QUANTUM GRAVITATIONAL CORRECTION TO SCALAR...
73
QUANTUM GRAVITATIONAL CORRECTION TO SCALAR FIELD EQUATIONS DURING INFLATION By EMRE ONUR KAHYA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 1
Transcript of QUANTUM GRAVITATIONAL CORRECTION TO SCALAR...
QUANTUM GRAVITATIONAL CORRECTION TO SCALAR FIELD EQUATIONSDURING INFLATION
By
EMRE ONUR KAHYA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2008
1
c© 2008 Emre Onur Kahya
2
To my dearest mother
3
ACKNOWLEDGMENTS
I am indebted to my advisor, Professor Richard Woodard. He is a very hardworking,
meticulous and a dedicated person as probably many know. But he also is a teacher who
really cares about his students. He would not hesitate to sacrifice his time and effort for
the benefit of his students. Generosity and self-discipline are two words that come into my
mind when I think of him. I hope that one day his contributions to the Bessel Functions
will be acknowledged.
I would like to thank my mother, Hurmet Kahya; my father, Orbay Kahya; and my
sister, Hilal Kahya, for their constant support. I would like to thank my wife, Selva; and
my daughter, Hafsa Asude, for making my life enjoyable.
I also would like to thank Professor Pierre Sikivie and Professor Atalay Karasu for
writing letters of recommendation on my behalf.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 FEYNMAN RULES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 ONE LOOP SELF-MASS-SQUARED . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 Contributions from the 4-Point Vertices . . . . . . . . . . . . . . . . . . . . 163.2 Contributions from the 3-Point Vertices . . . . . . . . . . . . . . . . . . . . 19
3.2.1 Local Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2 Logarithm Contributions . . . . . . . . . . . . . . . . . . . . . . . . 223.2.3 Normal Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 RENORMALIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 EFFECTIVE MODE EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.1 The Schwinger-Keldysh Formalism . . . . . . . . . . . . . . . . . . . . . . 445.2 Restrictions on Our Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 475.3 Local Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.4 Nonlocal Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
APPENDIX
A EXTRACTING DERIVATIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5
LIST OF TABLES
Table page
2-1 Three-Point Vertex Operators V αβI contracted into φ1φ2 hαβ. . . . . . . . . . . . 12
2-2 Four-Point Vertex Operators UαβρσI contracted into φ1φ2hαβhρσ. . . . . . . . . . 12
3-1 Nonlocal Logarithm Contributions from relation (3–37) with x ≡ y4. . . . . . . . 27
3-2 Divergent Normal Contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3-3 Finite Normal Contributions in terms of x ≡ y4. . . . . . . . . . . . . . . . . . . 33
4-1 Local Normal Contributions from Table 3-2. . . . . . . . . . . . . . . . . . . . . 39
4-2 Finite Normal Contributions from Table 3-2 with x = y4. . . . . . . . . . . . . . 40
4-3 Normal Contributions to Counterterms from Table 4-1. All terms are multiplied
by iκ2HD−4
(4π)D2
Γ(D2
)
(D−3)(D−4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4-4 Other Local Normal Contributions from Table 4-1. . . . . . . . . . . . . . . . . 42
4-5 All Finite Nonlocal Contributions with x ≡ y4. . . . . . . . . . . . . . . . . . . . 43
5-1 Integrals with a′3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5-2 Integrals with a′4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5-3 Integrals with a′5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5-4 The −u(0,k)a4
iH8
(4π)4
∫d4x′ (Ext. Operator)× f(y++
4)− f(y+−
4) terms. . . . . . . . 55
A-1 Contributions acted upon by α = (aa′)D¤2. . . . . . . . . . . . . . . . . . . . . 65
A-2 Contributions acted upon by β = (aa′)D−2(a2 + a′2)H2¤. . . . . . . . . . . . . . 66
A-3 Contributions acted upon by γ1 = (aa′)DH4. . . . . . . . . . . . . . . . . . . . . 66
A-4 Contributions acted upon by γ2 = (aa′)D−1(a2 + a′2)H4. . . . . . . . . . . . . . 66
A-5 Contributions acted upon by γ3 = (aa′)D−1(a + a′)2H4. . . . . . . . . . . . . . . 67
A-6 Contributions acted upon by δ = (aa′)D−2(a2 + a′2)∇2¤. . . . . . . . . . . . . . 67
A-7 Contributions acted upon by ε1 = (aa′)D−1H2∇2. . . . . . . . . . . . . . . . . . 68
A-8 Contributions acted upon by ε2 = (aa′)D−2(a2 + a′2)H2∇2. . . . . . . . . . . . . 68
A-9 Contributions acted upon by ε3 = (aa′)D−2(a + a′)2H2∇2. . . . . . . . . . . . . 69
A-10 Contributions acted upon by ζ = (aa′)D−2∇4. . . . . . . . . . . . . . . . . . . . 69
6
LIST OF FIGURES
Figure page
3-1 Contribution from 4-point vertices. . . . . . . . . . . . . . . . . . . . . . . . . . 16
3-2 Contribution from two 3-point vertices. . . . . . . . . . . . . . . . . . . . . . . . 19
4-1 Contribution from counterterms. . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
QUANTUM GRAVITATIONAL CORRECTION TO SCALAR FIELD EQUATIONSDURING INFLATION
By
Emre Onur Kahya
August 2008
Chair: Richard P. WoodardMajor: Physics
We computed the one loop corrections from quantum gravity to the self-mass-squared
of a massless, minimally coupled scalar on a locally de Sitter background. The calculation
was done using dimensional regularization and renormalized by subtracting fourth order
BPHZ counterterms. We used this computation of the self-mass-squared from quantum
gravity to include quantum corrections to the scalar evolution equation. The plane wave
mode functions are shown to receive no significant one loop corrections at late times. This
result probably applies as well to the inflaton of scalar-driven inflation. If so, there is no
significant correction to the ϕϕ correlator that plays a crucial role in computations of the
power spectrum.
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CHAPTER 1INTRODUCTION
My study was understanding quantum loop effects to massless minimally coupled
scalars during inflation. To do that I computed the self-mass-squared of massless
minimally coupled scalars coupled with gravitons at one-loop order. Then I used that
to solve the quantum corrected equations of motion in order to see whether this one-loop
effect alters the dynamics of a scalar inflaton.
Quantum loop effects can be understood as the reaction of classical field theory to
virtual particles. The energy-time uncertainty principle tells us that the number density
of these virtual particles, which persist forever during primordial inflation, is enhanced[1]
for the ones which are massless and not conformally invariant. Massless, minimally
coupled scalars and gravitons are the only two particles which have zero mass and are
not conformally invariant. As a result of inflation quantum effects for these particles are
vastly enhanced which is the origin of the primordial scalar [2] and tensor [3] perturbations
predicted by inflation [4, 5]. Weinberg has recently shown that loop corrections to these
perturbations are also enhanced by logarithms of the ratio of the scale factor to its value
at first horizon crossing, although not enough to make them observable [6, 7]. However,
much larger loop corrections (to other things) can be obtained either from interactions
with larger loop counting parameters, or by studying things we would perceive as spatially
constant such as the vacuum energy or particle masses.
Explicit computations have been made on de Sitter background in different models
which involve either scalars or gravitons. A massless, minimally coupled scalar with a
quartic self-interaction is pushed up its potential by inflationary particle production,
thereby inducing a violation of the weak energy condition [8, 9] and a nonzero scalar
mass [10, 11]. The vacuum polarization from a charged, massless, minimally coupled
scalar induces a nonzero photon mass [12, 13] and a small negative shift in the vacuum
energy [14]. The inflationary creation of massless, minimally coupled scalars which are
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Yukawa-coupled to a massless fermion gives the fermion mass [15, 16] and induces a
negative vacuum energy that grows without bound [17, 18].
It is natural to extend these studies by combining general relativity with a massless,
minimally coupled scalar to probe the effects of gravitons on scalars. Such an investigation
has great phenomenological interest because the inflaton potential of scalar-driven
inflation is so flat that inflaton mode functions are effectively those of a massless,
minimally coupled scalar. Indeed, it is standard to compute the power spectrum of scalar
perturbations by setting the scalar-scalar correlator to its value for a massless, minimally
coupled scalar on de Sitter background [4, 5]. The subject of quantum gravitational
corrections to the scalar effective potential has a long history [19, 20] but we will here
look at all one loop corrections to the linearized, effective scalar field equation, including
corrections to the derivative terms and also the fully nonlocal corrections. This is what
one must do to fix the scalar field strength in addition to its mass.
Our result is that the scalar mode functions experience no significant corrections at
one loop. We prove this by solving the linearized, effective scalar field equation. We derive
the Feynman Rules of MMCS+GR in chapter 2. Chapter 3 gives a review of one loop
computation of the scalar self-mass-squared [21], which represents the quantum correction
to the linearized effective field equation. In chapter 4 we extract finite terms from the
one-loop results using BPHZ renormalization scheme. In chapter 5 we solve the equation
in the relevant regime of late times. Our conclusions comprise chapter 6.
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CHAPTER 2FEYNMAN RULES
To facilitate dimensional regularization we work in D spacetime dimensions. Our
Lagrangian is,
L ≡ −1
2∂µφ ∂µφgµν
√−g +1
16πG
(R− (D−2)Λ
)√−g . (2–1)
Here G is Newton’s constant and Λ ≡ (D− 1)H2 is the cosmological constant. Because our
scalar is a spectator to Λ-driven inflation, its background value is zero. Our background
geometry is the conformal coordinate patch of D-dimensional de Sitter space,
ds2 = a2(−dη2 + d~x · d~x
)where a(η) = − 1
Hη. (2–2)
Perturbation theory is expressed using the graviton field hµν(x),
gµν(x) ≡ a2(ηµν + κhµν(x)
)where κ2 ≡ 16πG . (2–3)
The inverse metric and the volume element have the following expansions,
gµν =1
a2
(ηµν − κhµν + κ2hµ
ρhρν − . . .
), (2–4)
√−g = aD(1 +
1
2κh +
1
8κ2h2 − 1
4κ2hρσhρσ + . . .
). (2–5)
This computation requires the φ2h and φ2h2 interactions which derive from expanding the
scalar kinetic term,
−1
2∂µφ∂νg
µν√−g = −1
2∂µφ∂νφaD−2
ηµν − κhµν +
1
2ηµνκh +
1
8ηµνκ2h2
−1
4ηµνκ2hρσhρσ − 1
2κ2hhµν + κ2hµρhν
ρ + O(κ3)
. (2–6)
We represent the 3-point and 4-point interaction terms as vertex operators acting on
the fields. For example, the first of the 3-point vertices is,
− 1
2κaD−2∂αφ∂βφhαβ =⇒ V αβ
1 = iκaD−2∂α1 ∂β
2 . (2–7)
11
I V αβI
1 iκ aD−2 ∂α1 ∂β
2
2 − i2κ aD−2 ηαβ∂1 ·∂2
Table 2-1. Three-Point Vertex Operators V αβI contracted into φ1φ2 hαβ.
I UαβρσI
1 − i8κ2aD−2ηαβηρσ∂1 ·∂2
2 i4κ2aD−2ηαρηβσ∂1 ·∂2
3 i2κ2 aD−2ηαβ∂ρ
1∂σ2
4 −iκ2 aD−2∂α1 ηβρ∂σ
2
Table 2-2. Four-Point Vertex Operators UαβρσI contracted into φ1φ2hαβhρσ.
We number the fields “1”, “2”, “3”, etc, starting with the two scalars and proceeding
to the gravitons. Although we extract a factor of 12
for the two identical scalars, it is
more efficient, for our computation, not to extract a similar factor of 12
for the identical
gravitons of the 4-point vertices. Then we can dispense with the symmetry factor. So our
first 4-point vertex is,
− κ2
16aD−2∂µφ ∂µφ h2 =⇒ Uαβρσ
1 = − i
8κ2aD−2ηαβηρσ∂1 ·∂2 . (2–8)
The 3-point vertices are listed in Table 2-1; Table 2-2 gives the 4-point vertices.
Three notational conventions will simplify our discussion of propagators. The first is
to denote the background geometry with a hat,
gµν = a2ηµν , gµν =1
a2ηµν ,
√−g = aD and R = D(D−1)H2 . (2–9)
Second, because time and space are treated differently in the gauge we shall employ, it
is useful to have expressions for the purely spatial parts of the Lorentz metric and the
Kronecker delta,
ηµν ≡ ηµν + δ0µδ
0ν and δ
µ
ν ≡ δµν − δµ
0 δ0ν . (2–10)
12
Finally, the various propagators have simple expressions in terms of y(x; x′), a function of
the de Sitter invariant length `(x; x′) from xµ to x′µ,
y(x; x′) = 4 sin2(1
2H`(x; x′)
)= aa′H2
‖~x−~x′‖2 −
(|η−η′|−iδ
)2, (2–11)
where a ≡ a(η) and a′ ≡ a(η′).
The massless minimally coupled scalar propagator obeys,
∂µ
(√−g gµν∂ν
)i∆A(x; x′) = i δD(x− x′). (2–12)
It has long been known that there is no de Sitter invariant solution [22]. The de-Sitter
breaking solution which is relevant for cosmology is the one which preserves homogeneity
and isotropy. This is known as the “E(3)” vacuum [23], and the minimal solution takes the
form [8, 9],
i∆A(x; x′) = A(y) + k ln(aa′) where k ≡ HD−2
(4π)D2
Γ(D−1)
Γ(D2)
. (2–13)
The de Sitter invariant function A(y) is [9],
A(y) ≡ HD−2
(4π)D2
Γ(D
2−1)
D2−1
(4
y
)D2−1
+Γ(D
2+1)
D2−2
(4
y
)D2−2
−π cot(πD
2
)Γ(D−1)
Γ(D2)
+∞∑
n=1
[1
n
Γ(n+D−1)
Γ(n+ D2)
(y
4
)n
− 1
n−D2+2
Γ(n+ D2+1)
Γ(n+2)
(y
4
)n−D2
+2]
. (2–14)
To get the graviton propagator, we add the following gauge fixing term to the
invariant Lagrangian [24],
LGF = −1
2aD−2ηµνFµFν , Fµ ≡ ηρσ
(hµρ,σ − 1
2hρσ,µ + (D−2)Hahµρδ
0σ
). (2–15)
13
We can partially integrate the quadratic part of the gauge fixed Lagrangian to put it in
the form 12hµνD ρσ
µν hρσ, where the kinetic operator is,
D ρσµν ≡
1
2δ
(ρ
µ δσ)
ν − 1
4ηµνη
ρσ − 1
2(D−3)δ0µδ
0νδ
ρ0δ
σ0
DA
+δ0(µδ
(ρ
ν)δσ)0 DB +
1
2
(D−2
D−3
)δ0µδ
0νδ
ρ0δ
σ0 DC , (2–16)
The three scalar differential operators are,
DA ≡ ∂µ
(√−g gµν∂ν
), (2–17)
DB ≡ ∂µ
(√−g gµν∂ν
)− 1
D
(D−2
D−1
)R
√−g , (2–18)
DC ≡ ∂µ
(√−g gµν∂ν
)− 2
D
(D−3
D−1
)R
√−g . (2–19)
The graviton propagator in this gauge has the form of a sum of constant tensor
factors times scalar propagators,
i[
µν∆ρσ
](x; x′) =
∑I=A,B,C
[µνT
Iρσ
]i∆I(x; x′) . (2–20)
We can get the scalar propagators by inverting the scalar kinetic operators,
DI × i∆I(x; x′) = iδD(x− x′) for I = A,B, C . (2–21)
The tensor factors are,
[µνT
Aρσ
]= 2 ηµ(ρησ)ν −
2
D−3ηµνηρσ , (2–22)
[µνT
Bρσ
]= −4δ0
(µην)(ρδ0σ) , (2–23)
[µνT
Cρσ
]=
2
(D−2)(D−3)
[(D−3)δ0
µδ0ν + ηµν
][(D−3)δ0
ρδ0σ + ηρσ
]. (2–24)
With these definitions and equation (2–21) we can see that the graviton propagator
satisfies the following equation,
D ρσµν × i
[ρσ∆αβ
](x; x′) = δ(α
µ δβ)ν iδD(x− x′) . (2–25)
14
The most singular part of the scalar propagator is the propagator for a massless,
conformally coupled scalar [25],
i∆cf(x; x′) =HD−2
(4π)D2
Γ(D
2−1
)(4
y
)D2−1
. (2–26)
The A-type propagator obeys the same equation as that of a massless, minimally coupled
scalar. The de Sitter invariant B-type and C-type propagators are,
i∆B(x; x′) = i∆cf(x; x′)− HD−2
(4π)D2
∞∑n=0
Γ(n+D−2)
Γ(n+ D2)
(y
4
)n
−Γ(n+ D2)
Γ(n+2)
(y
4
)n−D2
+2
, (2–27)
i∆C(x; x′) = i∆cf(x; x′) +HD−2
(4π)D2
∞∑n=0
(n+1)
Γ(n+D−3)
Γ(n+ D2)
(y
4
)n
−(n−D
2+3
)Γ(n+ D2−1)
Γ(n+2)
(y
4
)n−D2
+2
. (2–28)
They can also be expressed as hypergeometric functions [26, 27],
i∆B(x; x′) =HD−2
(4π)D2
Γ(D−2)Γ(1)
Γ(D2)
2F1
(D−2, 1;
D
2; 1− y
4
), (2–29)
i∆C(x; x′) =HD−2
(4π)D2
Γ(D−3)Γ(2)
Γ(D2)
2F1
(D−3, 2;
D
2; 1− y
4
). (2–30)
These propagators might look complicated but they are actually simple to use since the
sums vanish in D = 4, and every term in these sums goes like a positive power of y(x; x′).
Therefore only a small number of terms in the sums can contribute when multiplied by a
fixed divergence.
15
CHAPTER 3ONE LOOP SELF-MASS-SQUARED
This is the heart of the paper. We first evaluate the contribution from the 4-point
vertices of Table 2-2. Then we compute the vastly more difficult contributions from
products of two 3-point vertices from Table 2-1. We do not renormalize at this stage,
although we do take D = 4 in finite terms. Renormalization is postponed until the next
section.
Figure 3-1. Contribution from 4-point vertices.
3.1 Contributions from the 4-Point Vertices
The generic diagram topology is depicted in Fig. 3-1. The analytic form is,
− iM24pt(x; x′) =
4∑I=1
UαβρσI × i
[αβ∆ρσ
](x; x) δD(x−x′) . (3–1)
In reading off the various contributions from Table 2-2 one should note that, whereas “∂2”
acts upon x′µ, the derivative operator “∂1” must be partially integrated back onto the
entire contribution. For example, the contribution from Uαβρσ1 is,
− i
8κ2aD−2ηαβηρσ∂1 ·∂2 × i
[αβ∆ρσ
](x; x) δD(x−x′)
=⇒ +i
8κ2∂µ
aD−2i
[αα∆ρ
ρ
](x; x)∂′µδ
D(x−x′)
. (3–2)
16
Reading off the other terms from Table 2-2 gives,
−iM24pt(x; x′) = − i
8κ2∂µ
aD−2i
[αα∆ρ
ρ
](x; x)∂µδ
D(x−x′)
+
i
4κ2∂µ
aD−2
×i[
αβ∆αβ
](x; x)∂µδ
D(x−x′)
+
i
2κ2∂ρ
aD−2i
[αα∆ρσ
](x; x)∂σδ
D(x−x′)
−iκ2∂α
aD−2i
[αρ∆
ρσ](x; x)∂σδ
D(x−x′)
. (3–3)
It is apparent from expression (3–3) that we require the coincidence limits of each of
the three scalar propagators [28],
limx′→x
i∆A(x; x′) =HD−2
(4π)D2
Γ(D−1)
Γ(D2)
−π cot
(π
2D
)+ 2 ln(a)
, (3–4)
limx′→x
i∆B(x; x′) =HD−2
(4π)D2
Γ(D−1)
Γ(D2)
×− 1
D−2−→ − H2
16π2, (3–5)
limx′→x
i∆C(x; x′) =HD−2
(4π)D2
Γ(D−1)
Γ(D2)
× 1
(D−2)(D−3)−→ H2
16π2. (3–6)
Note that the B-type and C-type propagators are finite for D = 4. The four contractions
of the coincident graviton propagator we require are [28],
i[
αα∆ρ
ρ
](x; x) −→ −4
(D−1
D−3
)i∆A(x; x) + 4× H2
16π2, (3–7)
i[
αβ∆αβ
](x; x) −→ (D−1)(D2−3D−2)
D−3i∆A(x; x)− 2× H2
16π2, (3–8)
i[
αα∆ρσ
](x; x) −→ − 4
D−3ηρσi∆A(x; x) +
[2δρ
0δσ0 + 2ηρσ
]× H2
16π2, (3–9)
i[
αρ∆ρσ
](x; x) −→
(D2−3D−2
D−3
)δ
σ
αi∆A(x; x) + 2δ0αδσ
0 ×H2
16π2. (3–10)
To save space we have taken D = 4 in the finite contributions from the B-type and C-type
propagators.
17
Substituting these relations into expression (3–3) and performing some trivial algebra
gives the final result,
−iM24pt(x; x′) =
iκ2HD−2
(4π)D2
Γ(D−1)
Γ(D2)
×−π cot(π
2D
)1
4D(D−1)
×∂µ(aD−2∂µδ
D(x−x′))−DaD−2∇2δD(x−x′)
+iκ2H2
4π2
3∂µ
(a2 ln(a)∂µδ
4(x−x′))− 4 ln(a)a2∇2δ4(x−x′)
−∂µ(a2∂µδ
4(x−x′))
+ a2∇2δ4(x−x′)
+ O(D−4) . (3–11)
Note that each of these terms vanishes in the flat space limit of H → 0 with the comoving
time t ≡ ln(a)/H held fixed. The reason for this is that the coincidence limit of the flat
space graviton propagator vanishes in dimensional regularization.
In order to combine −iM24pt with the 3-point contributions it is useful to introduce
notation for the scalar d’Alembertian in de Sitter background,
¤ ≡ 1√−g
∂µ
(√−g gµν∂ν
)=
1
aD∂µ
(aD−2∂µ
). (3–12)
We also extract the logarithm from inside the d’Alembertian,
∂µ(a2 ln(a)∂µδ
4(x−x′))
=1
2ln(aa′)a4¤δ4(x−x′) +
3
2H2a4δ4(x−x′) . (3–13)
With these conventions the final result takes the form,
−iM24pt(x; x′) =
iκ2HD−2
(4π)D2
Γ(D−1)
Γ(D2)
[−1
4D(D−1)π cot
(π
2D
)− 2
+3 ln(aa′)]aD¤ +
[Dπ cot
(π
2D
)+2−4 ln(aa′)
]aD−2∇2
+9H2aD + O(D−4)
δD(x−x′) . (3–14)
18
Figure 3-2. Contribution from two 3-point vertices.
3.2 Contributions from the 3-Point Vertices
In this section we calculate the contributions from two 3-point vertex operators. It
is diagrammatically represented in Fig. 3-2. Consulting Table 2-1 and remembering to
partially integrate any derivative that acts upon an outer leg gives,
−iM23pt(x; x′) =
2∑I=1
V αβI (x)
2∑J=1
V ρσJ (x′)× i
[αβ∆ρσ
](x; x′) i∆A(x; x′) , (3–15)
= −κ2∂α∂′ρ
(aa′)D−2i[
αβ∆ρσ]∂β∂′σi∆A
+
κ2
2∂µ∂′ρ
(aa′)D−2
×i[
αα∆ρσ
]∂µ∂
′σi∆A
+
κ2
2∂α∂
′ν
(aa′)D−2i[
αβ∆ρρ
]∂β∂′νi∆A
−κ2
4∂µ∂
′ν
(aa′)D−2i[
αα∆ρ
ρ
]∂µ∂
′νi∆A
. (3–16)
Upon substituting the graviton propagator, performing the contractions and segregating
terms with the same scalar propagators, one finds three generic sorts of terms. The first
are those which involve two A-type propagators,
κ2∇·∇′[(aa′)D−2 i∆A∇·∇′ i∆A
]− κ2
(D−1
D−3
)∂0∂
′0
[(aa′)D−2 i∆A ∂0∂
′0 i∆A
]
+κ2∂i∂′0
[(aa′)D−2 i∆A ∂i∂
′0 i∆A
]+ κ2∂0∂
′i
[(aa′)D−2 i∆A ∂0∂
′i i∆A
]. (3–17)
The second kind of term involves one A-type and one B-type propagator,
− κ2∂0∂′0
[(aa′)D−2 i∆B∇·∇′ i∆A
]− κ2∂i∂
′0
[(aa′)D−2 i∆B ∂0∂
′i i∆A
]
−κ2∂0∂′i
[(aa′)D−2 i∆B ∂i∂
′0 i∆A
]− κ2∇·∇′
[(aa′)D−2 i∆B ∂0∂
′0 i∆A
]. (3–18)
19
Finally, there is the case of one propagator of A-type and the other of C-type,
2κ2(D−2
D−3
)∂0∂
′0
[(aa′)D−2 i∆C ∂0∂
′0 i∆A
]. (3–19)
Each of the nine terms in expressions (3–17-3–19) has the form,
κ2∂µ∂′ν
[(aa′)D−2i∆I(x; x′)∂ρ∂
′σi∆A(x; x′)
], (3–20)
where “I” might be A, B or C. Note that the three propagators can be written almost
entirely as functions of y(x; x′) defined in (2–11),
i∆A(x; x′) = A(y) + k ln(aa′) , i∆B(x; x′) = B(y) and i∆C(x; x′) = C(y) . (3–21)
The functions A(y), B(y) and C(y) can be read off from expressions (2–14), (2–27) and
(2–28), respectively. Note also that the inner derivatives eliminate the de Sitter breaking
term of i∆A,
∂ρ∂′σi∆A(x; x′) = δ0
ρδ0σ
i
aD−2δD(x−x′) + A′′(y)
∂y
∂xρ
∂y
∂x′σ+ A′(y)
∂2y
∂xρ∂x′σ. (3–22)
It follows that the analysis breaks up into three parts:
• Local contributions from the delta function in (3–22);
• Logarithm contributions from the factor of k ln(aa′) in the A-type propagator when
I = A in expression (3–20); and
• Normal contributions to expression (3–20) of the form,
κ2∂µ∂′ν
(aa′)D−2I(y)
[A′′ ∂y
∂xρ
∂y
∂x′σ+ A′ ∂2y
∂xρ∂x′σ
]. (3–23)
We shall devote a separate part of this subsection to each.
3.2.1 Local Contributions
These are the simplest contributions. They only come from the 2nd term of (3–17),
the 4th term of (3–18) and from (3–19). To avoid overlap with the logarithm contributions
of the next part we define the local contribution from the 4th term of (3–17) without the
20
logarithm,
−κ2(D−1
D−3
)∂0∂
′0
[(aa′)D−2A(y)× i
aD−2δD(x−x′)
]=
iκ2HD−2
(4π)D2
Γ(D)
(D−3)Γ(D2)
×− π cot(D
2π)
−aD¤δD(x−x′) + aD−2∇2δD(x−x′)
. (3–24)
Note that we have chosen to convert primed derivatives into unprimed, and to absorb the
temporal derivatives into a covariant d’Alembertian ¤,
− ∂0
(aD−2∂′0δ
D(x−x′))
=−∂µ(aD−2∂µδ
D(x−x′))
+aD−2∇2δD(x−x′) , (3–25)
≡−aD¤δD(x−x′) + aD−2∇2δD(x−x′) . (3–26)
This will facilitate renormalization.
The other two local contributions are finite. The 4th term of (3–18) gives,
−κ2∇·∇′[(aa′)D−2B(y)× i
aD−2δD(x−x′)
]
= −iκ2H2
16π2× a2∇2δ4(x−x′) + O(D−4) . (3–27)
And (3–19) gives,
2κ2(D−2
D−3
)∂0∂
′0
[(aa′)D−2C(y)× i
aD−2δD(x−x′)
]
=iκ2H2
4π2
a4¤δ4(x−x′)− a2∇2δ4(x−x′)
+ O(D−4) . (3–28)
Summing the three local contributions gives,
−iM23ptloc
(x; x′) =iκ2HD−2
(4π)D2
Γ(D−1)
Γ(D2)
[(D−1
D−3
)π cot
(π
2D
)+ 2
]aD¤
+[−
(D−1
D−3
)π cot
(π
2D
)− 7
2
]aD−2∇2 + O(D−4)
δD(x−x′) . (3–29)
21
3.2.2 Logarithm Contributions
These all come from expression (3–17). They can be simplified by using the
propagator equation (2–12),
∂0
(aD−2∂0i∆A(x; x′)
)= −iδD(x−x′) + aD−2∇2A(y) , (3–30)
∂′0(a′D−2∂′0i∆A(x; x′)
)= −iδD(x−x′) + a
′D−2∇2A(y) . (3–31)
One can also take the limit D = 4 because all the logarithm contributions are finite. For
example, the function A(y) is,
A(y) =H2
16π2
4
y− 2 ln
(y
4
)− 1 + O(D−4)
. (3–32)
The first term of (3–17) gives,
κ2∇·∇′[(aa′)D−2 × k ln(aa′)×∇·∇′i∆A(x; x′)
]
=κ2H2
8π2ln(aa′)(aa′)2∇4A(y) + O(D−4) . (3–33)
The second term of (3–17) has the most complicated reduction,
−κ2(D−1
D−3
)∂0∂
′0
[(aa′)D−2 × k ln(aa′)× ∂0∂
′0i∆A(x; x′)
]
=i3κ2H2
8π2ln(aa′)
−a4¤ + 2a2∇2
δ4(x−x′)
−3κ2H2
8π2ln(aa′)(aa′)2∇4A(y)− i9κ2H4
8π2a4δ4(x−x′)
−3κ2H3
8π2(aa′)2(a∂0+a′∂′0)∇2A(y) + O(D−4) . (3–34)
The third term of (3–17) gives,
κ2∂i∂′0
[(aa′)D−2 × k ln(aa′)× ∂i∂
′0i∆A(x; x′)
]
= −iκ2H2
8π2ln(aa′)a2∇2δ4(x−x′) +
κ2H2
8π2ln(aa′)(aa′)2∇4A(y)
+κ2H3
8π2(aa′)2a′∂′0∇2A(y) + O(D−4) . (3–35)
22
A very similar contribution derives from the final term of (3–17),
κ2∂0∂′i
[(aa′)D−2 × k ln(aa′)× ∂0∂
′ii∆A(x; x′)
]
= −iκ2H2
8π2ln(aa′)a2∇2δ4(x−x′) +
κ2H2
8π2ln(aa′)(aa′)2∇4A(y)
+κ2H3
8π2(aa′)2a∂0∇2A(y) + O(D−4) . (3–36)
Combining all four terms results in some significant cancellations,
−iM23ptlog
(x; x′) =κ2H2
8π2
ln(aa′)
[−3a4¤ + 4a2∇2
]iδ4(x−x′)
−9H2a4iδ4(x−x′)− 2H(aa′)2(a∂0+a′∂′0)∇2A(y) + O(D−4)
. (3–37)
Each of the local terms in (3–37) cancels a similar finite, local 4-point contribution in
(3–14), leaving only the nonlocal contribution involving derivatives of A(y). It is possible
to eliminate the temporal derivatives in this expression. However, the procedure is best
explained in the final part of this section.
3.2.3 Normal Contributions
These contributions are the most challenging. Our strategy for reducing them is to
first extract the ∂ρ and ∂′σ derivatives from (3–23) generically, without exploiting the
functional forms of A(y), B(y) and C(y). We also convert all primed derivatives into
unprimed ones and express the final result in terms of ten “External Operators”. This not
only makes it possible to perceive general relations, it also reduces the superficial degree
of divergence of the terms we must eventually expand. And it leaves functions of the de
Sitter invariant variable y(x; x′) for which an improved expansion procedure is possible
[29].
This step of extracting derivatives is still quite involved so we shall describe only the
essentials in the body of the thesis and consign the details to an appendix. The appendix
also gives tabulated results for each of the ten External Operators. The final reduction of
these generic tabulated results is straightforward. This subsection closes with a description
23
of the technique and a pair of tables giving the final potentially divergent and manifestly
finite contributions, respectively.
Our generic method for extracting derivatives requires one to carry out many
indefinite integrations of functions of y. We define this operation by the symbol I[f ](y),
I[f ](y) ≡∫ y
dy′f(y′) . (3–38)
If the function F (y) is the product of two propagator functions, then acting two
derivatives on it can never produce a delta function,
∂ρ∂′σF (y) = F ′′(y)
∂y
∂xρ
∂y
∂x′σ+ F ′(y)
∂2y
∂xρ∂x′σ. (3–39)
It follows that we can express the inner part of the basic normal contribution (3–23) in
terms of integrals of such products,
f(y)
A′′(y)
∂y
∂xρ
∂y
∂x′σ+A′(y)
∂2y
∂xρ∂x′σ
=∂ρ∂
′σI
2[fA′′](y)+∂2y
∂xρ∂x′σI[f ′A′](y) . (3–40)
We must still deal with the final term of (3–40). In conformal coordinates the mixed
second derivative of y(x; x′) is [30],
∂2y
∂xρ∂x′σ= H2aa′
yδ0
ρδ0σ − 2aδ0
ρH∆xσ + 2a′H∆xρδ0σ − 2ηρσ
. (3–41)
Breaking this up into spatial and temporal components gives,
∂2y
∂x0∂x′0=H2aa′
[2−y+2aa′H2‖∆~x‖2
],
∂2y
∂x0∂x′j=H2aa′×−2aH∆xj , (3–42)
∂2y
∂xi∂x′0=H2aa′ × 2a′H∆xi ,
∂2y
∂xi∂x′j=H2aa′×−2ηij . (3–43)
One consequence is,
aa′H2‖∆~x‖2f(y) = −1
2(D−1)I[f ](y)− ∇ · ∇′
4aa′H2I2[f ](y) . (3–44)
24
Another consequence is the relations,1
f(y)∂0∂′0A(y) = ∂0∂
′0I
2[fA′′](y)− 1
2∇·∇′I3[f ′A′](y)
+H2aa′
(2−y)I[f ′A′](y)− (D−1)I2[f ′A′](y)
, (3–45)
f(y)∂0∂′jA(y) = ∂0∂
′jI
2[fA′′](y) + Ha∂′jI2[f ′A′](y) , (3–46)
f(y)∂i∂′0A(y) = ∂i∂
′0I
2[fA′′](y) + Ha′∂iI2[f ′A′](y) , (3–47)
f(y)∂i∂′jA(y) = ∂i∂
′jI
2[fA′′](y)− 2H2aa′ηijI[f ′A′](y) . (3–48)
Using these identities it is possible to extract the derivatives from the first of the
A-terms,
∇·∇′[(aa′)D−2A(y)∇·∇′A(y)
]
= (aa′)D−2(∇·∇′)2I2[AA′′](y)− 2(D−1)H2(aa′)D−1∇·∇′I[A′2](y) , (3–49)
= (aa′)D−2∇4I2[AA′′](y) + 2(D−1)H2(aa′)D−1∇2I[A′2](y) . (3–50)
Only the first term in the expansion of I2[AA′′](y) contributes a divergence; we can set
D =4 in the higher terms. Similarly, only the first two terms in the expansion of I[A′2](y)
can diverge.
It is very simple to convert the primed spatial derivatives to unprimed ones,
∂′if(y) = −∂if(y) . (3–51)
1 On the left hand side of relation (3–45) we mean the naive second derivative, withoutthe delta function.
25
We already used this relation in reducing the first of the A-terms. For time derivatives it
is useful to note,
∂y
∂x0= Ha
(y − 2a′H∆η
)= Ha
(y − 2 + 2
a′
a
), (3–52)
∂y
∂x′0= Ha′
(y + 2aH∆η
)= Ha′
(y − 2 + 2
a
a′
). (3–53)
From this follow three important identities. The simple one is,
(∂0 + ∂′0
)f(y) = H(a+a′)yf ′(y) . (3–54)
Another result is,
(a′∂0 + a∂′0
)f(y) = 2Haa′ × aa′H2‖∆~x‖2f ′(y) , (3–55)
= −(D−1)Haa′f(y) +∇2
2HI[f ](y) . (3–56)
The final identity results from combining (3–54) and (3–56),
(a∂0 + a′∂′0
)f(y) = (a+a′)(∂0+∂′0)f(y)− (a′∂0+a∂′0)f(y) , (3–57)
= H(a+a′)2yf ′(y)+(D−1)Haa′f(y)−∇2
2HI[f ](y) . (3–58)
We can now reduce the nonlocal logarithm contribution from equation (3–37).
Applying (3–58) gives,
κ2H2
8π2×−2H(aa′)2(a∂0+a′∂′0)∇2A(y) =
κ2H2
16π2
−12(aa′)3H2∇2A
−4(aa′)2(a+a′)2∇2(yA′) + 2(aa′)2∇4I[A]
. (3–59)
The derivative and the integral are straightforward using the D = 4 expansion for A(y)
given in (3–32). The final result is reported in Table 3-1. Of course we have neglected
terms which eventually vanish such as ∇4y.
26
External Operator Coefficient of κ2H4
(4π)4
(aa′)3H2∇2 −12x
+ 24 ln x
(aa′)2(a + a′)2H2∇2 4x
(aa′)2∇4 8 ln x− 16x ln x
Table 3-1. Nonlocal Logarithm Contributions from relation (3–37) with x ≡ y4.
We eventually want to absorb all double time derivatives into covariant d’Alembertian’s,
¤ = − 1
a2∂2
0 −(D−2)H
a∂0 +
1
a2∇2 . (3–60)
This is most effectively done with the internal factors of (aa′)D−2. For example, consider
reducing one of the mixed A-terms,
∂0∂′i
[(aa′)D−2A(y)∂0∂
′iA(y)
]
= ∇′2∂0
[(aa′)D−2∂0I
2[AA′′](y)]
+ H∇′2∂0
[aD−1a
′D−2I2[A′2](y)
], (3–61)
= −aDa′D−2∇2¤I2[AA′′](y) + (aa′)D−2∇4I2[AA′′](y)
+HaD−1a′D−2∇2∂0I
2[A′2](y) + (D−1)H2aDa
′D−2∇2I2[A′2](y) . (3–62)
Note also that we can convert a primed covariant d’Alembertian to an unprimed one if it
acts on a function of just y(x; x′),
¤f(y) = H2[(4y−y2)f ′′(y) + D(2−y)f ′(y)
]= ¤′f(y) . (3–63)
27
This is used in reducing the other mixed A-term,
∂i∂′0
[(aa′)D−2A(y)∂i∂
′0A(y)
]
= ∇2∂′0[(aa′)D−2∂′0I
2[AA′′](y)]
+ H∇2∂′0[aD−2a
′D−1I2[A′2](y)
], (3–64)
= −aD−2a′D∇2¤′I2[AA′′](y) + (aa′)D−2∇4I2[AA′′](y)
+HaD−2a′D−1∇2∂′0I
2[A′2](y) + (D−1)H2aD−2a
′D∇2I2[A′2](y) , (3–65)
= −aD−2a′D∇2¤I2[AA′′](y) + (aa′)D−2∇4I2[AA′′](y)
−HaD−3a′D∇2∂0I
2[A′2](y) +
1
2aD−3a
′D−1∇4I3[A′2](y) . (3–66)
The previous point can be summarized by the relations,
∂0
[(aa′)D−2∂0f(y)
]= −aDa
′D−2¤f(y) + (aa′)D−2∇2f(y) , (3–67)
∂′0[(aa′)D−2∂′0f(y)
]= −aD−2a
′D¤f(y) + (aa′)D−2∇2f(y) . (3–68)
Another important point is that it is almost always best to write any single factor of the
mixed product ∂0∂′0 as follows,
∂0∂′0 =
1
2(∂0 + ∂′0)
2 − 1
2∂2
0 −1
2∂′20 . (3–69)
So we find the ubiquitous reduction,
∂0∂′0
[(aa′)D−2f(y)
]= (aa′)D−2∂0∂
′0f(y)
+(D−2)H(aa′)D−2(a′∂0+a∂′0)f(y) + (D−2)2H2(aa′)D−1f(y) , (3–70)
=1
2(aa′)D−2(a2+a
′2)[¤f(y)+H2yf ′(y)
]− (aa′)D−2∇2f(y)
+1
4(D−2)(aa′)D−2∇2I[f ](y) +
1
2(D−2)(D−3)H2(aa′)D−1f(y)
+1
2H2(a+a′)2(aa′)D−2
[(D−1)yf ′(y)+y2f ′′(y)
]. (3–71)
28
Another example is the two B-terms,
∂i∂′0
[(aa′)D−2B(y)∂0∂
′iA(y)
]+ ∂0∂
′i
[(aa′)D−2B(y)∂i∂
′0A(y)
]
= ∂i(∂0+∂′0)[(aa′)D−2B(y)(∂0+∂′0)∂
′iA(y)
]
−∂i∂0
[(aa′)D−2B(y)∂0∂
′iA(y)
]− ∂i∂
′0
[(aa′)D−2B(y)∂′0∂
′iA(y)
], (3–72)
= −(a2+a′2)(aa′)D−2∇2
¤I2[A′′B](y) + H2I[A′B+yA′′B](y)
+(aa′)D−2∇4
2I2[A′′B](y)− 1
2I3[A′B′](y)
−H2(a+a′)2(aa′)D−2∇2
×
(D−2)I[A′B+yA′′B](y)+yA′(y)B(y) + y2A′′(y)B(y)
−yI[A′B′](y)
+ (D−1)H2(a2+aa′+a′2)(aa′)D−2∇2I2[A′B′](y) . (3–73)
Extracting derivatives in this way from the various normal contributions results in
functions of y which are acted upon by ten external operators,
α ≡ (aa′)D¤2 , (3–74)
β ≡ (aa′)D−1(a2 + a′2)H2¤ , (3–75)
γ1 ≡ (aa′)DH4 , (3–76)
γ2 ≡ (aa′)D−1(a2 + a′2)H4 , (3–77)
γ3 ≡ (aa′)D−1(a + a′)2H4 = 2γ1 + γ2 , (3–78)
δ ≡ (aa′)D−2(a2 + a′2)∇2¤ , (3–79)
ε1 ≡ (aa′)D−1H2∇2 , (3–80)
ε2 ≡ (aa′)D−2(a2 + a′2)H2∇2 , (3–81)
ε3 ≡ (aa′)D−2(a + a′)2H2∇2 = 2ε1 + ε2 , (3–82)
ζ ≡ (aa′)D−2∇4 . (3–83)
Tables A-1-A-10 of the Appendix give explicit results for each of these ten operators. Note
that in addition to the three propagator functions A(y), B(y) and C(y), we also employ
29
the following less singular differences:
∆B ≡ B − A and ∆C ≡ 2(D − 2
D − 3
)(C − A) . (3–84)
The next step is substituting the explicit forms (2–14), (2–27), (2–28) for the
propagator functions into the results of Tables A-1-A-10 and expanding to the required
order. To understand what this is, note that we will be integrating the result with respect
to x′µ against a smooth function (the zeroth order mode solution) with the derivatives
of the “External Operators” acted outside the integrals. Because y(x; x′) vanishes like
(x − x′)2 at coincidence, it is only necessary to retain the dimensional regularization for
terms which would go like 1/y2 and higher for D = 4.
Although these tables involve a bewildering variety of different integrals and
derivatives, careful examination of the results shows that they derive from just eight
products of the propagator functions,
A′2 , AA′′ , A′B′ , A′′B , A′∆B′ , A′′∆B , A′∆C ′ and A′′∆C . (3–85)
The most singular products of A′2 and AA′′ always appear either doubly integrated —
e.g., I2[AA′′] in Table A-1 — or else integrated once and then multiplied by y — e.g.,
−12yI[A
′2] in Table A-2. Hence we need only retain the dimensional regularization for the
1/yD terms of these expansions,
A′2 =
Γ2(D2)
16
H2D−4
(4π)D
(4
y
)D
+ 4(4
y
)3
+ 4(4
y
)2
+ O(D−4
y3
), (3–86)
AA′′ =Γ2(D
2)
16
H2D−4
(4π)D
D
D−2
(4
y
)D
− 4(4
y
)3
ln(y
4
)
−4(4
y
)2
ln(y
4
)− 2
(4
y
)2
+ O(D−4
y3
). (3–87)
The product A′B′ can appear with only a single integration — e.g., DI[A′B′] in Table A-2
— or multiplied by a single factor of y — e.g., DyA′B′ in Table A-4. We must therefore
30
retain the dimensional regularization for the 1/yD−1 term,
A′B′ =Γ2(D
2)
16
H2D−4
(4π)D
(4
y
)D
+ (D−2)(4
y
)D−1
+ O(D−4
y2
). (3–88)
However, the product A′′B is always shielded by two or more powers of y, so the
expansion we require for it is,
A′′B =Γ2(D
2)
16
H2D−4
(4π)D
D
D−2
(4
y
)D
+ 2(4
y
)3
+ O(D−4
y3
). (3–89)
The products involving ∆B and ∆C are less singular,
A′∆B′ =Γ2(D
2)
16
H2D−4
(4π)D
−2
(4
y
)D−1
− 4(4
y
)2
+ O(D−4
y2
), (3–90)
A′′∆B =Γ2(D
2)
16
H2D−4
(4π)D
4(4
y
)3
ln(y
4
)+ 2
(4
y
)3
+4(4
y
)2
ln(y
4
)+ 2
(4
y
)2
+ O(D−4
y3
), (3–91)
A′∆C ′ =Γ2(D
2)
16
H2D−4
(4π)D
−8
(4
y
)D−1
− 16(4
y
)2
+ O(D−4
y2
), (3–92)
A′′∆C =Γ2(D
2)
16
H2D−4
(4π)D
16
(4
y
)3
ln(y
4
)+ 8
(4
y
)3
+16(4
y
)2
ln(y
4
)+ 8
(4
y
)2
+ O(D−4
y3
). (3–93)
One next substititues these expansions into the totals of Tables A-1-A-10 and
performs the necessary integrations, differentiations, multiplications and summations.
We must also multiply by the overall factor of κ2. For example, the result for “External
31
Ext. Op. Coef. of κ2H2D−4
(4π)D Γ2(D2)( 4
y)D−1 Coef. of κ2H2D−4
(4π)D Γ2(D2)( 4
y)D−2
α 0 D(D−1)(D−2)2
β − D4(D−1)
−D3−3D2−4D+84(D−1)(D−2)
γ1 −D(D−2)4
−D3−3D2−4D+84
γ2D4
D3−3D2−4D+84(D−1)
γ3 −D4
0
δ 0 0
ε1 0 (D2−6D+4)2(D−1)(D−2)
ε2 0 (1−2D)(D−1)(D−2)
ε3 0 0
ζ 0 0
Table 3-2. Divergent Normal Contributions.
Operator” α is,
κ2
I2[AA′′] + I2[A′′∆C]
=Γ2(D
2)
16
κ2H2D−4
(4π)DI2
[D
D−2
(4
y
)D
+ 12(4
y
)3
ln(y
4
)+ 8
(4
y
)3
+12(4
y
)2
ln(y
4
)+ 6
(4
y
)2
+ O(D−4
y3
)], (3–94)
=κ2H2D−4
(4π)DΓ2
(D
2
)D
(D−1)(D−2)2
(4
y
)D−2
+ 6(4
y
)ln
(y
4
)+ 13
(4
y
)
−6 ln2(y
4
)− 18 ln
(y
4
)+ O
(D−4
y
). (3–95)
We have tabulated the results for each of the ten “External Operators”. Table 3-2 gives
the quadratically and logarithmically divergent terms; Table 3-3 gives the terms which are
manifestly finite. In all cases the expressions were worked out by hand and then checked
with Mathematica [31].
32
Ext. Op. Coefficient of κ2H4
(4π)4
α 6 ln xx
+ 13x− 6 ln2 x− 18 ln x
β 5x− 18 ln x
γ130x− 108 ln x− 36
γ2 − 5x− 18
γ3 − 5x− 36
δ −4 ln xx− 49
6x+ 4 ln2 x + 10 ln x + 12x ln x
ε1 − 263x
+ 60 ln x− 120x ln x + 72x
ε2136x− 12 ln x + 12x ln x
ε3 − 116x
+ 36x ln x + 12x
ζ 10 ln x3
− 24x ln x + 24x2 ln x− 36x2
Table 3-3. Finite Normal Contributions in terms of x ≡ y4.
33
CHAPTER 4RENORMALIZATION
In this section we obtain a completely finite result for the self-mass-squared
by subtracting 4th-order BPHZ counterterms [32]. We first identify two invariant
counterterms which can contribute to this 1PI (One Particle Irreducible) function at
one loop. Because our gauge fixing functional (2–15) breaks de Sitter invariance [24], we
must also consider noninvariant counterterms. We identify the only possible candidate
based on a careful discussion of the residual symmetries of our gauge fixing functional. It
remains to collect and compute the actual divergences. Contributions from the 4-point
vertices are already local, as are the “local contributions” from the 3-point vertices. Using
a now standard technique of partial integration [8] we segregate the divergences from the
“normal contributions” of Table 3-2. In the end we identify the divergent parts of the
three counterterms and report a completely finite result.
Figure 4-1. Contribution from counterterms.
One renormalizes the scalar self-mass-squared by subtracting diagrams of the form
depicted in Fig. 4-1. Because our scalar-graviton interactions have the form κnhn∂φ∂φ,
compared to the κnhn∂h∂h interactions of pure gravity, the superficial degree of divergence
at one loop order is four, the same as that of pure quantum gravity. Of course the
corresponding counterterms must contain two scalar fields, each of which has the
dimension of a mass. Because we are dealing with one loop corrections from quantum
gravity, all these counterterms must also carry a factor of the loop counting parameter
κ2 = 16πG, which has the dimension of an inverse mass-squared. Each counterterm must
therefore have an additional mass dimension of four, either in the form of explicit masses
34
or else as derivatives. The term with no derivatives is,
κ2m4φ2√−g . (4–1)
There is no way to obtain an invariant with one derivative. Two derivatives can act either
on the scalars or on the metric to produce a curvature. We can take the distinct terms to
be,
κ2m2∂µφ∂νφgµν√−g and κ2m2φ2R
√−g . (4–2)
There are no invariants with three derivatives. By judicious partial integration and use of
the Bianchi identity we can take the distinct terms with four derivatives to be,
κ2φ;µνφ;ρσgµνgρσ
√−g , κ2∂µφ∂νφRgµν√−g , κ2∂µφ∂νφRµν
√−g ,
κ2φ2R2√−g and κ2φ2RµνRµν
√−g . (4–3)
Because our scalar is massless and mass is multiplicatively renormalized in dimensional
regularization, we can dispense with (4–1) and (4–2). The last two counterterms of
(4–3) cannot occur because the unrenormalized Lagragnian (2–1) is invariant under
φ → φ + const. The second and third terms of (4–3) become degenerate when one uses
the background equation, Rµν = (D−1)H2gµν . In the end just two independent invariant
counterterms survive, each with its own coefficient,
1
2α1κ
2¤φ¤φaD and − 1
2α2κ
2H2∂µφ∂µφaD−2 . (4–4)
The associated vertices are,
1
2α1κ
2¤φ¤φaD −→ iα1κ2aD¤2δD(x−x′) , (4–5)
−1
2α2κ
2H2∂µφ∂µφaD−2 −→ iα2κ2H2aD¤δD(x−x′) . (4–6)
Had our gauge condition respected de Sitter invariance, all the divergences in
−iM2(x; x′) could have been absorbed using (4–5) and (4–6) with appropriate choices
for the divergent parts of the coefficients α1 and α2. Although the reasons for it are not
35
completely understood, there seems to be an obstacle to adding a de Sitter invariant gauge
fixing functional [24, 33, 34]. This is why we employed the noninvariant functional (2–15).
We must therefore describe how de Sitter transformations act in our conformal coordinate
system and which subgroup of them is respected by our gauge condition. The 12D(D+1)
de Sitter transformations can be decomposed as follows:
• Spatial translations — (D−1) distinct transformations.
η′ = η , x′i = xi + εi . (4–7)
• Rotations — 12(D−1)(D−2) distinct transformations.
η′ = η , x′i = Rijxj . (4–8)
• Dilatation — 1 distinct transformation.
η′ = k η , x′i = k xi . (4–9)
• Spatial special conformal transformations — (D−1) distinct transformations.
η′ =η
1−2~θ·~x+‖~θ‖2x·x, x′i =
xi − θix·x1−2~θ·~x+‖~θ‖2x·x
. (4–10)
It turns out that our gauge choice breaks only spatial special conformal transformations
(4–10) [18]. Hence we can use the other symmetries to restrict possible noninvariant
counterterms. Spatial translational invariance means that there can be no dependence
upon xi except through the fields. Rotational invariance implies that spatial indices
on derivatives must be contracted into one another. Dilatation invariance implies that
derivatives and the conformal time η can only occur in the form a−1∂µ.
We can always use the invariant counterterms (4–5-4–6) to absorb a ∂20 in favor of ∇2
and a single ∂0,
¤ =1
a2
[−∂2
0 − (D−2)Ha∂0 +∇2]
=⇒ 1
a2∂2
0 = −¤− (D−2)H
a∂0 +
∇2
a2. (4–11)
36
We can also avoid (∂0ϕ)2,
∂µϕ∂νϕgµν = −(∂0ϕ)2
a2+
1
a2∇ϕ · ∇ϕ =⇒ (∂0ϕ)2
a2= −∂µϕ∂νϕgµν +
1
a2∇ϕ · ∇ϕ . (4–12)
One might think we need HaD−1∂0ϕ¤ϕ, but a partial integration allows it to be written in
terms of an invariant counterterm and one with purely spatial derivatives,
HaD−1∂0ϕ¤ϕ −→ −HaD−3∂µ∂0ϕ∂νϕηµν −H2aD−2(∂0ϕ)2 , (4–13)
= −1
2HaD−3∂0(∂µϕ∂νϕ)ηµν + H2∂µϕ∂νϕgµν
√−g −H2aD−2∇ϕ·∇ϕ , (4–14)
−→ 1
2(D − 1)H2∂µϕ∂νϕgµν
√−g −H2aD−2∇ϕ · ∇ϕ . (4–15)
Another term one might consider is HaD−3∂0ϕ∇2ϕ, but it can be partially integrated
(twice) to give purely spatial derivatives,
HaD−3∂0ϕ∇2ϕ −→ −HaD−3∂0∇ϕ · ∇ϕ , (4–16)
= −1
2HaD−3∂0(∇ϕ · ∇ϕ) , (4–17)
−→ 1
2(D − 3)H2aD−2∇ϕ · ∇ϕ . (4–18)
Based on these considerations we conclude that only three noninvariant counterterms
might be needed in addition to the two invariant ones,
1
2κ2aD−2¤ϕ∇2ϕ ,
1
2κ2aD−4∇2ϕ∇2ϕ and − 1
2κ2H2aD−2∇ϕ · ∇ϕ . (4–19)
Because our gauge fixing term (2–15) becomes Poincare invariant in the flat space limit of
H → 0 with the comoving time held fixed, any noninvariant counterterm must vanish in
this limit. Hence we require only the final term of (4–19). The vertex it gives is,
− 1
2α3κ
2H2aD∇a
ϕ · ∇a
ϕ −→ iα3κ2H2aD−2∇2δD(x−x′) . (4–20)
The structure of the three possible counterterms serves to guide our further reduction
of −iM2(x; x′). First, we must convert all the factors of a′ into a on the local terms.
Second, we see that factors of HaD−3∇2∂0δD(x − x′) are not possible. Finally, it is not
37
possible to get a divergence proportional to H3aD−1∂0δD(x−x′) after using the delta
function to convert all the factors of a′ into factors of a.
It is now time to collect the divergent terms from the previous two sections. Those
from the 4-point contributions, and from the “local” 3-point contributions are already in a
form which can be absorbed into the three counterterms. However, we must still bring the
“normal” 3-point contributions of Table 3-2 to this form. Recall that these terms involve
powers of y that are not integrable for D = 4 dimensions,
(4
y
)D−1
and(4
y
)D−2
. (4–21)
Our procedure is to extract d’Alembertians from these terms until they become integrable
using the identity,
¤f(y) = H2[(4y−y2)f ′′(y) + D(2−y)f ′(y)
]
+Res[y
D2−2f
]× 4π
D2 H2−D
Γ(D2−1)
i√−gδD(x−x′) . (4–22)
Here Res[F ] stands for the residue of F (y); that is, the coefficient of 1/y in the Laurent
expansion of the function F (y) around y = 0.
The key identity (4–22) allows us to extract a covariant d‘Alembertian from each of
the nonintegrable terms,
(4
y
)D−1
=2
(D−2)2
¤H2
(4
y
)D−2
− 2
D−2
(4
y
)D−2
, (4–23)
(4
y
)D−2
=2
(D−3)(D−4)
¤H2
(4
y
)D−3
− 4
D−4
(4
y
)D−3
. (4–24)
We could use (4–24) on (4–23) to reduce them both to the power 1/yD−3. The power
1/yD−3 is integrable, so we could take D = 4 at this point were it not for the explicit
factors of 1/(D−4).
To segregate the divergence on the local term we add zero in the form,
0 =¤H2
(4
y
)D2−1
− D
2
(D
2−1
)(4
y
)D2−1
− (4π)D2 H−D
Γ(D2−1)
i
aDδD(x−x′) . (4–25)
38
External Operator Coef. of iκ2HD−4
(4π)D/2
Γ(D2
)
(D−3)(D−4)δD(x−x′)
aD
α D(D−1)(D−2)
β − D2(D−1)(D−2)
¤H2 − (D+2)(D−4)
4
γ1 − D2
2(D−2)¤H2 − (D4−5D3+16D−16)
4
γ2(D−2)(D3−3D2−4D+8)
4(D−1)
δ 0
ε1(D2−6D+4)
2(D−1)
ε2(1−2D)(D−1)
ζ 0
Table 4-1. Local Normal Contributions from Table 3-2.
Using (4–25) in (4–24) gives,
(4
y
)D−2
=2
(D−3)(D−4)
(4π)
D2 H−D
Γ(D2−1)
iδD(x−x′)aD
+¤H2
[(4
y
)D−3
−(4
y
)D2−1
]
− 4
D−4
(4
y
)D−3
− D(D−2)
8(D−3)
(4
y
)D2−1
, (4–26)
=iH−D(4π)
D2
(D−3)(D−4)Γ(D2)× (D−2)
δD(x−x′)aD
− ¤H2
4
yln
(y
4
)
+2(4
y
)ln
(y
4
)−
(4
y
)+ O(D−4) . (4–27)
The analogous result for the quadratically divergent term is,
(4
y
)D−1
=iH−D(4π)
D2
(D−3)(D−4)Γ(D2)
2
D−2
¤H2
−2
δD(x−x′)
aD− 1
2
¤2
H4
4
yln
(y
4
)
+¤H2
2(4
y
)ln
(y
4
)− 1
2
(4
y
)−2
(4
y
)ln
(y
4
)+
(4
y
)+O(D−4) . (4–28)
The divergent local terms that result from applying (4–27) and (4–28) to Table 3-2 are
reported in Table 4-1. Table 4-2 gives the corresponding finite terms. In each case we have
eliminated the redundant External Operators γ3 = 2γ1 + γ2 and ε3 = 2ε1 + ε2.
The next step is to reexpress the local terms of Table 4-1 as local counterterms.
This is done by using the delta function to convert all factors of a′ from the External
39
External Operator Coefficient of iκ2H4
(4π)4
α ¤H2 [− ln x
3x] + 2 ln x
3x− 1
3x
β ¤2
H4 [ln x6x
] + ¤H2 [− ln x
3x+ 1
6x]
γ1¤2
H4 [2 ln x
x] + ¤
H2 [−6 ln xx
+ 2x] + 4 ln x
x− 2
x
γ2¤H2 [−2 ln x
3x] + 4 ln x
3x− 2
3x
δ 0
ε1¤H2 [
ln x3x
]− 2 ln x3x
+ 13x
ε2¤H2 [
7 ln x6x
]− 7 ln x3x
+ 76x
ζ 0
Table 4-2. Finite Normal Contributions from Table 3-2 with x = y4.
Operators into factors of a, and then passing all factors of a to the left. In most cases this
is straightforward but β ¤H2 and β require the following identities:
(aa′)D−1(a2 + a′2)¤2[a−DδD(x−x′)
]=
[2aD¤2 − 12H2aD¤
+8aD−2H2∇2 + 2(D2−2D+2)H4aD]δD(x−x′) , (4–29)
(aa′)D−1(a2+a′2)H2¤[a−DδD(x−x′)
]= 2aD(H2¤−H4)δD(x−x′) . (4–30)
Our results for the three possible counterterms (4–5), (4–6) and (4–20) are reported
in Table 4-3. Note that the contribution to (4–5) vanishes, as it must because this
counterterm happens to be zero in flat space.
Another important consistency check comes from the local terms proportional to
iκ2H4aDδD(x−x′), which are reported in Table 4-4. Recall that a counterterm of this form
is forbidden by the symmetry φ → φ + const of the bare Lagrangian (2–1). Although three
of the four contributions to Table 4-4 diverge, their sum is finite for D = 4. It doesn’t
vanish because the A-type propagator equation implies,
iaDδD(x−x′) = (aa′)D
¤A(y)− (D−1)kH2
. (4–31)
Because the total for Table 4-4 is finite one can take D = 4 and then use (4–31) to
subsume the result into finite, nonlocal terms of the same form as have already been
40
reported in Table 3-3,
Table 4− 4 =iκ2HD−4
(4π)D2
(−D4+5D3−16D+16)Γ(D2)
4(D−2)(D−3)× aDH4δD(x−x′) , (4–32)
−→ κ2H4
8π2× ia4δ4(x−x′) , (4–33)
=κ2H4
(4π)4
(aa′)4H2¤
[2× 4
y− 4 ln
(y
4
)]− (aa′)4H4 × 12
. (4–34)
Table 4-5 includes this with the similarly finite results of Tables 3-1, 3-3 and 4-2.
Our final result for the regulated but unrenormalized, one loop self-mass-squared
derives from combining expressions (3–14), (3–29), and the local parts of (3–37), with
Tables 4-3 and 4-5. It takes the form,
− iM2reg(x; x′) = iκ2aD
(β1¤2 + β2¤ + β3
∇2
a2
)δD(x−x′) + Table 4− 5 + O(D−4) . (4–35)
The coefficients βi are
β1 = 0 , (4–36)
β2 =HD−4
(4π)D2
(−D3+D−4)Γ(D
2−1)
4(D−1)(D−3)− (D+1)(D−4)Γ(D)π cot(π
2D)
4(D−3)Γ(D2)
, (4–37)
=HD−4
(4π)D2
−61
3+ O(D−4)
, (4–38)
β3 =HD−4
(4π)D2
− 4
D−4+
58
3+ 2γ + O(D−4)
. (4–39)
(Here γ ∼ .577215 is Euler’s constant.) The obvious renormalization convention is to
choose each of the three αi’s to absorb the corresponding βi, leaving an arbitrary finite
term ∆αi,
αi = −βi + ∆αi . (4–40)
We can now take the unregulated limit (D = 4) to obtain the final renormalized result,
− iM2ren(x; x′) = iκ2a4
(∆α1¤2 + ∆α2¤ + ∆α3
∇2
a2
)δ4(x−x′) + Table 4− 5 . (4–41)
41
From aD¤2δD(x−x′) aDH2¤δD(x−x′) aD−2H2∇2δD(x−x′)
α D(D−1)(D−2)
0 0
β ¤H2 − D
(D−1)(D−2)6D
(D−1)(D−2)− 4D
(D−1)(D−2)
β 0 − (D+2)(D−4)2
0
γ1¤H2 0 − D2
2(D−2)0
ε1 0 0 (D2−6D+4)2(D−1)
ε2 0 0 (2−4D)(D−1)
Total 0 (D−4)(−D3+D−4)2(D−1)(D−2)
(D3−16D2+28D−16)2(D−1)(D−2)
Table 4-3. Normal Contributions to Counterterms from Table 4-1. All terms are multiplied
by iκ2HD−4
(4π)D2
Γ(D2
)
(D−3)(D−4).
Contrib. from Coef. of iκ2HD−4
(4π)D2
Γ(D2
)
(D−3)(D−4)× aDH4δD(x−x′)
β ¤H2 −D(D2−2D+2)
(D−1)(D−2)
β (D+2)(D−4)2
γ1 − (D4−5D3+16D−16)4
γ2(D−2)(D3−3D2−4D+8)
2(D−1)
Total (D−4)(−D4+5D3−16D+16)4(D−2)
Table 4-4. Other Local Normal Contributions from Table 4-1.
42
External Operator Coefficient of κ2H4
(4π)4
(aa′)4¤3/H2 − ln x3x
(aa′)4¤2 26 ln x3x
+ 383x− 6 ln2 x− 18 ln x
(aa′)4H2¤ −6 ln xx
+ 4x− 4 ln x
(aa′)4H4 4 ln xx
+ 18x− 120− 108 ln x
(aa′)3(a2 + a′2)¤3/H2 ln x6x
(aa′)3(a2 + a′2)¤2 − ln x3x
+ 16x
(aa′)3(a2 + a′2)H2¤ −2 ln x3x
+ 5x− 18 ln x
(aa′)3(a2 + a′2)H4 4 ln x3x
− 323x− 54
(aa′)3H2∇2 −2 ln x3x
− 16x
+ 84 ln x− 48x ln x + 96x
(aa′)3∇2¤ ln x3x
(aa′)2(a2 + a′2)H2∇2 −7 ln x3x
+ 112x− 12 ln x + 48x ln x + 12x
(aa′)2(a2 + a′2)∇2¤ −17 ln x6x
− 496x
+ 4 ln2 x + 10 ln x + 12x ln x
(aa′)2∇4 103
ln x− 24x ln x + 24x2 ln x− 36x2
Table 4-5. All Finite Nonlocal Contributions with x ≡ y4.
43
CHAPTER 5EFFECTIVE MODE EQUATION
This is the heart of the paper. We begin by clarifying what is meant by the effective
mode equation, then we explain the restricted sense in which we solve it. Finally we work
out the contributions from the local counterterms in (4–41) and from the nonlocal terms of
Table 4-5.
5.1 The Schwinger-Keldysh Formalism
We seek to find plane wave mode solutions to “the effective field equations.”
Although the quoted phrase is common parlance, it is nonetheless ambiguous because
there are different sorts of effective field equations whose solutions mean different things
in terms of the unique canonical operator formalism. Introductory courses in quantum
field theory typically concern the in-out effective field equations. The plane wave mode
solutions of these equations give in-out matrix elements of commutators of the full field
with a tree order creation operator of the in vacuum,
Φio(x;~k) =⟨Ωout
∣∣∣[ϕ(x), α†in(~k)
]∣∣∣Ωin
⟩. (5–1)
This quantity is of great interest for flat space scattering problems but it has little
relevance to cosmology where there may be an initial singularity and where particle
production precludes the in vacuum from evolving to the out vacuum.
The more interesting cosmological experiment is to release the universe from a
prepared state at finite time and let it evolve as it will. The mode solutions of interest to
this experiment are the expectation values of commutators of the full field with the tree
order creation operator of the initial vacuum,
Φ(x;~k) =⟨Ω
∣∣∣[ϕ(x), α†(~k)
]∣∣∣Ω⟩
. (5–2)
The effective field equation that Φ(x;~k) obeys is given by the Schwinger-Keldysh
formalism [18, 30]. This is a covariant covariant extension of Feynman diagrams which
44
produces true expectation values instead of in-out matrix elements [35–38]. Because there
are excellent reviews on this subject [39–42], we will confine ourselves to explaining how to
use the formalism.
The chief difference between the Schwinger-Keldysh and in-out formalisms is that
the endpoints of particle lines have a ± polarity. Therefore, every propagator i∆(x; x′)
of the in-out formalism gives rise to four Schwinger-Keldysh propagators: i∆++(x; x′),
i∆+−(x; x′), i∆−+(x; x′) and i∆−−(x; x′). Each of these propagators can be obtained
by making simple changes to the Feynman propagator. For our model, the Feynman
propagators of the scalar and graviton happen to depend upon the length function y(x; x′)
defined in expression (2–11), and also upon the two scale factors. The four polarities
derive from making the following substitutions for y(x; x′):
i∆++(x; x′) : y → y++(x; x′) ≡ a(η)a(η′)[‖~x−~x′‖2 − (|η−η′|−iδ)2
], (5–3)
i∆+−(x; x′) : y → y+−(x; x′) ≡ a(η)a(η′)[‖~x−~x′‖2 − (η−η′+iδ)2
], (5–4)
i∆−+(x; x′) : y → y−+(x; x′) ≡ a(η)a(η′)[‖~x−~x′‖2 − (η−η′−iδ)2
], (5–5)
i∆−−(x; x′) : y → y−−(x; x′) ≡ a(η)a(η′)[‖~x−~x′‖2 − (|η−η′|+iδ)2
]. (5–6)
Vertices in the Schwinger-Keldysh formalism either have all + lines or all − lines. The +
vertex is identical to that of the in-out formalism, whereas the − vertex is its conjugate.
Because any external line can be either + or −, each N-point one particle irreducible
(1PI) function of the in-out formalism gives rise to 2N 1PI functions in the Schwinger-Keldysh
formalism. The Schwinger-Keldysh effective action is the generating functional of these
1PI functions.
45
We can express the effective function in terms of fields ϕ+, to access the + lines, and
ϕ−, to access the − lines,
Γ[ϕ+, ϕ−] = S[ϕ+]− S[ϕ−]− 1
2
∫d4x
∫d4x′
ϕ+(x)M2++(x; x′)ϕ+(x′) + ϕ+(x)M2
+−(x; x′)ϕ−(x′)
+ϕ−(x)M2−+(x; x′)ϕ+(x′) + ϕ−(x)M2
−−(x; x′)ϕ−(x′)
+O(ϕ3±), (5–7)
where S[ϕ] is the classical scalar action.
At the order we are working, −iM2++(x; x′) is the same as the in-out self-mass-squared.
We can therefore read it off from (4–41). We get −iM2+−(x; x′) by dropping the delta
function terms, reversing the sign and replacing y(x; x′) by y+−(x; x′) in Table 4-5. The
other two 1PI 2-point functions derive from conjugating these two,
− iM2−−(x; x′) =
(−iM2
++(x; x′))∗
,−iM2−+(x; x′) =
(−iM2
+−(x; x′))∗
. (5–8)
To get the Schwinger-Keldysh effective field equations one varies the action with
respect to the field of either polarity, then sets the two polarities equal to Φ(x). At
linearized level this gives,
a4¤Φ(x)−∫ 0
ηi
dη′∫
d3x′
M2++(x; x′) + M2
+−(x; x′)
Φ(x′) = 0 . (5–9)
Here ηi = −1/H is the initial (conformal) time at which the universe is released in free
Bunch-Davies vacuum. One can see from relations (5–3) and (5–4), and from the extra
conjugated vertex in M+−(x; x′), that the bracketed term vanishes for η′ > η. The fact that
y+−(x; x′) is the complex conjugate of y++(x; x′) for η′ < η means that the bracketed term
is real. One also sees that it must involve the imaginary part of at least one propagator,
which means the only net effect comes from points x′µ on or inside the past light-cone of
xµ. Hence the Schwinger-Keldysh effective field equations are real and causal, unlike those
of the in-out formalism.
46
5.2 Restrictions on Our Solution
Two limitations on our knowledge impose important restrictions on the sense in which
we can solve (5–9):
1. We only know the scalar self-mass-squared at one loop order; and
2. We took the initial state to be free, Bunch-Davies vacuum.
The first limitation means we must solve (5–9) perturbatively. The full scalar self-mass-squared
can be expanded in powers of the loop-counting parameter κ2 = 16πG,
M2++(x; x′) + M2
+−(x; x′) =∞∑
`=1
κ2`M2`(x; x′) . (5–10)
A similar expansion applies for plane wave solutions to (5–9),
Φ(x;~k) =∞∑
`=0
κ2`Φ`(η, k)× ei~k·~x . (5–11)
To make Φ(x;~k) agree with (5–2) we must normalize the tree order solution appropriately,
Φ0(η, k) = u(η, k) ≡ H√2k3
(1− ik
aH
)exp
[ ik
aH
]. (5–12)
The ` ≥ 1 solutions obey,
a2[∂2
0 + 2H∂0 + k2]Φ`(η, k) = −
∑
k=1
∫ 0
ηi
dη′∫
d3x′M2k(x; x′)Φ`−k(η
′, k)ei~k·(~x′−~x). (5–13)
We know only M21(x; x′) so the sole correction we can compute is Φ1(η, k).
The second limitation means it only makes sense to solve for Φ1(η, k) at late times,
i.e., as η → 0−.1 Interactions result in important corrections to free vacuum on a
flat background and it is unthinkable that this does not happen as well for de Sitter
1 This raises the issue of how applicable our results are to using the mode functions forthe power spectrum of a realistic model in which inflation ceases before the conformal timereaches zero. We shall comment further at the end of this sub-section. For now, note thata mode which experiences horizon crossing 30 e-foldings before the end of inflation wouldstill experience the scale factor growing by an additional factor of about 1013.
47
background. In the Schwinger-Keldysh formalism these corrections would correspond
to vertices on the initial value surface [30, 43]. In the in-out formalism the free vacuum
is automatically corrected by time evolution. One can follow the progress of this in the
Schwinger-Keldysh formalism by isolating terms that decay with increasing time after the
release of the initial state. For example, the two loop expectation value of the stress tensor
of a massless, minimally coupled scalar with a quartic self-interaction gives the following
energy density and pressure [8, 9],
ρ =λH4
(2π)4
1
8ln2(a)+
1
18a3− 1
8
∞∑n=1
(n+2)a−n−1
(n+1)2
+ O(λ2) , (5–14)
p =λH4
(2π)4
−1
8ln2(a)− 1
12ln(a)− 1
24
∞∑n=1
(n2−4)a−n−1
(n+1)2
+ O(λ2) . (5–15)
We suspect that the (separately conserved) terms which fall like powers of 1/a can be
absorbed into an order λ correction of the initial state. On the other hand, the terms
which grow like powers of ln(a) represent the effect of inflationary particle production (in
this case, of scalars) pushing the field up its quartic potential.
Because we have not worked out the order κ and κ2 corrections to Bunch-Davies
vacuum, we cannot trust corrections to Φ1(η, k) that fall off at late times relative to the
tree order solution u(η, k),
u(η, k) =H√2k3
1 +
k2
2H2a2+
ik3
3H3a3+ O
( k4
H4a4
). (5–16)
To understand what this means, it is best to convert equation (5–13) for Φ1 from
conformal time η to comoving time t ≡ − ln(−Hη)/H,
[ ∂2
∂t2+ 3H
∂
∂t+
k2
a2
]Φ1 = − 1
a4
∫ 0
ηi
dη′ u(η′, k)
∫d3x′M2
1(x; x′)ei~k·(~x′−~x). (5–17)
Because the factor of k2/a2 on the left hand side redshifts to zero at late times, its effect
on the late time limit of Φ1 can be at most a constant (about which more shortly). Given
a putative form for the late time behavior of the right hand side it is easy to infer the
48
leading late time behavior of Φ1, for example,
r.h.s −→ C ln(a) =⇒ Φ1 −→ C
6H2ln2(a) , (5–18)
r.h.s −→ C =⇒ Φ1 −→ C
3H2ln(a) , (5–19)
r.h.s −→ Cln(a)
a=⇒ Φ1 −→ Constant− C
2H2
ln(a)
a, (5–20)
r.h.s −→ C1
a=⇒ Φ1 −→ Constant− C
2H2
1
a. (5–21)
The only effects which can be distinguished from (the currently unknown) corrections to
the initial state derive from contributions to the right hand side that fall off no faster than
1/ ln(a). It turns out that inverse powers of ln(a) cannot occur, so the practical dividing
line is between contributions to the right hand side which grow or approach a nonzero
constant and those which fall off.
Of course any correction to the left hand side of (5–17) is liable to induce a
time independent shift in the late time limit of Φ1. Shifts of this form which are also
independent of the wave number k could be absorbed into a field strength renormalization
and are unobservable. However, significant k-dependent shifts could induce observable
tilts into the power spectrum and are an important potential outcome from a study of this
sort. The possibility for such a k-dependent shift provides an excellent justification for the
laborious task of working out the order κ and κ2 corrections to Bunch-Davies vacuum.
It is easy to see that only a small range of k-dependent, constant shifts in Φ1 can be
significant. First, note that all one loop corrections are suppressed by the loop counting
parameter, whose largest value consistent with the current limit on the tensor-to-scalar
ratio is GH2 ≤ 10−12 [44]. Next, observe that any constant shift in Φ1 can be expressed as
the late time limit of the tree order mode function times a dimensionless function of the
ratio k/H,
Constant =H√2k3
× f( k
H
). (5–22)
49
Observable modes experienced first horizon crossing (which means k ∼ Ha) within the last
60 e-foldings of inflation, hence they must have huge values for k/H under our convention
that inflation begins with a = 1. One cannot get any more that (k/H)2 because then
the flat space limit would diverge. In fact, even (k/H)2 is ruled out because it would give
rise to a Gk2 correction in the flat space limit, which is not seen. So the only chance for a
significant effect seems to be a single power of k/H, possibly times logarithms of k/H. We
will see an example of such a correction in the next sub-section, although it seems likely
that this particular correction can be nulled by an appropriate correction to the initial
state.
5.3 Local Corrections
Because the scalar d’Alembertian annihilates the tree order solution, only the third,
noncovariant counterterm makes any contribution to (5–17),
1
a4
∫d4x′ a4
(∆α1¤2 + ∆α2¤ + ∆α3
∇2
a2
)δ4(x−x′)
×u(η′, ~k)ei~k·(~x′−~x) = −∆α3k2
a2u(η, k) . (5–23)
Because this term rapidly redshifts to zero, we see from the preceding discussion that it
can only shift Φ1 by a (possibly k-dependent) constant at late times. It is easy to work
this out explicitly.
Substituting (5–23) into (5–17) gives,
[ ∂2
∂t2+ 3H
∂
∂t+
k2
a2
]∆Φ1 = ∆α3 k2 × u(η, k)
a2. (5–24)
The solution can be written as an integral over co-moving time,
∆Φ(η, k) = ∆α3k2
∫ t
0
dt′ G(t, t′; k)u(η′, k)
a2(η′), (5–25)
where the Green’s function is,
G(t, t′; k) =θ(t−t′)W (t′, k)
[u(η, k)u∗(η′, k)− u∗(η, k)u(η′, k)
]. (5–26)
50
Here the Wronskian is,
W (t′, k) ≡ u(η′, k)u∗(η′, k)− u(η′, k)u∗(η′, k) =H2
2k3× −2ik3
H2a3(η′), (5–27)
and a dot denotes differentiation with respect to co-moving time.
The integrals in (5–25) are straightforward and the result is,
∆Φ1(η, k)
∆α3
=i
2
H
k
u(η, k)
[a(η′)− k2
H2a(η′)
]− u∗(η, k)
[a(η′)− ik
2Ha(η′)
]∣∣∣∣∣
t
0
, (5–28)
=H√2k3
[3
4− i3k
4Ha− k2
2H2a2
]eik/Ha
+( ik
2H− iH
2k
)u(η, k) +
(1
4+
iH
2k
)e2ik/Hu∗(η, k) . (5–29)
Taking the late time limit reveals a correction of order k/H,
limt→∞
∆Φ1(η, k) = ∆α3H√2k3
ik
2H+
3
4− iH
2k+
1
4e2ik/H +
iH
ke2ik/H
. (5–30)
The difficult but crucial question is, can this correction be subsumed into a redefinition of
the initial state? Although we have no proof, the strong suspicion is that it can be. The
order k/H term in (5–30) derives entirely from the homogeneous, lower limit terms on the
last line of (5–29), and this is precisely the property that an initial state correction would
have.
5.4 Nonlocal Corrections
Although Table 4-5 might seem to present a bewildering variety of nonlocal
contributions to (5–17), a series of seven straightforward steps suffices to evaluate each
one:
51
1. Eliminate any factors of 1/y using the identities,
4
y=
¤H2
ln
(y
4
)+ 3 , (5–31)
4
yln
(y
4
)=
¤H2
1
2ln2
(y
4
)−ln
(y
4
)+ 3 ln
(y
4
)− 2 . (5–32)
2. Extract the factors of ¤ and ∇2 from the integration over x′µ using the identities,
¤H2
−→ −[a2 ∂2
∂a2+ 4a
∂
∂a+
k2
a2H2
], (5–33)
∇2 −→ −k2 . (5–34)
3. Combine the ++ and +− terms to extract a factor of i and make causality manifest,
ln(y++
4
)− ln
(y+−
4
)= 2πiθ
(∆η−∆x
), (5–35)
ln2(y++
4
)− ln2
(y+−
4
)= 4πiθ
(∆η−∆x
)ln
(1
4aa′H2(∆η2−∆x2)
). (5–36)
Here we define a ≡ a(η), a′ ≡ a(η′), ∆η ≡ η − η′ and ∆x ≡ ‖~x − ~x′‖. Note that any
positive powers of y become,
y+± −→ −1
4aa′H2(∆η2−∆x2) . (5–37)
4. Make the change of variables ~r = ~x′ − ~x, perform the angular integrations and make
the further change of variable r = ∆η · z,
∫d3x′θ
(∆η−∆x
)F
(1
4aa′H2(∆η2−∆x2)
)ei~k·(~x′−~x)
= 4πθ(∆η)
∫ ∆η
0
dr r2F(1
4aa′H2(∆η2−r2)
)sin(k∆x)
k∆x, (5–38)
= 4πθ(∆η)∆η3
∫ 1
0
dz z2F
(aa′
( 1
a′− 1
a
)2(1−z2
4
))sin(k∆ηz)
k∆ηz. (5–39)
5. Reduce the z integration to a combination of elementary functions and sine and
cosine integrals [45].
52
6. Make the change of variables a′ = −1/Hη′, expand the integrand and perform the
integration over a′.
7. Act any derivatives with respect to a.
Much of the labor involved in implementing these steps derives from the spacetime
dependence of the zeroth order solution, u(η′, k)ei~k·~x′ . For example, one can see from
(5–39) that only elementary functions would result from the z integration if the zeroth
order solution were constant. Because we do not yet possess corrections to the initial state
there is no reason to avoid making the simplification,
− 1
a4
∫ 0
ηi
dη′∫
d3x′M21(x; x′)u(η′, k)ei~k·(~x′−~x)
−→ −u(0, k)
a4
∫ 0
ηi
dη′∫
d3x′M21(x; x′) ; (5–40)
To see why, note from expressions (5–16) and (5–39) that the deviation of the zeroth
order solution from u(0, k) introduces at least two factors of 1/a′ or 1/a. Because the
constant mode function can at best result in powers of ln(a), and because we cannot trust
contributions to (5–17) which fall off, we may as well discard any terms which acquire an
extra factor of 1/a. Extra factors of 1/a′ effectively restrict the integration to early times,
which again causes the net result to fall off at late times.
One consequence of the simplification (5–40) is that we can neglect any contribution
from Table 4-5 which contains a factor of ∇2. We therefore need only compute terms of
the form,
− u(0, k)iH8a4−K
(4π)4
( ¤H2
)N∫ 0
ηi
dη′ a′K∫
d3x′
f(y++
4
)−f
(y+−
4
), (5–41)
where the constant K takes the values of 3, 4 and 5, and the functions f(x) are 1/x,
ln x/x, ln x and ln2 x. The action of the d’Alembertian derives from setting k = 0 in
(5–33). The integrations for K = 3 (given in Table 5-1) and K = 4 (given in Table 5-2)
were worked out in a previous paper [29]. Applying the same technique — which is just
the 7-step procedure given above for k = 0 — gives the results for K = 5 in Table 5-3.
53
f(x) − iH4
16π2 ×∫
d4x′a′3f(y++
4)− f(y+−
4)
1x
− ln(a)a
+ 1a
+ O( 1a2 )
ln(x)x
− ln2(a)2a
+ 2 ln(a)a
− 3a
+ π2
3a+ O( ln(a)
a2 )
ln(x)16− ln(a)
2a+ 1
4a+ O( 1
a2 )
ln2(x)13ln(a)− 11
9− ln2(a)
2a+ 2 ln(a)
a− 9
4a+ π2
3a+O( ln(a)
a2 )
Table 5-1. Integrals with a′3.
f(x) − iH4
16π2 ×∫
d4x′a′4f(y++
4)− f(y+−
4)
1x
−12
+ O( 1a)
ln(x)x
34
+ O( 1a)
ln(x)16ln(a)− 11
36+ O( 1
a)
ln2(x)16ln2(a)− 8
9ln(a) + 7
4− π2
9+ O( ln(a)
a)
Table 5-2. Integrals with a′4.
It remains just to act the d’Alembertians and sum the results for each of the
contributions from Table 4-5. That is done in Table 5-4. Although individual contributions
can grow as fast as ln2(a), all growing or even finite terms cancel. It follows that the right
hand side of (5–17) falls off at least as fast as 1/a times powers of ln(a). Hence there are
no significant corrections to the mode function at one loop order.
f(x) − iH4
16π2 ×∫
d4x′a′5f(y++
4)− f(y+−
4)
1x
− 16a + O( 1
a)
ln(x)x
1736
a + O( 1a)
ln(x)124
a− 16
+ O( 1a)
ln2(x) − 13144
a− 13ln(a)+ 5
9+O( ln(a)
a)
Table 5-3. Integrals with a′5.
54
External Operator× f(x) Coefficient of u(0, k)× H4
16π2
(aa′)4¤3/H2 ×− ln x3x
0
(aa′)4¤2 × 26 ln x3x
0
(aa′)4¤2 × 383x
0
(aa′)4¤2 ×−6 ln2 x −18
(aa′)4¤2 ×−18 ln x 0
(aa′)4H2¤×−6 ln xx
0
(aa′)4H2¤× 4x
0
(aa′)4H2¤×−4 ln x 2
(aa′)4H4 × 4 ln xx
3
(aa′)4H4 × 18x
−9
(aa′)4H4 ×−108 ln x 33 − 18 ln(a)
(aa′)3(a2 + a′2)¤3/H2 × ln x6x
(−325+12π2
27) + 14
3ln(a)− 2
3ln2(a)
(aa′)3(a2 + a′2)¤2 ×− ln x3x
(85−12π2
27)− 4 ln(a) + 2
3ln2(a)
(aa′)3(a2 + a′2)¤2 × 16x
89− 2
3ln(a)
(aa′)3(a2 + a′2)H2¤×−2 ln x3x
(160−12π2
27)− 10
3ln(a) + 2
3ln2(a)
(aa′)3(a2 + a′2)H2¤× 5x
553− 10 ln(a)
(aa′)3(a2 + a′2)H2¤×−18 ln x −15 + 18 ln(a)
(aa′)3(a2 + a′2)H4 × 4 ln x3x
(−91+12π2
27) + 8
3ln(a)− 2
3ln2(a)
(aa′)3(a2 + a′2)H4 ×− 323x
−809
+ 323
ln(a)
Total 0
Table 5-4. The −u(0,k)a4
iH8
(4π)4
∫d4x′ (Ext. Operator)× f(y++
4)− f(y+−
4) terms.
55
CHAPTER 6CONCLUSION
We have computed one loop quantum gravitational corrections to the scalar
self-mass-squared on a locally de Sitter background. We used that to solve the one
loop-corrected, linearized effective field equation for a massless, minimally coupled
scalar. The computation was done using dimensional regularization and renormalized by
subtracting the three possible BPHZ counterterms. Because our gauge condition (2–15)
breaks de Sitter invariance, one of these counterterms is noninvariant.
Unlike previous analysis of the scalar effective potential [20], our technique uses
the full self-mass-squared, including corrections to the derivative terms and nonlocal
corrections. It should therefore be sensitive not only to the scalar’s mass but also to
its field strength. We do not yet possess the computational tools needed to search for
k-dependent but time independent shifts of the field strength but we find no growing
corrections at one loop order.
The point of this exercise is to discover whether or not the inflationary production
of gravitons has a significant effect upon minimally coupled scalars as it does on fermions
[18]. In order to check this we computed one loop corrections to the scalar mode functions
using the effective field equation,
∂µ
(√−g gµν∂νΦ(x)
)−
∫d4x′M2
ren(x; x′)Φ(x′) = 0 . (6–1)
Similar studies have already probed the effects of scalar self-interactions [10, 11], fermions
[45] and photons [21], but none has so far considered the effects of gravitons. Although our
scalar is a spectator to Λ-driven inflation, the near flatness of inflaton potentials suggests
that the result we shall obtain may apply as well to the inflaton of scalar-driven inflation.
A significant difference between this and previous scalar studies [10, 11, 21, 45] is
that quantum gravity is not renormalizable. Although we could absorb divergences with
quartic, BPHZ counterterms, no physical principle fixes the finite coefficients ∆αi of these
56
counterterms. That ambiguity is one way of expressing the problem of quantum gravity.
However, a little thought reveals that we will be able to get unambiguous results for
late time corrections to the mode functions. The reason is that the scalar d’Alembertian
annihilates the tree order mode solution,
Φ0(x;~k) = u(η, k)ei~k·~x where u(η, k) =H√2k3
[1− ik
Ha
]exp
[ ik
Ha
]. (6–2)
Hence only the third counterterm makes a nonzero contribution, and its effect rapidly
redshifts away,
∫d4x′ κ2a4
(∆α1¤2 + ∆α2¤ + ∆α3
∇2
a2
)δ4(x−x′)× Φ0(x
′;~k)
= −κ2a4 ×∆α3k2
a2Φ0(x;~k) . (6–3)
It is instructive to compare our de Sitter background result (4–41) with its flat space
analogue. In the flat space limit of H → 0 with fixed comoving time, the scalar and
graviton propagators become,
i∆flatA (x; x′) =
Γ(D2−1)
4πD2
1
∆xD−2, (6–4)
i[
αβ∆flatρσ
](x; x′) =
[2ηα(ρησ)β− 2
D−2ηαβηρσ
]Γ(D2−1)
4πD2
1
∆xD−2. (6–5)
Here ∆x2 is the Poincare length function analogous to y(x; x′),
∆x2(x; x′) ≡ ‖~x−~x′‖2 −(|η−η′| − iδ
)2
. (6–6)
Two features of the flat space propagators deserve comment:
1. Both propagators are manifestly Poincare invariant; and
2. The coincidence limits of both propagators vanish in dimensional regularization.
We exploited the first property to drop all but the final noninvariant counterterm on
the list (4–19). And the second property explains why our 4-point contribution (3–14)
57
vanishes in the flat space limit,
−iM2flat4pt
(x; x′) = −iκ2
8∂µ
i[
αα∆flat
ρσ
](x; x) ηρσ∂µδ
D(x−x′)
+iκ2
4∂µ
i[
αβ∆flatαβ
](x; x) ∂µδ
D(x−x′)
+
iκ2
2∂ρ
i[
αα∆flat
ρσ
](x; x) ∂σδD(x−x′)
−iκ2∂α
i[
αρ∆flatρσ
](x; x) ∂σδD(x−x′)
= 0 . (6–7)
An only slightly less trivial computation reveals that the flat space limit of our total
3-point contribution should also vanish,
−iM2flat3pt
(x; x′) = −κ2∂α∂′ρ
i[
αβ∆flatρσ
](x; x′)∂β∂′σi∆flat
A (x; x′)
+κ2
2∂µ∂′ρ
i[
αα∆flat
ρσ
]∂µ∂
′σi∆flatA
+
κ2
2∂α∂′ν
i[
αβ∆flatρσ
]ηρσ∂β∂′νi∆
flatA
−κ2
4∂µ∂′ν
i[
αα∆flat
ρσ
](x; x′) ηρσ∂µ∂
′νi∆
flatA (x; x′)
, (6–8)
= −κ2Γ2(D2−1)
16πD∂α∂ρ
1
∆xD−2∂β∂σ 1
∆xD−2
[2ηα(ρησ)β − ηαβηρσ
], (6–9)
= −κ2Γ2(D2−1)
16πD∂2
1
∆xD−2∂2 1
∆xD−2
, (6–10)
= 0 . (6–11)
This result has a number of consequences:
1. It explains why the highest dimension counterterm (4–5) fails to appear;
2. It explains why all the entries of Table 4-5 vanish in the flat space limit except the
first line,
κ2H2
(4π)4(aa′)4¤3
−1
3× 4
yln
(y
4
)
−→ −1
3× κ2
(4π)4∂6
4
∆x2ln
(1
4H2∆x2
), (6–12)
58
and the fifth line,
κ2H2
(4π)4(aa′)3(a2 + a′2)¤3
1
6× 4
yln
(y
4
)
−→ +1
3× κ2
(4π)4∂6
4
∆x2ln
(1
4H2∆x2
); (6–13)
3. It explains why (6–12) and (6–13) cancel in the flat space limit; and
4. It means that any physical effect we find must derive entirely from the nonzero
Hubble constant.
This is the right point to comment on accuracy. This has been a long and tedious
computation, involving the combination of many distinct pieces. It is significant when
these pieces join together to produce results that can be checked independently, such
as the vanishing of the flat space limit. One sees that in the way the α and β ¤H2
contributions to the α1 counterterm cancel in Table 4-3. Another example is the way
three individually divergent terms combine in Table 4-4 to produce a finite result for a
counterterm that is forbidden by the shift symmetry of the bare Lagrangian (2–1).
Although the α1 counterterm had to vanish by the flat space limit, we do not yet
understand why the coefficient the α2 counterterm is finite. The contribution of this
term to the scalar self-mass-squared vanishes in flat space, but it would seem to affect
the φ + h → φ + h scattering amplitude. The divergences on this were explored in the
classic paper of ’t Hooft and Veltman [46]. Unfortunately, their on-shell analysis makes no
distinction between R(∂φ)2 — which we have — and (∂φ)4 — which we do not have.
We should also comment on what subset of the full de Sitter group is respected by
our result (4–41). Recall that our gauge fixing term breaks spatial special conformal
transformation (4–10). This is why the noninvariant counterterm (4–20) occurs. It is also
responsible for the noninvariant factors of ∇2 and a2 + a′2 in Table 4-5. Because these
breakings derive entirely from the gauge condition, we expect them to have no physical
consequence.
59
The graviton and scalar propagators also break the dilatation symmetry (4–9). Unlike
the violation of spatial special conformal transformations, the breaking of dilatation
invariance is a physical manifestation of inflationary particle production and can have
important consequences. Dilatation breaking comes in the ln(aa′) term of the A-type
propagator (2–14). These logarithms were responsible for the secular growth that was
found in the fermion field strength [18], so one might expect them to drive any effect
on scalars as well. However, it turns out that the factors of ln(aa′) all drop out. For
the scalar propagator this is a trivial consequence of the fact that it always carries one
primed and one un-primed derivative. Logarithms from the graviton propagator do appear
in the 4-point contributions (3–14), and in the 3-point logarithm contributions (3–37).
But all factors of ln(aa′) cancel in the final result (4–41), which turns out to respect
dilatation invariance. Because of this we suspect that there will be no significant late time
corrections to the mode functions at one loop order.
Physically our result means that the sea of infrared gravitons produced by inflation
has little effect on the scalar. Although our scalar is a spectator to Λ-driven inflation,
typical scalar inflaton potentials are so flat that the inflaton is also unlikely to suffer
significant one loop corrections from quantum gravity. If so, it is consistent to use the tree
order scalar mode functions to compute the power spectrum of cosmological perturbations.
There are good reasons why the scalar should not acquire a mass from quantum
gravity [20] but there seems to be no reason why its field strength cannot suffer a time
dependent renormalization of the type experienced by massless fermions [18]. Weinberg’s
result for the power spectrum allows for ln(a) corrections [6], and they do arise from
individual terms in Table 5-4. However, there is no net correction of this form at one loop.
In the end, our null result may not be as surprising as it seems. It is very simple to
show that the scalar self-mass-squared vanishes at one loop order on a flat background
[21]. It is also wrong to think of the scalar as weaker than the fermion; what is weaker is
the fractional one loop correction. Although massless fermions experience a time-dependent
60
field strength renormalization, this represents a ln(a) enhancement of tree order mode
functions which fall like 1/a32 . By contrast, the tree order scalar mode functions approach
a nonzero constant. Indeed, it is the fact that derivatives of this constant vanish which
makes the fractional correction of the κh∂ϕ∂ϕ coupling so small. The sea of infrared
gravitons is present, but it can only couple, at this order, to the scalar’s stress-energy, and
the stress-energy of a single scalar redshifts to zero at late times.
61
APPENDIX AEXTRACTING DERIVATIVES
We group the various normal contributions into seven parts:
P1 ≡ ∇·∇′[(aa′)D−2A∇·∇′A
], (A–1)
P2 ≡ ∂i∂′0
[(aa′)D−2A∂i∂
′0A
]+ ∂0∂
′i
[(aa′)D−2A∂0∂
′iA
], (A–2)
P3 ≡ ∂0∂′0
[(aa′)D−2A∂0∂
′0A
], (A–3)
P4 ≡ −∂0∂′0
[(aa′)D−2B∇·∇′A
], (A–4)
P5 ≡ −∂i∂′0
[(aa′)D−2B∂0∂
′iA
]− ∂0∂
′i
[(aa′)D−2B∂i∂
′0A
], (A–5)
P6 ≡ −∇·∇′[(aa′)D−2B∂0∂
′0A
], (A–6)
P7 ≡ ∂0∂′0
[(aa′)D−2∆C∂0∂
′0A
]. (A–7)
In these definitions the exprssion “∂0∂′0A(y)” means the naive derivative, without the delta
function. Also note that we have suppressed the unbiquitous factors of κ2.
An important simplification in reducing P2 is to achieve a symmetric form which has
no ∂0. This can be done by adding equations (3–62) and (3–65) and then using equation
(3–58),
P2 = (−δ + 2ζ)I2[AA′′] + (D−1)ε2I2[A′2]
+H(aa′)D−2(a∂0+a′∂′0)∇2I2[A′2] , (A–8)
= (−δ + 2ζ)I2[AA′′] + (D−1)ε2I2[A′2]
+ε3yI[A′2] + (D−1)ε1I2[A′2]− ζ
2I3[A′2] . (A–9)
Another organizational point concerns removing factors of y from inside integrals.
This is desirable because it reduces the number of distinct integrals which appear. It can
always be accomplished by partial integration. We will illustrate using the function acted
upon by −ε3 in equation (3–73),
F (y) ≡ (D−2)I[A′B+yA′′B] + yA′B + y2A′′B − yI[A′B′] . (A–10)
62
Note the relations,
A′B + yA′′B = A′B + y∂
∂yI[A′′B] , (A–11)
=∂
∂y
yI[A′′B]
+ A′B − I[A′′B] , (A–12)
=∂
∂y
yI[A′′B]
+ I[A′B′] . (A–13)
We can therefore write,
F (y) = y2A′′B + (D−1)yI[A′′B] + (D−2)I2[A′B′] . (A–14)
With (A–13) and (A–14) we can read off the following result for P5 from equation
(3–73),
P5 = δI2[BA′′]− (D−1)ε1I2[B′A′]− (D−1)ε2I
2[B′A′]
+ζ−2I2[BA′′] +
1
2I3[B′A′]
+ ε2
yI[A′′B] + I2[A′B′]
+ε3
y2A′′B + (D−1)yI[A′′B] + (D−2)I2[A′B′]
. (A–15)
Many terms involving A in P2 combine with cognate terms involving B in P5 to produce
the less singular propagator function ∆B = B − A. Summing expressions (A–9) and
(A–15) gives,
P2+5=δI2[∆BA′′]− (D−1)ε1I2[∆B′A′]− (D−1)ε2I
2[∆B′A′] + ε3yI[A′2]
+ζ−2I2[∆BA′′] +
1
2I3[∆B′A′]
+ ε2
yI[A′′B] + I2[A′B′]
+ε3
y2A′′B + (D−1)yI[A′′B] + (D−2)I2[A′B′]
. (A–16)
63
In contradistinction to P2 and P5, the reduction of the other parts is facilitated by
further sub-division immediately after employing identities (3–45) and (3–48),
f(y)∂0∂′0A(y) = ∂0∂
′0I
2[fA′′] + 2aa′H2I[f ′A′]
−aa′H2
(D−1) + y∂
∂y
I2[f ′A′]− 1
2∇·∇′I3[f ′A′] , (A–17)
f(y)∇·∇′A(y) = ∇·∇′I2[fA′′]− 2(D−1)aa′H2I[f ′A] . (A–18)
One employs (A–17) on P3 (from which we can read off the result for P7) and P6 to give
the sub-parts,
P3a ≡ ∂0∂′0
[(aa′)D−2∂0∂
′0I
2[AA′′]]
, (A–19)
P3b ≡ 2∂0∂′0
[(aa′)D−1H2I[A′2]
], (A–20)
P3c ≡ −∂0∂′0
[(aa′)D−1H2
(D−1) + y
∂
∂y
I2[A′2]
], (A–21)
P3d ≡ −1
2∂0∂
′0
[(aa′)D−2∇·∇′I3[A′2]
], (A–22)
P7a ≡ ∂0∂′0
[(aa′)D−2∂0∂
′0I
2[∆CA′′]]
, (A–23)
P7b ≡ 2∂0∂′0
[(aa′)D−1H2I[A′∆C ′]
], (A–24)
P7c ≡ −∂0∂′0
[(aa′)D−1H2
(D−1) + y
∂
∂y
I2[A′∆C ′]
], (A–25)
P7d ≡ −1
2∂0∂
′0
[(aa′)D−2∇·∇′I3[A′∆C ′]
], (A–26)
P6a ≡ −∇·∇′[(aa′)D−2∂0∂
′0I
2[BA′′]]
, (A–27)
P6b ≡ −2∇·∇′[(aa′)D−1H2I[A′B′]
], (A–28)
P6c ≡ ∇·∇′[(aa′)D−1H2
(D−1) + y
∂
∂y
I2[A′B′]
], (A–29)
P6d ≡ 1
2∇·∇′
[(aa′)D−2∇·∇′I3[A′B′]
]. (A–30)
64
Part Contribution Acted upon by α
P3a I2[AA′′]
P7a I2[∆CA′′]
Total I2[AA′′] + I2[∆CA′′]
Table A-1. Contributions acted upon by α = (aa′)D¤2.
Applying the second identity (A–18) to P1 and P6 gives their sub-parts,
P1a ≡ ∇·∇′[(aa′)D−2∇·∇′I2[AA′′]
], (A–31)
P1b ≡ −2(D−1)∇·∇′[(aa′)D−1H2I[A′2]
], (A–32)
P4a ≡ −∂0∂′0
[(aa′)D−2∇·∇′I2[BA′′]
], (A–33)
P4b ≡ 2(D−1)∂0∂′0
[(aa′)D−1H2I[A′B′]
]. (A–34)
Of course there is no problem further reducing the spatial derivatives. The following
generic reductions serve to reduce terms involving the operator ∂0∂′0,
∂0∂′0
[(aa′)D−2∂0∂
′0f(y)
]=
α− δ + ζ
f(y) , (A–35)
∂0∂′0
[(aa′)D−1H2f(y)
]=
β
2+
1
2(D−1)(D−2)γ1 +
γ2
2y
∂
∂y
+γ3
2
[(D−1)y
∂
∂y+ y2 ∂2
∂y2
]+ ε1
[−1 +
D
4I]
f(y) , (A–36)
∂0∂′0
[(aa′)D−2∇·∇′f(y)
]=
− δ
2− 1
2(D−2)(D−3)ε1 − ε2
2y
∂
∂y
−ε3
2
[(D−1)y
∂
∂y+ y2 ∂2
∂y2
]+ ζ
[1− 1
4(D−2)I
]f(y) , (A–37)
∇·∇′[(aa′)D−2∂0∂
′0f(y)
]=
− δ
2− 1
2(D−1)(D−2)ε1 − ε2
2y
∂
∂y
−ε3
2
[(D−1)y
∂
∂y+ y2 ∂2
∂y2
]+ ζ
[1 +
1
4(D−2)I
]f(y) . (A–38)
Tables A-1-A-10 give the results for each of the ten External Operators.
65
Part Contribution Acted upon by β
P3b I[A′2]
P3c −12yI[A′2]− (D−1
2)I2[A′2]
P4b (D−1)I[A′B′]
P7b I[A′∆C ′]
P7c −12yI[A′∆C ′]− (D−1
2)I2[A′∆C ′]
Total DI[A′B′]− I[A′∆B′]− 12yI[A′2]− (D−1
2)I2[A′2]
+I[A′∆C ′]− 12yI[A′∆C ′]− (D−1
2)I2[A′∆C ′]
Table A-2. Contributions acted upon by β = (aa′)D−2(a2 + a′2)H2¤.
Part Contribution Acted upon by γ1
P3b (D−1)(D−2)I[A′2]
P3c −12(D−1)(D−2)yI[A′2]− 1
2(D−1)2(D−2)I2[A′2]
P4b (D−1)2(D−2)I[A′B′]
P7b (D−1)(D−2)I[A′∆C ′]
P7c −12(D−1)(D−2)yI[A′∆C ′]− 1
2(D−1)2(D−2)I2[A′∆C ′]
D(D−1)(D−2)I[A′B′]− (D−1)(D−2)I[A′∆B′]− 12(D−1)(D−2)yI[A′2]
Total −12(D−1)2(D−2)I2[A′2] + (D−1)(D−2)I[A′∆C ′]
−12(D−1)(D−2)yI[A′∆C ′]− 1
2(D−1)2(D−2)I2[A′∆C ′]
Table A-3. Contributions acted upon by γ1 = (aa′)DH4.
Part Contribution Acted upon by γ2
P3b yA′2
P3c −12y2A′2 − D
2yI[A′2]
P4b (D−1)yA′B′
P7b yA′∆C ′
P7c −12y2A′∆C ′ − D
2yI[A′∆C ′]
Total DyA′B′ − yA′∆B′ − 12y2A′2 − D
2yI[A′2]
+yA′∆C ′ − D2yI[A′∆C ′]− 1
2y2A′∆C ′
Table A-4. Contributions acted upon by γ2 = (aa′)D−1(a2 + a′2)H4.
66
Part Contribution Acted upon by γ3
P3b (D−1)yA′2 + y2(A′2)′
P3c −Dy2A′2 − 12y3(A′2)′ − 1
2(D−1)DyI[A′2]
P4b (D−1)2yA′B′ + (D−1)y2(A′B′)′
P7b (D−1)yA′∆C ′ + y2(A′∆C ′)′
P7c −Dy2A′∆C ′ − 12y3(A′∆C ′)′ − 1
2(D−1)DyI[A′∆C ′]
(D−1)DyA′B′ + Dy2(A′B′)′ − (D−1)yA′∆B′ − y2(A′∆B′)′
Total −Dy2A′2 − 12y3(A′2)′ − 1
2(D−1)DyI[A′2] + (D−1)yA′∆C ′
+y2(A′∆C ′)′ − Dy2A′∆C ′ − 12y3(A′∆C ′)′ − 1
2(D−1)DyI[A′∆C ′]
Table A-5. Contributions acted upon by γ3 = (aa′)D−1(a + a′)2H4.
Part Contribution Acted upon by δ
P2+5 I2[∆BA′′]
P3a −I2[AA′′]
P3d14I3[A′2]
P4a12I2[BA′′]
P6a12I2[BA′′]
P7a −I2[∆CA′′]
P7d14I3[A′∆C ′]
Total 2I2[∆BA′′] + 14I3[A′2]− I2[∆CA′′] + 1
4I3[A′∆C ′]
Table A-6. Contributions acted upon by δ = (aa′)D−2(a2 + a′2)∇2¤.
67
Part Contribution Acted upon by ε1
P1b 2(D−1)I[A′2]
P2+5 −(D−1)I2[A′∆B′]
P3b −2I[A′2] + D2I2[A′2]
P3c yI[A′2] + (D−1)I2[A′2]− D4yI2[A′2]− 1
4D(D−2)I3[A′2]
P3d14(D−2)(D−3)I3[A′2]
P4a12(D−2)(D−3)I2[BA′′]
P4b −2(D−1)I[A′B′] + 12(D−1)DI2[A′B′]
P6a12(D−1)(D−2)I2[BA′′]
P6b 2I[A′B′]
P6c −yI[A′B′]− (D−1)I2[A′B′]
P7b −2I[A′∆C ′] + D2I2[A′∆C ′]
P7c yI[A′∆C ′] + (D−1)I2[A′∆C ′]− D4yI2[A′∆C ′]− 1
4D(D−2)I3[A′∆C ′]
P7d14(D−2)(D−3)I3[A′∆C ′]
−2(D−2)I[A′∆B′]−yI[A′∆B′]−(5D−42
)I2[A′∆B′]+ D2
2I2[A′B′]
Total −D4yI2[A′2] + (D−2)2I2[A′′B]− 3
4(D−2)I3[A′2]− 2I[A′∆C ′]
+yI[A′∆C ′]+(3D−22
)I2[A′∆C ′]−D4yI2[A′∆C ′]− 3
4(D−2)I3[A′∆C ′]
Table A-7. Contributions acted upon by ε1 = (aa′)D−1H2∇2.
Part Contribution Acted upon by ε2
P2+5 −(D−1)I2[A′∆B′] + yI[BA′′] + I2[A′B′]
P3d14yI2[A′2]
P4a12yI[BA′′]
P6a12yI[BA′′]
P7d14yI2[A′∆C ′]
Total 2yI[BA′′] + I2[A′B′]−(D−1)I2[A′∆B′] + 1
4yI2[A′2] + 1
4yI2[A′∆C ′]
Table A-8. Contributions acted upon by ε2 = (aa′)D−2(a2 + a′2)H2∇2.
68
Part Contribution Acted upon by ε3
P2+5 y2A′′B + (D−1)yI[A′′B] + (D−2)I2[A′B′] + yI[A′2]
P3d (D−14
)yI2[A′2] + 14y2I[A′2]
P4a (D−12
)yI[A′′B] + 12y2A′′B
P6a (D−12
)yI[A′′B] + 12y2A′′B
P7d (D−14
)yI2[A′∆C ′] + 14y2I[A′∆C ′]
Total 2y2A′′B + 2(D−1)yI[A′′B] + (D−2)I2[A′B′] + yI[A′2]+(D−1
4)yI2[A′2] + 1
4y2I[A′2] + (D−1
4)yI2[A′∆C ′] + 1
4y2I[A′∆C ′]
Table A-9. Contributions acted upon by ε3 = (aa′)D−2(a + a′)2H2∇2.
Part Contribution Acted upon by ζ
P1a I2[AA′′]
P2+5 −2I2[∆BA′′] + 12I3[A′∆B′]
P3a I2[AA′′]
P3d −12I3[A′2] + (D−2
8)I4[A′2]
P4a −I2[A′′B] + (D−24
)I3[A′′B]
P6a −I2[A′′B]− (D−24
)I3[A′′B]
P6d12I3[A′B′]
P7a I2[∆CA′′]
P7d −12I3[A′∆C ′] + (D−2
8)I4[A′∆C ′]
Total −4I2[A′′∆B] + I3[A′∆B′] + (D−28
)I4[A′2]+I2[∆CA′′]− 1
2I3[A′∆C ′] + (D−2
8)I4[A′∆C ′]
Table A-10. Contributions acted upon by ζ = (aa′)D−2∇4.
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BIOGRAPHICAL SKETCH
Emre Kahya was born in Turkey and got his B.S. and M.S. at Middle East Technical
University. He decided to pursue a doctoral study in physics at USA and admitted by the
University of Florida. Been interested in theoretical physics, after taking a quantum field
theory course of Prof. Richard Woodard, he decided study Quantum Gravity with him.
He was awarded his Ph.D. in summer 2008.
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