Asymptotically Safe Gravity
From Euclidean to Lorentzian SignatureFrank Saueressig
Group for Theoretical High Energy Physics (THEP)
Institute of Physics

E. Manrique, S. Rechenberger and F.S., PRL 106 (2011) 251302

Renormalization Group Approach – from Ultra Cold Atoms to the Hot QGP
Kyoto, September 2nd, 2011

– p. 1/4

Outline• Motivation for Quantum Gravity• Foundations of Asymptotic Safety• Functional Renormalization Group Equations: Part I◦ covariant construction◦ Einstein-Hilbert results◦ Higher-derivative summary• Functional Renormalization Group Equations: Part II◦ causal construction◦ Einstein-Hilbert results• Conclusion and perspectives– p. 2/4

Introduction
standard model of particle physics:• describes: electromagnetic/strong/weak force + interactions with matter• extremely well tested– p. 3/4

Introduction
standard model of particle physics:• describes: electromagnetic/strong/weak force + interactions with matter• extremely well tested
theoretical basis: quantum field theory in flat, non-dynamical space-time• includes only relevant and marginal couplings=⇒ renormalizable quantum field theory– p. 3/4

Introduction
General Relativity:• describes: gravity + interactions with matter• extremely well tested– p. 4/4

Introduction
General Relativity:• describes: gravity + interactions with matter• extremely well tested
theoretical basis: classical theory in curved, dynamical space-timeRµν − 12 gµν R|{z}space-time curvature= −Λgµν + 8πGN Tµν|{z}matter content• Newton constant GN has negative mass-dimension– p. 4/4

Gravity: Perturbative quantization
Length scale for Quantum Gravity Effects:

Planck scale:

`Pl =



~GN
c3

1/2

≈ 10−33 cm ; mPl = 1019 GeV• Theoretically: expect strong quantum corrections to GN , Λ– p. 5/4

Gravity: Perturbative quantization
Length scale for Quantum Gravity Effects:

Planck scale:

`Pl =



~GN
c3

1/2

≈ 10−33 cm ; mPl = 1019 GeV• Theoretically: expect strong quantum corrections to GN , Λ
perturbative quantization of General Relativity
Z1√d4 x g {−R + 2Λ}S EH =
16πGN•

GN has negative mass-dimension ⇔ infinite number of counterterms

gravity + scalar:∆S 1−loop ∝pure gravity:∆S 2−loop ∝RR√
d4 x g
Cαβµν C αβµν√ 
d4 x g Cµν ρσ Cρσ αβ Cαβ µν– p. 5/4

Gravity: Perturbative quantization
Length scale for Quantum Gravity Effects:

Planck scale:

`Pl =



~GN
c3

1/2

≈ 10−33 cm ; mPl = 1019 GeV• Theoretically: expect strong quantum corrections to GN , Λ
perturbative quantization of General Relativity
Z1√d4 x g {−R + 2Λ}S EH =
16πGN•

GN has negative mass-dimension ⇔ infinite number of counterterms

gravity + scalar:∆S 1−loop ∝pure gravity:∆S 2−loop ∝RR√
d4 x g
Cαβµν C αβµν√ 
d4 x g Cµν ρσ Cρσ αβ Cαβ µνGeneral Relativity is perturbatively non-renormalizable– p. 5/4

General Relativity: perturbatively non-renormalizable
perturbative quantization of General Relativity:•

GN has negative mass-dimension:◦ infinite number of counterterms◦ General Relativity is perturbatively non-renormalizable
Possible conclusions:a) Treat General Relativity as effective field theory:2  1 (independent of UV-completion)• compute corrections in E 2 /MPl2
• breaks down at E 2 ≈ MPl– p. 6/4

General Relativity: perturbatively non-renormalizable
perturbative quantization of General Relativity:•

GN has negative mass-dimension:◦ infinite number of counterterms◦ General Relativity is perturbatively non-renormalizable
Possible conclusions:a) Treat General Relativity as effective field theory:2  1 (independent of UV-completion)• compute corrections in E 2 /MPl2
• breaks down at E 2 ≈ MPlb) UV-completion requires new physics:• supersymmetry, extra dimensions, . . .
• possibly: extension of QFT-framework– p. 6/4

General Relativity: perturbatively non-renormalizable
perturbative quantization of General Relativity:•

GN has negative mass-dimension:◦ infinite number of counterterms◦ General Relativity is perturbatively non-renormalizable
Possible conclusions:a) Treat General Relativity as effective field theory:2  1 (independent of UV-completion)• compute corrections in E 2 /MPl2
• breaks down at E 2 ≈ MPlb) UV-completion requires new physics:• supersymmetry, extra dimensions, . . .
• possibly: extension of QFT-frameworkc) Gravity makes sense as Quantum Field Theory:• UV-completion requires going beyond perturbation theory– p. 6/4

General Relativity: perturbatively non-renormalizable
perturbative quantization of General Relativity:•

GN has negative mass-dimension:◦ infinite number of counterterms◦ General Relativity is perturbatively non-renormalizable
Possible conclusions:a) Treat General Relativity as effective field theory:2  1 (independent of UV-completion)• compute corrections in E 2 /MPl2
• breaks down at E 2 ≈ MPlb) UV-completion requires new physics:• supersymmetry, extra dimensions, . . .
• possibly: extension of QFT-frameworkc) Gravity makes sense as Quantum Field Theory:• UV-completion requires going beyond perturbation theory– p. 7/4

Renormalizing
the Non-Renormalizable– p. 8/4

Wilson’s modern picture of renormalization
central idea: integrate out quantum fluctuations shell-by-shell in momentum-space– p. 9/4

Wilson’s modern picture of renormalization
central idea: integrate out quantum fluctuations shell-by-shell in momentum-space

implementation:• action with scale-dependent couplings (GN , Λ, . . .):gi (k)• scale-dependence governed by β -functions:k∂k gi = βgi ({gi })– p. 9/4

Ensuring good UV-behavior: fixed points of the RG-flow
amplitudes depend on dimensionless couplings only

• RG-flow for dimensionless running couplings:

gi (k)

Fixed points gi∗ :•β -functions vanish:!

βgi ({gi∗ }) = 0
gi∗ remain finite• RG-trajectory captured by fixed point in UV:=⇒ physical quantities remain free of unphysical divergences– p. 10/4

Ensuring good UV-behavior: fixed points of the RG-flow
amplitudes depend on dimensionless couplings only

• RG-flow for dimensionless running couplings:

gi (k)

Fixed points gi∗ :•β -functions vanish:!

βgi ({gi∗ }) = 0
gi∗ remain finite• RG-trajectory captured by fixed point in UV:=⇒ physical quantities remain free of unphysical divergences

Concepts associated with UV-fixed points:• trajectories emanating from fixed point in UV≡ span UV critical surface• predictivity:≡ UV critical surface has finite dimension– p. 10/4

Renormalization: asymptotic freedom and asymptotic safety
Wilsonian formulation:• UV fixed points allow two classes of renormalizable Quantum Field Theories• Gaussian Fixed Point (GFP):◦ perturbatively renormalizable field theories◦ UV-limit: free theory◦ asymptotic freedom(example: QCD)

– p. 11/4

Renormalization: asymptotic freedom and asymptotic safety
Wilsonian formulation:• UV fixed points allow two classes of renormalizable Quantum Field Theories• Gaussian Fixed Point (GFP):◦ perturbatively renormalizable field theories◦ UV-limit: free theory◦ asymptotic freedom(example: QCD)
• non-Gaussian Fixed Point (NGFP):◦ non-perturbatively renormalizable field theories◦ UV-limit: interacting theory◦ asymptotic safety– p. 11/4

Renormalization: asymptotic freedom and asymptotic safety
Wilsonian formulation:• UV fixed points allow two classes of renormalizable Quantum Field Theories• Gaussian Fixed Point (GFP):◦ perturbatively renormalizable field theories◦ UV-limit: free theory◦ asymptotic freedom(example: QCD)
• non-Gaussian Fixed Point (NGFP):◦ non-perturbatively renormalizable field theories◦ UV-limit: interacting theory◦ asymptotic safety

Wilsonian picture: generalization of perturbative renormalization
asymptotic safety as predictive as asymptotic freedom– p. 11/4

Renormalizing gravity
Wilsonian formulation:• UV fixed points allow two classes of renormalizable Quantum Field Theories• Gaussian Fixed Point (GFP):◦ perturbatively renormalizable field theories◦ UV-limit: free theory◦ asymptotic freedom• non-Gaussian Fixed Point (NGFP):◦ non-perturbatively renormalizable field theories◦ UV-limit: interacting theory◦ asymptotic safety
Weinberg’s asymptotic safety conjecture (1979):
gravity in d = 4 has non-Gaussian UV fixed point– p. 12/4

Functional Renormalization Group Equations I
covariant construction– p. 13/4

Covariant functional RG equation for gravity
Theory:degrees of freedom:

metric field gµνsymmetries:

coordinate transformations (diffeomorphisms)

Functional Renormalization group equation:• Wetterich equation for effective average action Γk[C. Wetterich, Phys. Lett. B301 (1993) 90]• adapted to gravity[M. Reuter, Phys. Rev. D 57 (1998) 971, hep-th/9605030]– p. 14/4

Covariant functional RG equation for gravity
Theory:degrees of freedom:

metric field gµνsymmetries:

coordinate transformations (diffeomorphisms)

Construction of the flow equation:• starting point: generic diffeomorphism invariant action S grav [gµν ]– p. 15/4

Covariant functional RG equation for gravity
Theory:degrees of freedom:

metric field gµνsymmetries:

coordinate transformations (diffeomorphisms)

Construction of the flow equation:• starting point: generic diffeomorphism invariant action S grav [gµν ]• background covariance ⇐⇒ background field formalism◦ quantum field is split into (fixed) background value + arbitrary fluctuationgµν = ḡµν + hµν◦ implement gauge-fixing term:
1
S gf = 2αZ√
d4 x ḡFµ ḡ µν Fν ,Fµ = D̄µ hµν − β D̄µ h◦ and corresponding ghost action√gh
S [h, C, C̄; ḡ] = − 2Z√
d4 x ḡ C̄µ Mµ ν C ν

Mµ ν = ḡ µρ ḡ σλ D̄λ (gρν Dσ + gσν Dρ ) − ḡ ρσ ḡ µλ D̄λ (gσν Dρ )– p. 15/4

Covariant functional RG equation for gravity
Theory:degrees of freedom:

metric field gµνsymmetries:

coordinate transformations (diffeomorphisms)

Construction of the flow equation:• starting point: generic diffeomorphism invariant action S grav [gµν ]• background covariance ⇐⇒ background field formalismgµν = ḡµν + hµν• add: k-dependent IR cutoff:∆k S[h; ḡ] =Z√
d4 x ḡ {hµν Rk [ḡ]µνρσ hρσ + . . .}◦ Rk [ḡ] ∝ Zk k2 R(0) = k-dependent mass term
◦ discriminates low/high- D̄2 -eigenmodes
R

(0)

2

2

(p /k ) =


 0
 1

UV

p2  k 2
p2  k 2• high momentum modes: integrated out• low momentum modes: suppressed by mass termIRp2 = k̂2?
?
? k2
p2 = 0– p. 16/4

Covariant functional RG equation for gravity
Theory:degrees of freedom:

metric field gµνsymmetries:

coordinate transformations (diffeomorphisms)

Construction of the flow equation:• starting point: generic diffeomorphism invariant action S grav [gµν ]• background covariance ⇐⇒ background field formalismgµν = ḡµν + hµν• add: k-dependent IR cutoff:∆k S[h; ḡ] =Z√
d4 x ḡ {hµν Rk [ḡ]µνρσ hρσ + . . .}• exact RG equation for Γk :k∂k Γk [h; ḡ] = 12 STr



(2)

Γk + Rk−1k∂k Rk◦ Γ(2)
k = Hessian with respect to fluctuation fields◦ “extra” ḡ -dependence necessary for formulating exact equation– p. 17/4

Theory space underlying the Functional Renormalization Group– p. 18/4

Non-perturbative approximation: derivative expansion of Γk• caveat: FRGE cannot be solved exactly⇐⇒ gravity: need non-perturbative approximation scheme– p. 19/4

Non-perturbative approximation: derivative expansion of Γk• caveat: FRGE cannot be solved exactly⇐⇒ gravity: need non-perturbative approximation scheme• expand Γk in derivatives and truncate series:Γk [Φ] =NXi=1ūi (k) Oi [Φ]=⇒ substitute into FRGE=⇒ projection of flow gives β -functions for running couplingsk∂k ūi (k) = βi (ūi ; k)

– p. 19/4

Non-perturbative approximation: derivative expansion of Γk• caveat: FRGE cannot be solved exactly⇐⇒ gravity: need non-perturbative approximation scheme• expand Γk in derivatives and truncate series:Γk [Φ] =NXi=1ūi (k) Oi [Φ]=⇒ substitute into FRGE=⇒ projection of flow gives β -functions for running couplingsk∂k ūi (k) = βi (ūi ; k)

• testing the reliability:◦ within a given truncation:
cutoff-scheme dependence of physical quantities (= vary Rk )◦ stability of results within extended truncations– p. 19/4

Letting things flow
The Einstein-Hilbert truncation– p. 20/4

The Einstein-Hilbert truncation: setup
Einstein-Hilbert truncation: two running couplings: G(k), Λ(k)Z
1√d4 x g [−R + 2Λ(k)] + S gf + S ghΓk =16πG(k)• project flow onto G-Λ−plane– p. 21/4

The Einstein-Hilbert truncation: setup
Einstein-Hilbert truncation: two running couplings: G(k), Λ(k)Z
1√d4 x g [−R + 2Λ(k)] + S gf + S ghΓk =16πG(k)• project flow onto G-Λ−plane
explicit β -functions for dimensionless couplings gk := k2 G(k) , λk := Λ(k)k−2• Particular choice of Rk (optimized cutoff)k∂k gk = (ηN + 2)gk ,gkk∂k λk = − (2 − ηN ) λk − 2π

i
h
5
1
1
5 1−2 λ − 4 − 6 1−2λ ηNkk• anomalous dimension of Newton’s constant:
gB1
1 − gB2ηN =1
B1 = 3πhi

1
1− 9 (1−2λ)5 1−2λ2 −71, B2 = − 12πhi11
5 1−2λ + 6 (1−2λ)2– p. 21/4

Einstein-Hilbert truncation: Fixed Point structure
β -functions for gk := k2 G(k) , λk := Λ(k)k−2
k∂k gk = (ηN + 2)gk ,gkk∂k λk = − (2 − ηN ) λk − 2πh1
ηN
5 1−21 λ − 4 − 56 1−2λkkimicroscopic theory ⇐⇒ fixed points of the β -functions
βg (g ∗ , λ∗ ) = 0 ,βλ (g ∗ , λ∗ ) = 0– p. 22/4

Einstein-Hilbert truncation: Fixed Point structure
β -functions for gk := k2 G(k) , λk := Λ(k)k−2
k∂k gk = (ηN + 2)gk ,gkk∂k λk = − (2 − ηN ) λk − 2πh1
ηN
5 1−21 λ − 4 − 56 1−2λkkimicroscopic theory ⇐⇒ fixed points of the β -functions
βg (g ∗ , λ∗ ) = 0 ,βλ (g ∗ , λ∗ ) = 0• Gaussian Fixed Point:◦ at g ∗ = 0, λ∗ = 0 ⇐⇒ free theory◦ UV-repulsive for g > 0– p. 22/4

Einstein-Hilbert truncation: Fixed Point structure
β -functions for gk := k2 G(k) , λk := Λ(k)k−2
k∂k gk = (ηN + 2)gk ,gkk∂k λk = − (2 − ηN ) λk − 2πh1
ηN
5 1−21 λ − 4 − 56 1−2λkkimicroscopic theory ⇐⇒ fixed points of the β -functions
βg (g ∗ , λ∗ ) = 0 ,βλ (g ∗ , λ∗ ) = 0• Gaussian Fixed Point:◦ at g ∗ = 0, λ∗ = 0 ⇐⇒ free theory◦ UV-repulsive for g > 0∗ = −2):• non-Gaussian Fixed Point (ηN◦ at g ∗ > 0, λ∗ > 0 ⇐⇒ “interacting” theory◦ UV attractive in gk , λk

– p. 22/4

Einstein-Hilbert truncation: Fixed Point structure
β -functions for gk := k2 G(k) , λk := Λ(k)k−2
k∂k gk = (ηN + 2)gk ,gkk∂k λk = − (2 − ηN ) λk − 2πh1
ηN
5 1−21 λ − 4 − 56 1−2λkkimicroscopic theory ⇐⇒ fixed points of the β -functions
βg (g ∗ , λ∗ ) = 0 ,βλ (g ∗ , λ∗ ) = 0• Gaussian Fixed Point:◦ at g ∗ = 0, λ∗ = 0 ⇐⇒ free theory◦ UV-repulsive for g > 0∗ = −2):• non-Gaussian Fixed Point (ηN◦ at g ∗ > 0, λ∗ > 0 ⇐⇒ “interacting” theory◦ UV attractive in gk , λk

Asymptotic safety: non-Gaussian Fixed Point is UV completion for gravity– p. 22/4

Einstein-Hilbert truncation: Stability properties
Ref.

g∗

λ∗

g ∗ λ∗

θ0 ± iθ00

gauge

Rk

BMS

0.902

0.109

0.099

2.52 ± 1.78i

geometricII, opt

RS

0.403

0.330

0.133

1.94 ± 3.15i

harmonicI, sharp

LR

0.272

0.348

0.095

1.55 ± 3.84i

harmonicI, exp

0.344

0.339

0.117

1.86 ± 4.08i

LandauI, exp

L

1.17

0.25

0.295

1.67 ± 4.31i

LandauI, opt

CPR

0.707

0.193

0.137

1.48 ± 3.04i

harmonicI, opt

0.556

0.092

0.051

2.43 ± 1.27i

harmonicII, opt

0.332

0.274

0.091

1.75 ± 2.07i

harmonic

III, opt

BMS:
RS:
LR:
L:
CPR:

Benedetti, Machado, Saueressig, 2009.
Reuter, Saueressig, 2002.
Lauscher, Reuter, 2002.
Litim, 2004.
Codello, Percacci, Rahmede, 2009.

– p. 23/4

PSfrag replacements
Einstein-Hilbert truncation: NGFP in d = 2 + 
β -functions continuous in d ⇐⇒ reproduce perturbative fix point in d = 2 + 


0.5

4

0.4

3:5

0.3

0.2

2

2:5
2:5

4
3:5

3
0.1

3:5

sc s = 1
sc s = 30
Exp s = 1

3
0.1

0.2

0.3

4

0.4

0.5

g

-0.1

NGFP in d = 4 ⇐⇒ analytic continuation of NGFP in d = 2 + – p. 24/4

Einstein-Hilbert-truncation: the phase diagram
g
1

0.75

0.5

Type IIa
0.25

Type Ia−0.2Type IIIa
−0.1

0.1

0.2

0.3

0.4

0.5

λ

−0.25

Type IIIb−0.5

Type Ib−0.75– p. 25/4

Letting things flow
Higher-derivative terms– p. 26/4

Charting the theory space of gravity..
.
Einstein-Hilbert truncation

R8

...

R7

...

R6

...

R5

...

R4

...

R3

Cµν ρσ Cρσ κλ Cκλ µν

R R

R2

Cµνρσ C µνρσRµν Rµν

+ 7 moreR1– p. 27/4

Charting the theory space of gravity
..
.



Einstein-Hilbert truncation

R8

...

R7

...

R6

...

R5

...

R4

...

R3

Cµν ρσ Cρσ κλ Cκλ µνR2Cµνρσ C µνρσ



 
R

polynomial f (R)-truncation
R2 + C 2 -truncation




R R

+ 7 moreRµν Rµν1– p. 28/4

Exploring the gravitational theory space
Some key results . . .

• evidence for asymptotic safety⇒ non-Gaussian fixed point provides UV completion of gravity• finite dimensional UV-critical surface⇒ possibly: 3 relevant parameters• perturbative counterterms:◦ gravity + matter: asymptotic safety survives 1-loop counterterm– p. 29/4

Exploring the gravitational theory space
Some key results . . .

• evidence for asymptotic safety⇒ non-Gaussian fixed point provides UV completion of gravity• finite dimensional UV-critical surface⇒ possibly: 3 relevant parameters• perturbative counterterms:◦ gravity + matter: asymptotic safety survives 1-loop counterterm. . . and open questions:• Existence of NGFP in extended truncations?
• Dimension of its UV-critical surface?

– p. 29/4

Exploring the gravitational theory space
Some key results . . .

• evidence for asymptotic safety⇒ non-Gaussian fixed point provides UV completion of gravity• finite dimensional UV-critical surface⇒ possibly: 3 relevant parameters• perturbative counterterms:◦ gravity + matter: asymptotic safety survives 1-loop counterterm. . . and open questions:• Existence of NGFP in extended truncations?
• Dimension of its UV-critical surface?
How does the signature of space-time
affect asymptotic safety?

– p. 29/4

Functional Renormalization Group Equations II
foliated space-times– p. 30/4

Foliation structure via ADM-decomposition
Preferred “time”-direction via foliation of space-time• foliation structure Md+1 = S 1 × Md with yµ 7→ (τ, xa ):ds2 = N 2 dt2 + σij dxi + N i dt

dxj + N j dt

• fundamental fields: gµν 7→ (N, Ni , σij )N 2 + Ni N igµν = 
Ni

Nj
σij




– p. 31/4

Foliation structure via ADM-decomposition
Preferred “time”-direction via foliation of space-time• foliation structure Md+1 = S 1 × Md with yµ 7→ (τ, xa ):
2

2

2

i

i

ds = N dt + σij dx + N dt



j

j

dx + N dt

• fundamental fields: gµν 7→ (N, Ni , σij )gµν = N 2 + Ni N i

Nj

Ni

σij




Allows to include signature parameter  = ±1

– p. 32/4

Foliated functional renormalization group equation
Theory:degrees of freedom:

component fields N, Ni , σij

symmetries:

diffeomorphisms (full or foliation preserving)

Construction of the flow equation:• starting point: generic diffeomorphism invariant action S grav [N, Ni , σij ]◦ diffeomorphism invariant or Horava-type– p. 33/4

Foliated functional renormalization group equation
Theory:degrees of freedom:

component fields N, Ni , σij

symmetries:

diffeomorphisms (full or foliation preserving)

Construction of the flow equation:• starting point: generic diffeomorphism invariant action S grav [N, Ni , σij ]◦ diffeomorphism invariant or Horava-type
gauge-fixing: Background field formalism for (N, Ni , σij ):N = N̄ + h ,Ni = N̄i + hi ,• choice of backgrounds:N̄ = 1, N̄i = 0=⇒ admit temporal gauge:
1 √S gf = 2ασij = σ̄ij + hijZ

h = 0, hi = 0
dτ

Z

√ 
d3 x σ̄ h2 + σ̄ ij hi hj– p. 33/4

Foliated functional renormalization group equation
Theory:degrees of freedom:

component fields N, Ni , σij

symmetries:

diffeomorphisms (full or foliation preserving)

Construction of the flow equation:• starting point: generic diffeomorphism invariant action S grav [N, Ni , σij ]◦ diffeomorphism invariant or Horava-type
gauge-fixing: Background field formalism for (N, Ni , σij ):N = N̄ + h ,Ni = N̄i + hi ,• choice of backgrounds:N̄ = 1, N̄i = 0=⇒ admit temporal gauge:
1 √S gf = 2ασij = σ̄ij + hij

h = 0, hi = 0

Z

dτ

Z

dτ

Z

√ 
d x σ̄ C̄∂τ C + C̄i ∂τ C i√ 
d3 x σ̄ h2 + σ̄ ij hi hj

ghost action:
S

gh

=

√



Z

3

– p. 33/4

Foliated functional renormalization group equation
Theory:degrees of freedom:

component fields N, Ni , σij

symmetries:

diffeomorphisms (full or foliation preserving)

Construction of the flow equation:• starting point: generic diffeomorphism invariant action S grav [N, Ni , σij ]gauge-fixing: Background field formalism for (N, Ni , σij ):N = N̄ + h ,Ni = N̄i + hi ,σij = σ̄ij + hij

k-dependent IR-cutoff ∆k S
∆k S[h; σ̄] =

•

√



Z

dτ

Z

√ 
d3 x σ̄ hij Rk [σ̄]hij + . . .

Rk : depends on spatial Laplacian ∆ only!

◦ ∆ = −σ̄ ij D̄i D̄j is positive definite◦ Time-like fluctuations: regulated by circle◦ cutoff respects foliation-preserving diffeomorphisms only
⇒ explore RG-flows in Horava gravity– p. 34/4

Foliated functional renormalization group equation
Flow equation: formally the same as in covariant construction


−1
(2)
k∂k Γk [h, hi , hij ; σ̄ij ] = 21 STr Γk + Rkk∂k Rk• covariant: M4

STr ≈• foliated: S 1 × M3
STr ≈

√



XZ

√
d4 y ḡfieldsX

X

component fields KK−modesZ√
d x σ̄
3◦ structure resembles: quantum field theory at finite temperature!

– p. 35/4

Foliated functional renormalization group equation
Flow equation: formally the same as in covariant construction


−1
(2)
k∂k Γk [h, hi , hij ; σ̄ij ] = 21 STr Γk + Rkk∂k Rk• covariant: M4

STr ≈• foliated: S 1 × M3
STr ≈

√



XZ

√
d4 y ḡfieldsX

X

component fields KK−modesZ√
d x σ̄
3◦ structure resembles: quantum field theory at finite temperature!
Advantages of the foliated flow equation:• limits: same as covariant equation•

-dependence: keep track of signature effects• structure: same as lattice approach of CDT– p. 35/4

signature-dependent renormalization group flows
Einstein-Hilbert truncation– p. 36/4

ADM-decomposed Einstein-Hilbert truncation: setup
ADM-decomposed Einstein-Hilbert truncation: running couplings: Gk , Λk
√


=
ΓADM
k
16πGk

Z

n
o


√
dτ d3 xN σ −1 Kij σ ik σ jl − σ ij σ kl Kkl − R(3) + 2Λk +S gf +S gh•

Kij : extrinsic curvature• = ±1: signature parameter

R(d) : intrinsic curvature– p. 37/4

ADM-decomposed Einstein-Hilbert truncation: setup
ADM-decomposed Einstein-Hilbert truncation: running couplings: Gk , Λk
√


=
ΓADM
k
16πGk

Z

n
o


√
dτ d3 xN σ −1 Kij σ ik σ jl − σ ij σ kl Kkl − R(3) + 2Λk +S gf +S gh
R(d) : intrinsic curvature•

Kij : extrinsic curvature• = ±1: signature parameter

Structure of the flow equationk∂k Γk = T TT + T 0√

k3 d2T X
TTT=(4π)3/2 nZi√ h 1,0
2 2,0
1 1,0
R̄
d x σ̄ q3/2 (w2T ) + k2 6 q1/2 (w2T ) − 3 q3/2 (w2T )3– p. 37/4

ADM-decomposed Einstein-Hilbert truncation: setup
ADM-decomposed Einstein-Hilbert truncation: running couplings: Gk , Λk
√


=
ΓADM
k
16πGk

Z

n
o


√
dτ d3 xN σ −1 Kij σ ik σ jl − σ ij σ kl Kkl − R(3) + 2Λk +S gf +S gh
R(d) : intrinsic curvature•

Kij : extrinsic curvature• = ±1: signature parameter

Structure of the flow equationk∂k Γk = T TT + T 0√

k3 d2T X
TTT=(4π)3/2 nZi√ h 1,0
2 2,0
1 1,0
R̄
d x σ̄ q3/2 (w2T ) + k2 6 q1/2 (w2T ) − 3 q3/2 (w2T )3

β -functions:k∂k gk = βg (g, λ; m),k∂k λk = βλ (g, λ; m)• depend parametrically on dimensionless Kaluza-Klein-mass m = T2πk

– p. 37/4

Analyticity properties of β -functions
Kaluza-Klein sums: carry out analytically:
X 1,0
X
qd/2 (w2T ) ∝
n

n

1
1
m2 n2 − 2λk
1 + 2

Summation: depends on signature :
X

π
1
,
=
n2 + x2
x tanh(πx)

x2 > 0(hyperbolic functions)

X

1
π
,
=
n2 + x2
i x tan(πx)

x2 < 0(trigonometric functions)nn– p. 38/4

Analyticity properties of β -functions
Kaluza-Klein sums: carry out analytically:
X 1,0
X
qd/2 (w2T ) ∝
n

n

1
1
m2 n2 − 2λk
1 + 2

Summation: depends on signature :
X

π
1
,
=
n2 + x2
x tanh(πx)

x2 > 0(hyperbolic functions)

X

1
π
,
=
n2 + x2
i x tan(πx)

x2 < 0(trigonometric functions)nnanalytic structure of β -functions: determined by , λ:

λ < λ(1) < 0

λ(1) < λ < λ(2) = 1/2

λ(2) < λ+1

hyperbolic

mixturetrigonometric−1

trigonometricmixturehyperbolic– p. 38/4

NGFP (Part I): fixed Kaluza-Klein mass
Scenario I:

T = const⇐⇒

limk→∞ mk = T2πk → 0– p. 39/4

NGFP (Part I): fixed Kaluza-Klein mass
Scenario I:

T = const⇐⇒

limk→∞ mk = T2πk → 0

trigonometric terms in β -functions diverge:

λ < λ(1) < 0

λ(1) < λ < λ(2) = 1/2

λ(2) < λ+1

hyperbolic

mixturetrigonometric−1

trigonometricmixturehyperbolic– p. 39/4

NGFP (Part I): fixed Kaluza-Klein mass
Scenario I:

T = const⇐⇒

limk→∞ mk = T2πk → 0

trigonometric terms in β -functions diverge:

λ < λ(1) < 0

λ(1) < λ < λ(2) = 1/2

λ(2) < λ+1

hyperbolic

mixturetrigonometric−1

trigonometricmixturehyperbolic

No analogue of the Non-Gaussian fixed Point!

– p. 39/4

NGFP (Part II): running Kaluza-Klein mass
T ∝ k−1

Scenario II:⇐⇒

limk→∞ mk = m∗ 6= 0

(sequel : m = 2π)

β -functions well-defined in all regions:λ < λ(1) < 0

λ(1) < λ < λ(2) = 1/2

λ(2) < λ+1

hyperbolic

mixturetrigonometric−1

trigonometricmixturehyperbolic– p. 40/4

NGFP (Part II): running Kaluza-Klein mass
T ∝ k−1

Scenario II:

limk→∞ mk = m∗ 6= 0⇐⇒

(sequel : m = 2π)

β -functions well-defined in all regions:λ < λ(1) < 0

λ(1) < λ < λ(2) = 1/2

λ(2) < λ+1

hyperbolic

mixturetrigonometric−1

trigonometricmixturehyperbolic

Obtain: NGFP for both signatures:


g∗

λ∗

g∗ λ∗

θ1,2

+1

0.19

0.31

0.059

1.07 ± 3.31i

−1

0.21

0.30

0.063

0.94 ± 3.10i– p. 40/4

NGFP (Part II): running Kaluza-Klein mass
T ∝ k−1

Scenario II:

limk→∞ mk = m∗ 6= 0⇐⇒

(sequel : m = 2π)

β -functions well-defined in all regions:λ < λ(1) < 0

λ(1) < λ < λ(2) = 1/2

λ(2) < λ+1

hyperbolic

mixturetrigonometric−1

trigonometricmixturehyperbolic

Obtain: NGFP for both signatures:


g∗

λ∗

g∗ λ∗

θ1,2

+1

0.19

0.31

0.059

1.07 ± 3.31i

−1

0.21

0.30

0.063

0.94 ± 3.10i

stability coefficients: almost the same!

– p. 40/4

Comparison: phase diagrams
g
1.0
0.8
0.6
0.4
0.2
Λ
-0.4

0.0

-0.2

0.2

0.4

covariant computation

g

g

0.35

0.35

0.30

0.30

0.25

0.25

0.20

0.20

0.15

0.15

0.10

0.10

0.05

0.05
Λ

-0.3

-0.2

-0.1

0.0

0.1

Euclidean

0.2

0.3

0.4

Λ
-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Lorentzian– p. 41/4

Conclusion and Perspectives
novel causal functional renormalization group equation• symmetries: foliation-preserving diffeomorphism• applications:◦ RG flows of Euclidean and Lorentzian signature metrics◦ analytic complement to causal dynamical triangulations◦ Horava-type gravitational theories– p. 42/4

Conclusion and Perspectives
novel causal functional renormalization group equation• symmetries: foliation-preserving diffeomorphism• applications:◦ RG flows of Euclidean and Lorentzian signature metrics◦ analytic complement to causal dynamical triangulations◦ Horava-type gravitational theories
Asymptotic Safety• ADM-decomposed Einstein-Hilbert action:◦ Euclidean and Lorentzian signature: similar non-Gaussian fixed points◦ phase portraits identical to covariant computation– p. 42/4

Conclusion and Perspectives
novel causal functional renormalization group equation• symmetries: foliation-preserving diffeomorphism• applications:◦ RG flows of Euclidean and Lorentzian signature metrics◦ analytic complement to causal dynamical triangulations◦ Horava-type gravitational theories
Asymptotic Safety• ADM-decomposed Einstein-Hilbert action:◦ Euclidean and Lorentzian signature: similar non-Gaussian fixed points◦ phase portraits identical to covariant computation
gravity in UV
signature does not affect asymptotic safety– p. 42/4