The renormalization group
and Weyl invariance
Roberto Percacci
SISSA Trieste
Based in part on: New J. Phys. 13 125013 (2011)
arXiv:1110.6758 [hep-th]
FRGE I: D EFINITIONS
e−Wk [J] =
Z
Z
(Dφ) exp(−(S + ∆Sk +
∆Sk (φ) =
1
2
Z
Jφ))
d 4 qφ(−q)Rk (q 2 )φ(q)
with Rk (q 2 ) → 0 for k → ∞, Rk (q 2 ) → k 2 for k → 0
Z
Γk [φ] = Wk [J] − Jφ − ∆Sk
note limk →0 Γk = Γ
FRGE II
1
∂t Γk = Tr
2
δ 2 Γk
+ Rk
δφδφ
−1
∂t Rk
C. Wetterich, Phys. Lett. B 301 (1993) 90.
Trace is over momenta and all quantum numbers
FRGE III: O NE LOOP
Γ(1) = S +
1
ln det O ;
2
1
(1)
Γk = S+∆Sk + ln det Ok −∆Sk ;
2
(1)
∂t Γk =
O=
Ok =
δ2S
δφδφ
δ 2 (S + ∆Sk )
= O+Rk
δφδφ
1
1
(det Ok )−1 ∂t det Ok = TrOk−1 ∂t Ok
2
2
One–loop RG equation
(1)
∂t Γk
1
= Tr
2
δ2S
+ Rk
δφδφ
−1
∂t Rk
FRGE IV: B ETA FUNCTIONS
Γk (φ) =
X
gi (k )Oi (φ)
i
∂t Γk =
X
∂t gi Oi =
i
X
βgi Oi
i
compare with
1
∂t Γk = Tr
2
δ 2 Γk
+ Rk
δφδφ
−1
∂t Rk
use to calculate Γ
or to investigate the UV properties of the theory
Dimensional analysis
Choose dimensionless coordinates.
Invariance under global Weyl transformations
gµν → Ω2 gµν , ψa → Ωda ψa , gi → Ωdi gi
S(gµν , ψa , gi ) = S(Ω2 gµν , Ωda ψa , Ωdi gi )
fixes da , di . ALWAYS true.
Global scale transformations
S(gµν , ψa , gi ) = S(Ω2 gµν , Ωda ψa , gi )
not always true, must be di = 0.
Weyl’s idea
Interpret scale transformations as changes of the unit of length.
Allow choice of unit to depend on position.
“It is evident that two rods side by side, stationary with respect to
each other, can be intercompared....this cannot be done for....rods
with either a space- or time-like separation”.
“A statement such as “a hydrogen atom on Sirius has the same
diameter as one on the Earth” is either a definition or else
meaningless” (Dicke 1962)
Physics must be formulated in a way that is invariant under
local changes of units, i.e. under local rescalings of the metric.
Furthermore: allow parallel transport to affect norm of vectors.
Weyl gauging
Abelian gauge field bµ 7→ bµ + Ω−1 ∂µ Ω
scalar field φ transforming as φ → Ωd φ
Dµ φ = ∂µ φ − dbµ φ
More generally
Γ̂µ λ ν = Γµ λ ν − δµλ bν − δνλ bµ + gµν bλ
is invariant under local Weyl transformations, hence for a tensor
of dimension d
ˆ µ t − dbµ t
Dµ t = ∇
is diffeomorphism and Weyl covariant.
Weyl curvature
[Dµ , Dν ]v ρ = Rµν ρ σ v σ
Rµνρσ = Rµνρσ + (d − 1)Fµν gρσ
+gµρ (∇ν bσ + bν bσ ) − gµσ (∇ν bρ + bν bρ )
−gνρ (∇µ bσ + bµ bσ ) + gνσ (∇µ bρ + bµ bρ )
− (gµρ gνσ − gµσ gνρ ) b2
Fµν = ∂µ bν − ∂ν bµ
Weyl’s failed unification idea
Weyl regarded bµ as e.m. field.
Einstein’s critique: spectral lines would be blurred.
Weyl’s theory still viable if bµ interpreted as a piece of the
spacetime connection and its curvature is sufficiently weak.
Assume curvature of bµ is zero. Then
bµ = −χ−1 ∂µ χ
where the field χ transforming as χ → Ω−1 χ is....
...the dilaton
Starting from any action S(gµν , ψa , gi ), replace
gi → χdi ĝi , ∇ → D and all curvatures R → R.
Resulting action Ŝ(gµν , χ, ψa , ĝi ) is Weyl invariant and agrees
with original action in the gauge where χ is constant
Ŝ(gµν , χ0 , ψa , ĝi ) = S(gµν , ψa , gi )
Dilaton acts as classical compensator field (Stückelberg)
Motivation: scalar-tensor theory
Z
4
√
d x g
1
(∇φ)2 + V (φ2 ) + F (φ2 )R
2
V (φ2 ) = λ0 + λ2 φ2 + λ4 φ4 + . . .
F (φ2 ) = ξ0 + ξ2 φ2 + . . .
Z
1
4 √
4
2
2
d x g λ4 φ + (∇φ) + ξ2 φ R
2
1
.
Weyl-invariant for ξ2 = 12
Not preserved by RG flow
D. Perini and R.P. Phys. Rev. D68 044018 (2003) hep-th/0304222;
G. Narain and R.P. Cl. Q. Grav. 27 075001 (2010) arXiv:0911.0386 [hep-th])
The conformal anomaly
Cutoff breaks scale invariance, but anomaly can be avoided in
theories with dilaton
Englert, C. Truffin and R. Gastmans, “Conformal invariance in quantum
gravity”, Nucl. Phys. B117, 407 (1976)
R.Floreanini and R. P., “Average effective potential for the conformal factor”,
Nucl. Phys. B436, 141 (1995)
M. Shaposhnikov and I. Tkachev, “Quantum scale invariance on the lattice”,
Phys. Lett. B675, 403 (2009)
Example: matter in background metric and dilaton
1
S(φ, gµν ) =
2
Z
√
d 4 x g φ∆(1/6) φ ,
Z
√
d 4 x g ψ̄Dψ
Z
√
d 4 x g Fµν F µν
S(ψ, gµν ) =
1
S(A, gµν ) =
4
∆(1/6) = − +
R
6
Trace anomaly
2
δΓ
hT µ µ i = √ g µν µν = b C 2 + b0 E
g
δg
E = Rµνρσ R µνρσ − 4Rµν R µν + R 2
C 2 = Cµνρσ C µνρσ
b=
1
(nS + 6nD + 12nM )
120(4π)2
b0 = −
1
(nS + 11nD + 62nM )
360(4π)2
The scalar and fermion measure
(dφ) =
Y dφ(x)
x
Z
Γ(gµν ) =
(dφ)e−
R
µ
√
d 4 x gφ∆(1/6) φ
(dψ) =
1
= ln det
2
Y dψ(x)
x
µ3/2
∆(1/6)
µ2
!
The Maxwell theory measure
(dAµ ) =
Y
dAµ (x)
x
S + SGF
Z
√
1
d 4 x g(∇µ Aµ )2
SGF =
2α
Z
p
1
d 4 x |g| Aµ −∇2 g µν + R µν Aν
=
2
Z
p
Sgh = d 4 x |g|C̄(−∇2 )C
Z
(dAd C̄dC)e−Sem (A,g)−SGF (A,g)−Sgh (C̄,C,g)
!
!
1
log ∆(1)
∆(0)
Tr
− Tr log
2
µ2
µ2
Γ(g) = log
=
The Weyl-invariant scalar measure
Y φ(x) (dφ) =
d
χ(x)
x
Z
Γ(gµν ) =
(dφ)e
−
R
√
d 4 x gφ∆(1/6) φ
1
= ln det
2
1 (1/6)
∆
χ2
1 (1/6)
∆
φ
Ω−2 χ2
χ2
eigenvalues are Weyl invariant → det χ12 ∆(1/6) is Weyl
invariant
1
(1/6)
∆Ω2 g (Ω−1 φ) = Ω−1
Weyl-covariant gauge fixing
SGF
S + SGF
1
=
2
Z
√
d 4 x g(D µ Aµ )2
√
1 d 4 x gχ2 g µν Aµ 2 −D 2 δνσ + Rσν Aσ
χ
1
(−D 2 δνµ + Rµν ) are Weyl
χ2
1
(−D 2 δνµ + Rµν ) is Weyl invariant
χ2
eigenvalues of
→ det
Z
1
=
2α
invariant
Weyl invariant quantization
dilaton acts as compensator (Stückelberg) field in the quantum
effective action
µ has been promoted to a field
Stückelberg trick commutes with quantization
Another point of view
ΓI (g χ ) − ΓI (g) = ΓWZ (g, χ)
Wess-Zumino consistency condition:
ΓWZ (g Ω , χΩ ) − ΓWZ (g, χ) = −ΓWZ (g, Ω)
where g Ω = Ω2 g, χΩ = Ω−1 χ
If we identify ΓI (g) with ΓII (g, χ = µ),
ΓII (g, χ) = ΓI (g) + ΓWZ (g, χ)
Dynamical metric and dilaton
Z
S=
√
1 d x g λZ χ − Z ξχ2 R + g µν ∂µ χ∂ν χ
2
4
2 4
for ξ = 1/6
Z
S=
√
1
2
d x g λZ χ −
Zχ R
12
4
2 4
can choose Weyl gauge (i.e. units) where
Z χ2 =
12
;
16πG
S reduces to Hilbert action.
For ξ 6= 1/6 scalar field is physical.
λ=
2π
GΛ
9
Expansion of action
Let δgµν = hµν , δχ = η.
S (2) =
=
1
Z H((h, η), (h, η))
2
Z
√
1
Z d 4 x g hµν η
2
µνρσ
µν
Hhh
Hhη
ρσ
Hηh
Hηη
!
hρσ
η
For ξ = 1/6
µνρσ
Hhh
"
1 2
1
1
1
= χ − 1µνρσ D 2 + g (ν|σ D |µ) D ρ − g µν D ρ D σ − g ρσ D (µ D ν)
12
2
2
2
1
1
+ g µν g ρσ D 2 − Rµρνσ − g (µ|ρ R|ν)σ + (g µν Rρσ + Rµν g ρσ )
2
2
#
+ R − 12λZ χ2 K µνρσ
µν
µν
Hhη
=Hηh
=
1
1
χ g µν D 2 − D µ D ν + Rµν − Rg µν + 2λZ χ3 g µν
6
2
1
Hηη =D 2 − R + 12λZ χ2 .
6
background Weyl transformations
gµν 7→ Ω2 gµν ;
χ 7→ Ω−1 χ ;
hµν 7→ Ω2 hµν ;
η 7→ Ω−1 η
Wave operators
Z
G((h1 , η1 ), (h2 , η2 )) =
S (2) =
i
√ h
d 4 x g χ4 h1µν g µρ g νσ h2ρσ + χ2 η1 η2
1
1
Z H(θ, θ) = Z G(θ, Oθ)
2
2
αβρσ
(Ohh )µν ρσ = χ−4 gµα gνβ Hhh
,
αβ
(Ohη )µν = χ−4 gµα gνβ Hhη
,
ρσ
ρσ
Oηh
= χ−2 Hηh
,
Oηη = χ−2 Hηη .
Weyl covariance of wave operators
(Ohh(Ω2 gµν ,Ω−1 χ) )µν ρσ (Ω2 hρσ ) = Ω2 (Ohh(gµν ,χ) )µν ρσ hρσ ,
(Ohη(Ω2 gµν ,Ω−1 χ) )µν (Ω−1 η) = Ω2 (Ohη(gµν ,χ) )µν η ,
(Oηh(Ω2 gµν ,Ω−1 χ) )ρσ (Ω2 hρσ ) = Ω−1 (Oηh(gµν ,χ) )ρσ hρσ ,
Oηη(Ω2 gµν ,Ω−1 χ) (Ω−1 η) = Ω−1 Oηη(gµν ,χ) η .
Gauge fixing
SGF =
1
2α
Z
where
√ 1
d 4 x g Z ξχ2 Fµ ḡ µν Fν ,
2
β+1
Dν h
4
C̄, Ogh C .
Fν = Dµ hµ ν −
Sgh = Ggh
Z
√
Ggh (A, B) = d 4 x g χ2 Aµ g µν Bν
(Ogh )νµ = −
1
χ2
Weyl-gauge-fixing η = 0
1−β
δµν D 2 +
Dµ D ν + Rµ ν
2
Weyl invariance of one loop effective action
1
Γ(1) (gµν , χ) = S(gµν , χ) + Tr log Ohh − Tr log Ogh
2
If ∆(gµν ,χ) hµν = λhµν then
∆(Ω2 gµν ,Ω−1 χ) Ω2 hµν = Ω2 ∆(gµν ,χ) hµν = λΩ2 hµν
The spectrum of ∆ is Weyl invariant, so Γ(1) is Weyl invariant.
FRGE
∆Sk
=
1
1 1
1
2
2
Z G h,
Rk (χ Ohh )h + Ggh C̄, 2 Rk (χ Ogh )C
2
12 χ2
χ
∂t Γk =
1
TrF (Ohh ) − TrF (Ogh )
2
r.h.s. of FRGE is Weyl invariant if initial point is Weyl invariant
also Γ is Weyl invariant
Beta functions
Example:
Z
Γk =
dZ
du
dλ
u
du
u
√
1
d 4 x g λZ 2 χ4 −
Z χ2 R
12
=
=
23 2
u
4π 2
u2 2
u
−
184Z
λ
16π 2 Z 2
where u = k /χ is assumed constant
Solutions
Z (u) = Z0 +
λ(u) =
23 2
23 2
u →
u
2
8π
8π 2
π 2 (u 4 + 64π 2 Z02 λ0 )
π2
→
529
(8π 2 Z0 + 23u 2 )2
note: Z is redundant coupling and does not reach a FP,
λ is essential coupling and reaches a FP.
m̃P2 = Z
χ2
Z
23
= 2 →
2
k
u
8π 2
Summary
in presence of a dilaton, it is possible to quantize theory
preserving Weyl invariance.
RG flow also preserves Weyl invariance.
trace anomaly still present
ξ = 1/6 is a fixed point of RG flow.
fixed point of the Einstein-Hilbert formulation appears in
different guise
since all couplings are dimensionless, situation similar to
usual renormalizable QFT
Further work
cutoff and renormalization scale can depend on position.
Theory with nonconstant cutoff equivalent to theory with
constant cutoff but conformally related metric.
relation between f (R) and scalar-tensor theories
generalize to nonflat Weyl connection