Post on 26-Sep-2020
Asymptotic safety 1-loop FRGE Matter Conclusions
Constraints on matter
from asymptotic safety
Roberto Percacci
SISSA, Trieste
Padova, February 19, 2014
Asymptotic safety 1-loop FRGE Matter Conclusions
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
Asymptotic safety 1-loop FRGE Matter Conclusions
General picture
Asymptotic safety 1-loop FRGE Matter Conclusions
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?
Asymptotic safety 1-loop FRGE Matter Conclusions
Outline
1 Asymptotic safety
2 1-loop
3 FRGE
4 Matter
5 Conclusions
Asymptotic safety 1-loop FRGE Matter 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
Asymptotic safety 1-loop FRGE Matter Conclusions
Perturbative beta functions
βG = 2G − 46G 2
6π,
βΛ = −2Λ +2G
4π− 16G Λ
6π
Λ∗ =3
62G∗ =
12π
46
Asymptotic safety 1-loop FRGE Matter Conclusions
Perturbative flow
-0.5 0.0 0.5 1.0
-0.5
0.0
0.5
1.0
L�
G�
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. Class. Quant. Grav. 31 015024 (2014); arXiv:1308.3398
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
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
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 .
Asymptotic safety 1-loop FRGE Matter Conclusions
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�
Asymptotic safety 1-loop FRGE Matter Conclusions
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
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
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.
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 no
Gaussian FP. The beta functions diverge on the Λ axis.
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.
Asymptotic safety 1-loop FRGE Matter Conclusions
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)]
Asymptotic safety 1-loop FRGE Matter Conclusions
Single-field truncation
In the FRGE identify Zh with 116πG . Consequently identify
ηh = −kd log Zh
dk
with
ηh = kd logG
dk
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
Asymptotic safety 1-loop FRGE Matter Conclusions
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
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)
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]
Asymptotic safety 1-loop FRGE Matter Conclusions
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)]
Asymptotic safety 1-loop FRGE Matter Conclusions
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.
Asymptotic safety 1-loop FRGE Matter Conclusions
Exclusion plots NV = 0, 6, 12, 24, 45
0 50 100 150 2000
20
40
60
80
100
Asymptotic safety 1-loop FRGE Matter Conclusions
Position of FP for NV = 12
0 20 40 60 800
10
20
30
40
0 20 40 60 800
10
20
30
40
Asymptotic safety 1-loop FRGE Matter Conclusions
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 + . . .
Asymptotic safety 1-loop FRGE Matter Conclusions
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 )
]
Asymptotic safety 1-loop FRGE Matter Conclusions
Graviton+ghost contributions to graviton η
∂tγ(2,0,0;0)k = − 2− 1
2
+
Asymptotic safety 1-loop FRGE Matter Conclusions
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)
Asymptotic safety 1-loop FRGE Matter Conclusions
Matter contribution to graviton η
Asymptotic safety 1-loop FRGE Matter Conclusions
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
]
Asymptotic safety 1-loop FRGE Matter Conclusions
Graviton+ghost contributions to ghost η
∂tγ(0,1,1;0)k = +
Asymptotic safety 1-loop FRGE Matter Conclusions
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
Asymptotic safety 1-loop FRGE Matter Conclusions
Graviton contribution to matter η
Asymptotic safety 1-loop FRGE Matter Conclusions
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
)
]
Asymptotic safety 1-loop FRGE Matter Conclusions
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
Asymptotic safety 1-loop FRGE Matter Conclusions
Gravity+scalars
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0.10
0.15
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-60
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-30
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Asymptotic safety 1-loop FRGE Matter Conclusions
Gravity+fermions
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à à à à àà
à
à
à
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-10
-5
5
10
G�*, L�*
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2
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1.0
1.5
2.0
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-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
Ηc
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-0.4
-0.3
-0.2
-0.1
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Asymptotic safety 1-loop FRGE Matter Conclusions
Gravity+vectors
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æ
æ
æ
æ
æææææææææææææææææææææææææææææææææææææææææææææ
10 20 30 40 50NV
0.1
0.2
0.3
0.4
0.5
0.6
0.7G�*
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æ
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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
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æ
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0.5
1.0
1.5
Ηh
æ
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æ
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æ
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-0.88
-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
Asymptotic safety 1-loop FRGE Matter Conclusions
Exclusion plot NV = 0
5 10 15 20 25 30NS
5
10
15
N D
Asymptotic safety 1-loop FRGE Matter Conclusions
Exclusion plot NV = 12
10 20 30 40 50 60 70NS
10
20
30
40ND
Asymptotic safety 1-loop FRGE Matter Conclusions
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
Asymptotic safety 1-loop FRGE Matter Conclusions
Exclusion plot NS = 0
10 20 30 40N D
10
20
30
40
50
NV
Asymptotic safety 1-loop FRGE Matter Conclusions
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
Asymptotic safety 1-loop FRGE Matter Conclusions
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 - - - - -
Asymptotic safety 1-loop FRGE Matter Conclusions
Exclusion plot NV = 12, d = 5
10 20 30 40 50 60NS
5
10
15
20
25
N D
Asymptotic safety 1-loop FRGE Matter Conclusions
Exclusion plot NV = 12, d = 6
10 20 30 40 50 60NS
2468
10N D
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)]
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
)
.
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+ . . .
Asymptotic safety 1-loop FRGE Matter Conclusions
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) ]
Asymptotic safety 1-loop FRGE Matter Conclusions
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
Asymptotic safety 1-loop FRGE Matter Conclusions
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