.
Quantum Gravitational Contributions to
Gauge Theories
.
..
.
.
.
Yong Tang(汤勇)
Institute of Theoretical Physics,
Chinese Academy of Sciences.
中国科学院理论物理研究所
2011-4-1
Y.Tang (ITP-CAS)
Quantum Gravitational Contributions
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Outline
.
.1. Introduction
.
.2. Diagrammatical Calculation
.
.3. Background Field Method
.
.4. Vilkovisky-DeWitt Formalism
.
.5. Summary
Y.Tang (ITP-CAS)
Quantum Gravitational Contributions
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Reference
S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 96,
231601 (2006).
A. R. Pietrykowski, Phys. Rev. Lett. 98, 061801
(2007).
D. J. Toms, Phys. Rev. D 76, 045015 (2007), Phys.
Rev. Lett. 101, 131301(2008), Phys. Rev. D 80,
064040(2009).
D. Ebert, J. Plefka and A. Rodigast, Phys. Lett.
B660, 579(2008).
Y. Tang and Y. L. Wu, Comm. Theo. Phys. 54,
1040(2010)[arXiv:0807.0331v2], arXiv:1012.0626.
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Y.Tang (ITP-CAS)
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Introduction
Introduction
Einstein gravity theory is not renormalizable from
the perspective of quantum field theory.
Quantum effects of gravity become significant only at
Planck scale, and it may be possible to treat it as an
effective field theory at low energy scale.
Robinson and Wilczek claimed that quantum gravity can
correct gauge couplings with power-law running and
render all gauge theories asymptotically free. Later,
further analysis by other authors suggested that the
previous result was gauge dependent.
Y.Tang (ITP-CAS)
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Diagrammatical Calculation
The Lagrangian
.
We begin with the action of Einstein-Yang-Mills theory
.
[
]
∫
.
√
1
1 µα νβ a a
4
S = d x −g 2 R − g g Fµν Fαβ
(1)
κ
4
.
..
.
a
where R is Ricci scalar and Fµν
is the Yang-Mills fields
strength Fµν = ∇µ Aν − ∇ν Aµ − ig[Aµ , Aν√
]. With the
minus-dimension coupling constant κ = 16πG. Usually,
one expands the metric tensor around a background metric
ḡµν and treats graviton field as quantum fluctuation hµν
propagating on the background space-time, ḡµν .
gµν = ḡµν + κhµν , gµν = ḡµν − κhµν + κ2 hµα hαν + ... (2)
√
√
1
1
1
−g =
−ḡ[1 + κh − κ2 (hµν hµν − h2 )...]
(3)
2
4
2
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Diagrammatical Calculation
Gauge-fixing condition
Let us set ḡµν = ηµν , where ηµν is the Minkowski metric.
hµν is interpreted as graviton field, fluctuating in flat
space-time. The lagrangian can be arranged to different
orders of hµν or κ. In the de Donder harmonic gauge
1
χµ = ∂ν hµν − ∂ µ hνν = 0
2
Graviton propagator has a simple form
Pµνρσ
(k) =
G
i νρ µσ
[g g + gµρ gνσ − gµν gρσ ]
k2
(4)
For simplicity, in the following, the metric gµν is
understood as η µν .
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Diagrammatical Calculation
Faddeev-Popov factor
Gauge fixing condition generally is accompanied with
ghost. For instance, gauge condition
χα [gµν ] = 0
introduces ghost with its action as
Lgh = c̄α Qα β cβ ,
(5)
where Qα β is the Faddeev-Popov factor,
Qα β =
δχα
.
δβ
With harmonic gauge, the ghosts do not interact with
other field, we can ignore them.
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Diagrammatical Calculation
β function and Counterterms
Using the traditional Feynman diagram calculations, we
can get the β function by evaluating two and three point
functions of gauge fields. These Green functions are
generally divergent, so counter-terms are needed to
cancel these divergences. The relevant counter-terms to
the β function are
Tµν = iδab Qµν δ2 , Tµνρ = gfabc Vµνρ
qkp δ1
µν
Q ≡ qµ qν − q2 gµν
νρ
µ
ρµ
ν
µν
ρ
Vµνρ
qkp ≡ g (q − k) + g (k − p) + g (p − q)
(6)
The β function is given by
β(g) = gµ
Y.Tang (ITP-CAS)
∂ 3
( δ2 − δ1 )
∂µ 2
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Diagrammatical Calculation
Feynman Diagrams
At one loop level, diagrams are listed below. Also, we
only have to keep the quadratic divergences, because
logarithmic ones will only contribute to high order
operator and then not to β function.
(a)
(c)
Y.Tang (ITP-CAS)
(b)
(d)
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(e)
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Diagrammatical Calculation
Two-point
The two point functions from Fig. 1(a) and Fig. 1(b) are
found in terms of ILIs to be
[
∫
[
]
(a)µν
2
T
= 2κ
dx Qµν I2 + q2 (3x2 − x)I0
]
νρ
µρ
ρσ
µ
ν
µν
2 µν
+q qρ I2 + q qρ I2 − g qρ qσ I2 − q I2 (M2q )
T(b)µν = −3κ2 Qµν I2 (0)
where we have defined
(8)
∫
1
,
2
k
−
M
∫
kµ kν
2
I2µν (M ) ≡
d4 k 2
(k − M2 )2
I2 (M ) ≡
2
Y.Tang (ITP-CAS)
d4 k
2
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(9)
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Diagrammatical Calculation
Three-piont
Three point functions from Fig.1(d) and 1(e) are found,
when keeping only the quadratically divergent terms, to be
{
(d)µνρ
2
T
= igκ − Vµνρ
qkp I2 (0) +
[(
)
∫
ρσ
µσ
µν
νρ
ρ µν
µ νρ
dx g qσ I2 − g qσ I2 + q I2 − q I2 (M2q )
(
)
µσ
µ νρ
ν ρµ
νρ
ρµ
νσ
+ g kσ I2 − g kσ I2 + k I2 − k I2 (M2k )
(
)
]}
ρσ
ρµ
νσ
µν
ν ρµ
ρ µν
2
+ g pσ I2 − g pσ I2 + p I2 − p I2 (Mp )
T(e)µνρ = 3igκ2 Vµνρ
qkp I2 (0)
(10)
with M2q = x(x − 1)q2 . Contraction is performed by using
FeynCalc.
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Diagrammatical Calculation
Regularization schemes
To handle with the divergent loop momentum integral, we
need a regularization scheme. The lesson from quantum
field theory tells us two guide rules.
Regularization
schemes
Cut-off R
Dimensional R
Loop R
Y.Tang (ITP-CAS)
Gauge
invariance
×
√
√
Quantum Gravitational Contributions
Divergence
behaviour
√
×
√
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Diagrammatical Calculation
Cut-off Regularization
Now we shall apply the different regularization schemes
to the divergent ILIs. In cut-off regularization, when
keeping only quadratically divergent terms, one has
1
I2Rµν = gµν I2R ,
4
I2R '
i µν 2
g Λ
16π 2
(11)
The resulting two and three point functions are
(a+b)µν
(a)µν
(b)µν
(d)µνρ
(e)µνρ
Tcutoff
≡ Tcutoff + Tcutoff
[
[
]]
∫
1 i
i
3 i
µν 2
2
2
2
≈ 2Q κ
dx
Λ +
Λ −
Λ
=0
2 16π 2
16π 2
2 16π 2
(d+e)µνρ
≡ Tcutoff + Tcutoff ≈ 0
Tcutoff
which agrees with the result obtained by Ebert et.al.
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Diagrammatical Calculation
Dimensional Regularization
In dimensional regularization, where I2R (0) = 0 and
R
I2µν
= 21 gµν I2R , the two and three point functions are found
to be
∫
(a+b)µν
2 µν
TDR
≈ 4κ Q
dxI2R (M2q ),
(12)
[
∫
(d+e)µνρ
2
dx (gµν qρ − qµ gνρ )I2R (M2q )
TDR
= 2igκ
(13)
]
ν ρµ
νρ µ
R
2
ρµ ν
ρ µν
R
2
+(g k − k g )I2 (Mk ) + (g p − p g )I2 (Mp )
where the regularized quadratic divergence in dimensional
regularization behaves as the logarithmic one
2
−i
M2q [ − γE + 1 + O(ε)]
(14)
I2R (M2q )|DR = −
2
16π
ε
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Diagrammatical Calculation
Loop Regularization
We now make a calculation by using loop regularization.
R
With the consistency condition I2µν
= 12 gµν I2R , we obtain
for two and three point functions
[
∫
3
(a+b)µν
2 µν
dx − I2R (0) + 2I2R (M2q )
TLR
= 2κ Q
2
]
2
2
R
2
+ q (3x − x)I0 (Mq )
(15)
[
∫
1
(d+e)µνρ
2
TLR
= 2igκ
dx Vµνρ
I R (0) + (gµν qρ − qµ gνρ )I2R (M2q )
2 qkp 2
]
ν ρµ
2
ρµ ν
ρ µν
R
2
νρ µ
R
+ (g k − k g )I2 (Mk ) + (g p − p g )I2 (Mp )
(16)
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Diagrammatical Calculation
β function
from which we can directly read off the two-point and
three-point counter-terms δ2κ and δ1κ respectively(only keep
the leading quadratically divergent part):
[
]
[
]
2
2
κ
2 1
2
κ
2 1
2
δ2 = κ
M − µ , δ1 = κ
M −µ
16π 2 c
16π 2 c
we obtain the gravitational corrections to the gauge β
function
∆β κ = −gκ2
Y.Tang (ITP-CAS)
µ2
16π 2
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Diagrammatical Calculation
Running of couplings
gi
1.2
1.0
0.8
0.6
0.4
0.2
Μ
5
10
15
20
Log10
GeV
Figure: An illustration of gravitational contributions to the
running of gauge couplings in the MSSM
Y.Tang (ITP-CAS)
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Background Field Method
Background Field Method
The method is used to calculate the effective action, Γ
[
]
∫
i
i
δΓ i
i
exp Γ[ϕ̄] =
Dµ[ϕ] exp
S[ϕ] − i (ϕ − ϕ̄ )
(18)
~
~
δ ϕ̄
Γ[ϕ̄] = S[ϕ̄] − ln det Qαβ [ϕ̄]
)
(
1
1 i
α
+ ln det S[ϕ̄],ij +
K [ϕ̄]Kj [ϕ̄]
(19)
2
2Ω α
In Einstein-Gauge system,
gµν = ḡµν + κhµν ,
Aµ = Āµ + aµ
(20)
coupling renormalization constant Zg is connected to the
gauge field renormalization constant ZA = 1 + δA with
1/2
Zg ZA = 1 in Background-Field Gauge, then β function is
∂
∂
1
∂
0
βgκ = µ g = µ Z−1
δA
(21)
g g = gµ
∂µ
∂µ
2 ∂µ
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Vilkovisky-DeWitt Formalism
Vilkovisky-DeWitt Formalism
The diagrammatic approach and traditional background
field method share a gauge condition dependent problem.
To overcome the gauge condition problem, we shall us
Vilkovisky-DeWitt Formalism. The effective action is
modified to
[
]
∫
i
i
δ
Γ̂
exp Γ̂[ν i ; ϕ∗ ] = Dµ[ϕ] exp
S[ϕ] − i (σ i (ϕ∗ ; ϕ) − ν i )
~
~
δν
(22)
where ν i ≡ hσ i (ϕ∗ ; ϕ)i and the world function,
1
σ[ϕ? ; ϕ] = (length of geodesic from ϕ? to ϕ)2 .
2
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Vilkovisky-DeWitt Formalism
The metric, Gij[ϕ]
let Gij [ϕ] denote the metric of the field space, then at
one-loop order, the effective action is given by
Γ̂[ϕ̄] = S[ϕ̄] − ln det Qαβ [ϕ̄]
(
)
1
1 ,i
i
α
+
ln det ∇ ∇j S[ϕ̄] +
χ [ϕ̄]χ,j [ϕ̄]
2
2Ω α
(23)
with ∇i ∇j S[ϕ̄] = S,ij [ϕ̄] − Γ̄kij S,k [ϕ̄]. Here the connection Γ̄kij
is determined by Gij [ϕ]. The metric is chosen to be
(
)
1
1 µν ρσ
1
µ(ρ
σ)ν
Ggµν (x)gρσ (x0 ) = 2 |g(x)| 2 g g − g g
δ(x, x0 ) (24)
κ
2
GAµ (x)Aν (x0 ) = |g(x)| 2 gµν (x)δ(x, x0 )
1
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Vilkovisky-DeWitt Formalism
Landau-DeWitt gauge
The Vilkovisky-DeWitt formalism is applicable for any
gauge condition. In Landau-DeWitt gauge, the calculation
is much simpler. Landau-DeWitt gauge conditions (ω = 1)
is determined by the gauge transformation and in U(1)
gauge it reads
2 µ
1
(∂ hµλ − ∂λ h) + ω(Āλ ∂ µ aµ + aµ F̄µλ ),
κ
2
µ
χ = − ∂ aµ .
χλ =
(26)
And in this case we can replace Γ̄kij with Γkij , where Γkij is
the Christoffel connection determined by
∇i Gjk = 0
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Vilkovisky-DeWitt Formalism
Parameter, ω and v
ω is a parameter introduced for a comparison with the
traditional background-field method. When ω = 0, it goes
to harmonic gauge. Also a parameter, v, is introduced for
the connection terms,
∇i ∇j S[ϕ̄] = S,ij [ϕ̄] − vΓkij S,k [ϕ̄]
(27)
At one-loop order with Landau-DeWitt gauge, the effective
action is
Γ[ϕ̄] = S[ϕ̄] − ln det Qαβ [ϕ̄]
(
)
1
1 i
α
i
+ lim ln det ∇ ∇j S[ϕ̄] +
K [ϕ̄]Kj [ϕ̄] . (28)
2 Ω→0
2Ω α
Use
R
I2µν
= a2 gµν I2R
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Vilkovisky-DeWitt Formalism
One-loop correction
The total quadratically divergent one-loop gravitational
contribution to the effective action is given
∫
∫
1
4
2
2
R1
Γ̂[Āµ ] =
d xF̄ + κ CI2
d4 xF̄ 2
(30)
4
4
where the constant C is given by
]
4a2 − 1 ( [
C =
v (2ζ − 1) − 4(κ2 ξ − 1)
8
)
2
+ 8(κ ξ − 1) − 16ω − 4 + ω(−1 + 6a2 )
(31)
obtain the gravitational correction to the β function
1
∂
βgκ = gµ δA ,
2 ∂µ
Y.Tang (ITP-CAS)
δA ' −κ2 CI2R
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(32)
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Vilkovisky-DeWitt Formalism
Results
Note that we used the notation for the quadratic
divergences,
R
I2µν
= a2 gµν I2R .
(33)
For different regularization schemes, a2 is different.
We list the results
C
v = 1, ω = 1,
ζ → 0, ξ → 0
v = 0, ω = 0,
ζ = 1/2, ξ = 1/κ2
Y.Tang (ITP-CAS)
a2 =
1
4
a2 =
1
2
− 98
0
− 12
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1
2
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Vilkovisky-DeWitt Formalism
gi
1.2
1.0
0.8
0.6
0.4
0.2
5
10
15
20
Log10@
Μ
D
GeV
Figure: An illustration of gravitational contributions to the
running of gauge couplings in the MSSM
Y.Tang (ITP-CAS)
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Summary
Summary
We considered the gravitational contributions to the
runnning of gauge couplings.
Both the traditional Diagrammatical and Background
Field Method give the same result that the gravity
tends to render all gauge theories asymptotically
free. However the result is gauge condition
dependent.
We present the gauge condition independent result in
the framework of the Vilkovisky-DeWitt formalism.
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Summary
THANK YOU!
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