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EFT Asymptotic safety 1-loop FRGE Matter Conclusions
Asymptotic safety:
motivations and results
Roberto Percacci
1SISSA, Trieste
IPMU, September 2013
EFT Asymptotic safety 1-loop FRGE Matter Conclusions
Outline
1 EFT
2 Asymptotic safety
3 1-loop
4 FRGE
5 Matter
6 Conclusions
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 .
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
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
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.
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.
EFT Asymptotic safety 1-loop FRGE Matter Conclusions
Newtonian potential
∼ Gm1m2q2
V (r) =
∫
d3q
(2π)3Gm1m2
q2e iqr = −Gm1m2
r
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)
EFT Asymptotic safety 1-loop FRGE Matter Conclusions
One loop graphs
(from Bjerrum Bohr, Donoghue, Holstein, 2002)
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
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
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
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
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
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
EFT Asymptotic safety 1-loop FRGE Matter Conclusions
General picture
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
EFT Asymptotic safety 1-loop FRGE Matter Conclusions
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. arXiv:1308.3398
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
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
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 .
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�
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
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
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.
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.
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.
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)]
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
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
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)
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]
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
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]
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
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
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
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]
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
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]
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]
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
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)]
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
)
.
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+ . . .
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)]
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) ]
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
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
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
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