Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add...

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EFT Asymptotic safety 1-loop FRGE Matter Conclusions Asymptotic safety: motivations and results Roberto Percacci 1 SISSA, Trieste IPMU, September 2013

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Page 1: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Asymptotic safety:

motivations and results

Roberto Percacci

1SISSA, Trieste

IPMU, September 2013

Page 2: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Outline

1 EFT

2 Asymptotic safety

3 1-loop

4 FRGE

5 Matter

6 Conclusions

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EFT Asymptotic safety 1-loop FRGE Matter Conclusions

The gospel according to de Witt

Expandgµν = gµν + hµν

Gauge fix using a background gauge fixing condition e.g.

SGF (g , h) =1

2

dx√−g gµνχµχν ; χµ = ∇νhνµ − 1

2∇µh

Add ghost Lagrangian

Sghost(g , c , c) =

dx√−g cµ(−δνµ∇2 − Rν

µ)cν

Compute Γ(g , h).Formalism preserves background gauge invariance:δǫgµν = Lǫgµν , δǫhµν = Lǫhµν .

Page 4: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Issues

Non-renormalizable (Goroff and Sagnotti 1985)

interaction strength grows like G = Gk2

violation of unitarity

lack of predictivity

Page 5: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

General EFT recipe

fix the level of precision that is required in the calculation

fix the ratio E/M. This determines the order of the expansionthat will be required

at the given order in the expansion, determine all the terms inthe action that can contribute to the given process.By power counting there can only be finitely many.

if the fundamental theory is known, their coefficients can inprinciple be calculated

otherwise, measure them with as many experiments.

use the EFT Lagrangian to compute the amplitudes to thedesired precision

Page 6: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Chiral perturbation theory

Strong interactions at low energy described by chiral model

S =

dx

[

f 2

π

4tr(U−1∂U)2 + ℓ1tr((U

−1∂U)2)2 + ℓ2tr((U−1∂U)2)2 + O(∂6)

]

Expansion parameter E/4πfπ.For low energy meson physics need fπ, ℓ1, ℓ2 and perhaps a fewothers. One loop in fπ, tree level in ℓ1, ℓ2.Successfully describe a rich phenomenology.

Page 7: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Gravity

S =

dx√g[

2m2

PΛ−m2

PR + ℓ1RµνρσRµνρσ + ℓ2RµνR

µν + ℓ3R2 + O(∂6)

]

mP similar to fπ.

Page 8: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Newtonian potential

∼ Gm1m2q2

V (r) =

d3q

(2π)3Gm1m2

q2e iqr = −Gm1m2

r

Page 9: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Leading quantum correction

For dimensional reasons the leading quantum correction is

V (r) = −Gm1m2

r

[

1 + βG~

r2c3+ . . .

]

and∫

d3q

(2π)3log

(

q2

µ2

)

e iqr = − 1

2π2r3

Clearly distinct from contributions of local counterterms that giveanalytic corrections to the amplitude. E.g.

d3q

(2π)3ℓe iqr = ℓδ(r)

Page 10: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

One loop graphs

(from Bjerrum Bohr, Donoghue, Holstein, 2002)

Page 11: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Evaluation

- J.F. Donoghue, P.R.L. 72, 2996 (1994); P.R.D50, 3874 (1994)- H.W. Hamber, S. Liu, Phys. Lett. B357, 51 (1995)- A. Akhundov, S. Bellucci, A. Shiekh, Phys. Lett. B395, 16 (1997)- N.E.J. Bjerrum-Bohr (2002) Phys. Rev. D66, 084023- I.B. Khriplovich, G.G. Kirilin (2002) J. Exp. Theor. Phys. 95, 981-986(Zh. Eksp. Teor. Fiz. 95, 1139-1145 (2002))- N.E.J. Bjerrum-Bohr, J.F. Donoghue, B.R. Holstein Phys. Rev. D68,084005; Erratum-ibid.D71, 069904 (2005)- N.E.J. Bjerrum-Bohr, J.F. Donoghue and B.R. Holstein Phys. Rev. D67, 084033 (2003) [Erratum-ibid. D 71 (2005) 069903- I.B. Khriplovich, G.G. Kirilin (2004) J. Exp. Theor. Phys. 98,1063-1072

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EFT Asymptotic safety 1-loop FRGE Matter Conclusions

A prediction of quantum gravity

V (r) = −Gm1m2

r

[

1 +41

10π

G~

r2c3+ . . .

]

EFT makes predictions!Comes from non-local terms in the effective action, e.g.

dx√g [RF1(�)R + RµνF2(�)Rµν ]

Local terms cannot be predicted

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EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Lessons

no clash between QM and GR

experimentally indistinguishable from classical GR

agrees with all experimental data

to some extent, background independent

open issues in the UV, IR, strong field

Page 14: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

The issues of QG, and the RG solution

Try to extend beyond Planck scale

interaction strength grows like G = Gk2

lack of predictivity

first problem solved if G = G (k)k2 → G∗

more generally, if

Γk(g , h) =∑

i

λi(k)Oi (g , h)

require λi → λi∗, where λi = k−diλi

Page 15: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

the RG solution cont’d

Define a theory space (fields, symmetry, action functionals)parameterized by λi .

RG trajectory is “renormalizable” or “asymptotically safe” if itflows to a FP in the UV

The RG trajectories that flow into the FP for k → ∞ form theUV critical surface SUV

Predictivity demands dim(SUV ) is finite

Page 16: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Examples

QCD:

Gaußian Fixed Point at λi∗ = 0.

λi = k−diλi

βi = ∂t λi = −di λi + k−diβi

Mij

∗= ∂βi

∂λj

∗= −diδij

relevant couplings=renormalizable couplings

Non-renormalizable examples: 4-Fermi interactions in d = 3

Page 17: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

General picture

Page 18: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Gravity in 2 + ǫ

d − 2 = ǫ

G = Gkǫ

βG = ǫG − 38

3G 2

0.05 0.1 0.15 0.2 0.25

-0.15

-0.1

-0.05

0.05

Page 19: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Higher derivative gravity

Γk =

d4x√g

[

2ZΛ− ZR +1

(

C 2 − 2ω

3R2 + 2θE

)]

Z =1

16πG

K.S. Stelle, Phys. Rev. D16, 953 (1977).J. Julve, M. Tonin, Nuovo Cim. 46B, 137 (1978).E.S. Fradkin, A.A. Tseytlin,Phys. Lett. 104 B, 377 (1981).I.G. Avramidi, A.O. Barvinski,Phys. Lett. 159 B, 269 (1985).G. de Berredo–Peixoto and I. Shapiro, Phys.Rev. D71 064005 (2005).A. Codello and R. P., Phys.Rev.Lett. 97 22 (2006)N. Ohta and R.P. arXiv:1308.3398

Page 20: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Beta functions I

βλ = − 1

(4π)2133

10λ2

βω = − 1

(4π)225 + 1098ω + 200ω2

60λ

βθ =1

(4π)27(56 − 171θ)

90λ

λ(k) =λ0

1 + λ01

(4π)213310 log

(

kk0

)

ω(k) → ω∗ ≈ −0.0228

θ(k) → θ∗ ≈ 0.327

Page 21: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Beta functions II

βΛ = −2Λ +1

(4π)2

[

1 + 20ω2

256πGω2λ2 +

1 + 86ω + 40ω2

12ωλΛ

]

−1 + 10ω2

64π2ωλ+

2G

π− q(ω)G Λ

βG= 2G − 1

(4π)23 + 26ω − 40ω2

12ωλG−q(ω)G 2

where q(ω) = (83 + 70ω + 8ω2)/18π

Page 22: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Flow in Λ–G plane I

βΛ = −2Λ +2G

π− q∗G Λ

βG = 2G − q∗G2

where q∗ = q(ω∗) ≈ 1.440

Λ∗ =1

πq∗≈ 0.221 , G∗ =

2

q∗≈ 1.389 .

Page 23: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Flow in Λ–G plane II

-0.4 -0.2 0.2 0.4 L�

0.5

1

1.5

2

G�

Page 24: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Topologically massive gravity

Action

S(g) = Z

d3x√g(

2Λ− R + 12µ ελµνΓρλσ

(

∂µΓσνρ +

23Γ

σµτΓ

τνρ

)

)

Z = 116πG

Dimensionless combinations of couplings

ν = µG ; τ = ΛG 2 ; φ = µ/√

|Λ|

R.P., E. Sezgin, Class.Quant.Grav. 27 (2010) 155009, arXiv:1002.2640 [hep-th]

Recently extended to TM SUGRA: R.P., M. Perry, C. Pope, E. Sezgin, arXiv 1302.0868

Page 25: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Beta functions of

βν = 0 ,

βG = G + B(µ)G 2 ,

βΛ = −2Λ +1

2G(

A(µ, Λ) + 2B(µ)Λ)

Since ν = µG = µG is constant

can replace µ by ν/G

Page 26: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

L�

G�

-0.10 -0.05 0.00 0.05 0.10

-0.10

-0.05

0.00

0.05

0.10

L�

G�

Figure : The flow in the Λ-G plane for α = 0, ν = 5. Right:enlargement of the region around the origin, showing the Gaussian FP.The beta functions become singular at |G | = 1.9245.

Page 27: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

L�

G�

-0.10 -0.05 0.00 0.05 0.10

-0.10

-0.05

0.00

0.05

0.10

L�

G�

Figure : The flow in the Λ-G plane for α = 0, ν = 0.1. Right:enlargement of the region around the origin, showing that there is noGaussian FP. The beta functions diverge on the Λ axis.

Page 28: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Functional renormalization

Define Γk(φ). limk→0 Γk(φ) = Γ(φ). It satisfies a FRGE

kdΓk(φ)

dk= β(φ)

The quantity β is UV and IR finite.

Use FRGE to calculate the effective action.

Page 29: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Application to gravity

FRGE can only be defined for Γk(gµν , h). Expand

Γk(gµν , h) = Γk(gµν) + Γ(1)k (gµν , h) + Γ

(2)k (gµν , h) + . . .

Γk(gµν) = Γk(gµν , 0)

Single-field truncations: neglect Γ(n)k (gµν , h), n > 0.

[M. Reuter, Phys. Rev. D 57 971(1998)]

[D. Dou and R. Percacci, Class. and QuantumGrav. 15 3449 (1998)]

Page 30: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Einstein–Hilbert truncation I

Γk(g , h) = SEH(gµν + h) + SGF (g , h) + Sghost(g , Cµ,Cν)

SEH(gµν) =

dx√gZ (2Λ− R) ; Z =

1

16πG

βΛ =−2(1 − 2Λ)2Λ + 36−41Λ+42Λ2

−600Λ3

72π G + 467−572Λ288π2 G 2

(1 − 2Λ)2 − 29−9Λ72π G

βG =2(1 − 2Λ)2G − 373−654Λ+600Λ2

72π G 2

(1 − 2Λ)2 − 29−9Λ72π G

Page 31: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Einstein–Hilbert truncation III

-0.2 -0.1 0.1 0.2 0.3 0.4 0.5L�

0.2

0.4

0.6

0.8

1

G�

Cutoff type Ia

-0.2 -0.1 0.1 0.2 0.3 0.4 0.5L�

0.2

0.4

0.6

0.8

1

G�

Cutoff type Ib

-0.2 -0.1 0.1 0.2 0.3 0.4 0.5L�

0.2

0.4

0.6

0.8

1

G�

Cutoff type II

-0.2 -0.1 0.1 0.2 0.3 0.4 0.5L�

0.2

0.4

0.6

0.8

1

G�

Cutoff type III

Page 32: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Fourth order gravity

R2 O.Lauscher, M. Reuter, Phys. Rev. D 66, 025026 (2002)arXiv:hep-th/0205062

R2 + C 2 D. Benedetti, P.F. Machado, F. Saueressig, Mod.Phys. Lett. A24, 2233-2241 (2009) arXiv:0901.2984 [hep-th]Nucl. Phys. B824, 168-191 (2010), arXiv:0902.4630 [hep-th]

M. Niedermaier, Nucl. Phys. B833, 226-270 (2010)

Page 33: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Bi-field truncation I

Γk(g , h) = SEH(g + h) + SGF (g , h) + Sghost(g , c , c)

hµν → Z1/2h hµν , c

µ → Z1/2c cµ and cν → Z

1/2c cν

Anomalous dimensions ηh = −∂tZh

Zhηc = −∂tZc

Zc

computed from 〈hµνhρσ〉, 〈cµcν〉FP at Λ = −0.00812786, G = 1.44627scaling exponents −3.32286, −1.95403anomalous dimensions ηh = 0.0720708, ηc = −1.50325.A. Codello, G. d’Odorico, G. Pagani, arXiv:1304.4777 [gr-qc]

Page 34: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Bi-field truncation II

-0.2 0.0 0.2 0.4

-0.5

0.0

0.5

1.0

1.5

2.0

Page 35: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

f (R) gravity

Γk(gµν) =

d4x√gf (R)

f (R) =

n∑

i=0

gi (k)Ri

n=6A. Codello, R.P. and C. Rahmede Int.J.Mod.Phys.A23:143-150 arXiv:0705.1769 [hep-th];n=8A. Codello, R.P. and C. Rahmede Annals Phys. 324 414-469 (2009) arXiv: arXiv:0805.2909;P.F. Machado, F. Saueressig, Phys. Rev. D arXiv: arXiv:0712.0445 [hep-th]n=35K. Falls, D.F. Litim, K. Nikolakopoulos, C. Rahmede, arXiv:1301.4191 [hep-th]n=∞

Dario Benedetti, Francesco Caravelli, JHEP 1206 (2012) 017, Erratum-ibid. 1210 (2012) 157arXiv:1204.3541 [hep-th]Juergen A. Dietz, Tim R. Morris,JHEP 1301 (2013) 108 arXiv:1211.0955 [hep-th]Dario Benedetti, arXiv:1301.4422 [hep-th]

Page 36: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

f (R) gravity n = 8

Position of FP

-0,025000

-0,020000

-0,015000

-0,010000

-0,005000

0,000000

0,005000

0,010000

1 2 3 4 5 6 7 8g0

g1

g2

g3

g4

g5

g6

g7

g8

Critical exponents

-14

-12

-10

-8

-6

-4

-2

0

2

4

1 2 3 4 5 6 7 8

Retheta1

theta2

theta3

Retheta4

theta6

theta7

theta8

Page 37: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

f (R) gravity n = 8 predictions

Critical surface:

g3 = 0.00061243 + 0.06817374 g0 + 0.46351960 g1 + 0.89500872 g2

g4 = −0.00916502 − 0.83651466 g0 − 0.20894019 g1 + 1.62075130 g2

g5 = −0.01569175 − 1.23487788 g0 − 0.72544946 g1 + 1.01749695 g2

g6 = −0.01271954 − 0.62264827 g0 − 0.82401181 g1 − 0.64680416 g2

g7 = −0.00083040 + 0.81387198 g0 − 0.14843134 g1 − 2.01811163 g2

g8 = 0.00905830 + 1.25429854 g0 + 0.50854002 g1 − 1.90116584 g2

Page 38: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

f (R) gravity n = 35

0 5 10 15 20 25 30 35

0.5

1.0

1.5

2.0

2.5

3.0

N

Page 39: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

f (R) gravity n = 35

0 5 10 15 20 25 30 351.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

N

−θ3

1 + θ′′

θ′

θ2

n=35

K. Falls, D.F. Litim, K. Nikolakopoulos, C. Rahmede, arXiv:1301.4191 [hep-th]

Page 40: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

f (R) gravity n = 35

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0 5 10 15 20 25 30 35

0

20

40

60

353025151052

Page 41: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

f (R) gravity, functional treatment

Do not expand f (R) but write flow equation for f

∂t f (R) = β(f , f ′, f ′′, f ′′′)

where R = R/k2, f = f /k4.For large R

f (R) = AR2

(

1 +∑

n>0

dnR−n

)

Dario Benedetti, Francesco Caravelli, JHEP 1206 (2012) 017, Erratum-ibid. 1210 (2012) 157arXiv:1204.3541 [hep-th]Juergen A. Dietz, Tim R. Morris,JHEP 1301 (2013) 108 arXiv:1211.0955 [hep-th]

Page 42: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

f (R) gravity, functional treatment

Theorem 1: Γ∗(gµν) = A∗

d4x√gR2, A∗ 6= 0

Theorem 2: if f∗ exists, the spectrum of perturbations is discrete,real, and there are at most finitely many relevant direction.

D. Benedetti, arXiv:1301.4422 [hep-th]

Page 43: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

One loop, with matter

n M = 0

n M =12

n M = 24

0 20 40 60 80 100

0

20

40

60

80

100

n S

nD

Page 44: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Gravity+scalar

Γk [g , φ] =

ddx√g

(

V (φ2)− F (φ2)R +1

2gµν∂µφ∂νφ

)

[R.P., D.Perini, Phys. Rev. D68, 044018(2004)][G. Narain, R.P., Class. and Quantum Grav. 27,075001 (2010)]

Γ[g , φ] =

ddx√g

(

L(φ2,R) +1

2gµν ∂µφ∂νφ

)

[G. Narain, C. Rahmede, Class. and QuantumGrav. 27, 075002 (2010)]

Page 45: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Functional flow of F , V

∂tV =k4

192π2

{

6 +30 V

Ψ+

6(k2 Ψ + 24φ2 k2 F ′ Ψ′ + k2 FΣ1)

∆+

(

4

F+

5 k2

Ψ+

k2 Σ1

)

∂tF +24φ2 k2 Ψ′

∆∂tF

}

,

∂tF =k2

2304π2

{

150 +120 k2 F (3 k2F − V )

Ψ2−

24

(

24φ2k2F′Ψ′+ k

2Ψ + k

2FΣ1

)

36

∆2

[

−4φ2(6 k

4F′2

+ Ψ′2) ∆ + 4φ

2ΨΨ

′(7 k

2F′− V

′) (Σ1 − k

2) + 4φ

2Σ1 (7 k

2F′− V

′) (2Ψ V

′− V Ψ

′)

+2 k4Ψ

2Σ2 + 48 k

4F′φ

2ΨΨ

′Σ2 − 24 k

4F φ

2Ψ′2

Σ2

]

∂tF

F

[

30 −

10 k2F (7Ψ + 4 V )

Ψ2+

6

∆2

(

k2F Σ1 ∆ + 4φ

2V

′Ψ′∆ − 24 k

4F φ

2Ψ′2

Σ2

−4φ2k2F Ψ

′Σ1(7 k

2F′− V

′)

)]

+ ∂tF′

24 k2 φ2

∆2

[

(k2F′+ 5 V

′)∆ − 12 k

2ΨΨ

′Σ2 − 2 (7 k

2F′− V

′) ΨΣ1

]}

where we have defined the shorthands:

Ψ = k2F − V ; Σ1 = k

2+ 2V

′+ 4φ

2V

′′; Σ2 = 2 F

′+ 4φ

2F′′

; ∆ =(

12φ2Ψ′2

+ ΨΣ1

)

.

Page 46: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Polynomial truncations

V (φ2) = λ0 + λ2φ2 + λ4φ

4 + . . .

F (φ2) = ξ0 + ξ2φ2 + . . .

∂t λ4 =9λ2

4

2π2+

Gλ4

π+ . . .

Page 47: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Gauge fields + gravity

0

0.2

0.4

0.6

0.8

1

1.2

1.4

g

2 4 6 8 10 12 14 16 18 20

log[10](E/GeV)

[S.P. Robinson, F. Wilczek Phys. Rev. Lett.96:231601, (2006)]

Page 48: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

SM+gravity

dh

dt= βSM

h +ah

8πG (µ)h

βSM1

=1

16π2

41

6g3

1; βSM

2= − 1

16π2

19

6g3

2; βSM

3= − 1

16π27g3

3;

βSMy =

1

16π2

[

9

2y3 − 8g2

3 y − 9

4g2

2 y − 17

21g2

1 y

]

βSMλ =

1

16π2

[

24λ2+ 12λy2− 9λ

(

g2

2+

1

3g2

1

)

− 6y4+9

8g4

2+

3

8g4

1+

3

4g2

2g2

1

]

If a1 = a2 = a3 < 0, ay < 0, aλ > 0, predict mh=126GeV[M. Shaposhnikov and C. Wetterich, Phys.Lett.B683, 196 (2010) ]

Page 49: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Summary

use continuum covariant QFT

bottom up approach

guaranteed to give correct low energy limit

growing evidence

hints of agreement with CDT

Page 50: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Outlook

FRGE with more complicated truncations

other approximations, e.g. two loops

matter coupled to gravity

observable consequences

describe as CFT

Page 51: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

One Loop Corrections in Einstein’s Theory

kd

dk

1

16πG (k)= ckd−2

kdG

dk= −16πcG 2kd−2

G = Gkd−2

kdG

dk= (d − 2)G − 16πcG 2

fixed point at G = (d − 2)/16πc

Page 52: Asymptotic safety: motivations and resultsresearch.ipmu.jp/seminar/sysimg/seminar/1008.pdf · Add ghost Lagrangian S ghost(¯g,c¯,c) = Z dx √ −g¯c¯µ(−δν µ ∇¯2 −R¯ν

EFT Asymptotic safety 1-loop FRGE Matter Conclusions

Exorcizing the ghosts

Julve-Tonin-Salam-Strathdee argument(m2

phys = m2(p2 = m2phys))

artifact of polynomial truncation (g2p2 + g4p

4 + . . . = 0)

Bonanno and Reuter arXiv:1302.2928 [hep-th]: expandingaround wrong vacuum

Mukohyama arXiv:1303.1409 [hep-th]: −+++ is emergent