

|η|

k-kc

UV

G

IR

η=-2

k2

k-3/2

Taking into account the higher order terms in the curvature of the QEG action

we get the exponents

UV: RG: ν ≈ 1/2− 2/3, lattice calculations: ν = 1/3

G: RG: ν = 1/2 exact

IR: RG: ν = 1/2 exact


Models and their fixed points

model UV crossover IR

3d O(N) Gaussian Wilson-Fisher IR

3d non-linear σ non-Gaussian non-Gaussian Gaussian

3d Gross-Neveu Gaussian non-Gaussian IR

2d sine-Gordon Gaussian and non-Gaussian Kosterlitz-Thouless Gaussian and IR

4d QEG non-Gaussian Gaussian IR


Thank You


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