Automatic Calculation of Graviton Contributions and Hadronic Light

Transcrição

Automatic Calculation of Graviton
Contributions and Hadronic
Light-by-Light Scattering
Contributions by Spin-2 Mesons to
the Muon Anomalous Magnetic
Moment
Diplomarbeit
zur Erlangung des wissenschaftlichen Grades
Diplom-Physiker
vorgelegt von
Ulrik Günther
geboren am 19.05.1987 in Dresden
Institut für Kern- und Teilchenphysik
der Technischen Universität Dresden
2013
Eingereicht am 04.03.2013
1. Gutachter:
Prof. Dr. Dominik Stöckinger
2. Gutachter:
PD Dr. Günter Plunien
iii
Abstract
Extra-dimensional models provide rich and interesting phenomenology both in high-energy collider
experiments and in low-energy observables (e.g., the anomalous magnetic moment of the muon
g 2). In recent years, the g 2 was established as an essential observable for verifying or excluding
models in particle physics, complementary to high-energy collider experiments.
In this work, we present a calculation pipeline for the automatic calculation of graviton contributions to the g 2 by utilizing the computer algebra systems Mathematica and FORM and compare
the obtained results to existing literature in the light of the recent Large Hadron Collider (LHC)
outcomes.
Furthermore, we present a novel ansatz, based on the AdS/CFT duality, to calculate resonance
contributions from Spin-2 particles to the hadronic light-by-light scattering part of the anomalous
magnetic moment of the muon.
Zusammenfassung
Modelle mit zusätzlichen Raumdimensionen besitzen viele interessante phänomenologische Aspekte, die sich sowohl in Hochenergie-Beschleunigerexperimenten, als auch in Niedrigenergieobservablen (wie dem anomalen magnetischen Moment des Myons g 2) zeigen können. In den
letzten Jahren wurde das g 2 als wichtige Observable zum verifizieren und falsifizieren von
Modellen in der Teilchenphysik etabliert, da es Experimente an Hochenergie-Beschleunigern
ergänzt.
In dieser Arbeit präsentieren wir eine Methode zum automatischen Berechnen von GravitonBeiträgen zum g 2 unter Benutzung der Computeralgebra-Systeme Mathematica und FORM. Wir
vergleichen die Resultate mit existierender Literatur und diskutieren sie mit besonderer Beachtung
neuer Resultat vom Large Hadron Collider (LHC).
Weiterhin präsentieren wir einen neuen Ansatz, basierend auf der AdS/CFT-Korrespondenz,
zur Berechnung von Resonanz-Beiträgen durch Spin-2-Teilchen zum hadronischen light-by-light
scattering-Anteil des g 2.
v
Contents
Contents
ix
List of Figures
xi
1
Introduction
1
2
The Standard Model of Particle Physics and its problems
2.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Particle Content . . . . . . . . . . . . . . . . . . . . . .
2.3 Electroweak Symmetry Breaking . . . . . . . . . . . .
2.3.1 Fermion Masses . . . . . . . . . . . . . . . . . .
2.4 Assorted Problems of the Standard Model . . . . . . .
2.4.1 Theoretical Problems . . . . . . . . . . . . . . .
2.4.2 Phenomenological Problems . . . . . . . . . .
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Extra dimensions - Solving the Gauge/Gravity Hierarchy Problem
4.1 Solving the Hierarchy Problem with Extra Dimensions . . . . . . . . . . . . . . . . .
4.1.1 ADD/Large Extra Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Randall-Sundrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Randall-Sundrum Models
5.1 Solving the Gauge/Gravity Hierarchy Problem . . . . . . . . . .
5.2 Problems of the Original Model and their Solutions . . . . . . .
5.2.1 Super-GZK Events, Neutrino Masses and Proton Decay .
5.2.2 Impact on Electroweak Precision Observables . . . . . .
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The Anomalous Magnetic Moment Of The Muon
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 How to calculate the Anomalous Magnetic Moment of the Muon
3.2.1 QED Contribution . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 g 2 Projection Operator . . . . . . . . . . . . . . . . . . .
3.2.3 Electroweak Contributions . . . . . . . . . . . . . . . . . .
3.2.4 Hadronic Vacuum Polarization Contributions . . . . . . .
3.2.5 Hadronic Light-by-Light Scattering Contributions . . . . .
3.3 How to measure the Anomalous Magnetic Moment of the Muon .
3.3.1 Principle of the Experiment . . . . . . . . . . . . . . . . . .
3.3.2 Design of the Experiment . . . . . . . . . . . . . . . . . . .
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vii
6
Analysis - g 2 Spin-2 Graviton Contributions
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Relation to Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Calculation Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Details of the Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 Model Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Unit Testing with Mathematica . . . . . . . . . . . . . . . . . . . . .
6.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.2 Compatibility with the Results from Beneke, et al. . . . . . . . . . .
6.5.3 Compatibility with Results from the ATLAS and CMS experiments
6.5.4 Compatibility with E821 Results . . . . . . . . . . . . . . . . . . . .
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A Feynman Rules for Gravitons
A.1 Propagator for Massive Gravitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Feynman Rules for Fermion and Vector Boson Interaction with Massive Gravitons .
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B Model Files
B.1 Minimal Randall-Sundrum Model File for FeynArts with SM on TeV brane . . . . .
B.1.1 Generic Model File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.1.2 Model Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
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7
8
viii
The AdS/CFT Correspondence
7.1 Sketch of the Derivation . . . . .
7.1.1 SUGRA Perspective . . .
7.1.2 String Theory Perspective
7.1.3 Symmetry Argument . . .
7.2 Recipe for use . . . . . . . . . . .
7.2.1 Fields and Operators . . .
7.2.2 Correlation Functions . .
7.3 Drawbacks . . . . . . . . . . . . .
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.
Analysis - Hadronic light-by-light Scattering Contributions by Spin-2 Mesons
8.1 Introduction - The Pion Exchange Contribution . . . . . . . . . . . . . . . . .
8.1.1 Pµnrs ( p1 , p2 , p3 ) from Three-Point Functions . . . . . . . . . . . . . .
8.1.2 Pion Pole Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 The Spin-2 Meson Off-Shell Form Factor in AdS/CFT . . . . . . . . . . . . .
8.3.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.3 Three-Point Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3.4 Pµnrs Contribution from Spin-2 mesons . . . . . . . . . . . . . . . . .
8.4 Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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C Software Packages used
C.1 Third-party software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 Software written by the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
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Bibliography
71
ix
List of Figures
2.1
Self-energy contributions to the Higgs mass by fermions . . . . . . . . . . . . . . . .
3.1
Leading-order QED contribution to g
2 . . . . . . . . . . . . . . . . . . . . . . . . .
11
3.2
One-loop electroweak contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.3
Example LO hadronic vacuum polarization contributions . . . . . . . . . . . . . . . .
15
3.4
Hadronic light-by-light scattering contributions to g 2, the shaded region represents
the actual hadronic part (see Figure 3.5 for the pion pole and exchange contribution).
16
(a) Pion-pole/other resonance contribution for hadronic light-by-light scattering; (b)
charged pion loop contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.6
Muon injection and storage in the experiment, reproduced from [24] . . . . . . . . .
18
5.1
Space geometry in Randall-Sundrum models. Opposite points on the S1 of the warped
extra dimension are identified by S1 /Z2 orbifolding. . . . . . . . . . . . . . . . . . .
22
3.5
6.1
Gr,g
7
The triangle graviton-muon contributions, called aµ
in the text. . . . . . . . . . . .
27
6.2
The flight-into-sunset four-vertex contributions, called
Gr,µ,g
aµ
in the text. . . . . . . .
27
6.3
The triangle graviton-muon contributions, called
Gr,µ
aµ
in the text. . . . . . . . . . . .
28
6.4
Calculation pipeline of the gminus3 package developed for and used in this work
.
30
6.5
RS cutoff L as function of k/MPlanck and the mass of the lightest graviton. Experimentally favoured is the high-mGr , high-k region. . . . . . . . . . . . . . . . . . . . .
38
ATLAS limits for exotic particles (ATLAS Experiment ©2012 CERN, reproduced from
ATLAS exotics group TWiki page). The most important section for this work is the
upper, cyan part, where limits on particles from extra dimensions are given. In this
plot, the current constraints on the ADD model are also given and one can see that
while not being excluded yet, the MD parameter of ADD gets shifted to higher and
higher regions, where the model becomes phenomonologically uninteresting. Results
marked in red are already from the 8 TeV run. . . . . . . . . . . . . . . . . . . . . . . .
39
6.6
6.7
aGr
µ
Dependence of
with 70 and 100 contributing graviton KK states on k/MPlanck
and the mass of the lightest graviton mGr . The region right of the blue dotted line is
excluded by collider searches (linearly extrapolated from ATLAS data), the region
between the red dashed lines is the region where aGr
µ is in the region of the deviation
from the SM in the E821 experiment. See Section 6.5.4 for a full discussion. . . . . . .
41
7.1
The QCD phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
7.2
Geometry used in the AdS/CFT correspondence (reproduced from [60]) . . . . . . .
44
xi
7.3
8.1
8.2
xii
Schematic illustration of the AdS/CFT correspondence. While (super)gravity resides
in the 5D negatively curved AdS space, a conformal field theory – without gravity
– is fixed to the shell of the 5D AdS space, which is 4-dimensional. In its current
formulation, the AdS/CFT correspondence is not limited to string theory anymore.
Hadronic light-by-light scattering contributions to g 2, the shaded region represents
the actual hadronic part. This graph gives a definition of the momenta used in the
text. Apart from ks , all photon momenta are flowing into the hadronic blob. . . . . .
Masses of the neutral Spin 2 mesons. The red line represents the cutoff for Chiral
Perturbation Theory. The green line represents the muon mass. . . . . . . . . . . . .
46
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1 Introduction
For this work, two principal problems have been investigated: First, the calculation of contributions
by gravitons to the anomalous magnetic moment of the muon in Randall-Sundrum models; second,
the calculation of hadronic light-by-light scattering contributions by Spin-2 mesons to the anomalous
magnetic moment of the muon g 2.
In recent years, the g 2 has received a considerable amount of interest, due to its capacity
of augmenting measurements performed at high-energy particle colliders, like the LHC or its
predecessors LEP or Tevatron. The discrepancy between measurement and theory found in the E821
experiment at Brookhaven National Laboratory has further sparked curiosity about the contributions
the g 2 might receive from physics beyond the Standard Model.
Recently, models featuring additional space dimensions became more and more popular with
theorists and phenomenologists owing to their predictive power at current and near-future collider
experiments and their ability to solve one of the biggest theoretical problems of the Standard Model,
the gauge-gravity hierarchy problem.
Hence, we investigate contributions to the g 2 stemming from massive gravitons arising in the
Randall-Sundrum model, a model with an additional, warped space dimension.
Furthermore, we will develop a new ansatz based on holography for calculating resonance
contributions by Spin-2 mesons in the hadronic light-by-light scattering sector of the g 2, which is
notorious for dependence on low-energy observables from Quantum Chromodynamics.
Organization of this work
We start with a general introduction to the Standard Model of Particle Physics, the physics of
the g 2, and various contributions to it. We will then establish extra-dimensional models as a
possible solution to the Standard Model’s problems and an explaination of the discrepancy between
experiment and theory concerning the g 2. After that, our analysis of graviton contributions to
the g 2 will be presented and conclusions drawn.
Next, we change topics a bit, introducing the AdS/CFT correspondence as a means to calculate
low-energy observables in Quantum Chromodynamics, like meson form factors in chapter 7.
In the chapter following, we will present our ansatz for the analysis of light-by-light scattering
contributions by Spin-2 mesons, utilizing the AdS/CFT correspondence.
1
2 The Standard Model of Particle Physics
and its problems
The Standard Model (SM) of Particle Physics is one of the best-verified theories to be conceived so
far. Nevertheless, it is afflicted by quite a few problems. In this section we want to show the basics
of the Standard Model as well as sketch its problems.
2.1 Symmetries
As a relativistic quantum field theory, the Standard Model has various underlying symmetries,
which can be split into internal and external symmetries of the theory. The external symmetries of the
SM consist of invariance under the 4-dimensional pseudo-Euclidean translation group E3,1 and the
6-dimensional Lorentz group SO(3,1). Both groups are combined via the semi-direct product into
the 10-dimensional Poincare group,
E3,1 o SO(3, 1).
Every element of this group can therefore be represented by the pair
(L, a) with L 2 SO(3, 1), a 2 E3,1
and is subject to the following Lie algebra:
[ Pµ , Pn ] = 0
i [ Mµn , Pr ] = hµr Pn
i [ Mµn , Mrs ] = hµr Mns
hnr Pµ
hµs Mnr
hnr Mµs + hns Mµr ,
(2.1)
where Mµn are the generators of the Lorentz group and Pµ are the generators of the translations.
Group multiplication is defined as
(L, a) · (X, b) = (L · X, a + L(b)), where L, X 2 SO(3, 1); a, b 2 E3,1
Even more important, the theory includes local internal symmetries, called gauge symmetries. The
full gauge symmetry group of the SM is
SU (3)C ⇥ SU (2) L ⇥ U (1)Y .
3
2 The Standard Model of Particle Physics and its problems
SU (3)C is the non-abelian symmetry group of Quantum Chromodynamics (QCD), whose generators
are the 8 Gell-Mann matricies li . SU (2) L ⇥ U (1)Y is the complex 3-dimensional non-abelian
symmetry group of Quantum Flavourdynamics (QFD). The generators of SU (2) are the Pauli
matrices si . This product group is broken into U (1) of Quantum Electrodynamics (QED) - whose
generator is the complex one-dimensional rotation around the origin - by the mechanism of
electroweak symmetry breaking.
2.2 Particle Content
The SM contains the fermionic (Spin-1/2) leptons and quarks and the (Spin-1) interaction bosons, as
well as the Spin-0 Higgs boson.
Their quantum numbers determine in which interactions they take part in. Leptons, for example,
do not carry the SU (3)C colour charge, but carry SU (2) L weak isospin as well as U (1)Y hypercharge.
In contrast, quarks do have colour charge as well as weak isospin and hypercharge.
A further division is made in the weak sector between particles of left and right chirality: Only lefthanded particles take part in weak interactions, while right-handed particles only carry hypercharge
and are therefore exempt from SU (2) L interactions. An overview of the charges of the various
particles is given in Table 2.1.
Class
Leptons
Quarks
1st gen.
(ne e) L
eR
(u d) L
uR , dR
2nd gen.
nµ µ L
µR
(c s) L
cR , sR
3rd gen.
(nt t ) L
tR
(t b) L
t R , bR
colour
0
0
r/g/b
r/g/b
T3
±1/2
0
±1/2
0
Y/2
1/2
1
1/6
2/3, 1/3
Table 2.1: Fermion content of the Standard Model
Apart from the fermions, the SM contains gauge bosons as well as the infamous scalar Higgs
boson. The gauge bosons are the carriers of the force described by the gauge symmetry and are
necessarily massless. A list of gauge bosons in the unbroken SM is given in Table 2.2
Gauge group
SU (3)C
SU (2) L
U ( 1 )Y
gauge bosons
8 gluons gi
3 W bosons W i
B boson
generators
li /2
si /2
Y/2
coupling
gs
g
gY
Table 2.2: Gauge bosons of the Standard Model
So far, the Standard Model is described by a classical Lagrangian of the form
L = Lmatter + Lgauge
4
(2.2)
2.3 Electroweak Symmetry Breaking
where
/ L + i ū R Du
/ R + i d¯R Dd
/ R + i l¯L Dl
/ L + i ē R De
/ R
Lmatter = i q̄ L Dq
1
1
1
Lgauge =
Gµn,a G µn,a
Wµn,a W µn,a
Bµn Bµn
2
2
4
The field strength tensors G, W and B are defined as
Gµn,a = ∂µ Gn,a
∂n Gµ,a + gs f abc Gµ,b Gn,c
Wµn,a = ∂µ Wn,a
∂n Wµ,a + ge abc Wµ,b Wn,c
∂n Bµ .
Bµn = ∂µ Bn
The covariant derivative of the gauge fields is given as
Dµ = ∂µ + igs
la
sb
Y
Gµ,a + ig Wµ,b + igY Bµ .
2
2
2
The Higgs boson comes into play at the point where massive gauge bosons are needed: From the
experimental perspective, no massless W or B bosons have been observed, what we see are massive
W ± and Z0 bosons. Theoretically, mass terms break gauge invariance.
2.3 Electroweak Symmetry Breaking
Breaking of electroweak symmetry is achieved via the Higgs mechanism, which was developed by
Englert, Brout, Kibble, Guralnik, Hagen, and Higgs in the early 1960’s [1].
p
Symmetry breaking is realized by modifying a f4 field theory to have a mass term of
µ2 :
V ( h ) = | Dµ h | 2 + µ 2 | h 2 |
LHiggs = LHiggs,kin
l 4
|h| .
4
The Higgs field h transforms as doublet under SU (2) L transformations:
h=
f+
f0
!
As the Higgs field does have SU (2) L or U (1)Y charges, the vacuum is not invariant anymore - its
symmetry is spontaneously broken. Via minimization of the potential V (h), the Higgs field acquires
a non-zero vacuum expectation value (vev) of
| h0| h |0i |2 =
2µ
v2
=
6= 0.
l
2
It is then possible to expand h around zero to give masses to the gauge bosons:
h=
p1
2
G+
v + H ( x ) + iG0
!
5
where the physical Higgs field H ( x ) and the Goldstone bosons G0 and G + have been introduced.
The Goldstone bosons can then be removed via a gauge transformation so that only
h=
p1
2
0
(v + H ( x ))
!
is left.
After transformation to mass eigenstates, three massive and one massless electroweak gauge
bosons are found:
g2
v = m Z cos QW
2q
v
with m Z =
g12 + g22
2
with m A = 0,
Wµ± = (Wµ1 ⌥ iWµ2 )
Zµ = cos QW Wµ3
with mW =
sin QW Bµ
Aµ = sin QW Wµ3 + cos QW Bµ
where QW is the Weinberg angle.
2.3.1 Fermion Masses
So far, in the Lagrangian (2.2), fermions are massless as well, as mass terms would break gauge
invariance. After electroweak symmetry breaking, they acquire masses via Yukawa interactions
with the Higgs field:
LYukawa =
e⇤ ¯
yik
liL he jR
d⇤
yik
q̄iL hd jR
u⇤
yik
q̄iL hC u jR ,
where hC = it 2 h⇤ is the charge-conjugated Higgs field.
The mass matrices are then given by
yijk v
mijk = p .
2
2.4 Assorted Problems of the Standard Model
Despite the enormous successes of the Standard Model - the extremely precise prediction of the
anomalous magnetic moment of the electron [2] or the prediction of the masses of the gauge bosons
[3] shall serve as important examples here - it has quite a number of problems. In this section, we
will explore a few.
2.4.1 Theoretical Problems
First the SM contains a few theoretical problems. These are problems in the structure of the theory,
where for example mass ratios are inexplicably large or a certain elegance is missing.
6
f
H
H
f¯
Figure 2.1: Self-energy contributions to the Higgs mass by fermions
Hierarchy Problem
Comparing the coupling constants of gravity and the weak interaction, one arrives at
M2
GF
= Planck
⇡ 1036 ,
2
GN
mW
where MPlanck is the Planck mass and mW is the mass of the W boson.
While puzzling that the strength of those forces lies so much apart, it does not seem to be troubling
immediately. This, however, changes when one considers loop corrections to the Higgs mass which
arise after renormalizing the Standard Model. As the resonance found at the ATLAS [4] and CMS
[5] experiments is the Higgs particle with relative certainty, we are relatively safe to say its mass
is approximately 125 GeV. Nevertheless, it will receive quantum corrections from every particle it
couples to. The diagram in fig. 2.1 will contribute with
Dm2H =
| y f |2 2
L +···
8p 2 UV
where L2UV is the scale where the Standard Model’s predictions break down. Assuming the Standard
Model to be valid up to the Planck scale, this will produce corrections to the Higgs mass of about 30
orders of magnitude higher than the mass itself. Even considering the TeV scale as cutoff - which is
compatible with all recent results - yields corrections of several times the Higgs mass itself.
Now even if one repeats this calculation in dimensional regularization, the cutoff is only hidden,
and large logarithms exposing the same problem remain as long as there are particles coupling to
the Higgs which are heavier than the Higgs itself.
These potentially massive corrections have somehow to be canceled or eliminated. One possibility
for this is the introduction of supersymmetry (SUSY), an extension of the symmetries of the Poincare
group, where operators relating fermions and bosons are introduced and bosonic contributions
cancel the dangerously large fermionic contribution to the Higgs mass. Another option is the
introduction of additional space dimensions. The latter option will be used in this work and shall be
explored further after the next chapter.
2.4.2 Phenomenological Problems
The second class of problems to be discussed are phenomenological problems - this means, for
example, particles which would expected to be in the SM, but are not, or discrepancies in actual
7
measurements of particle properties.
Dark Matter
Observations on galaxy clusters made by Oort and Zwicky in the early 1930’s [6, 7] already led to
the idea that galaxies have to consist of more than just normal baryonic matter. More evidence of
“invisible” mass was provided by measurements in the cosmic microwave background (CMB) by the
COBE and WMAP experiments [8, 9], Lyman a-forest measurements [10], and gravitational lensing
[11], to name a few examples. The Standard Model in its current form does not accommodate for that,
as it only contains matter which interacts via the strong, weak, or electromagnetic forces. For dark
matter, one or more particles would be required which are “dark” and interact only gravitationally.
Baryon Asymmetry
No substantial amounts of antimatter have been detected in the universe so far [12], which is
puzzling, as this means that at some point in the universe’s evolution, all matter and antimatter
must have been annihilated but for a minute amount of matter. Andrei Sacharov provided criteria a
theory needs to satisfy so that a sufficient baryon asymmetry can be created [13].
While the Standard Model satisfies Sacharov’s criteria for the creation of a baryon asymmetry
qualitatively, it leaves much to be desired from the quantitative viewpoint:
1. The Kobayashi-Maskawa phase in the electroweak sector does violate CP, but the violation is
orders of magnitude too small
2. The non-perturbative sphaleron process [14] violates baryon and lepton number. It is suppressed by a factor exp( Sinstanton ) = exp( 4p/aW ), which numerically amounts to about
10 170
3. The electroweak phase transition at 250 GeV could be a phase transition of first order, which
would then occur outside of thermal equilibrium. Unfortunately, it is a second order phase
transition and these do always happen in equilibrium.
The Anomalous Magnetic Moment of the Muon
The anomalous magnetic moment of the muon is defined as the deviation of the particle’s magnetic
moment from the prediction of the Dirac theory:
aµ =
1
( gµ
2
2).
As magnetic moments are loop-level effects (the operators generating them are non-renormalizable
and therefore forbidden to enter the Lagrangian of the theory), it is possible that new physics might
contribute to them. This might actually be the case, as the current experimental value (from the E821
experiment at Brookhaven [15]) differs from the SM prediction with a statistical significance of 3.6s
[16]:
Daµ (Experiment SM) = (255 ± 80) ⇥ 10 11
8
The muon g
2 is the main concern of this work. It will be explored further in the next chapter.
9
3 The Anomalous Magnetic Moment Of The
Muon
3.1 Motivation
Magnetic moments are related to the spin of the particle by the Lande factor gµ :
~µ = gµ
⇣ q ⌘
~S
2m
For particles of mass m and spin-1/2, Dirac theory predicts a value of g = 2. As the operators
generating magnetic moments are forbidden in the Lagrangian of the theory, because they are
non-renormalizable, corrections to the Dirac value can only be generated on loop-level. Examples
are diagrams like the one in fig. 3.1, which constitutes the simplest QED contribution (a sketch of
the QED calculation will be given in the following section).
Currently, further and more precise experiments for measurement of the g 2 are in approval
stage. The benefit of these measurements is that several models of physics beyond the standard
model (BSM) predict similar signatures at LHC: For example, the Randall-Sundrum (RS) and
Universal Extra Dimensions (UED) extensions of the SM both predict heavy spin-2 Kaluza-Klein
excitations. Such models can be distinguished with the g 2, as UED models do not provide
significant corrections, while RS models do. The same is the case with various SUSY models, to the
extent that the Fittino collaboration considers g 2 to be the most important constraint for SUSY
[17]. In [16] it is also outlined that further investigating g 2 might provide an insight of BSM
physics, even if the LHC in its energy range “just” finds the SM-predicted Higgs boson.
µ
µ
µ
g
µ
Figure 3.1: Leading-order QED contribution to g
2
11
3 The Anomalous Magnetic Moment Of The Muon
3.2 How to calculate the Anomalous Magnetic Moment of the
Muon
We now turn to the theoretical calculation of the g
2 and what different parts contribute.
3.2.1 QED Contribution
In QED, it is quite simple to decompose the µµg matrix element into Lorentz covariant quantities:
The possible candidates for constructing these are the momenta p1 and p2 of the muons, the Dirac
gamma matrices gµ and the Pauli tensor sµn = 12 [gµ , gn ]:
iGa ( p1 , p2 ) = i ū( p2 )Pa u( p1 )

i ab
= ū( p2 ) FE (q)2 ga + FM (q2 )
s q b u ( p1 ).
2mµ
FE and FM are the electric and magnetic form factor, respectively. They both only depend on the
momentum difference squared q2 = ( p1 p2 )2 and in the classical limit q2 ! 0,
FE (0) = 0, FM (0) = aµ .
Finally, one finds
a
,
2p
which is the result obtained by Schwinger that served as the first major success of QED and Quantum
Field Theory in general. Most impressively, this value already accounts for 99% of the radiative
corrections!
aQED
=
µ
Further contributions from QED arise in higher loop orders, with each order being suppressed by
one more order of a. So far, these contributions have been computed up to tenth order(!) - where
9080 diagrams contribute - [18], with a value of
aQED,10L
= 116584718.951(9)(19)(7)(77) ⇥ 10
µ
14
,
where a was determined outside any QED calculations by measuring h/mRb [19], combined with
the very precisely known Rydberg constant and mRb /me . The uncertainties here arise from the
lepton mass ratios, the eighth and tenth order term, and the uncertainty in a.
So far, all QED contributions have gotten smaller and smaller when proceeding to higher loop
orders – significant changes here are therefore unexpected.
12
3.2.2 g
2 Projection Operator
In the general situation of the SM, the situation is slightly more complicated: Now with g5 , another
covariant enters the equation, creating four more terms in the vertex function Pa ,
qa
i( p
p2 )a g5
+ A4 ga g5 + A5 qa g5 + A6 1
.
2m
2m
P a = A1 g a + A2 ( p1 + p2 ) a + A3
By means of Gordon identities, this equation is transformed to
✓
Pa = FE (q2 )ga + FA (q2 ) ga
qa
q2
2m
◆
g5 + FM (q2 )
i ab
1 ab
s q b + FD (q2 )g5
s qb .
2m
2m
(3.1)
Here, we have introduced FD , the form factor for a possible dipole moment, as well as FA , the
form factor for a possible anapole moment.
Now we can understand why the anomalous magnetic moment cannot be generated on tree level
- this would require a term µ aµ meµ sµn Fµn , whose coupling constant is of mass dimension 1,
rendering it non-renormalizable.
Turning back to the vertex function Pa in equation (3.1), we need to find a way to extract aµ . This
can be done by applying a projection operator P a to the vertex function:
FM (q2 ) = Tr [Pa Pa ]
A reasonable ansatz for this projection operator is

a
P = (/
p 1 + m ) C1 g a + C2
( p1 + p2 ) a
(p
p2 ) a
+ C3 1
(/
p 2 + m ),
m
m
the coefficients C1 , · · · , C3 can then be determined by tracing them with the vertex function Pa ,
keeping the d explicitly for using Dimensional Regularization:
Tr[P a Pa ] =
h⇣
8m2 + 2(d

+ 2q2 (d

q2
+ 2q2
⌘
2) q2 C1 + 2( q2
1)C1 + q2
4m2
m2
q2
i
4m2 )C2 FE (q2 )
4m2
C2 FM (q2 )
2m2
C3 FD (q2 ).
such that
h
8m2 + 2q2 (d
2q2 (d
i
2) C1 + 2( q2
1)C1 + q2
q2
4m2 )C2 = 0
4m2
C2 = 1
2m2
!
C3 = 0,
13
which can be solved for the coefficients. We find that
2m2
FM (q2 ) =
2) q2 ( q2
(d
4m2 )

✓
◆
4m2 + (d 2)q2
a
⇥ Tr (/
p 1 + m) g +
( p1 + p2 ) a ( /
p 2 + m)Pa .
m(q2 4m2 )
To extract FM (0) directly, a bit more work is neccessary: First, we expand Pa to first order in
( p1 p2 ) a ,
P a ( p1 + p2 , p1
p2 ) ⇡ Pa ( p, 0) + ( p1
p2 ) b
∂
∂ ( p1
Second, we have to average over all orientations of p1
Z
dW( p1
1
4p
p2 )( p1
Z
dW( p1
p2 ) a ( p1
p2 )( p1
p2 ) b =
1
d
Pa ( p, p1
p2 )
( p1 p2 )=0
p2 ) b Tba ( p1 + p2 ).
⌘ Va ( p1 + p2) + ( p1
1
4p
p2 ) b
1
( p1
p2 , using
✓
p2 )2 hab
( p1 + p2 ) a ( p1 + p2 ) b
( p1 + p2 )2
◆
p2 ) a = 0
to execute the limit q2 ! 0. The result is
1
1)(d 2)m2
d 2⇣ 2
⇥ Tr
m ga + d ( p 1 + p 2 ) µ ( /
p1 + /
p2 )
2
aµ = FM (0) =
2( d
"
(d
⌘
1) m ( p1 + p2 ) a V a
(3.2)
#
m
ab
+
( p + p ) + m [ g b , ga ] ( /
p1 + /
p2 ) + m T
.
4 /1 /2
To calculate g 2, one simply has to split the amplitude for a given contribution into the parts V a
and T ab according to eq. (3.2.2) and apply the projector as in eq. (3.2).
This strategy bears the merit that after all Dirac traces are calculated, only scalar integrals will
remain, which simplifies further calculations, especially with computer algebra systems.
3.2.3 Electroweak Contributions
Electroweak contributions to the g 2 include exchanges of W ± , Z0 and Higgs bosons at one-loop
level (see Figure 3.2). Due to the large mass of these bosons compared to the muon, the structure of
these one-loop corrections is rather simple:
aEW,1L
µ
G m2
= pF
8 2p 2
"
10 1 ⇣
+
5 + 1(1
3
3
4s2w )2
⌘
+O
m2µ
2
mW,H
!#
Problems then arise at the two-loop level, where one may easily underestimate the contribution
due to suppression by a/p ⇡ 10 3 – in fact, the large Z mass creates numerically large logarithms,
14
enhancing the contribution by an order of magnitude (see c.f. [20], chapter 4). Further problems
arise as light quarks appear in the two-loop diagrams, contributing primarily in the low-momentum
region, where the description of perturbative QCD (PQCD) breaks down. Technical difficulties
are troubling this calculation even more, for example the “curse of combinatorics” – about 1700
diagrams contribute to the two-loop calculation.
g
µ
g
µ
W±
µ
W±
µ
Z, H
nµ , n̄µ
µ
µ
Figure 3.2: One-loop electroweak contributions
3.2.4 Hadronic Vacuum Polarization Contributions
g
µ
µ
g
g
µ
Hadrons
µ
Figure 3.3: Example LO hadronic vacuum polarization contributions
Hadronic vacuum polarization contributions are the largest hadronic contributions to g 2,
unfortunately also being governed by low-momentum contributions, where the description by
PQCD breaks down. For the calculation, it is possible to use the dispersion representation of
the photon propagator, which enables the use of experimentally measured cross sections for the
e+ e ! gg process to be of use. Despite the problems, the large NC ’t Hooft-limit of QCD provides
theoretical insight into these contributions.
3.2.5 Hadronic Light-by-Light Scattering Contributions
In addition to the hadronic vacuum polarization contributions, the g 2 receives further corrections
by hadronic light-by-light scattering. These are notoriously difficult to calculate, as they are primarily
governed by low-energy QCD, where the perturbative expansion fails. The situation is complicated
as three of the four photons are off-shell and have to be integrated over the full momentum space –
therefore, no experimental input is available on the corresponding correlator [21].
15
g
µ
µ
µ
µ
Figure 3.4: Hadronic light-by-light scattering contributions to g 2, the shaded region represents
the actual hadronic part (see Figure 3.5 for the pion pole and exchange contribution).
p±
p±
p0
p±
p±
Figure 3.5: (a) Pion-pole/other resonance contribution for hadronic light-by-light scattering; (b)
charged pion loop contribution
Currently, the pion pole and pion exchange contributions (see Figure 3.5) are assumed to be the
leading order contribution. For the pion form factor, chiral perturbation theory (ChPT), a low-energy
QCD approximation, valid to about 1 GeV yields good results. Problematically, it only contains the
pion as Goldstone bosons, as well as the h and h 0 resonances, but to look further into HLbL, higher
resonances have to be considered as well. In this regime, the Vector Meson Dominance (VMD),
Resonance Lagrangian Approach (RLA) or the extended Nambu-Jona-Lasinio Model give valuable
insight. Recently, Dyson-Schwinger equations have also been employed to calculate the HLbL part
of g 2 [22, 23], leading to a bit of controversy in the community concerning the size, theory error
and sign of the contributions.
As Spin-2 HLbL contributions to the g
explored more in-depth in chapter 8.
2 are the second main concern of this work, they will be
3.3 How to measure the Anomalous Magnetic Moment of the
Muon
This section explains how to measure the anomalous magnetic moment of the muon based on the
planned E989 experiment at Fermilab [16], which is currently in approval phase at the United States
Department of Energy.
16
3.3 How to measure the Anomalous Magnetic Moment of the Muon
3.3.1 Principle of the Experiment
The E989 experiment is based on measuring the muonic spin precession with muons in a superconducting storage ring. Assuming the muon momentum vector ~b perpendicular to the magnetic field,
~b · ~B = 0, the spin precession ws and cyclotron precession wc frequencies are given as
q~B
mg
gq~B
~c =
w
~s =
w
(1
2m
g)
q~B
,
mg
so the anomalous precession frequency can be determined by subtraction,
~a = w
~s
w
aµ
~c =
w
q~B
.
m
In the experiment, electric quadrupoles are used for vertical focusing in the muon storage ring.
This field appears in the muon rest frame as an additional magnetic field which affects the precession
frequency. In the case that the momentum vector ~b is perpendicular to both the magnetic and electric
fields ~B and ~E, the precession frequency is given by
~a = w
~s
w
~c =
w
q
m
aµ ~B
✓
aµ
1
g2
1
◆
~b ⇥ ~E
c
!
.
At a “magic momentum” of 3.094 GeV, g = 29.4, the coefficient of the ~b ⇥ ~E vanishes and only the
magnetic field determines the precession frequency.
Now, the main quantity determining the precession is the integrated magnetic field seen by the
muons during their flight, hence the magnetic field should be as uniform as possible.
In the weak decay of the muon, parity is violated, thus there exists a correlation between the
direction of the muon spin and the flight direction of the resulting electron – in the muon rest frame
the angular distribution of the electrons rotates with the same frequency wa of the muon spin.
aµ is then given by
aµ =
wa
wL
wa
=
wa /w̃p
R
=
,
wL /w̃p wa /w̃p
l R
where w̃p is the average magnetic field seen by the muons and wL is the Larmor frequency of the
muon. R = wa /w̃p is the ratio determined experimentally and l = µµ /µp is the muon-to-proton
magnetic moment ratio, which is determined from measurements in muonic hydrogen and theory.
3.3.2 Design of the Experiment
In the E989 experiment, bunches of 1011 protons from a linear accelerator with an average energy of
8 GeV are shot onto a target to produce 3.1 GeV positively charged pions. These pions decay further
via the dominant p + ! µ+ nµ process into highly relativistic, polarized muons. These muons are
injected into the muon storage ring from the older E821 experiment at Brookhaven, where they are
17
Figure 3.6: Muon injection and storage in the experiment, reproduced from [24]
vertically focused with electric quadrupoles. An overview of the experiment is shown in fig. 3.6.
18
4 Extra dimensions - Solving the
Gauge/Gravity Hierarchy Problem
The idea to introduce additional space dimensions is not a new one. In the beginning 1920s,
Oskar Klein and Theodor Kaluza already considered an extension of General Relativity to 5 space
dimensions in an attempt to unify the forces of gravity and electromagnetism [25, 26].
In his 1926 paper, Klein put forward the idea for compactification: The additional dimension
would not be a non-compact dimension as the four others, it would be a closed loop. This means
that a particle traveling along in the 5th dimension would always return to its starting point. As
particles are still modeled as plane waves, the equations of motion can be factored in a 4D and an
additional 1D part (or how many extra dimensions in your theory).
4.1 Solving the Hierarchy Problem with Extra Dimensions
Let us consider the 4D Newton potential for gravitation:
V (r ) =
GN
m1 m2
=
r
1
m1 m2
2
r
MPlanck
The extension to 4 + d dimensions reads as
V (r ) =
m1 m2
,
02+ d 1+ d
MPlanck r
0
where MPlanck
is the 4 + d-dimensional Planck mass. The influence of this modified Newton’s law
will only be substantial if we go to r ⇡ r 0 , where r 0 is the size of the extra dimension. For r
r0
the force flux into the extra dimension will be constant as the extra dimension is of limited volume.
Therefore, the flux will be proportional to r 0d . This leads us to
2
02+ d 0 d
MPlanck
= MPlanck
r .
Now the 4D MPlanck is not the fundamental scale for gravity anymore. Gravity is now actually strong,
its strength is simply not felt in 4D as most of the flux escapes into the additional dimension(s).
4.1.1 ADD/Large Extra Dimensions
In 1998, Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali published their proposal [27]: in the
ADD model, the electroweak scale MEW is taken as nature’s fundamental scale. The Standard Model
19
4 Extra dimensions - Solving the Gauge/Gravity Hierarchy Problem
is confined to a 4D brane of thickness 1/MEW in 4 + d dimensional space-time. The model is also
know as Large Extra Dimensions, as the extra dimensions are thought to be of a size of about 100 µm.
The “pure” ADD scenario is now mostly ruled out, as gravity has now been measured to a distance
of about 1 µm and was found to be compatible with the 4D Newton’s law. Recent measurements at
the LHC in events with one high-energy jet and missing transverse energy also turned out to be
fully consistent with the SM and made ADD more unlikely, by shifting its mass scale up to 4.37 TeV
for two extra dimensions and up to 2.53 TeV for six extra dimensions at Next-to-Leading-Order
(NLO) at 95% confidence level [28].
4.1.2 Randall-Sundrum
In 1999, Lisa Randall and Raman Sundrum published their solution to the hierarchy problem [29,
30]. The Randall-Sundrum (RS) models are based on the premise that one not only introduces an
Planck-scale size extra dimension, but also a warp factor which enables the derivation of the weak
scale from the Planck scale, when parameters are chosen accordingly.
The additional extra dimension is compactified on a circle, which opposite points identified,
which is called S1 /Z2 compactification.
Randall-Sundrum models are the main concern of this work, we will investigate them in more
depth in the following chapter.
20
5 Randall-Sundrum Models
Randall-Sundrum models are based upon a non-factorizable metric with an additional 4th space
dimension [29, 30]
ds2 = e
2s (f)
hµn dx µ dx n
r2 df2 ,
(5.1)
where hµn is the 4D Minkowski space metric tensor with signature ( 1, 1, 1, 1) and s (f) = kr |f|.
k and r are the curvature and radius of the 4th warped space dimension. Furthermore, opposite
points in the 5th dimension are identified. This is called S1 /Z2 -orbifolding (see Figure 5.1).
On both f = 0 and f = p, 4-dimensional subspaces are attached. These subspaces are called
3-branes (from membrane, 3 denotes the count of space dimensions). The warp factor acts as loupe,
scaling up length dimensions when moving from the Planck brane towards the TeV brane – MPlanck
for example, would be modified by factor of e0 at the brane located at f = 0. As one increases f up
to f = p, MPlanck is rescaled to e 2krp MPlanck . Because of that, the branes at f = 0 and f = p are
called Planck brane and TeV brane.
On these 3-branes, it is possible to have (3 + 1)-dimensional field theories, which will then couple
to the purely 4D metric tensor:
Planck
gµn
= Gµn ( x µ , f = 0)
TeV
gµn
= Gµn ( x µ , f = p )
The elegance of this model lies in the fact that nothing more but the existence of 3-branes and 4D
Poincare invariance is needed.
The classical action is
SRS = Sgravity + STeV + SPlanck
Z
Z p
h
i
p
Sgravity = d4 x
df
G
L + 2M3 R
STeV =
SPlanck =
Z
Z
4
d x
d4 x
p
p
(5.2)
p
gTeV [LTeV
VTeV ]
gPlanck [LPlanck
VPlanck ] ,
where VPlanck and VTeV are called brane tensions in the literature and act as vacuum energy on the
respective 3-branes.
21
π
0
r
4D UV brane
4D IR brane
Figure 5.1: Space geometry in Randall-Sundrum models. Opposite points on the S1 of the warped
extra dimension are identified by S1 /Z2 orbifolding.
Upon variation of (5.2) with respect to the metric, one arrives at the Einstein equations:
p
✓
1
G MN R
2
G R MN
◆
p
1
LG MN
G
3
4M
p
TeV µ n
+ VTeV
gTeV gµn
d M dN d ( f p )
p
Planck µ n
+ VPlanck
gPlanck gµn
d M dN d ( f
=
0).
The non-factorizable metric (5.1) satisfies the requirement of 4D Poincare invariance. Using this
ansatz for solving the Einstein equations, we arrive at
6s02
L
=
r2
4M3
3s00
V
= TeV
d(f
2
r
4M3 r
p) +
VPlanck
d(f
4M3 r
0),
where the primes denote the derivative with respect to f.
These equations are solved by
s(f) = r |f|
r
L
.
24M3
(5.3)
Now it becomes clear that the bulk in the interval [0, p ] is simply a slice of Anti-deSitter space,
which is a space with negative constant curvature, as L has to be negative.
The second derivative of (5.3) connects the brane tensions with the fundamental scale k:
00
s = 2r
r
) VTeV
22
L
(d(f 0) d(f p ))
24M3
= VPlanck = L = 24M3 k2 .
5.1 Solving the Gauge/Gravity Hierarchy Problem
Replacing
hµn ! hµn + h̄µn ,
in the metric (5.1), where h̄µn is the physical 4D graviton, we can now go to construct a 4D effective
theory with Kaluza-Klein excitations and derive Feynman rules from that. The result of this
derivation can be found in Appendix A.
5.1 Solving the Gauge/Gravity Hierarchy Problem
A 4D effective action can be obtained by inserting (5.1) with s according to 5 into the original 5D
action (5.2)
Z
Z p
p
Seff = d4 x
df2M3 re 2kr|f|
gPlanck RPlanck ,
p
and integrating over the f coordinate. We find for MPlanck ,
2
MPlanck
= M3 r
Z p
p
dfe
2kr |f|
=
M3 h
1
k
e
2krp
i
.
Let us now consider the Higgs field, which is bound to the TeV brane in the original model:
For the action it follows that

Z
⇣
⌘2
p
µn
4
STeV
d x
gTeV gTeV Dµ H † Dn H l | H |2 v20
.
By continuity, we know that
TeV
gµn
=e
such that
STeV
Z
4
d x
which after rescaling H ! e
STeV
Z
d4 x
p
gPlanck e
2krp H
p
4krp

2krp Planck
gµn
gPlanck Dµ H † Dn H
µn
becomes

µn
gPlanck gPlanck Dµ H † Dn H
⇣
l | H |2
⇣
l | H |2
e
v20
⌘2
,
2krp 2
v0
⌘2
.
If now ekrp ⇡ 1015 , physical masses of O(TeV) can be created as the mass scale is set by
v=e
2krp
v0 ,
which is only governed by parameters of O( MPlanck ) – like k, M and r. To solve the gauge-gravity
hierarchy problem, only kr ⇡ 10 is necessary.
Despite this great success, some problems remain to be solved. In the next section, problems of
the original Randall-Sundrum model will be discussed and solutions will be proposed.
23
5.2 Problems of the Original Model and their Solutions
5.2.1 Super-GZK Events, Neutrino Masses and Proton Decay
In 1966, Greisen, Zatsepin, and Kuzmin noted that particles carrying an energy above ⇡ 1020 GeV,
the GZK cutoff, would be absorbed by the Cosmic Microwave Background (CMB), as photons,
protons and nuclei would have been depleted of their energy over a distance of ⇡ 100 Mpc by
interaction with the 2.3 eV photons of the CMB [31, 32].
Over a few years lateron, about 20 events with an energy above the GZK cutoff have been
observed. Their source has been theorized to be ultra-high energy neutrinos, which can travel the
universe with virtually no energy loss. These can produce Z bosons through annihilation with the
relic neutrino background. This mechanism is called Z-burst.
In 2000, Davoudiasl, Hewitt and Rizzo published their analysis centered around a KK gravitonenhanced Z-burst mechanism, where a Z-burst peak in the energy spectrum would be followed
by numerous and widening KK graviton peaks [33]. While the SM expectation for the count of
Z-burst events over a timeframe of five years at the Auger Observatory would be n ⇡ 0.04 21, the
mechanism proposed by Davoudiasl, et al. predicts an event count of over 1000.
Facing the experimental results, the Randall-Sundrum model with SM particles confined to
the TeV brane is ruled out, since the Auger Collaboration only reported a few events of energy
1020 GeV in their 5-year results [34], completely consistent with the SM prediction.
Further, as argued by Gherghetta and Pomarol [2000NuPhB.586..141G], having the SM confined
to the TeV brane introduces the problem that higher-dimensional operators excerbating the unobserved proton decay are not Planck-scale suppressed anymore, but are scaled down to MEW . This
problem can be solved by moving the SM into the bulk as well.
Consequently, the classical action (5.2) has to be modified. While we will keep the Higgs boson
on the TeV brane to keep our model solving the hiearchy problem, the remaining part of the SM will
be allowed to propagate in the bulk:
STeV = SSM + SHiggs =
+
Z
Z
d4 x
d4 x
Z p
p
p
df
p

G LSM
gTeV gTeV Dµ H † Dn H
µn
⇣
l | H |2
v20
⌘2
VTeV
5.2.2 Impact on Electroweak Precision Observables
For a parametrization of the effects from new physics regarding electroweak precision observables
like W and Z mass in the SM, the Peskin-Takeuchi parameters can be used [35]. These are based
upon the assumptions that the SM gauge group only consists of SU (2)L ⇥ U (1)Y , that the new
physics scale is substantially higher than the electroweak scale of 250 GeV and that there are no
unsuppressed additional couplings to light fermions.
24
5.2 Problems of the Original Model and their Solutions
Then, these paramters, called S, T, and U, can be written as

16p 2 s2w c2w
s2w c2w 0
0
0
P
(
0
)
+
PZA (0) PAA
(0)
ZZ
sw cw
e2
i
4p 2 h
T = 2 2 2 PWW (0) c2w PZZ (0) 2sw cw PZA (0) s2w PAA (0)
e cw m Z
i
16p 2 s2w h 0
2 0
0
2 0
U=
P
(
0
)
c
P
(
0
)
2s
c
P
(
0
)
s
P
(
0
)
.
w
w
w ZZ
w AA
WW
ZA
e2
S=
Problems arise in the T parameter when working with the original RS model: while PAA is
protected from corrections by gauge invariance, and the other self-energies behave normally, PZZ
receives a correction µ log k/mKK , which is not consistent with experimental observations [36].
This problem can be remedied by adding SU (2) R ⇥ U (1) X to the gauge group, which is then
broken down to the SM gauge group by the Higgs VEV on the TeV brane and additionally, by
symmetry breaking on the UV brane.
The additional symmetry then protects the Peskin-Takeuchi parameters from dangerously large
corrections, very much like the custodial symmetry in the SM.
25
6 Analysis - g 2 Spin-2 Graviton
Contributions
6.1 Introduction
As discussed in the previous chapters, models with warped extra dimensions, like Randall-Sundrum,
may provide significant contributions to the muon g 2.
In this chapter, we discuss contributions from KK gravitons. The contributions on one-loop level
are shown in figs. 6.1, 6.2, and 6.3. The Feynman rules for this calculation can be found in appendix
A, they were derived from the Fierz-Pauli linearisation of the Einstein-Hilbert action. As an effective
field theory, whose UV completion is currently unknown, it is non-renormalizable, which can be
directly seen in the Feynman rules: the couplings all have a mass dimension of 1.
g
Gr
g
g
Gr
g
µ
µ
µ
µ
µ
µ
Gr,g
Figure 6.1: The triangle graviton-muon contributions, called aµ
g
Gr
in the text.
Gr
g
µ
µ
µ
µ
µ
µ
Gr,µ,g
Figure 6.2: The flight-into-sunset four-vertex contributions, called aµ
in the text.
27
6 Analysis - g
2 Spin-2 Graviton Contributions
g
µ
µ
Gr
µ
µ
Gr,µ
Figure 6.3: The triangle graviton-muon contributions, called aµ
in the text.
6.2 Relation to Previous Work
In several previous publications, graviton contributions were discussed, with quite different model
assumptions, and different results, However, no consensus has been found so far:
1. Berends and Gastmans[37] first calculated the contribution of a massless graviton to the g 2
in 1975. While the numerical value was miniscule due to Planck-scale suppression, the more
surprising result was that it actually is finite.
2. Graesser [38] made the first calculation of the muon g 2 in extra-dimensional models, in
his case in the ADD model. His results include a cancellation of the divergences when all
contributing diagrams are summed up. Furthermore, the KK states in ADD have a large
multiplicity; therefore, primarily KK gravitons near the theory’s cutoff will contribute.
3. Kim, et. al [39] also calculated the muon g 2 in warped extra dimensions, but in their case
with the SM confined to the TeV brane. They report a non-divergent result of
aµG
5
=
16p 2
✓
mµ
Lp
◆2
nc ,
where nc is the number of KK states contributing. They fix nc to be of O(100), via unitarity
bounds from the gg ! gg, which is domininated by KK graviton contributions, as SM
contributions are absent on tree-level.
4. Davoudiasl, et. al [40] extended Graesser’s analysis to warped extra dimensions with the SM
propagating in the bulk, noting that the divergence calcellation found in ADD does not occur
with Randall-Sundrum models, as the couplings of the KK modes are not universal anymore.
They expect the graviton contributions to be
aµG
=Â
ijk
ffGr C ffGr m m
C0ik
ij µ
0jk
16p 2 L2p
"
#
L2
log
,
m2KK
ffGr are the fermion-fermion-graviton coupling constants and i, j, k are the flavour
where the Cijk
indices.
5
2
5. Park and Song [41] found a contribution of 16p
2 ( m µ /MPlanck ) , argumenting the factor of 5
arises from the five degrees of freedom a spin-2 particle has. While this seems to be intuitive,
the same argumentation fails completely in the cases of the photon or of massive gauge bosons.
28
6.3 Strategy
6. Very recently, Beneke, et al. [42] calculated g 2 in theories with warped extra dimensions
in a novel way: instead of relying on factorisation of the 4D and 5D parts via Kaluza-Klein
decomposition, they developed a wholly 5D formalism on curved space-time. With this
approach, they are able to omit somehow arbitrary limits on the number of KK modes summed
over. In contrast to the aforementioned papers, Beneke et al. do not treat gravitons, instead
referring to the already existent literature.
6.3 Strategy
To calculate the graviton contributions to g 2, a combination of FeynArts [43], TwoCalc [44–46],
and FormCalc [47] based on Mathematica [48] was used. Cross-checks were additionally performed
with FeynRules [49]. The program was originally developed by Dominik Stöckinger for two-loop
calculation in SUSY models [45, 46] and partially rewritten by Christoph Gnendiger. It has been
extended further for this work, in order to include spin-2 particles and their interactions as well as
to cope with the extended Lorentz structures required for calculations with these particles.
6.3.1 Calculation Pipeline
An overview of the calculation pipeline is given in Figure 6.4. Let us now describe what the gminus3
package does in detail:
1. FeynArts needs two types of model files: a generic model file and a model file with the
definitions of the particle couplings. For SM or MSSM calculations, generic model files already
exist – these determine the Lorentz structure of the couplings and the propagators of the fields.
In our calculation, an extended generic model file, based on QED.gen from FeynArts, was
constructed to accommodate the Lorentz structures needed by the Feynman rules (given in
Appendix A). As FeynArts has particular requirements on how these have to be written, this
proved to be not trivial. Additionally, a model file containing the couplings was written. Both
can be found in Appendix B.
2. FeynArts then generates a list of scattering amplitudes conforming to the desired topology,
which consists of, in our case, diagrams for the process µg ! µ with one loop and only muons,
(KK-)gravitons, and photons allowed in the loop.
3. These full scattering amplitudes are then fed to MDMProjector [45, 46], a package realising the
projection operator introduced in (3.2). MDMProjector will return a new scattering amplitude,
which effectively looks like a self energy for the muon, containing multiple Dirac traces. For
this work, MDMProjector was extended to accept multiple products of metric tensors and
additional mass terms occuring with spin-2 particles.
4. Next, a calculation of the self-energy scattering amplitude returned by MDMProjector is
possible both with FormCalc, as well as with TwoCalc.
a) TwoCalc evaluates Dirac traces in the scattering amplitudes, then reduces any occuring Lorentz structures as much as possible. With these results, integral reduction by
Integration-by-Parts (IBP) is performed [45, 46], so that the end result only contains A0
29
6 Analysis - g
Mathematica
Generic Model File
Generate Feynman
Diagrams
with FeynArts
Model File
Full scattering
amplitudes for
µɣ ➝ µ process
Project out magnetic
form factor with
MDMProjector
Calculation
Calculation
with TwoCalc
Large Mass
Expansion
Integral
Reduction
MathLink
Calculation
with FormCalc
Expansion in ϵ
g-2
FORM
Figure 6.4: Calculation pipeline of the gminus3 package developed for and used in this work
integrals. The results from TwoCalc are further processed with large mass expansion and
are then expanded in e, finally yielding a result for aµ .
b) FormCalc effectively is a wrapper for the program FORM [50], which is much faster
and more efficient then Mathematica, but harder to use and less generally applicable.
30
6.4 Details of the Calculation
FormCalc provides an easy-to-use Mathematica interface for FORM for the calculation of
scattering amplitudes. Similarily to TwoCalc, it includes routines for common tasks, such
as the reduction of Dirac traces. Differently than TwoCalc, FormCalc performs integral
reduction in the end, via Passarino-Veltman reduction – if requested.
FormCalc was introduced into the project at a late stage, primarily to enable crosschecking of the results. As MDMProjector produces a non-standard FeynArts amplitude,
where the amplitude looks like a scalar self-energy after projection, some custom modifications had to be made to FormCalc. This was performed with the help of Thomas
Hahn.
The fact that FormCalc now understands both the output of FeynArts and the output of
MDMProjector renders it an invaluable tool for checking our calculations. If a calculation
produces unanticipated results, it is possible to cross-check with the combinations
FeynArts/MDMProjector/TwoCalc, FeynArts/MDMProjector/FormCalc and FeynArts/FormCalc. If those deliver the same results and errors still persist, the error is most
likely to be found in the (generic) model file.
5. Both TwoCalc and FormCalc yield a file containing the final expression for aµ . The result can
then be checked for consistency. For better error checking options, single diagrams are marked
in the final result. This mark can then easily be removed by pattern matching and the results
simplified further.
6.4.1 Model Files
For the generation of Feynman diagrams, both a generic model file and a model definition were
handcrafted. Both can be found in Appendix B. In this section, we want to highlight a few details.
Propagator Definition
The massive graviton propagator is defined as
Pµn,rs (k) =
( Gµn Gns + Gµs Gnr )
iDµn,rs
=i
k 2 m2
k 2 m2
2
D 1 Gµn Grs
.
(6.1)
Into FeynArts, this translates as
1
2
3
4
5
6
7
8
AnalyticalPropagator[Internal][ s T[i, mom, {li1, li2} -> {li3, li4}] ] ==
1/2 I PropagatorDenominator[mom, Dummy[GMassTerm] Mass[T[i]]] *
(
(MetricTensor[li1, li3] Dummy[KFactor] FourVector[mom, li1] * FourVector[mom, li3]/Mass[T[i]]^2)
31
6 Analysis - g
9
10
11
12
13
14
15
16
17
18
19
+(MetricTensor[li1, li4] Dummy[KFactor] FourVector[mom, li1] * FourVector[mom, li4]/Mass[T[i]]^2)
- 2/($D-2 + Dummy[GMassTerm])Dummy[vDVZ]
(MetricTensor[li1, li2] Dummy[KFactor] FourVector[mom, li1] FourVector[mom, li2]/Mass[T[i]]^2)
)
FeynArts differentiates between internal and external propagators. In our case, only a definition
for the internal propagator is needed, as we do not consider gravitons as external particles.
We want to emphasize here the Dummy[vDVZ], Dummy[GMassTerm] and Dummy[KFactor]
variables. These were introduced to be able to discern divergences and finite parts in the final results
arising from the various parts of the graviton propagator:
1. Dummy[GMassTerm] for example enables to do a calculation for massless and massive
gravitons simultaneously (for consistency in the massless case, Dummy[KFactor] has to
be set to 0 as well). The variable can then be set to 1 or 0 after the calculation is finished. This
was especially useful to explore how divergences arise in the massless and massive cases.
2. Dummy[KFactor] was used to investigate whether gravitational Ward identities render the
k µ k n terms in the graviton propagator 0 in the final result or not.
3. Dummy[vDVZ] was used to show that the last term of the graviton propagator, which we call
the vDVZ term, does have an impact on the generated divergences or not.
Vertex Definition
As the vector-vector-tensor vertex is the vertex with the most involved structure,
Abb (k2 )
=
Grµn
Aaa (k1 )
Wµnab (k1 , k2 ) =
i
⇤
1 ab ⇥
d Wµnab (k1 , k2 ) + Wnµab (k1 , k2 )
L
1
hµn (k1a k2b
2
+ hab k1µ k2n
+ hµa (k1 k2 hnb
k1 k2 hab )
k1b k2b )
hµb k1n k2a ,
it shall serve as the example here. Into FeynArts, it translates as
1
2
3
AnalyticalCoupling[
s1 T[j1, mom1, {li1, li2}],
s2 V[j2, mom2, {li3}],
32
4
5
6
7
8
9
10
s3 V[j3, mom3, {li4}]
] ==
G[+1][s1 T[j1], s2 V[j2], s3 V[j3]] .
{
Wgamma[li1, li2, li3, li4, mom2, mom3] +
Wgamma[li2, li1, li3, li4, mom2, mom3]
}
AnalyticalCoupling[particles] defines the couplings of the particles listed in the sequence
particles. It may contain (as of FeynArts 3.7) scalars, pseudoscalars, fermions, vector and tensor
particles. particles also defines the names of the Lorentz indices (here li1 and li2 for the
graviton and li3 and li4 for the vectors, respectively) and four-momenta of the particles taking
part in the interaction.
G[+1][] is also worth mentioning – it defines whether the vertex is closed (corresponding to
+1) or unclosed (corresponding to 1) under permutations of fermions. This will not lead to any
problems if wrongly defined for vertices not containing fermions – but with fermions, it should be
checked carefully.
Wgamma[] is the tensor Wµnab , with the corresponding Lorentz indices, defined outside the vertex
definition as
1
2
3
4
5
6
7
8
9
10
11
12
Wgamma[mu_, nu_, alpha_, beta_, k1_, k2_] :=
(1/2 MetricTensor[mu,nu] *
(FourVector[-k1, beta] FourVector[-k2, alpha]
- ScalarProduct[-k1, -k2] MetricTensor[alpha, beta])
+ MetricTensor[alpha, beta]
FourVector[-k1, mu] FourVector[-k2, nu]
+ MetricTensor[mu, alpha]
(ScalarProduct[-k1, -k2] MetricTensor[nu, beta]
- FourVector[-k1, beta] FourVector[-k2, nu])
- MetricTensor[mu, beta]
FourVector[-k1, nu] FourVector[-k2, alpha]
)
Particular care must be taken with objects defined outside of AnalyticalCoupling[]: FeynArts
model files are valid Mathematica code. Therefore, all cautions that apply when writing Mathematica
code apply here as well. As Mathematica does not produce errors if a "lone" expression is
encountered, it is easily possible to miss that a tensor as Wgamma[] has been defined wrongly.
Ideally, the whole model file should be unit-tested meticulously, as this has been the single most
important cause of errors in the calculation.
6.4.2 Unit Testing with Mathematica
In the context of computer programming, unit testing refers to a method where individual parts of a
program are tested independently against a set of already-known result data to determine whether
this part of the program is working correctly.
A unit testing framework for Mathematica is available in the context of the Wolfram Workbench
Integrated Development Environment (IDE) from Wolfram Research. Workbench is an Eclipse-based
33
6 Analysis - g
IDE with graphical user interface. As our calculation pipeline is designed to run on clusters as well
as personal notebooks, it is undesirable to be forced to use a graphical interface for unit testing. To
solve this problem, a lean unit testing framework named UnitTest for Mathematica was built⇤ .
With the UnitTest package, unit tests can be set up as simple as
UnitTest[test, expected_result]
For example,
UTDeriveXSquared = UnitTest[D[x^2,x], 2x]
which will test whether the derivative function D[] really produces 2x when deriving x2 with
respect to x.
If multiple unit tests are defined in a Mathematica notebook or source file, the function UnitTestSummary[]
comes in handy: It scans the whole Global‘ namespace of Mathematica for symbols with the head
UnitTest and summarizes their results.
As test in the first argument of UnitTest[] can be arbitrarily complicated, this provides a
flexible framework for testing functions, model files and even whole calculations.
6.5 Results and Discussion
6.5.1 Results
In this section, log(m2 ) = log(m2 /µ2 ) is understood.
Massless Gravitons
The results for the case of massless gravitons, to order # are
Gr,µ
aµ
Gr,g
aµ
Gr,µ,g
aµ
⇤ UnitTest
34
1 ⇣ m µ ⌘2 1
#
48p⇣2 ⌘L
2
3
2
⇣ m ⌘2
log mµ
61
µ
5
+4
#0 ,
L
48p 2
288p 2
=
1 ⇣ m µ ⌘2 1
8p 2 L ⇣# ⌘
2
3
2
⇣ m ⌘2
log
m
µ
7
µ
5
+4 2
#0 ,
2
L
8p
4p
=
11 ⇣ mµ ⌘2 1
96p 2 ⇣ L ⌘ #
2
3
⇣ m ⌘2
11 log m2µ
4
µ
5
+4
#0 .
L
48p 2
9p 2
=
can be found on the author’s GitHub page
(6.2)
In total, the divergences cancel, leaving a finite result of
Gr,µ
Gr,µ,g
+ 2aGr,g
+ 2aµ
µ
⇣
⌘
2
mµ
7
=
,
32p 2 L
aµ = aµ
in agreement with the results from Berends and Gastmans [37]. As the massless graviton is the zero
mode of the KK graviton tower in extra dimensional models, its contribution is not suppressed by
LRS ⇡ O(TeV), but by MPlanck , leaving only a miniscule contribution to g 2.
From this result, one can immediately understands why Park/Song’s interpretation [41] of the
5
factor 16p
2 (in the case of massive KK gravitons) arises because Spin-2 particles have 5 degrees of
freedom is flawed – while intuitive, it would give the massless graviton 3.5 degrees of freedom,
while it only has two.
The results we present in the following are relatively model-independent, they work for ADD
models, as well as for RS models - the cutoff just has to be adjusted appropriately.
Massive KK Gravitons - no k µ k n terms (deDonder gauge)
These results have been obtained in the harmonic/de Donder gauge,
Gaµn gµn = 0.
This equation is the generalization of the d’Alembert equation ⇤f = 0 to space-time and essentially
drops the non-transversal part of the massive graviton propagator.
We find,
Gr,µ
aµ
=
+
h
1 ⇣ m µ ⌘2 1
#
48p 2 L
16m6µ
9p 2 L2 m4Gr
m4µ log m2Gr
13m4µ
72p 2 L2 m2Gr
m4µ log
⇣
m2µ
4m6µ
⌘
log m2Gr
3p 2 L2 m4Gr
m2µ log m2Gr
+
⇣ ⌘
4m6µ log m2µ
3p 2 L 2 m4Gr
7m2µ
i
#0 ,
48p 2 L 2
288p 2 L2
12p 2 L2 m2Gr
12p 2 L2 m2Gr
1 ⇣ m µ ⌘2 1
Gr,g
aµ =
#
8p 2 L
⇣ ⌘
"
4 log m2
4
4
2
m
mµ
mµ log mGr
µ
µ
+
+
2
2
2
2
2
2
2
2
2
6p L mGr
3p L mGr
3p L mGr
#
2
2
2
mµ log mGr
11mµ
+
#0 ,
4p 2 L2
24p 2 L2
11 ⇣ mµ ⌘2 1
Gr,µ,g
aµ
=
#
96p 2 L
⇣ ⌘
"
6 log m2
6
4
6
2
31m
155mµ
29mµ
31mµ log mGr
µ
µ
+
+
2
4
4
4
2
2
2
2
2
2
2
2
432p L mGr
432p L mGr
72p L mGr
72p L mGr
+
+
+
35
6 Analysis - g
+
⇣ ⌘
2m4µ log m2µ
2m4µ log m2Gr
9p 2 L2 m2Gr
9p 2 L2 m2Gr
+
11m2µ log m2Gr
49m2µ
48p 2 L2
288p 2 L2
#
#0 .
In the case of massive KK gravitons, with k µ k n gauged away from the graviton propagator (6.1),
the divergences remain the same as in the massless case (6.2) and, therefore, cancel. We find for the
contribution per KK graviton n,
Gr,µ
Gr,µ,g
+ 2aGr,g
+ 2aµ
µ
⇣
⌘
95m2µ ⇣ mµ ⌘2
835m4µ
mµ 2
5
=
+
16p 2 L
432m2Gr p 2 L
432m4Gr p 2
⇣
⌘
⇣
⌘2
⇣ m ⌘2
m2µ 10m2Gr 113m2µ log mmGr
µ
µ
+
,
4
2
L
72mGr p
aGr,n
= aµ
µ
in agreement with Graesser’s results in [38]. However, we provide an extension of Graesser’s
results, as the finite parts receive additional, sub-leading contributions of at least O(m2µ /m2Gr ), which
were not calculated in Graesser’s work. Numerically, however, these can be safely neglected, due to
being suppressed by at least a factor of (mµ /mGr )2 .
Using our technique of separating the different contributions from the graviton propagator, we
were also able to verify that the contributions arising from the vDVZ term in the propagator remain
finite, as already noted by Graesser.
Massive KK Gravitons - k µ k n terms included
A theoretically interesting effect can be observed when the longitudinal parts of the massive graviton
propagator are included:
Gr,µ
aµ
+
+
Gr,g
aµ
m2µ
=
"
197m6µ
48p 2 L2 m2Gr
⇣ ⌘
m4µ log m2µ
108p 2 L2 m4Gr
72p 2 L2 m2Gr
12p 2 L2 m2Gr
"
15m2Gr ⇣ mµ ⌘2 1
L
#
144p 2 m2Gr
25m6µ log m2Gr
18p 2 L2 m4Gr
#
5m2µ
+
#0 ,
72p 2 L2
2m2µ
2m4µ
27p 2 L2 m2Gr
5m2µ log m2Gr
24p 2 L2
36
7m4µ
5m4µ log m2Gr
=
+
192p 2 m2Gr
⇣ m ⌘2 1
µ
L
#
5m4µ log
18p 2 L2
+
m2Gr
m2Gr
53m2µ
144p 2 L2
#
#0 ,
+
⇣ ⌘
11m4µ log m2µ
36p 2 L2 m2Gr
+
⇣ ⌘
25m6µ log m2µ
18p 2 L2 m4Gr
Gr,µ,g
aµ
=
+
+
m2µ
"
15m2Gr ⇣ mµ ⌘2 1
L
#
144p 2 m2Gr
5m6µ
16p 2 L2 m4Gr
5m4µ log m2Gr
24p 2 L2 m2Gr
7m4µ
216p 2 L2 m2Gr
⇣ ⌘
2m4µ log m2µ
9p 2 L2 m2Gr
+
+
3m6µ log m2Gr
8p 2 L2 m4Gr
5m2µ log m2Gr
24p 2 L 2
⇣ ⌘
3m6µ log m2µ
8p 2 L 2 m4Gr
#
m2µ
#0 .
8p 2 L2
For the final contribution per KK graviton, we get
Gr,µ
Gr,µ,g
+ 2aGr,g
+ 2aµ
µ
95m2µ ⇣ mµ ⌘2
835m4µ
5 ⇣ m µ ⌘2
=
+
16p 2 L
432m2Gr p 2 L
432m4Gr p 2
⇣
⌘
⇣
⌘2
⇣ m ⌘2
m2µ 10m2Gr 113m2µ log mmGr
µ
µ
+
,
L
72m4Gr p 2
aGr,n
= aµ
µ
While the divergences still cancel, the effect is fascinating: The divergences here are augmented
by additional powers of k in the numerator, creating different coefficients than in the previous
case. However, they still cancel altogether and leave the same finite part. This is the effect of the
gravitational Ward identity.
This extends the claim made by Graesser [38] that the longitudinal parts do not change the result
in any way. They in fact do change the constituents of the result, but leave the same sum in the end
as in the case where the de Donder gauge was used.
The vDVZ term does not contribute to the divergent part in this case, too.
6.5.2 Compatibility with the Results from Beneke, et al.
In [42], Beneke, et al. found the correction to aµ in the minimal RS model, calculated fully in 5D, to
be
✓
◆
1 TeV
Daµ ⇡ 8.8 · 10 11 ⇥
,
T
with T = 500 GeV, corresponding to KK resonances beginning at 1.3 TeV. At this value of T, the
correction to aµ is of the order of the theoretical uncertainty and well below the magnitude of
graviton contributions. However, the calculation by Beneke, et al. is subject to change, as the
minimal RS model is ruled out by electroweak precision measurements (see section 5.2.2) and they
already indicated they might extend it to the custodial RS model, which has a changed particle
content.
37
6 Analysis - g
6.5.3 Compatibility with Results from the ATLAS and CMS experiments
Currently, the strongest constraint on RS models comes from diphoton events in proton-proton
collisions at the LHC. With 2011 data from the ATLAS Collaboration, the limit is set to 1.00 (2.06) TeV
mass for the lightest KK graviton for k/MPlanck = 0.1 (0.01) [51]. Further limits are imposed by
searches in final states with leptons and jets from ZZ and WW resonance production [52, 53], but
these are less severe than the aforementioned limit from the diphoton channel.
The CMS Collaboration found similar results, setting the exclusion limit for KK gravitons at 95%
confidence level between 0.86 TeV and 1.84 TeV, depending on k/MPlanck [54].
Figure 6.5: RS cutoff L as function of k/MPlanck and the mass of the lightest graviton. Experimentally favoured is the high-mGr , high-k region.
A plot showing the RS cutoff scale L in dependence of the mass of the lightest graviton and
k/MPlanck is shown in Figure 6.5
6.5.4 Compatibility with E821 Results
The E821 experiment at Brookhaven found for aµ [15]
aE821
= 11659208.0(6.3) ⇥ 10
µ
10
,
leaving a difference to the SM of
Daµ (Experiment
SM) = (255 ± 80) ⇥ 10
11
unexplained. The contribution of a single massive graviton was determined to be
(n)
aµ =
38
5 ⇣ m µ ⌘2
16p 2 L
Extra dimensions
CI
LQ
V'
ferm.
Excit. New quarks
Large ED (ADD) : monojet + E T ,miss
Large ED (ADD) : monophoton + E T ,miss
Large ED (ADD) : diphoton & dilepton, mγ γ / ll
UED : diphoton + E T ,miss
1
S /Z 2 ED : dilepton, mll
RS1 : diphoton & dilepton, mγ γ / ll
RS1 : ZZ resonance, mllll / lljj
RS1 : WW resonance, mT ,lν lν
RS g →tt (BR=0.925) : tt → l+jets, m
KK
tt,boosted
ADD BH ( M TH /M D =3) : SS dimuon, N ch. part.
ADD BH ( M TH /M D =3) : leptons + jets, Σ p
T
Quantum black hole : dijet, Fχ(mjj )
qqqq contact interaction : χ(m )
jj
qqll CI : ee & µµ, m
ll
uutt CI : SS dilepton + jets + E T ,miss
Z' (SSM) : mee/ µ µ
Z' (SSM) : mττ
W' (SSM) : mT,e/µ
W' (→ tq, g =1) : mtq
R
W'R ( → tb, SSM) : m
tb
W* : mT,e/µ
Scalar LQ pair (β =1) : kin. vars. in eejj, eν jj
Scalar LQ pair (β =1) : kin. vars. in µµjj, µν jj
Scalar LQ pair (β=1) : kin. vars. in ττjj, τν jj
th
4 generation : t't'→ WbWb
th
4 generation : b'b'(T T 5/3 )→ WtWt
5/3
New quark b' : b'b'→ Zb+X, mZb
Top partner : TT → tt + A 0A 0 (dilepton, M )
T2
Vector-like quark : CC, mlν q
Vector-like quark : NC, mllq
Excited quarks : γ -jet resonance, m
γ jet
Excited quarks : dijet resonance, mjj
Excited lepton : l-γ resonance, m
lγ
Techni-hadrons (LSTC) : dilepton,mee/ µ µ
Techni-hadrons (LSTC) : WZ resonance (ν lll), m
T ,WZ
Major. neutr. (LRSM, no mixing) : 2-lep + jets
W R (LRSM, no mixing) : 2-lep + jets
H±L± (DY prod., BR(H±L±→ll)=1) : SS ee (µµ), m
ll
H±L± (DY prod., BR(H±±→eµ)=1) : SS eµ, meµ
L
Color octet scalar : dijet resonance, mjj
-1
1.93 TeV
4.37 TeV
M D (δ =2)
M D (δ =2)
10-1
L =4.8 fb , 7 TeV [ATLAS-CONF-2012-072]
-1
-1
L =4.7 fb , 7 TeV [1211.1150]
1
1.41 TeV
∫
10
102
Mass scale [TeV]
ATLAS
4.18 TeV M S (HLZ δ =3, NLO)
Preliminary
Compact. scale R -1
-1
-1
L =4.9-5.0 fb , 7 TeV [1209.2535]
4.71 TeV MKK ~ R
-1
L =4.7-5.0 fb , 7 TeV [1210.8389]
2.23 TeV Graviton mass (k / M Pl = 0.1)
-1
L =1.0 fb , 7 TeV [1203.0718]
845 GeV Graviton mass (k / M Pl = 0.1)
-1
Ldt = (1.0 - 13.0) fb-1
L =4.7 fb , 7 TeV [1208.2880]
1.23 TeV Graviton mass (k / M Pl = 0.1)
-1
1.9 TeV g mass
KK
s = 7, 8 TeV
-1
L =1.3 fb , 7 TeV [1111.0080]
1.25 TeV M D (δ =6)
-1
L =1.0 fb , 7 TeV [1204.4646]
1.5 TeV M D (δ =6)
-1
L =4.7 fb , 7 TeV [1210.1718]
4.11 TeV M D (δ =6)
-1
7.8 TeV Λ
-1
L =4.9-5.0 fb , 7 TeV [1211.1150]
13.9 TeV Λ (constructive int.)
-1
L =1.0 fb , 7 TeV [1202.5520]
1.7 TeV Λ
-1
L =5.9-6.1 fb , 8 TeV [ATLAS-CONF-2012-129]
2.49 TeV Z' mass
-1
L =4.7 fb , 7 TeV [1210.6604]
1.4 TeV Z' mass
-1
L =4.7 fb , 7 TeV [1209.4446]
2.55 TeV W' mass
-1
L =4.7 fb , 7 TeV [1209.6593]
430 GeV W' mass
-1
L =1.0 fb , 7 TeV [1205.1016]
1.13 TeV W' mass
-1
L =4.7 fb , 7 TeV [1209.4446]
2.42 TeV W* mass
st
-1
L =1.0 fb , 7 TeV [1112.4828]
660 GeV 1 gen. LQ mass
nd
-1
L =1.0 fb , 7 TeV [1203.3172]
rd
-1
L =4.7 fb , 7 TeV [Preliminary]
-1
L =4.7 fb , 7 TeV [1210.5468]
656 GeV t' mass
-1
670 GeV b' (T ) mass
5/3
-1
L =2.0 fb , 7 TeV [1204.1265]
400 GeV b' mass
-1
L =4.7 fb , 7 TeV [1209.4186]
483 GeV T mass (m(A ) < 100 GeV)
0
-1
1.12 TeV VLQ mass (charge -1/3, coupling κ qQ = ν /mQ)
-1
1.08 TeV VLQ mass (charge 2/3, coupling κ qQ = ν /mQ)
-1
L =2.1 fb , 7 TeV [1112.3580]
2.46 TeV q* mass
-1
3.84 TeV q* mass
-1
2.2 TeV l* mass ( Λ = m(l*))
-1
L =4.9-5.0 fb , 7 TeV [1209.2535]
850 GeV ρ / ωT mass ( m(ρ / ωT) - m(πT) = M )
T
T
W
-1
L =1.0 fb , 7 TeV [1204.1648]
483 GeV ρ mass ( m(ρ ) = m(πT) + mW , m(a ) = 1.1 m(ρ ))
T
T
T
T
-1
L =2.1 fb , 7 TeV [1203.5420]
1.5 TeV N mass (m(W ) = 2 TeV)
R
-1
L =2.1 fb , 7 TeV [1203.5420]
2.4 TeV W R mass ( m(N) < 1.4 TeV)
±±
-1
L =4.7 fb , 7 TeV [1210.5070]
409 GeV HL mass (limit at 398 GeV for µ µ )
±±
-1
L =4.7 fb , 7 TeV [1210.5070]
375 GeV HL mass
-1
L =4.8 fb , 7 TeV [1210.1718]
1.86 TeV Scalar resonance mass
L =4.6 fb , 7 TeV [1209.4625]
-1
L =4.7 fb , 7 TeV [1210.4491]
*Only a selection of the available mass limits on new states or phenomena shown
Other
ATLAS Exotics Searches* - 95% CL Lower Limits (Status: HCP 2012)
Figure 6.6: ATLAS limits for exotic particles (ATLAS Experiment ©2012 CERN, reproduced from
ATLAS exotics group TWiki page). The most important section for this work is the upper, cyan part,
where limits on particles from extra dimensions are given. In this plot, the current constraints on
the ADD model are also given and one can see that while not being excluded yet, the MD parameter
of ADD gets shifted to higher and higher regions, where the model becomes phenomonologically
uninteresting. Results marked in red are already from the 8 TeV run.
39
6 Analysis - g
in leading order. What is unusual about this result, which includes five diagrams involving KK
graviton exchanges, is the independence from the mass of the KK graviton. Usually, these 1/mGr
contributions arising from KK graviton exchanges can be reformulated using the warp function† of
the given geometry[55], as they are in essence a remnant of the KK decomposition.
Two interpretations are possible:
1. We interpret our model as incomplete and discard the results, hoping that in some extension,
some mechanism might limit the number of KK states contributing. Note that in purely 5D
calculations (like in the one from Beneke, et al.), these KK towers are absent.
2. We accept the infinite tower of KK modes as fact, trying to impose a cutoff on the number of
KK states contributing. This is the approach taken by Kim/Kim/Song in [39] - they impose a
cutoff from unitarity relations from the gg ! gg process, concluding for the g 2 that 10 to
120 states can contribute without violating unitarity.
Taking the latter approach, we show in Figure 6.7 the dependence of aGr
µ on the coefficient
k/MPlanck and the mass of the lightest graviton mGr , which are both related to the cutoff L. For
the analysis, we have linearily extrapolated between both data points available from the ATLAS
experiment (for k/MPlanck = 0.1, 0.01 and mGr = 1.00, 2.06 TeV (it is unfortunate that no data is
available in the intermediate region or for higher k). Nevertheless, with this limited dataset, we can
already see that for a low number of contributing KK gravitons, we slip out of the region where KK
graviton contributions to g 2 can explain the E821 difference fully.
The compatiblity region between ATLAS and E821 is greater for a higher number of KK states,
with the upper limit given by the unitarity bound n ⇡ 100 for L = 3 TeV from Kim/Kim/Song [39],
and the lower limit given by this analysis, n ⇡ 38 for L = 2.6 TeV, k = 0.1.
6.6 Conclusions
To our knowledge, this work is the first fully automatic calculation of graviton contributions to the
muon anomalous magnetic moment using computer algebra systems (CAS). CAS have already been
employed by Berends and Gastmans [37], yet no work utilizing a complete calculation pipeline
reaching from diagram creation, over contraction of the occuring fermion traces, to calculation of
the integrals and final output is known to the author.
We have verified the results obtained for the case of massless gravitons by Berends and Gastmans
in 1975 [37].
Furthermore, this work is the first published independent verification of the results obtained by
Graesser in 1999 for the ADD model [38] – Park/Song, Kim/Kim/Song and Hewitt/Davoudiasl did
no independent calculation, but only referred to Graesser’s result, but obtaining different cutoffs for
the number of KK states involved in contributing.
Beneke, et al. noted in his conclusions (chapter 5 in [42]) that the contributions to g 2 in the
custodial RS model remain to be computed. Our result however is independent of this custodial
† exp(
40
krc p ), in case of the Randall-Sundrum model
6.6 Conclusions
Figure 6.7: Dependence of aGr
µ with 70 and 100 contributing graviton KK states on k/MPlanck and
the mass of the lightest graviton mGr . The region right of the blue dotted line is excluded by
collider searches (linearly extrapolated from ATLAS data), the region between the red dashed lines
is the region where aGr
µ is in the region of the deviation from the SM in the E821 experiment. See
Section 6.5.4 for a full discussion.
41
6 Analysis - g
symmetry, so there are no constraints on the results coming from dangerous corrections to the
Peskin-Takeuchi parameters.
As can be seen in Figure 6.7, the original RS model still provides a viable scenario for explaining
the difference from the SM in the g 2 – if enough KK states contribute. If the LHC rules out even
higher KK graviton masses, the contribution to g 2 will get smaller and smaller, meaning that
the original RS model cannot provide a full explanation for the muon anomaly anymore. For the
time being, the lower limit on the number of KK states that need to contribute is set to n = 38 for
k = 0.1, L = 2.6 TeV, corresponding to a mass for the lowest KK graviton of mGr = 1 TeV.
The result verified here is also applicable to other theories with extra dimensions and not confined
only to RS or ADD models. Just the couplings have to be adjusted appropriately for other models.
6.7 Next Steps
In this section, we want to make a few remarks where room for improvement is still left.
1. A complete pipeline containing the definition of an extra-dimensional model, solving the
Einstein equations, and automatically creating a model file for FeynRules [56], which then
creates a FeynArts model file for further calculation, would be desireable. For solving the
Einstein equations, the Mathematica package xAct [57, 58] might prove very useful. This
package has already been used for a number of difficult calculations in General Relativity and
is capable of solving tensorial equations, also containing spinors, on Riemannian manifolds.
2. A more rigorous testing regime for model files should be implemented in gminus3: As
designing the generic model file, which contains the Lorentz structure of the vertices proved
to be the biggest hurdle in the development process, there should be a program which tests for
closure under index contractions, existence of all indices, absence of any superfluous output
when evaluation the model file and structural integrity. Mathematica’s error messages are not
very helpful in the development process. Meaningful errors from such a tool could therefore
prove to speed up the development of new models significantly.
3. Various components, especially TwoCalc, have lots of potential for parallelization: At the
moment, diagram calculation via TwoCalc is mostly a serial process due to the way it is
written. The same calculation run with FormCalc will be parallelized automatically and can
also be cluster-parallelized using Parallel FORM. A few ansätze for parallelization have been
introduced to TwoCalc with the hope that this can be expanded further.
4. While the calculation done is independent of whether Randall-Sundrum custodial symmetry
is in effect or not, no explicit calculation for KK muons in the graviton loops has been done
so far. Davoudiasl and Hewitt only give a crude estimate [40] while Beneke, et al. calculated
muon contributions in 5D, but without coupling to gravitons [42].
42
7 The AdS/CFT Correspondence
We now turn to a different subject, yet still concerning the muon g 2. As mentioned in section
3.2.5, the hardest-to-calculate part of the g 2 are the hadronic light-by-light (HLbL) corrections. In
this chapter we will give an overview of the AdS/CFT conjecture - one of the options to shed light
on the low-energy region of QCD, where the perturbative expansion fails. The AdS/CFT duality
will then be used in the next chapter to calculate HLbL corrections by Spin-2 mesons.
When performing high-energy calculations in QCD, one can rely on the property of asymptotic
freedom. Asymptotic freedom means that the coupling strength of a theory decreases with increasing
energy. This can immediately be seen from the b-function of QCD, which reads on one-loop level
b QCD ( g) =
g3
16p 2
✓
11
C ( A)
3
◆
4
NF T ( F ) ,
3
where C ( A) = 3, NF = 3, T ( F ) = 1/2.
Much more problematic are calculations in the low-energy regime. Here, QCD becomes a strongly
coupled theory. For this regime, methods like lattice gauge theory or Chiral Perturbation Theory
(ChPT), as a low-energy effective theory, have been employed to circumvent the problems of normal
QCD. Nevertheless, these theories are also unsatisfactory, because for example in the case of ChPT,
only a particle nonet is contained in the theory and Lattice QCD is computationally extremely
intensive, so usually supercomputers have to be used. For illustration, the QCD phase diagram is
shown in Figure 7.1.
Figure 7.1: The QCD phase diagram
43
An idea providing some remedy for this problem was suggested in 1998 by Maldacena, who discovered a fundamental duality between type IIB string theory⇤ on AdS5 ⇥ S5 space and supergravity
(SUGRA) in 10D Minkowski space [59]. This duality enables calculations done in a weakly-coupled
gravitational theory to be transferred to a strongly-coupled conformal field theory (CFT).
In this chapter, we want to explore the fundamentals of what now is known as the AdS/CFT
correspondence and how it can be utilized for calculations in the regime where perturbative QCD
(PQCD) fails.
7.1 Sketch of the Derivation
For the original argument the reader is referred to [59]. A more complete introduction to AdS/CFT
including current developments can be found for example in [60], [61], or [62]. Here, we want to
present a brief version sufficient for the calculations needed in the next chapter.
We consider three perspectives: first, the SUGRA side; second, the string theory side; finally, we
will introduce a symmetry argument.
7.1.1 SUGRA Perspective
Figure 7.2: Geometry used in the AdS/CFT correspondence (reproduced from [60])
We consider a 3-brane solution to supergravity,
ds2 = f
4
1/2
⇣
⌘
⇣
⌘
dt2 + dx12 + dx22 + dx32 + f 1/2 dr2 + r2 dW25 ,
where f = 1 + Rr4 and R is the size of the brane. For r
R, 10-dimensional Minkowski space
5
remains, for r ⌧ R, AdS5 ⇥ S is left, both with radius R. The latter region is called the throat region
⇤ Type
IIB string theory is a supersymmetric string theory in 10 space-time dimensions, where only closed strings are bound
to D-branes, no tachyons exist and massless fermions are chiral
44
7.1 Sketch of the Derivation
(see Figure 7.2). Now one can consider low-energy excitations with an observer at infinity, which
leads to a decoupling of the theories as these low-energy excitations cannot be absorbed by the
brane due to their too high wavelength. On the other hand, excitations near the brane are confined
to the throat region. Hence, we are left with 10-dimensional type IIB SUGRA in one limit and with
type IIB superstring theory on AdS5 ⇥ S5 in the other.
7.1.2 String Theory Perspective
On this side, we take the limit of N coinciding D3-branes† , and integrate out massive string modes.
The low-energy action is given as (see [63])
S = Sbulk + Sbrane + Sint ,
with
Sbulk = SIIB,SUGRA + Sc,corr
Sbrane = SSYM + So,corr .
The interaction Sint , the closed string corrections Sc,corr , and the open string corrections So,corr are all
proportional to a0 , where a0 = lS2 and lS is the string energy scale.
In the low energy limit of a0 ! 0, while keeping the string coupling gS and N fixed, two decoupled
theories remain: In the bulk, type IIB SUGRA is left, while on the branes, a conformal N = 4‡
SU ( N ) Super Yang-Mills (SYM) theory remains.
Then, one can take the limit gS ! 0, while keeping gS · N fixed. This leads to the ’t Hooft
limit of Quantum Field Theory and classical string theory. Therefore, type IIB string theory on
AdS5 ⇥ S5 is equivalent to the large-N limit of N = 4 SU ( N ) SYM in 4D Minkowski space. This
limit alsoqproduces corrections to the SUGRA limit on the string scale depending on powers of
a0 /R2 =
2 N.
1/gYM
2 N ! •) is taken, which means that type IIB
Finally, the limit a0 /R2 ! 0 (or equivalently, gYM
SUGRA on AdS5 ⇥ S5 is dual to the large-N, strongly coupled limit of N = 4 SU ( N ) SYM in 4D
Minkowski space.
In this limit, the radius of S5 is large, so we have a small curvature and the strongly-coupled
N = 4 SU ( N ) SYM theory on AdS5 is accurately approximated by the type IIB SUGRA, which is
tractable more easily.
So far, we have considered AdS5 ⇥ S5 on the gravity side of the correspondence. However, it
is possible to compactify AdSd such that the boundary is always a d 1 dimensional Minkowski
space. This is based on a Kaluza-Klein decomposition, where fields on all of AdS5 ⇥ S5 are split into
spherical harmonics on S5 and fields with effective masses on AdS5 .
† branes
are lower-dimensional, extended subspaces in string theory. There are D and Dp branes, where D branes (their
dimension is usually indicated by a number after the D, e.g. D3) are branes on which open strings can end, subject to
Dirichlet boundary conditions. If in p spatial dimensions von Neumann boundary conditions are satisfied, the brane is
called Dp brane.
‡ this means there are 4 supersymmetric generators in the theory
45
Schematically, this is shown in Figure 7.3.
Figure 7.3: Schematic illustration of the AdS/CFT correspondence. While (super)gravity resides
in the 5D negatively curved AdS space, a conformal field theory – without gravity – is fixed to
the shell of the 5D AdS space, which is 4-dimensional. In its current formulation, the AdS/CFT
correspondence is not limited to string theory anymore.
7.1.3 Symmetry Argument
In a 4D CFT, there is a set of conformal symmetries generated by 15 operators. Of these, ten are
the Poincaré group generators Pµ and Mµn given in (2.1). The five remaining generators are four
generators of special conformal transformations Kµ plus one scale transformation D, defined by
commutation relations as
[ D, Kµ ] =
iKµ ,
[ D, Pµ ] = iPµ ,
[Kµ , Pn ] = hµn D
[Kµ , Mnr ] = i hµn Kr
iMµn ,
hnr Kµ ,
while all other commutators vanish. These 15 operators define the 4D spacetime conformal Lie
algebra (see c.f. [64]), which contains the Poincaré algebra (2.1) as subalgebra. On the CFT, these act
as field transformations, which also have to appear on the string theory side. Coincidentally, the
isometries of AdS5 spacetime are generated by 15 operators subject to the same algebra as the CFT.
This also works for the S5 part: The scalar and fermion fields in the N = 4 SU ( N ) SYM are
related by SUSY transformations that match the isometries of S5 .
So the isometries of AdS5 ⇥ S5 are the symmetries of the field theory on its boundary. This
argument is presented in Chapter 4 of [60] more rigorously.
46
7.2 Recipe for use
7.2 Recipe for use
Shortly after Maldacenas paper, Witten published details on how to map fields between the
two theories [65]. This mapping can be formulated without the need for any string-theoretical
ingredients.
7.2.1 Fields and Operators
The following identifications are be made:
1. T µn on the gauge theory side is dual to gµn on the gravity side,
2. Tr( F µn Fµn ) on the gauge theory side is dual to a scalar field f on the gravity side,
3. a current jµ on the gauge theory side is dual to a vector field Aµ on the gravity side.
If fields other than gµn are required on the gravity side, two possibilities exist:
1. The field causes a backreaction on the metric, thus changing the geometry. For this, the
energy-momentum tensor of General Relativity has to be modified, resulting in a different
solution to Einstein’s equations.
2. We consider a static metric on which the fields reside. Here, the Einstein equations do not
have to be solved. Instead, we solve the equations of motion for the field.
7.2.2 Correlation Functions
How gauge theory correlators are to be matched was first introduced by Witten [65] and Gubser,
Klebanov, and Polyakov [66]. For the partition functions, after Wick-rotating to Euclidean space,
one obtains
ZGravity [F(z, x ); lim F(z, x ) = F0 ( x )] = ZGauge [F0 ( x )] =
z !0
Z
DA e
R
S[A]+ d4 x F0 ( x )A
.
Here, A is the gauge theory operator corresponding to the source F0 on the brane.
Usually, n-point correlation functions are obtained in the path integral formalism via a multiple
functional derivative
Gn ( x1 , . . . , xn ) ⌘
=
Z
DA e
S[A]
A( x1 ) . . . A( xn )
dn
Z
[F0 ( x )]
dF0 ( x1 ) . . . dF0 ( xn ) Gauge
F0 ( xi )=0
.
(7.1)
Using the steepest ascent approximation, it is then possible to identify
ZString [F] ⇡ ZGravity,cl. [F] = e
SGravity [Fcl. (z,xi ),F0 ( xi )]
,
47
where SGravity [Fcl. (z, xi ), F0 ( xi )] is evaluated on Fcl. with the boundary condition F(z ! 0, xi ) =
F0 ( xi ). Inserting the approximation into (7.1), we arrive at
Gn ( x1 , . . . , xn ) =
dn
Z
[F(z, xi ), F0 ( xi )]
dF0 ( x1 ) . . . dF0 ( xn ) Gravity,cl.
F0 ( xi )=0
.
(7.2)
The equations of motion for the fields contained in SGravity can then be derived the standard way,
imposing the conditions that for z ! 0, F(z, x ) ! F0 ( x ) and that for z ! •, z remains regular. The
solution F(z, x ) can be obtained by either directly solving the equations of motion or by using the
bulk-to-boundary propagator K (z, x; x 0 ) defined as
lim D (F)K (z, x; x 0 ) = d(4) ( x
z !0
x 0 ),
with D (F) being the differential operator of the kinetic term of F, arising in SGravity . F is then
calculated by integration,
F(z, x ) =
Z
d4 x 0 K (z, x; x 0 )F0 ( x 0 ).
Afterwards, SGravity is evaluated using the obtained equations of motion and integrating by parts.
A regularization scheme has to be introduced, as the AdS metric diverges for z = 0. The desired
correlation function can then be calculated using (7.2) and removing the regulator afterwards.
In the last step, divergences might occur. These can be cancelled by adding covariant local
boundary counterterms to SGravity , using a scheme called holographic renormalization (developed
in [67], see [68] for an accessible introduction).
7.3 Drawbacks
All the positive aspects of the AdS/CFT correspondence notwithstanding, some problems remain:
1. The correspondence is only valid in the ’t Hooft large-N limit of QFT. This poses a serious
problem as in QCD, N = 3. Taking gYM ! •, while keeping N fixed also provides no viable
solution, as in this case, the SUGRA limit will not be weakly coupled [69].
2. A strict mathematical verification of the AdS/CFT has not been presented so far. A rigorous
proof is complicated because AdS/CFT also works vice-versa: For strong coupling on the
gravity side, weak coupling on the field theory side arises – this time complicating the
calculations on the gravity side.
3. The duality is only useful in the strong-coupling limit on the CFT side. Therefore, high-energy
QCD results cannot be verified using AdS/CFT
4. N = 4 SYM is not QCD. QCD is neither conformal nor supersymmetric. This problem can be
somewhat alleviated by using only non-supersymmetric General Relativity instead of SUGRA.
The conformal invariance can also be broken on the AdS side, then yielding a non-constant b
function for QCD, as required.
48
7.3 Drawbacks
5. So far, no model has been discovered which completely reproduces QCD with the AdS/CFT
duality. While various quantities, like the pion off-shell form factor [70]or the lower meson
and baryon resonances [71, 72] can be very successfully reproduced, an exact limit was not yet
discovered. It is therefore difficult to estimate how much a calculation in AdS/CFT diverges
from the “true” QCD.
49
8 Analysis - Hadronic light-by-light
Scattering Contributions by Spin-2 Mesons
Now that we have seen in chapter 7 how the AdS/CFT conjecture can be used to gain knowledge
about the strong-coupling regime of QCD, we want to put it into practice. First, we review the pion
exchange contribution; second we cover how Spin-2 mesons can be handled using AdS/CFT.
ks
p2n
p
µ
r
p3
p1
p0
Figure 8.1: Hadronic light-by-light scattering contributions to g 2, the shaded region represents
the actual hadronic part. This graph gives a definition of the momenta used in the text. Apart from
ks , all photon momenta are flowing into the hadronic blob.
8.1 Introduction - The Pion Exchange Contribution
Following the discussion in [21], to calculate the amplitude given by the diagrams in Figure 8.1
(see also for momenta definitions) one has to evaluate the hadronic light-by-light (HLbL) scattering
tensor Pµnrs ( p1 , p2 , p3 ), defined as
Pµnrs ( p1 , p2 , p3 ) =
Z
d4 x1 d4 x2 d4 x3 ei( p1 x1 + p2 x2 + p3 x3 ) h0| Tjµ ( x1 ) jn ( x2 ) jr ( x3 ) js (0) |0i ,
which represents the hatched blob in the diagrams. The currents jµ ( x ) = ȳ( x ) Qgµ y( x ), with
Q = diag(2, 1, 1)/3 and y = (u, d, s), represent the electromagnetic current for the three lightest
quarks.
As the electromagnetic current is conserved via Noether’s theorem, Pµnrs ( p1 , p2 , p3 ) obeys the
Ward-Takahashi identities
µ
r
{ p1 , p2n , p3 , ks }Pµnrs ( p1 , p2 , p3 ) = 0, k = p1 + p2 + p3 .
(8.1)
51
8 Analysis - Hadronic light-by-light Scattering Contributions by Spin-2 Mesons
These identities imply that
Pµnrl ( p1 , p2 , k
p1
p2 ) = k
s
✓
∂
∂kl
◆
Pµnrs ( p1 , p2 , k
p1
p2 ),
which in turn implies that the tensor has to be linear in k when the limit k ! 0 is calculated, in
which the g 2 is defined. The electromagnetic vertex amplitude then takes the form Ms ( p0 , p) =
k r Mrs ( p0 , p). For aµ follows, applying the same projection technique we used in chapter 3,
aµ =
with
Mrs ( p0 , p) =
ie6
Z
⇥ gµ (/
p0
⇥
⇤
1
Tr (/
p + mµ )[gr , gs ](/
p + mµ ) Mrs ( p, p) ,
48mµ
d4 p1 d4 p2
1
2
2
4
4
(2p ) (2p ) p1 p2 ( p1 + p2
p 1 + mµ )gn (/
p
/
p1
/
(8.2)
1
1
( p0 p1 )2 m2µ ( p p1 p2 )2 m2µ
✓
◆
∂
l
p
+
m
)
g
Pµnls ( p1 , p2 , k p1 p2 ).(8.3)
µ
/2
∂kr
k )2
We now turn to the evaluation of Pµnrs ( p1 , p2 , p3 ): In general, this tensor has a very involved
Lorentz structure [73],
Pµnrs ( p1 , p2 , p3 ) = P1 ( p1 , p2 , p3 )hµn hrs + P2 ( p1 , p2 , p3 )hµr h ns
+ P3 ( p1 , p2 , p3 )hµs hnr
j
j
j
j
j
j
+ P1jk ( p1 , p2 , p3 )hµn pr pks + P2jk ( p1 , p2 , p3 )hµr pn pks
+ P3jk ( p1 , p2 , p3 )hµs pn pkr + P4jk ( p1 , p2 , p3 )hnr pµ pks
+ P5jk ( p1 , p2 , p3 )hns pµ pkr + P6jk ( p1 , p2 , p3 )hrs pµ pkn
j
+ Pijkm ( p1 , p2 , p3 ) piµ pn pkr pm
s.
Here, i, j, k, m 2 {1, 2, 3} and the summation convention is implied. In total, we now have 138 P
functions. We can reduce them using the following facts:
1. The Ward-Takahashi identities in (8.1) interrelate them hence, they are not all independent.
2. We only need the antisymmetric part of Mrs , so all symmetric functions are not needed in our
case.
3. In Mrs , we only need the derivative of the HLbL tensor with respect to p3 , at p3 = 0.
In the end, we only need the functions
P3jkm ( p1 , p2 , p3 ),
Pi3km ( p1 , p2 , p3 ),
⇣
∂
Pijk1 ( p1 , p2 , p3 )
∂p3l
Pij3m ( p1 , p2 , p3 ),
⌘
Pijk2 ( p1 , p2 , p3 ) p =0 ,
3
where i, j, k, m 2 {1, 2}, thus “only” 32 functions out of the original 138 are required.
52
8.1 Introduction - The Pion Exchange Contribution
An additional possibility for obtaining an expression for Pµnrs ( p1 , p2 , p3 ) is to construct it using
three-point functions. The explicit construction via the 32 remaining contributing functions may
then serve as a possibility for cross-checking the results.
We end our short review of the construction of the HLbL tensor here and proceed with a
verification of the calculation of the HLbL tensor at 2-loop level. First, we will investigate how to
reconstruct the HLbL tensor from three-point functions for the pion resonance.
8.1.1 Pµnrs ( p1 , p2 , p3 ) from Three-Point Functions
Schematically, the approach to construct the HLbL tensor for a specific resonance contribution is
P = (three-point function) (propagator) (reverse three-point function)(glue),
(8.4)
where (glue) contains the metric tensors required to paste the tree-level diagram into the two-loop
tensor (8.3).
For example, in the case of a scalar spin-0 particle,
⇣
⌘
S
SVV
abcd,µnrs
Pµnrs ( p1 , p2 , p3 ) = PVVS
(
p
,
p
+
p
)
g
1
+
g
P
(
p
+
p
)
( p1 , p2 , p3 ),
3 S
2
3 Pcd ( p2 , p3 )V
1 2
S
ab
where
V abcd,µnrs ( p1 , p2 , p3 ) = g aµ gbn gcr gds .
This is the simplest case. When models like VMD or ENJL are used, V still contains metric tensor
for contracting the tree-level diagram into the two-loop diagram, but might have additional term⇤ .
8.1.2 Pion Pole Contribution
We proceed now with the verification of the calculation of the pion pole contribution to the g
2, using the approach to construct the HLbL tensor for a specific resonance contribution from
three-point functions.
For the pion pole contribution, the three-point functions VVP (vector-vector-pseudoscalar)
and PVV (pseudoscalar-vector-vector) are needed, which introduce the model dependent, but
experimentally-constrained p 0 g⇤ g⇤ off-shell form factor
s
2 2
PPVV
µn ( p1 , p2 ) = i# µnrs p1 p2 Fp 0 g⇤ g⇤ ( p1 , p2 ),
r
(8.5)
where Fp0 g⇤ g⇤ ( p21 , p22 ) is the aforementioned form factor. Jegerlehner and Nyffeler recently remarked
that for a correct calculation of the pion exchange contribution, the off-shell form factor has to be
used, as the on-shell form factors violate four-momentum conservation at the external vertex [24].
⇤ in
[73] a calculation using the ENJL model is done, where this can be seen
53
From the expression (8.5), the VVP function can be obtained by symmetry,
PVV
PVVP
µn ( p1 , p2 ) = Pµn ( p1
p2 , p2 ).
To the HLbL tensor, three diagrams contribute (tree-level diagrams with photons as external legs,
s, t, and u channel), such that for the derivative of the HLbL tensor the expression
✓
∂
∂kr
◆
Pµnls ( p1 , p2 , k
=
p1
p2 )
i
( p1 + p2 )2
m2p
i
+
p21
+
p22
Fp0 g⇤ g⇤ ( p21 , p22 )Fp0 g⇤ g⇤ (( p1 + p2 )2 , 0)# µnab p1a p2 # lsrt ( p1 + p2 )t
b
m2p
Fp0 g⇤ g⇤ ( p21 , 0)Fp0 g⇤ g⇤ ( p22 , ( p1 + p2 )2 )# µstr p1t # nlab p1a p2
m2p
Fp0 g⇤ g⇤ ( p21 , ( p1 + p2 )2 )Fp0 g⇤ g⇤ ( p22 , 0)# µlab p1a p2 # nsrt p2t
b
i
b
is found.
After performing the trace in (8.2), the expression
0
e6
ap
µ =
Z
d4 p1 d4 p2
1
1
2
2
4
4
2
(2p ) (2p ) p1 p2 ( p1 + p2 ) ( p + p1 )2
⇥
"
+
Fp0 g⇤ g⇤ p21 , p22 Fp0 g⇤ g⇤ ( p1 + p2 )2 , 0
Fp0 g⇤ g⇤ p21 , ( p1 + p2 )2 Fp0 g⇤ g⇤ p22 , 0
p22
( p1 + p2 )2
m2p
m2p
m2µ
(p
1
p2 )2
m2µ
T1 ( p1 , p2 ; p)
#
T2 ( p1 , p2 ; p) ,
can be acquired, where T1 and T2 are defined as
16
16
8
( p · p1 )( p · p2 )( p1 · p2 )
( p · p2 )2 p21
( p · p1 )( p1 · p2 ) p22
3
3
3
16
16
16 2
+ 8( p · p2 ) p21 p22
( p · p2 )( p1 · p2 )2 + m2µ p21 p22
m ( p · p2 )2 ,
3
3
3 µ 1
16
16
T2 ( p1 , p2 ; p) = ( p · p1 )( p · p2 )( p1 · p2 )
( p · p1 )2 p22
3
3
8
8
8
8 2
+ ( p · p1 )( p1 · p2 ) p22 + ( p · p1 ) p21 p22 + m2µ p21 p22
m ( p · p2 )2 .
3
3
3
3 µ 1
T1 ( p1 , p2 ; p) =
Contributions to T1 are arising from the s and t channel diagrams, while the u channel diagram
only contributes to T2 .
This expression was originally derived by Knecht and Nyffeler [74] and independently verified
for this work, using a custom extension† of the TRACER package for Mathematica [75] to allow for
four-derivatives in fermion traces, and the open source CAS REDUCE. With this package we were
† The
54
package is to be published on the author’s GitHub page after the publication of this work
8.2 Idea
able to verify the expressions for the derivative of the HLbL tensor occuring in 8.3 and to extract
the function T2 afterwards with the help of REDUCE. T1 remains to be verified‡ . The expression
T2 has been simplified with the symmetry of the T2 -proportional form factors under the exchange
p1 $ p2 in mind, where terms proportional to an even-odd combination of p1 and p2 vanish.
The resulting two 4D integrals can then be solved by using the method of Gegenbauer polynomials,
which is beyond the scope of this work (see p.e. [21] or [74]).
8.2 Idea
In Figure 8.2, the masses of the neutral spin-2 mesons discovered so far are plotted. The lightest
meson of these is the f 2 meson, an excited state with a mass of 1270 ± 1.2 MeV [2]. Not considering
the spin for a moment, this mass is well beyond what ChPT can reliably describe. PQCD also fails
in this regime, as the region around resonances is needed explicitly and in it, a flat continuum is
predicted by PQCD.
Spin 2 meson masses
Χb2 2P
10
Mass GeV
8
6
Χc2 1P
4
2
f2 1270
Η2 1645
X 1870
Π2 2005
Π2 2245
0
Figure 8.2: Masses of the neutral Spin 2 mesons. The red line represents the cutoff for Chiral
Perturbation Theory. The green line represents the muon mass.
As we need predictions about the low-energy, non-perturbative region of QCD, we use the
AdS/CFT gauge-gravity correspondence to calculate the F Tg⇤ g⇤ off-shell form factor needed in the
VVT (vector-vector-tensor) and TVV (tensor-vector-vector) three-point functions, which contribute
to the HLbL tensor Pµnrs .
So far, AdS/CFT has been used very successfully for spin-2 mesons, for example for calculating
the mass of the f 2 meson or it’s decay constant for decay into g’s [76]. In this work, the f 2 mass as
predicted by AdS/CFT is 1236 MeV - only a 5% deviation from the experimental value. Furthermore,
‡A
discussion with the original author via email is in progress, but as this result has been used in several reviews already, it
is more probably that the error is on our side.
55
the decay rate G( f 2 ! gg) was calculated to be 2.54 keV, which is fully compatible with the
experimentally measured value of 2.60 ± 0.24 keV.
8.3 The Spin-2 Meson Off-Shell Form Factor in AdS/CFT
8.3.1 Equations of Motion
On the AdS5 side we already have a spin-2 field propagating in the bulk: the graviton – obtained by
linearly expanding the metric around the 4D Minkowski metric hµn :
ds2 =
1
hµn + hµn dx µ dx n
z2
1 2
dz .
z2
For the 5D action follows
SAdS =
=
Z
p
d5 x gR5
Z
d5 x
1 1
∂z hµn ∂z hµn
2 z3
hµn ⇤hµn . . . ,
where R5 is the 5D Ricci scalar. This action will produce a spin-2 field h MN in the bulk, with a
number of KK resonances, where the first one will be the f 2 meson.
Using plane wave solutions (according to the prescription given in 7.2) for the tensor particle, we
obtain the equation of motion
3
z ∂z

1
∂z hµn + ∂a ∂a hµn = 0,
z3
where hµn = h(q, z)h0µn (q) is the Fourier transform of the solution. The equation is solved by
hn (z) = Cn z2 (J2 (q z) + b n Y2 (q z)) ,
where J2 and Y2 are the second Bessel functions of first and second type, respectively. Imposing the
boundary conditions h0 (zm ) = h(0) = 0 then leads to b n = 0. On-shell, we get quantized masses by
J1 (q = mnh ) = 0:
h(z) = 3.51
z2
z
J2 (3.83 ).
zm
zm
From this rather simple setup, an already important result can be obtained [76] – the mass of the
f 2 meson:
m( f 2 ) = 3.83 zm 1 = 1236 MeV,
where zm ⇡ 1/LQCD has been used. This is called a hard-wall model, where a hard cut-off exists,
above which the CFT will not be strongly coupled anymore. Soft-wall models, replacing the
hard-wall with a soft Gaussian are also possible, but beyond the scope of this work.
56
8.3 The Spin-2 Meson Off-Shell Form Factor in AdS/CFT
As we are interested in the three-point function TVV, we also need a gauge field Aµ in the AdS
bulk, which corresponds to the quark current jµ = i ȳ∂µ y on the QCD side. Its action is given as
Z
p 1 MN
g F
FMN
4

Z
1 1 2
2
= d5 x
F
Fµ5
.
z 4 µn
S5D
A =
d5 x
Like in the case of the Spin-2 field discussed above, the equation of motion is obtained using a
plane wave expansion, leading to
1
q2
z∂z ∂z Aµ (q, z) + Aµ (q, z) = 0,
z
2
which is also solved by the Bessel functions Ji and Yi , but this time for i = 1 and Aµ = A(q, z) A0µ (q):
h
i
Aµ (q, z) = NnA z J1 (q z) + b nA Y1 (q z) .
(8.6)
Aµ (q, z) is subject to the same boundary conditions as hµn (q, z), namely A0µ (q, zm ) = Aµ (q, 0) = 0,
which leads to b nA = 0.
8.3.2 Interactions
For three-point functions to be meaningful, some interactions are needed. For the TVV function, we
are interested in the interaction of a massive Spin-2 field with a Spin-1 vector field, both in the bulk.
The most natural coupling in this case is the coupling of the Spin-2 field hµn to the Spin-1 part of the
energy-momentum tensor Tµn :
Z
p
1
A
d5 x g hrs Trs
g5
Z
h
i
1
1
=
d5 x h rg h sd hgd
Fsz Frz + h ab Fsa Frb .
g5
z
5D
Sint
=
With the definition of the field-strength tensor Fµn for an Abelian field Aµ ,
Fµn = ∂µ An
∂n Aµ ,
we can expand this expression by using integration by parts:
"
1
∂
5D
Sint
d5 x h rg h sd hgd
Az ∂[r Az]
z
∂x r
✓
h ab
∂
∂
+
Aa ∂[r A b]
Ar ∂[r A b]
2
∂x s
∂x a
1
=
g5
where ∂[r A b] = ∂r A b
Z
∂
Ar ∂[r Az]
∂x z
⇣
A[a ∂s] ∂[r A b]
⇣
⌘
A[z ∂r] ∂[r Az]
⌘◆
#
,
∂ b Ar .
57
8.3.3 Three-Point Function
The required three-point function TVV can then be derived by functional derivation and then
Fourier-transforming to momentum space,
d3
S5D .
dA0# ( x, p1 )dhµn (y, q)dA0c (w, p2 ) int
After a tedious calculation, we arrive at a rich Lorentz structure constituted by the terms T1 . . . T6 ,
where #, c are the indices of the vector fields and µ, n are the indices of the tensor field:
1
T#cµn
( p1 , p2 ) =
Ihyy
1n
· p p2µ h#c
p1n p2# hµc
+ p1µ p2n h#c
p1c p2n hµ#
2n
p2n p2# hµc
1n
p1n p1c hµ# ,
+ p p2µ h#c
+ p p1µ hc#
2
T#cµn
( p1 , p2 ) = Ihyy
· h n# p1c p2µ
hn# ( p1 p2 )hµc
+ h nc ( p2# p1µ
p2 p1 hµ# )
n#
p2 p2 hµc )
nc
p1 p1 hµ# ) ,
+ h ( p2c p2µ
+ h ( p1# p1µ
3
T#cµn
( p1 , p2 ) =
2n
Ihyy
p p2µ h#c
p2n p2# hµc
+ p1n p1µ hc#
p1n p1c hµ#
2c
p p2µ hn# + hn# p22 hµc
h nc p1# p1µ + hnc hµ# p21
4
T#cµn
( p1 , p2 ) =
p
1n
Ihyy
hc# p2µ
+ p2n hc# p1µ
p2# hµc ,
p1c hµ#
+p
2n
p2µ hc#
p2# hµc
+p
1n
p1µ hc#
p1c hµ#
5
T#cµn
( p1 , p2 ) =
Ihyy
n#
· h ( p2µ p1c
p1 p2 hµc )
+ h nc ( p1µ p2#
p1 p2 hµ# )
µ#
p2 p2 hµc )
nc
p1 p1 hµ# )
+ h ( p2µ p2c
+ h ( p1µ p1#
6
T#cµn
( p1 , p2 ) =
Ihyy
h n# p2µ p2c + hn# p22 hµc
h nc p1µ p1# + hn# p21 hµc .
58
,
8.4 Next Steps
The overlap integral Ihyy is
Ihyy =
Z
dz
h(q, z)yn (z)yn (z),
z
where yn are the vector wave functions obtained from (8.6) as
yn (z) =
1
zJ (mn z)
zm J1 (mn zm ) 1
With the T terms, the three-point function VVT can be constructed as (see (7.2))
1
6
PTVV
µnrs ( p1 , p2 ) = i ( Tµnab ( p1 , p2 ) + . . . Tµnab ( p1 , p2 )).
The three-point function TVV can be constructed from this definition analogous to how VVP can be
constructed from PVV.
8.3.4 Pµnrs Contribution from Spin-2 mesons
Using the schema in (8.4), we can now proceed to construct the contribution to the HLbL tensor as
PTµnrs ( p1 , p2 ) = VVTµnab ( p1 , p2 )
= PVVT
µnab ( p1 , p2 )
⇣
tensor propagatorabgd ( p1 + p2 )
iD abgd ( p1 + p2 ) TVV
P
( p1 , p2 ),
( p1 + p2 )2 m2T gdrs
⌘
TVVgdrs ( p1 , p2 )
where m T is the mass of the Spin-2 particle and D abgd is the numerator of the graviton propagator
as in (A.1).
To obtain the contribution to g 2, this expression can be put into (8.3), the traces evaluated, the
remaining integrals solved with the method of Gegenbauer polynomials, like in the case of the pion
pole contribution.
8.4 Next Steps
The AdS/CFT ansatz for the calculation of Spin-2 meson contributions to the HLbL tensor has now
been established. To decide about the true value of this calculation, a few more things are necessary,
but beyond the scope of this work:
1. For a few Spin-2 mesons, like the f 2 , the decay rates to photons have already been measured.
These depend also on the form factors, much like in the case with the pion. For a quantitative
statement of the correctness of this calculation, comparison to experiment is the first priority.
2. If experimental confirmation for this form factor calculation can be found, a full calculation of
its contribution to the g 2 should be done as outlined. If no experimental input is available,
a full calculation might still prove useful to determine whether the contribution has the right
order of magnitude.
59
A.1 Propagator for Massive Gravitons
As massive spin-2 particle, the graviton does have 2 ⇥ 2 + 1 = 5 degrees of freedom, which requires
(i )
five polarization tensors # µn with
(i )
kµ # µn (k) = 0,
(i )
gµn # µn (k) = 0.
The normalization of these is fixed by
(i )
(i )
Â # µn (k)# rs (k) = 1.
i
Poincare invariance then dictates
(i )
2
(i )
Â # µn (k)# rs (k) = (Gµn Gns + Gµs Gnr )
i
Gµn = gµn
D
1
Gµn Grs ,
kµ kn
,
m2
such that the D-dimensional massive graviton propagator in momentum space is given by
Pµn,rs (k) =
( Gµn Gns + Gµs Gnr )
iDµn,rs
=i
2
2
k
m
k 2 m2
2
D 1 Gµn Grs
.
(A.1)
We want to keep the D here explicitly as we are going to use dimensional regularization for the
integrations arising in our calculation.
Note that the prefactor of the last term is 2/( D 1), in contrast to the case of the massless graviton,
with 2/( D 2). This is known as the van Dam-Veltman-Zakharov (vDVZ) discontinuity, which
basically means that linearized massive gravity gives different predictions than linearized General
Relativity (see c.f. [77, 78]).
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A.2 Feynman Rules for Fermion and Vector Boson Interaction
with Massive Gravitons
The Feynman rules for massive (KK) gravitons can be derived via the Fierz-Pauli action, which in
fact is linearized General Relativity. For the details of the derivation, the reader is referred to [79] or
[80], which both provide a very accessible approach to the derivation.
We finally arrive at these Feynman rules (particle momenta are indicated as required, all momenta
are inbound):
FFT :
f (k2 )
=
i
=
i
Grµn
f (k1 )
VVT :
Abb (k2 )
Grµn
Aaa (k1 )
Wµnab (k1 , k2 ) =
1 ⇥
(k1
4L
k 2 ) µ gn + ( k 1
k 2 ) n gµ
⇤
⇤
1 ab ⇥
d Wµnab (k1 , k2 ) + Wnµab (k1 , k2 )
L
1
hµn (k1a k2b
2
+ hab k1µ k2n
+ hµa (k1 k2 hnb
k1 k2 hab )
k1b k2b )
hµb k1n k2a
f i (k2 )
FFVT :
Grµn =
f j (k1 )
i
Tjia
2L
gµ gna + gn gµa
Aaa (k1 )
Tjia and d ab are the structure constants of the Lie algebra of the underlying symmetry group and
constants. For QED, Tjia = eQ and d ab = 1.
In the model files shown in Appendix B, only the QED version is implemented for simplicity.
62
B Model Files
The model files printed here, as well as more tools can also be found in the author’s BitBucket repository at
https://bitbucket.org/ulrikguenther/gminus3.
B.1 Minimal Randall-Sundrum Model File for FeynArts with SM
on TeV brane
B.1.1 Generic Model File
The generic model file (*.gen) for FeynArts defines the Lorentz structures of the couplings.
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(*
MinimalRS.gen
Generic model file for ADD and RS calculations, based on QED.gen
(by A. Denner, H. Eck, O. Hahn, S. Kueblbeck)
last modified 25 Feb 13 by Ulrik Guenther
*)
KinematicIndices[ F ] = {};
KinematicIndices[ V ] = {Lorentz}
KinematicIndices[ T ] = {Lorentz, Lorentz}
IndexStyle[ Index[Lorentz, i_Integer] ] := Greek[i + 11]
Attributes[ MetricTensor ] = Attributes[ ScalarProduct ] = {Orderless}
FourVector/: -FourVector[ mom_, mu___ ] := FourVector[Expand[-mom], mu]
FourVector[ 0, ___ ] = 0
If[$MinimalRSUseGravitationalWardIdentity,
Dummy[KFactor] = 0;Dummy[vDVZ]=1;,
Dummy[KFactor] = 1;Dummy[vDVZ]=1;
];
If[$MinimalRSMasslessGraviton,
Dummy[GMassTerm] = 0,
Dummy[GMassTerm] = 1
];
M$GenericPropagators = {
(* general fermion propagator: *)
AnalyticalPropagator[External][ s F[i, mom] ] ==
NonCommutative[
DiracSpinor[-mom, Mass[F[i]], Sequence@@ Drop[{i}, 1]] ],
AnalyticalPropagator[Internal][ s F[i, mom] ] ==
NonCommutative[ DiracSlash[-mom] + Mass[F[i]] ] *
I PropagatorDenominator[mom, Mass[F[i]]],
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(* general vector boson propagator: *)
AnalyticalPropagator[External][ s V[i, mom, {li2}] ] ==
PolarizationVector[V[i], mom, li2],
AnalyticalPropagator[Internal][ s V[i, mom, {li1} -> {li2}] ] ==
-I PropagatorDenominator[mom, Mass[V[i]]] *
(MetricTensor[li1, li2] - (1 - GaugeXi[V[i]]) *
FourVector[mom, li1] FourVector[mom, li2] *
PropagatorDenominator[mom, Sqrt[GaugeXi[V[i]]] Mass[V[i]]]),
(* massive Spin 2 boson propagator *)
AnalyticalPropagator[Internal][ s T[i, mom, {li1, li2} -> {li3,
1/2 I PropagatorDenominator[mom, Dummy[GMassTerm] Mass[T[i]]]
(
(MetricTensor[li1, li3] Dummy[KFactor] FourVector[mom, li1] * FourVector[mom,
(MetricTensor[li2, li4] Dummy[KFactor] FourVector[mom, li2] * FourVector[mom,
+(MetricTensor[li1,
Dummy[KFactor]
(MetricTensor[li2,
Dummy[KFactor]
li4}] ] ==
*
li3]/Mass[T[i]]^2)
li4]/Mass[T[i]]^2)
li4] FourVector[mom, li1] * FourVector[mom, li4]/Mass[T[i]]^2)
li3] FourVector[mom, li2] * FourVector[mom, li3]/Mass[T[i]]^2)
- 2/($D-2 + Dummy[GMassTerm])Dummy[vDVZ]
)
}
Wgamma[mu_, nu_, alpha_, beta_, k1_, k2_] :=
(1/2 MetricTensor[mu,nu] *
(FourVector[-k1, beta] FourVector[-k2, alpha]
- ScalarProduct[-k1, -k2] MetricTensor[alpha, beta])
+ MetricTensor[alpha, beta]
FourVector[-k1, mu] FourVector[-k2, nu]
+ MetricTensor[mu, alpha]
(ScalarProduct[-k1, -k2] MetricTensor[nu, beta]
- FourVector[-k1, beta] FourVector[-k2, nu])
- MetricTensor[mu, beta]
FourVector[-k1, nu] FourVector[-k2, alpha]
)
M$GenericCouplings = {
(* F-F: *)
AnalyticalCoupling[
s1 F[i, mom1],
s2 F[j, mom2]
] ==
G[1][s1 F[i], s2 F[j]] .
{
NonCommutative[DiracSlash[mom1], ChiralityProjector[-1]],
NonCommutative[DiracSlash[mom2], ChiralityProjector[+1]],
NonCommutative[ChiralityProjector[-1]],
NonCommutative[ChiralityProjector[+1]]
},
(* V-V: *)
AnalyticalCoupling[
s1 V[i, mom1, {li1}],
s2 V[j, mom2, {li2}]
] ==
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G[1][s1 V[i], s2 V[j]] .
{
MetricTensor[li1, li2] ScalarProduct[mom1, mom2],
MetricTensor[li1, li2],
FourVector[mom1, li2] FourVector[mom2, li1]
},
(* F-F-V: *)
AnalyticalCoupling[
s1 F[i, mom1],
s2 F[j, mom2],
s3 V[k, mom3, {li3}]
] ==
G[-1][s1 F[i], s2 F[j], s3 V[k]] .
{
NonCommutative[DiracMatrix[li3], ChiralityProjector[-1]],
NonCommutative[DiracMatrix[li3], ChiralityProjector[+1]]
},
(* F-F-T *)
AnalyticalCoupling[
s1 F[j1, mom1],
s2 F[j2, mom2],
s3 T[j3, mom3, {li1, li2}]
] ==
G[-1][s1 F[j1], s2 F[j2], s3 T[j3]] .
{
FourVector[mom1 - mom2, li1]
NonCommutative[DiracMatrix[li2], ChiralityProjector[-1]] +
FourVector[-mom2 + mom1, li2]
NonCommutative[DiracMatrix[li1], ChiralityProjector[-1]],
FourVector[mom1 - mom2, li1]
NonCommutative[DiracMatrix[li2], ChiralityProjector[+1]] +
FourVector[-mom2 + mom1, li2]
},
(* T-F-F-V *)
AnalyticalCoupling[
s1 F[j1, mom1],
s2 F[j2, mom2],
s4 V[j4, mom4, {li5}],
s3 T[j3, mom3, {li3, li4}]
] ==
G[-1][ s1 F[j1], s2 F[j2], s4 V[j4], s3 T[j3] ] .
{
NonCommutative[DiracMatrix[li3], ChiralityProjector[-1]]
MetricTensor[li4, li5] +
NonCommutative[DiracMatrix[li4], ChiralityProjector[-1]]
MetricTensor[li3, li5],
MetricTensor[li4, li5] +
MetricTensor[li3, li5]
},
(* T-V-V *)
AnalyticalCoupling[
s1 T[j1, mom1, {li1, li2}],
s2 V[j2, mom2, {li3}],
s3 V[j3, mom3, {li4}]
] ==
G[+1][s1 T[j1], s2 V[j2], s3 V[j3]] .
{
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Wgamma[li1, li2, li3, li4, mom2, mom3] +
Wgamma[li2, li1, li3, li4, mom2, mom3]
}
}
M$FlippingRules =
NonCommutative[dm:_DiracMatrix | _DiracSlash, ChiralityProjector[pm_]] ->
-NonCommutative[dm, ChiralityProjector[-pm]]
M$TruncationRules = {
_PolarizationVector -> 1,
_DiracSpinor -> 1
}
M$LastGenericRules = {
(* relicts of the truncation of spinors: *)
Dot[1, line__, 1] :> Dot[line],
Dot[1, 1] :> 1,
(* outgoing vector bosons: throw away signs of momenta *)
PolarizationVector[p_, _. k:FourMomentum[Outgoing, _], li_] :>
Conjugate[PolarizationVector][p, k, li]
}
Format[
FermionChain[
NonCommutative[_[s1_. mom1_, mass1_]],
r___,
NonCommutative[_[s2_. mom2_, mass2_]]] ] :=
Overscript[If[FreeQ[mom1, Incoming], "u", "v"], "_"][mom1, mass1] .
r . If[FreeQ[mom2, Outgoing], "u", "v"][mom2, mass2]
Format[ DiracSlash ] = "gs"
Format[ DiracMatrix ] = "ga"
Format[ ChiralityProjector[1] ] = SequenceForm["om", Subscript["+"]]
Format[ ChiralityProjector[-1] ] = SequenceForm["om", Subscript["-"]]
Format[ GaugeXi[a_] ] := SequenceForm["xi", Subscript[a]]
Format[ PolarizationVector ] = "ep"
Unprotect[Conjugate];
Format[ Conjugate[a_] ] := SequenceForm[a, Superscript["*"]];
Protect[Conjugate]
Format[ MetricTensor ] = "g"
Format[ ScalarProduct[a__] ] := Dot[a]
Format[ FourVector[a_, b_] ] := a[b]
Format[ FourVector[a_] ] := a
B.1.2 Model Definition
In the model definition file (*.mod), the coupling constants for the Lorentz structures defined in the
generic model file are set.
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(*
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MinimalRS.mod
Generic model file for ADD and RS calculations, based on QED.mod
(by A. Denner, H. Eck, O. Hahn, S. Kueblbeck)
last modified 25 Feb 13 by Ulrik Guenther
*)
IndexRange[ Index[Generation] ] = {1, 2, 3}
IndexStyle[ Index[Generation, i_Integer] ] := Alph[i + 8]
ViolatesQ[ q__ ] := Plus[q] =!= 0
M$ClassesDescription = {
(* Leptons (e, mu, tau) *)
F[1] == {
SelfConjugate -> False,
Indices -> {Index[Generation]},
Mass -> MLE,
QuantumNumbers -> -Charge,
PropagatorLabel -> ComposedChar["e", Index[Generation]],
PropagatorType -> Straight,
PropagatorArrow -> Forward },
(* Photon *)
V[1] == {
SelfConjugate -> True,
Mass -> 0,
PropagatorLabel -> "\\gamma",
PropagatorType -> Sine,
PropagatorArrow -> None },
(* Massive Graviton *)
T[1] == {
SelfConjugate -> True,
Mass -> MGr,
PropagatorLabel -> "G",
PropagatorType -> Straight,
PropagatorArrow -> None }
}
MLE[1] = ME;
MLE[2] = MM;
MLE[3] = ML
TheLabel[ F[1, {1}] ] = "e";
TheLabel[ F[1, {2}] ] = "\\mu";
TheLabel[ F[1, {3}] ] = "\\tau"
GaugeXi[ V[1] ] = GaugeXi[A]
mdZfLR1[ type_, j1_ ] :=
Mass[F[type, j1]]/2 *
(dZfL1[type, j1, j1] + Conjugate[dZfR1[type, j1, j1]])
mdZfRL1[ type_, j1_ ] :=
Mass[F[type, j1]]/2 *
(dZfR1[type, j1, j1] + Conjugate[dZfL1[type, j1, j1]])
dZfL1cc[ type_, j1_ ] :=
dZfL1[type, j1, j1]/2 + Conjugate[dZfL1[type, j1, j1]]/2
dZfR1cc[ type_, j1_ ] :=
dZfR1[type, j1, j1]/2 + Conjugate[dZfR1[type, j1, j1]]/2
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M$CouplingMatrices = {
(* F-F:
G(+) . { slash[mom1] omega[-], slash[mom2] omega[+],
omega[-], omega[+] } *)
C[ -F[1, {j1}], F[1, {j2}] ] == I IndexDelta[j1, j2] *
{ {0, -dZfL1cc[1, j1]},
{0, dZfR1cc[1, j1]},
{0, -mdZfLR1[1, j1] - dMf1[1, j1]},
{0, -mdZfRL1[1, j1] - dMf1[1, j1]} },
(* V-V:
G(+) . { -g[mu, nu] mom^2, g[mu, nu], -mom[mu] mom[nu] } *)
C[ V[1], V[1] ] == I *
{ {0, dZAA1},
{0, 0},
{0, -dZAA1} },
(* F-F-V:
G(-) . { gamma[mu3] omega[-], gamma[mu3] omega[+] } *)
C[ -F[1, {j1}], F[1, {j2}], V[1] ] ==
I EL IndexDelta[j1, j2] *
{ {1, dZe1 + dZAA1/2 + dZfL1cc[1, j1]},
{1, dZe1 + dZAA1/2 + dZfR1cc[1, j1]} },
(* F-F-T *)
C[ -F[1, {j1}], F[1, {j2}], T[1] ] ==
I 1/(4 MPlanck) * IndexDelta[j1, j2] * {{1,0},{1,0}},
(* F-F-V-T *)
C[ -F[1, {j1}], F[1, {j2}], V[1], T[1] ] ==
-EL I 1/(2 MPlanck) * IndexDelta[j1, j2] * {{1,0},{1,0}},
(* T-V-V *)
C[ T[1], V[1], V[1] ] ==
-1 I/(MPlanck) * {{1,0}}
}
M$LastModelRules = {}
(* some short-hands for excluding classes of particles *)
NoGeneration1 = ExcludeParticles -> F[_, {1}]
68
C Software Packages used
C.1 Third-party software
1. Wolfram Mathematica, Version 8.0.4 and 9.0.1, commercial http://www.wolfram.com/mathematica
2. FeynArts, Version 3.7, GNU Public License, http://www.feynarts.de
3. FORM, Version 4.0, GNU Public License, http://www.nikhef.nl/˜form/
4. FormCalc, Version 8.0, GNU Public License, http://www.feynarts.de/formcalc
5. MDMProjector
6. TwoCalc
C.2 Software written by the Author
1. UnitTest, MIT license, http://github.com/skalarproduktraum/UnitTest/
2. gminus3, extended from previous work by Dominik Stöckinger and Christoph Gnendiger,
GNU Public License, to be published on the author’s GitHub page, if possible
3. tracer2, extended from previous work by Jamin, et. al [75] and made compatible with
Mathematica 9, license status unknown at the moment, to be published on the author’s
GitHub page, if possible
69
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Danksagung
Mein besonderer Dank gilt Holger Steinfurth, Felix Socher, Prof. Dr. Roland Ketzmerick, Prof. Dr.
Dominik Stöckinger und meinen Eltern, ohne deren Bestärkung, Hilfe und Unterstützung es nie zur
Anfertigung dieser Arbeit gekommen wäre.
Prof. Dr. Dominik Stöckinger danke ich für die umfassende Betreuung der zwei interessanten
Themen, von denen ja eigentlich nur eines behandelt werden sollte, letztlich aber doch beide zu
interessant waren, um das andere aussen vor zu lassen.
Meinen Kollegen Markus Bach, Philip Diessner, Christoph Gnendiger, Marcus Sperling und
Alexander Voigt danke ich für die angenehme Büroatmosphäre, so manch angeregte Diskussion
über Physik, das Fundstück der Woche oder auch die Dogmen des Richard Stallman. Oder den
fruchtbaren und informativen Gesprächen zum Thema vim vs. Emacs bei einer kühlen Club Mate ;)
Philip Diessner, Dr. Marco Giovanni di Pruna, Marcus Sperling und Markus Bach danke ich für
das auffinden zahlreicher Typos und für hilfreiche Hinweise die korrekte Wortwahl betreffend.
Herzlichen Dank auch an Roman Yaresko und Marcus Sperling, die beide bei der “Last MinuteBerechnung” des Spin-2-Mesonen-Formfaktors per AdS/CFT-Korrespondenz eine grosse Hilfe
waren.
Erklärung
Hiermit versichere ich, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter und ohne
Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe. Die aus fremden Quellen
direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht. Die Arbeit
wurde bisher weder im Inland noch im Ausland in gleicher oder ähnlicher Form einer anderen
Prüfungsbehörde vorgelegt.
Ulrik Günther
Dresden, März 2013

Automatic Calculation of Graviton Contributions and Hadronic Light

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