Dilaton Quantum Gravity - A Functional Renormalization ... · Asymptotically Safe Quantum Gravity...

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Introduction & Physical Context FRG Analysis Summary and Outlook Dilaton Quantum Gravity A Functional Renormalization Group Approach Tobias Henz with Jan Martin Pawlowski, Andreas Rodigast & Christof Wetterich Institute for Theoretical Physics, University of Heidelberg based on arXiv:1304.7743 Asymptotic Safety Seminar, June 2013 1 / 22

Transcript of Dilaton Quantum Gravity - A Functional Renormalization ... · Asymptotically Safe Quantum Gravity...

Page 1: Dilaton Quantum Gravity - A Functional Renormalization ... · Asymptotically Safe Quantum Gravity FRGTechnicalities RegulatorDependence BackgroundDependence TruncationStability CouplingtoSM

Introduction & Physical Context FRG Analysis Summary and Outlook

Dilaton Quantum GravityA Functional Renormalization Group Approach

Tobias Henzwith Jan Martin Pawlowski, Andreas Rodigast & Christof Wetterich

Institute for Theoretical Physics, University of Heidelberg

based on arXiv:1304.7743

Asymptotic Safety Seminar, June 2013

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Introduction & Physical Context FRG Analysis Summary and Outlook

Outline

1 Introduction & Physical ContextNonrenormalizibility and Asymptotic SafetyChallenges & Open Questions

2 FRG AnalysisFlow EquationsFixed Point SolutionsScaling Solution

3 Summary and Outlook

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Introduction & Physical Context FRG Analysis Summary and Outlook

Renormalizibility and Quantum Gravity

Einstein-Hilbert action

Γ [gµν ] =1

16πGN

∫ddx√g (2Λ− R[gµν ])

mass dimensions of the couplings

dim(Λ) = 2

dim(GN) = 2− d

=⇒ Perturbatively not renormalizable if d > 2.

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Introduction & Physical Context FRG Analysis Summary and Outlook

The Flow Diagram of Quantum Einstein Gravity

Reuter, 1998Coupling Constants approach a nontrivial UV Fixed Point

=⇒ Prospect of Gravity being Asymptotically Safe

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Introduction & Physical Context FRG Analysis Summary and Outlook

The Flow Diagram of Quantum Einstein Gravity

!2 !0.5 0 0.1 0.2 0.3 0.4 0.45 0.5!"

0

0.1

0.2

0.5

1

2

10

"

0.45 0.5

0

0.015

Ia

Ib

A

B CD

C

λ

gN

Christiansen, Litim, Pawlowski, Rodigast, 2012Stable Infrared Scenarios

=⇒ Prospect of UV and IR consistent theory5 / 22

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Introduction & Physical Context FRG Analysis Summary and Outlook

Asymptotically Safe Quantum Gravity

FRG TechnicalitiesRegulator DependenceBackground DependenceTruncation Stability

Coupling to SMYang Mills TheoryBackground DependenceAsymptotic Freedom

Infrared LimitTrajectory UV → IRGR as limiting case?

Hierarchy ProblemMSM ≈ 102 GeVMPlanck ≈ 1019 GeV

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Introduction & Physical Context FRG Analysis Summary and Outlook

Setup & Action

Γk [gµν ] =1

16πGN,k

∫ddx√g (2Λk − R[gµν ])

⇓ ⇓ ⇓

Γk [gµν , χ] =

∫ddx√g

(Vk [χ]− 1

2Fk [χ]R[gµν ] +12gµν∂ µχ∂ νχ

)

First investigated by Narain & Percacci, 2010Extensions:

investigation of IR limittrajectory UV ↔ IRimproved UV analysisall investigations in d = 4 & deDonder gauge

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Introduction & Physical Context FRG Analysis Summary and Outlook

Dilatation Symmetry

Dilatations ↔ Conformal Transformationsgµν(x) 7→ Ω(x)gµν(x)with Ω = const.Dilatation ↔ global resetting of the physical scaleDilatation Symmetry ↔ Physical Scale is introduced only byexpectation value of the scalar fieldArising Goldstone Boson: Dilaton

Dilatation Symmetric ActionsΓ is invariant under dilatations

mall couplings have scaling dimension 0.

(d = 4 : F ∝ χ2, V ∝ χ4)

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Introduction & Physical Context FRG Analysis Summary and Outlook

Physical Motivation

Infrared LimitTrajectory UV → IRGR as limiting case?

Hierarchy ProblemMSM ≈ 102 GeVMPlanck ≈ 1019 GeV

a scale invariant IR limit for Einstein-Hilbert Quatum Gravityb generation of Planck mass via breaking of dilatation symmetryc quintessence cosmology scenarios with vanishing cosmologicalconstant

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Introduction & Physical Context FRG Analysis Summary and Outlook

Fixed Point Action

Let y = χ2

k2 , Vk(χ2) = k4y2 vk(y), Fk(χ2) = k2y fk(y)

⇒ Dilatation Symmetric parts factored out

proposed fixed point for large y

limy→∞

f (y) = ξ limy→∞

v(y) = 0.

⇒ Γ =∫

d4x√g(12gµν∂µχ∂νχ− 1

2ξχ2 R)

(Jordan)

⇒ Γ =∫

d4x√g(12gµν∂µφ∂νφ− 1

2M2 R)

(Einstein)

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Introduction & Physical Context FRG Analysis Summary and Outlook

Properties of the Fixed Point

Let f0 = limy→∞ fk(y)

⇒ Gravitational Interaction scales like f −10 χ−2

⇒ Gravity induced flow from f −10 y−1 with limy→∞ f −10 y−1 = 0⇒ For limy→∞ vk(y) = 0 free scalar theory

Flow of free scalar theory can at most produce a constantterm ∝ v−2 in V andFor gN = 0 no flow of gravitational interaction

⇒ Asymptoticallylim

y→∞v(y) = v−2 y−2 + . . .

limy→∞

f (y) = ξ + f−1 y−1 + . . .

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Introduction & Physical Context FRG Analysis Summary and Outlook

Flow Equation

O(R) truncationno wave function renormalizationcurved background spacetimeoptimized cut-offs with

Γ2k(p2) +Rk(p2) = Γ2k

(p2 + rk(p2)

)rk(p2) = (k2 − p2)×Θ(k2 − p2)

set of flow equations at fixed y∂tvk(y) = 2 y v ′k(y) +

1y2 ζV ,

∂t fk(y) = 2 y f ′k(y) +1y ζF .

deDonder gauge fixing and mostly d = 4

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Introduction & Physical Context FRG Analysis Summary and Outlook

Flow Equation II

ζV =1

192π2

6 +

30 VΣ0

+3(2Σ0 + 24 y F ′ Σ′0 + FΣ1)

∆+ δV (∂ t F )

ζF =

11152π2

150 +

30 F (3 F − 2V )

Σ20

−12∆

(24 y F ′ Σ′0 + 2Σ0 + FΣ1

)− 6y (3 F ′2 + 2Σ′20 )

−36∆2

[2y Σ0 Σ′0 (7 F ′ − 2V ′) (Σ1 − 1) + 2Σ2

0 Σ2

+2 yΣ1 (7 F ′ − 2V ′) (2Σ0 V ′ − V Σ′0)

+24 y F ′ Σ0 Σ′0 Σ2 − 12 y F Σ′20 Σ2

]+ δF (∂ t F )

V = y2 vk (y) , F = y fk (y),

Σ0 =12

F − V , ∆ =(12 y Σ′20 + Σ0 Σ1

),

Σ1 = 1 + 2 V ′ + 4 y V ′′ , Σ2 = F ′ + 2 y F ′′

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Introduction & Physical Context FRG Analysis Summary and Outlook

Fixed Point Solutions

Aim: Find Fixed Point Solutions

a for y → 0 or ∞Taylor Expansions in y or y−1free parameter ξ = limy→∞ f0

Ultraviolet: y → 0χ→ 0 or k →∞Percacci at al, 2010

Infrared: y →∞χ→∞ or k → 0current project, 2013

b globallyconnecting UV & IRPadé ressumationexponential improvement

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Introduction & Physical Context FRG Analysis Summary and Outlook

The Limit y →∞

Expandv(y) = v0 + v−1y−1 + v−2y−2 + . . . ,

f (y) = f0 + f−1y−1 + . . .

Finite limits for ζV and ζF

⇒ Flow of v0, v−1 and f0 vanishesFor v0 = v−1 = 0ζV = lim

y→∞ζV =

332π2 +

5 + 33f096π2(1 + 6f0)

∂t f0f0

ζF = limy→∞

ζF =77 + 534f0

192π2(1 + 6f0)+

17 + 186f0 + 720f 20576π2(1 + 6f0)2

∂t f0f0

closed set of flow equations guaranteed by derivative structure⇒ fixed point solutions

limy→∞

f ∗(y) = ξ +ζF2y , lim

y→∞v∗(y) =

ζV4y2 .

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Introduction & Physical Context FRG Analysis Summary and Outlook

The Limit y → 0

Expand

F = yf = F0 + F1y + F2y2 + . . .

V = y2v = V0 + V1y + V2y2 + . . .

system of flow equations not closed

⇒ Treat V0 and F0 as “free” parameters

extension from Narain, Percacci 2010

Apparent good convergence of the series

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Introduction & Physical Context FRG Analysis Summary and Outlook

Closing the Gap: Possible Global Solutions

a Einstein-Hilbert Gravity + Scalar Field

⇔ V = y2v(y) = 0.008620 and F = yf (y) = 0.04751

b Continuations of Taylor Expansions around y =∞

y-1

y-3

y-4

y-5

y-7

y-8

F

Padé

1 ´ 10-7

5 ´ 10-7

1 ´ 10-6

5 ´ 10-6

1 ´ 10-5

5 ´ 10-51 ´ 10

-40.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

y

F

HyL

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Introduction & Physical Context FRG Analysis Summary and Outlook

Scaling Solution

y-1

y-3

y-4

y-5

y-7

y-8

F

Padé

1 ´ 10-7

5 ´ 10-7

1 ´ 10-6

5 ´ 10-6

1 ´ 10-5

5 ´ 10-51 ´ 10

-40.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

y

F

HyL

Define an error via

V ′′(y) = GV (y , F , F ′, V , V ′),

F ′′(y) = GF (y , F , F ′, V , V ′)

ε =12

(GV − V ′′

)2G2

V+ V ′′2

+12

(GF − F ′′

)2G2

F+ F ′′2

E =

∫ y0

0ε(y)dy

Taylor expansions around y =∞ have ε ≤ 10−2 for

y ≤ y0 ≈ 1/(100ξ)

Brakedown of y−1 expansions expected for y → 0

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Introduction & Physical Context FRG Analysis Summary and Outlook

Improvement I: Padé ressumation

Strategy:stick with Taylor Solution for as long as possiblesmoothen solution for y → 0connect to known solution for small y

Padé resummation:Expand both numerator and denominator in powers of y−1dependency on parameter ξ remains

V

exp

V

Padé

V

Taylor

0 1. ´ 10-6

2. ´ 10-6

3. ´ 10-6

4. ´ 10-6

5. ´ 10-6

6. ´ 10-6

7. ´ 10-6

0.003

0.004

0.005

0.006

0.007

0.008

y

V

HyL

F

exp

F

Padé

F

Taylor

0 1. ´ 10-6

2. ´ 10-6

3. ´ 10-6

4. ´ 10-6

5. ´ 10-6

6. ´ 10-6

7. ´ 10-6

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

y

F

HyL

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Introduction & Physical Context FRG Analysis Summary and Outlook

Improvement II: Exponential Functions

V

exp

V

Padé

V

Taylor

0 1. ´ 10-6

2. ´ 10-6

3. ´ 10-6

4. ´ 10-6

5. ´ 10-6

6. ´ 10-6

7. ´ 10-6

0.003

0.004

0.005

0.006

0.007

0.008

y

V

HyL

F

exp

F

Padé

F

Taylor

0 1. ´ 10-6

2. ´ 10-6

3. ´ 10-6

4. ´ 10-6

5. ´ 10-6

6. ´ 10-6

7. ´ 10-6

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

y

F

HyL

Propagators Σ0, ∆ produce singularities in intermediate regionSee y as a coupling & use exponential parts ∝ exp(−c/y)

Fexp(y) = FTaylor(y) + e−c/y Fe(y); Vexp(y) = VTaylor(y) + e−c/y Ve(y)

parameter optimisation to connect solutions & minimiseintegrated error E

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Introduction & Physical Context FRG Analysis Summary and Outlook

Physical Implications

Dilatation symmetric infrared limit due tolim

y→∞V = 0, lim

y→∞F = ξχ2

Planck mass is generated by spontaneous breaking ofDilatation Symmetry via 〈χ〉 = 0

⇒ Realistic gravity possible on a pure fixed point trajectoryquintessence cosmology for small deviations from fixed pointand y 1

Γ =

∫d4x√g

M2

2R +

12

k2(ϕ)∂µϕ∂µϕ+ V (ϕ)

k2(ϕ) =

14ξ

[1 + 6ξ +

(M2

Fexp(ϕ

M

)− 1)−1]

V (ϕ) = V exp(−2ϕM

)⇒ cosmological constant vanishes for ϕ→∞

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Introduction & Physical Context FRG Analysis Summary and Outlook

Summary and Outlook

prospect of scaling solution with scale invariant IR limit

cosmological constant vanishes on and away from FP in the IR

for small deviation from the fixed point trajectory admits

quintessence cosmology

Thank you!

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