Constraints on matter from asymptotic...

Asymptotic safety 1-loop FRGE Matter Conclusions

Constraints on matter

from asymptotic safety

Roberto Percacci

SISSA, Trieste

Padova, February 19, 2014


Fixed points

Γk(φ) =∑

i

λi(k)Oi (φ)

βi (λj , k) =dλi

dt, t = log(k/k0)

λi = k−diλi

βi =dλj

dt= −di λi + k−diβi

Fixed point: βi (λj∗) = 0

“Asymptotically safe” RG trajectory: λi → λi∗

UV attractive=IR repulsive=relevant

UV-repulsive=IR attractive=irrelevant

predictivity demands finitely many relevant directions


General picture


Examples

Gaußian Fixed Point at λi∗ = 0.

Mij

∣

∣

∗= ∂βi

∂λj

∣

∣

∗= −diδij

relevant=renormalizable

- QCD- non-AF examples: 4-Fermi interactions in d = 3- maybe gravity?


Outline

1 Asymptotic safety

2 1-loop

3 FRGE

4 Matter

5 Conclusions


One Loop Corrections in Einstein’s Theory

kd

dk

1

16πG (k)= ck2

kdG

dk= −16πcG 2k2

G = Gk2

kdG

dk= 2G − 16πcG 2

if c > 0 fixed point at G = 1/8πcone calculation: c = 23/48π2


Perturbative beta functions

βG = 2G − 46G 2

6π,

βΛ = −2Λ +2G

4π− 16G Λ

6π

Λ∗ =3

62G∗ =

12π

46


Perturbative flow

-0.5 0.0 0.5 1.0

-0.5

0.0

0.5

1.0

L�

G�


Higher derivative gravity

Γk =

∫

d4x√g

[

2ZΛ− ZR +1

2λ

(

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. Class. Quant. Grav. 31 015024 (2014); arXiv:1308.3398


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


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


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 .


Flow in Λ–G plane II

-0.5 0.0 0.5 1.0-0.5

0.0

0.5

1.0

1.5

2.0

L�

G�


Topologically massive gravity

Action

S(g)=1

16πG

∫

d3x√g(

2Λ− R + 12µ ε

λµνΓρλσ(

∂µΓσνρ +

23Γ

σµτΓ

τνρ

)

)

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


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


-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.


-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 no

Gaussian FP. The beta functions diverge on the Λ axis.


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.


Application to gravity

Background field method: qµν = gµν + hµν

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

Γk(g , h) = Γk(g) + Γ(1)k (g , h) + Γ

(2)k (g , h) + . . .

Γk(g) = Γk(g , 0)

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

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

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


Single-field truncation

In the FRGE identify Zh with 116πG . Consequently identify

ηh = −kd log Zh

dk

with

ηh = kd logG

dk


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


Einstein–Hilbert truncation 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 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 II


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)


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]


Enter matter

because it’s there

because it may help (large N limit)

because pure gravity has no local observables

because experimental constraints more likely

e.g. M. Shaposhnikov and C. Wetterich, Phys.Lett. B683, 196(2010)

[L. Griguolo, R.P. Phys. Rev. D 52, 5787 (1995)]

[R.P., D.Perini, Phys.Rev. D 67 081503 (2003)]

[R.P., D.Perini, Phys. Rev. D68, 044018 (2004)]

[G. Narain. R.P. Class. Quant. Grav. 27 075001 (2010)]

[P. Donà, A. Eichhorn, R.P. arXiv:1311.2898 [hep-th](2013)]


Perturbative beta functions with matter

βG = 2G +G 2

6π(NS + 2ND − 4NV − 46) ,

βΛ = −2Λ +G

4π(NS − 4ND + 2NV + 2)

+G Λ

6π(NS + 2ND − 4NV − 16)

Λ∗ = −3

4

NS − 4ND + 2NV + 2

NS + 2ND − 4NV − 31,

G∗ = − 12π

NS + 2ND − 4NV − 46.


Exclusion plots NV = 0, 6, 12, 24, 45

0 50 100 150 2000

20

40

60

80

100


Position of FP for NV = 12

0 20 40 60 800

10

20

30

40

0 20 40 60 800

10

20

30

40


Truncated FRGE, bimetric formalism

Γk(g , h) =1

16πG

∫

ddx√g(

−R + 2Λ)

+Zh

2

∫

ddx√g hµνK

µναβ((−D2 − 2Λ)1ρσαβ +W

ρσαβ )hρσ

−√

2Zc

∫

ddx√g cµ

(

DρgµκgκνDρ + DρgµκgρνDκ − DµgρσgρνDσ

)

cν

SS =ZS

2

∫

ddx√g gµν

NS∑

i=1

∂µφi∂νφ

i

SD = iZD

∫

ddx√g

ND∑

i=1

ψi /∇ψi ,

SV =ZV

4

∫

ddx√g

NV∑

i=1

gµνgκλF iµκF

iνλ + . . .


Gravitational beta functions

dΛ

dt= −2Λ +

8πG

(4π)d/2d(d + 2)Γ[d/2]

[

d(d + 1)(d + 2 − ηh)

1 − 2Λ− 4d(d + 2 − ηc )

+2NS (2 + d − ηS ) − 2ND2[d/2]

(2 + d − ηD ) + 2NV (d2− 4 − d ηV )

]

−

4πG Λ

3d(4π)d/2Γ[d/2]

[

d(5d − 7)(d − ηh)

1 − 2Λ+ 4(d + 6)(d − ηc )

−2NS (d − ηS ) − ND2[d/2]

(d − ηD ) + 2NV (d (8 − d) − (6 − d)ηV )

]

dG

dt= (d − 2)G −

4πG2

3d(4π)d/2Γ(d/2)

[ d(5d − 7)(d − ηh)

1 − 2Λ+ 4(d + 6)(d − ηc )

−2NS (d − ηS ) − ND2[d/2]

(d − ηD ) + 2NV (d(8 − d) − (6 − d)ηV )

]


Graviton+ghost contributions to graviton η

∂tγ(2,0,0;0)k = − 2− 1

2

+


Graviton contributions to ηh

ηh = ηh∣

∣

gravity+ ηh

∣

∣

matter

ηh

∣

∣

∣

gravity=[

a(Λk ) + c(Λk )ηh + e(Λk )ηc

]

Gk

a(Λ) =a0 + a1Λ + a2 Λ

2 + a3Λ3 + a4Λ

4

(4π)d/2Γ(d/2)d2(d2− 4)(3d − 2)(1 − 2Λ)4

,

a0 = −4π (d − 2) (−896 + 264 d + 1076 d2− 434 d

3+ 21 d

4+ d

5) ,

a1 = 16π (d − 2) (−2048 + 2552 d − 318 d2− 125 d

3+ 2 d

4+ d

5) ,

a2 = −16π(12544 − 25760 d + 16968 d2− 4228 d

3+ 354 d

4− 17 d

5+ d

6) ,

a3 = 4096π(d − 2)(−32 + 50 d − 19 d2+ 2 d

3) ,

a4 = −2048π(d − 2)(−32 + 50 d − 19 d2+ 2 d

3) ;

c(Λ) =8π(d − 1)

[

128 + 720 d − 350 d2 + 29 d3 + 32(d − 2)(d + 4)Λ]

(4π)d/2Γ(d/2) d2(d + 2)(d + 4)(3d − 2)(1 − 2Λ)3

e(Λ) = −

128π(

32 − 50 d + 23 d2

)

(4π)d/2Γ(d/2)d2(d + 2)(d + 4)(3 d − 2)


Matter contribution to graviton η


Matter contributions to ηh

ηh

∣

∣

∣

matter= −NS

32πG

(4π)d/2Γ(d/2)

1

d2(d + 2)(3d − 2)

[

(d − 2)3+ 2

8 − 10 d + d2

d + 4ηS

]

+ND 2[d/2] 16πG

(4π)d/2Γ(d/2)

(d − 1)(d − 2)

d3(3d − 2)

[

2 +d − 2

d + 1ηD

]

−NV

32πG

(4π)d/2Γ(d/2)

(d − 1)(d − 2)

d2(d + 2)(3d − 2)

[

d2− 12 d + 8 − 2

16 − d

d + 4ηV

]


Graviton+ghost contributions to ghost η

∂tγ(0,1,1;0)k = +


Ghost ηc

ηc =[

b(Λk) + d(Λk)ηh + f (Λk)ηc

]

Gk

with

b(Λ) =64π

[

−8 + 4 d + 18 d2 − 7 d3 + 2(4 − 9 d2 + 3 d3)Λ]

(4π)d/2Γ(d/2)d2(d2 − 4)(d + 4)(1 − 2Λ)2

d(Λ) =−64π(4 − 4 d − 9 d2 + 4 d3)

(4π)d/2Γ(d/2)d2(d2 − 4)(d + 4)(1 − 2Λ)2

f (Λ) =−64π(4 − 9 d2 + 3 d3)

(4π)d/2Γ(d/2)d2(d2 − 4)(d + 4)(1 − 2Λ)2


Graviton contribution to matter η


Matter anomalous dimensions

ηS = −

32πG

(4π)d/2Γ(d/2)

[

2

d + 2

1

(1 − 2Λ)2

(

1 −

ηh

d + 4

)

+2

d + 2

1

1 − 2Λ

(

1 −

ηS

d + 4

)

+(d + 1)(d − 4)

2d(1 − 2Λ)2

(

1 −

ηh

d + 2

)

]

,

ηD =32πG

(4π)d/2Γ(d/2)

[

(d − 1)(d2 + 9 d − 8)

8d (d − 2)(d + 1)(1 − 2Λ)2

(

1 −

ηh

d + 3

)

+(d − 1)2

2d(d + 1)(d − 2)

1

1 − 2Λ

(

1 −

ηD

d + 2

)

−

(d − 1)(2d2− 3d − 4)

4d(d − 2)(1 − 2Λ)2

(

1 −

ηh

d + 2

)

]

ηV = −

32πG

(4π)d/2Γ(d/2)

[

(d − 1)(16 + 10 d − 9 d2 + d3)

2d2(d − 2)(1 − 2Λ)2

(

1 −

ηh

d + 2

)

+4(d − 1)(2d − 5)

d(d2− 4)(1 − 2Λ)

(

1 −

ηV

d + 4

)

+4(d − 1)(2d − 5)

d(d2− 4)(1 − 2Λ)2

(

1 −

ηh

d + 4

)

]


General structure of anomalous dimensions

For Ψ = h, c ,S ,D,V ,

ηΨ = − 1

ZΨkdZΨ

dk

~η = (ηh, ηc , ηS , ηD , ηV )

~η = ~η1(G , Λ) + A(G , Λ)~η

one loop anomalous dimensions ~η = ~η1

RG improved anomalous dimensions ~η = (1 − A)−1~η1


Gravity+scalars

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50100200

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0.10

0.15

0.20

0.25

0.30L�*

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100

1000

104

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30

40Ηh

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-80

-60

-40

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10

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1

2

3

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1.0

1.5

2.0

2.5Ηh

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2 4 6 8ND

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

Ηc

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2 4 6 8ND

-0.4

-0.3

-0.2

-0.1

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Gravity+vectors

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10 20 30 40 50NV

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0.2

0.3

0.4

0.5

0.6

0.7G�*

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10 20 30 40 50NV

0.05

0.10

0.15

0.20

0.25

0.30

L�*

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1

2

3

4

Θ1, Θ2

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0.5

1.0

1.5

Ηh

æ

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-0.86

-0.84

-0.82

-0.80

Ηc

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10 20 30 40 50NV

-0.15

-0.10

-0.05

0.05

ΗV


Exclusion plot NV = 0

5 10 15 20 25 30NS

5

10

15

N D


Exclusion plot NV = 12

10 20 30 40 50 60 70NS

10

20

30

40ND


Exclusion plot ND = 0

10 20 30 40 50 60 70NS

10

20

30

40

50

NV

10 20 30 40 50 60 70NS

10

20

30

40

50

NV


Exclusion plot NS = 0

10 20 30 40N D

10

20

30

40

50

NV


Standard model matter

1L-II full-II 1L-Ia full-Ia

Λ∗ −2.399 −2.348 −3.591 −3.504

G∗ 1.762 1.735 2.627 2.580

θ1 3.961 3.922 3.964 3.919

θ2 1.644 1.651 2.178 2.187

ηh 2.983 2.914 4.434 4.319

ηc −0.139 −0.129 −0.137 −0.125

ηS −0.076 −0.072 −0.076 −0.073

ηD −0.015 0.004 −0.004 0.016

ηV −0.133 −0.145 −0.144 −0.158


Specific models

model NS ND NV G∗ Λ∗ θ1 θ2 ηh

no matter 0 0 0 0.77 0.01 3.30 1.95 0.27

SM 4 45/2 12 1.76 -2.40 3.96 1.64 2.98

SM +dm scalar 5 45/2 12 1.87 -2.50 3.96 1.63 3.15

SM+ 3 ν’s 4 24 12 2.15 -3.20 3.97 1.65 3.71

SM+3ν’s+ axion+dm 6 24 12 2.50 -3.62 3.96 1.63 4.28

MSSM 49 61/2 12 - - - - -

SU(5) GUT 124 24 24 - - - - -

SO(10) GUT 97 24 45 - - - - -


Exclusion plot NV = 12, d = 5

10 20 30 40 50 60NS

5

10

15

20

25

N D


Exclusion plot NV = 12, d = 6

10 20 30 40 50 60NS

2468

10N D


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)]


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

)

.


Polynomial truncations

V (φ2) = λ0 + λ2φ2 + λ4φ

4 + . . .

F (φ2) = ξ0 + ξ2φ2 + . . .

∂t λ4 =9λ2

4

2π2+

Gλ4

π+ . . .


SM+gravity

dh

dt= βSMh +

ah

8πG (µ)h

βSM1 =

1

16π2

41

6g3

1 ; βSM2 = − 1

16π2

19

6g3

2 ; βSM3 = − 1

16π27g3

3 ;

βSMy =

1

16π2

[

9

2y3 − 8g2

3y − 9

4g2

2y − 17

21g2

1y

]

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) ]


AS: strengths and weaknesses

use continuum covariant QFT

bottom up approach

guaranteed to give correct low energy limit

highly predictive

but

strong coupling problem

scheme dependence

off shell: gauge dependence

no observables computed yet


Summary/outlook

Pure gravity:

truly functional truncations

other approximations, e.g. two loops

describe as CFT

Gravity+matter:

BSM physics strongly constrained

running gravitational mass from 2 point function

matter interactions: LPA for scalar+gravity

turn on gauge and Yukawa couplings

Constraints on matter from asymptotic...

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