Light from Dark Matter Michael Gustafsson

Light from Dark Matter Michael Gustafsson
Doctoral
Thesis
in
Theoretical
Physics
Light from Dark Matter
– Hidden Dimensions, Supersymmetry, and Inert Higgs
Michael Gustafsson
Thesis for the degree of Doctor of Philosophy in Theoretical Physics
Department of Physics
Stockholm University
Sweden
c Michael Gustafsson, Stockholm 2008
Copyright ISBN: 978 91-7155-548-9 (pp. i–xv, 1–184)
Figure 8.1, 9.1 and 9.2 have been adopted, with author’s permission, from
c Phys. Rev. D, c Astrophys J. and
the sources cited in the figure captions (
c Astron. Astrophys., respectively).
Typeset in LATEX
Printed by Universitetsservice US AB, Stockholm
Cover illustration: Artist’s impression of two massive Kaluza-Klein particles γ (1) which collide and annihilate into two gamma rays γ. Under the
magnifying glass, it is shown how these massive particles in reality are
just ordinary photons circling a cylindrically-shaped extra dimension.
c Andreas Hegert and Michael Gustafsson.
Abstract
Recent observational achievements within cosmology and astrophysics have lead to a concordance model in which the energy content in our Universe is dominated by presumably fundamentally
new and exotic ingredients – dark energy and dark matter. To reveal the nature of these ingredients is one of the greatest challenges
in physics.
The detection of a signal in gamma rays from dark matter annihilation would significantly contribute to revealing the nature of
dark matter. This thesis presents derived imprints in gamma-ray
spectra that could be expected from dark matter annihilation. In
particular, dark matter particle candidates emerging in models with
extra space dimensions, extending the standard model to be supersymmetric, and introducing an inert Higgs doublet are investigated.
In all these scenarios dark matter annihilation induces sizeable and
distinct signatures in their gamma-ray spectra. The predicted signals are in the form of monochromatic gamma-ray lines or a pronounced spectrum with a sharp cutoff at the dark matter particle’s
mass. These signatures have no counterparts in the expected astrophysical background and are therefore well suited for dark matter
searches.
Furthermore, numerical simulations of galaxies are studied to
learn how baryons, that is, stars and gas, affect the expected dark
matter distribution inside disk galaxies such as the Milky Way.
From regions of increased dark matter concentrations, annihilation
signals are expected to be the strongest. Estimations of dark matter
induced gamma-ray fluxes from such regions are presented.
The types of dark matter signals presented in this thesis will be
searched for with existing and future gamma-ray telescopes.
Finally, a claimed detection of dark matter annihilation into
gamma rays is discussed and found to be unconvincing.
List
of
Accompanying
Papers
Paper I
Cosmological Evolution of Universal Extra Dimensions
T. Bringmann, M. Eriksson and M. Gustafsson
Phys. Rev. D 68, 063516 (2003)
Paper II
Gamma Rays from Kaluza-Klein Dark Matter
L. Bergström, T. Bringmann, M. Eriksson and M. Gustafsson
Phys. Rev. Lett. 94, 131301 (2005)
Paper III
Two Photon Annihilation of Kaluza-Klein Dark Matter
L. Bergström, T. Bringmann, M. Eriksson and M. Gustafsson
JCAP 0504, 004 (2005)
Paper IV
Gamma Rays from Heavy Neutralino Dark Matter
L. Bergström, T. Bringmann, M. Eriksson and M. Gustafsson
Phys. Rev. Lett. 95, 241301 (2005)
Paper V
Is the Dark Matter Interpretation of the EGRET
Gamma Excess Compatible with Antiproton
Measurements?
L. Bergström, J. Edsjö, M. Gustafsson and P. Salati
JCAP 0605, 006 (2006)
Paper VI
Baryonic Pinching of Galactic Dark Matter Haloes
M. Gustafsson, M. Fairbairn and J. Sommer-Larsen
Phys. Rev. D 74, 123522 (2006)
Paper VII Significant Gamma Lines from Inert Higgs Dark Matter
M. Gustafsson, E. Lundström, L. Bergström and J. Edsjö
Phys. Rev. Lett. 99, 041301 (2007)
Published Proceedings Not Accompanying:
Paper A
Stability of Homogeneous Extra Dimensions
T. Bringmann, M. Eriksson and M. Gustafsson
AIP Conf. Proc. 736, 141 (2005)
Paper B
Gamma-Ray Signatures for Kaluza-Klein Dark Matter
L. Bergström, T. Bringmann, M. Eriksson and M. Gustafsson
AIP Conf. Proc. 861, 814 (2006)
Contents
Abstract
iii
List of Accompanying Papers
v
Preface
xi
Notations and Conventions
xiv
Part I: Background Material and Results
1 The Essence of Standard Cosmology
1.1 Our Place in the Universe . . . . . . . . .
1.2 Spacetime and Gravity . . . . . . . . . . .
Special Relativity . . . . . . . . . . . . . .
General Relativity . . . . . . . . . . . . .
1.3 The Standard Model of Cosmology . . . .
1.4 Evolving Universe . . . . . . . . . . . . .
1.5 Initial Conditions . . . . . . . . . . . . . .
1.6 The Dark Side of the Universe . . . . . .
Dark Energy . . . . . . . . . . . . . . . .
Dark Matter . . . . . . . . . . . . . . . .
All Those WIMPs – Particle Dark Matter
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2 Where Is the Dark Matter?
2.1 Structure Formation History . . . . . . . . .
2.2 Halo Models from Dark Matter Simulations
2.3 Adiabatic Contraction . . . . . . . . . . . .
A Simple Model for Adiabatic Contraction .
Modified Analytical Model . . . . . . . . . .
2.4 Simulation Setups . . . . . . . . . . . . . .
2.5 Pinching of the Dark Matter Halo . . . . .
2.6 Testing Adiabatic Contraction Models . . .
2.7 Nonsphericity . . . . . . . . . . . . . . . . .
Axis Ratios . . . . . . . . . . . . . . . . . .
Alignments . . . . . . . . . . . . . . . . . .
2.8 Some Comments on Observations . . . . . .
2.9 Tracing Dark Matter Annihilation . . . . .
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vii
viii
Contents
Indirect Dark Matter Detection . . . . . . . . . . . . . . . . . . 34
2.10 Halo Substructure . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Beyond the Standard Model: Hidden Dimensions and
3.1 The Need to Go Beyond the Standard Model . . . . . .
3.2 General Features of Extra-Dimensional Scenarios . . . .
3.3 Modern Extra-Dimensional Scenarios . . . . . . . . . . .
3.4 Motivations for Universal Extra Dimensions . . . . . . .
More
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39
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4 Cosmology with Homogeneous Extra Dimensions
4.1 Why Constants Can Vary . . . . . . . . . . . . . .
4.2 How Constant Are Constants? . . . . . . . . . . .
4.3 Higher-Dimensional Friedmann Equations . . . . .
4.4 Static Extra Dimensions . . . . . . . . . . . . . . .
4.5 Evolution of Universal Extra Dimensions . . . . . .
4.6 Dimensional Reduction . . . . . . . . . . . . . . . .
4.7 Stabilization Mechanism . . . . . . . . . . . . . . .
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Universal Extra Dimensions
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5 Quantum Field Theory in
5.1 Compactification . . . .
5.2 Kaluza-Klein Parity . .
5.3 The Lagrangian . . . . .
5.4 Particle Propagators . .
5.5 Radiative Corrections .
5.6 Mass Spectrum . . . . .
6 Kaluza-Klein Dark Matter
6.1 Relic Density . . . . . . . . . . . .
6.2 Direct and Indirect Detection . . .
Accelerator Searches . . . . . . . .
Direct Detection . . . . . . . . . .
Indirect Detection . . . . . . . . .
6.3 Gamma-Ray Signatures . . . . . .
Gamma-Ray Continuum . . . . . .
Gamma Line Signal . . . . . . . .
6.4 Observing the Gamma-Ray Signal
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7 Supersymmetry and a New Gamma-Ray Signal
7.1 Supersymmetry . . . . . . . . . . . . . . . . . . .
Some Motivations . . . . . . . . . . . . . . . . . .
The Neutralino . . . . . . . . . . . . . . . . . . .
7.2 A Neglected Source of Gamma Rays . . . . . . .
Helicity Suppression for Fermion Final States . .
Charged Gauge Bosons and a Final State Photon
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ix
Contents
8 Inert Higgs Dark Matter
8.1 The Inert Higgs Model . . . . . . . . . . . . . .
The New Particles in the IDM . . . . . . . . .
Heavy Higgs and Electroweak Precision Bounds
More Constraints . . . . . . . . . . . . . . . . .
8.2 Inert Higgs – A Dark Matter Candidate . . . .
8.3 Gamma Rays . . . . . . . . . . . . . . . . . . .
Continuum . . . . . . . . . . . . . . . . . . . .
Gamma-Ray Lines . . . . . . . . . . . . . . . .
9 Have Dark Matter Annihilations Been
9.1 Dark Matter Signals? . . . . . . . . . .
9.2 The Data . . . . . . . . . . . . . . . .
9.3 The Claim . . . . . . . . . . . . . . . .
9.4 The Inconsistency . . . . . . . . . . .
Disc Surface Mass Density . . . . . . .
Comparison with Antiproton Data . .
9.5 The Status to Date . . . . . . . . . . .
10 Summary and Outlook
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Observed?
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139
A Feynman Rules: The UED model
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A.1 Field Content and Propagators . . . . . . . . . . . . . . . . . . 143
A.2 Vertex Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Bibliography
159
Part II: Scientific Papers
185
Preface
This is my doctoral thesis in Theoretical Physics. During my years as a Ph.D.
student, I have been working with phenomenology. This means I live in the
land between pure theorists and real experimentalists – trying to bridge the
gap between them. Taking elegant theories from the theorist and making
firm predictions that the experimentalist can detect is the aim. My research
area has mainly been dark matter searches through gamma-ray signals. The
ultimate aim in this field is to learn more about our Universe by revealing
the nature of the dark matter. This work consists of quite diverse fields:
From Einstein’s general relativity, and the concordance model of cosmology,
to quantum field theory, upon which the standard model of particle physics is
built, as well as building bridges that enable comparison of theory with experimental data. I can therefore honestly say that there are many subjects only
touched upon in this thesis that in themselves deserve much more attention.
An Outline of the Thesis
The thesis is composed of two parts. The first introduces my research field
and reviews the models and results found in the second part. The second part
consists of my published scientific papers.
The organization for part one is as follows: Chapter 1 introduces the
essence of modern cosmology and discusses the concept of dark energy and
dark matter. Chapter 2 contains a general discussion of the dark matter distribution properties (containing the results of Paper VI), and its relevance
for dark matter annihilation signals. Why there is a need to go beyond the
standard model of particle physics is then discussed in Chapter 3. This is
followed by a description of general aspects of higher-dimensional theories,
and the universal extra dimension (UED) model is introduced. In Chapter 4,
a toy model for studying cosmology in a multidimensional universe is briefly
considered, and the discussion in Paper I is expanded. Chapter 5 then focuses on a detailed description of the field content in the UED model, which
simultaneously gives the particle structure of the standard model. After a
general discussion of the Kaluza-Klein dark matter candidate from the UED
model, special attention is placed on the results from Papers II-III in Chapter 6. This is then followed by a brief introduction to supersymmetry and the
results of Paper IV in Chapter 7. The inert Higgs model, its dark matter
candidate, and the signal found in Paper VII are then discussed in Chapter 8. Chapter 9 reviews Paper V and a claimed potential detection of a
dark matter annihilation signal, before Chapter 10 summarizes this thesis.
xi
xii
Preface
For a short layman’s introduction to this thesis, one can read Sections 1.1
and 1.6 on cosmology (including Table 1.1), together with Section 3.1 and
large parts of Section 3.2–3.3 on physics beyond the standard model. This
can be complemented by reading the preamble to each of the chapters and the
summary in Chapter 10 – to comprise the main ideas of the research results
in the accompanying papers.
My Contribution to the Accompanying Papers
As obligated, let me say some words on my contribution to the accompanying
scientific papers.
During my work on Papers I-IV, I had the privilege of closely collaborating with Torsten Bringmann and Martin Eriksson. This was a most democratic collaboration in the sense that all of us were involved in all parts of
the research. Therefore, it is in practice impossible to separate my work from
theirs. This is also reflected in the strict alphabetic ordering of author names
for these papers. If one should make one distinction, in Paper III I was
more involved in the numerical calculations than in the analytical (although
many discussions and crosschecks were made between the two approaches).
In Paper V, we scrutinized the claim of a potential dark matter detection by
de Boer et al. [1]. Joakim Edsjö and I independently implemented the dark
matter model under study, both into DarkSUSY and other utilized softwares.
I did the first preliminary calculations of the correlation between gamma-ray
and antiproton fluxes in this model, which is our main result in the paper. I
was also directly involved in most of the other steps on the way to the final
publication and wrote significant parts of the paper. For Paper VI Malcolm Fairbairn and I had similar ideas on how we could use Jesper-Sommer
Larsen’s galaxy simulation to study the dark matter distribution. I wrote
parts of the paper, although not the majority. Instead I did many of the
final calculations, had many of the ideas for the paper, and produced all the
figures (except Figure 4) for the paper. Regarding Paper VII, I got involved
through discussions concerning technical problems that appeared in implementing the so-called inert Higgs model into FeynArts. I found the simplicity
of the inert Higgs model very intriguing, and contributed many new ideas on
how to proceed with the paper, performed a majority of the calculations, and
wrote the main part of the manuscript.
Acknowledgments
Many people have influenced, both directly and indirectly, the outcome of this
thesis.
Special thanks go to my supervisor Professor Lars Bergström, who over
the last years has shown generous support, not only financial but also for his
sharing of fruitful research ideas.
xiii
My warmest thanks go to Torsten Bringmann and Martin Eriksson, who
made our collaboration such a rewarding and enjoyable experience, both scientifically and personally. Likewise, I want to thank Malcolm Fairbairn and
Erik Lundström for our enlightening collaborations. Many thanks also to my
collaborators Jesper-Sommer Larsen and Pierre Salati. Not the least, I want
to thank my collaborator Joakim Edsjö, who has often been like a supervisor
to me. His efficiency and sense of responsibility are truly invaluable.
Many of the people here at the physics department have had great impact
on my life during my years as a graduate student, and many have become my
close friends. Thank you, Sören Holst, Joachim Sjöstrand, Edvard Mörtsell,
Christofer Gunnarsson, Mia Schelke, Alexander Sellerholm and Sara Rydbeck.
Likewise, I want to thank Kalle, Jakob, Fawad, Jan, Emil, Åsa, Maria, Johan,
and all other past and present corridor members for our many interesting
and enjoyable discussions. The long lunches and dinners, movie nights, fun
parties, training sessions, and the many spontaneously cheerful moments here
at Fysikum have made my life much better.
I am also very grateful to my dear childhood friend Andreas Hegert for
helping me to produce the cover illustration.
All those not mentioned by name here, you should know who you are and
how important you have been. I want you all to know that I am extremely
grateful for having had you around and for your support in all ways during
all times. I love you deeply.
Michael Gustafsson
Stockholm, February 2008
Notations
and
Conventions
A timelike signature (+, −, −, · · · ) is used for the metric, except in Chapters 1
and 4, where a spacelike signature (−, +, +, · · · ) is used. This reflects my
choice of following the convention of Misner, Thorne and Wheeler [2] for
discussions regarding General Relativity, and Peskin and Schroeder [3] for
Quantum Field Theory discussions.
In a spacetime with d = 4 + n dimensions, the spacetime coordinates are
denoted by x̂ with capital Latin indices M, N, . . . ∈ {0, 1, . . . , d − 1} if it is a
higher dimensional spacetime with n > 0. Four-dimensional coordinates are
given by a lower-case x with Greek indices µ, ν, . . . (or lower-case Latin letters
i, j, . . . for spacelike indices). Extra-dimensional coordinates are denoted by
y p , with p = 1, 2, . . . , n. That is:
xµ
i
x
yp
≡
≡
≡
x̂M
M
x̂
x̂M
(µ = M = 0, 1, 2, 3) ,
(i = M = 1, 2, 3) .
(p = M − 3 = 1, 2, . . . , n) .
In the case of one extra dimension, the notation is slightly changed, so that
spacetime indices take the value {0, 1, 2, 3, 5} and the coordinate for the extra
dimension is denoted y (≡ y 1 ≡ x̂5 ). Higher-dimensional quantities such as
coordinates, coupling constants and Lagrangians will frequently be denoted
with a ‘hat’ (as in x̂, λ̂, L̂) to distinguish them from their four-dimensional
analogs (x, λ, L).
Einstein’s summation convention is always implicitly understood in expressions, i.e., one sums over any two repeated indices.
The notation ln is reserved for the natural logarithm (loge ), whereas log
is intended for the base-10 logarithm (log10 ).
Natural units, where c = ~ = kB = 1, are used throughout this thesis,
except occasionally where ~ and c appear for clarity.
Useful Conversion Factors (c = ~ = kB = 1)
1
1
1
1
GeV−1
=
GeV
=
barn (1 b) =
parsec (1 pc) =
6.5822 · 10−25 s = 1.9733 · 10−14 cm
1.6022 · 10−3 erg = 1.7827 · 10−24 g = 1.1605 · 1013 K
1012 pb = 10−24 cm2
3.2615 light yr = 2.0626·105 AU = 3.0856 · 1018 cm
xv
Useful Constants and Parameters
Speed of light:
Planck’s constant:
Boltzmann’s const:
Newton’s constant:
Planck mass:
c
~
~c
kB
G
Mpl
Electron mass:
Proton mass:
Earth mass:
Solar mass:
Hubble constant:
Critical density:
me
mp
M⊕
M⊙
H0
ρc
≡
=
=
=
=
≡
=
=
=
=
=
=
≡
=
=
=
2.99792458 · 1010 cm s−1
h/2π = 6.5821·10−25 GeV s
1.97 · 10−14 GeV cm
8.1674 · 10−14 GeV K−1
6.6726 · 10−8 cm3 g−1 s−2
(~c/G)1/2 = 1.2211 · 1019 GeV c−2
2.177 · 10−5 g
5.1100 · 10−4 GeV c−2
9.3827 · 10−1 GeV c−2
3.352 · 1054 GeV c−2 = 5.974 · 1030 g
1.116 · 1057 GeV c−2 = 1.989 · 1033 g
100h km s−1 Mpc−1 (h ∼ 0.7)
3H02 /8πG
1.0540h2 · 10−5 GeV c−2 cm−3
1.8791h2 · 10−29 g cm−3
2.7746h2 · 10−7 M⊙ pc−3
Acronyms Used in This Thesis
BBN
CDM
CERN
CMB
DM
EGRET
EWPT
FCNC
FLRW
GLAST
IDM
KK
LEP
LHC
LIP
LKP
MSSM
NFW
Ph.D.
SM
UED
WIMP
WMAP
Big Bang Nucleosynthesis
Cold Dark Matter
Conseil Européen pour la Recherche Nucléaire
(European Council for Nuclear Research)
Cosmic Microwave Background
Dark Matter
Energetic Gamma Ray Experiment Telescope
ElectroWeak Precision Tests
Flavor Changing Neutral Current
Friedmann Lemaı̂tre Robertson Walker
Gamma-ray Large Area Space Telescope
Inert Doublet Model
Kaluza-Klein
Large Electron-Positron Collider
Large Hadron Collider
Lightest Inert Particle
Lightest Kaluza-Klein Particle
Minimal Supersymmetric Standard Model(s)
Navarro Frenk White
Doctor of Philosophy
Standard Model (of particle physics)
Universal Extra Dimension
Weakly Interacting Massive Particle
Wilkinson Microwave Anisotropy Probe
Part
I
Background Material and Results
Chapter
1
The Essence of
Standard Cosmology
The Universe is a big place, filled with phenomena far beyond everyday experience. The scientific study of the properties and evolution of our Universe as
a whole is called cosmology. This chapter’s aim is to give a primary outline
of modern cosmology, present basic tools and notions, and introduce the dark
side of our Universe: the concepts of dark energy and dark matter.
1.1
Our Place in the Universe
For a long time, Earth was believed to be in the center of the Universe. Later∗
it was realized that the motion of the Sun, planets, and stars in the night sky
is more simply explained by having Earth and the planets revolving around
the Sun instead. The Sun, in turn, is just one among about 100 billion other
stars that orbit their mutual mass center and thereby form our own Milky
Way Galaxy. In a clear night sky, almost all of the shining objects we can
see by the naked eye are stars in our own Galaxy, but with current telescopes
it has been inferred that our observable Universe also contains the stars in
hundreds of billions of other galaxies.
The range of sizes and distances to different astronomical objects is huge.
Starting with our closest star, the Sun, from which it takes the light about
eight minutes to reach us here at Earth. This distance can be compared to
the distance around Earth, that takes mere one-tenth of a second to travel
at the speed of light. Yet these distances are tiny compared to the size of
our galactic disk – 100 000 light-years across – and the distance to our nearest
(large) neighbor, the Andromeda galaxy – 2 million light-years away. Still, this
is nothing compared to cosmological distances. Our own Milky Way belongs
to a small group of some tens of galaxies, the Local Group, which in turn
∗
The astronomer Nicolaus Copernicus (1473-1543) was the first to formulate the heliocentric view of the solar system in a modern way.
3
4
The Essence of Standard Cosmology
Chapter 1
belongs to a supercluster, the Virgo supercluster, including about one hundred
of such groups of clusters. The superclusters are the biggest gravitationally
bound systems and reach sizes up to some hundred million light-years. No
clusters of superclusters are known, but the existence of structures larger than
superclusters is observed in the form of filaments of galaxy concentrations,
thread-like structures, with a typical length scale of up to several hundred
million light-years, which form the boundaries between seemingly large voids
in the Universe.
This vast diversity of structures would make cosmology a completely intractable subject if no simplifying characteristic could be used. Such a desired,
simplifying feature is found by considering even larger scales, at which the Universe is observed to be homogeneous and isotropic. That is, the Universe looks
the same at every point and in every direction. Of course this is not true in
detail, but only if we view the Universe without resolving the smallest scales
and ‘smears out’ and averages over cells of 108 light-years, or more, across.
The hypothesis that the Universe is spatially isotropic and homogeneous at
every point is called the cosmological principle, and is one of the fundamental
pillars of standard cosmology. A more compact way to express the cosmological principle is to say that the Universe is spatially isotropic at every point, as
this automatically implies homogeneity [4]. The cosmological principle combined with Einstein’s general theory of relativity is the foundation of modern
cosmology.
1.2
Spacetime and Gravity
Since the study of the evolutionary history of our Universe is based on Einstein’s general theory of relativity, let us briefly go through the basic concepts
used in this theory and in cosmology. As the name suggests, general relativity
is a generalization of another theory, namely special relativity. The special
theory of relativity unifies space and time into a flat spacetime, and the general theory of relativity in turn unifies special relativity with Newton’s theory
of gravity.
Special Relativity
What does it mean to unify space and time into a four-dimensional spacetime theory? Obviously, already Newtonian mechanics involved three spatial
dimensions and a time parameter, so why not already call this a theory of
a four-dimensional spacetime? The answer lies in which dimensions can be
‘mixed’ in a meaningful way. For example, in Newtonian mechanics and in
a Cartesian coordinate system, defined by perpendicular directed x, y and z
axes, the Euclidian distance ds between two points is given by Pythagoras’
theorem
ds2 = dx2 + dy 2 + dz 2 .
(1.1)
Section 1.2.
Spacetime and Gravity
5
However, a rotation or translation into other Cartesian coordinate systems
(x′ , y ′ , z ′ ) could equally well be used and the distance would of course be
unaltered,
ds2 = dx′2 + dy ′2 + dz ′2
(1.2)
This invariance illustrates that the choice of axes and labels is not important
in expressing physical distances. The coordinate transformations that keep
Euclidian distances intact are the same that keep Newton’s laws of physics
intact, and they are called the Galileo transformations. The reference frames
where the laws of physics take the same form as in a frame at rest are called
inertial frames. In the Newtonian language, these are the frames where there
are no external forces, and particles remain at rest or in steady, rectilinear
motion.
The Galileo transformations do not allow for any transformations that mix
space and time; on the contrary, there is an absolute time that is independent of spatial coordinate choice. This, however, is not the case in another
classical theory – electrodynamics. The equations of electrodynamics are not
form-invariant under Galileo transformations. Instead, there is another class
of coordinate transformations that mix space and time and keep the laws
of electrodynamics intact. This new class of transformations, called Lorentz
transformations, leaves another interval dτ between two spacetime points invariant. This spacetime interval is given by
dτ 2 = −c2 dt2 + dx2 + dy 2 + dz 2 ,
(1.3)
where c2 is a constant conversion factor between three-dimensional Cartesian
space and time distances. That is, Lorentz transformations unifies space and
time into a four-dimensional spacetime (t, x, y, z), where space and time can
be mixed as long as the interval dτ in Eq. (1.3) is left invariant. Taking this
as a fundamental property, and say that all laws of physics must be invariant
with respect to transformations that leave dτ invariant, is the lesson of special
relativity.
Let me set up the notation that will be used in this thesis: x0 = ct, x1 = x,
x2 = y, and x3 = z. The convention will also be that Greek indices run from
0 to 3 so that four-vectors typically look like
(dx)α = (cdt, dx, dy, dz)
(1.4)
Defining the so-called Minkowski metric,
ηαβ

−1
 0

=
0
0
0
1
0
0
0
0
1
0

0
0 
,
0 
1
(1.5)
6
The Essence of Standard Cosmology
Chapter 1
allows for a very compact form for the interval dτ :
2
dτ =
3
X
ηαβ dxα dxβ = ηαβ dxα dxβ .
(1.6)
α,β=0
In the last step, the Einstein summation convention was used: Repeated indices appearing both as subscripts and superscripts are summed over. There
is one important comment to be made regarding the sign convention on ηαβ
used in this thesis: In Chapters 1 and 4, the sign convention of Eq. (1.5) is
adopted (as is the most common convention in the general relativity community), whereas in all other chapters ηαβ will be defined to have the opposite
overall sign (as is the most common convention within the particle physics
community).
In general, the allowed infinitesimal transformations in special relativity
are rotations, boosts, and translations. These form a ten-parameter nonabelian group called the Poincaré group.
The invariant interval (1.4), and thus special relativity, can be deduced
from the following two postulates [5]:
1. Postulate of relativity: The laws of physics have the same form in
all inertial frames.
2. Postulate of a universal limiting speed: In every inertial frame,
there is a finite universal limiting speed c for all physical entities.
Experimentally, and in agreement with electrodynamics being the theory of
light, the limiting speed c is equal to the speed of light in a vacuum. Today c
is defined to be equal to 2.99792458 × 108 m/s. Another way to formulate the
second postulate is to say that the speed of light is finite and independent of
the motion of its source.
General Relativity
In special relativity, nothing can propagate faster than the speed of light, so
Newton’s description of gravity, as an instant force acting between masses,
was problematic. Einstein’s way of solving this problem is very elegant. From
the observation that different bodies falling in the same gravitational field
acquire the same acceleration, he postulated:
The equivalence principle: There is no difference between gravitational and inertial masses (this is called the weak equivalence
principle). Hence, in a frame in free fall no local gravitational
force phenomena can be detected, and the situation is the same
as if no gravitational field was present. Elevate this to include
all physical phenomena; the results of all local experiments are
consistent with the special theory of relativity (this is called the
strong equivalence principle).
Section 1.2.
Spacetime and Gravity
7
From this postulate, you can derive many fundamental results of general relativity. For example, that time goes slower in the presence of a gravitational
field and that light-rays are bent by gravitating bodies. Due to the equivalence between gravitational and inertial masses, an elegant, purely geometrical
formulation of general relativity is possible: All bodies in a gravitational field
move on straight lines, called geodesics, but the spacetime itself is curved and
no gravitational forces exist.
We saw above that intervals in a flat spacetime are expressed by means
of the Minkowski metric (Eq. 1.5). In a similar way, intervals in a curved
spacetime can be express by using a generalized metric gµν (x) that describes
the spacetime geometry. The geometrical curvature of spacetime can be condensed into what is called the Riemann tensor, which is constructed from the
metric gµν (x) as follows:
σ
α
σ
α
α
α
Rα
βµν ≡ ∂µ Γβν − ∂ν Γβµ + Γσµ Γβν − Γσν Γβµ ,
(1.7)
where
1 αβ
g (∂ν gβµ + ∂µ gβν − ∂β gµν ) ,
(1.8)
2
and g µν (x) is the inverse of the metric gµν (x), i.e., g µσ (x)gνσ (x) = δνµ .
Having decided upon a description of gravity that is based on the idea of
a curved spacetime, we need a prescription for determining the metric in the
presence of a gravitational source. What is sought for is a differential equation
in analogy with Newton’s law for the gravitational potential:
Γα
µν =
∇φ = 4πGρ,
(1.9)
where ρ is the mass density and G Newton’s constant. If we want to keep
matter and energy as the gravitational source, and avoid introducing any
preferred reference frame, the natural source term is the energy-momentum
tensor Tµν (where the T00 component is Newton’s mass density ρ). A secondorder differential operator on the metric, set to be proportional to Tµν , can
be constructed from the Riemann curvature tensor:
8πG
1
Rµν − R gµν − Λgµν = 4 Tµν ,
2
c
(1.10)
αβ
where Rµν ≡ Rα
Rαβ . These are Einstein’s equations of genµαν and R ≡ g
eral relativity, including a cosmological constant Λ-term.† The left hand-side
of Eq. (1.10) is in fact uniquely determined if it is restricted to be divergence free (i.e., local source conservation ∇µ T µν = 0), be linear in the second
derivatives of the metric and free of higher derivatives, and vanish in a flat
spacetime [2]. The value of the proportionality constant 8πG in Eq. (1.10) is
†
Independently, and in the same year (1915), David Hilbert derived the same field equations from the action principle (see Eq. (4.4)) [6].
8
The Essence of Standard Cosmology
Chapter 1
obtained from the requirement that Einstein’s equations should reduce to the
Newtonian Eq. (1.9) in the weak gravitational field limit.
In summary, within general relativity, matter in free fall moves on straight
lines (geodesics) in a curved spacetime. In this sense, it is the spacetime that
tells matter how to move. Matter (i.e., energy and pressure), in turn, is the
source of curvature – it tells spacetime how to curve.
1.3
The Standard Model of Cosmology
We now want to find the spacetime geometry of our Universe as a whole, i.e.,
a metric solution gµν (xα ) to Einstein’s equations. Following the cosmological
principle, demanding a homogenous and isotropic solution, you can show that
the metric solution has to take the so-called Friedmann Lemaı̂tre Robertson
Walker (FLRW) form. In spherical coordinates r, θ, φ, t this metric is given
by:
dr2
2
2
2
2
dτ 2 = −dt2 + a(t)2
+
r
dθ
+
sin
θdφ
,
(1.11)
1 − kr2
where a(t) is an unconstrained time-dependent function, called the scale factor, and k = −1, +1, 0 depending on whether space is negatively curved,
positively curved, or flat, respectively. Note that natural units, where c is
equal to 1, have now been adopted.
To present an explicit solution for a(t) we need to further specify the
energy-momentum tensor Tµν in Eq. (1.10). With the metric (1.11), the
energy-momentum tensor must take the form of a perfect fluid. In a comoving
frame, i.e., the rest frame of the fluid, the Universe looks perfectly isotropic,
and the energy-momentum tensor has the form:
T µν = diag(ρ, p, p, p) ,
(1.12)
where ρ(t) represents the comoving energy density and p(t) the pressure of
the fluid. Einstein’s equations (1.10) can now be summarized in the so-called
Friedmann equation:
H2 ≡
and
2
ȧ
8πGρ + Λ
k
=
− 2
a
3
a
(1.13)
d
d
(ρa3 ) = −p a3 .
(1.14)
dt
dt
The latter equation should be compared to the standard thermodynamical
equation, expressing that the energy change in a volume V = a3 is equal to
the pressure-induced work that causes the volume change. Given an equation
of state p = p(ρ), Eq. (1.14) determines ρ as a function of a. Knowing
ρ(a), a solution a(t) to the Friedmann equation (1.13) can then be completely
Section 1.4.
9
Evolving Universe
specified once boundary conditions are given. This a(t) sets the dynamical
evolution of the Universe.
By expressing all energy densities in units of the critical density
ρc ≡
3H 2
,
8πG
(1.15)
the Friedmann equation can be brought into the form
1 = Ω + ΩΛ + Ωk ,
ρ
ρc ,
Λ
8πGρc
(1.16)
−k
ȧ2 .
ΩΛ ≡
and Ωk ≡
The energy density fraction Ω
where Ω ≡
is often further split into the contributions from baryonic matter Ωb (i.e.,
ordinary matter), cold dark matter ΩCDM , radiation/relativistic matter Ωr ,
and potentially other forms of energy. For these components, the equation of
state is specified by a proportionality constant w, such that p = wρ. Specifically, w ≈ 0 for (non-relativistic) matter, w = 1/3 for radiation, and if one so
prefers, the cosmological constant Λ can be interpreted as an energy density
with an equation of state w = −1. We can explicitly see how the energy density of each component‡ depends on the scale factor by integrating Eq. (1.14),
which gives:
ρi ∝ a−3(1+wi ) .
(1.17)
1.4
Evolving Universe
In 1929 Edwin Hubble presented observation that showed that the redshift
in light from distant galaxies is proportional to their distance [7]. Redshift,
denoted by z, is defined by
1+z ≡
λobs
,
λemit
(1.18)
where λemit is the wavelength of light at emission and λobs the wavelength
at observation, respectively. In a static spacetime, this redshift would presumably be interpreted as a Doppler shift effect: light emitted from an object
moving away from you is shifted to longer wavelengths. However, in agreement with the cosmological principle the interpretation should rather be that
the space itself is expanding. As the intergalactic space is stretched, so is
the wavelength of the light traveling between distant objects. For a FLRW
metric, the following relationship between the redshift and the scale factor
holds:
a(tobs )
1+z =
.
(1.19)
a(temit )
The interpretation of Hubble’s observation is therefore that our Universe is
expanding.
‡
Assuming that each energy component separately obeys local ‘energy conservation’.
10
The Essence of Standard Cosmology
Chapter 1
Proper distance is the distance we would measure withRa measuring tape
between two space points at given cosmological time (i.e. dτ ). In practice,
this is not a measurable quantity, and instead there are different indirect ways
of measuring distances. The angular distance dA is based on the flat spacetime
notion that an object of known size D, which subtends a small angle δθ, is at
a distance dA ≡ D/δθ. The luminosity distance dL instead makes use of the
fact that a light source appears weaker the further away it is, and is defined
by
r
S
dL ≡
,
(1.20)
4πL
where S is the intrinsic luminosity of the source and L the observed luminosity.
In flat Minkowski spacetime, these measures would give the same result, but
in an expanding universe they are instead related by dL = dA (1 + z)2. For an
object at a given redshift z, the luminosity distance for the FLRW metric is
given by
Z z
1
dz ′
dL = a0 (1 + z)f
,
(1.21)
a0 0 H(z ′ )

 sinh(x), if k = −1
x,
if k = 0
f (x) ≡
.

sin(x),
if k = +1
Here a0 represents the value of the scale factor today, and H(z) is the Hubble
expansion at redshift z:
sX
H(z) = H0
Ω0i (1 + z)−3(1+wi ) ,
(1.22)
i
where H0 is the Hubble constant, and Ω0i are the energy fraction in different
energy components today.
It is often convenient to define the comoving distance, the distance between
two points as it would be measured at the present time. This means that the
actual expansion is factored out, and the comoving distance stays constant,
even though the Universe expands. A physical distance d at redshift z corresponds to the comoving distance (1 + z) · d.
By measuring the energy content of the Universe at a given cosmological
time, e.g., today, we can, by using Eq. (1.17), derive the energy densities at
other redshifts. By naı̈vly extrapolating backwards in time we would eventually reach a singularity, when the scale factor a = 0. This point is sometimes
popularly referred to as the Big Bang. It should, however, be kept in mind that
any trustworthy extrapolation breaks down before this singularity is reached
– densities and temperatures will become so high that we do not have any
adequately developed theories to proceed with the extrapolation. A better
(and the usual) way to use the term Big Bang is instead to let it denote the
Section 1.5.
Initial Conditions
11
early stage of a very hot, dense, and rapidly expanding Universe. A brief
timeline for our Universe is given in Table 1.1.
This Big Bang theory shows remarkably good agreement with cosmological observations. The most prominent observational support of the standard
cosmological model comes from the agreement with the predicted abundance
of light elements formed during Big Bang nucleosynthesis (BBN), and the existence of the cosmic microwave background radiation. In the early Universe,
numerous photons, which were continuously absorbed, re-emitted, and interacting, constituted a hot thermal background bath for other particles. This
was the case until the temperature eventually fell below about 3 000 K. At this
temperature, electrons and protons combine to form neutral hydrogen (the socalled recombination), which then allows the photons to decouple from the
primordial plasma. These photons have since then streamed freely through
space and constitute the so-called cosmic microwave background (CMB) radiation. The CMB photons provides us today with a snapshot of the Universe
at an age of about 400 000 years or, equivalently, how the Universe looked
13.7 billion years ago.
1.5
Initial Conditions
The set of initial conditions required for this remarkable agreement between
observation and predictions in the cosmological standard model is however
slightly puzzling. The most well-known puzzles are the flatness and horizon
problems.
If the Universe did not start out exactly spatial flat, the curvature tends
to become more and more prominent. That means that already a very tiny
deviation from flatness in the early Universe would be incompatible with the
close to flatness observed today. This seemingly extreme initial fine-tuning is
what is called the flatness problem.
The horizon problem is related to how far information can have traveled
at different epochs in the history of our Universe. There is a maximal distance
that any particle or piece of information can have propagated since the Big
Bang at any given comoving time. This defines what is called the particle
horizon §
Z r(t)
Z t
dt′
dr′
√
=
a(t)
.
(1.23)
dH (t) =
′
1 − kr′2
0 a(t )
0
That is, in the past a much smaller fraction of the Universe was causally
connected than today. For example, assuming traditional Big Bang cosmology,
the full-sky CMB radiation covers about 105 patches that have never been in
causal contact. Despite this, the temperature is the same across the whole
§
There is also the notion of event horizon in cosmology, which is the largest comoving
distance from which light can ever reach the observer at any time in the future.
12
The Essence of Standard Cosmology
Table 1.1:
Chapter 1
The History of the Universe.
Time = 10−43 s
Size ∼ 10−60 × today
Temp = 1032 K
The Planck era: Quantum gravity is important; current theories are inadequate, and we cannot go any further back in time.
Time = 10−35 s
Size = 10−54→−26 × today
Temp = 1026→0→26 K
Inflation: A conjectured period of accelerating expansion; an inflaton field
causes the Universe to inflate and then decays into SM particles.
Time = 10−12 s
Size = 10−15 × today
Temp = 1015 K
Electroweak phase transition: Electromagnet and weak interactions become distinctive interactions below this temperature.
Time = 10−6 s
Size = 10−12 × today
Temp = 1012 K
Quark-gluon phase transition: Quarks and gluons become bound into
protons and neutrons. All SM particles are in thermal equilibrium.
Time = 100 s
Size = 10−8 × today
Temp = 109 K
Primordial nucleosynthesis: The Universe is cold enough for protons
and neutrons to combine and form light atomic nuclei, such as He, D and Li.
Time = 1012 s
Size = 3 · 10−4 × today
Temp = 104 K
Matter-radiation equality: Pressureless matter starts to dominate.
Time = 4 × 105 yrs
Size = 10−3 × today
Temp = 3 × 103 K
Recombination: Electrons combine with nuclei and form electrically neutral atoms, and the Universe becomes transparent to photons. The cosmic
microwave background is a snapshot of photons from this epoch.
Time = 108 yrs
Size = 0.1× today
Temp = 30 K
The dark ages: Small ripples in the density of matter gradually assemble
into stars and galaxies.
Time = 1010 yrs
Size = 0.5× today
Temp = 6 K
Dark energy: The expansion of the Universe starts to accelerate. A second
generation of stars, the Sun and Earth, are formed.
Time = 13.7 × 109 yrs Size = 1× today
Temp = 2.7 K
Today: ΩΛ ∼ 74%, ΩCDM ∼ 22%, Ωbaryons = 4%, Ωr ∼ 0.005%, Ωk ∼ 0
Section 1.6.
The Dark Side of the Universe
13
sky to a precision of about 10−5 . This high homogeneity between casually
disconnected regions is the horizon problem.
An attractive, but still not established, potential solution to these initial
condition problems was proposed in the beginning of the 1980’s [8–10]. By
letting the Universe go through a phase of accelerating expansion, the particle
horizon can grow exponentially and thereby bring all observable regions into
causal contact. At the same time, such an inflating Universe will automatically
flatten itself out. The current paradigm is basically that such an inflating
phase is caused by a scalar field Φ dominating the energy content by its
potential V (Φ). If this inflaton field is slowly rolling in its potential, i.e.,
1
2 φ̇ ≪ V (Φ), the equation of state is pΦ ≈ −V (Φ) ≈ −ρΦ . If V (Φ) stays
fairly constant for a sufficiently long time, it would mimic a cosmological
constant domination. From Eq. (1.13) it follows that H 2 ≈ constant and thus
that the scale factor grows as a(t) ∝ eHt . This will cause all normal matter
fields (w > −1/3) to dilute away†. During this epoch, the temperature drops
drastically, and the Universe super-cools due to the extensive space expansion.
Once the inflaton field rolls down in the presumed minimum of its potential, it
will start to oscillate, and the heavy inflaton particles will decay into standard
model particles. This reheats the Universe, and it evolves as in the ordinary
hot Big Bang theory with the initial conditions naturally‡ tuned by inflation.
During inflation, quantum fluctuations of the inflaton field will be stretched
and transformed into effectively classical fluctuations (see, e.g., [13]). When
the inflation field later decays, these fluctuations will be transformed to the
primordial power spectrum of matter density fluctuations. These seeds of
fluctuations will then eventually grow to become the large-scale structures,
such as galaxies etc, that we observe today. Today, the observed spectrum of
density fluctuations is considered to be the strongest argument for inflation.
1.6
The Dark Side of the Universe
What we can observe of our Universe are the various types of signals that
reach us – light of different wavelengths, neutrinos, and other cosmic rays.
This reveals the distribution of ‘visible’ matter. But how would we know if
there is more substance in the Universe, not seen by any of the above means?
The answer lies in that all forms of energy produce gravitational fields
(or in other words, curve the surrounding spacetime), which affect both their
local surroundings and the Universe as a whole. Perhaps surprisingly, such
gravitational effects indicate that there seems to be much more out there in
†
This would also automatically explain the absence of magnetic monopoles, which could
be expected to be copiously produced during Grand Unification symmetry breaking at
some high energy scale.
‡
A word of caution: Reheating after inflation drastically increases the entropy, and a very
low entropy state must have existed before inflation, see, e.g., [11, 12] and references
therein.
14
The Essence of Standard Cosmology
Chapter 1
our Universe than can be seen directly. It turns out that this ‘invisible stuff’
can be divided into two categories: dark energy and dark matter. Introducing
only these two types of additional energy components seems to be enough to
explain a huge range of otherwise unexplained cosmological and astrophysical
observations.
Dark Energy
In 1998 both the Supernova Cosmology Project and the High-z Supernova
Search Team presented for the first time data showing an accelerating expansion of the Universe [14, 15]. To accomplish this result, redshifts and
luminosity distances to Type Ia supernovae were measured. The redshift dependence of the expansion rate H(z) can then be deduced from Eq. (1.21).
The Type Ia supernovae data showed a late-time§ acceleration of the expansion of our Universe (ä/a > 0). This conclusion relies on Type Ia supernovae
being standard candles, i.e., objects with known intrinsic luminosities, which
are motivated both on empirical as well as theoretical¶ grounds.
These first supernova results have been confirmed by more recent observations (e.g., [16, 17]). The interpretation of a late-time accelerated expansion
of the Universe also fits well into other independent observations, such as data
from the CMB [18] and gravitational lensing (see, e.g., [19]).
These observations indicate that the Universe is dominated by an energy
form that i) has a negative pressure that today has an equation of state
w ≈ −1, ii) is homogeneously distributed throughout the Universe with an
energy density ρΛ ≈ 10−29 g/cm3 , and iii) has no significant interactions other
than gravitational. An energy source with mentioned properties could also be
referred to as vacuum energy, as it can be interpreted as the energy density
of empty space itself. However, within quantum field theory, actual estimates
of the vacuum energy are of the order of 10120 times larger than the observed
value.k
The exact nature of dark energy is a matter of speculation. A currently
viable possibility is that it is the cosmological constant Λ. That is, the Λ term
in Einstein’s equation is a fundamental constant that has to be determined by
observations. If the dark energy really is an energy density that is constant in
time, then the period when the dark energy and matter energy densities are
similar, ρΛ ∼ ρm , is extremely short on cosmological scales (i.e., in redshift
d
dt
ln ( aa ) =
d
dt
1
ln ( 1+z
)=
−1 dz
.
1+z dt
§
To translate between z and t, one can use H(z) =
¶
A Type Ia supernova is believed to be the explosion of a white dwarf star that has
gained mass from a companion star until reaching the so-called Chandrasekhar mass
limit ∼ 1.4M⊙ (where M⊙ is the mass of the Sun). At this point, the white dwarf
becomes gravitationally instable, collapses, and explodes as a supernova.
k
Inclusion of broken supersymmetry could decrease this disagreement to some 1060 orders
of magnitude.
0
Section 1.6.
The Dark Side of the Universe
15
range). We could wonder why we happen to be around to observe the Universe
just at the moment when ρΛ ∼ ρm ?
Another proposed scenario for dark energy is to introduce a new scalar
field, with properties similar to the inflaton field. This type of scalar fields
is often dubbed quintessence [20] or k-essence [21] fields. These models differ
from the pure cosmological constant in that such fields can vary in time (and
space). However, the fine-tuning, or other problems, still seems to be present
in all suggested models, and no satisfactory explanation of dark energy is
currently available.
Dark Matter
The mystery of missing dark matter (in the modern sense) goes back to at
least the 1930s when Zwicky [22] pointed out that the movements of galaxies
in the Coma cluster, also known as Abell 1656, indicated a mass-to-light ratio
of around 400 solar masses per solar luminosity, which is two orders of magnitude higher than in our solar neighborhood. The mass of clusters can also
be measured by other methods, for example by studying gravitational lensing
effects (see, e.g., [23] for an illuminating example) and by tracing the distribution of hot gas through its X-ray emission (e.g., [24]). Most observations on
cluster scales are consistent with a matter density of Ωmatter ∼ 0.2 − 0.3 [25].
At the same time the amount of ordinary (baryonic) matter in clusters can be
measured by the so-called Sunayaev-Zel’dovich effect [26], by which the CMB
gets spectrally distorted through Compton scattering on hot electrons in the
clusters. This, as well as X-ray observations, shows that only about 10% of
the total mass in clusters is visible baryonic matter, the rest is attributed to
dark matter.
At galactic scales, determination of rotation curves, i.e., the orbital velocities of stars and gas as a function of their distance from the galactic center,
can be efficiently used to determine the amount of mass inside these orbits. At
these low velocities and weak gravitational fields, the full machinery of general relativity is not necessary, and circular velocities should be in accordance
with Newtonian dynamics:
r
GM (r)
v(r) =
,
(1.24)
r
where M (r) is the total mass within radius r (and spherical symmetry has
been assumed). If there were no matter apart from the visible √
galactic disk,
the circular velocities of stars and gas should be falling off as 1/ r. Observations say otherwise: The velocities v(r) stay approximately constant outside
the bulk of the visible galaxy. This indicates the existence of a dark (invisible)
halo with M (r) ∝ r, and thus ρDM ∼ 1/r2 (see, e.g., [27]).
On cosmological scales, the observed CMB anisotropies combined with
other measurements are a powerful tool in determining the amount of dark
16
The Essence of Standard Cosmology
Chapter 1
matter. In fact, without dark matter, the cosmological standard model would
fail dramatically to explain the CMB observations [18]. Simultaneously, the
baryon fraction is determined to be about only 4%, which is in good agreement
with the value inferred, independently, from BBN to explain the abundance
of light elements in our Universe.
Other strong support for a large amount of dark matter comes from surveys
of the large-scale structures [28] and the so-called baryon acoustic peak in the
power spectrum of matter fluctuations [29]. These observations show how tiny
baryon density fluctuations, deduced from the CMB radiation, in the presence
of larger dark matter fluctuations have grown to form the large scale structure
of galaxies. The structures observed today would not even have had time to
form from these tiny baryon density fluctuations, if no extra gravitational
structures (such as dark matter) were present.
Finally, recent developments in weak lensing techniques have made it possible to produce rough maps of the dark matter distribution in parts of the
Universe [30].
Models that instead of the existence of dark matter suggest modifications
of Newton’s dynamics (MOND) [31, 32] have, in general, problems explaining
the full range of existing data. For example, the so-called ‘bullet cluster’
observation [33, 34] rules out the simplest alternative scenarios. The bullet
cluster shows a snapshot of what is interpreted as a galaxy cluster ‘shot’
through another cluster (hence the name bullet) – and is an example where the
gravitational sources are not concentrated around most of the visible matter.
The interpretation is that the dark matter (and stars) in the two colliding
clusters can pass through each other frictionless, whereas the major part of
the baryons, i.e., gas, will interact during the passage and therefore be halted
in the center. This explains both the centrally observed concentration of Xray-emitting hot gas, and the two separate concentrations of a large amount
of gravitational mass observed by lensing.
In contrast to dark energy, dark matter is definitely not homogeneously
distributed at all scales throughout the Universe. Dark matter is instead
condensed around, e.g., galaxies and galaxy clusters, forming extended halos.
To be able to condense, in agreement with observations, dark matter should
be almost pressureless and non-relativistic during structure formation. This
type of non-relativistic dark matter is referred to as cold dark matter.
The concordance model that has emerged from observations is a Universe
where about 4% is in the form of ordinary matter (mostly baryons in the form
of gas, w ≈ 0) and about 0.005% is in visible radiation energy (mostly the
CMB photons, w = 1/3). The remaining part of our Universe’s total energy
budget is dark and of an unknown nature. Of the total energy roughly 74%
is dark energy (w ∼ −1), and 22% is dark matter (w = 0). Most of the dark
matter is cold (non-relativistic) matter, but there is definitely also some hot
dark matter in the form of neutrinos. However, the hot dark matter can at
most make up a few percent [35,36]. Some fraction of warm dark matter, i.e.,
Section 1.6.
The Dark Side of the Universe
Dark Energy
74%
17
Dark Matter
22%
Baryonic Matter
4%
x
2% Luminous (Gas & Stars)
0.005% Radiation (CMB)
2% Dark Baryons (Gas)
Figure 1.1: The energy budget of our Universe today. Ordinary matter
(luminous and dark baryonic matter) only contributes some percent, while
the dark matter and the dark energy make up the dominant part of the
energy content in the Universe. The relative precisions of the quoted
energy fractions are roughly ten percent in a ΛCDM model. The figure is
constructed from the data in [18, 37–39].∗∗
particles with almost relativistic velocities during structure formation, could
also be present. This concordance scenario is often denoted the cosmological
constant Λ Cold Dark Matter (ΛCDM) model. Figure 1.1 shows this energy
composition of the Universe (at redshift z = 0).
Note that the pie chart in Fig. 1.1 do change with redshift (determined by
how different energy components evolve, see Eq. (1.17)). For example, at the
time of the release of the CMB radiation the dark energy part was negligible.
At that time the radiation contribution and the matter components were of
comparable size, and together made up more or less all the energy in the
Universe.
The wide range of observations presents very convincing evidence for the
existence of cold dark matter, and it points towards new, yet unknown exotic
physics. A large part of this thesis contain our predictions, within different
scenarios, that could start to reveal the nature of this dark matter.
All Those WIMPs – Particle Dark Matter
Contrary to dark energy, there are many proposed candidates for the dark
matter. The most studied hypothesis is dark matter in the form of some
∗∗
The background picture in the dark energy pie chart shows the WMAP satellite image
of the CMB radiation [40]. The background picture in the dark matter pie chart is a
photograph of the Bullet Cluster showing the inferred dark matter distribution (in blue)
and the measured hot gas distributions (in red) [41].
18
The Essence of Standard Cosmology
Chapter 1
yet undiscovered species of fundamental particle. To have avoided detection,
they should only interact weakly with ordinary matter. Furthermore, these
particles should be stable, i.e., have a life time that is at least comparable
to cosmological time scales, so that they can have been around in the early
Universe and still be around today.
One of the most attractive classes of models is that of so-called Weakly
Interacting Massive Particles – WIMPs. One reason for the popularity of these
dark matter candidates is the ‘WIMP miracle’. In the very early Universe,
particles with electroweak interactions are coupled to the thermal bath of
standard model particles, but at some point their interaction rate falls below
the expansion rate of the Universe. At this point, the WIMPs decouple, and
their number density freezes in, thereby leaving a relic abundance consistent
with the dark matter density today. Although the complete analysis can be
complicated for specific models, it is usually a good estimate that the relic
density is given by [42]
ΩWIMP h2 ≈
3 · 10−27 cm−3 s−1
,
hσvi
(1.25)
where h is the Hubble constant in units of 100 km s−1 Mpc−1 (h is today
observed to be 0.72 ± 0.03 [18]) and hσvi is the thermally averaged interaction rate (cross section times relative velocity of the annihilating WIMPs).
This equation holds almost independently of the WIMP mass, as long the
WIMPs are non-relativistic at freeze-out. The ‘WIMP miracle’ that occurs is that the cross section needed, hσvi ∼ 10−26 cm3 s−1 , is roughly
what is expected for particle masses at the electroweak scale. Typically
2
∼ 10−26 cm−3 s−1 , where α is the fine structure constant and
σv ∼ M 2α
WIMP
the WIMP mass MWIMP is taken to be about 100 GeV.
There are other cold dark matter candidates that do not fall into the
WIMP dark matter category. Examples are the gravitino and the axion. For
a discussion of these and other types of candidates, see for example [25] and
references therein.
Chapter
2
Where Is the
Dark Matter?
Without specifying the true nature of dark matter, one can still make general
predictions of its distribution based on existing observations, general model
building, and numerical simulations. Specifically, this chapter concentrates on
discussing the expected dark matter halos around galaxies like our own Milky
Way. For dark matter in the form of self-annihilating particles, the actual
distribution of its number density plays an extremely important role for the
prospects of future indirect detection of these dark matter candidates. An
effective pinching and reshaping of dark matter halos caused by the central
baryons in the galaxy, or surviving small dark matter clumps, can give an
enormously increased potential for indirect dark matter detection.
2.1
Structure Formation History
During the history of our Universe, the mass distribution has changed drastically. The tiny 10−5 temperature fluctuations at the time of the CMB radiation reflects a Universe that was almost perfectly homogeneous in baryon
density. Since then, baryons and dark matter have, by the influence of gravity,
built up structures like galaxies and clusters of galaxies that we can observe
today. To best describe this transition, the dark matter particles should be
non-relativistic (‘cold’) and experience at most very weak interactions with
ordinary matter. This ensures that the dark matter was pressureless and
separated from the thermal equilibrium of the baryons and the photons well
before recombination, and could start evolving from small structure seeds –
these first seeds could presumably originating from quantum fluctuations in
an even earlier inflationary epoch.
Perturbations at the smallest length scales – entering the horizon prior
to radiation-matter equality – will not be able to grow, but are washed out
due to the inability of the energy-dominating radiation to cluster. Later,
when larger scales enter the horizon during matter domination, dark matter
19
20
Where Is the Dark Matter?
Chapter 2
density fluctuations will grow in amplitude due to the absence of counterbalancing radiative pressure. This difference in structure growth, before and
after matter-radiation equality, is today imprinted in the matter power spectrum as a suppression in density fluctuations at comoving scales smaller than
roughly 1 Gpc, whereas on larger scales the density power spectrum is scale
invariant (in agreement with many inflation models). The baryons are, however, tightly coupled to the relativistic photons also after radiation-matter
equality and cannot start forming structures until after recombination. Once
released from the photon pressure, the baryons can then start to form structures rapidly in the already present gravitational wells from the dark matter.
Without these pre-formed potential wells, the baryons would not have the
time to form the structures we can observe today. This is a strong support
for the actual existence of cold dark matter.
As long as the density fluctuations in matter stay small, linearized analytical calculations are possible, whereas once the density contrast becomes
close to unity one has to resort to numerical simulations to get reliable results on the structure formation. The current paradigm is that structure is
formed in a hierarchal way; smaller congregations form first and then merge
into larger and larger structures. These very chaotic merging processes result
in so-called violent relaxation, in which the time-varying gravitational potential randomizes the particle velocities. The radius within which the particles
have a fairly isotropic distribution of velocities is commonly called the virial
radius. Within this radius, virial equilibrium should approximately hold, i.e.,
2Ek ≈ Ep , where Ek and Ep are the averaged kinetic energy and gravitational
potential, respectively.
By different techniques, such as those mentioned in Chapter 1, it is possible
to get some observational information on the dark matter density distribution. These observations are often very crude, and therefore it is common to
use halo profiles predicted from numerical simulations rather than deduced
from observations. In the regimes where simulations and observations can
be compared, they show reasonable agreement, although some tension might
persists [25].
2.2
Halo Models from Dark Matter Simulations
Numerical N -body simulations of structure formation can today contain up
to about 1010 particles (as, e.g., in the ‘Millennium simulation’ [43]) that
evolve under their mutual gravitational interactions in an expanding universe.
Such simulations are still far from resolving the smallest structures in larger
halos. Furthermore, partly due to the lack of computer power, many of these
high-resolution simulations include only gravitational interactions, i.e., dark
matter. These simulations suggest that radial density profiles of halos ranging
from masses of 10−6 [44] to several 1015 [45] solar masses have an almost
Section 2.2.
Halo Models from Dark Matter Simulations
21
Table 2.1: Parameters for some widely used dark matter density profile
models (see equation 2.1). The values of rs are for a typical Milky-Waysized halo of mass M200 ∼ 1012 M⊙ at redshift z = 0.
Model
NFW
Moore
Kra
Iso
α
1.0
1.5
2.0
2.0
β
3.0
3.0
3.0
2.0
γ
1.0
1.5
0.4
0.0
rs [kpc]
20
30
20
4
universal form.∗ A suitable parametrization for the dark matter density ρ
is to have two different asymptotic radial power law behaviors, i.e., r−γ at
the smallest radii and r−β at the largest radii, with a transition rate α by
which the profile interpolates between these two asymptotic powers around
the radius rs :
ρ0
(2.1)
ρ(r) =
β−γ .
(r/rs )γ [1 + (r/rs )α ] α
It is often convenient to define a radius r200 , sometimes also referred to as the
virial radius, inside which the mean density is 200 times the critical density
ρc . The total mass enclosed is thus:
3
4πr200
ρc .
(2.2)
3
For a given set of (α, β, γ) the density profile in Eq. (2.1) is completely specified
by only two parameters, e.g., the halo mass M200 and the scale radius rs . The
two parameters M200 and rs could in principle be independent, but numerical
simulations indicate that they are correlated. In that sense, it is sometimes
enough to specify only M200 for a halo (see, e.g., the appendix of [48]). Instead
of rs , the concentration parameter c200 = r200 /rs is also often introduced.
Although less dependent on halo size than rs , c200 also varies with a tendency
to increase for smaller halo size and larger redshifts (see, e.g., [49] and [47]).
Some of the most common values of parametrization parameters (α, β, γ)
found for dark matter halos are given in Table 2.1. From top to bottom, the
table gives the values for the Navarro, Frenk and White (NFW, [50]), Moore
et al. (Moore, [51]), and the Kravtsov et al. (Kra, [52]) profile. The modified
isothermal sphere profile (Iso, e.g., [53, 54]), with its constant density core, is
also included.
The most recent numerical simulations appear to agree on a slightly new
paradigm for the dark matter density. They suggest that the logarithmic
slope, defined as
d ln(ρ)
γ(r) ≡
,
(2.3)
d ln(r)
M200 = 200
∗
The density profiles are actually not found to be fully universal as the density slope in
the center of smaller halos is in general steeper than in larger halos [46, 47].
22
Where Is the Dark Matter?
Chapter 2
decreases continuously towards the center of the halos. In accordance with
this, the following, so-called Einasto density profile is suggested [55, 56]:
ρ(r) = ρ−2
2
exp −
α
r
r−2
α
−1
.
(2.4)
In this profile, ρ−2 and r−2 correspond to the density and radius where
ρ ∝ r−2 . (Note that the logarithmic slope converges to zero when r = 0.)
Typically α is found to be of the order of ∼ 0.2 [56].
The dark matter profile are sometimes referred to as cored, cuspy, or spiked
depending on whether the density in the center scales roughly as r−γ with
γ ≈ 0, γ & 0 or γ & 1.5, respectively.
All the above results stem from studies of dark matter dominated systems.
This should in many respects be adequate, as the dark matter makes up
∼80% [18] of all the matter and therefore usually dominates the gravitationally
induced structure formation. However, in the inner parts of e.g., galaxy halos,
the baryons, i.e., gas and stars, dominate the gravitational potential and
should be of importance also for dark matter distribution.
2.3
Adiabatic Contraction
The main difference between dark and baryonic matter is that the latter
will frequently interact and cool by dissipating energy. This will cause the
baryons, unless disturbed by major merges, to both form disk structures and
contract considerably in the centers. This behavior is indeed observed both
in simulations containing baryons and also in nature, where the baryons form
a disk and/or bulge at the center of apparently much more extended dark
matter halos. It has long been realized that this ability of baryons to sink
to the center of galaxies would create an enhanced gravitational potential
well within which dark matter could congregate, increasing the central dark
matter density. This effect is commonly modeled by the use of adiabatic
invariants [53, 54, 57–66].
A Simple Model for Adiabatic Contraction
The most commonly used model, suggested by Blumenthal et al. [60], assumes
a spherically symmetric density distribution and circular orbits of the dark
matter particles. From angular momentum conservation pi ri = pf rf and
p2
gravitational-centripetal force balance G M(r)
r 2 = m2 r , we obtain the adiabatic
invariant
rf Mf (rf ) = ri Mi (ri ) ,
(2.5)
where M (r) is the total mass inside a radius r and the lower indices i and
f indicate if a quantity is initial or final, respectively. Splitting up the final
Section 2.3.
Adiabatic Contraction
23
mass distribution Mf (r) into a baryonic part Mb (r) and an unknown dark
matter part MDM (r), we have Mf (r) = Mb (r) + MDM (r). Eq. (2.5) gives
rf =
ri Mi (ri )
.
Mb (rf ) + MDM (rf )
(2.6)
From mass conservation, the non-crossing of circular orbits during contraction,
and a mass fraction f of the initial matter distribution in baryons, we get
MDM (rf ) = Mi (ri )(1 − f ) ,
(2.7)
This means: From an initial mass distribution Mi , of which a fraction f (i.e.,
the baryons) forms a new distribution Mb , the remaining particles (i.e., the
dark matter) would respond in a such way that orbits with initial radius ri
end up at a new orbital radius rf . These new radii are given by Eq. (2.6),
and the dark matter mass inside these new radii is given by Eq. (2.7).
Modified Analytical Model
In reality, the process of forming the baryonic structure inside an extended
halo is neither a fully adiabatic process, nor spherically symmetric. Instead, it
is well established that typical orbits of dark matter particles inside simulated
halos are rather elliptical (see, e.g., [67]). This means that M (rorbit ) changes
around the orbit, and M (r)r in Eq. (2.5) is no longer an adiabatic invariant.
It has therefore been pointed out by Gnedin et al. [62] that Eq. (2.5) could be
modified to try to take this into account. In particular, they argue that using
the value of the mass within the average radius of a given orbit, r̄, should give
better results.
The average radius r̄ for a particle is given by
Z
2 ra r
dr
(2.8)
r̄ =
T rp vr
where vr is the radial velocity, ra (rp ) is the aphelion (perihelion) radius, and
T is the radial period. The ratio between r and r̄ will change throughout the
halo, but a suitable parametrization of hr̄i (i.e., r̄ averaged over the population
of orbits at a given radius r) is a power law with two free parameters [62]
w
r
hr̄i = r200 A
(2.9)
r200
The numerical simulations in [62] result in A = 0.85±0.05 and w = 0.8±0.02†‡
†
‡
In [62] they used r180 instead of r200 as used here, but the difference in A is very small.
In general, we have A180 = A200 (r200 /r180 )1−w , which for a singular isothermal sphere
(ρ ∝ 1/r 2 ) implies that A180 = A200 (180/200)(1−w)/2 ≈ 0.99A200 .
Another, almost identical, parametrization was used in [68]: hr̄i = 1.72y 0.82 /(1 +
5y)0.085 , where y = r/rs .
24
Where Is the Dark Matter?
Chapter 2
The modified adiabatic contraction model is given by
rf =
ri Mi (hr̄i i)
,
Mb (hr̄f i) + MDM (hr̄f i)
(2.10)
whereas the equation for the conservation of mass is unchanged
MDM (rf ) = Mi (ri )(1 − f ) .
(2.11)
These two equations can now be solved for any given A and w value. Thus
the model predicts the final dark matter distribution MDM (r) if one knows
the initial mass distribution together with the final baryonic distribution in a
galaxy (or some other similar system, like a cluster). How well these analytical
models work can now be tested by running numerical simulations including
baryons.
2.4
Simulation Setups
In Paper VI, we aimed at investigating the dark matter halos as realistically
as possible by using numerical simulations that included both dark matter
and baryons. These simulations were known from previous studies to produce
overall realistic gas and star structures for spiral galaxies [69–71]. Although
the numerical resolution is still far from being able to resolve many of the
small-scale features observed in real galaxies, the most important dynamical
properties such as the creation of stable disk and bulge structures both for
the gas and star components are accomplished.
Four sets of simulated galaxies were studied in Paper VI. The simulations
were performed by the Hydra code [72] and an improved version of the
Smoothed Particle Hydrodynamics code TreeSPH [73].§ In accordance with
the observational data, the simulations were run in a ΛCDM cosmology with:
ΩM = 0.3, ΩΛ = 0.7, H0 = 100h km s−1 Mpc−1 = 65 km s−1 Mpc−1 , a
matter power spectrum normalized such that the present linear root mean
square amplitude of mass fluctuations inside 8h−1 Mpc is σ8 = 1.0 and a
baryonic fraction f set to 0.15. By comparing simulations with different
resolutions, we could infer that the results are robust down to an inner radius
rmin of about 1 kpc. The simulations were run once including only dark
matter, then rerun with the improved TreeSPH code, incorporating star
formation, stellar feedback processes, radiative cooling and heating, etc. The
final results in the simulations including baryons are qualitatively similar to
observed disk and elliptical galaxies at redshift z = 0, a result that is mainly
possible by overcoming the angular momentum problem by an early epoch of
§
Hydra is a particle-particle, particle-mesh code that calculates the potential among
N point masses, and TreeSPH is for simulating fluid flows both with and without
collisionless matter. Each of the simulations in Paper VI took about 1 month of CPU
time on an Itanium II 1 GHz processor.
Section 2.5.
Pinching of the Dark Matter Halo
25
Table 2.2: The main properties (at redshift z=0) of the benchmark
galaxy and its dark matter halo.
Simulation
Virial radius r200 [kpc]
Total mass M200 [1011 M⊙ ]
Number of particles N200 [×105 ]
DM particle mass mDM [106 M⊙ ]
SPH particle mass mbaryon [106 M⊙ ]
Baryonic disk + bulge mass [1010 M⊙ ]
Baryonic bulge-to-disk mass ratio
DM+galaxy
209
8.9
3.6
6.5
1.1
7.17
0.19
DM only
211
9.3
1.2
7.6
...
...
...
strong, stellar energy feedback in the form of SNII energy being fed back to
the interstellar medium (see Paper VI and [69–71] for further details on the
numerical simulations).
In the following we will focus on the generic results in Paper VI. Although four galaxies were studied, I concentrate here on only one of them
to exemplify the generic results. [All examples will be from simulation S1
and its accompanying simulation DM1 found in Paper VI.] The simulated
galaxy resembles in many respects our own Milky Way, and some of its main
properties are found in Table 2.2.
2.5
Pinching of the Dark Matter Halo
With the baryonic disks and bulges formed fully dynamically, the surrounding
dark matter halo response should also be realistically predicted. Figure 2.1
shows the comparison of the simulation that includes the correct fraction of
baryons to the otherwise identical simulation with all the baryons replaced by
dark matter particles. It is clear how the effect of baryons – forming a central
galaxy – is to pinch the halo and produce a much higher dark matter density
in the central part.
For simulations including only dark matter, the density profile in Eq. (2.4),
with a continuously decreasing slope, turns out to be a good functional form.
The best fit values for the two free parameters in this profile are given in
Table 2.3. The simulation that includes baryons produces a dark matter cusp
Table 2.3: Best fit parameters to Eq. (2.4) for the spherical symmetrized
dark matter halo in the simulation with only dark matter.
α
0.247
a
r−2 [kpc]
18.5
χ2 /dof a
1.5
This χ2 fit was done with 50 bins, i.e., 48 degrees of freedom (dof).
26
Where Is the Dark Matter?
Chapter 2
DM in galaxy sim.
ρDM r2 / [ρc r2200]
100
DM only sim.
10
0.01
0.1
r / [r200]
1
Figure 2.1: Dark matter density for a galaxy simulation including
baryons (solid line) compared to an identical simulation including dark
matter only (dashed line). A clear steepening in the dark matter density
of the central part has arisen due to the presence of a baryonic galaxy.
The curves’ parameterizations are given in Table 2.4 and Table 2.3 for the
solid curve and the dashed curve, respectively. The data points, shown
as solid and open circles, are binned data directly from the simulations.
The arrows at the bottom indicate, respectively, the lower resolution limit
(rmin = 2 kpc) and the virial radius (r200 ≈ 200 kpc). These arrows also
indicate the range within which the curves have been fit to the data.
Table 2.4: Best fit parameters to Eq. (2.1) for the spherical symmetrized
dark matter halos in the simulation including baryons.
α
1.76
a
β
3.31
γ
1.83
rs [kpc]
44.9
χ2 /dof a
1.4
This χ2 fit was done with 50 bins, i.e., 46 degrees of freedom (dof).
and was therefore better fitted with Eq. (2.1), which allows for a steeper
logarithmic slope γ in the center. The best-fit parameter values are found in
Table 2.4. It should be realized that with four free parameters in the profile
(2.1), there are degeneracies in the inferred parameter values (see, e.g., [74]).
Although the numbers given in Table 2.4 give a good parametrization¶, they
¶
See, e.g., [75] for a nice introduction to statistical data analysis.
Section 2.6.
Testing Adiabatic Contraction Models
27
do not necessarily represent a profile that could be extrapolated to smaller
radii with confidence.
Let me summarize the result of all the dark matter halos studied in Paper VI. Simulations without baryons have a density slope continuously decreasing towards the center, with a density ρDM ∼ r−1.3±0.2 , at about 1%
of r200 . This is a result that lies between the NFW and the Moore profile
given in Table 2.1. The central dark matter cusps in the simulations that also
contain baryons become significantly steeper, with ρDM ∼ r−1.9±0.2 , with an
indication of the inner logarithmic slope converging to roughly this value.
2.6
Testing Adiabatic Contraction Models
The proposed adiabatic contraction models would, if they included all the
relevant physics, be able to foresee the true dark matter density profiles from
simulations that include only dark matter and known (i.e., observed) baryonic
distributions. Having in disposal simulations with identical initial conditions
except that in one case baryons are included and in the other not, we could
test how well these adiabatic contraction models work.
It turns out that the simpler contraction model by Blumenthal et al. [60]
significantly overestimates the contraction in the inner 10% of the virial radius
as compared to our numerical simulations; see Fig. 2.3. To continue and
test the modified adiabatic contraction proposed by Gnedin et al. [62] we
in addition need to first find out the averaged orbital eccentricity for the
dark matter (i.e., determine A and w in Eq. (2.9) for the pure dark matter
simulation). For our typical example model, we found that the averaged
orbital structure is well described by A = 0.74 and w = 0.69, as seen in
Fig. 2.2 (similar values were found for all our simulated dark matter halos).
From these A and w values, and the baryonic distribution in our corresponding (baryonic) galaxy simulation, the final dark matter distribution is
deduced from Eq. (2.10) and (2.11). Comparing the result from these two
equations with the dark matter density profile found in the actual simulation
including baryons showed that the Gnedin et al. model is a considerable improvement compared to the Blumenthal et al. model. However, this model’s
prediction also differed somewhat from the N -body simulation result that included baryons. To quantify this, A and w were taken as free parameters, and
a scan over different values was performed. With optimally chosen values of
A and w (no longer necessarily describing the orbital eccentricity structure of
the dark mater), it was always possible to obtain a good reconstruction of the
dark matter density profile.
Figure 2.3 shows the region in the (A,w)-plane that provides a good reconstruction of the dark matter halo for our illustrative benchmark simulation.
From these contour plots, it follows that the fits for (A,w)=(1,1) – which corresponds to circular orbits and therefore the original model of Blumenthal et
al. – are significantly worse than the fits for the optimal values (A ∼ 0.5 and
28
Where Is the Dark Matter?
30
Chapter 2
<r̄/r200 >= A(r/r200 )w
20
0.75
15
0.7
w
<r̄ > [kpc]
25
10
0.65
0.6
0.65
5
0.7
0.75
0.8
0.85
A
0
0
5
10
15
20
r [kpc]
Figure 2.2: hr̄i versus r for the halo simulation including only dark
matter. The best fit (solid line), corresponding to (A, w)=(0.74,0.69) in
Eq. (2.9), shows that the power law assumption is an excellent representation of the data. The large crosses represent the binned data, and the
smaller horizontal lines indicate the variance for hr̄i in each data point.
The smaller sub-figure shows in black the 1σ (68%) confidence region
whereas the lighter gray area is the 3σ (99.7%) confidence region in the
(A,w) plane. Figure from Paper VI.
w ∼ 0.6). We also see that although the Gnedin et al. model (marked by a
cross in Fig. 2.3) is a significant improvement it is not at all perfect.
All of our four simulations in Paper VI showed more or less significant
deviations from the model predictions. By changing the stellar feedback
strength, we could also find that this had an impact on the actual best fit
values of (A,w) (see Paper VI for more details). This difference between
(A,w) obtained directly from the relationship between hr̄i and r in Eq. (2.9)
and from the best fit values suggests (not surprisingly) that there is more
physics at work than can be described by a simple analysis of the dark matter
orbital structure.
2.7
Nonsphericity
We have just seen how the centrally concentrated baryons pinch the dark
matter. Since the dark matter particles have very elliptical orbits and the
baryons dominate the gravitational potential in the inner few kpc, it would
be interesting to see how the presence of the baryonic galactic disk influences
the triaxial properties of the dark matter halo. This was studied in Paper VI
Section 2.7.
Nonsphericity
29
1
0.8
w
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
A
Figure 2.3: Best fit parameters for reconstructing the baryon compressed
dark matter halo from its dark-matter-only halo. The black area is the 1σ
(68%) confidence region, and the larger gray is the 3σ (99.7%) confidence
region. The (A,w) value, expected from the analysis of orbit ellipticities
as proposed by Gnedin et al. [62], is marked by a cross, and the original
adiabatic contraction model, by Blumenthal et al. [60], by a circle. Figure
from Paper VI.
by relaxing the spherical symmetry assumption in the profile fitting, and
instead studying the halos’ triaxial properties. With a ellipsoidal assumption,
and studying the momentum of inertia tensor Iij , we determined the three
principal axes a, b, and c at different radii scales (see Paper VI for more
details). That is, we find how much we would need to stretch out the matter
distribution in three different directions to get a spherically symmetric density
profile.
Axis Ratios
Let the principal axes be ordered such that a ≥ b ≥ c and introduce the
parameters e = 1 − b/a (ellipticity) and f = 1 − c/a (flatness). Figure 2.4
shows how these quantities vary with radius. We can clearly see that ellipticity
and flatness differ between the simulation with only dark matter (left panel)
and the simulation including the formation of a baryonic disk galaxy (right
panel). The radius R on the horizontal axis gives the size of the elliptical shell
– that is semiaxes R, (b/a) · R and (c/a)R – inside which particles have been
used to calculate e and f .
Having obtained the semiaxes, we can determine whether a halo is prolate,
in other words, shaped like a rugby ball, or oblate, i.e., flattened like a Frisbee,
30
Where Is the Dark Matter?
Chapter 2
DM (incl baryons)
DM (DM only)
0.8
f,e
0.6
0.4
0.2
0
1
2
10
10
R [kpc]
1
2
10
10
R [kpc]
Figure 2.4: The triaxial parameters e = 1 − b/a (dashed line) and f =
1−c/a (solid line) of the dark matter halos in the simulation with only dark
matter (left panel) and the simulation including baryons (right panel).
Table 2.5: Values of the oblate/prolate-parameter T inside R = 10 kpc
for the dark matter halo.
Simulation:
T value
including baryons
0.076
only dark matter
0.74
by the measure
T =
a2 − b 2
.
a 2 − c2
(2.12)
If the halo is oblate, that is a and b are of similar size and larger than c, and
the measure is T < 0.5, whereas if T > 0.5 the halo is prolate. The T value
for the dark matter halo with and without baryons are listed in Table 2.5.
The general result, from all our four studied simulated halos, is that the
inclusion of the baryons causes the dark matter halo to change its shape from
being prolate in the pure dark matter simulations into a more spherical and
oblate form in simulations that include the formation of a central disk galaxy.
This result agrees and compliment the studies in [76–78].
Alignments
Given these results of nonsphericity, the obvious thing to check is whether the
principal axes of the dark matter and the baryon distributions are aligned.
Figure 2.5 shows this alignment between the stellar disk, the gaseous disk, and
the dark matter ‘disk’. The parameter ∆θ is the angle between each of these
vectors and a reference direction, defined to correspond to the orientation
vector of the gaseous disk with radius R = 10 kpc. The figure shows that the
orientation of the minor axes of the gas, stars, and dark matter is strongly
Section 2.8.
Some Comments on Observations
31
80
∆θ
60
40
20
0
1
2
10
10
R [kpc]
Figure 2.5: Diagram showing angular alignment of the gas (dotted), stars
(dot-dashed), and dark matter (solid line) in our four galaxy simulations.
The vertical scale is the difference in angle between the orientation of the
minor axis (around which the moment of inertia is the greatest) of the
component in question relative to the axis of the gas inside 10 kpc (by
definition zero and marked with a cross). The dashed line is the dark
matter in the simulation without baryons, showing that the baryonic disk
is formed aligned with the plane of the original dark matter halo.
correlated, and they line up with each other. However, at 50 kpc there is a
clear step in the stars’ alignment. The reason for this discrepancy is due to a
massive star concentration in a satellite galaxy outside the galactic disk.
Figure 2.5 also shows that the orientation of the baryonic disk is rather
correlated with the orientation of the flattest part of the dark matter halo in
the simulation without baryons. The dark matter therefore seems to have a
role in determining the orientation of the baryonic disk.
Let me summarize the nonsphericity results for all studied halos in Paper VI. All four galaxy simulations indicate that the inclusion of baryons
significantly influences the dark matter halos. Instead of being slightly prolate, they all became more spherical and slightly oblate, with their (modest)
flattening aligned with their galaxies’ gas disk planes.
2.8
Some Comments on Observations
The amount of triaxiality of dark matter halos seems to be a fairly generic
prediction in the hierarchial, cold dark matter model of structure formation,
and observational probes of halo shapes are therefore a fundamental test of
this model. Unfortunately, observational determination of halo shapes is a difficult task, and only coarse constraints exist. Probes of the Milky Way halo
indicate that it should be rather spherical with f . 0.2 and that an oblate
32
Where Is the Dark Matter?
Chapter 2
structure of f ∼ 0.2 might be preferable (see, e.g., [79] and references therein).
Milky-Way-sized halos formed in dissipationless simulations are usually predicted to be considerably more triaxial and prolate, although a large scatter is
expected [80–86]. Including dissipational baryons in the numerical simulation,
and thereby converting the halo prolateness into a slightly oblate and more
spherical halo, might turn out to be essential to produce good agreement with
observations [77].
Having determined the ellipsoidal triaxiality of the dark matter distribution, we can include this information in the profile fits. Including triaxiality
to the radial density profile fits would not change any results (see Paper VI).
This should not be surprising since the flattening of the dark matter halo is
very weak. The oblate structure of the dark matter would have some minor effects on the expected indirect dark matter signal [87]. However, the
baryonic effects found here have no indication of producing such highly diskconcentrated dark matter halo profiles as used in, e.g., [1] to explain the
excess of diffuse gamma-rays in the EGRET data by WIMP annihilation (see
Chapter 9 for more details).
Observations of presumably dark-matter-dominated systems, such as low
surface brightness dwarf galaxies, indicate that dark matter halos have constant density cores instead of the steep cusps found in numerical simulations
(see, eg., [88–93]). This could definitely be a challenge for the standard cold
dark matter scenario. Even if baryons are included in the N -body simulations,
and very explosive feedback injections are enforced, it seems unlikely that it
could resolve the cusp-core problem (see, e.g., [94] and references therein).
However, several studies also demonstrate that the cusp-core discrepancy
not necessarily implies a conflict. Observational and data processing techniques in deriving the rotation curves (see, e.g., [89–91]), and the neglected
complex effects on the gas dynamics due to the halos’ triaxiality properties [95], indicate that there might not even be a discrepancy between observation and theory. One should also note that the story is actually different
for galaxy halos where the baryons dominate the gravitational mass in the
inner parts (as the halos studied in Paper VI). Here the problem of separating the dark matter component from the dominant baryonic component
allows the dark matter profile to be more cuspy without any conflict with observation. As adiabatic contraction increases the central dark matter density
in such a way that the dark matter density only tends to track the higher
density baryonic component, strong adiabatic contraction of the dark matter
halo in these systems should not be excluded.
2.9
Tracing Dark Matter Annihilation
Improved knowledge about the dark matter distribution is essential for reliable
predictions of the detection prospects for many dark matter signals. For any
self-annihilating dark matter particle, the number of annihilations per unit
Section 2.9.
33
Tracing Dark Matter Annihilation
time and volume element is given by
dn
1
ρ2 (r)
= hσvi DM2 ,
dt
2
mDM
(2.13)
where v is the relative velocity of the two annihilating particles, σ is the total
cross section for annihilation, and ρDM (r) the dark matter mass density at the
position r where the annihilation take place.
Indirect detection of dark matter would be to detect particles produced
in dark matter annihilation processes, e.g., to find an excess in the amount
of antimatter, gamma rays, and/or neutrinos arriving at Earth [25]. Since
the expected amplitude of any expected signal depends quadratically on ρDM ,
it seems most promising to look for regions of expected high dark matter
concentrations. Unfortunately, charged particles will be significantly bent by
the magnetic fields in our Galaxy and will no longer point back to their source.
On the other hand, this is not the case for neutrinos and gamma rays as these
particles propagate more or less unaffected through our Galaxy.
When looking for gamma rays towards a region of enhanced dark matter
density, the expected differential photon flux along the line of sight (l.o.s.) in
a given direction ψ is given by
hσtot vi dNγeff
dΦγ (ψ)
=
dEγ
8πm2DM dEγ
Z
dℓ(ψ)ρ2DM (ℓ) ,
(2.14)
l.o.s
where dNγeff /dEγ is the energy differential number of photons produced per
dark matter pair annihilation.
Expected dark matter induced fluxes are still very uncertain. Any attempt
to accurately predict such fluxes is still greatly hampered by both theoretical uncertainties and lack of detailed observational data on the dark matter
distribution. It can be handy to separate astrophysical quantities (ρDM ) from
particle physics properties (mDM , hσvi and dNγ /dEγ ). It is therefore convenient to define the dimensionless quantity [96]
hJi∆Ω (ψ) ≡
1
8.5 kpc
1
0.3 GeV cm−3
2
1
∆Ω
Z
dΩ
∆Ω
Z
dℓ(ψ)ρ2DM (ℓ) ,
l.o.s.
(2.15)
which embraces all the astrophysical uncertainties. The normalization values
3
8.5 kpc and 0.3 GeV/cm are chosen to correspond to commonly adopted
values for the Sun’s galactocentric distance and the local dark matter density,
respectively. For a detector of angular acceptance ∆Ω, the flux thus becomes
dNγeff
dΦγ
= 9.4 · 10−13
dEγ
dEγ
hσtot vitot
10−26 cm3 s−1
1 TeV
mDM
2
· ∆Ω hJi∆Ω cm−2 s−1 TeV−1 . (2.16)
34
Where Is the Dark Matter?
Chapter 2
For example, looking towards the galactic center with an angular acceptance of ∆Ω = 10−5 sr (which is comparable to the angular resolution of, e.g.,
the H.E.S.S. or GLAST telescope) it is convenient to write
∆Ω hJi∆Ω (0) = 0.13 b sr .
(2.17)
where b = 1 if the dark matter distribution follows a NFW profile as given in
Table 2.1 [97]. On the other hand, we have just seen that taking into account
the effect of baryonic compression due to the dense stellar cluster observed to
exist very near the galactic center, could very well enhance the dark matter
density significantly. A simplifying and effective way to take into account such
an increase of the dark matter density is to simply allow the so-called boost
factor b to take much higher values. Exactly how high this boost factor can
be in the direction of the galactic center, or in other directions, is still not
well understood, and we will discuss this in some more detail in what follows.
Indirect Dark Matter Detection
It is today impossible for galaxy simulations to get anywhere near the length
resolution corresponding to the very center of the galaxy. Despite this, different profile shapes from numerical simulations have frequently been extrapolated into the galactic center. This enables, at least naı̈vely, to predict expected fluxes from dark matter annihilation in the galactic center. In the
spirit of comparing with the existing literature we performed the baryonic
contraction with our best fit values on A and w. For typical values of the
Milky Way baryon density (see Paper VI for details) and an initial Einasto
dark matter profile (given in Table 2.3) it is straightforward to apply the
contraction model. The local dark matter density is here normalized to be
ρDM (r = 8.5kpc) ∼ 0.3 GeV cm−3 . Figure 2.6 shows both results if a 2.6 × 106
M⊙ central supermassive black hole is included in the baryonic profile and if
it is not.
No attempt is made to model the complicated dynamics at subparsec
scales of the galaxy, other than trying to take into account the maximum
density due to self-annihilation. In other words, a galactic dark matter halo
unperturbed by major mergers or collisions for a time scale τgal cannot contain
stable regions with dark matter densities larger than ρmax ∼ mDM /hσviτgal .
In Fig. 2.6 and in Table 2.6, it is assumed that τgal = 5×109 years and for the
WIMP property a dark matter mass of mDM = 1 TeV, with an annihilation
cross section of hσvi = 3 × 10−26 cm3 s−1 , is adopted.
Table 2.6 shows the energy flux obtained with the values of (A,w) found
in the previous section, as well the Blumenthal et al. estimate. This is the
total energy luminosity, not in some specific particle species, and is hence the
flux given in Eq. (2.13) multiplied by 2 times the dark matter mass. The
Blumenthal et al. adiabatic contraction model gives fluxes far in excess of the
modified contraction model. Even with the modified contraction model, and
Section 2.9.
Tracing Dark Matter Annihilation
ba
10
ry
x
ma
on
s
(A
,w
)
=(
3
↓
ρ
2
[ (GeV/cm kpc) ]
10
1,1
)
← (αβγ)−profile
5
10
(A,w)=(0
.5
ρ r
2 2
1,0.6)
Einasto →
←r
0
10
BH
−10
10
−5
10
0
10
r [kpc]
Figure 2.6: Diagram showing the contraction of a Einasto dark matter
density profile, given in Table 2.3, (dot-dashed line) by a baryon profile
(dotted line) as described in Paper VI. The resulting dark matter profiles (dashed lines) are plotted for (A, w) = (1, 1) and (0.51, 0.6), each
splitting into two at low radii, the denser corresponding to the density
profile achieved from a baryon profile that includes a central black hole.
The extrapolated density profile from Table 2.4 from our simulation is
shown for comparison (solid line). Remember the numerical simulation is
only robust into rmin ≈ 1 kpc. Also shown are the maximum density line
and the radius corresponding to the lowest stable orbit around the central
black hole. Figure adapted from Paper VI.
Table 2.6: Luminosity in erg s−1 from dark matter annihilation for different contraction model parameters (A,w). The initial dark matter profiles
are an NFW profile, given in Table 2.1, or the Einasto profile, given in
Table 2.3. The baryon profile includes a super massive black hole, as described in the text. Quoted values are for the flux from the inner 10 pc
and 100 pc. Note that this is the total luminosity, and not of some specific
particle species. A dark matter particle mass of 1 TeV is assumed.
A
w
Initial profile →
1
1
0.51 0.6
NFW (Table 2.1)
Einasto (Table 2.3)
L10pc
L100pc
L10pc
L100pc
3.9 × 1033 3.9 × 1034 2.7 × 1031 4.7 × 1033
2.8 × 1040 2.8 × 1040 9.8 × 1039 9.8 × 1039
2.8 × 1037 3.5 × 1037 7.9 × 1035 3.5 × 1036
35
36
Where Is the Dark Matter?
Chapter 2
the values of (A, w) found in Paper VI, we find a large flux enhancement
compared to the traditional NFW profile (as given in Table. 2.1). This stays
true also if the initial profile is the Einasto profile, which does not initially
possess a cusp at all. From Table 2.6, the boost of the luminosity compared
to the standard NFW profile takes values of about 102 – 104 in the direction
of the galactic center. This extrapolation to very small radii neglects many
potential effects, such as the scattering of dark matter particles on stars or
the effect from a supermassive black hole not exactly in the galactic center.
Case studies of the dark matter in the galactic center [68, 98–103] show that
compared to an NFW density profile the expected flux from self-annihilation
can be boosted as much as 107 , but also that the opposite effect might be
possible leading to a relative depletion of an initial dark matter cusp.
2.10
Halo Substructure
So far the dark matter density has been described as a smooth halo profile with
a peak concentration in the center. Presumably, the halos contain additional
structure. Numerical simulations find a large number of local dark matter
concentrations (clumps) within each halo (see, e.g., [47, 104–112]). Also substructures within substructures etc. are found. This should not be surprising
as the hierarchal structure formation paradigm predicts the first formed structures to be numerous small dark matter halos. These first formed (WIMP)
halos could still be around today as clumps of about the Earth mass and with
sizes similar the solar system [44, 113–115]. In the subsequent processes, accretion to form larger structures by the merging of smaller progenitors is not
always complete, the cores of subhalos could survive as gravitationally bound
subhalos orbiting within a larger host system. More than 1015 of this first
generation of dark halo objects could potentially be within the halo of the
Milky Way [111], but gravitational disruption during the accretion process as
well as late tidal disruption from stellar encounters can significantly decrease
this number [116].
This additional substructure could be highly relevant for indirect detection of dark matter. As the dark matter annihilation rate, into for example
gamma rays, increases quadratically with the dark matter density, the internal substructure may enhance not only the total diffuse gamma-ray flux
compared to the smooth halo, but also individual clumps of dark matter
could be detectable with, e.g., gamma-ray telescopes such as GLAST. The
prospects for detecting these subhalos, however, depend strongly on the assumptions (see, e.g., [117]). With surviving microhalos down to masses of
10−6 M⊙ , the dark matter fraction in a galaxy could be as large as about
50% of the total mass [117]. Translated into an enhancement of the total
dark matter annihilation rate for whole galaxies, this gives a boost factor of
a few, up to perhaps some hundred [112, 117, 118]. The local annihilation
boost, compared to the smooth background halo profile, due to subhalos is,
Section 2.10.
Halo Substructure
37
however, expected to strongly depend on the galactocentric distance. In the
outer regions, the clumps can boost the rates by orders of magnitude. On
the other hand, in the inner regions the increase in annihilation rates due to
clumps might be negligible. This is both because expected tidal disruption
could have destroyed many clumps in the center and that the smooth component already give larger annihilation rates. Spatial variations of the local
annihilation boost mean that different dark matter signals are expected to
depend on both species and energy [119] of the annihilation products. For
example, positrons are most sensitive to the local boost factors. Positrons are
strongly affected by magnetic fields – they quickly lose energy and directional
information – and become located within some kpc to their source before
diffusing outside the galactic disk and escape from the Galaxy. On the contrary, gamma rays propagate almost freely in our Galaxy, and are therefore
affected also by more distant dark matter density boosts. The intermediate
case are antiproton signals, which, like the positrons, are sensitive to clumps
in all sky directions, but due to their much higher mass they are less deflected
by magnetic fields and can therefore travel longer distances in the disk (see,
e.g., [119]).
Chapter
3
Beyond the
Standard Model:
Hidden Dimensions
and More
Why is there a need to go beyond today’s standard model of particle physics?
This chapter presents motivations and introduces possible extensions, which
will be discussed further in this thesis. Special focus is here put on the possibility that hidden extra space dimensions could exist. General features expected in theories with extra space dimensions are discussed, together with
a short historical review. The chapter concludes by giving motivations to
study the specific model of so-called universal extra dimensions, as this will
be the model with which we will start our discussions on dark matter particle
phenomenology in the following chapters.
3.1
The Need to Go Beyond the Standard Model
Quantum field theory is the framework for today’s standard model (SM) of
particle physics – a tool box for how to combine three major themes in modern
physics: quantum theory, the field concept, and special relativity. Included
in the SM is a description of the strong, weak, and electromagnetic forces
as well as all known fundamental particles. The theoretical description has
been a great success and agrees, to a tremendous precision, with practically
all experimental results up to the highest energies reached (i.e., some hundred
GeV). However, it is known that the SM is not a complete theory as it stands
today. Perhaps the most fundamental drawback is that it does not include a
quantum description of gravitational interactions. There are also a number
of reasons directly related to particle physics for why the SM needs to be
extended. For example, the SM does not include neutrino masses (neutrinos
masses are by now a well-accepted interpretation of the observed neutrino
39
40
Beyond the Standard Model: Hidden Dimensions and More
Chapter 3
oscillations [120–125]), and extreme fine-tuning is required in the Higgs sector
if no new divergence canceling physics appears at TeV energies [with no new
physics between the electroweak scale (102 GeV) and the Planck scale (1019
GeV) is usually called hierarchy problem]. Physics beyond the SM is also
very attractive for cosmology, where new fields in the form of scalar fields
driving inflation are discussed, and since the SM is incapable of explaining
the observed matter-antimatter asymmetry and the amount of dark energy.
The strongest reason for new fundamental particle physics, however, comes
perhaps from the need for a viable dark matter candidate.
Despite the necessity of replacing (or extending) the SM, its great success
hints also that new fundamental physics could be closely tied to some of its
basic principles, such as its quantization and symmetry principles. As the SM
is a quantum field theory with an SU (3) × SU (2) × U (1) gauge symmetry
and an SO(1, 3) Lorentz symmetry, it is tempting to investigate extensions of
these symmetries.
This is the idea behind grand unified theories (GUTs), where at high
energies (typically of the order of 1016 GeV) all gauge couplings have the
same strength and all the force fields are fused into a unified field. This is
the case for the SU (5) GUT theory. This larger symmetry group is then
thought to be spontaneously broken at the grand unification scale, down to
the SU (3) × SU (2) × U (1) gauge group of the SM that we observe at todays
testable energies. However, the simplest SU (5) theory predicts a too-short
proton lifetime, and is nowadays excluded.
Another type of symmetry extension (with generators that anticommutate) is a spacetime symmetry that mixes bosons and fermions. These are the
supersymmetric extensions of the SM, where every fundamental fermion has
a bosonic superpartner of equal mass and vice versa, that every fundamental boson has a fermionic superpartner. This symmetry must, if it exists, be
broken in nature today, so as to give all superpartners high enough masses to
explain why they have evaded detection. This possibility is further discussed
in Chapter 7.
Yet another possibility is a Lagrangian with extended Lorentz symmetry,
achieved by including extra dimensions. The most obvious such extension is
to let the SM have the Lorentz symmetry SO(1, 3 + n), with n ≥ 1 an integer. This implies that all SM particles propagate in n extra spatial dimensions endowed with a flat metric. These are called universal extra dimensions
(UEDs) [126]∗ Other extra-dimensional scenarios also exist where all or part
of the matter and SM gauge fields are confined to a (3+1)-dimensional brane
on which we are assumed to live. The aim of the many variations of extradimensional models is usually to propose different solutions, or new perspectives, to known problems in modern physics. Most such scenarios are therefore
∗
Along similar lines, studies of possible effects of extra dimensions felt by SM particles
were also done earlier in [127–129].
Section 3.2.
General Features of Extra-Dimensional Scenarios
41
of a more phenomenological nature, but a notable exception is string theory.
String theory (see, e.g., [130] for a modern introduction) certainly aims to be a
fundamental theory, and by replacing particles with extended strings, it offers
a consistent quantum theory description of gravity (how much this theory is
related to reality is, however, still an open question). The requirement that
the theory should be anomaly free leads canonically to a critical value of the
spacetime dimensionality. In the case of supersymmetric strings, the number
of dimensions must be d = 10 (or d = 11 for M-theory) [130]. Performing a
fully consistent extra dimensional compactification within string theory that
leads to firm observational predictions at accessible energies is at the moment
very challenging. Therefore, fundamental string theory is not yet ready for
making unique (or well constrained) phenomenological predictions in a very
rigorous way. However, as string theory is perhaps the most promising candidate for a more fundamental theory today, it is of interest to try to anyway
investigate its different aspects, such as extra space dimensions, from a more
phenomenological perspective.
Finally, another approach to go beyond the SM could be to try to extend
it as minimally as possible to incorporate only new physics that can address
specific known drawbacks. One such approach, which extends only the scalar
sector of the SM, is discussed in Chapter 8. There it is shown that such an
extension, besides other advantages, gives rise to an interesting scalar dark
matter particle candidate with striking observational consequences.
3.2
General Features of Extra-Dimensional Scenarios
Although we are used to thinking of our world as having three spatial dimensions, there is an intriguing possibility that space might have more dimensions.
This might at first sound like science fiction, and seemingly ruled out by observations, but in the beginning of the 20th century Nordström [131], and
more prominently Kaluza [132] and Klein [133] asked whether extra dimensions could say something fundamental about physics. To allow for an extra
dimension, without violating the apparent observation of only three space
dimensions, it was realized that the extra dimension could be curled up on
such a small length scale that we have not yet been able to resolve the extra
dimension. As an analogy, imagine you are looking at a thin hose from a long
distance. The hose then seems to be just a one-dimensional line, but as you
get closer you are able to resolve the thickness of the hose, and you realize it
has an extended two-dimensional surface.
In the original idea by Kaluza [132], and rediscovered by Klein, the starting point was a five-dimensional spacetime with the dynamics governed by
the Einstein-Hilbert action (i.e. general relativity). After averaging over the
(assumed static) extra dimension, and retaining an ordinary four-dimensional
effective theory, the result was an action containing both Einstein’s general
relativity and Maxwell’s action for electro-magnetism. Although it at first
42
Beyond the Standard Model: Hidden Dimensions and More
Chapter 3
seemed to be a very ‘magic’ unification, it can be traced back to the fact that
the compactification of one extra dimension on a circle automatically gives a
U (1) symmetry, which is exactly the same key symmetry as in the electromagnetic (abelian) gauge theory. There is, however, a flaw in the five dimensional
Kaluza-Klein (KK) theory, even before trying to include the weak and the
strong interactions. Once matter fields are introduced, and with the U (1)
symmetry identified with the usual electromagnetism, the electric charge and
mass of a particle must be related and quantized. With the quantum of charge
being the charge of an electron, all charged particles must have masses on the
Planck scale Mpl ∼ 1019 GeV. This is not what is observed – all familiar
charged particles have very much smaller masses.
Today much of the phenomenological studies of extra dimensions concern
the generic features that can be expected. To illustrate some of these features, let us take the spacetime to be a direct product of the ordinary (fourdimensional) Minkowski spacetime and n curled up, flat extra dimensions.
We can then show that in the emerging effective four-dimensional theory:
(1) a tower of new massive particles appears
(2) Newton’s 1/r law is affected at short distances
(3) fundamental coupling ‘constants’ will vary with the volume spanned by
the extra dimensions
Feature (1) can qualitatively be understood quite easily. Imagine a particle
moving in the direction of one of the extra dimensions. Even if the particle’s
movement cannot be directly observed, the extra kinetic energy will still contribute to its total energy. For an observer, not aware of the extra dimensions,
this additional kinetic energy will be interpreted as a higher mass (E = mc2 )
for that particle compared to an identical particle that is not moving in the
extra dimensions. To do this more formally, let us denote local coordinates
by
{x̂M } = {xµ , y p } ,
(3.1)
where M = 0, 1, . . . , 3 + n, µ = 0, 1, 2, 3 and p = 1, 2, . . . , n, and consider
a scalar field Φ̂(x̂) with mass m in five dimensions. Its dynamics in a flat
spacetime are described by the Klein-Gordon equation:
ˆ (5) + m2 Φ̂(xµ , y) = ∂t2 − ∇2 − ∂y2 + m2 Φ̂(xµ , y) = 0 .
(3.2)
With the fifth dimension compactified on a circle with circumference 2πR,
i.e.,
y ∼ y + 2πR,
(3.3)
any function of y can be Fourier series expanded, and the scalar field is decomposed as
X
n
(3.4)
Φ̂(xµ , y) =
Φ(n) (xµ ) e−i R y .
n
Section 3.2.
General Features of Extra-Dimensional Scenarios
43
Each Fourier component Φ(n) (x) thus separately fulfills the four-dimensional
Klein-Gordon equation:
+ m2n Φ(n) (xµ ) = ∂t2 − ∇2 + m2n Φ(n) (xµ ) = 0 ,
(3.5)
where the masses are
m2n = m2 +
n 2
.
(3.6)
R
That is, a single higher-dimensional field appears in the four-dimensional description as an (infinite) tower of more massive KK states Φ(n) . This fits well
with the expectation that very small extra dimensions should not affect low
energy physics (or large distances), as it would take high energies to produce
such new heavy states. In general, the exact structure of the KK tower will
depend on the geometry of the internal dimensions.
Feature (2), that the gravitational 1/r potential will be affected at small
distances, is also straightforward to realize. At small distances r ≪ R, the
compactification scale is of no relevance, and the space should be fully (3 +n)dimensional rotation invariant. At a small distance rn from a mass m the
ˆ 2 V (r) =
gravitational potential is thus determined by the Laplace equation ∇
2
2
2
2
ˆ
0, where ∇ = ∂x̂1 + ∂x̂2 + . . . + ∂x̂n+3 . The solution is
V (r) ∼ −Ĝ4+n
m
rnn+1
rn ≪ R ,
(3.7)
where Ĝ4+n is the fundamental (4 + n)-dimensional gravitational constant
and rn2 = r2 + y12 + y22 + . . . + yn2 is the radial distance. The potential will
thus qualitatively behave as in Eq. (3.7) out to the compactification radius
m
R, where the potential becomes ∼ Ĝ4+n Rn+1
. At larger distances, space is
effectively three-dimensional and Newton’s usual 1/r-law is retained:
Ĝ4+n m
r ≫ R.
(3.8)
Rn r
At small distances the presence of the extra dimensions thus steepens the
gravitational potential. Such a deviation from Newton’s law is conventionally
parameterized as [134]:
1
1 + αe−r/λ .
(3.9)
V (r) ∝
r
Experiments have today tested gravity down to sub-millimeter ranges and
have set upper limits in the (λ, α)-plane. At 100 µm the deviation from
Newton’s law cannot be larger than about 10%, i.e., (λ, |α|) . (100 µm, 0.1)
[135].
By comparing Eq. (3.8) with Newton’s law V (r) = −G m
r we find that the
ordinary Newton’s constant scales with the inverse of the extra-dimensional
volume according to
Ĝ4+n
G ≡ G4 ∝
.
(3.10)
Rn
V (r) ∼ −
44
Beyond the Standard Model: Hidden Dimensions and More
Chapter 3
This is exactly feature (3) for Newton’s constant G – fundamental constants
vary with the inverse of the volume of the internal space ∼ R−n .
Another instructive way of seeing the origin of the features (1)–(3) is to
work directly with the Lagrangian [136]. Take for simplicity a φ4 -theory in a
(4+n)-dimensional Minkowski spacetime
"
#
Z
p
1
m2 2 λ̂4+n 4
4+n
M
S= d
x̂ −ĝ
∂M φ̂ ∂ φ̂ −
φ̂ −
φ̂ .
(3.11)
2
2
4!
With the extra n dimensions compactified on an orthogonal torus, with all
radii equal to R, the higher-dimensional real scalar field φ̂ can be Fourier
expanded in the compactified directions as
n ~n · ~y o
1 X (~n)
φ (x) exp i
φ̂(x, y) = √
.
R
Vn ~n
(3.12)
Here Vn = (2πR)n is the volume of the torus, and ~n = {n1 , n2 , . . . , nn } is
a vector of integers ni . The coefficients φ(~n) (x) are the KK modes, which
in the effective four-dimensional theory constitute the tower of more massive
particle fields. Substituting the KK mode expansion into the action (3.11)
and integrating over the internal space, we get
(
Z
2 m2
√
1
(φ(0) )2
S =
d4 x −g
∂µ φ(0) −
2
2
∗
i
X h
+
∂µ φ(~n) ∂ µ φ(~n) − m~n2 φ(~n) φ(~n)∗
~
n>0
−
)
λ4 (0) 4 λ4 (0) 2 X (~n) (~n)∗
(φ ) − (φ )
φ φ
+ ... ,
4!
4
(3.13)
~
n>0
where the dots stand for the additional terms that do not contain any zero
modes (φ(0) ) of the scalar field. The masses of the modes are given by
m~n2 = m2 +
~n2
.
R2
(3.14)
The coupling constant λ4 of the four-dimensional theory is identified to the
coupling constant λ̂5 of the initial multidimensional theory by the formula
λ4 =
λ̂4+n
.
Vn
(3.15)
We thus again find that the four-dimensional coupling constant is inversely
proportional to the volume Vn spanned by the internal dimensions. The same
is true for any coupling constant connected to fields in higher dimensions.
Section 3.3.
3.3
Modern Extra-Dimensional Scenarios
45
Modern Extra-Dimensional Scenarios
The original attempt by Kaluza and Klein of unifying general relativity and
electromagnetism, the only known forces at the time, by introducing a fifth
dimension was intriguing. After the discovery of the weak [137–139] and
strong forces [140, 141] as gauge fields, it was investigated whether these two
forces could be fit into the same scheme. It was found that with more extra
dimensions these new forces could be incorporated [142, 143]. This developed
into a branch of supergravity in the 1970s, which combined supersymmetry
and general relativity into an 11 dimensional theory. The 11-dimensional
spacetime was shown to be the unique number of dimensions to be able to
contain the gauge groups of the SM [143,144]†. The initial excitement over the
11-dimensional supergravity waned as various shortcomings were discovered.
For example, there was no natural way to get chiral fermions as needed in the
SM, nor did supergravity seem to be a renormalizable theory.
For some time, the ideas of extra dimensions then fell into slumber, before
the rise of string theory in the 1980s. Due to consistency reasons, all string
theories predict the existence of new degrees of freedom that are usually taken
to be extra dimensions. The reason for the popularity of string theory is its
potential to be the correct long time searched for quantum theory for gravity.
The basic entities in string theory are one-dimensional strings, instead of the
usual zero-dimensional particles in quantum field theory, and different oscillation modes of the strings correspond to different particles. One advantage
of having extended objects, instead of point-like particles, is that ultra-violet
divergences, associated with the limit of zero distances, get smeared out over
the length of the string. This could solve the problem of unifying quantum
field theory and general relativity into a firm physical theory. For superstring
theories, it was shown that the number of dimensions must be 10 in order
for the theory to be self-consistent, and in M-theory the spacetime is 11 dimensional. The extra dimensions beyond the four observed, which have to
be made unobservable, and are commonly compactified on what is called a
Calabi-Yau manifold. It might also be possible that non-perturbative lower
dimensional objects called branes can host the four-dimensional world that
we experience.
With the hope that string theory will eventually turn out to be a more
fundamental description of our world, many string-inspired phenomenological
scenarios have been developed. For example, the concept of branes in string
theory gave room for addressing the strong hierarchy problem from a new
geometrical perspective. Branes are membranes in the higher dimensional
spacetime to which open strings, describing fermions and vector gauge fields,
are attached, but closed strings, describing gravitons, are not. In 1998 ArkaniHamed, Dimopoulos, and Dvali (ADD) [145] proposed a string-inspired model,
†
Today, many techniques exist to embed the SM gauge group in supergravity in any
number of dimensions, by, e.g., the introduction of D-branes [130].
46
Beyond the Standard Model: Hidden Dimensions and More
Chapter 3
where all the SM particle fields are confined to a four-dimensional brane in a
higher dimensional flat spacetime. Only gravity is diluted into the additional
extra dimensions, and therefore the gravitational force is weakened compared
to the other known forces. With the extra dimensions spanning a large enough
volume (see Eq. (3.10)), the gravitational scale could be brought down to
the electroweak scale – explaining the strong hierarchy of forces. For two
extra dimensions spanning a volume Vn=2 ∼ (1 µm)2 the fundamental energy
scale for gravity M̂pl is brought down to the electroweak scale, i.e., M̂pl ∼
2
(Mpl
· Vn )1/(2+n) ∼ (1 TeV)2 .
Even with small extra dimensions, the strong hierarchy problem can be
addressed in a geometrical way. In 1999 Randall and Sundrum proposed a
model with one extra dimension that ends at a positive and a negative tension
brane (of which the latter is assumed to contain our visible SM). This model
is often referred to as the Randall and Sundrum I model, or RS I model [146].
The five-dimensional metric is not separable in this scenario, but has a warp
factor e−w|y| connecting the fifth dimension to the four others:
ds2 = e−w|y| ηµν dxµ dxν + dy 2 .
(3.16)
The bulk is a slice of an anti de Sitter space (AdS5 ), i.e., a slice of a spacetime with constant negative curvature. After integrating out the extra dimension in this model, the connection between the four-dimensional and the
five-dimensional fundamental Planck mass (that is the relationship between
the scale for gravity at the brane and in the bulk) is found to be
2
Mpl
=
3
M̂pl
1 − e−2wL
w
(3.17)
where L is the separation between the branes. Here Mpl depends only weakly
on the size L of the extra dimension (at least in the large L limit); this is
a completely different relation than found in the ADD model. To address
the strong hierarchy problem in the RS I model, they looked at how the mass
parameters on the visible brane are related to the physical higher-dimensional
masses. In general, the mass parameter m0 in the higher-dimensional theory
will correspond to a mass
m = e−wL m0
(3.18)
when interpreted with the metric on our visible brane (see [146]). This means
that with wL of about 50 it is possible to have all fundamental mass parameters of the order of the Planck mass, and by the warp factor still produce
masses of the electroweak scale. Instead of having large flat extra dimensions,
the large hierarchy between the electroweak and the Planck energy scales is
here induced by the large curvature of the extra dimension, i.e., the warp
factor e−wL. In a follow-up paper [147], Randall and Sundrum demonstrated
that the metric (3.16) could also allow for a non-compact extra dimension.
This possibility is partly seen already in Eq. (3.17), where it is no problem to
Section 3.4.
Motivations for Universal Extra Dimensions
47
take the indefinitely large L limit. If the curvature scale of the AdS space is
smaller than a millimeter, then Newton’s gravitational law is retained within
experimental uncertainties [147,148]. The reason why the extra dimension can
be non-compact is that the curved AdS background supports a localization
of the higher dimensional gravitons in the extra dimensions. In this so-called
RS II case, the hierarchy problem is not addressed.
Another more recent extra-dimensional scenario is the model by Dvali,
Gabadadze, and Porrati (DGP) [149] where gravity gets modified at large
distances. The action introduced is one with two gravity scales: one fivedimensional bulk and one four-dimensional brane gravity scale. This model
could be used to discuss an alternative scenario for the cosmological problem
of a late-time acceleration of our expanding Universe. However, problems
such as violation of causality and locality make it theoretically less attractive.
For a recent review of braneworld cosmology, see [150].
As mentioned above, another approach for extra-dimensional phenomenology is to look at models where all SM particles can propagate in a higher
dimensional space. This is the case in the UED model [126]. The UED model
will be of special interest in this thesis, as this model can give rise to a new
dark matter candidate. This dark matter candidate will not only be discussed in detail, but the UED model will also serve as the starting point to go
through multidimensional cosmology, the particle SM structure, dark matter
properties, and dark matter searches in general.
3.4
Motivations for Universal Extra Dimensions
Even though the idea of the UED model is a conceptually simple extension
of the SM – basically just add extra dimensions – it provides a framework
to discuss a number of open questions in modern physics. Theoretical and
practical motivations to study the UED model include:
• Simplicity: only 2 new parameters in its minimal version (R and Λcut ).
• A possibility to achieve electroweak symmetry breaking without any
need to add an explicit Higgs field [151].
• Proton stability can be achieved even with new physics coming in at
low-energy scales. With the SM applicable up to an energy scale ΛSM ,
the proton would in general only be expected to have a lifetime of
4
τp ∼ 10−30 years (ΛSM /mp ) ,
(3.19)
where mp is the proton mass [152]. In [153] it was shown that global
symmetries within UED instead can lead to a proton lifetime of
τp ∼ 1035 years
1/R
500 GeV
12 Λcut R
5
12
,
(3.20)
48
Beyond the Standard Model: Hidden Dimensions and More
Chapter 3
where Λcut is the cut-off energy scale of the UED model. This illustrates that the UEDs model can (for relevant R, Λcut values) fulfill the
constraint on proton stability of τp & 1033 years [152, 154].
• It has addressed the (unanswered) question of why we observe three
particle generations. In order to cancel gauge anomalies‡ that appear
in an even number of UEDs, it has been shown, in the case of two extra
dimensions that the number of generations must be three [155].
• Unlike most other extra-dimensional scenarios, single KK states cannot
be produced, but must come in pairs. This means that indirect constraints, such as those coming from electroweak precision observables,
are not particularly strong. As a result, KK states of SM particles can
be much lighter than naı̈vely expected. Such ‘light’ new massive particles should, if they exist, be produced as soon as the upcoming Large
Hadron Collider (LHC) operates. This is especially true in the region of
parameter space favored by having the dark matter in the form of KK
particles.
• Studies of the UED could lead to insights about supersymmetry, which
today is the prime candidate for new physics at the TeV scale. There are
many similarities between supersymmetry and the UED model, which
have even led some people to dub the UED scenario ‘bosonic supersymmetry’ [156]. With two such similar models in hand, it gives a
great possibility to study how to experimentally distinguish different,
but similar, models. In particular, such studies have triggered work on
how to measure spin at LHC. By studying the gamma-ray spectrum
from annihilating dark matter particles in the UED scenario, we have
learned more about similar phenomena within supersymmetry (see, e.g.,
Paper II versus Paper IV).
• The UED model naturally encompasses a dark matter particle candidate. Although this was not the original motivation for the model, this
is perhaps one of the most attractive reasons to study it.
The following three chapters will be devoted to discussions of cosmological
aspects of UEDs and its dark matter properties in detail.
‡
A gauge anomaly occurs when a quantum effect, such as loop diagrams, invalidates the
classical gauge symmetry of the theory.
Chapter
4
Cosmology with
Homogeneous
Extra Dimensions
For every physical theory, it is crucial that it is consistent with observational
constraints. A generic prediction in extra-dimensional models is that at least
some of the fundamental coupling constants vary with the volume of the extradimensional space. Due to the tight observational constraints on the potential
variability of fundamental coupling ‘constants’, it is necessary that the size
of the extra dimensions stays close to perfectly static during the cosmological
history of our Universe. If the extension of the extra dimensions is considerably larger than the Planck scale, it is reasonable that their dynamics should
be governed by classical general relativity. However, in general relativity it is
nontrivial to obtain static extra dimensions in an expanding universe. Therefore, the evolution of the full spacetime in a multidimensional universe must
be scrutinized in order to see if general relativity can provide solutions that
are consistent with current observational constraints. The results presented
in this chapter, coming partly from Paper I, show that a homogeneous multidimensional universe only can have exactly static extra dimensions if the
equations of state in the internal and external space are simultaneously finetuned. For example, in the case of the UED model, it is not expected that the
extra dimensions stay static, unless some stabilization mechanism is included.
A brief discussion of the requirements of such stabilization mechanisms concludes this chapter.
4.1
Why Constants Can Vary
With coupling constants defined in a higher-dimensional theory, the effective
four-dimensional coupling ‘constants’ will vary with the size of the extradimensional volume. In a UED scenario, where all particles can propagate in
the bulk, all force strengths (determined by their coupling constants) pick up
49
50
Cosmology with Homogeneous Extra Dimensions
Chapter 4
a dependence on the internal volume. This is not the case in general, since
some or all of the force carrying bosons might be confined to a membrane and
therefore insensitive to the full bulk. However, since gravity is associated with
spacetime itself the gravitational coupling constant (i.e., Newton’s constant)
will inevitably depend on the size of the internal space.
Consider specifically (4+n)-dimensional Einstein gravity in a separable
spacetime M4 (x) × Kn (x̂), with the internal space Kn compactified to form a
n-dimensional torus with equal radii R. If the matter part is confined to our
four dimensions, then the extra spatial volume ffects only the gravitational
part of the action:
Z
p
1
SE =
d4+n x̂ −ĝR̂ ,
(4.1)
16π Ĝ4+n
where R̂ is the Ricci scalar (calculated from the higher dimensional metric ĝMN ) and Ĝ4+n is the higher dimensional gravitational coupling constant. By Fourier expanding the metric in KK modes, with the zero (i.e.,
(0)
y-independent) mode denoted by gMN , the Ricci scalar can be expanded
(0)
as R̂[ĝMN ] = R[gµν ] + . . .. The missing terms, represented by the dots,
are the nonzero KK modes, and in Section 4.6 it is shown that they correspond to new scalar fields, so-called radion fields, appearing in the effective,
four-dimensional theory. After integrating over the internal dimensions in
Eq. (4.1), the four-dimensional action takes the form
Z
1
(0)
4 √
SE = d x −g
R[gµν ] + . . . .
(4.2)
16πG
In analogy with Eq. (3.10), the four-dimensional Newton’s constant G is given
by
Ĝ4+n
G=
(4.3)
Vn
R 4 p
and, as before, Vn = d x −g(n) ∝ (2πR)n is the volume of the internal
space and g(n) is the determinant of the metric on the internal manifold. Also
in the previous chapter we saw, in the example of a φ4 -theory, how the volume
Vn of the extra dimensions rescale higher dimensional coupling constants λ̂4+n
into the dynamical (i.e. Vn dependent) four-dimensional coupling constant
λ=
λ̂4+n
.
Vn
(3.15)
Similar relations hold also for other types of multidimensional theories.
4.2
How Constant Are Constants?
Numerous experimental and observational bounds exist on the allowed time
variation of fundamental constants, and thus on the size variation of extra
Section 4.2.
How Constant Are Constants?
51
dimensions. Some of these constraints are summarized below (for a more
complete review see, e.g., [157, 158]).
The constancy of Newton’s constant G has been tested in the range from
laboratory experiments to solar system and cosmological observations.∗ Laboratory experiments have mainly focused on testing the validity of Newton’s
1/r2 force law down to sub-millimeter distances, but so far no spatial (or
temporal) variation has been detected [161]. In the solar system, monitoring of orbiting bodies, such as the Moon, Mercury and Venus, sets an upper
limit of |∆G/G| . 10−11 during the last decades of observations [157]. On
cosmological scales, the best limit comes perhaps from BBN, which puts a
constraint of |∆G/G| . 0.2 between today and almost 14 billion years ago
(i.e., zBBN ∼ 108 − 1010 ) [160]. The limit from BBN is derived from the
effect a change in Newton’s constant has on the expansion rate of the Universe, and accordingly on the freeze-out temperature, which would affect the
abundance of light elements observed today. It is worth noting that some
multidimensional models might retain the same expansion rate as in conventional cosmology, despite an evolving gravitational constant G, and therefore
some stated constraints on G might not be directly applicable [162].
The constraints on the possible variation of the electromagnetic coupling
constant, or rather the fine structure constant αEM , are both tight and cover
much of the cosmological history. One fascinating terrestrial constraint comes
from studies of the isotopic abundances in the Oklo uranium mine, a prehistorical natural fission reactor in central Africa that operated for a short time,
about 2 × 109 yr ago. From the αEM dependence on the capture rate of neu−7
trons of, e.g., 149
has been derived
62 Sm, an upper limit of |∆αEM /αEM | . 10
[163,164]. Another very suitable way of testing the constancy of the fine structure constant is by analyzing light from astrophysical objects, since the atomic
spectra encode the atomic energy levels. Analyses of different astrophysical
sources has put limits of |∆αEM /αEM | . 10−3 up to a redshift z ∼ 4 [157].
Worth noticing is that in the literature there have even been claims of an
observed variation in αEM . Webb et al. [165, 166], and also later by Murphy
et al. [167], studied relative positions of absorption lines in spectra from distant quasars, and concluded a variation ∆αEM /αEM = (−5.4 ± 1.2) × 10−6 at
redshift 0.2 < z < 3.7. However, this is inconsistent with other analyses of
quasar spectra. For example, Chand et al. [168] and Srianand et al. [169] get
∆αEM /αEM = (−0.6 ± 0.6) × 10−6 at redshift 0.4 < z < 2.3. In cosmology,
CMB [170, 171] and BBN [160, 172] set the constraint |∆αEM /αEM | . 10−2 at
redshift zCMB ∼ 1000 and zBBN ∼ 1010 , respectively.
∗
Strictly speaking, it makes no sense to consider variations of dimensionful constants,
such as G. We should therefore give limits only on dimensionless quantities, like the
gravitational coupling strength between protons Gm2p /(~c), or specify which other coupling constants that are assumed to be truly constant. The underlying reason is that
experiments in principle can count only number of events or compare quantities with
the same dimensionality [157, 159, 160].
52
Cosmology with Homogeneous Extra Dimensions
Chapter 4
Less work has been put into constraining the weak (αw ) and the strong
(αs ) coupling constants. This is partly due to the more complex modeling; for
example, in the weak sector there are often degeneracies between the Yukawa
couplings and the Higgs vacuum expectation value, and for strong interactions
there is the strong energy dependence on αs . Nonetheless, existing studies of
the BBN – where a change in the weak and the strong interactions should
have observable effects – indicate no changes in αw or αs [157].
Of course, we should keep in mind that there are always some underlying
assumptions in deriving constraints, and that the entire cosmological history
has not been accessible for observations (accordingly, much less is known at
times between epochs of observations). Despite possible caveats, it is still fair
to say that there exists no firm observational indication of any variation of
any fundamental constant ranging from the earliest times of our Universe until
today. These constraints are directly translated into the allowed variation in
size of any extra dimension which coupling constants depend on. Therefore,
in conclusion, observations restrict the volume of extra spatial dimensions to
be stabilized, and not vary by more than a few percentages throughout the
history of our observable Universe.
4.3
Higher-Dimensional Friedmann Equations
We now turn to the equations of motion that describe the evolution of spacetime in a multidimensional universe. Standard cosmology is well described by
an isotropic and homogeneous (four-dimensional) FLRW model. Any scenario
with internal spatial dimensions must therefore mimic this four-dimensional
FLRW model and at the same time be in agreement with the above constraints
on the size stability of the extra dimensions. The gravitational dynamics of a
multidimensional cosmology will be assumed to be governed by the ordinary
Einstein-Hilbert action with n extra dimensions
Z
p 1
S = 2 d4+n x̂ −ĝ R̂ + 2Λ̂ + 2κ̂2 L̂matter ,
(4.4)
2κ̂
where the notation κ̂2 = 8π Ĝ has been introduced. By varying the action
with respect to the metric we derive Einstein’s field equations in d = 3 + n + 1
dimensions:
1
matter
R̂AB − R̂ ĝAB = κ̂2 TAB
+ Λ̂ ĝAB ≡ κ2 TAB .
(4.5)
2
The higher dimensional cosmological constant Λ̂ is here taken to be a part of
the energy momentum tensor TAB .
To address the question whether it is possible to retain ordinary cosmology together with static extra dimensions, let us consider a toy-model† ,
†
Some similar studies of multidimensional cosmologies can be found in, e.g., [162, 173–
181].
Section 4.4.
Static Extra Dimensions
53
where the multidimensional metric is spatially homogeneous, but has two
time-dependent scale factors a(t) and b(t):
ĝMN dx̂M dx̂N
=
gµν (x) dxµ dxν + b2 (x)g̃pq (y) dy p dy q
=
dt2 − a2 (t)γij dxi dxj − b2 (t)γ̃pq dy p dy q .
(4.6)
Here γij is the usual spatial part of the FLRW metric (1.11) for the ordinary,
large dimensions and γ̃pq is a similar maximally symmetric metric for the internal extra-dimensional space. The most general form of the energy-momentum
tensor, consistent with the metric, is, in its rest frame:
T00 = ρ̂ ,
Tij = −p̂a a2 γij ,
T3+p 3+q = −p̂b b2 γ̃pq .
(4.7)
This describes a homogeneous, but in general anisotropic, perfect fluid with
a 3D pressure pa and a common pressure pb in the n directions of the extra
dimensions.
With the above ansatz, the nonzero components of the higher-dimensional
Friedmann equations (4.5) can be written as:
 !

" #
2
2
ȧ
ka
ȧ ḃ n(n − 1)  ḃ
kb
3
+ 2 + 3n
+
+ 2  = κ̂2 ρ̂
(4.8a)
a
a
ab
2
b
b
 !

2
2
ä
ȧ
ka
ȧ ḃ
b̈ n(n − 1)  ḃ
kb
2 +
+ 2 + 2n +n +
+ 2  = κ̂2 p̂a
(4.8b)
a
a
a
ab
b
2
b
b
 !

2
ȧ ḃ
ḃ
kb
κ̂2
b̈
+3
+ (n − 1) 
+ 2 =
(ρ̂ − 3p̂a + 2p̂b )
(4.8c)
b
ab
b
b
n+2
where dots (as in ȧ) denote differentiation with respect to the cosmic time t
and, as usual, the curvature scalars ka,b are +1, 0, −1 depending on whether
the ordinary/internal spatial space is positively, flat, or negatively curved.
With the extra dimensions exactly static (ḃ ≡ 0) the first two equations
(4.8a,4.8b) reduce to the ordinary Friedmann equation (1.11), with an effective
vacuum energy due to the internal curvature kb . If also the third equation
(4.8c) can be simultaneously satisfied, this seems to be what was looked for –
a solution to Einstein’s field equations that has static extra dimensions and
recovers standard cosmology.
4.4
Static Extra Dimensions
Since the internal curvature parameter kb is just a constant in Eq. (4.8c),
exactly static extra dimensions (ḃ ≡ 0) are only admitted if also
kb
ρ̂ − 3p̂a + 2p̂b ≡ C = (n + 2)(n − 1) 2 2 = constant .
(4.9)
κ̂ b
54
Cosmology with Homogeneous Extra Dimensions
Chapter 4
stays constant. This equation will severely restrict the possible solutions if we
do not allow the internal pressure to be a freely adjustable parameter.
Similarly to standard cosmology, the energy content will be taken to be a
multicomponent perfect fluid. Each matter type will be specified by a constant
equation of state parameter w(i) , such that
(i) (i)
p̂(i)
a = wa ρ̂
and
(i)
(i)
p̂b = wb ρ̂(i)
(4.10)
This permits the equation T A0;A = 0 to be integrated to
(i)
(i)
ρ̂(i) ∝ a−3(1+wa ) b−n(1+wb
)
.
(4.11)
The total energy ρ̂ and pressures p̂a and p̂b are the sum of the individual
P
P (i)
matter components, i.e., ρ̂ = i ρ̂(i) and p̂a,b = i wa,b ρ̂(i) . With a multiP (i)
(i)
component fluid allowed in Eq. (4.9) [that is C ≡
≡
i C , where C
(i)
(i)
(i)
ρ̂ − 3p̂a + 2p̂b ], we could in principle imagine a cancellation of the time
dependency of individual C (i) such that C still is time independent. However,
in the case of static extra dimensions this requires that canceling terms have
(i)
the same wa , and therefore it is convenient to instead define this as one
(i)
(i)
matter component, with the given wa , and then adopt an effective wb .
With this nomenclature, each matter component (i) must separately fulfill
Eq. (4.9) to admit static extra dimensions:
(i)
ρ̂(i) 1 − 3wa(i) + 2wb
≡ C (i) = constant ,
(4.12)
From Eq. (4.11) and (4.12), it is clear that static internal dimensions in
(i)
an evolving universe (ȧ 6= 0) requires that either wa = −1, so that ρ̂(i) is
(i)
(i)
constant, or that the equations of state fulfill (1 − 3wa + 2wb ) = 0. Thus,
in summary:
In a homogeneous, non-empty, and evolving multidimensional cosmology with n exactly static extra dimensions, the equation of state
for each perfect fluid must fulfill either
I.
wa(i) = −1
or
II.
(i)
wb =
(i)
3wa − 1
.
2
(4.13)
Note that a matter component that fulfills case I. (and not II.) implies
that the internal space must be curved, since it implies C 6= 0 in Eq. (4.9).
Similarly, it is not possible to have a multidimensional cosmological constant
(i)
(i)
(wa = wb = −1) together with flat extra dimensions (kb = 0).
To actually determine the expected equations of state in a multidimensional scenario, we have to further specify the model. In the next section, we
take a closer look at the scenario of UED and KK states as the dark matter.
Section 4.5.
4.5
Evolution of Universal Extra Dimensions
55
Evolution of Universal Extra Dimensions
In the UED model, introduced in the previous chapter, all the SM particles
are allowed to propagate in the extra dimensions. Momentum in the direction
of the compactified dimensions gives rise to massive KK states in the effective
four-dimensional theory. Furthermore, if the compactification scale is in the
TeV range, then the lightest KK particle (LKP) of the photon turns out to be
a good dark matter candidate. The exact particle field content of the UED
model and the properties of the dark matter candidate are not important at
this point, and further discussion on these matters will be postponed to the
following chapters. However, from the mere fact that the KK dark matter particles gain their effective four-dimensional masses from their own momentum
in the extra dimensions, it is possible to predict the pressure.
The classical pressure, in a direction x̂A , is defined as the momentum flux
through hypersurfaces of constant x̂A . In case of isotropy in the 3 ordinary
dimensions and also – but separately – in the n extra dimensions, we find that
(see Paper I):
2
m
3p̂a + np̂b = ρ̂ −
.
(4.14)
E
For the SM particles, with no momentum in the extra dimensions, there is no
contribution to the pressure in the direction of the extra dimensions (pb = 0),
whereas for KK states, with momentum in the extra dimensions, we can
always ignore any SM mass m compared to their total energy E ∼ TeV (i.e.
momentum in the extra dimensions). Thus, with KK states being the dark
matter, the energy in the universe will not only be dominated by relativistic
matter (m2 /E ≪ ρ̂) during the early epoch of radiation domination but also
during matter domination in the form of LKPs. Thus, Eq. (4.14) gives the
equation of states:
wa =
1
,
3
wa = 0,
wb = 0
(4D Radiation dominated)
(4.15a)
1
n
(4D Matter dominated)
(4.15b)
wb =
During four-dimensional radiation domination, the requirement II. in (4.13)
for static extra dimensions is actually satisfied, whereas this is clearly not the
case during what looks like matter domination from a four-dimensional point
of view.
With exactly static extra dimensions being ruled out, a numerical evolution of the field equations (4.8) was performed in Paper I, to test if nearly
static extra dimensions could be found. The freeze-out of the LKPs takes place
during four-dimensional radiation domination, and a tiny amount of the energy density in the universe is deposited in the form of the LKPs (which much
later will dominate the energy during matter domination). Consequently, the
initial condition for the numerical evolution is a tiny amount of LKPs with
56
Cosmology with Homogeneous Extra Dimensions
Chapter 4
0.7
10
0.6
8
6
0.4
a
4
0.3
ρ̂DM
ρ̂
ρ̂DM /ρ̂
log a, log b
0.5
0.2
2
0.1
b
0
0
5
10
15
20
0.0
log t
Figure 4.1: Evolution of the ordinary and internal scale factors (a and
b, respectively) as well as the fractional LKP energy density (ρ̂DM /ρ̂) for
n =1 (thin), 2 (medium), and 7 (thick) extra dimensions. The evolution
is within a model with Λ = ka = kb = 0 and initially ρ̂DM /ρ̂ = 10−7 .
During a long period of radiation domination, the extra dimensions stay
nearly static, and a evolves in accordance with standard cosmology. When
ρ̂DM /ρ̂ > 0.1, however, neither a nor b show the desired behavior. a, b, and
t are dimensionless arbitrarily scaled to unity as their initial conditions.
Figure from Paper I.
stable extra dimensions in the regime of radiation domination. However, not
surprisingly, as soon as the relative amount of energy in dark matter becomes
significant (∼10%) the size of the extra dimensions start escalating, as is shown
in Fig. 4.1. Such an evolution would severely violate the constraints on both
αEM and G presented in Section 4.2. In conclusion, not even approximately
static extra dimensions can be found in this setup, and therefore some extra
mechanism is needed in order to stabilize the extra dimensions and reproduce
standard cosmology.
4.6
Dimensional Reduction
To study dynamical stabilization mechanisms, it is more practical to consider
the equations of motion after dimensional reduction of the action. In this
section, such a dimensional reduction is performed before we return to the
stabilization of internal spaces in the next section.
Integrating over the internal dimensions in Eq. (4.4), with the metric
Section 4.6.
Dimensional Reduction
57
ansatz (4.6), gives (see, e.g., [182, 183]):
S=
1
2κ̄2
Z
h
√
d4 x −gbn R + b−2 R̃ + n(n − 1)b−2 ∂µ b∂ µ b+
i
2Λ + 2κ̂2 L̂matter , (4.16)
R
√
where κ̄2 ≡ κ̂2 / dn y −g̃, and R and R̃ are the Ricci scalars constructed
from gµν and g̃pq , respectively. With the assumed metric, the internal Ricci
scalar can also be written as R̃ = n(n − 1)kb . A conformal transformation to
the new metric ḡµν = bn gµν takes the action to the standard Einstein-Hilbert
form (so-called Einstein frame), i.e., the four-dimensional Ricci scalar appears
with no multiplicative scalar field [182]:
Z
√
1
1
µ
S = d4 x −ḡ
R̄[ḡ
]
−
∂
Φ∂
Φ
−
V
(Φ)
.
(4.17)
µν
µ
eff
2κ̄2
2
This is our dimensionally reduced action. It gives ordinary four-dimensional
general relativity coupled to a new scalar field (the radion field):
r
n(n + 2)
Φ≡
ln b ,
(4.18)
2κ̄2
with an effective potential
Veff (Φ) = −
R̃ −
e
2κ̄2
q
2(n+2)
κ̄Φ
n
+
√ 2n
1 Λ − κ̂2 L̂matter e− n+2 κ̄Φ .
2
κ̄
(4.19)
Although the equations of motion for the metric ḡµν and Φ, derived from this
new action, can be directly translated to the higher-dimensional Friedmann
equations (4.8), the equation of motion for the radion field,
¯ = − ∂ Veff
Φ
∂Φ
(4.20)
¯ = √1 ∂ µ √−ḡ∂µ , now has a much more intuitive interpretation
where −ḡ
regarding when stable extra dimensions are expected. As the internal scale
¯ = Φ̈ + 3H Φ̇. We have thus found
factor depends only on t, we have Φ
that the scalar field Φ has the same equation of motion as a classical particle
moving in a potential Veff , with a friction term given by three times the Hubble
expansion rate H. If the effective potential Veff (Φ) has a stationary minimum,
then the radion field, and thus the extra dimension, has a static and stable
solution.
Before we continue the discussion in the next section on how to obtain
stable extra dimensions, it is interesting to note a subtlety regarding conformal transformations. From the higher-dimensional perspective the naı̈ve
58
Cosmology with Homogeneous Extra Dimensions
Chapter 4
guess would be that the four-dimensional metric part gµν with the scale factor
a(t) describes the effective four-dimensional space. However, after the conformal transformation into the Einstein frame (i.e., the action written as in
Eq. (4.17)) suggests that it should rather be the effective four-dimensional
metric ḡµν with the scale factor ā(t̄) = a(t)bn/2 (t) that describes the physical four-dimensional space. The problem of which of the conformally related
frames should be regarded as the physical one, or if they both are physically
equivalent, is to some extent still debated [184–186]. However, for the discussion in this chapter the distinction between a(t) and ā(t̄) is not of any
importance, because the tight observational constraints on the variation of
coupling constants imply that the extra dimensions can always be assumed
to be almost static. In that case the different frames are in practice identical.
4.7
Stabilization Mechanism
Let us briefly investigate what is needed to stabilize the extra dimensions.
The effective potential for Φ in Eq. (4.19) can be rewritten as
Veff (b(Φ)) = −
R̃ −(n+2) X (i) −3(1+wa(i) ) − n (1−3wa(i) +2w(i) )
b
. (4.21)
b
+
ρ0 ā
b 2
2κ̄2
i
This
P expression follows from the definition in Eq. (4.18), that L̂matter = −ρ̂ =
− i ρ̂(i) , and that each fluid component dependency on ā = ab−n/2 and Φ
is given by
(i)
(i)
(i)
ρ̂(i) = ρ̂0 a−3(1+wa ) b−n(1+wb
)
=
κ̄2 (i) −3(1+wa(i) ) n/2(1+3wa(i) −2w(i) )
b
ρ ā
b
,
κ̂2 0
(i)
(i)
(i)
where the four-dimensional densities ρ0 are defined by ρ0 ≡ ρ̂0 Vn ≡
(i)
ρ̂0 κ̄2 /κ̂2 .
We can now easily verify that ∂µ Φ ≡ 0 in Eq. (4.20) implies the same
condition (4.9) as obtained earlier for exactly static extra dimensions. However, in the perspective of the radion field, in its effective potential, it is also
obvious that even the static extra-dimensional solution is only stable if the
potential has a stationary minimum. In fact, the static solution found earlier
(for a radiation-dominated universe with wa = 1/3 and wb = 0) is not really
a stable minimum, since the effective potential is totally flat (when kb = 0).
During matter domination in the UED model, with wa = 0 and wb = 1/n,
the radion field is of course not in a stationary minimum either. In the case
of a higher dimensional cosmological constant, wa = wb = −1, there could
in principle be a stationary minimum if ρΛ (ā) < 0 and R̃ < 0, but then the
effective four-dimensional vacuum energy from the cosmological constant and
the internal curvature contribution will sum up to be negative. This would
give an anti de Sitter space, which is not what is observed.
Section 4.7.
Stabilization Mechanism
59
At this point let us, ad hoc, introduce a stabilization mechanism in the
form of a background potential V bg (Φ) for the radion field. If this potential
has a minimum, say at Φ0 , the contribution to the potential is to a first-order
expansion around this minimum given by:
m2
∂ 2 Vbg 2
2
Vbg (Φ) ≃ Vbg (Φ0 ) +
(Φ − Φ0 ) ,
where m ≡ 2 .
(4.22)
2
∂ Φ Φ0
′
′
′
The minima of the total potential are found from Vtot
= Veff
(Φ) + Vbg
(Φ) = 0,
and can implicitly be written as:
r
r
nκ2
R̃
2 + n −(n+2)
(1 − 3wa + 2wb )
Φmin ≃ Φ0 +
ρ(ā, bmin)− 2
b
. (4.23)
2
m
2+n
m
2nκ2 min
Here the sum over different fluid components has been suppressed for brevity,
and only one (dominant) contributor ρ(ā, bmin ) is included. Due to the coupling to the matter density ρ(ā, b) the minimum Φmin will thus in general be
time dependent. For small shifts of the minimum, it can be expressed as
r
(1 − 3wa + 2wb )
nκ2
∆Φmin ≃
× ∆ρ(ā, bmin ) ,
(4.24)
2
m
2+n
Thus, a change in b(Φ) of less than 1% between today and BBN, corresponding
to ∆ρ ∼ 1019 eV4 , would only require the mass in the stabilization potential
to be m & 10−16 eV – a very small mass indeed.‡
In fact, since a light mass radion field would mediate a new long-range
(fifth) force with the strength of about that of gravity, sub-millimeter tests of
Newton’s law impose a lower mass bound of m & 10−3 eV [187] (see also [188]
for more cosmological aspects on radions in a UED scenario).
Still, only a fairly shallow stabilization potential is needed to achieve approximately static extra dimensions for a radion field that tracks its potential
minimum. The stabilization mechanism could be a combination of different
(i)
(i)
perfect fluids with tuned wa and wb to produce a potential minimum, or one
of many proposed mechanisms. Examples are the Freund-Rubin mechanism,
with gauge-fields wrapped around the extra dimensions [189]; the GoldbergerWise mechanism, with bulk fields interacting with branes [190]; the Casimir
effect in the extra dimensions [191]; quantum corrections to the effective potential [192]; or potentially something string theory related (like the KachruKallosh-Linde-Trivedi (KKLT) model with nonperturbative effects and fluxes
that stabilizes a warped geometry [193]). Whichever the mechanism might
be, it seems necessary to introduce an additional ingredient to stabilize extra
dimensions. In the following discussions of the UED model, it will be implicitly assumed that the extra dimensions are stabilized, and that the mechanism
itself does not induce further consequences at the level of accuracy considered.
‡
In general, a small ∆b/b shift requires m &
P
√
i
(i)
(i)
2(1−3wa +2wb ) κ2 ∆ρ(i)
(2+n)
∆b/b
1/2
Chapter
5
Quantum Field Theory
in Universal Extra
Dimensions
We now turn to the theoretical framework for the UED model. Proposed
by Appelquist, Cheng, and Dobrescu [126] in 2001, the UED model consists
of the standard model in a higher dimensional spacetime. As all standard
model particles are allowed to propagate into the higher-dimensional bulk in
this model, this means that, when it is reduced to a four-dimensional effective
theory, every particle will be accompanied by a Kaluza-Klein tower of identical but increasingly more massive copies. Conservation of momentum in
the direction of the extra space dimensions imply that heavier Kaluza-Klein
states can only be produced in pairs. This chapter develops the effective fourdimensional framework, and will simultaneously give an overview of the field
structure of the standard model of particle physics.
5.1
Compactification
The simplest reasonable extension of spacetime is to add one extra flat dimension, compactified on a circle S 1 (or a flat torus in more extra dimensions) with
radius R. However, such a compactification of a higher-dimensional version
of the SM would not only give new massive KK particles but also unwanted
light scalar degrees of freedom and fermions with the wrong chirality.
The extra scalar degrees of freedom that appear at low energies for a
four-dimensional observer are simply the fifth component of any higher dimensional vector fields, which transform as scalars under four-dimensional
Lorentz transformations. Such massless scalar fields (interacting with usual
gauge strengths) are not observed and are strongly ruled out by the extra
‘fifth’ force interaction they would give rise to [194].
In the SM, the fermions are chiral, meaning that the fermions in the SU (2)
doublet are left-handed, whereas the singlets are right-handed (and the oppo61
62
Quantum Field Theory in Universal Extra Dimensions
Chapter 5
site handedness for anti-fermions). With one extra dimension, fermions can
be represented by four-component spinors, but the zero modes will consist of
both left-handed and right-handed fermions. Technically, the reason behind
this is that the four-dimensional chiral operator is now a part of the higher
dimensional Dirac algebra, and higher-dimensional Lorentz transformations
will in general mix spinors of different chirality.
If the extra dimensions instead form an orbifold, then the above problems
can be avoided. With one extra dimension the simplest example is an S 1 /Z2
orbifold, where Z2 is the reflection symmetry y → −y (y being the coordinate
of the extra dimension). Fields can be set to be even or odd under this Z2
symmetry, which allows us to remove unwanted scalar and fermionic degrees
of freedom, and thereby reproduce the particle content of the SM. From a
quantum field theory point of view, the orbifold can be viewed as projecting
the circular extra dimension onto a line segment of length πR stretching between the two fixpoints at y = 0 and y = πR. Fields, say Φ, are then given
Neumann (or Dirichlet) boundary conditions ∂5 Φ = 0 (or Φ = 0), so that
they become even (or odd) functions along the y-direction.
5.2
Kaluza-Klein Parity
The circular compactification S 1 breaks global Lorentz invariance, but local
invariance is preserved. By Noether’s theorem, local translational invariance
corresponds to momentum conservation along the extra dimension. A set
n
of suitable base functions are thus ei R y , which are the eigenstates of the
momentum operator i∂y (the integer n is a conserved quantity called the KK
number). For example, a scalar field Φ, is expanded as
Φ(xµ , y) =
∞
X
n
Φ(n) (xµ )ei R y ,
(5.1)
n=−∞
We might expect that the fifth component of the momentum should be a
conserved quantity in UED. However, the orbifold compactification S 1 /Z2 ,
with its fixpoints, breaks translational symmetry along the extra dimension,
and KK number is no longer a conserved quantity. However, as long as the
fixpoints are identical there is a remnant of translation invariance, namely
translation of πR, which takes one fixpoint to the other. Rearranging the
sum in Eq. (5.1) into terms that are eigenstates to the orbifold operator PZ2
(i.e., Φ(y) is even or odd) and simultaneously KK mass eigenstates (i.e.,
momentum squared, or ∂y2 , eigenstates) gives:
µ
Φ(xµ , y) = Φ(0)
even (x ) +
∞
X
n=1
Φ(n)
even cos
ny
ny
(n)
+ Φodd sin
.
R
R
(5.2)
Section 5.3.
The Lagrangian
63
Depending on KK level, the terms behave as follows under the translation
y → y + πR:
ny
n(y + πR)
ny
→ cos
= (−1)n cos
,
R
R
R
ny
n(y + πR)
ny
sin
→ sin
= (−1)n sin
.
R
R
R
cos
(5.3a)
(5.3b)
Hence, a Lagrangian invariant under the πR translation can only contain
terms that separately sum their total KK level to an even number. In other
words, every term in the UED Lagrangian must have (−1)ntot = 1, where ntot
is the sum of all the KK levels in a particular term. In an interaction term,
split ntot into ingoing and outgoing particles, such that ntot = nin + nout , and
we have (−1)nin = (−1)nout . This is known as the conservation of KK parity
(−1)n , and will be essential in our discussion of KK particles as a dark matter
candidate.
5.3
The Lagrangian
In the UED model, all the SM particles, with its three families of fermions,
force carrying gauge bosons and one Higgs boson, are allowed to propagate
in the extra S 1 /Z2 dimension. In such a higher dimensional Lagrangian, the
gauge, Yukawa and quartic-Higgs couplings have negative mass dimensions,
and the model is non-renormalizable [3]. Therefore, should the UED model be
viewed as an effective theory, applicable only below some high-energy cutoff
scale Λcut . With the compactification scale 1/R distinctly below the cutoff
Λcut , a finite number of KK states appears in the effective four-dimensional
theory. If only KK states up to Λcut are considered, the UED model is from
this perspective a perfectly valid four-dimensional field theory. In the minimal
setup, all coupling strengths in the UED model are fixed by the measured fourdimensional SM couplings, and the only new parameters are the cutoff and
the compactification scale, Λcut and R, respectively.
For later chapters, mainly the electroweak part of the Lagrangian is relevant. Therefore the fermions, the U (1) × SU (2) gauge bosons, the Higgs
doublet, and their interactions will be discussed in some detail, whereas the
Quantum Chromo Dynamics (QCD) sector, described by the SU (3) gauge
group, is left out.∗ Furthermore, with the expectation that much of the interesting phenomenology can be captured independently of the number of extra
space dimensions, we will in the following only consider the addition of one
extra space dimension.
∗
The reason for the QCD sector being of little importance in this thesis is because it
contains neither any dark matter candidate, nor do the gluons (i.e., gauge bosons of
QCD) interact directly with photons (which means they are not relevant for the gammaray yield calculations made in Paper II,III).
64
Quantum Field Theory in Universal Extra Dimensions
Chapter 5
The UED Lagrangian under consideration will be split into the following
parts:
L̂UED = L̂Bosons + L̂Higgs + L̂Fermions .
(5.4)
In general, the inclusion of gauge fixing terms will additionally result in a
L̂ghost -term including only unphysical ghost fields. In what follows, all these
parts are reviewed separately.
Gauge Bosons
The first four components of the SU (2) and U (1) gauge fields, denoted ArM
and BM respectively, must be even under the orbifold projection to retain the
ordinary four-dimensional gauge fields as zero modes in their Fourier series:
PZ2 Arµ (xµ , y) = Arµ (xµ , −y) .
(5.5)
A four-dimensional gauge transformation of such a gauge field is given by
Arµ (xµ , −y) = Arµ (xµ , y) → Arµ (xµ , y) +
1
∂µ αr (xµ , y) + f rst Asµ αt , (5.6)
ĝ
where f rst is the structure constant, defining the lie algebra for the generators, [σr , σs ] = if rst σt , and ĝ is a five-dimensional coupling constant. From
Eq. (5.6) we read off that the function α(xµ , y) has to be even under Z2 .
Therefore ∂y αr (xµ , y) is odd, and hence
Ar5 (xµ , y) = −Ar5 (xµ , −y)
(5.7)
to keep five-dimensional gauge invariance. The Fourier expansions of the
gauge fields in the extra dimension are then:
Aiµ (x̂)
∞
1
1 X i (n) µ
ny
i (0)
=√
Aµ (x) + √
Aµ (x ) cos
,
R
2πR
πR n
Ai5 (x̂) = √
∞
1 X i (n)
ny
A5 (x) sin
.
R
πR n
(5.8a)
(5.8b)
Note that the five-dimensional gauge invariance automatically implies the
absence of the unwanted zero mode scalars Ar5 and B5 in the four-dimensional
theory.
The kinetic term for the gauge fields in the Lagrangian reads
1
1
L̂gauge = − FM N F M N − FMr N F r M N ,
4
4
(5.9)
with the U (1) and SU (2) field strength tensors given by
FM N = ∂M BN − ∂N BM ,
i
i
i
FM N = ∂M AN − ∂N AM + ĝǫ
(5.10a)
ijk
j
k
AM AN .
(5.10b)
Section 5.3.
The Lagrangian
65
Inserting the field expansions (5.8) and integrating out the extra dimension
results in the SM Lagrangian accompanied by a KK tower of more massive
states. For example, the U (1) part of L̂gauge becomes
L4D
gauge
Z
1 2πR
dy FM N F M N
4 0
1
= − ∂µ Bν(0) − ∂ν Bµ(0) ∂ µ B (0)ν − ∂ ν B (0)µ
4
∞
1X
−
∂µ Bν(n) − ∂ν Bµ(n) ∂ µ B (n)ν − ∂ ν B (n)µ
4 n=1
⊃ −
+
∞
1X
n
n
(n)
(n)
∂µ B5 + Bµ(n) ∂ µ B5 + B (n)µ ,
2 n=1
R
R
(5.11)
and with a similar expression for the kinetic part of the SU (2) gauge fields.
n
At each KK level, a massive vector field B (n) with mass R
appears. The
(n)
scalars B5 are, however, not physical, because by a gauge transformation
(n)
(n)
(n)
Bµ → Bµ − (R/n)∂µ B5 the scalar field terms ∂µ B5 can be removed.
This actually becomes apparent already from a naı̈ve counting of degrees of
freedom: A massive four-dimensional vector field has three degrees of freedom,
which is the same as the number of polarization directions for a massless
five-dimensional gauge field. Once the Higgs mechanism with electroweak
symmetry breaking is added (discussed in the next section), the additional
Goldstone scalar fields will form linear combinations with the vector field’s
fifth component to form both physical as well as unphysical scalar fields.
For the non-abelian gauge fields there will also be cubic and quartic interaction terms, and we can identify the ordinary four-dimensional SU (2)
coupling constant with
1
g≡ √
ĝ .
(5.12)
2πR
This mapping between the four-dimensional couplings in the SM and the fivedimensional couplings is general and will hold for all coupling constants.
The Higgs Sector
The electroweak masses of the SM gauge fields are generated by the usual
Higgs mechanism† . The Higgs field is a complex SU (2) doublet that is a
scalar under Lorentz transformations:
1 χ2 + iχ1
1
φ≡ √
, χ± ≡ √ (χ1 ∓ iχ2 ) .
(5.13)
3
H
−
iχ
2
2
†
Sometimes also called the Brout-Englert-Higgs mechanism, Higgs-Kibble mechanism or
Anderson-Higgs mechanism.
66
Quantum Field Theory in Universal Extra Dimensions
Chapter 5
To have the SM Higgs boson as the zero mode, the expansion must be even
under the Z2 orbifold
φ(x̂) = √
∞
1
1 X (n)
ny
φ(0) (x) + √
φ (x) cos
.
R
2πR
πR n
(5.14)
To make the H field electromagnetically neutral, the hypercharge is set to
Y = 1/2 for the Higgs doublet and its covariant derivative is
DM = ∂M − iĝArM
σr
− iY ĝY BM ,
2
(5.15)
where ĝY and ĝ are the higher dimensional U (1) and the SU (2) coupling
constants, respectively, and σr are the usual Pauli matrices:
0 1
0 −i
1 0
σ1 =
,
σ2 =
,
σ3 =
.
(5.16)
1 0
i 0
0 −1
The full Higgs Lagrangian is written as
†
L̂Higgs = (DM φ) (DM φ) − V (φ) ,
(5.17)
where the potential V (φ) is such that spontaneous symmetry breaking occurs.
That is
V (φ) = µ̂2 φ† φ + λ̂(φ† φ)2 ,
(5.18)
where the values of the parameters are such that −µ̂2 , λ̂ > 0. This ‘Mexican
hat’ potential has a (degenerate) minimum at
|φ|2 =
−µ̂2
2λ̂2
≡
v̂ 2
.
2
(5.19)
By choosing any specific point in the minimum as the vacuum state, around
which the physical fields then are expanded, the symmetry is said to be spontaneously broken. With
the vacuum expectation value chosen to lie along the
0
real axis, hφi = √12
, the Higgs field H should be replaced by h + v̂ so
v̂
that h is a perturbation around the vacuum and hence represents the Higgs
particle field.
Electroweak gauge boson mass terms will now emerge from the kinematic
part of Eq. (5.17). As in the standard Glashow-Weinberg-Salam electroweak
theory, three of the gauge bosons become massive,
WM± =
√1
2
A1M ∓ iA2M
ZM = cW A3M − sW BM
ĝv̂
,
2
mW
with mass mZ =
,
cW
with mass mW =
(5.20a)
(5.20b)
Section 5.3.
The Lagrangian
67
and one, orthogonal to Z, is massless
AM = sW A3M + cW BM .
(5.21)
The Weinberg angle θW that appears above is (at tree level) given by
ĝY
sW ≡ sin θW = p
,
2
ĝ + ĝY2
ĝ
cW ≡ cos θW = p
.
2
ĝ + ĝY2
(5.22)
This relates the electromagnetic and weak coupling constants in such a way
that the gauge field Aµ becomes the photon with its usual coupling to the
electric charge e = sW g = cW gY .
In the quadratic part of the kinetic Higgs term
h1
2 i
2 1
2 (kin)
L̂Higgs ⊃
∂M h + ∂M χ3 − mZ ZM + ∂M χ+ − mW WM+ , (5.23)
2
2
there are unwanted cross-terms that mix vector fields and scalar fields (e.g.,
Z µ ∂µ χ3 ). This is the same type of unwanted terms that appeared in the
Lagrangian of Eq. (5.11). To remove these unwanted terms properly, five
dimensional gauge-fixing terms are added to the Lagrangian:
2 X 1 i 2
1
L̂gaugefix = − G Y −
G
,
(5.24)
2
2
i
1
Gi = √
ξ
1
GY = √
ξ
µ i
∂ Aµ − ξ −mW χi + ∂5 Ai5 ,
(5.25a)
µ
∂ Bµ − ξ sW mW χ3 + ∂5 B5i .
(5.25b)
which is a generalization of the Rξ gauge [3]. These are manifestly fivedimensional Lorentz breaking terms. However, this should not be worrying
since in the effective theory we are restricted to four-dimensional Poincaré
transformations anyway, and the above expressions are still manifestly fourdimensionally Lorentz invariant. If we add up the scalar contributions from
the gauge bosons in Eq. (5.9), the kinetic part of the Higgs sector in Eq. (5.17)
and the gauge fixing terms in Eq. (5.24), and then integrate over the internal
dimensions, the result is:
(
∞
X
2
1
(kin)
(n) 2
Lscalar =
∂µ h(n) ∂ µ h(n) − Mh h(n)
2
n=0
+
1
(n)
(n)
(n) 2 (n) 2
∂µ G0 ∂ µ G0 − ξMZ G0
2
(n)
(n)
(n) 2 (n) (n)
+ ∂µ G+ ∂ µ G− − ξMW G+ G−
)
+
68
Quantum Field Theory in Universal Extra Dimensions
∞
X
n=1
(
Chapter 5
1
(n)
(n)
(n) 2 (n) 2
∂µ a0 ∂ µ a0 − MZ a0
2
(n)
(n)
(n) 2 (n) (n)
+ ∂µ a+ ∂ µ a− − MW a+ a−
)
n2 (n) 2
(n) µ (n)
∂µ A5 ∂ A5 − ξ 2 A5
.
R
1
+
2
(5.26)
The above appearing mass eigenstates in the four-dimensional theory are
given by:
a0
(n)
a±
(n)
G0
(n)
G±
(n)
=
=
=
=
where
M (n)
χ3
(n)
χ±
(n)
χ3
(n)
χ±
MZ
M (n)
MW
MZ
MZ
MW
MW
r
(n)
(n)
(n)
(n)
(n)
+
+
−
−
(n)
(n)
Z5 ,
(n)
W5±
(n)
Z5 (n) ,
(n)
W5±
MZ
MW
MW
M (n)
MZ
M (n)
MW
(n)
(n)
(5.27a)
,
(5.27b)
(5.27c)
,
(5.27d)
n
.
(5.28)
R
R
Now, let us count the number of physical degrees of freedom in the bosonic
sector. At KK zero-level, we recover the SM with one physical Higgs field h(1)
together with the three massive gauge fields Z (0) , W ±(0) that have eaten the
(0)
three degrees of freedom from the three unphysical Goldstone bosons G0,± .
At each higher KK mode, there are four additional scalar degrees of freedom
coming from the fifth components of the gauge bosons. These scalars form,
together with four KK mode scalars from the Higgs doublet, four physical
(n)
(n)
(n)
(n)
scalars h(n) , a0 and a± and four unphysical Goldstone bosons A5 , G0
(n)
and G± that have lost their physical degrees of freedoms to the massive KK
(n)
(n)
± (n)
vector modes Aµ , Zµ , and Wµ , respectively.
(n)
MX ≡
m2X +
n 2
MZ
,
M (n) =
Ghosts
In the case of non-Abelian vector fields, or when Abelian vector fields acquire masses by spontaneous symmetry breaking, the gauge-fixing terms in
Eq. (5.24) are accompanied by an extra Faddeev-Popov ghost Lagrangian
term [3]. These ghost fields can be interpreted as negative degrees of freedom
that serve to cancel the effects of unphysical timelike and longitudinal polarization states of gauge bosons. The ghost Lagrangian is determined from the
Section 5.3.
69
The Lagrangian
gauge fixing terms in Eq. (5.24) as
L̂ghost = −c̄a
δG a b
c ,
δαb
(5.29)
where a, b ∈ {i = 1, 2, 3, Y }. The ghost fields ca are complex, anticommuting
Lorentz scalars. The bar in the expression denotes Hermitian conjugation.
The ghosts are set to be even under the Z2 orbifold. The functional derivatives
in Eq. (5.29) are found by studying infinitesimal gauge transformations:
1
∂M αi + ǫijk AjM αk ,
ĝ
1
=
∂M αY ,
ĝY
δAiM =
(5.30)
δBM
(5.31)
and
i
α σi
αY
1 δχ2 + iδχ1
δφ = i
+i
φ≡ √
3 ,
2
2
2 δH − iδχ
(5.32)
with
i
1h 1
α H − α2 χ3 + α3 χ2 + αY χ2 ,
2
i
1h
2
δχ = α1 χ3 + α2 H − α3 χ1 − αY χ1 ,
2
i
1h
3
δχ =
− α1 χ2 + α2 χ1 + α3 H − αY H .
2
δχ1 =
(5.33a)
(5.33b)
(5.33c)
The ghost Lagrangian√
then becomes, after the ghost fields have been rescaled
according to ca → (ĝa ξ)1/2 ca ,
− c̄a
where
h
i
δG a b
c = c̄a − ∂ 2 δ ab − ξ(M ab − ∂52 δ ab ) cb
b
δα
h
ĝ a ĝ b v̂ ab i b
+ c̄a ĝǫijk δ kb δ ia (ξ∂5 Aj5 − ∂ µ Ajµ ) − ξ
I c , (5.34)
4

h
 χ3
I =
−χ2
χ2
−χ3
h
χ1
−χ1
χ2
−χ1
h
−h

χ2
−χ1 
,
−h 
h
ĝ 2
2 
v̂  0
M=
4 0
0

0
ĝ 2
0
0
0
0
ĝ 2
−ĝĝY

0
0 
.
−ĝĝY 
ĝY2
Integrating out the extra dimensions, the kinetic part of the ghost Lagrangian becomes
(kin)
Lghost =
∞
X
n=0
n
o (n)
(n)
c̄a (n) −∂ 2 δ ab − ξM ab
cb ,
(5.35)
70
Quantum Field Theory in Universal Extra Dimensions
Chapter 5
(n)
where the mass matrix M ab
now is given by


(n) 2
MW
0
0
0


(n) 2
 0

MW
0
0
.
M (n) = 
2


(n)
 0
0
MW
− 14 v 2 ggY 
n 2
0
0
− 14 v 2 ggY 41 v 2 gY2 + R
(5.36)
The same rotation as in Eq. (5.20)-(5.21) diagonalizes this matrix and the
ghosts mass eigenstates become c± ≡ √12 (c1 ∓ c2 ), cZ and cγ , with the cor√
√
(n) √
(n)
responding masses ξMW , ξMZ and ξMγ(n) .‡ As for the Goldstone
bosons, the gauge dependent masses indicate the unphysical nature of these
fields.
Fermions
Fermions can appear both as singlets and as components of doublets under
SU (2) transformations. Furthermore, in the massless limit fermions have a
definite chirality in even spacetime dimensions. In the SM, all fermions in
(0)
doublets ψd have, from observation, negative chirality, whereas all singlets
(0)
ψs have positive chirality:
(0)
(0)
γ 5 ψd = ψd
and γ 5 ψs(0) = −ψs(0) ,
(5.37)
where γ 5 ≡ iΓ0 Γ1 Γ2 Γ3 is the four-dimensional chirality operator constructed
from the generators of the Clifford algebra. The d-dimensional Clifford algebra
reads:
{ΓM , ΓN } = 2η M N .
(5.38)
For an even number of spacetime dimensions (d = 2k + 2) the fundamental
representation of the Clifford algebra’s generators are 2k+1 × 2k+1 -matrices,
whereas for odd spacetime dimensions (d = 2k +3) they are the same matrices
as for the case of one less spacetime dimension and with the addition of
Γ2k+2 ≡ −i1+k Γ0 Γ1 . . . Γ2k+1
(5.39)
Note that our four-dimensional chirality operator γ 5 is the same as iΓ4 in five
dimensions, i.e., the chirality operator γ 5 is a part of the Clifford algebra’s
generators in five dimensions.
For an even number of spacetime dimensions we can always reduce the
Dirac representation into two inequivalent Weyl representations, characterized
by their spinors’ chirality. For odd spacetime dimensions, this is not possible,
since the ΓM matrices then form an irreducible Dirac representation and they
‡
Note that c+ and c− are not each other’s Hermitian conjugates.
Section 5.3.
The Lagrangian
71
mix under Lorentz transformations. However, we can still always artificially
split up spinors with respect to their four-dimensional chirality operator. That
is
ψ = PR ψ + PL ψ ≡ ψR + ψL .
(5.40)
where PR,L are the chirality projection operators:
PR,L ≡
1
1 ± γ5 ,
2
2
PR,L
= PR,L ,
PL PR = PR PL = 0 .
(5.41)
Five-dimensional Lorentz transformations mix ψR and ψL , but if restricted to
only four-dimensional Lorentz transformations, as in the effective KK theory,
they do not mix. Therefore it makes sense to assign different orbifold projections depending on a fermion’s four-dimensional chirality when constructing
an effective four-dimensional model. To recover the chiral structure of the SM,
the fermion fields should thus be assigned the following orbifold properties:
PZ2 ψd (y) = −γ 5 ψd (−y) and PZ2 ψs (y) = γ 5 ψs (−y) .
(5.42a)
With these orbifold projections, the doublets and singlets have the following
expansions in KK modes
∞
1
1 X (n)
ny
ny (0)
(n)
ψd = √
ψdL + √
ψdL cos
+ ψdR sin
,
R
R
2πR
πR n=1
∞
1 X (n)
ny
1
ny (n)
√
ψs(0)
+
ψ
cos
ψs = √
+
ψ
sin
,
sR
sL
R
R
2πR R
πR n=1
(5.43a)
(5.43b)
where (as wanted) the zero mode doublets are left-handed and the zero mode
singlets are right-handed.
The fermion Lagrangian has the following structure:
ψd,U
(Yukawa)
+ iψ̄s 6Dψs + L̂fermion ,
(5.44)
L̂fermion = i ψ̄d,U ψ̄d,D 6D
ψd,D
where U and D denote up-type (T3 = +1/2) and down-type (T3 = −1/2)
fermions in the SU (2) doublet (where T3 is the eigenvalue to the SU (2) generator σ3 operator), respectively, and the covariant derivative is
σr
6D ≡ ΓM ∂M − iĝArM
− iY ĝY BM ,
(5.45)
2
where Y ∈ {Yd,U , Yd,D , Ys } is the hypercharge of the fermion in question. Note
that the ArM term is absent for the singlet part, since ψs does not transform
under SU (2) rotations.
The fermions have Yukawa-couplings to the Higgs field,
(Yukawa)
L̂fermion = −λ̂D (ψ̄d,U , ψ̄d,D ) · φ ψs,D − λ̂U (ψ̄d,U , ψ̄d,D ) · φ̃ ψs,U + h.c. ,
(5.46)
72
Quantum Field Theory in Universal Extra Dimensions
Chapter 5
from which, after the spontaneous symmetry breaking in the Higgs sector,
the fermions get their electroweak fermions masses. The conjugate of the
Higgs doublet is defined by φ̃a ≡ ǫab φ†b . The electroweak masses will become
√
= (λ̂U,D v̂)/ 2.
mU,D
EW
When several generations of quarks are present, there can be couplings
that mix quark generations. It is still always possible to diagonalize the Higgs
couplings, which is also the base that diagonalizes the mass matrix. However,
this diagonalization causes complications in the gauge couplings regarding
quarks and will lead to the need for introducing a quark-mixing matrix V ij .
Let
i
u
c
t
ψd,
U = (ψd,U , ψd,U , ψd,U )
and
i
u
c
t
ψd,
D = (ψd,D , ψd,D , ψd,D ) ,
(5.47)
denote the up- and down-type quarks for the three families in the mass basis.
These mass eigenstates ψ (i.e., the base that diagonalize the Higgs sector) are
related to the interaction eigenstates ψ ′ by unitary matrices UU and UD :
j
′i
ψd,U
= UUij ψd,U
ij j
′i
ψd,D
= UD
ψd,D .
and
(5.48)
This difference between ψ ′ and ψ will appear only in the interaction with W ± .
The covariant derivative in the mass eigenstate base becomes [3]:
ĝ
ij
M
6D ≡ Γ δ ij ∂M − ieAM Qδ ij − i ZM (σ3 − s2W Q)δ ij
cW
ĝ
−i √ (W + σ+ + W − σ− )V ij , (5.49)
2
where V ≡ UU† UD is known as the Cabbibo-Kobayashi-Maskawa (CKM) mixing
matrix, σ± = (σ1 ± iσ2 ) are the usual step operators in SU (2), and the charge
Q is related to the weak isocharge and the hypercharge by Q = T3 + Y . For
the terms containing the photon and the Z boson, the unitary matrices UD
and UU effectively disappear because they appear only in the combination
U † U = 13×3 . In the absence of right-handed neutrinos (ψs,U ) the CKM
matrix can also be made to vanish for the couplings to the W± and the
replacement Vij → δij should be done in the leptonic sector [3].
Let us now finally integrate out the extra dimension and see how the KK
masses arise from the kinetic term:
Z 2πR
dy iψ̄d ΓM ∂M ψd + iψ̄s ΓM ∂M ψs = iψ̄ (0) γ µ ∂µ ψ (0)
0
+
∞ h
X
n (n)
n (n) i
(n)
ψ̄d iγ µ ∂µ −
ψd + ψ̄s(n) iγ µ ∂µ +
ψ
, (5.50)
R
R s
n=1
where
(0)
ψ (0) ≡ ψdL + ψs(0)
,
R
(n)
(n)
(n)
ψd ≡ ψdL + ψdR ,
ψs(n) ≡ ψs(n)
+ ψs(n)
.
L
R
(5.51)
Section 5.4.
Particle Propagators
73
(n)
Hence, for each SM fermion ψ (0) there are two fermions at each KK level ψs
(n)
and ψd (this can also be seen directly from Eq. (5.43)). Note that the singlet
fields get the ‘wrong’ sign for their KK masses. Adding the Yukawa masses
mEW to the KK masses, the mass matrix becomes
n

!
(n)
(n)
+ δmd
mEW
ψ
(n)
(n)  R
d
i
hn
,
(5.52)
ψ̄d
ψ̄s
(n)
(n)
ψs
+ δms
mEW
−
R
(n)
(n)
where δmd and δms are additional radiative loop corrections that appear
(these corrections will be further discussed in Section 5.5). By a rotation,
and a correction for the ‘wrong’ sign of the singlets’ mass terms, the following
mass eigenstates are found:
! !
(n)
(n)
ξd
1
0
cos α(n)
sin α(n)
ψd
=
,
(5.53)
(n)
(n)
0 −γ 5
− sin α(n) cos α(n)
ξs
ψs
where the mixing angle is
tan 2α(n) =
2mEW
n
2R
(n)
(n)
+ δmd + δms
.
The physical masses for these states are
rh
i2
n
1
1
(n)
(n)
(n)
(n)
(n)
md,s = ± (δmd − δms ) +
+ (δmd + δms ) + m2EW ,
2
R 2
(5.54)
(5.55)
For all fermions, except for the top quark, the mixing angles α(n) are very
close to zero.
5.4
Particle Propagators
The five-dimensional propagators get modified by the orbifold compactification. There are two differences compared to an infinite Lorentz invariant
spacetime. The first is trivial and comes from the compactification itself,
which implies that the momentum in the extra dimension is quantized and is
given by py = n/R, where n takes only integer values. The other is related to
the reflection symmetry of the orbifold boundaries, which breaks momentum
conservation in the direction of the extra dimension and therefore allows for
a sign flip of py in the propagator.
By the introduction of unconstrained auxiliary fields, the momentum space
propagators can be found [195, 196]. For example, a scalar field Φ can be
expressed in terms of an unconstrained auxiliary field χ as:
Φ(xµ , y) =
1
(χ(xµ , y) ± χ(xµ , −y)) ,
2
(5.56)
74
Quantum Field Theory in Universal Extra Dimensions
Chapter 5
where the ±-sign depends on whether the scalar is constrained to be even
or odd under the orbifold transformation. Note that since the unconstrained
fields contain both y and −y, the propagator hΦ(xµ , y)Φ(x′µ , y ′ )i will (after a
Fourier transformation into momentum space) depend on py − p′y , as well as
py + p′y . The five-dimensional propagator for a massless scalar field becomes
[195, 196]
i δpy ,p′y ± δ−py ,p′y
Φ(pµ , py )Φ∗ (p′µ , y) =
.
(5.57)
2
p2 − p2y
In the effective four-dimensional theory, the propagators for the KK particles
take the usual form (see also Appendix A):
Φ(pµ )Φ∗ (p′µ ) =
i
,
p 2 − m2
(5.58)
where m is the mass of the KK particle.
5.5
Radiative Corrections
A main characteristic of KK theories is that at each KK level the particle
states are almost degenerate in mass. This is true even after generation of
electroweak masses mEW , at least as long as the compactification scale 1/R
is much larger than mEW . The mass degeneracy implies that all momentumconserving decays are close to threshold and radiative corrections will determine if certain decay channels are open or not. For cosmology, these loop
corrections are essential in that they determine which of the fields is the lightest KK particle (LKP) and hence if the model has a natural dark matter
candidate.
For example, in the massless limit of five-dimensional QED, the reaction
e(1) → e(0) + γ (1)
(5.59)
is exactly marginal at three level. After inclusion of the electroweak
electron
q
(1)
2
mass me , the reaction becomes barely forbidden as Me ≡ 1/R + m2e <
1/R + me . Radiative mass corrections are, however, naı̈vely expected to be
larger than mEW , because they are generically of the order of α ∼ 10−2 ,
which is much larger than mEW /m(n) (which ranges from 10−12 for electrons
to 10−2 for the top quarks at the first KK level). The study in [196] of oneloop radiative corrections show that radiative corrections generically open up
decay channels, like the one in Eq. (5.59), so that all first-level KK modes can
decay into the LKP and SM particles.
Radiative corrections to masses arise from Feynman loop diagrams contributing to the two-point correlation functions. These contributions can in
the UED model be artificially split into three types: (1) five-dimensional
Lorentz invariant loops (2) winding modes and (3) orbifold contributions.
Section 5.5.
75
Radiative Corrections
y
xµ
Figure 5.1: Winding modes around the compactified extra dimension
give Lorentz violating loop corrections to Kaluza-Klein masses.
I will only briefly outline what goes into the calculation of the radiative
mass corrections (the interested reader is referred to [196], and references
therein). The five-dimensional dispersion relation reads
pµ pµ = m2 + Z5 p2y ≡ m2phys + Z5 /Zm2(n) ,
(5.60)
where Z and Z5 are potential radiative quantum corrections. In the SM, the
(divergent) quantum correction Z is absorbed into the physically observed
masses mphys . Hence, any extra mass corrections to the KK masses must come
from extra contributions to Z5 . In general, both Z and Z5 receive (divergent)
loop radiative quantum corrections, but in a five-dimensional Lorentz invariant theory they are protected to stay equal, Z = Z5 , to preserve the usual
dispersion relation. At short distances (away from the orbifold fixpoints),
the compactification is actually not perceptible, and localized loops should
therefore preserve Lorentz invariance. In fact, all five-dimensional Lorentz
invariance preserving self-energy contributions can therefore be absorbed into
the renormalization of the zero mode mass m → mphys /Z so these do not give
any additional radiative mass correction to the KK modes.
Although local Lorentz invariance still holds under a S 1 compactification,
global Lorentz invariance is broken. Such non-local effects appear in those
Feynman diagrams that have an internal loop that winds around the compact
dimension, as shown in Fig. 5.1. The radiative corrections from such winding
propagators can be isolated by the following procedure. Because of the S 1
compactification, the momenta py in the compact dimension are quantized
in units of 1/R, and therefore the phase space integral over internal loop
momenta reduces to a sum over KK levels. This sum can be translated into
a sum over net winding n around the compactified dimension, where the first
term with n = 0 exactly corresponds to an uncompactified five-dimensional
theory. The non-winding loops (n = 0) can be dealt with in the same way as
described in the previous paragraph. The sum of the remaining winding modes
(n 6= 0) turns out to only give finite and well-defined radiative corrections to
each KK mass.
76
Quantum Field Theory in Universal Extra Dimensions
Chapter 5
Finally, the third kind of radiative correction appears due to the orbifold compactification S 1 /Z2 . This is a local effect, caused by the orbifold’s
fixpoints that break translational invariance in the y-direction. As shown
in Section 5.4, this leads to modified five-dimensional propagators. Thus we
must redo the two previously described calculations, but now instead with the
correct orbifold propagators. The finite contribution stated in the previous
paragraph remains the same (although divided by two because the orbifold
has projected out half of the states), but also new, logarithmically divergent
terms localized at the orbifold fixpoints appear. This means that counter
terms should be included at the boundaries to cancel these divergences and
that these calculations strictly speaking determine only the running behavior. In order to keep unknown contributions at the cutoff scale Λcut under
control, they are assumed to be small at that high energy cutoff scale – a selfconsistent assumption since the boundary terms are generated only by loop
corrections. If large boundary terms were present, they could induce mixing
between different KK modes [196].
5.6
Mass Spectrum
Including these radiative corrections, we can calculate the mass spectra of the
KK particles. In most cases, the electroweak mass can be ignored, with the
prominent exceptions of the heavy Higgs and gauge bosons, and the top quark.
In the case of fermions, these radiative corrections were already included in
the fermionic mass matrix Eq. (5.52). Similarly this can be done also for
the bosons. Let us take a closer look at the cosmologically important mass
matrix for the neutral gauge bosons. In the B, A3 basis, including radiative
corrections δmB(n) and δmA3 (n) , the mass matrix reads
!
n 2
+ δm2B(n) + 14 v 2 gY2
− 41 v 2 ggY
R
.
(5.61)
n 2
+ δm2A3 (n) + 14 v 2 g 2
− 41 v 2 ggY
R
By diagonalizing this matrix, the physical KK photon mass and Z-boson mass
can be found. The nth KK level ‘Weinberg’ angle for the diagonalizing rotation
is given by
tan 2θ(n) =
v 2 ggY
h
2 δm2A3 (n) − δm2B(n) +
v2
2
4 (g
i.
− gY2 )
(5.62)
For small compactifications scales, this Weinberg angle is driven to zero since
generally δm2A3 (n) − δm2B(n) ≫ v 2 ggY . For example, for R−1 & 500 GeV and
Λcut R & 20, θ(n) is less than 10−2 . Therefore, the KK photon γ (n) is very
well approximated by the weak hypercharge gauge boson B (n) and these two
states are often used interchangeably.
Figure 5.2 shows the spectrum for the first KK excitations of all SM particles, both at (a) tree level and (b) including one-loop radiative corrections.
(a) Tree level
650
600
(b) One-loop level
650
g
td
600
550
Q
u
d
bs b
ts d
L
e
τd ,ντ
τs
550
Z,W ±
td ,ts
500
77
Mass Spectrum
Mass (GeV)
Mass (GeV)
Section 5.6.
a0
a±
H0
Z ±
W
γ,g
Q,u,d bd ,bs ,τd ,τs ,ντ
L,e
500
H0
a0
a±
γ
Figure 5.2: The mass spectrum of the first Kaluza-Klein states at (a) tree
level and (b) including one-loop radiative corrections. A compactification
radius of R−1 = 500 GeV, Higgs mass of 200 GeV, and cutoff of Λcut =
20R−1 have been used. The first column shows the Higgs and gauge
bosons. The second column shows the quark SU (2) doublets (Q) and the
singlets (u,d) for the two first families. In the last column, this is repeated
for the third family, taking into account the larger electroweak masses of
the tau lepton (τs,d , ντ ), the bottom quark (bs,d ), and the top quark (ts,d ).
We find that the lightest SM KK particle is the first KK level photon γ (1) .
Since unbroken KK parity (−1)(n) guarantees the LKP to be stable, it provides a possible dark matter candidate. Due to the loop corrections, the mass
degeneracy will be lifted enough so that all other first KK level states will
promptly cascade down to the γ (1) .
Instead of the minimal UED model – defined by using the described oneloop expressions, with boundary terms negligible at the cutoff scale Λcut (unless specified otherwise, Λcut = 20R−1 ) – you could take a more phenomenological approach and treat all the KK mass corrections as independent input
to the theory (see, e.g., [197]). With such an approach, other KK particles than the γ (1) can become the LKP. With the γ (1) not being the LKP,
the LKP still has to be electrically neutral to be a good dark matter candidate [198, 199]. Therefore, the only other potentially attractive LKP options
are h(1) , Z (1) , ν (1) , g (1) and if going beyond SM particles also the first KK
level of the graviton. At least naı̈vely, some of these can be directly excluded:
the high electroweak Higgs mass naturally excludes the Higgs particles; the
gluons are color charged and thus excluded by their strong interactions [199];
the Z (1) is expected to be heavier than γ (1) since usually both electroweak
and one-loop order correction contribute to the mass matrix in a way that
78
Quantum Field Theory in Universal Extra Dimensions
Chapter 5
the lighter state is almost B (1) (i.e., γ (1) ). When it comes to the graviton, it
is interesting to note that for R−1 . 800 GeV the first KK graviton G(1) in
the minimal UED model becomes lighter than the γ (1) . This result is under
the reasonable assumption that the radiative mass corrections to the graviton
is very small, since it only couples gravitationally, and therefore the mass of
G(1) is well approximated by R−1 . The G(1) will be a superweakly interacting particle, as it interacts only gravitationally. Therefore, it is hard, if not
impossible, to detect it in conventional dark matter searches. However, the
presence of G(1) could cause effects both on the BBN and the CMB radiation
predictions [200] (see also [201–203]). In the case of a KK neutrino, thermal
relic calculations show that the favored mass range is between 0.8 TeV and
1.3 TeV [197]. This is in strong conflict with direct detection searches that
exclude ν (1) masses below 1000 TeV [204, 205] [206, 207].
For the remaining chapters, γ (1) ≈ B (1) will be taken to be the LKP.
Chapter
6
Kaluza-Klein
Dark Matter
Numerous phenomenological constraints must be fulfilled in order to make
a particle a viable dark matter candidate. In the UED model, it turns out
that the first Kaluza-Klein excitation of the photon γ (1) is an excellent dark
matter candidate. If γ (1) (≈ B (1) ) is the lightest Kaluza-Klein state and its
mass is about 1 TeV, then this candidate can make up all the dark matter.
It is therefore of importance to find direct and indirect ways to detect such
a candidate. The properties of this Kaluza-Klein dark matter particle are
discussed, and observational constraints are reviewed. The chapter concludes
by discussing the results of Paper II and Paper III, where characteristic
indirect detection signals were predicted from pair-annihilation of these dark
matter particles. So-called final state radiation, producing very high energy
gamma rays, and a monocromatic gamma-ray line, with an energy equal to
the mass of the dark matter particle, produce distinctive signatures in the
gamma spectrum.
6.1
Relic Density
In the early Universe, the temperature T was higher than the compactification scale (T > R−1 ∼ 1 TeV), and KK particles were freely created and
annihilated and kept in thermal and chemical equilibrium by processes like:
(n)
(n)
Xi
Xj
(n)
Xi
xj
(m)
Xi
↔ xk xl
(6.1a)
↔
(6.1b)
→
(n)
(n)
Xk xl
(n)
Xj xk
...
(6.1c)
where Xi is the nth -level excitation of a SM particle xi (i, j, k, l = 1, 2, . . .).
Processes of the type (6.1a) are usually referred to as annihilation when i = j
and coannihilation when i 6= j (in fact, coannihilation processes usually refer
79
80
Kaluza-Klein Dark Matter
Chapter 6
to all pair annihilations that not exclusively include the dark matter candidate
under consideration).
As the Universe expands, the temperature drops and the production of KK
states soon becomes energetically suppressed, leading to their number density
dropping exponentially. This happens first for the heavier KK particles, which
by inelastic scattering with numerous SM particles (Eq. (6.1b)) and decays
(Eq. (6.1c)) transform into lighter KK particles and SM particles. Due to
conservation of KK parity (−1)n (n being the KK level), the lightest of the
KK particles is not allowed to decay and can therefore only be destroyed by
pairwise annihilation (or potentially by coannihilation if there is a sufficient
amount of other KK particles around). Hence, at some point the LKPs annihilation rate cannot keep up with the expansion rate, the LKP density leaves
its chemical equilibrium value, and their comoving number density freezes.
What is left is a thermal remnant of non-relativistic LKPs that act as cold
dark matter particles. That only the LKPs contribute to the dark matter
content today is of course only valid under the assumption that all level-1
KK states are not exactly degenerate in mass and can decay into the LKP.
Generically this is the case for typical mass splittings, as in e.g., the minimal
UED model at the one-loop level.
As stated in the previous chapter, the lightest of the KK particles is in the
minimal UED model the first KK excitation of the photon γ (1) , which to a
good approximation is equal to B (1) . In practice, there is a negligible difference
between using γ (1) and B (1) as the LKP (See Section 5.6). Henceforth, I switch
notation from γ (1) to B (1) for the KK dark matter particle, as this is the state
actually used in the calculations.
Quantitatively, the number density of a dark matter particle is described
by the Boltzmann equation [208, 209]:
h
i
dn
2
+ 3Hn = −hσeff vi n2 − (neq ) ,
(6.2)
dt
P
where n =
ni is the total number density including both the LKP and
heavier states that will decay into the LKP. H is the Hubble expansion rate
and neq the sum of the chemical equilibrium number densities for the KK
particles, which in the non-relativistic limit are given by
!3/2
−m (1) mX (1) T
X
eq
i
i
exp
,
(6.3)
n (1) ≃ gi
Xi
2π
T
where gi is the internal number of degrees of freedom for the particle in question. Finally, the effective annihilation (including coannihilation) cross section
times the relative velocity (or more precisely, the Møller velocity [210]) of the
annihilating particles hσeff vi is given by
eq
hσeff vi =
X
i,j
hσij vij i
nj
neq
i
eq
n neq
(6.4)
Section 6.1.
Relic Density
81
where the thermal average is taken for the quantity within brackets hi.
In general, a lower effective cross section means that the comoving number
density freezes out earlier, when the density is higher, and therefore leaves a
larger relic abundance today. A high effective cross section, on the other hand,
means that the particles under consideration stay in chemical equilibrium
longer and the number density gets suppressed. The exact freeze-out temperature and relic densities are determined by solving Eq. (6.2), but a rough
estimate is that the freeze-out occurs when the annihilation rate Γ = nhσvi
fall below the Hubble expansion H. As mentioned in Chapter 1, a rule of
thumb is that the relic abundance is given by
ΩWIMP h2 ≈
3 · 10−27 cm−3 s−1
.
hσeff vi
(6.5)
At freeze-out, typically around T ∼ mWIMP /25, there are roughly 1010 SM
particles per KK excitation that by reaction (6.1b) can keep the relative
eq
abundance among KK states in chemical equilibrium (ni /nj = neq
i /nj ∝
exp (−∆M/T )). Coannihilation is therefore only of importance when the
mass difference between the LKP and the other KK states is smaller or comparable to the freeze-out temperature. Since the masses of the KK states are
quasi-degenerated by nature, we could expect that coannihilation are especially important for the UED model. In general, the presence of coannihilations can both increase and decrease relic densities. In the case of UED, we
could also expect that second-level KK states could significantly affect the
relic density if the appear in s-channel resonances, see [211, 212].
To see how differences in the KK mass spectrum affect the relic density,
let us take a look at three illustrative examples. These cases set the mass
range for which B (1) is expected to make up the observed dark matter.
First, consider a case where we artificially allow only for self-annihilation.
This would mimic the case when all other KK states are significantly heavier
than the LKP. The effective cross section in Eq. (6.4) then reduces to σeff =
σB (1) ,B (1) . The relic density dependence on the compactification scale, or
equivalently the LKP mass, is shown by the dotted line in Fig. 6.1. In this
case the B (1) mass should be in the range 700 GeV to 850 GeV to coincide
with the measured dark matter density.
Imagine, as a second case, that coannihilation with the strongly interacting
KK quarks and gluons are important. This can potentially cause a strong
enhancement of the effective cross section if these other states are not much
heavier than the LKP. Physically, this represents the case when other KK
particles deplete the LKP density by keeping it longer in chemical equilibrium
through coannihilations. Thus this effect will allow for increased B (1) masses.
As an illustrative example, take the mass spectrum to be that of the minimal
UED model, except that the KK gluon mass mg(1) is treated a free parameter.
For a mass difference ∆g(1) ≡ (mg(1) − mB (1) )/mB (1) as small as 2% an allowed
relic density can be obtained for B (1) masses up to about 2.5 TeV (see the
82
Kaluza-Klein Dark Matter
Chapter 6
0.3
mUED
0.25
ΩB (1) h2
0.2
B (1) B (1)
0.15
0.1
B (1) , g (1) : ∆g(1) = 2%
0.05
0
0
0.5
1
1.5
2
2.5
3
R−1 [TeV]
Figure 6.1: Relic density of the lightest Kaluza-Klein state B (1) as a
function of the inverse compactification radius R−1 . The dotted line is
the result from considering B (1) B (1) annihilation only. The dashed line is
the when adding coannihilation with the Kaluza-Klein gluon g (1) , using a
small mass split to the B (1) : ∆g(1) ≡ (mg(1) − mB(1) )/mB(1) = 0.02. The
solid line is from a full calculation in the minimal UED model, where all
coannihilations have been taken into account. The gray horizontal band
denotes the preferred region for the dark matter relic density 0.094 <
ΩCDM h2 < 0.129. Figure constructed from results in [213].
dashed line in Fig. 6.1). The small distortion in the curve around 2.3 TeV
is due to the details concerning the change of the total number of effectively
massless degrees of freedom that affects the Hubble expansion during freezeout (see [213]).
As a final case, let the mass spectrum be that of the minimal UED model,
and include all coannihilation effects with all the first level KK states. The
minimal UED model is here set to have a cutoff scale equal to Λcut = 20R−1 .
The relic density result is shown as the solid line in Fig. 6.1. At first sight the
displayed result might look surprising, since in the previous example including
coannihilations with quarks and gluons reduced the relic density. However,
in the minimal UED scenario the KK quarks and gluons get large radiative
corrections, and therefore they are more than 15% heavier than B (1) and are
not very important in the coannihilation processes. Here, a different effect
becomes important. Imagine that the coannihilation cross section σB (1) ,X (1)
with a state X (1) of similar mass is negligibly small. Then the effective cross
Section 6.2.
Direct and Indirect Detection
83
section in Eq. (6.4) for the two species, in the limit of mass degeneration,
tends towards
2
2
σeff ≈ σB (1) ,B (1) gB
(1) /(gX (1) + gB (1) ) + σX (1) ,X (1) gX (1) /(gX (1) + gB (1) ) .
(6.6)
If σX (1) ,X (1) is small then σeff may be smaller than the self-annihilation cross
section σB (1) ,B (1) . This is exactly what happens in the minimal UED model:
the mass splittings to the leptons are small (about 1% to the SU (2)-singlets
and 3% to the SU (2)-doublets). Contrary to the case of coannihilating KK
quarks and gluons (and the usual case in supersymmetric models with the
neutralino as dark matter), the coannihilation processes are here weak (or
actually of similar strength as the self-annihilation) and the result is therefore
an increase of the relic density. The physical understanding of this situation is
that the two species quasi-independently∗ freeze-out, followed by the heavier
state decaying thus enhancing the LKP relic density.
In summary, relic density calculations [197, 213, 214] show that if the KK
photon B (1) is to make up the observed amount of dark matter, it must have
a mass roughly in the range from 500 GeV up to a few TeV.
While freeze-out is the process of leaving chemical equilibrium, it is not
the end of the B (1) interacting with the much more abundant SM particles.
Elastic and inelastic scattering keeps the LKPs in thermal equilibrium until
the temperature is roughly somewhere between 10 MeV and a few hundred
MeV (see, e.g., [205, 215, 216]). The kinetic decoupling that occurs at this
temperature sets a distance scale below which dark matter density fluctuations get washed out. In other words, there is a cutoff in the matter power
spectrum at small scales. This means that there is a lower limit on the size
of the smallest protohalos created, whose density perturbations later go nonlinear at a redshift between 40 and 80. The consequence is that the smallest
protohalos (or dark matter clumps) should not be less massive than about
10−3 to 103 Earth masses. Whether these smallest clumps of dark matter
have survived until today depends critically on to which extent these structures are tidally disrupted through encounters with, e.g., stars, gas disks, and
other dark matter halos [217–219].
6.2
Direct and Indirect Detection
With the B (1) as a dark matter candidate, the experimental exclusion limits
on its properties and its prospect for detection should be investigated.
∗
Species never really freeze-out completely independently of each other because of the
presence of the processes (6.1b) that usually efficiently transform different species into
each other.
84
Kaluza-Klein Dark Matter
Chapter 6
Accelerator Searches
High-energy colliders are in principle able to produce heavy particles below
their operating center of mass energy. The absence of a direct discovery of any
non-SM particle therefore sets upper limits on production cross sections, which
in the case of UED translates into a lower limit on KK masses. Because of KK
parity conservation, KK states can only be produced in pairs. For the Large
Electron-Positron Collider (LEP) at CERN, this means that only masses less
than ECM ∼ 100 GeV could be reached. The circular proton-antiproton
Tevatron accelerator at the Fermi laboratory can reach much higher energies
as it is running at a center of mass energy of ECM ∼ 2 TeV. Non-direct
detection (i.e., no excess of unexpected missing energy events) in the Tevatron
sets an upper limit on one extra-dimensional radius of R . (0.3 TeV)−1 [126,
220–222]. The future LHC experiment will be able to probe KK masses up to
about 1.5 TeV [205]. Suggested future linear electron-positron colliders could
significantly improve measuring particle properties such as masses, couplings
and spins of new particles discovered at the LHC [156,223], but will probably
not be a discovery machine for UED unless they are able to probe energies
above ∼ 1.5 TeV.
Physics beyond the SM can manifest itself not only via direct production,
but also indirectly by its influence on other observables such as magnetic
moments, rare decays, or precision electroweak data [205]. For a ∼ 100 GeV
Higgs mass, electroweak precision data limits 1/R to be & 800 GeV, which
weakens to about 300 GeV for a ∼ 1 TeV Higgs. Strong indirect constraints
also come from data related to the strongly suppressed decay b → sγ, which
is less dependent on the Higgs mass, and gives 1/R & 600 GeV [224].
Direct Detection
Direct detection experiments are based on the idea of observing elastic scattering of dark matter particles that pass through the Earth’s orbit. The
searched signal is that a heavy WIMP depositing recoil energy to a target nucleus. There are three common techniques to measure this recoil energy (and
many experiments actually combine them). (i) Ionization: in semiconductor
targets, like germanium (Ge) [206, 225, 226], silicon (Si) [206, 226] or xenon
(Xe) [207], the recoil energy can cause the surrounding atoms to ionize and
drift in an applied electric field out to surrounding detectors. (ii) Scintillation:
for example, sodium iodide (NaI) crystals [227–229] or liquid/gas Xenon (Xe)
scintillators [207, 230] can produce fluorescence light when a WIMP interaction occurs. This fluorescence light is then detected by surrounding photon
detectors. (iii) Phonon production: cryogenic (i.e., low temperature) crystals
like germanium and silicon detectors, as in [206,225,226], look for the phonon
(vibration) modes produced by the impulse transfer due to WIMP scattering.
To achieve the required sensitivity for such rare scattering events, a low
background is necessary. Operating instruments are therefore well shielded
Section 6.2.
Direct and Indirect Detection
85
and often placed in underground environments. Features that are searched
for are the recoil energy spectral shape, the directionality of the nuclear recoil,
and possible time modulations. The time modulation in the absolute detection
rate is expected due to the Earth’s spin and velocity through the dark matter
halo (see Section 9.1 for a short comment on the DAMA experiment’s claim
of such a detected annual modulation).
The elastic scattering of a WIMP can be separated into spin-independent
and spin-dependent contributions. The spin-independent scattering can take
place coherently with all the nucleons in a nucleus, leading to a cross section
proportional to the square of the nuclei mass [42]. Due to the available phase
space, there is an additional factor proportional to the square of the reduced
nuclei mass mr = mDM mN /(mN +mDM ) [where mN and mDM are the nuclei and
dark matter particle mass, respectively]. The present best limits on the spinindependent cross section comes from XENON [207] and is for WIMP masses
around 1 TeV of the order of σSI . 10−6 pb per nucleon (that is per proton or
neutron, respectively), whereas for WIMP masses of 100 GeV it is somewhat
better, σSI . 10−7 pb. This translates roughly to mB (1) & 0.5 TeV [204, 231],
when assuming a mass shift of about 2% to the KK quarks (the limits gets
weakened for increased mass shifts).
The spin-dependent cross-section limits are far weaker, and in the same
WIMP mass range they are at present roughly σSD . 0.1 − 1 pb. These limits
are set by CDMS (WIMP-neutron cross section) [226] and NaIAD (WIMPproton cross section) [229].
Indirect Detection
Indirect searches aim at detecting the products of dark matter particle annihilation. The most promising astrophysical indirect signals seem to come
from excesses of gamma-rays, neutrinos or anti-matter. As the annihilation
rate is proportional to the number density squared, nearby regions of expected
enhanced dark matter densities are the obvious targets for studies.
WIMPs could lose kinetic energy through scattering in celestial bodies,
like the Sun or the Earth, and become gravitationally trapped. The concentration of WIMPs then builds up until an equilibrium between annihilation
and capture rate is obtained. In the case of B (1) dark matter, equilibrium
is expected inside the Sun, but not in the center of the Earth. The only
particles with a low enough cross section to directly escape the inner regions
of the Sun and Earth, where the annihilations rate into SM particles is the
highest, are neutrinos. Current experiments are not sensitive enough to put
any relevant limits, and at least kilometer size detectors, such as the IceCube
experiment under construction, will be needed [204, 231, 232]. Although only
the neutrinos can escape the inner parts of a star, it has been proposed that
in optimistic scenarios the extra energy source from dark matter annihilation
in the interior of stars and white dwarfs could change their temperatures in a
86
Kaluza-Klein Dark Matter
Chapter 6
detectable way [233–235].
Another way to reveal the existence of particle dark matter would be to
study the composition of cosmic rays, and discover products from dark matter
annihilations. Unfortunately, charged products are deflected in the galactic
magnetic fields, and information about their origin is lost. However, dark
matter annihilation yields equal amount of matter and antimatter, whereas
anti-matter in conventionally produced cosmic rays is expected to be relatively
less abundant (this is because anti-matter is only produced in secondary processes, where primary cosmic-ray nuclei – presumably produced in supernova
shock fronts – collide with the interstellar gas). A detected excess in the antimatter abundance in cosmic rays could therefore be a sign of dark matter
annihilation.
In this manner, the positron spectrum can be searched for dark matter
signals. In the UED model, 20% of the B (1) annihilations are into monochromatic electron-positron pairs, and 40% into muons or tau pairs. Muons and
taus can subsequently also decay into energetic electrons and positrons. A
sizable positron flux from KK dark matter, with a much harder energy spectrum than the expected background, could therefore be looked for [231, 236].
No such convincing signal has been seen, but the High-Energy Antimatter
Telescope (HEAT) [237], covering energies up to a few 10 GeV, has reported
a potential small excess in the cosmic positron fraction around 7-10 GeV (see,
e.g., [238] for a dark matter interpretation, including KK states, of the HEAT
data).
In the UED scenario, the large branching ratio into leptonic states makes
the expected antiprotons yield – mainly produced from the quark-antiquark
final states – relatively low compared to the positron signal. Today antiproton observations do not provide any significant constraints on the UED
model [239, 240]. However, the PAMELA (Payload for Antimatter Matter
Exploration and Light-nuclei Astrophysics) [241] satellite, already in orbit,
might find a convincing energy signature that deviates from the conventionally expected antiproton or positron spectrum, as it will collect more statistics
as well as probe higher energies (up to some 100 GeV).
6.3
Gamma-Ray Signatures
In addition to the above-mentioned ways to detect WIMP dark matter, there
is the signal from annihilation into gamma rays. In general, this signal has
many advantages: (i) gamma rays point directly back to their sources, (ii) at
these energies the photons basically propagate through the galactic interstellar
medium without distortion [242] [243,244], (iii) they often produce characteristic spectral features that differ from conventional backgrounds, (iv) existing
techniques for space and large ground-based telescopes allow to study gamma
rays in a large energy range – up to tens of TeV.
Dark matter annihilations into photons can produce both a continuum of
Section 6.3.
Gamma-Ray Signatures
87
gamma-ray energies, as well as monochromatic line signals when γγ, γZ or
γh are the final states.
Gamma-Ray Continuum
At tree level, with all first KK levels degenerated in mass, B (1) dark matter
annihilates into charged lepton pairs (59%), quark pairs (35%), neutrinos
(4%), charged (1%) and neutral (0.5%) gauge bosons, and the Higgs boson
(0.5%).
The fraction of B (1) B (1) that annihilates into quark pairs will in a subsequent process of quark fragmentation produce gamma-ray photons, mainly
through the decay of neutral π 0 mesons:
B (1) B (1) → q q̄ → π 0 + . . . → γγ + . . . .
(6.7)
These chain-processes result in differential photon multiplicities, i.e., the number density of photons produced per annihilation, and is commonly obtained
from phenomenological models of hadronization. In Paper III, we use a
parametrization of the differential photon multiplicity dNγq /dEγ published
in [245] for a center of mass energy of 1 TeV. This parametrization was based
on the Monte Carlo code Pythia [246], which is based on the so-called Lund
model for quark hadronization. Since Pythia are able to reproduce experimental data well, and dNγq /dEγ is fairly scale invariant at testable high energies, we judge it to be reliable to use Pythia results up to TeV energies.
The massive Higgs and gauge bosons can also decay into quarks that produce photons in their hadronization process. For B (1) dark matter, this gives
a negligible contribution due to the small branching ratios into these particles.
The remaining and majority part of the B (1) dark matter annihilations
result in charged lepton pairs: electrons (e), muons (µ) and tau (τ ) pairs, each
with a branching ratio of about 20%.† The only lepton heavy enough to decay
into hadrons is the τ . Hence, τ pairs generate quarks that, as described above,
subsequently produce gamma rays in the process of quark fragmentation.
The photon multiplicity dNγτ /dxγ for this process is taken from reference
[245]. Due to the high branching ratio into τ -pairs for B (1) annihilation, this
contribution is very significant at the highest photon energies (see Fig. 6.3 at
the end of this section).
An even more important contribution to the gamma-ray spectrum at the
highest energies comes from final state radiation.‡ This process is a threebody final state, where one of the charged final state particles radiates a
†
This is radically different from the neutralino dark matter candidate in supersymmetric
theories, where annihilation into light fermions are helicity suppressed (see the discussion
in Section 7.2).
‡
Pythia partly takes final state radiation into account, but certainly not as specifically
as in Paper II and Paper IV (see also [247]). This is particularly true for internal
bremsstrahlung from charged gauge bosons, treated in Chapter 7 and in Paper IV.
88
Kaluza-Klein Dark Matter
B (1)
ℓ−
ℓ−
Chapter 6
B (1)
ℓ(1)
ℓ(1)
B (1)
ℓ
+
ℓ−
ℓ−
B (1)
ℓ+
ℓ(1)
ℓ+
ℓ(1)
γ
B (1)
γ
B (1)
ℓ+
γ
Figure 6.2: Tree-level diagrams contributing to final state radiation in
lepton pair production: B (1) B (1) → ℓ+ ℓ− γ. Figure from Paper II.
photon. The relevant Feynman diagrams for B (1) annihilation with a final
state photon is depicted in Fig. 6.2. In principle, diagrams with an s-channel
Higgs bosons also exists. These diagram can safely be neglected since the
Higgs boson couple very weakly to light leptons (∝ mℓ ) and are typically far
enough from resonance. Typically, the galactic velocity scale for WIMPs is
10−3 c, and the cross section can be calculated in the zero velocity limit of
B (1) . As found in Paper II, the differential photon multiplicity can be well
approximated by:
2
dNγℓ
mB(1)
d(σℓ+ ℓ− γ v)/dx
α (x2 − 2x + 2)
≡
≃
ln
(1 − x) ,
(6.8)
dx
σℓ+ ℓ− v
π
x
m2ℓ
where x ≡ Eγ /mB(1) and mℓ is the mass of the lepton species in consideration.
The factor α/π arises due to the extra vertex and the phase space difference
between two- and three-body final states. The large logarithm ln(m2B(1) /m2ℓ )
appears due to a collinear divergence behavior of quantum electrodynamics.
This effect is easy to see from the kinematics. Consider the propagator of the
lepton that emits a photon in the first (or third) diagram of Fig. 6.2. Denoting
the outgoing photon [lepton] momentum by k µ = (Ek , ~k) [pµ = (Ep , p~)], the
denominator of the propagator is
(p + k)2 − m2ℓ = 2p · k = 2Ek (Ep − |~
p| cos θ)
(6.9)
where θ is the angle between the outgoing photon and lepton. This expression
shows that for a highly relativistic lepton (|~
p| → Ep ) and a collinear (θ → 0)
outgoing photon the lepton propagator diverges, meaning that leptons tend
to rapidly lose their energy by emission of forward-directed photons.
Let us be slightly more quantitative and investigate the cross section. In
next to lowest order, cos θ → 1 + θ2/2 and |~
p| → Ep (1 − m2ℓ /2Ep2 ), and the denominator of the propagator becomes Ek Ep (θ2 + m2ℓ /Ep2 ). The photon vertex
itself gives in this limit a contribution ū(p)γ µ u(p + k), where the approximation consists of treating the virtual, almost on shell, electron with momentum
p + k, as a real incoming electron.§ Squaring this part of the amplitude, and
Section 6.3.
Gamma-Ray Signatures
89
taking the spin sum, leads to:
X
|ǫµ ū(p)γ µ u(p + k)|2 = −Tr[(6p + 6k + mℓ )γ µ (6p + mℓ )γµ ]
spin
= 8(p + k)µ pµ ≈ 8(Ek Ep − ~k~
p) ≈ 8Ek Ep (1 − cos θ) ≈ 8Ek Ep θ2 ,
where the standard notation that u(p) and ū ≡ u† γ 0 are dirac spinors, and ǫµ
is the photon polarization. In the second line, the lepton mass is setR to zero,
mℓ = 0. The final ingredient including θ is the phase space factor d3 k. In
the small θ limit we have d3 k = 2πp⊥ dkk dk⊥ → 2πEk θdEk dθ and therefore
the photon multiplicity should scale as:
Z θmax
dNγℓ
θ3
1
2
2
∝
dθ (6.10)
2 ≈ ln(Ep /mℓ ) ,
2
2
2
dx
2
0
θ + mℓ /Ep
where only the leading logarithm is kept, and θmax is an arbitrary upper
limit for which the used collinear approximations hold. In the colinear limit,
energy√conservation√ in the vertex of the radiating final state photon gives
Ep = s/2 − Eγ ( s being the center of mass energy), which for incoming
non-relativistic (dark matter) particles reduces to Ep ≈ mWIMP (1 − x). This is
qualitatively the result we obtained for the UED model above. I would like to
stress that these arguments are very general. Thus, for any heavy dark matter
candidate, with unsuppressed couplings to fermions, high-energy gamma rays,
as in Eq. (6.8), are expected. This means that a wide class of dark matter
particles should by annihilation produce very hard gamma spectra with a
sharp edge feature, with the flux dropping abruptly at an energy equal to the
dark matter mass.
The general behavior of final state radiation from dark matter annihilations was later studied also in [247], where they further stress that annihilation
into any charged particles, X and X̄, together with a final state radiated photon takes a universal form:
dσ(χχ → X X̄γ)
αQ2X
s(1 − x)
≈
FX (x) ln
σ(χχ → X X̄),
(6.11)
dx
π
m2X
where QX and mX are the electric charge and the mass of the X particle,
respectively. The splitting function F depends only on the spin of the X
particles. When X is a fermion:
Ff (x) =
§
1 + (1 − x)2
,
x
(6.12)
I am definitely a bit sloppy here. A full kinematically and gauge invariant calculation
(including all contributing Feynman diagrams) would, however, give the same result.
If we so wish, we could be a bit more correct and imagine the electron-positron pair
(momentum p1 = p + k and p2 ) to be produced directly from a scalar interaction,
where the p1 particle radiates a photon with momentum k. The exact spinor part of the
amplitude then becomes ǫµ ū(p)γ µ (6p1 )u(p2 ), which again leads to −Tr[(6p2 6p1 γ µ 6pγµ 6p1 ] ∝
θ 2 in lowest order in θ.
90
Kaluza-Klein Dark Matter
Chapter 6
x2 dNγeff /dx
0.1
0.03
0.01
0.01
0.1
1
x ≡ Eγ /mB(1)
Figure 6.3: The total number of photons per B (1) B (1) annihilation (solid
line), multiplied by x2 = (Eγ /mB(1) )2 . Also shown is what quark fragmentation alone would give (dashed line), and adding to that τ lepton
production and decay (dotted line). Here a B (1) mass of 0.8 TeV and a
5% mass split to the other particles first Kaluza-Klein level are assumed –
the result is, however, quite insensitive to these parameters. Figure from
Paper II.
whereas if X is a scalar particle,
Fs (x) =
1−x
.
x
(6.13)
If X is a W boson, the Goldstone boson equivalence theorem implies that
FW (x) ≈ Fs (x) [248]. Unfortunately, a sharp endpoint is not obvious in
the scalar or W boson case. According to Eq. (6.13), limx→1 Fs (x) = 0 and
the flux near the endpoint might instead be dominated by model-dependent
non-collinear contributions [247] (see Section 7.2 and Paper IV for internal
bremsstrahlung in the case of neutralino annihilations).
Including also this final state radiation, the total observable gamma spectrum per B (1) B (1) annihilation is finally given by:
X
dNγeff /dx ≡
κi dNγi /dx ,
(6.14)
i
where the sum is over all processes that contribute to primary or secondary
gamma rays, and κi are the corresponding branching ratios. The result is
shown as the solid line in Fig. 6.3.
Section 6.3.
Gamma-Ray Signatures
91
There are also other sources of photon production accompanying the LKP
annihilation. For example, the induced high-energy leptons will Compton
scatter on the CMB photons and starlight. Although these processes produce
gamma rays, they are expected to give small fluxes [249]. Another source of
photon fluxes emerge when light leptons lose energy by synchrotron radiation
in magnetic fields [249]. With moderate, although uncertain, assumptions for
the magnetic fields in the galactic center, the synchrotron radiation could give
a significant flux of photons, both at radio [250] and X-ray [251] wavelengths.
Gamma Line Signal
Since the annihilating dark matter particles are highly non-relativistic, the
processes B (1) B (1) → γγ, B (1) B (1) → Zγ, and B (1) B (1) → Hγ result in
almost mono-energetic gamma-ray lines with energies Eγ = mB(1) , Eγ =
mB(1) (1 − m2Z /4m2B(1) ) and Eγ = mB(1) (1 − m2h /4m2B(1) ), respectively. With
the expectation that dark matter being electrically neutral, these annihilation processes are bound to be loop suppressed (since photons couple only to
electric charge). On the other hand, such gamma-ray lines would constitute
a ‘smoking gun’ signature for dark matter annihilations if they were to be
observed, since it is hard to imagine any astrophysical background that could
mimic such a spectral feature.
In the case of B (1) dark matter, with unsuppressed couplings to fermions,
it could be expect that loops with fermions should dominate the cross section.
This is a naı̈ve expectation from the tree-level result of a 95% branching ratio
into charged fermions (but this expectation should be true if no particular
destructive interference or symmetry suppressions are expected to occur at
loop-level).
In Paper III, we calculated the line signal B (1) B (1) → γγ. The pure
fermionic contributions give in total 2×12 different diagrams¶ for each charged
SM fermion that contributes to B (1) B (1) → γγ; see Fig. 6.4. The calculation of
the Feynman amplitude of B (1) B (1) → γγ, within the QED sector, is described
in detail in Paper III. The basic steps in obtaining the analytical result are
as follows:
1. The total amplitude
M = ǫµ1 1 (p1 )ǫµ2 2 (p2 )ǫµ3 3 (p3 )ǫµ4 4 (p4 ) Mµ1 µ2 µ3 µ4 (p1 , p2 , p3 , p4 ),
(6.15)
is formally written down from the Feynman rules given in Appendix A.
2. Charge invariance of the in and out states and a relative sign between
vector and axial couplings,
C ψ̄γ µ ψC −1 = −ψ̄γ µ ψ,
¶
C ψ̄γ µ γ 5 ψC −1 = ψ̄γ µ γ 5 ψ,
(6.16)
Remember, at a given KK level the fermionic field content is doubled as compared to
the SM.
92
Kaluza-Klein Dark Matter
B (1)
ψ (0)
(1)
ψ (0)
ξd,s
ψ (0)
B (1)
γ
γ
Chapter 6
γ
B (1)
ψ (0)
ψ (0)
B (1)
(1)
ξd,s
(1)
ξd,s
γ
B (1)
(1)
ξd,s
(1)
ψ (0)
B (1)
γ
ξd,s
(1)
ξd,s
γ
Figure 6.4: Fermion box contributions to B (1) B (1) → γγ including the
first KK levels. Not shown are the additional nine diagrams that are
obtained by crossing external momenta. Figure from Paper III.
means that an odd number of axial couplings in the Feynman amplitudes must automatically vanish (i.e no γ 5 in the trace). Splitting the
amplitude into a contribution that contains only vector-like couplings
Mv and terms that contains only axial vector couplings Ma is therefore
possible. In this specific case, where the axial part has equal strength as
the vector part in couplings between B (1) and fermions (see Appendix A
or Paper III) we also have Ma = Mv and the full amplitude can be
written as
Mµ1 µ2 µ3 µ4 = −2iαem αY Q2 (Ys2 + Yd2 )Mµv 1 µ2 µ3 µ4 ,
(6.17)
where αEM ≡ e2 /4π, αY ≡ gY2 /4π, and Q and Y are the electric and
hypercharge, respectively, of the KK fermions in the loop.
3. From momentum conservation in the zero velocity limit of the B (1) s (p =
p1 = p2 = (mB(1) , 0)), and transversality of the polarization vectors, we
can decompose the amplitude into the following Lorentz structure:
B1
A
pµ1 pµ2 pµ3 pµ4 + 2 g µ1 µ2 pµ3 pµ4
m4B(1) 3 4
mB(1)
B3
B4
g µ1 µ3 pµ4 2 pµ4 + 2 g µ1 µ4 pµ4 2 pµ3 + 2 g µ2 µ3 pµ3 1 pµ4
mB(1)
mB(1)
B5
B6
+ 2 g µ2 µ4 pµ3 1 pµ3 + 2 g µ3 µ4 pµ3 1 pµ4 2
mB(1)
mB(1)
Mµv 1 µ2 µ3 µ4 =
+
B2
m2B(1)
+ C1 g µ1 µ2 g µ3 µ4 + C2 g µ1 µ3 g µ2 µ4 + C3 g µ1 µ4 g µ2 µ3 . (6.18)
4. The expressions for the coefficients A, B, and C are determined by
identification of the amplitude found in step 1 above. However, by
Bose symmetry and gauge invariance of the external photons, the whole
amplitude can be expressed by means of only three coefficients, which
are chosen to be B1 , B2 and B6 .
Section 6.3.
93
Gamma-Ray Signatures
5. These coefficients, B1 , B2 , and B6 , are linear combinations of tensor
integrals
D0 ; Dµ ; Dµν ; Dµνρ ; Dµνρσ (k1 , k2 , k3 ; m1 , m2 , m3 , m4 )
Z n
d q
1; qµ ; qµ qν ; qµ qν qρ ; qµ qν qρ qσ
, (6.19)
=
iπ 2 q12 − m21 q22 − m22 q32 − m23 q42 − m24
where
q1 = q,
q2 = q + k1 ,
q3 = q + k1 + k2 ,
q4 = q + k1 + k2 + k3 . (6.20)
All of these can in turn be reduced to scalar loop integrals [252], for
which closed expressions exist [253]. However, for incoming particles
with identical momenta the original reduction procedure [252] of Passarino and Veltman breaks down, and we therefore used the LERG
program [254], which has implemented an extended Passarino-Veltman
scheme to cope with such a case.
6. With the first-level fermions degenerate in mass, we finally found
(σv)γγ =
o
4 n
α2Y α2EM geff
2
2
2
3 |B1 | + 12 |B2 | + 4 |B6 | − 4Re [B1 (B2∗ + B6∗ )] ,
2
144πmB(1)
(6.21)
where
2
geff
≡
X
SM
Q2 (Ys2 + Yd2 ) =
52
.
9
(6.22)
The sum runs over all charged SM fermions, and the analytical expressions of B1 , B2 , and B6 can be found in the appendix of Paper III.
Figure 6.5 shows the annihilation rate (σv)γγ as a function of the mass shift
between the B (1) and the KK fermions.
In addition to the fermion box diagrams, there will be a large number of
Feynman diagrams once SU (2) vectors and scalar fields are included. There
are 22 new diagram types that are not related by any obvious symmetry, and
they are shown in Fig. 6.6. Obviously, it would be a tedious task to analytically calculate all these contributions by hand. Instead, we took another
approach, and implemented the necessary Feynman rules into the FeynArts
software [255]. FeynArts produces a formal amplitude of all contributing diagrams, that can then be numerically evaluated with the FormCalc [256]
package (which in turn uses the Form code [257] and LoopTools [256] to
evaluate tensor structures and momentum integrals). This numerical method
could also be used to check our analytical result of the fermion loop contribution. Adding the Feynman diagrams including bosons as internal propagators,
it was numerically found that they make only up a slight percentage of the
94
Kaluza-Klein Dark Matter
Chapter 6
(σv)γγ [10−6 pb]
160
140
120
100
80
60
1.01
1.02
1.05
1.1
1.2
mξ(1) /mB(1)
Figure 6.5: The annihilation rate into two photons as a function of the
mass shift between the B (1) and first Kaluza-Klein level fermions ξ (1) . This
is for mB(1) = 0.8 TeV, but the dependence on the B (1) mass is given by
the scaling (σv)γγ ∝ m−2
. A convenient conversion is σv = 10−4 pb =
B (1)
−4
−30
3 −1
10 c pb ≈ 3 · 10
cm s . Figure from Paper III.
total cross section. This is in agreement with the naı̈ve expectation, previously mentioned, that the fermionic contribution should dominate. Higher
KK levels also contributes; however, the larger KK masses in their propagators suppress these contributions. By adding second-level fermions, we could
confirm that our previous result only changed by a few percent. In conclusion, the analytical expression (6.21) is a good approximation for two photon
production and (σv)γγ ∼ few × 10−30 cm3 /s (1 TeV/mB(1) )2 .
The two other processes that can give mono-energetic photons, B (1) B (1) →
Zγ and B (1) B (1) → Hγ, have so far not been fully investigated. The fermionic
contribution to B (1) B (1) → Zγ can at this stage easily be calculated. The only
difference to the two photon case is here that the Z boson has both a vector
and axial part in its coupling to fermions. From the values of these couplings,
we have analytically, and numerically, found that this Zγ line should have a
cross section of about 10% compared to the γγ line.
The Hγ linek could also enhance the gamma line signal. If the Higgs mass
is very heavy, it could also potentially be resolved as an additional line at
energy Eγ = mB(1) (1 − m2h /4m2B(1) )). The contributing diagrams can have
significantly different structure than in the B (1) B (1) → γγ process. Although
k
In supersymmetry, the Hγ line is inevitably very week as its forbidden in the limit of
zero velocity annihilating neutralinos.
Section 6.4.
B (1)
γ
B (1)
a,G
a,G
a,G
a,G
γ
a,G
B (1)
B (1)
a,G
B (1)
W
a,G
γ
a,G
B (1)
γ
γ
a,G
B (1)
W
γ
a,G
γ
W
B (1)
B
B
B
γ
B
G B
W
a,G
B (1)
γ
B (1)
a,G
a,G
B
γ
B
a,G
B (1)
γ
B (1)
γ
γ
W
a,G
B (1)
a,G
a,G
B (1)
γ
G
W
γ
a,G
γ
γ
γ
c
a,G B
a,G
γ
a,G
H
(1)
B
γ
W
c
B
c
(1)
H
(1)
B
γ
W
W
(1)
γ
γ
a,G
H
(1)
a,G
γ
B (1)
H
(1)
W
a,G
γ
(1)
B (1)
W
a,G
γ
γ
(1)
B (1)
γ
γ
B (1)
γ
W
H
(1)
a,G
a,G
ψ (0) B
ψ (0)
(1)
γ
B
B (1)
γ
W
a,G
H
(1)
G
(1)
B (1)
γ
W
a,G
a,G
γ
ψ (0)
a,G
(1)
W
W
B (1)
γ
B (1)
B (1)
γ
B (1)
γ
a,G
W
a,G
B (1)
95
Observing the Gamma-Ray Signal
G
B (1)
W
B
(1)
H
γ
B
(1)
W
γ B (1)
γ
H
B (1)
a,G
γ
γ
H
B
(1)
W
γ
a,G
γ
Figure 6.6: The 22 different types of bosonic diagrams, in addition to
the fermion loop type in Fig. 6.4, that contribute to B (1) B (1) → γγ.
it is not expected to give any particularly strong signal∗∗ , an accurate analysis
of this processes has not yet been carried out.
6.4
Observing the Gamma-Ray Signal
The characteristic dark matter signature in the gamma-ray spectrum – a hard
spectrum with a sharp drop in the flux, and potentially even a visible gamma
line, at an energy equal to the mass of the B (1) – is something that can be
searched for in many experiments. Due to the large uncertainties in the dark
matter density distribution, the expected absolute flux from different sources,
such as the galactic center, small dark matter clumps and satellite galaxies,
∗∗
Estimates including only fermion propagators indicate that these contributions are not
very significant.
96
Kaluza-Klein Dark Matter
Chapter 6
as well as the diffuse extragalactic, is still very difficult to predict with any
certainty.
The velocity scale for cold dark matter particles in our Galaxy halo is
of the order of v ∼ 10−3 c. For an ideal observation of the monochromatic
gamma line, this would lead to a relative smearing in energy of ∼ 10−3 due
to the Doppler shift. This narrow line is much too narrow to be fully resolved
with the energy resolution of current gamma-ray telescopes. To compare a
theoretically predicted spectrum to experimental data, the predicted signal
should first be convolved with the detector’s response. The actual detector
response is often unique for each detector and is often rather complicated.
To roughly take the convolution into account, it is reasonable to use a simple
convolution/smearing of the theoretical energy spectrum before comparing
with published data. For a Gaussian convolution function with an energy
resolution σ(E ′ ), the predicted experimental flux is given by
Z ∞
e−(E ′ −E)2 /(2σ2 )
dΦexp ′ dΦtheory √
=
dE
.
(6.23)
dE E
dE E ′
2πσE ′
−∞
As an illustrative example, let us compare our theoretically predicted spectrum with the TeV gamma-ray signal observed from the direction of the
galactic center, as observed by the air Čerenkov telescopes H.E.S.S. [258],
Magic [259], VERITAS [260], and CANGAROO [261]. The nature of this
source is partly still unknown. Because the signal does not show any apparent time variation, is located in a direction where a high dark matter density
concentration could be expected, and is observed to be a hard spectrum up
to high gamma-ray energies (i.e., the flux does not drop much faster than
E −2 ), it has been discussed if the gamma flux could be due to dark matter
annihilations (see, e.g., [262] and references therein).
In Fig. 6.7, the predicted gamma spectrum from annihilating B (1) s with
masses of 0.8 TeV, smeared with an energy resolution of σ = 0.15E ′ , is shown
together with the latest H.E.S.S. data. Annihilation of such low mass dark
matter particles can certainly not explain the whole range of data. However,
it is interesting to note that the flux comes out to be of the right order of
magnitude for reasonable assumptions about the dark matter density distribution. For the flux prediction, an angular acceptance of ∆Ω = 10−5 sr and a
boost factor b ≈ 200 to the NFW density profile were used in the flux equation
(2.16). With a better understanding of backgrounds and more statistics, it
might be possible to extract such a dark matter contribution; especially since
there is a sharp cutoff signature in annihilations spectrum to look for. When
the first data were presented from the H.E.S.S. collaboration, it was noticed
that the spectral energy distribution of gamma rays showed a very similar
hard spectrum as predicted from final state radiation from light leptons. To
point this out, we suggested in Paper II a hypothetical case with MB(1) ∼ 10
TeV. A good match to the 2003 year data (solid boxes in Fig. 6.7) [263] was
then found. Later observations during 2004 [258] did not match the prediction
Section 6.4.
Observing the Gamma-Ray Signal
97
Eγ2 dΦγ /dEγ [m−2 s−1 TeV]
10−7
10−8
10−9
0.1
1
10
Eγ [TeV]
Figure 6.7: The H.E.S.S. data (open boxes: 2003 data [263]; solid triangles: 2004 data [258]) compared to the gamma-ray flux expected from
a region of 10−5 sr encompassing the galactic center, for a B (1) mass of
0.8 TeV, a 5% mass splitting at the first Kaluza-Klein level, and a boost
factor b ∼ 200 (dotted line). The solid line corresponds to a hypothetical
9 TeV WIMP with similar artificial couplings, a total annihilation rate
given by the WMAP relic density bound, and a boost factor of around
1000. Both signals have been smeared to simulate an energy resolution of
15%, appropriate for the H.E.S.S. telescope.
for such a heavy B (1) particle; neither was any cutoff at the highest energies
found. Although it is in principle possible to reconstruct the observed spectral
shape with dark matter particles of tens of TeV, as illustrated in [262] with
non-minimal supersymmetry models, this does not work for the most simple
and common models of dark matter [258, 262]. With more statistics in the
2004 data, a search for any significant hidden dark matter signal on top of a
simple power law astrophysical background was performed, but no significant
typical dark matter component could be found [258]. When it comes to suggested astrophysical explanations of the observed TeV signal from the galactic
center, the main proposed sources are particle accelerations in the Sagittarius
A supernova remnant, processes in the vicinity of the supermassive black hole
Sagittarius A∗ , and the detected nearby pulsar wind nebula (see [264] and
references therein).
When it comes to experimental searches for the monochromatic gamma
line signal from B (1) annihilation, a much better energy resolution would
be needed. Nevertheless, with an energy resolution close to the mentioned
natural Doppler width of 10−3 the line could definitely be detectable. In
Kaluza-Klein Dark Matter
dΦγ /dEγ [10−8 m−2 s−1 TeV−1 ]
98
Chapter 6
4
3
2
1
780
790
800
810
Eγ [GeV]
Figure 6.8: The gamma line signal as expected from B (1) dark matter
annihilations. The line-signal is superimposed on the continuous gammaray flux (solid line) and is roughly as it would be resolved by a detector
with a Gaussian energy resolution of 1% (dashed), 0.5% (dotted) and
0.25% E’ (dash-dotted), respectively. The actual line width of the signal is
about 10−3 , with a peak value of 1.5 · 10−7 m−2 s−1 TeV−1 . The example
here is for a B (1) mass mB(1) = 0.8 TeV, and a mass shift mξ(1) /mB(1) =
1.05. An angular acceptance of ∆Ω = 10−5 sr and a boost factor of
b = 100 to a NFW profiles were assumed. Figure from Paper III.
Fig. 6.8, the expected gamma-ray spectrum around a 0.8 TeV mass B (1) is
shown for three different detector resolutions; an energy resolution better than
1% would be needed to resolve the line signal. Typically the energy resolution
of today’s detectors is a factor of ten larger.
Chapter
7
Supersymmetry
and a New
Gamma-Ray Signal
The most well known dark matter candidate is the neutralino, a WIMP that
appears in supersymmetry theories. Although the detection signals for this
dark matter candidate have been well studied, the internal bremsstrahlung
contribution to the gamma-ray spectrum for typical heavy neutralinos has
previously not been investigated. In our study in Paper IV we found that
internal bremsstrahlung produce a pronounced signature in the gamma-ray
spectrum, in the form of a very sharp cutoff, or even a peak, at the highest
energies. This signal can definitely have a positive impact on the neutralino
detection prospects, as it not only possesses a striking signature, but also significantly enhances the expected total gamma-ray flux at the highest energies.
With the energy resolution of current detectors, this signal can even dominate
the monochromatic gamma-ray lines (γγ and Zγ) that previously been shown
to provide exceptional strong signals for heavy neutralinos.
7.1
Supersymmetry
Supersymmetry relates fermionic and bosonic fields, and is thus a symmetry
mixing half-integer and integer spins. The generators of such a symmetry must
carry a half integer spin and commute with the Hamiltonian. These generators
transform non-trivially under Lorentz transformations and the internal symmetries interact non-trivially with the spacetime Poincaré symmetry. This
might seem to violate a no-go theorem by Coleman and Mandula, stating
that any symmetry group of a consistent four-dimensional quantum field theory can only be a direct product of internal symmetry groups and the Poincaré
group (otherwise the scattering-matrix is identically equal to 1, and no scattering is allowed). However, the new feature of having anticommuting, spin
99
100
Supersymmetry and a New Gamma-Ray Signal
Chapter 7
half, generators for a symmetry turn out to allow for a nontrivial extension
of the Poincaré algebra. The details of the construction of a supersymmetric
theory are beyond the scope of this thesis, and only some basic facts and the
most important motivations for supersymmetry will be mentioned here.
Since it is not possible to relate the fermions and bosons within the SM,
each SM fermion (boson) is instead given new bosonic (fermionic) supersymmetric partner particles. The nomenclature for superparners is to add a prefix
‘s’ to the corresponding fermion name (e.g., the superpartner to the electron is
called the selectron), whereas superpartners to bosons get their suffix changed
to ‘ino’ (e.g., the superpartner to the photon is called the photino).
Superpartners inherit mass and quantum numbers from the SM particles.
Only the spin differs. Since none of these new partners have been observed,
supersymmetry must be broken, and all supersymmetric particles must have
obtained masses above current experimental limits. The actual breaking of
supersymmetry might introduce many new unknown parameters. Explicit
symmetry breaking terms can introduce more than 100 new parameters [265].
In specific constrained models, where the supersymmetry is broken spontaneously, the number of parameters is usually much smaller. For example, in
the minimal supergravity (mSUGRA) model [266] the number of supersymmetry parameters is reduced to only five, specified at a high energy grand
unification scale.
In the following, only minimal supersymmetric standard models (MSSM)
are considered. They are minimal in the sense that they contain a minimal
number of particles: the SM fields (now with two Higgs doublets∗ ) and one
supersymmetric partner to each of these.
Some Motivations
A main motivation for having a supersymmetric theory, other than for the
mathematical elegance of a symmetry relating fermions and bosons, is that
it presents a solution to the fine-tuning problem within the SM. Within the
SM, the Higgs mass mh gets radiative corrections that diverge linearly with
any regulating ultraviolet cutoff energy. With the cutoff of the order of the
Planck scale (1019 GeV), the required fine-tuning is of some 17 orders of
magnitude to reconcile the Higgs mass mh with the indirectly measured value
mh ∼ 100 GeV. This naturalness problem (or fine-tuning problem) of the SM
is elegantly solved within supersymmetry by the fact that supersymmetric
partners exactly cancel these divergent quantum contribution to the Higgs
mass. Even if supersymmetry is broken, there is no need for extreme finetuning, at least not as long as the breaking scale is low and the supersymmetric
partners have masses not much higher than the Higgs mass.
∗
This is a type II two Higgs doublet model (to be discussed more in Section 8.1), and
is needed in supersymmetry to generate masses to both the up- and down-type quarks
[267, 268]
Section 7.2.
A Neglected Source of Gamma Rays
101
A second motivation for supersymmetry is that the required spontaneous
symmetry breaking of the electroweak unification can be obtained through
radiative quantum corrections, where the quadratic Higgs mass parameter is
driven to a negative value.
A third intriguing attraction has been that the running of the three gauge
coupling constants of the electromagnetic, weak, and strong forces are modified in such a way that they all become equal, within a unification scheme, at
an energy of about 1016 GeV. That this force strength unification occurs at
an energy scale significantly below the Plank scale and where the theory still
is perturbatively reliable is far from trivial.
Finally, supersymmetry can provide dark matter candidates. Usually this
is the case in models with an additional symmetry called R-parity, which is an
imposed conserved multiplicative quantity. Every SM particle is given positive R-parity, whereas all supersymmetric particles are given negative. This is
very important for cosmology, because R-parity guarantees that the lightest
supersymmetric particle is stable since it cannot decay into any lighter state
having negative R-parity. The introduction of such a symmetry can be further motivated as it automatically forbids interactions within supersymmetry
that otherwise would lead to proton lifetimes much shorter than experimental
limits.
The Neutralino
In many models, the lightest stable supersymmetric particle is the lightest
neutralino, henceforth just ‘the neutralino’. It is a spin-1/2 Majorana particle
and a linear combination of the gauginos and the Higgsinos
χ ≡ χ̃01 = N11 B̃ + N12 W̃ 3 + N13 H̃10 + N14 H̃20 .
(7.1)
With R-parity conserved, the neutralino is stable and a very good dark matter
candidate. This is the most studied dark matter candidate, and there are
many previous studies on its direct and indirect detection possibilities (see,
e.g., [25, 42], and references therein). We will here focus on a new type of
gamma-ray signature first discussed in Paper IV.
7.2
A Neglected Source of Gamma Rays
Previous studies of the gamma-ray spectrum from neutralino annihilations
have mainly focused on the continuum spectrum, arising from the fragmentation of produced quarks and τ -leptons, and the second order, loop-induced
γγ and Zγ line signals [96, 269, 270]. For high neutralino masses, the almost
monochromatic γγ and Zγ photon lines can be exceptionally strong, with
branching ratios that reach percent level despite the naı̈ve expectation of being two to three orders of magnitude smaller. The origin of this enhancement
102
Supersymmetry and a New Gamma-Ray Signal
Chapter 7
is likely due to nonperturbative, binding energy effects in the special situation of very small velocities, large dark matter masses, as well as small mass
differences between the neutralino and the lightest chargino [271, 272].
The contribution from radiative processes, i.e., processes with one additional photon in the final state, should naı̈vely have a cross section two
orders of magnitude larger than the loop-suppressed monochromatic gamma
lines, since they are one order lower in the fine structure constant αem . As
investigated in Paper IV, internal bremsstrahlung, in the production of
charged gauge bosons from annihilating heavy neutralinos, results in highenergy gamma rays with a clearly distinguishable signature. This is partly
reminiscent of the case of KK dark matter, where final state radiation in annihilation processes with charged lepton final states dominates the gamma-ray
spectrum at the highest energies.
Helicity Suppression for Fermion Final States
In Paper II, we found that final state radiation from light leptons produced
in B (1) annihilation gave an interesting signature in the gamma-ray spectrum.
Neutralino annihilations into only two light fermion pairs have an exceptionally strong suppression, and typical branching ratios into electrons are often
quoted to be only of the order of 10−5 . The reason for this so-called helicity
suppression can be understood as follows. The neutralinos are self-conjugate
(Majorana) fermions obeying the Pauli principle and must therefore form an
antisymmetric wave function. In the case of zero relative velocity† , the spatial
part of the two particles’ wave function is symmetric – i.e., an orbital angular
momentum L = 0 state (s-wave) – and the spin part must form an antisymmetric (S = 0) singlet state. Thus the incoming state has zero total angular momentum. The contributing neutralino annihilation processes conserve
chirality‡ , so that massless (or highly relativistic) fermions and antifermions
come with opposite helicities (i.e., handedness). Therefore the spin projection
in the outgoing direction is one, which precludes s-wave annihilation§ . The
conclusion must be that the annihilation cross section into monochromatic
massless fermions is zero; for massive fermions it is instead proportional to
m2f /m2χ . An interesting way to circumvent this behavior of helicity suppres†
Typical dark matter halo velocities are v ∼ 10−3 , and p-wave annihilations would be
suppressed by v2 ∼ 10−6 .
‡
Both the Z-fermion-antifermion and fermion-sfermion-gaugino vertices conserve helicity.
Contributions from Higgs-boson exchange, from Higgsino-sfermion-fermion Yukawa interactions, and from sfermion mixing violate chirality conservation, but they all include
an explicit factor of the fermion mass mf [273]
§
Orbital angular momentum of the outgoing fermions can never cancel the spin component in the direction of the outgoing particles (this is clear because orbital angular
momentum of two particles can never have an angular momentum component in the
same plane as their momentum vectors lie in). Therefore the total angular momentum
must be nonzero, in contradiction to the initial state of zero angular momentum.
Section 7.2.
A Neglected Source of Gamma Rays
χ
W−
W−
χ
χ−
1
γ
χ−
1
W
W−
χ
χ−
1
+
χ
W
+
W−
χ
γ
χ−
1
+
χ
103
γ
W+
W+
−
Figure 7.1: Contributions to χχ → W W γ for a pure Higgsino-like
neutralino (crossing fermion lines are not shown). Figure from Paper IV.
sion is to have a photon accompanying the final state fermions [274]. This
open up the possibility of a significant photon and fermion spectrum, where
the first-order corrected cross section can be many orders of magnitude larger
than the tree-level result. I will not pursue this interesting possibility here
(see, however, [274] and the recent work of [275]), but instead discuss internal
bremsstrahlung when W ± gauge bosons are produced by annihilating heavy
Higgsinos [Paper IV].
Charged Gauge Bosons and a Final State Photon
In order not to overclose the Universe, a TeV-mass neutralino must in general
2
2
have a very large Higgsino fraction¶ Zh (Zh ≡ |N13 | + |N14 | ), ensuring a
significant cross section into massive gauge bosons. A pure bino state with
a TeV mass, on the other hand, does not couple to W at all in lowest order.
This usually excludes TeV binos as they would freeze-out too early and overproduce the amount of dark matter. Let us therefore focus on a Higgsino-like
neutralino with N11 ≈ N12 ≈ 0 and N13 ≈ ±N14 . The annihilation rate into
charged gauge bosons often dominates, and radiation of a final state photon
should be of great interest to investigate.
For a pure Higgsino, the potential s-channel exchanges of Z and Higgs
bosons vanish, and the only Feynman diagrams contributing to the W + W − γ
final states are shown in Fig. 7.1.
For the analytical calculation of these Feynman diagrams, there is one
technicality worth noticing. Due to the Majorana nature of the neutralinos,
the Feynman diagrams can have crossing fermion lines, and special care must
be taken to deal with the spinor indices correctly. Proper Feynman rules
have been developed (see, e.g., [276]), which also have been implemented in
different numerical code packages (e.g., FeynArts and FormCalc [277]). For
manual calculations a practical simplifying technique can be adopted: in the
limit of zero relative velocity, the two ingoing annihilating neutralinos must
¶
This is the case if the usual GUT condition M1 ∼ M2 /2 is imposed; otherwise a heavy
wino would also be acceptable. For a pure wino the results are identical to what is
found for the anti-symmetric N13 = −N14 Higgsinos considered here; apart from a
multiplicative factor of 16 in all cross sections.
104
Supersymmetry and a New Gamma-Ray Signal
Chapter 7
form a 1 S0 state (as explained above) and the sum over all allowed spin state
configurations of the two incoming Majorana particles can be replaced by the
projector [278]
1
P1 S0 ≡ − √ γ 5 (mχ − p
6 ),
(7.2)
2
where p is the momentum of one of the incoming neutralinos. P1 S0 is simply inserted in front of the gamma-matrices originating from the Majorana
fermion line, and then the trace is taken over the spinor indices. All analytical
calculations in Paper IV were performed both by this technique of using the
P1 S0 projector operator, and direct calculations by explicitly including all the
diagrams with their crossing fermion lines. The calculations were also further
checked by numerical calculations with the FeynArts/FormCalc numerical
package.
The analytical result is rather lengthy, but up to zeroth order in ǫ ≡
mW /mχ and retaining a leading logarithmic term, the resulting photon multiplicity is given by
dNγW d(σv)W W γ /dx
αem 4(1 − x + x2 )2 ln(2/ǫ)
≡
≃
dx
(σv)W W
π
(1 − x)x
2
3
2(4 − 12x + 19x − 22x + 20x4 − 10x5 + 2x6 )
−
(2 − x)2 (1 − x)x
2(8 − 24x + 42x2 − 37x3 + 16x4 − 3x5 ) ln(1 − x)
+
(2 − x)3 (1 − x)x
2x(2 − (2 − x)x) 8(1 − x) ln(1 − x)
+ δ2
+
(2 − x)2 (1 − x)
(2 − x)3
x(x − 1) (x − 1)(2 − 2x + x2 ) ln(1 − x)
+ δ4
+
,
(7.3)
(2 − x)2
(2 − x)3
where x ≡ Eγ /mχ and δ ≡ (mχ± −mχ )/mW , with mχ± denoting the chargino
1
1
mass.
Figure 7.2 shows the photon multiplicity together with a concrete realized
minimal supersymmetric model example as specified in Table 7.1k .
Two different effects can be singled out to cause increased photon fluxes
at the highest energies. The first occurs for large mass shifts δ between the
neutralino and the chargino, whereby the last two terms in Eq. (7.3) dominate.
These terms originate from the longitudinal polarization modes of the charged
gauge bosons. Such polarization modes are not possible for a 1 S0 state with
k
The MSSM parameters specify the input to DarkSUSY [279]. M2 , µ, mA , and mf˜
are the mass scales for the gauginos, Higgsinos, supersymmetry scalars, and fermions,
respectively. Af (= At = Ab ) is the trilinear soft symmetry breaking parameter, and
tan β = vu /vrmd is the ratio of vacuum expectation values of the two neutral Higgs
doublet. All values are directly given at the weak energy scale. For more details of this
7-parameter MSSM, see [280].
Section 7.2.
105
A Neglected Source of Gamma Rays
dNγW /dx
1
0.1
0.01
0
1
0.5
x ≡ Eγ /mχ
Figure 7.2: The photon multiplicity for the radiative process χχ →
W + W − γ. The dots represent the minimal supersymmetric model given
in Table 7.1, as computed with the FormCalc package [256] for a relative
neutralino velocity of 10−3 . The thick solid line shows the full analytical
result for the pure Higgsino limit of the same model but with zero relative
neutralino velocity. Also shown, as dashed and dotted lines, are two pure
Higgsino models with a lightest neutralino (chargino) mass of 10 TeV (10
TeV) and 1.5 TeV (2.5 TeV), respectively. Figure from Paper IV.
Table 7.1: MSSM parameters for the example model shown in Fig. 7.27.3, and the resulting neutralino mass (mχ ), chargino mass (mχ± ), Hig1
gsino fraction (Zh ), branching ratio into W pairs (W ± ), and neutralino
relic density (Ωχ h2 ), as calculated with DarkSUSY [281] and micrOMEGAs
[282]. Masses are given in units of TeV.
M2
3.2
µ
1.5
mA
3.2
mf˜
3.2
Af
0.0
tan β
10.0
mχ
1.50
mχ±
1
1.51
Zh
0.92
W±
0.39
Ωχ h 2
0.12
only two vector particles (remember that the initial state must be in this
state due to the Majorana nature of the low velocity neutralino); but when
an additional photon is added to the final state, this channel opens up and
enhances the photon flux at high energies. Typically MSSM models are,
however, not expected to have very large mass shifts δ. The other effect is,
on the other hand, dominated by transversely polarized photons. For heavy
neutralino masses the W bosons can be treated as light, and the cross section
is thus expected to be enhanced in a similar way to the infrared divergence that
appears in QED when low-energy photons are radiated away. For kinematical
Supersymmetry and a New Gamma-Ray Signal
104
d(vσ)γ /dEγ [10−29 cm2 s−1 TeV−1 ]
d(vσ)γ /dEγ [10−29 cm2 s−1 TeV−1 ]
106
103
102
10
1
0.1
0.1
0.5
Eγ [TeV]
1
2
Chapter 7
30
10
2
0.6
1
2
Eγ [TeV]
Figure 7.3:
Left panel: The total differential photon distribution
from χχ annihilations (solid line) for the minimal supersymmetric model
of Table 7.1. Also shown separately is the contribution from internal bremsstrahlung χχ → W + W − γ (dashed), and the fragmentation of
mainly the W and Z bosons, together with the χχ → γγ, Zγ lines (dotted). Right panel: A zoom in of the same spectra as it would approximately appear in a detector with a relative energy resolution of 15 percent.
Figures from Paper IV.
reasons, each low energy W boson is automatically accompanied by a high
energy photon. The resulting peak in the spectrum at the highest energies
is hence an amusing reflection of QED infrared behavior also for W bosons.
The two different effects are illustrated in Fig. 7.2 by the dotted and dashed
curves, respectively.
In addition to the internal bremsstrahlung discussed above, secondary
gamma rays are produced in the fragmentation of the W pairs, mainly through
the production and subsequent decay of neutral pions. Similarly, production
of Z-bosons (or quarks) results in secondary gamma rays; altogether producing a continuum of photons dominating the gamma flux at lower energies.
Previous studies have also shown that there are strong line signals from the
direct annihilation of a neutralino pair into γγ [269] and Zγ [270]. Due to the
high mass of the neutralino (as studied here), the two lines cannot be resolved
but effectively add to each other at an energy almost equal to the neutralino
mass. Adding all contributions, and using the model of Table 7.1, the total
spectrum is shown in the left panel of Fig. 7.3.
The practical importance of internal bremsstrahlung is even clearer when
taking into account an energy resolution of about 15%, which is a typical
value for current atmospheric Čerenkov telescopes in that energy range. The
result is a smeared spectrum as shown in the right panel of Fig. 7.3. We can
Eγ2 d(σv)γ /dEγ [10−29 cm3 s−1 TeV]
Section 7.2.
A Neglected Source of Gamma Rays
107
103
102
10
0.1
0.5
1
2
Eγ [TeV]
Figure 7.4: Comparing the shape of the total gamma-ray spectrum that
can be expected from B (1) Kaluza-Klein (dashed line) and neutralino (solid
line) dark matter annihilations, as seen by a detector with an energy
resolution of 15%. In both cases, the dark matter particle has a mass of
1.5 TeV and both spectra are normalized to have a total annihilation cross
section of hσvi0 = 3 · 10−26 cm3 s−1 . Figure from Paper B.
see that the contribution from the internal bremsstrahlung enhances the flux
in the peak at the highest energies by a factor of about two. The signal is also
dramatically increased, by almost a factor of 10, at slightly lower energies,
thereby filling out the previous ‘dip’ just below the peak. This extra flux at
high energies improves the potential to detect a gamma-ray signal. It is worth
pointing out that this example model is neither tuned to give the most extreme
enhancements, nor is it only pure Higgsinos with W final states that should
have significant contributions from this type of internal bremsstrahlung. On
the contrary, this type of internal bremsstrahlung can contribute 10 times
more to high-energy photon flux than the gamma-ray line, see [275] for a
extensive scan of MSSM parameters.
As in the case of the KK dark matter candidate B (1) , the internal radiation
of a photon in the neutralino annihilation case also gives a very characteristic
signature in the form of a very sharp cutoff in the gamma-ray spectrum at
an energy equal to the neutralino mass. This is a promising signal to search
for and with current energy resolution and with enough statistics the shape
of the gamma-ray spectra could even provide a way to distinguish between
different dark matter candidates. Figure 7.4 illustrates this by comparing the
gamma-ray spectrum of a 1.5 TeV neutralino (specified in Table 7.1) and a
1.5 TeV B (1) KK dark matter candidate.
Chapter
8
Inert Higgs
Dark Matter
A possible, and economical, way to incorporate new phenomenology into the
standard model would be to enlarge its Higgs sector. One of the most minimal
way to do this, which simultaneously gives rise to a dark matter candidate, is
the so-called inert doublet model (or inert Higgs model), obtained by adding a
second scalar Higgs doublet with no direct coupling to fermions. The lightest
of the new appearing inert Higgs particles could, if its mass is between 40 and
80 GeV, give the correct cosmic abundance of cold dark matter. One way to
unambiguously confirm the existence of particle dark matter and determine
its mass would be to detect its annihilation into monochromatic gamma rays
by current or upcoming telescopes. In Paper VII, we showed that for the
inert Higgs dark matter candidate the annihilation signal into such monochromatic γγ and Zγ final states is exceptionally strong. The energy range and
rates for these gamma-ray line signals therefore make them ideal to search
for with upcoming telescopes, such as the GLAST satellite. This chapter reviews the inert Higgs dark matter candidate and discusses the origin of these
characteristic gamma line signals.
8.1
The Inert Higgs Model
Let us start by shortly reviewing why there is a need for a Higgs sector in the
first place. In the SM of particle physics, it is not allowed to have any explicit
gauge boson or fermion mass terms since that would spoil the underlying
SU (2) × U (1) gauge invariance and lead to a non-renormalizable theory∗ . To
circumvent this, we start with a fully gauge invariant theory – with no gauge
field or fermion mass terms – and adds couplings to a complex, Lorentz scalar,
∗
A fundamentally non-renormalizable theory would lack predicability as the canonically
appearing divergences from quantum corrections can not be cured by the renormalization
procedure of absorbing them into a finite number of measurable quantities.
109
110
Inert Higgs Dark Matter
Chapter 8
SU (2) doublet φ, which spontaneously develops a non-vanishing vacuum expectation value, and thereby breaks the full SU (2)×U (1) gauge structure and
generates particle masses. This is exactly what was technically described in
Section 5.3 of the UED model (but, of course, now without the complications
of having an extra spatial dimension). The Higgs Lagrangian written down in
Eq. (5.17), with the potential (5.18), is in four dimensions the most general
setup we can have with one Higgs field φ. As explained in Section 5.3, this
Higgs field has four scalar degrees of freedom. Three of the degrees of freedom
are absorbed by the new polarization modes of the now massive gauge bosons,
and thus only one degree of freedom is left as a physical particle h (the Higgs
particle). The mass of the Higgs particle h is a completely free parameter in
the SM (which can be measured and constrained).
Another way to express the need for the Higgs particle is that, without
it, certain cross sections would grow with the center of mass energy (denoted
by Ecm ) beyond the unitarity limit for large enough Ecm . An example is
the process f f¯ → W + W − into longitudinally polarized W ± (those W ’s that
arose by the Higgs mechanism). If the Higgs particle is not included, the only
contributing Feynman diagrams are from s-channel gauge bosons and t- or
2
u-channel fermions, which give rise to a term that grows as m2f Ecm
. This is
the piece that is exactly canceled by the s-channel Higgs boson that couples
proportionally to mf . This should make it clear that a physical Higgs, or a
similar scalar interaction, must be included to have a sensible theory.
So why is there only one Higgs doublet in the standard model? The inclusion of just one Higgs doublet is the most economical way of introducing
masses into the SM, but in principle nothing forbids models with more complicated Higgs sectors. This might at first sound as a deviation from the
principle of Occam’s razor, but we will soon see that such extensions can be
motivated by its potential to address several shortcomings of the SM and still
satisfy theoretical and existing experimental constraints.
A minimal extension of the SM Higgs sector would be to instead have two
Higgs doublets, H1 and H2 . The Lagrangian for such so-called two-Higgsdoublet models can formally be written as
|Dµ H1 |2 + |Dµ H2 |2 − V (H1 , H2 ) ,
(8.1)
The most general gauge invariant, renormalizable potential V (H1 , H2 ) that
is also invariant under the discrete Z2 symmetry
H2 → −H2
and H1 → H1 ,
(8.2)
can be written as
2
2
4
4
V (H1 , H2 ) = µ21 |H1 | + µ22 |H2 | + λ1 |H1 | + λ2 |H2 |
2
2
+λ3 |H1 | |H2 | +
λ4 |H1† H2 |2
+
λ5 Re[(H1† H2 )2 ],
(8.3)
Section 8.1.
The Inert Higgs Model
111
where µ2i , λi are real parameters.† The latter constraint of a discrete Z2
symmetry is related to the experimental necessity to diminish flavor-changing
neutral currents (FCNCs) and CP-violations in the Higgs sector [139,267,268,
283].
The experimental limits on FCNCs are very strong and come from, e.g.,
studies of the neutral K 0 meson. K 0 is a bound state (containing a down
quark and a strange anti-quark) that would, if FCNCs were mediated by schannel Z or Higgs bosons at tree level, rapidly oscillate into its antiparticle
K¯0 or decay directly into lepton pairs. These processes are so rare that they
are only expected to be compatible with loop-level suppressed reactions (or in
some other way protected, as for example pushing the FCNC mediator to very
high masses). In the SM, FCNCs are naturally suppressed as they are forbidden at tree level.‡ Technically, this comes about since the diagonalization
of the mass matrix automatically also flavor diagonalizes the Higgs-fermion
couplings, as well as the fermion couplings to the photon and the neutral Z
gauge boson. This lack of FCNCs would in general no longer be true in the
Higgs sector once additional scalar doublets are included that have Yukawa
couplings to fermions. In a theorem by Glashow and Weinberg [284], it was
shown that FCNCs mediated by Higgs bosons will be absent if all fermions
with the same electric charges do not couple to more than one Higgs doublet.
Adopting this approach to suppress FCNCs, the scalar couplings to fermions
are constrained, but not unique. To specify a model, it is practical to imposes
a discrete symmetry. Any such discrete symmetry must necessarily be of the
form H2 → −H2 and H1 → H1 (or vice versa) [267]. The Z2 symmetry can
then be used to design which Yukawa couplings are allowed or not. This is
exactly the Z2 symmetry that was already incorporated in the potential given
in Eq. (8.3).
In, what is called, a Type I two-Higgs-doublet model, the fermions couple
only to the first Higgs doublet H1 , and there are no couplings between fermions
and H2 . This is the same as saying that the Lagrangian is kept invariant
under the Z2 symmetry that takes H2 → −H2 and leaves all other fields
unchanged. This is the type of model we will be interested in here. A Type II
model is when the down-type fermions only couple directly to H1 and up-type
fermions only couple directly to H2 (corresponding to the appropriate choice
for the Z2 transformation of the right-handed fermion fields, uR → −uR ).
The minimal supersymmetric models belongs to this Type II class of models.
Other choices where quarks and leptons are treated in some asymmetrical way
†
An additional term is actually possible, but can always be eliminated by redefining the
phases of the scalars [267, 268].
‡
Actually, even the loop-level FCNCs in the SM need to be suppressed. In 1970 Glashow,
Iliopoulos, and Maiani realized that this could be achieved if quarks come in doublets
for each generation (the GIM mechanism) [139]. Their work was before the detection of
the charm quark, and therefore predicted this new quark to be the doublet companion
for the already known strange quark.
112
Inert Higgs Dark Matter
Chapter 8
could in principle also be possible.
Coming back to the Type I model, where only Yukawa terms involving H1
are allowed. This means that H1 must develop a nonzero vacuum expectation
value v 6= 0 to generate fermion and gauge masses. In other words, the
potential must have a minimum for H1 6= 0. For H2 there are, however, two
choices for its potential. Either H2 also develops a vacuum expectation value
vH2 6= 0 by spontaneous symmetry breaking and the potential has a global
minimum for H2 6= 0, or the Z2 symmetry H2 → −H2 is unbroken and the
potential has a global minimum at H2 = 0. (Note that in general this latter
case is not the vH2 → 0 limit of the vH2 6= 0 case.)
The model that contains our dark matter candidate is the latter of the
two Type I two-Higgs-doublet models that have vH2 = 0. In other words,
this is an ordinary two-Higgs-doublet model with the H2 → −H2 symmetry
unbroken. The H1 field is identified as essentially the SM Higgs doublet – it
gets a vacuum expectation value and gives masses to the W , Z and fermions
exactly as in the SM. On the other hand the H2 does not get any vacuum
expectation value and does not couple directly to fermions. This H2 will be
called the inert Higgs doublet and the model the inert doublet model (IDM).§
The origin of this IDM goes back to at least the 1970s [285] when the
different possibilities for the two-Higgs-doublet models were first investigated.
The IDM has recently received much new interest. Besides providing a dark
matter candidate [286, 287], this type of model has the potential to allow
for a high Higgs mass [286], generate light neutrinos and leptogenesis (see,
e.g., [288] and references therein), as well as break electroweak symmetry
radiatively [289].
The New Particles in the IDM
Let us set up some notation and at the same time present how many free
parameters and physical fields this inert doublet model contains. The two
Higgs doublets will be parameterized according to
1
G+
√
H1 = √
(8.4)
2v + h + iG0
2
1
H+
H2 = √
(8.5)
0
H + iA0
2
where G± , H ± are complex scalar fields while h, G0 , H 0 and A0 are real.¶
We can always use the freedom of SU (2) × U (1) rotations to get the vacuum
§
The name inert Higgs doublet might be found misleading as it is neither completely
inert (since it has ordinary gauge interactions) nor contributes to the Higgs mechanism
to generate masses. The name dark scalar doublet has later been proposed, but we will
here stick to the nomenclature used in Paper VII and call it the inert Higgs or inert
scalar.
¶
I have here slightly changed the notation for the scalar fields compared to Section 5.3
Section 8.1.
The Inert Higgs Model
113
expectation value v for H1 to be real valued and in the lower component of the
doublet. G+ and G0 constitute Goldstone fields that in unitarity gauge can
be fixed to zero. After giving mass to the gauge bosons, five out of the original
eight degrees of freedom in H1 and H2 remain. Besides the SM Higgs particle
(h), the physical states derived from the inert doublet H2 are thus two charged
states (H ± ) and two neutral; one CP-even (H 0 ) and one CP-odd (A0 ).k The
h field in the H1 doublet will be referred to as the SM Higgs particle and
the particle fields in H2 as the inert Higgs particles. The corresponding (tree
level) masses are given by:
m2h
m2H 0
m2A0
m2H ±
=
=
=
=
−2µ21
µ22 + (λ3 + λ4 + λ5 )v 2
µ22 + (λ3 + λ4 − λ5 )v 2
µ22 + λ3 v 2 .
(8.6)
Measurements of the gauge boson masses determine v = 175 GeV, and we are
left with only 6 free parameters in the model. A convenient choice is to work
with mh , mH 0 , mA0 , mH + , µ2 and λ2 .
Heavy Higgs and Electroweak Precision Bounds
One of the original motivations for the IDM was that it could incorporate a
heavy SM Higgs particle. Let us briefly review why this might be of interest
and how this is possible, in contrast to the SM and the MSSM.
In the SM, the Higgs boson acquires an ultraviolet divergent contribution
from loop corrections which will be, at least, of the same size as the energy scale where potential new physics comes in to cancel divergences. With
no such new physics coming in at TeV energies, a tremendous fine-tuning
is required to keep the Higgs mass below the upper limit of 144 GeV (95%
confidence level), determined by electroweak precision tests (EWPT). [290]∗∗ .
Low-energy supersymmetry provides such new divergence-canceling physics;
and this is one of the strongest reasons to expect that physics beyond the
SM will be found by the LHC at CERN. However, in the MSSM, the lightest
Higgs particle is naturally constrained to be lighter than ∼135 GeV [152], and
some amount of fine-tuning [291] is actually already claimed to be needed to
fulfill the experimental lower bound of roughly 100 GeV from direct Higgs
searches [152, 290]. This suggestive tension has motivated several studies on
k
∗∗
and Appendix A discussing the UED model. This is for consistency with the notation
of Paper VII and our implementation of the IDM into FeynArts [255]. The translation
√
between the notations is trivial: G0 ≡ −χ3(0) , G± ≡ ±iχ±(0) and vIDM ≡ vUED / 2.
If both doublets develop vacuum expectation values, the physical fields will be linear
combinations with contributions from both H1 and H2 [267].
The central value for the SM of the Higgs mass is 76 GeV from EWPT alone and there is
a lower limit of 114 GeV from direct searches. Note that if direct searches are included,
the upper mass limit on the Higgs increases to 182 GeV according to [290].
114
Inert Higgs Dark Matter
Chapter 8
the theoretical possibilities to allow for large Higgs masses both within supersymmetry and other extensions of the SM (see, e.g., [286, 291] and references
therein). In [286] it was shown that the IDM can allow for a heavy SM-like
Higgs (i.e., h). This was a basic motivation for the model, as it meant that
the need for divergence canceling physics could be pushed beyond the reach of
the upcoming LHC accelerator without any need for fine-tuning. While this
argument of less fine-tuning (or improved naturalness) [286] has been disputed [292], the mere fact that the IDM allows for the SM Higgs mass to be
pushed up to about 500 GeV is interesting in itself, as it might provide a clear
distinction from the SM and the MSSM Higgs (as well as having an impact
on the expected gamma-ray spectrum from annihilation of H 0 s as discussed
in Section 8.2).
To allow for a heavy SM-like Higgs, the upper mass limit of about 144
GeV from electroweak precision tests must be avoided. The so-called PeskinTakeuchi parameters, denoted S, T , and U , are measurable quantities constructed to parameterize contributions (including beyond SM physics) to electroweak radiative corrections, such as the loop-diagram induced contribution
to self-energies of the photon, Z boson, and W boson, and the Weinberg angle [293]. These S, T , and U parameters are defined such that they vanish
for a reference point in the SM (i.e., a specific value for the top-quark and
Higgs masses). Deviations from zero would then signal the existence of new
physics, or set a limit on the Higgs mass when the SM is assumed. Instead
of the T parameter the ρ parameter is sometimes used, which is defined as
ρ = m2W / m2Z cos θw (mW ) . A deviation of ρ from 1 measures how quantum corrections alter the tree level SM link between the W and Z boson
masses [267, 293]. In fact, in most cases T represents just the shift of the ρ
parameter ∆ρ ≡ ρ − 1 = αT .
Electroweak precision measurements of the S, T and U parameters limit
the Higgs boson mass. A heavy h of a few hundred GeV would produce a too
small value for the observable T , whereas the S and U parameters are less
sensitive to the Higgs mass [286], see Fig. 8.1. However, a heavy Higgs can
be consistent with the electroweak precision tests if new physics produce a
compensating positive ∆T . For a mh = 400 − 600 GeV the compensation ∆T
must be ∆T ≈ 0.25 ± 0.1 to bring the value back near the central measured
point and within the experimental limits [286]. In [286] it was found that
neither the S nor the U parameter is affected much by the extra contribution
from the IDM, but that the T parameter is approximately†† shifted according
to:
1
∆T ≈
(mH + − mA0 ) (mH + − mH 0 ) .
(8.7)
24π 2 αv 2
We thus see that a heavy SM Higgs that usually produces too negative values
of ∆T can be compensated for by the proper choice of masses for the inert
††
This approximation is within a few percent accuracy for
1 ≤ mH ± /mH 0 , mH ± /mA0 , mA0 /mH 0 ≤ 3
Section 8.1.
115
The Inert Higgs Model
0.4
mt= 172.7 ± 2.9 GeV
mh = 114...1000 GeV
T
0.2
U=0
0
mt
−0.2
mh
−0.4
−0.4
−0.2
0
68 % CL
0.2
0.4
S
Figure 8.1: Dependence of the S, T parameters on the Higgs mass (mh )
within the standard model. The thick black band marks mh = 400 −
600 GeV. The top quark mass mt range within the experimental bounds.
Figure adapted from [286].
Higgs particles. For example, for a mh = 500 GeV the required compensation
is ∆T ≈0.25±0.1, and the masses of the inert scalar masses in Eq. (8.7) should
satisfy
(mH + − mA0 ) (mH + − mH 0 ) = M 2 ,
M = 120+20
−30 GeV .
(8.8)
This means that a Higgs mass mh of up to about 500 GeV can be allowed in
the IDM if only the inert Higgs masses are such that they fulfill Eq. (8.8).
As the Higgs mass is increased, the quartic scalar interactions become
stronger, and the maximal scale at which perturbation theory can be used
decreases. To have a natural perturbative theory up to, say, 1.5 TeV (which
is about the highest new energy scale we can have without fine-tuning the
Higgs mass [286]) the Higgs mass cannot be heavier than about mh = 600
GeV [286].
More Constraints
There are several other constraints that must be imposed, besides the electroweak precision measurements bounds discussed above. The following constraints on the six free parameters are also used (which are the same constraints as used in Paper VII):
116
Inert Higgs Dark Matter
Chapter 8
• Theoretically, the potential needs to be bounded from below in order to
have a stable vacuum, which requires:
λ1,2 >
λ3 , λ3 + λ4 − |λ5 | >
0, p
−2 λ1 λ2 .
(8.9)
• To trust perturbation theory calculations, at least up to an energy scale
of some TeV, the couplings strengths cannot be allowed to become too
large. Here and in Paper VII we followed the constraints found in [286],
which can be summarized, as a rule of thumb, in that no couplings
should become larger than λi ∼ 1 (see [286] for more details).
• In order not to be in conflict with the observed decay width of the
Z boson we should impose that mH 0 + mA0 & mZ (see Paper VII
and [294]).
• No full analysis of the IDM has been done with respect to existing
collider data from the LEP and the Tevatron experiments. However,
comparison with similar analyses of supersymmetry enable at least some
coarse bounds to be found. The summed mass of H 0 and A0 should
be greater than about 130 GeV [295], or the mass split must be less
than roughly 10 GeV [286, 295]. Similarly, the mass of the charged
Higgs scalars H ± is constrained by LEP data to be above about 80
GeV [295, 296].
• To explain the dark matter by the lightest inert particle (LIP), its relic
abundance should fall in the range 0.094 < ΩCDM h2 < 0.129. See Section 8.2 for more details.
• Direct detection searches of dark matter set limits on scattering cross
sections with nucleons. At tree level, there are two spin-independent
interactions whereby H 0 could deposit kinetic energy to a nuclei q in
Z
h
direct search detectors: H 0 q −
→ A0 q and H 0 q −
→ H 0 q. The former
process, with a Z exchange, is very strong and is forbidden by current
experiment limits [286, 296, 297]. However, this process becomes kinematically forbidden if the mass splitting is more than a few 100 keV, as
typically the kinetic energy of the dark matter candidate H 0 would then
not be enough to produce an A0 (this thus excludes the λ5 → 0 limit;
see Eq. 8.6). With this process kinematically excluded, the signals from
Higgs-mediated scattering is roughly two orders of magnitude below any
current limits. The next generation of detectors could potentially reach
this sensitivity [296].
Also naturalness could be imposed, i.e., parameters should not be tuned
to extreme precision. In Paper VII, we applied the naturalness constraints
found in [286], but to be less restrictive, we relaxed their parameter bounds
Section 8.2.
Inert Higgs – A Dark Matter Candidate
117
by a factor two because of the somewhat arbitrariness in defining naturalness (this constraint is not crucial for any of the general results here or in
Paper VII).
When it comes to the upcoming LHC experiment, the IDM should be seen
in the form of both missing transverse energy and an increased width of the
SM Higgs [286, 294, 295].
8.2
Inert Higgs – A Dark Matter Candidate
The existence of the unbroken Z2 symmetry in the IDM, where the inert Higgs
doublet are attributed negative Z2 -parity and all SM particles have positive
parity, means that none of the inert Higgs particles can directly decay into
only SM particles. The lightest inert particle (LIP) is therefore a (cosmologically) stable particle, which is a first necessity for a dark matter candidate.
Furthermore, the LIP should be electric and color neutral to not violate any
of the strict bounds on charged dark matter [198,199]. The only choices for an
inert Higgs dark matter particle are therefore H 0 or A0 . Although the roles
of H 0 and A0 are interchangeable for all the results, let us for definiteness
choose H 0 as the LIP.
The next crucial step is to see if this H 0 candidate can give the right
relic density to constitute the dark matter. In [296] it was shown that H 0
can constitute all the dark matter if its mass is roughly 10 − 80 GeV (or
above 500 GeV if parameters are particularly fine-tuned). However, this study
was made for SM Higgs masses of 120 and 200 GeV which, although giving
higher gamma rates, deviates from one of the motivation for the model – a
raised Higgs mass [286]. The setup we had in Paper VII is based on a 500
GeV SM Higgs. H 0 relic density calculations were therefore performed. This
was done by implementing the proper Feynman rules from the Lagrangian in
Eq. (8.1) into the Feynman diagram calculator FormCalc [256]. Cross-section
calculations with FormCalc were then interfaced with the DarkSUSY [298] relic
density calculator. This allowed us to accurately calculate the relic density
for any given choice of IDM parameters after imposing existing experimental
constraints. The correct relic density is still roughly obtained for masses in
the range of 40 − 80 GeV, and we will next see why this result is almost
independent of the SM Higgs mass.
Typically the relic density is governed by the cross section for annihilating
two H 0 . For masses mH 0 above the SM Higgs mh (> mZ , mW ) the annihilation channels are given by the diagrams in Fig. 8.2. If mH 0 is above the W
mass, then the cross sections from the middle row diagrams dominate. These
diagrams produce very large annihilation cross sections, and therefore the H 0
relic density becomes too small to constitute the dark matter. For masses
below the W mass, only the diagrams in the bottom line will contribute (as
the heavier W and Z bosons are generically not energetically allowed to be
produced during freeze-out). For these ‘low’ H 0 masses, the tree-level anni-
118
Inert Higgs Dark Matter
Chapter 8
h
h
H
H
h
H
H0
0
H0
H0
λ2L
H
0
W − (Z)
0
H ± (A0 )
W + (Z)
W + (Z)
H0
λL λ1
W − (Z)
W − (Z)
H0
H0
g2
g2
h
H0
λL yf
h
W + (Z)
H0
λL g
f (f ± )
H0
f¯(νf )
A0 (H ± )
H0
h
h
h
0
λL
H
h
H0
0
f (f ± )
Z(W ± )
f¯(νf )
g2
Figure 8.2: Feynman diagrams for different annihilation (and coannihilation) channels for the inert Higgs H 0 . The two top rows show the
contributing diagrams into standard model Higgs h and W ± /Z, respectively. The bottom row shows the H 0 annihilation channel into fermions
(left diagram), and its possible coannihilations processes (right diagram).
Displayed under each diagram is the total coupling strength from the
vertex factors, where g is the gauge coupling strength (g ∼ gY ∼ e),
λL = (λ3 + λ4 + λ5 )/2 and yf = mf /v. In general the H 0 mass is below
the gauge bosons’; otherwise the annihilation strength into W and Z is
too large (if no fine-tuned cancellation occurs) for H 0 to constitute the
dark matter.
hilation rates are small, especially for high SM Higgs masses, and you could
tend to assume that the relic density would be far too high. However, coannihilations with the next-to-lightest inert scalar allow us to reach the correct
relic abundance (the right-hand-side diagram in the bottom row of Fig. 8.2).
It is mainly this coannihilation process which regulates the relic density, and
this process is completely independent of the SM Higgs mass.
This is an interesting aspect of the IDM – that the H 0 mass generically has
to be just below the charged gauge boson mass because the relatively strong
coupling to W + W − would otherwise give a too low relic density to account
for the dark matter – and, as we next will see, this will also affect the indirect
detection signal from gamma rays.
Section 8.3.
8.3
Gamma Rays
119
Gamma Rays
Continuum
The dark matter particle in this model is thus the H 0 , with a mass below
mW . This means that only annihilations into fermions lighter than mH 0 are
accessible at tree level, and the only contributing Feynman diagram is the
bottom left one of Fig. 8.2. The annihilation rate is calculated to be
4m2
Nc πα2 m2f (1 − s f )3/2 (m2H 0 − µ22 )2
vrel σf f¯ =
,
(s − m2h )2 + m2h Γ2h
sin4 θW m4W
(8.10)
√
where Nc is a color factor (which equals 1 for leptons and 3 for quarks), s
is the center of mass energy, α the fine-structure constant, mW the W boson
mass, θW the weak mixing angle, Γh the decay width of h, and mf the final
state fermion mass.
The heaviest kinematically allowed fermion state will dominate the treelevel annihilation channels, since the cross section is proportional to m2f .
Hence, in our case of interest, the bottom quark final states are the most important process at tree level, and with some contributions from charm quarks
and τ pairs. Although the running of lepton masses can be safely neglected,
the QCD strong interaction corrections to quark masses might be substantial,
and we therefore take the leading order correction into account by adjusting
the running quark masses [267, 299] to their values at the energy scale of the
physical process (∼ 2mH 0 ). Quark pairs will, as already described in the case
of the KK and supersymmetric dark matter, hadronize and produce gamma
rays with a continuum of energies. Because of the much harder gamma spectrum from the decay of τ -leptons, these could also contribute significantly at
the highest energies, despite their much lower branching ratio. Pythia (version 6.4) [300] was used to calculate the photon spectrum in the process of
hadronization.
Gamma-Ray Lines
As has been said, the H 0 couplings are relatively strong to W + W − (i.e.,
ordinary gauge couplings), which forces the mass of H 0 to be below mW if it is
to explain the dark matter. Virtual gauge bosons close to threshold could, on
the other hand, significantly enhance loop processes producing monochromatic
photons (see Fig. 8.3). In Paper VII, we showed that this is indeed correct
and found ‘smoking gun’ line signals for the H 0 dark matter from the final
states γγ and, when kinematically allowed, Zγ. This, in combination with
small tree-level annihilation rates into fermions, makes the gamma lines a
most promising indirect detection signal.
Let us see what these important line signals from direct annihilation of
H 0 pairs into γγ and Zγ look like. First of all these spectral lines would show
120
Inert Higgs Dark Matter
W
γ
γ
H0
γ
H0
H0
0
4
W
g
γ
H
H0
γ
H0
γ
H0
4
γ
W
γ
γ
W
W
W
g
H
W
H0
H
W
H
W
g
W
H0
W
g
Chapter 8
γ
4
H0
γ
H
W
W
H0
γ
H0
4
g
4
g
4
γ
W
H0
h
H0
W
λL g
3
γ H0
γ
h
H0
W
W
W
λL g
+ symmetry related
and non-W contributions
γ
3
Figure 8.3: Typical contributing Feynman diagrams for the annihilation
process H 0 H 0 → γγ. Due to unsuppressed couplings to W ± , virtual W ±
in the intermediate states are expected to give the largest contribution to
this process.
up as characteristic dark matter fingerprints at the energies mH 0 and mH 0 −
m2Z /4mH 0 , respectively. The Zγ line might not be strictly monochromatic due
to the Breit-Wigner width of the Z mass, but can still be strongly peaked. The
potential third gamma line from hγ is forbidden for identical scalar particle
annihilation, as in the IDM, due to gauge invariance.
We could also note that when the branching ratio into Zγ becomes large,
the subsequent decay of the Z boson significantly contributes to the continuum
gamma-ray spectrum. The full one-loop Feynman amplitudes were calculated
by using the numerical FormCalc package [256] – after the Feynman rules for
the IDM had been derived and implemented.
To show the strength of the gamma-ray lines and the continuum spectrum for different parameter choices, four IDM benchmark models are defined, shown in Table 8.1. The two models III and IV have a low Higgs mass
and could therefore be directly comparable to the relic density calculations
done in [296]. Annihilation rates, branching ratios and relic densities for these
models are given in Table 8.2. As an illustrative example, Fig. 8.4 shows the
predicted gamma spectrum for model I.
The spectral shape with its characteristic peaks in the hitherto unexplored
energy range between 30 and 100 GeV is ideal to search for with the GLAST
Section 8.3.
Gamma Rays
121
γγ
102
x2 dNγ /dx
Zγ
0
10−2
τ +τ −
bb̄
10−4
0.01
1
0.1
x = Eγ /mH 0
Figure 8.4: The total differential photon distribution from annihilations
of an inert Higgs dark matter particle (solid line). Shown separately are
the contributions from H 0 H 0 → bb̄ (dashed line), τ + τ − (dash-dotted line)
and Zγ (dotted line). This is for the benchmark model I in Table 8.1.
Figure from Paper VII.
experiment [301]. In Fig. 8.5, this is illustrated by showing the predicted fluxes
from a ∆Ω = 10−3 sr region around the direction of the galactic center together
with existing observations in the same sky direction. In this figure, a standard
NFW density profile, as specified in Table 2.1 and with a normalization density
3
of 0.3 GeV/cm at 8.5 kpc, is the underlying assumption for the dark matter
halo for our Galaxy. With the notation of Eq. (2.15), this correspond to
J × ∆Ω ∼ 1 for ∆Ω = 10−3 sr. Processes such as adiabatic compression,
which we discussed in Chapter 2, could very well enhance the dark matter
density significantly near the galactic center. Therefore, the predicted flux
compared to a pure NFW profile could very well be scaled up by a large
‘boost factor’. The boost factors used for the shown signals are also displayed
in Fig. 8.5. Since the continuum part of the expected spectrum is within
the energy range covered by EGRET satellite, there is an upper limit on
Table 8.1: IDM benchmark models. (In units of GeV.)
Model
I
II
III
IV
mh
500
500
200
120
mH 0
70
50
70
70
m A0
76
58.5
80
80
mH ±
190
170
120
120
µ2
120
120
125
95
λ2 ×1 GeV
0.1
0.1
0.1
0.1
122
Inert Higgs Dark Matter
log(Eγ2 Φγ [cm−2 s−1 GeV])
-4
Chapter 8
IDM: NFW, DW~10-3 , ΣEΓ =7%
-5
50 GeV, boost ~104
EGRET:DW=2‰10
-6
-3
70 GeV, boost ~100
-7
-8
-9
HESS:DW=10-5
vity
siti
en
Ts
S
LA
-10
G
0
-1
1
2
3
4
log(Eγ [GeV])
Figure 8.5: Predicted gamma-ray spectra from the inert Higgs benchmark models I and II as seen by GLAST (solid lines). The predicted
gamma flux is from a ∆Ω = 10−3 sr region around the direction of the
galactic center assuming an NFW halo profile (with boost factors as indicated in the figure) and convolved with a 7% Gaussian energy resolution.
The boxes show EGRET data (which set an upper limit for the continuum signal) and the thick line H.E.S.S. data in the same sky direction.
The GLAST sensitivity (dotted line) is here defined as 10 detected events
within an effective exposure of 1 m2 yr within a relative energy range of
±7%. Figure from Paper VII.
the allowed flux in the continuum part of our spectrum. The EGRET data
are taken from [97]. For example, for benchmark model II we find that an
optimistic, but not necessarily unrealistic [101], boost of 104 could be allowed.
In that case, there would be a spectacular γγ line signal waiting for GLAST.
However, to enable detection, boost factors of such magnitudes are not at all
necessary. For H 0 masses closer to the W threshold the γγ annihilation rates
become even higher and in addition Zγ production becomes important. In
Table 8.2: IDM benchmark model results.
Model
I
II
III
IV
v→0
vσtot
3 −1
[cm s ]
1.6 × 10−28
8.2 × 10−29
8.7 × 10−27
1.9 × 10−26
Branching ratios [%]:
γγ
Zγ
bb̄
cc̄
36
33
26
2
29
0.6
60
4
2
2
81
5
0.04
0.1
85
5
+ −
τ τ
3
7
9
10
ΩCDM h2
0.10
0.10
0.12
0.11
Section 8.3.
Gamma Rays
123
fact, these signals would potentially be visible even without any boost at all
(especially if the background is low, as might be the case if the EGRET signal
is a galactic off-center source as indicated in [302]). Also shown in Fig. 8.5 are
the data from the currently operating air Čerenkov telescope H.E.S.S. [258].
The H.E.S.S. data are within a solid angle of only ∆Ω = 10−5 sr, but since
the gamma-ray flux is dominated by a point source in the galactic center, a
larger solid angle would not affect the total flux much. Future air Čerenkov
telescopes with lower energy thresholds and much larger effective area than
GLAST are planned and will, once operating, be able to cover the entire
region of interest for this dark matter candidate.
To go beyond just a few example models, we performed in Paper VII a
systematic scan over the parameters in the IDM, for a SM higgs mass mh =
500 GeV, and calculated the cross section into gamma lines. The constraints
mentioned in Section 8.1 allowed us to scan the full parameter space for dark
matter masses below the W threshold of 80 GeV. The dependence on mH ±
and λ2 is small, and we chose to set these equal to mH 0 + 120 GeV (to
fulfill precision tests) and 0.1, respectively. Importantly, we note that the
right relic density is obtained with a significant amount of early Universe
coannihilations with the inert A0 particle. The resulting annihilation rates
into γγ and Zγ are shown in Fig. 8.6. The lower and upper mH 0 mass bounds
come from the accelerator constraints and the effect on the relic density by the
opening of the W + W − annihilation channel, respectively. For comparison, the
same figure also shows the corresponding annihilation rates for the neutralino
(χ) within MSSM. The large lower-right region is the union of the range of
cross sections covered by the annihilation rates 2σvγγ and σvZγ as obtained
with a large number of scans within generous MSSM parameter bounds with
the DarkSUSY package [298]. The stronger line signal and smaller spread
in the predicted IDM flux are caused by the allowed unsuppressed coupling
to W pairs that appear as virtual particles in contributing Feynman loop
diagrams. In the MSSM, on the other hand, high γγ and Zγ rates are harder
to achieve [301, 303–305], at least while still satisfying both relic density and
LEP constraints for the masses of interest here.
The IDM’s true strength lies in its simplicity and its interesting phenomenology. The lightest new particle in the model typically gives a WIMP
dark matter candidate, once coannihilations are included, and the model allows a SM Higgs mass of up to at least a few hundred GeV without contradicting LEP precision tests. These are two typical features of the model,
but the IDM also shows the typical dark matter properties of having weak
interactions and electroweak masses. The main reasons why this scalar dark
matter model gives such particularly strong gamma lines are that: (1) The
dark matter mass is just below the kinematic threshold for W production in
the zero velocity limit. (2) The dark matter candidate almost decouples from
fermions (i.e., couples only via SM Higgs exchange), while still having ordinary gauge couplings to the gauge bosons. In fact, these two properties by
Inert Higgs Dark Matter
10
Chapter 8
1
H0 H0 → γγ
Nγ (σv) [10
-29
3 -1
cm s ]
124
0
10
H0 H0 → Zγ
10
χχ → Zγ, γγ
-1
-2
10
40
50
60
70
80
WIMP Mass [GeV]
Figure 8.6: Annihilation strengths into gamma-ray lines 2vσγγ (upper
band) and vσZγ (middle band) from the scan over the IDM parameter
space. For comparison the lower-right region indicates the corresponding
results within the minimal supersymmetric standard model as obtained
with the DarkSUSY package [298]. This lower region is the union of Nγ vσ
from χχ → γγ and χχ → Zγ. Figure from Paper VII.
themselves could define a more general class of models for which the IDM is
an attractive archetype because of its simplicity with only six free parameters
(including the SM Higgs mass).
Chapter
9
Have Dark Matter
Annihilations Been
Observed?
Over the past several years, observed anomalies in the spectra from cosmic
photons and anti-particles have been suggested to originate from dark matter
annihilations. One strongly promoted claim of a dark matter annihilation
signal is based on the anomaly that the EGRET experiment found in the diffuse galactic gamma-ray emission. For gamma-ray energies above roughly 1
GeV the data seems to show, in all sky directions, an excess of flux compared
to what is conventionally expected. It has been realized that this excesses
in the spectrum could be due to dark matter annihilations. De Boer and
collaborators [1, 306–308] have therefore proposed a dark matter distribution
in our Galaxy to explain this observed gamma-ray anomaly. Internal consistency of such a dark matter explanation must, however, be investigated.
Generically, the same physical process producing the diffuse gamma rays also
produces antiprotons. In Paper V, we therefore studied this proposed dark
matter model to see if it is compatible with measured antiproton fluxes. Using
current, and generally employed, propagation models for the antiprotons, we
showed that this dark matter explanation is excluded by a wide margin when
checked against measured antiproton fluxes.
9.1
Dark Matter Signals?
Observations that have been proposed to be the product of dark matter annihilations include the cosmic positron spectrum measured by HEAT, the
511 keV emission from the galactic Bulge measured by INTEGRAL, the microwave excess from the galactic Center observed by WMAP, and the diffuse
galactic and extragalactic gamma-ray spectra measured by EGRET. All of
these potential dark matter signals are still very speculative. For a recent
review, and references, see, e.g., [309]. There has also been a claim of a direct
125
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Have Dark Matter Annihilations Been Observed?
Chapter 9
detection signal of dark matter by the DAMA collaboration [227, 228], but
this result is very controversial as other similar experiments have not been
able to reproduce their result [206, 207, 225, 226, 229, 230].
It is beyond the scope of this thesis to go through all these potential hints
of a dark matter signal in detail. We will only focus on scrutinizing (as in
Paper V) the perhaps most strongly promoted claim in the last few years –
that the GeV anomaly in the diffuse galactic gamma-ray spectrum, measured
by the EGRET satellite, could be well explained by a signal from WIMP dark
matter annihilations.
9.2
The Data
Between the years 1991 and 2000, the Energetic Gamma Ray Emission Telescope EGRET [310], onboard the Compton gamma ray observatory, took
data. During this period, it made an all-sky survey of the gamma-ray flux
distribution for energies mainly between 0.03 and 10 GeV.
Diffuse emission from the Milky Way completely dominates the gammaray sky. The main part of the emission originates from interactions of cosmic
rays (mostly protons and electrons) with the gas and radiation fields in the
interstellar medium. Any calculation of the galactic diffuse emission is therefore primarily dependent on the understanding of the cosmic-ray spectra and
interstellar gas and radiation fields throughout our Galaxy. Cosmic rays are
believed to originate mainly from acceleration processes in supernovae, and
propagate through large parts of the Galaxy, whereas the radiation fields
mainly come from the CMB and photons from stars inside the Galaxy. The
physical processes involved in the cosmic-ray interactions, which produce the
gamma rays, are mainly the production and subsequent decay of π 0 , inverse
Compton scattering, and bremsstrahlung.
The first detailed analysis of the diffuse gamma rays was done by Hunter
et al. [311] (using EGRET data in the galactic plane: latitudes |b| ≤ 10◦ in
galactic coordinates). The main assumptions in their analysis were that the
cosmic rays are of galactic origin, that there exists a correlation between the
interstellar matter density and the cosmic-ray density, and that the cosmicray spectra throughout our Galaxy are the same as measured in the solar
vicinity. Their result confirmed that the agreement between the EGRET
observed diffuse gamma rays and the expectations are overall good. However,
at energies above 1 GeV the measured emission showed an excess over the
expected spectrum. This excess is known as the EGRET ‘GeV anomaly’.
Later Strong, Moskalenko, and Reimer [312–315] developed a numerical
code, GALPROP [316], for calculating the cosmic-ray propagation and diffuse
gamma-ray emission in our Galaxy. Their code includes observational data on
the interstellar matter, and a physical model for cosmic-ray propagation. The
model parameters are constrained by the different existing observations, such
as cosmic-ray data on B/C (i.e., the Boron to Carbon ratio, which relates
Section 9.2.
The Data
E 2 × Flux [GeV cm−2 s−1 sr−1 ]
10−4
10−5
127
10−4
10−5
total
total
IC
IC
10−6
10−6
EB
bremss
EB
πο
bremss
10−7 −3
10−2 10−1
10
1
10
102
Energy [GeV]
103
πο
10−7 −3
10−2 10−1
10
1
10
102
103
Energy [GeV]
Figure 9.1: Gamma-ray spectrum models compared to data. Left
panel. The conventional model for the gamma-ray spectrum for the inner galactic disk. The model components are π 0 -decay (dots, red), inverse
compton (dashes, green), bremsstrahlung (dash-dot, cyan), extragalactic
background (thin solid, black), total (thick solid, blue). EGRET data:
red vertical bars. COMPTEL data: green vertical bars. Right panel.
The same but for the optimized model fitting the observed gamma-ray
spectrum. Figures adopted from [315].
secondary to primary cosmic rays). This makes it possible to derive a diffuse gamma-ray spectrum in all sky directions. In the ‘conventional scenario’
in [315] the existence of the EGRET GeV anomaly was confirmed. However,
by allowing for a spatial variation of the electron and proton injection spectra,
it was pointed out that an ‘optimized scenario’ gives a good description of the
diffuse gamma-ray sky [315]. To explain the GeV anomaly, this optimized
model allows for a deviation of the cosmic-ray spectrum (within observational
uncertainties) from what is measured in the solar vicinity. The electron injection spectrum is made slightly harder, with a drastic drop at 30 GeV, and
at the same time normalized upward with a factor of about 5 compared to
the measured spectrum in the solar vicinity. The proton injection spectrum
is also made harder, and the normalization is increased by a factor 1.8 at
100 GeV. The derived spectra, in the conventional and the optimized model,
compared to observational data are shown in Fig. 9.1.
The origin of the potential GeV anomaly is still a matter of debate. There
are mainly three proposed explanations of its origin: (i) it is of conventional
astrophysical origin, like in the mentioned optimized cosmic-ray model or due
to unresolved conventional sources, (ii) it is an instrumental artefact due to
uncertainties in the instrument calibration, or (iii) it is caused by dark matter
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Have Dark Matter Annihilations Been Observed?
Chapter 9
annihilations.
That the GeV anomaly could be due to a systematic instrumental artefact
has recently been discussed by Stecker et al. [317]. They argue that the lack
of spatial structure in the excess related to the galactic plane, galactic center,
anti-center, or halo, indicates that the GeV anomaly above ∼1 GeV is more
likely due to a systematic error in the EGRET calibration. Although not at
all in contradiction with a calibration problem, it is notably that in a recent
reanalysis [318] of the EGRET instrument response the GeV anomaly was
found to be even larger. This reanalysis was done by modifying the GLAST
simulation software, to model the EGRET instrument, and indicated that
previously unaccounted instrumental effects mistakenly lead to the rejection
of some gamma-ray events.
The alternative explanation that the GeV anomaly, in all sky directions, is
a result of dark matter annihilations has been promoted in a series of papers
by de Boer et al., e.g., [1,306–308]. The idea to use the gamma-ray excess as a
dark matter annihilations signal has a long history (at least [97,305,319,320]),
but de Boer et al. have extended this idea to claim that all the diffuse galactic
gamma rays detected above 1 GeV by EGRET, irrespective of the direction,
has a sizeable dark matter contribution.
9.3
The Claim
Specific supersymmetric models have been proposed as examples of viable
candidates that can explain the EGRET GeV anomaly [307]. The precise
choice of dark matter candidate is in itself not crucial, as long as its dark
matter particles are non-relativistic, have a mass between 50 and 100 GeV, and
annihilate primarily into quarks that then produce photons in their process
of hadronization. In these cases, the predicted gamma-ray spectrum has the
right shape to be added to the ‘conventional’ cosmic-ray model in [315] in
order to match the the GeV anomaly; see Fig. 9.2.
The price to pay is, however, a rather peculiar dark matter halo of the
Milky Way, containing massive, disk concentrated rings of dark matter besides
the customary smooth halo. The dark matter distribution de Boer et al.
propose is a profile with 18 free parameters. With the given proposal, a
best fit to the EGRET data is performed. This is possible to do because
gamma rays have the advantage of pointing back directly to their sources
in the Galaxy, and since the gamma-ray spectral shape from dark matter
annihilations is presumed to be known (and distinct from the conventional
background). The sky-projected dark matter distribution can therefore be
extracted from the EGRET observations. The deduced dark matter profile
in [1] has the following main ingredients:
• a triaxial smooth halo in the form of a modified isothermal sphere, but
somewhat flattened in the direction of the Earth and in the z-direction
(i.e., the height above the galactic plane),
The Claim
E 2 × Flux [GeV cm−2 s−1 sr−1 ]
Section 9.3.
10
Dark Matter
Pion decay
Inverse Compton
Bremsstrahlung
EGRET
background
signal
3.5/6
χ2:
χ2 (bg only): 178.8/7
−4
10−5
129
-5
10−1
1
10
102
Energy [GeV]
Figure 9.2: Fit of the shapes of background and dark matter annihilation
signal to the EGRET data in the inner part of the galactic disk. The
light shaded (yellow) areas indicate the background using the shape of
the conventional GALPROP model [315], while the dark shaded (red)
areas are the signal contribution from dark matter annihilation for a 60
GeV WIMP mass. The reduced χ2 [75] for the background only and the
corresponding fit including dark matter is indicated in the figure. Note
the smaller error bars in this figure compared to in Fig. 9.1 – this is due
to the disagreement in how to take into account systematic and correlated
errors (see Paper V for details). Figure adopted from [1].
• an inner ring at about 4.15 kpc with a density falling off as ρ ∼ e−|z|/σz,1 ,
where σz,1 = 0.17 kpc, and
• an outer ring at about 12.9 kpc with a density falling off as ρ ∼ e−|z|/σz,2 ,
where σz,2 = 1.7 kpc.
Fig. 9.3 shows this dark matter profile: The strong concentration of dark
matter to the disk (upper panel), as well as the ring structure of the model
(lower panel), is clearly seen.
A 50-100 GeV dark matter candidate, with a distribution as described,
constitutes the claimed explanation of the EGRET GeV anomaly. In addition, which will become important later, this model also has to boost the
predicted gamma-ray flux, in all sky directions, by a considerable ‘boost factor’ of around 60. With these ingredients, a good all sky fit to the gamma-ray
spectra, as in Fig. 9.2, is obtained.
130
Have Dark Matter Annihilations Been Observed?
3
3
Dark matter density, ρ [MSun/pc ]
0.14
0.12
0.12
0.1
0.1
0.08
0
0.06
−10
−20
−20
−10
0
x [kpc]
10
/pc3]
0.14
Sun
z [kpc]
10
Dark matter density, ρ [MSun/pc ]
ρ [M
20
Chapter 9
Outer
Sun
Inner
Center
0.08
0.06
0.04
0.04
0.02
0.02
0
−5
20
2
0
z [kpc]
5
2
Surface density, Σ0.8 kpc [MSun/pc ]
Surface density, Σ0.8 kpc [MSun/pc ]
20
100
40
/pc2]
80
60
40
Σ
−10
Sun
60
0
[M
80
100
0.8 kpc
y [kpc]
10
20
−20
−20
−10
0
x [kpc]
10
20
20
0
−20
−10
0
x [kpc]
10
20
Figure 9.3: The dark matter distribution in the halo model of de Boer et
al. [1]. Upper panel: The concentration of dark matter along the galactic
disk. The right figure displays the density dependence as a function of the
vertical distance from the galactic plane – at the position of the outer
ring (dotted/green), solar system (solid/black), inner ring (dashed/red)
and galactic center (dash-dotted/blue). Lower panel: The dark matter
surface mass density within 0.8 kpc from the galactic disk. The Earth’s
location is marked with a ×-sign. Figure from Paper V.
9.4
The Inconsistency
Even though the dark matter halo profile by de Boer et al. explains the
EGRET data very well, it is of great importance to check its validity with
other observational data.
Disc Surface Mass Density
Note that the distribution of the dark matter in this model seems very closely
correlated to the observed baryon distribution in the Milky Way – containing
a thin and a thick disk and a central bulge (see, e.g., [321]). Since the dark
halo is much more massive than the baryonic one, one of the first things we
should investigate is whether there is room to place as much unseen matter
Section 9.4.
The Inconsistency
131
Table 9.1: Derived local surface densities Σ|z| , within heights |z|, compared to the amount of dark matter in the model of de Boer et al. [1].
The amount of dark matter exceeds the allowed span for unidentified
gravitational matter in the inner part of the galactic disk (i.e., around
z = 0). [323, 324]
surface density:
Σ50 pc
Σ350 pc
Σ800 pc
Σ1100 pc
dynamical
M⊙ /pc2
9 – 11
36 – 48
59 – 71
58 – 80
identified
M⊙ /pc2
∼9
∼ 34
∼ 46
∼ 49
unidentified
M⊙ /pc2
0–2
2 – 14
13 – 25
9 – 32
DM in [1]
M⊙ /pc2
4.5
19
29
35
in the vicinity of the disk as in the model by de Boer et al.
Observations of the dynamics and density distribution of stars in the disk
give a measure of the gravitational pull perpendicular to the galactic plane.
This can be translated into an allowed disk surface mass density (a method
pioneered in [322]). Observational data from the local surroundings in the
galactic disk sets fairly good limits on the disk surface mass density at the
solar system location [323]. Observations are well described by a smooth dark
matter halo and a disk of identified matter (mainly containing stars, white
and brown dwarfs and interstellar matter in form of cold and hot gases).
Therefore, there is little room for a concentration of dark matter in the disk.
Table 9.1 shows the observed local surface mass density in both identified
components and the total dynamical mass within several heights. Their differences give an estimate of the allowed amount of dark matter in the local
disk – the result is an exclusion of such strong concentrations of unidentified/dark matter as used in the model of [1] to explain the EGRET data. For
example these observations give room for only about 0.01M⊙/pc3 in unidentified matter, which should be compared to the dark matter density of 0.05
M⊙ /pc3 in the model of de Boer et al. [1]. We should keep in mind that the
estimates of the possible amount of dark matter are somewhat uncertain and
that the disk models also have uncertainties of the order of 10% in their star
plus dwarf components and uncertainties as large as about 30% in their gas
components. Also the de Boer et al. halo model could easily be modified to
give a lower disk surface mass density at the solar vicinity. However, such a
modification just to circumvent this problem seems fine-tuned. The model, as
it now stands, already has made fine-tuning modifications, i.e., the rings are
constructed so that they can be very massive, while keeping the local density
low. Figure 9.3 clearly shows that our Sun is already located in a region with
relatively low disk mass surface density.
The dark matter distribution does not at all resemble what would be
132
Have Dark Matter Annihilations Been Observed?
Chapter 9
expected from dissipationless cold dark matter. The distribution should be
much more isotropic than that of the baryonic disk material, which supposedly
forms dissipatively with energy loss but very little angular momentum loss
[325] (see also Chapter 2).
Comparison with Antiproton Data
Any model based on dark matter annihilations into quark-antiquark jets inevitably also predicts a primary flux of antiprotons (and an equal amount of
protons∗ ) from the same jets. As discussed in Section 9.2, the propagation
models of the antiprotons (i.e., cosmic rays) are observationally constrained
to enable reasonably reliably predict antiproton fluxes at Earth. To find out
what the antiproton flux would be in the model proposed by de Boer et al.,
we calculated the expected antiproton fluxes in detail in paper Paper V.
Calculating the Antiproton Flux
There is no need to repeat here the details of the procedure to calculate the
antiproton flux, which can be found in Paper V. The main point is that
we followed, as closely as possible, how de Boer et al. found the necessary
annihilation rates to explain the EGRET data on gamma rays, and then
calculated the antiproton flux based on these same annihilation rates.
For the background gamma flux, we used, as de Boer et al., both the
conventional diffuse gamma-ray background and the optimized background
shown in Fig. 9.1.
For predicting the signals, we used DarkSUSY [281] to calculate the dark
matter annihilation cross sections, gamma-ray and antiproton yields. We
normalized our boost factors to fit the diffuse gamma-ray data from the inner
region of our Galaxy, from where most observational data exist. The dark
matter profile was fixed, and defined by the 18 parameters found in the de
Boer et al. paper [1]. A least χ2 -fit [75] was made to the EGRET data in 8
energy bins in the energy range 0.07 to 10 GeV.
By this procedure we were able to reproduce the result in [1]. That is,
we find a good fit to the EGRET data for WIMP masses between roughly 50
and 100 GeV, and that the required boost factors can be less than the order
of 100. Our best fit χ2 -values for different dark matter masses are shown
in Fig. 9.4 for both the conventional (triangles) and the optimized (circles)
diffuse gamma-ray background.
Note that the fits with the optimized background never get very bad for
higher masses, as there is no real need for a signal with this model. One
should also note that we used relative errors of only 7% for the gamma fluxes,
although the overall uncertainty is often quoted to be 10-15% [315, 326]. The
∗
The protons produced in dark matter annihilations would be totally swamped by the
much larger proton flux from conventional sources.
The Inconsistency
χ
2
Section 9.4.
133
100
Fit with standard background
90
Fit with optimized background
80
70
60
50
40
30
20
10
0
10
2
3
10
10
Neutralino mass (GeV)
4
Figure 9.4: The best χ2 for a fit of background and a dark matter
signal to the EGRET data for different dark matter masses. The fit is
to eight energy bins, where the two free parameters are the normalization
of the dark matter and the background spectrum (i.e., 8-2=6 degrees of
freedom). Figure from Paper V.
true errors are still under debate, and we chose to followed de Boer et al.
using their estimate of 7% for the relative errors in our χ2 -fits. Our results
are, however, not sensitive to this choice (apart from the actual χ2 values
of course). The optimized background model produce in this case a reduced
χ2 ∼ 22/6, which corresponds to a probability of P ∼ 0.1% (P-value) that
the data would give this or a worse (i.e., greater) χ2 value if the hypothesis
were correct [75]. If instead conventional/larger uncertainties of ∼ 15% for
EGRET’s observed gamma fluxes were adopted, the reduced χ2 was decreased
to ∼ 5/6, and a P-value of P = 56%.
Boost factors were determined model-by-model. This means, for each supersymmetric model we demanded it to give an optimized fit to the gammaray spectrum, which thus gave us an optimized boost factor for each model.
Based on the determined boost factors, the antiproton fluxes could be directly
calculated for each model. Sine the boost factor is assumed to be independent of location in the Galaxy, the same boost factor could be used for the
antiproton flux as that found for the gamma rays. To be concrete, the analysis
in Paper V was done within the MSSM, but, as mentioned, the correlation
between gamma rays and antiprotons is a generic feature and the results are
more general. We used DarkSUSY [281] to calculate the antiproton fluxes for
a generous set of supersymmetry models with the halo profile of de Boer et
Have Dark Matter Annihilations Been Observed?
Chapter 9
1
Boost factors < 100
-2 -1
-1
-1
Boosted antiproton flux (m s sr GeV )
134
χ < 10
2
10
-1
10 < χ < 25
2
25 < χ < 60
2
low
er
bo
un
da
2
10
ry
inc
l. u
nc
ert
.
60 < χ
-2
BESS 98 1σ
BESS 98 2σ
T = 0.40-0.56 GeV
2
10
10
Neutralino mass (GeV)
3
Figure 9.5: The antiproton fluxes boosted with the same boost factor
as found for the gamma rays compared to the measured BESS data. The
solid line indicate how far down we could shift the models by choosing
an extreme minimal propagation model (see Section 9.4). Figure from
Paper V.
al. Once we had the calculated antiproton fluxes at hand, we could compare
it with antiproton measurements.
We chose to primarily compare antiproton data in the energy bin at 0.40–
0.56 GeV and using BESS (Balloon-borne Experiment with a Superconducting
Spectrometer) data from 1998 [327]. The reason for using the BESS 98 data
is because the solar modulation parameter is estimated to be relatively low
(φF = 610 MV) at this time, and the low-energy bin correspond to an energy
range where the signal is expected to be relatively high compared to the
background.
Figure 9.5 shows (using the conventional background) the antiproton flux
when enlarged with the boost factor found from the fit to the EGRET data.
Models with the correct mass, i.e., low χ2 , clearly overproduce antiprotons.
In the figure, we have imposed a cut on the boost factor, to only allow models
with reasonably low boost factors. To be conservative, we have allowed the
boost factor to be as high as 100, which is higher than expected from recent
analyses (see e.g., [116, 328]). It is fairly evident that all the models with
good fits to the EGRET data give far too high antiproton fluxes. We find
that low-mass models (masses less than 100 GeV) overproduce antiprotons by
Section 9.4.
The Inconsistency
135
a factor of around ten. Higher-mass models (above a few hundred GeV) have
a lower antiproton rate, so the overproduction is slightly less. However, they
hardly give any improvements to the fits to the gamma-ray spectrum.
Other dark matter candidates, like KK dark matter, would also give a
similar behavior since the gamma rays and antiprotons are so correlated.
However, for, e.g. KK dark matter in the UED model one would not improve
the fits to EGRET data as only heavier models are favored by the relic density
constraint. For the IDM the boost factor would have to be very large in all
sky directions, at least as long as its gamma-ray continuum part is strongly
suppressed by having only heavy Higgs coupling to quarks. In fact, since
antiprotons and gamma rays are so strongly correlated in general, our results
should be valid for any typical WIMP.
Antiproton Propagation Uncertainties
We could be worried about the well-known fact that the antiproton flux from
dark matter annihilations is usually beset with large uncertainties relating
to unknown diffusion parameters combined with uncertainties in the halo
distribution. In [329] it is pointed out that the estimated flux may vary by
almost a factor of 10 up or down, for models that predict the correct cosmicray features. However, the results of such a large uncertainty are only valid
for a relatively smooth halo profile, where much of the annihilation occurs
away from the disk and propagation properties are less constrained.
The main reason for the large uncertainties found in [329] is a degeneracy
(for the secondary signal) between the height of the diffusion box and the
diffusion parameter. If we increase the height of the diffusion box, we would
get a larger secondary signal because cosmic rays can propagate longer in
the diffusion box before escaping. This can be counterbalanced by increasing
the diffusion coefficient to make the cosmic rays diffuse away faster from
the galactic disk. Hence, for the secondary signal, which originates in the
galactic disk, we can get acceptable fits by changing these parameters. For
the dark matter in a smooth halo the effect of these changes is different.
If we increases the height of the diffusion box, we also increase the volume
in which annihilations occur, and the total flux increases more than can be
counterbalanced by an increase in the diffusion coefficient. This is, however,
not true to the same extent in the de Boer et al. profile where most of the
dark matter is concentrated to the disk.
To investigate this effect on the antiproton flux from variations in the
propagation models, we recalculated the expected antiproton fluxes with the
propagation code in [329]. For illustration, let us look at a supersymmetric
configuration for which the agreement with the EGRET data is good – a reduced χ2 of roughly 3/6, a neutralino mass of 50.1 GeV and a derived boost
factor of 69. By varying the propagation parameters to be as extreme as allowed from other cosmic-ray data (details on what this correspond to can be
136
Have Dark Matter Annihilations Been Observed?
Chapter 9
Figure 9.6: A supersymmetric model that provides a good fit to the
EGRET data has been selected and its antiproton yield has been carefully derived. It is featured by the red solid line in the case of the median
cosmic-ray configuration. Predictions spread over the yellow band as the
cosmic-ray propagation parameters are varied from the minimal to maximal configurations (see Table 2 in Paper V). The long-dashed black curve
is calculated with DarkSUSY for a standard set of propagation parameters [330]. The narrow green band stands for the conventional secondary
component. As is evident from this figure, the antiproton fluxes for this
example model clearly overshoots the data. Figure from Paper V.
found in Paper V), we get the range of allowed predicted antiproton fluxes.
In Fig. 9.6, the yellow band delimits the whole range of extreme propagation
model configurations. This gives an indication on how well the flux of neutralino induced antiprotons can be derived in the case of the de Boer et al.
dark matter distribution. The red solid curve is a median cosmic-ray propagation model configuration which could be compared to the long–dashed black
curve computed with the DarkSUSY package † . The conventional secondary
Section 9.5.
The Status to Date
137
background, producing antiprotons, is indicated as the narrow green band as
it was derived in [331] from the observed B/C ratio. For maximal cosmic-ray
configuration the dark matter induced antiproton flux is observed to increase
by a factor of 2.5, and for the minimal configuration a decrease of a factor
of 2.6. The total uncertainty in the expected dark matter induced antiproton
flux corresponds therefore to an overall factor of only ∼ 6.5, to be compared
to a factor of ∼ 50 in the case of an NFW dark matter halo.
Even after these propagation uncertainties are included the yellow uncertainty band is at least an order of magnitude above the secondary green
component. This was for one example model, but this argument can be made
more general. In Fig. 9.5, a solid line represents how far down we would shift
the antiproton fluxes by going to the extreme minimal propagation models.
As can be seen, the antiprotons are still overproduced by a factor of 2 to 10
for the models with good fits to EGRET data. It is therefore difficult to see
how this dark matter interpretation of the EGRET data could be compatible
with the antiproton measurements.
Our conclusion is therefore that the proposal of de Boer et al. [1] is not
viable, at least not without further fine-tuning of the model or by significant
changes in generally employed propagation models.
9.5
The Status to Date
Currently, the uncertainties regarding the data of the EGRET GeV anomaly
can still be debated. In any case, the dark matter model proposed by de Boer
et al. to explain the potential GeV anomaly has a problem of severe overproduction of antiprotons. With the usual cosmic propagation models there does
not seem to be a way out of this problem. However, attempts with anisotropic
diffusion models have been proposed by de Boer et al. as a way to circumvent
these antiproton constraints. These models feature anisotropic galactic winds
which transport charged particles to outer space, and places our solar system
in an underdense region with overdense clouds and magnetic walls causing
slow diffusion. In such a scenario, it is claimed that the antiproton flux due
to dark matter annihilations could very well be decreased by an order of magnitude [308, 332] (see also [333, 334]). This seems somewhat contrived at the
moment, especially since the dark matter distribution itself is not standard.
There are, of course, also other ways to tune the model to reduce the antiproton flux. For example, if we take away the inner ring, the antiproton fluxes
goes down a factor of 2.0. This indicates that fine-tuning the distribution even
more, by having the high dark matter density as far away as possible from
our solar system, could potentially reduce the antiproton significantly without reducing the gamma-ray flux. Remember though that the disk density
(although notably still quite uncertain) has already been tuned to have a dip
†
Because diffusive reacceleration is not included in DarkSUSY, the flux falls more steeply
close to the neutralino mass than in the median model of [329].
138
Have Dark Matter Annihilations Been Observed?
Chapter 9
in order not to be in large conflict with stellar motion measurements in the
solar neighborhood. All such attempts to avoid the antiproton contradiction
should be judged against optimized cosmic-ray propagation models, that also
have been shown to enable an explanation of the GeV anomaly, without the
need of dark matter. The actual density profiles of the rings, with exponential
fall-off away from the disk, is also not what is expect from WIMP dark matter,
although mergers of dwarf galaxies could leave some traces of minor ring-like
dark matter structures in the galactic plane [308]‡ . In fact, the de Boer et
al. model seems most likely to be a fit of the baryonic matter distribution of
our Galaxy, and not the dark matter density. This is not at all to say that
there cannot be any hidden dark matter annihilation signal in the gamma-ray
sky, instead we have shown that standard propagation models and antiproton
measurements constrain the possibilities for an all-sky WIMP dark matter
signal in gamma rays.
What can be said for certain is that to date there is no dark matter
annihilation signal established beyond reasonable doubt.
Many open questions, especially regarding the EGRET GeV anomaly, will
presumably be resolved once the GLAST satellite has data of the gamma-ray
sky. For example, we will then know more about the actual disk concentration
of the gamma-ray distribution, if the GeV anomaly persists, and how the
spectrum is continued up to higher energies. Before GLAST, the PAMELA
satellite [241] will collect data on, e.g., antiprotons and positrons, which could
even further enhance the understanding of the cosmic-ray sky and potential
dark matter signals in it.
‡
It has been put forward that rotation curve [1] and gas flaring [335] data support the
existence of a very massive dark matter ring at a galactocentric distance of about 10-20
kpc. However, this is controversial and is, e.g., not supported by the derived rotation
curve in [336] and the recent analysis in [337].
Chapter
10
Summary and Outlook
The need to explain a wide range of cosmological and astrophysical phenomena
has compelled physicists to introduce the concept of dark matter. Once dark
matter, together with dark energy (in the simplest form of a cosmological
constant), is adopted, conventional laws of physics give a remarkably good
description of a plethora of otherwise unexplained cosmological observations.
However, what this dark matter is made of remains one of the greatest puzzles
in modern physics.
Any experimental signals that could help to reveal the nature of dark matter are therefore sought. This could either be in the form of an observational
discovery of an unmistakable feature or the detection of several different kinds
of signals that all can be explained by the same dark matter model.
In this thesis I have presented the background materials and research results regarding three different types of dark matter candidates. These candidates all belong to the class of weakly interacting massive particles, and are:
the lightest Kaluza-Klein particle γ (1) , the lightest neutralino within supersymmetry χ, and the lightest inert Higgs H 0 . They could be characterized as
dark matter archetypes for a spin 1, 1/2, and 0 particle, respectively.
The first dark matter candidate that was studied originates from the fascinating possibility that our world possesses more dimensions than the observed
four spacetime dimensions. After a discussion of multidimensional universes –
where it was concluded that a stabilization mechanism for extra dimensions is
essential – the particle content within the particular model of universal extra
dimensions (UED) was investigated. The cosmologically most relevant aspect
of this UED model is that it naturally gives rise to a dark matter candidate
γ (1) (≈ B (1) ). This candidate is a massive Kaluza-Klein state of an ordinary
photon – i.e. a photon with momentum in the direction of an extra dimension. In this thesis, prospects for indirect dark matter detection by gamma
rays from annihilating γ (1) particles were explored. It was discovered that by
internal bremsstrahlung, from charged final state fermions, very high-energy
gamma rays collinear with the fermions are frequently produced. An expected
signature from γ (1) annihilations is therefore a gamma-ray energy spectrum
139
140
Summary and Outlook
Chapter 10
that is very prominent at high energies, and has a characteristic sharp cutoff
at the energy equal to the dark matter particle’s mass. As a by-product it was
realized that this is a quite generic feature for many dark matter candidates.
The amplitude of a potentially even more characteristic signal, a monochromatic gamma line, was also calculated. Although in principle a very striking
signal, its feebleness seems to indicate that a new generation of detectors are
needed for a possible detection.
The second dark matter candidate studied was the neutralino, appearing
from a supersymmetric extension of the standard model. Also here it was
found that internal bremsstrahlung in connection to neutralino annihilation
can give characteristic signatures in the gamma-ray spectrum. The reason for
these gamma-ray signals is somewhat different from that of the UED model.
High-mass neutralinos annihilating into W ± can give rise to a phenomenon
similar to the infrared divergence in quantum electrodynamics, that significantly enhances the cross section into low-energy W ± bosons and high-energy
photons. Another possible boosting effect is that the (helicity) suppression of
neutralinos annihilating into light fermions will no longer be present if a photon accompanies the final state fermions. For neutralino annihilations, both
these effects result in the same type of characteristic signature, a pronounced
high energy gamma-ray spectrum with a sharp cutoff at the energy equal to
the neutralino mass.
The third candidate arises from considering a minimal extension of the
standard model by including an additional Higgs doublet. With an unbroken Z2 symmetry, motivated by experimental constraints on neutral flavor
changing currents, an inert Higgs emerges that constitutes a good dark matter candidate. This dark matter candidate turns out to have the potential
to produce a ‘smoking gun’ signal in the form of a strong monochromatic
gamma-ray line in combination with a low gamma-ray continuum. Such a
tremendous signal could be waiting just around the corner and be detected
by the GLAST satellite, to be launched later this year.
The anticipated absolute fluxes of gamma rays from dark matter annihilations, and thus the detection prospects of predicted signals, are still accompanied by large uncertainties. This is because of the large uncertainty
in the dark matter density distribution. To learn more about the expected
dark matter density distribution, we used numerical N -body/hydrodynamical
simulations. The effect of baryons, i.e., the ordinary matter particles constituting the gas and stars in galaxies, upon the dark matter was found to be
significant. In the center of the galaxy the dark matter gets pinched, and the
overall halo changes shape from somewhat prolate to more oblate. The pinching of dark matter in the center of galaxies could cause a boost of dark matter
annihilations which might be needed for detection of annihilation signals.
The claimed observation of a dark matter annihilation signal by de Boer et
al. was also scrutinized. It was concluded that to date there is no convincing
observational evidence for dark matter particle annihilations.
141
If dark matter consists of massive and (weakly) interacting particles, there
is a chance that in the near future some or many of the existing and upcoming
experiments will start to reveal the nature of the dark matter. The types of
signal presented in this thesis have the potential to contribute significantly to
this process as they might be detected both with the GLAST satellite and with
existing and upcoming ground based air Čerenkov telescopes. Other experiments – like PAMELA, measuring anti-particle fluxes – will simultaneously
continue to search the sky for dark matter signals. Once the Large Hadron
Collider at CERN is running, if not earlier, many theories beyond standard
model physics will be experimentally scrutinized to see whether they give a
fair description of how nature behaves and whether they are able to explain
dark matter.
Appendix
A
Feynman Rules:
The UED model
This Appendix collects a complete list of Feynman rules for all physical fields
and their electroweak interactions in the five-dimensional UED model. The extra dimension is compactified on a S 1 /Z2 orbifold (the fields’ orbifold boundary conditions are specified in Chapter 5). All interaction terms are located in
the bulk and are KK number conserving. In principle, radiative corrections at
the orbifold fixpoints could give rise to interactions that violate KK number
conservation [231]. These types of interactions are loop-suppressed, and it is
self-consistent to assume they are small; they will therefore not be considered
here.
A.1
Field Content and Propagators
In the four-dimensional theory, the mass eigenstates of vector fields are A(0) µ ,
(0) µ
Z (0) µ , and W±
at the SM level, whereas their KK level excitations are
(n) µ
(n) µ
3 (n) µ
B
,A
, and W±
(n ≥ 1) (for the heavy KK masses the ‘Weinberg’
angle is taken to be zero as it is essentially driven to zero). Depending on
the gauge, there will also be an unphysical ghost field c associated to every
vector field. From the Higgs sector, there is one physical scalar h(0) present
(n)
(n)
at the SM-level, and three physical scalars h(n) , a0 and a± at each KK
level. Depending on the gauge, the Higgs sector also contains the unphysical
(n)
(n)
(n)
Goldstone bosons χ3 (0) and χ± (0) at SM-level, and G0 , G± , and A5 at
each KK level. These generate the longitudinal spin modes for the massive
(0)
vector fields. Finally, for every SM fermion ξs there are two towers of KK
(n)
(n)
fermions ξs and ξd – except for the neutrinos, which only appear as a
component in the SU (2) doublet, both at zero and higher KK levels.
Following the conventions in [3], the propagator for an internal particle is
143
144
Feynman Rules: The UED model
Appendix A
given by
i
,
q 2 − m2 + iǫ
i(6q + m)
,
2
q − m2 + iǫ
−i
qµqν
µν
η − 2
(1 − ξ)
q 2 − m2 + iǫ
q − ξm2
for scalars :
for fermions :
for vectors :
(A.1)
(A.2)
(A.3)
where ǫ is a small positive auxiliary parameter, which is allowed to tend to zero
after potential integrations over q, and ξ is a gauge parameter (ξ = 0 being
the Landau gauge and ξ = 1 being the Feynman-’t Hooft gauge). The same
propagators apply also for the
√ unphysical ghosts and Goldstone bosons, but
with the masses replaced by ξmV (where mV is the mass of the associated
vector boson).
In Chapter 5, mass eigenstates were expressed in the fields appearing directly in the Lagrangian, see Eq. (5.20), (5.21), (5.27), and (5.53). Sometimes
the inverse of these relationships are also convenient to have at hand:
A3M
=
sw AM + cw ZM
(A.4a)
BM
=
cw AM − sw ZM
(A.4b)
Z5
(n)
=
χ3 (n)
=
± (n)
=
W5
χ± (n)
=
ψs(n)
=
(n)
ψd
A.2
=
mZ
(n)
a −
(n) 0
MZ
M (n)
(n)
a
(n) 0
+
(n)
a
(n) ±
−
(n)
a
(n) ±
+
MZ
mW
MW
M (n)
MW
M (n)
(n)
(A.5a)
(n)
(A.5b)
(n)
(A.5c)
(n)
(A.5d)
(n)
G0
(n)
G0
(n)
G±
(n)
G±
MZ
mZ
MZ
M (n)
MW
mW
MW
(n)
sin α(n) ξd − cos α(n) γ 5 ξs(n)
(n)
cos α(n) ξd
+
sin α(n) γ 5 ξs(n)
(A.6a)
(A.6b)
Vertex Rules
All vertex rules can be expressed in terms of five independent quantities, e.g.,
the electron charge e (= −|e|), the Weinberg angle θw , the W gauge boson
mass mW , the Higgs mass mh , and the compactification size of the extra
Section A.2.
Vertex Rules
145
dimension R. To shorten some of the vertex-rule expressions, the following
shorthand notations will be frequently used:
cw
sw
≡
≡
M (1)
≡
(1)
≡
MX
cos(θw ) ,
sin(θw ) ,
(A.7a)
(A.7b)
1/R ,
q
2
M (1) + m2X ,
(A.7c)
(A.7d)
together with the quantities:
mZ
gY
g
λ
= mW /cw ,
(A.8a)
= e/cw ,
(A.8b)
= e/sw ,
g 2 m2h
=
.
8m2W
(A.8c)
(A.8d)
In addition, all the fermion Yukawa couplings are free parameters, i.e., the
fermion masses mξ . In the case of charged gauge boson interactions, there are
also additional independent parameters in the Cabibbo-Kobayahi-Maskawa
(CKM) Vij matrix∗ , whose elements contain information on the strengths of
flavor-changing interactions. All momenta are ingoing in the vertex rules.
All vertex rules in the UED model for the physical fields, up to the first
KK level, will now follow.† Additional vertex rules, including unphysical
Goldstone and ghost fields, are displayed if they were explicitly used in our
numerical calculation of the one-loop process B (1) B (1) → γγ, Zγ which was
done in the Feynman-’t Hooft gauge (ξ → 1) in Paper III. In unitarity gauge
(ξ → ∞) all such unphysical fields disappear.
Vector-Vector-Vector Vertices
These couplings between tree vector fields originating from the cubic terms
in gauge fields that appear in the field strength part of the Lagrangian (5.9).
The following four-dimensional Feynman vertex rules are derived:
∗
In the SM Vij is a 3×3 complex unitary matrix. Unitarity (9 conditions) and the fact
that each quark field can absorb a relative phase (5 parameters reduction) leaves only
2 × 32 − 9 − 5 = 4 free parameters in the CKM matrix.
†
Some of these vertex rules can also be found in [214, 338, 339]. Note: A few typos were
identified in [214, 338, 339] [Personal communication with Graham Kribs and Torsten
Bringmann].
146
Feynman Rules: The UED model
Appendix A
V2µ
q2
q1
V1ρ
igV1 V2 V3 (q1 − q2 )ν · η ρµ
+(q2 − q3 )ρ η µν + (q3 − q1 )µ η ρν
q3
V3ν
where
gA(0) W (1,0) W (1,0)
=
−e
gZ (0) W (0,1) W (0,1)
=
−cw g
(A.10)
gA(1) W (0,1) W (1,0)
=
−g
(A.11)
−
+
−
+
3
−
+
(A.9)
Vector-Vector-Scalar Vertices
Couplings between two gauge fields and one scalar field appear both in the
field strength part of the Lagrangian (5.9) and the kinetic part of the Higgs
Lagrangian (5.17). The following four-dimensional Feynman vertex rules are
derived:
V1µ
S
igV1 V2 S · η µν
V2ν
where
g
mZ
cw
g mW
(A.12)
gZ (0) Z (0) h(0)
=
gW (0) W (0) h(0)
=
gZ (0) W (1) a(1)
=
±ig mZ
gZ (0) A3 (1) h(1)
gZ (0) B (1) h(1)
=
=
g mZ
−gY mZ
gW (0) A3 (1) a(1)
=
∓ig mW
gW (0) B (1) a(1)
=
∓igY mW
+
−
±
±
±
∓
∓
∓
(A.13)
M (1)
(1)
MW
(A.14)
(A.15)
(A.16)
M (1)
(1)
MW
M (1)
(1)
MW
(A.17)
(A.18)
Section A.2.
Vertex Rules
M (1)
147
gW (0) W (1) a(1)
=
±ig mW
gW (0) W (1) h(1)
=
g mW
(A.20)
gA3 (1) A3 (1) h(0)
gA3 (1) B (1) h(0)
=
=
(A.21)
(A.22)
gB (1) B (1) h(0)
=
gW (1) W (1) h(0)
=
g mW
−gY mW
s2
g 2w mW
cw
g mW
±
∓
±
0
∓
−
+
(A.19)
(1)
MZ
(A.23)
(A.24)
In unitarity gauge (ξ → ∞) unphysical scalar fields are not present. However, in a general gauge there are also additional vertices including unphysical
scalar fields. I here choose to include only those vertex rules that were explicitly used in Paper III to calculate the one-loop process B (1) B (1) → γγ in the
Feynamn-’t Hooft gauge (ξ → 1):
gA(0) W (0) χ∓ (0)
=
∓ie mW
(A.25)
gZ (0) W (0) χ∓ (0)
=
±igs2w mZ
(A.26)
gA(0) W (1) G(1)
=
∓ie MW
gW (0) B (1) G(1)
=
∓igY mW
gW (1) B (1) χ∓ (0)
=
∓igY mW
±
±
±
±
∓
∓
±
(1)
(A.27)
mW
(A.28)
(1)
MW
(A.29)
Vector-Scalar-Scalar Vertices
Couplings between one gauge field and two scalar fields appear both in the
field strength part of the Lagrangian (5.9) and the kinetic part of the Higgs
Lagrangian (5.17). The following four-dimensional Feynman vertex rules are
derived:
S2
q1
S1
Vµ
igV S1 S2 · (q1 − q2 )µ
q2
where
gA(0) a(1) a(1)
+
−
= e
(A.30)
148
Feynman Rules: The UED model
Appendix A
2
gZ (0) a(1) a(1)
+
−
gZ (0) h(1) a(1)
0
gW (0) a∓(1) a(1)
0
±
1
M (1)
mW 2
(gcw − gY sw )
+ gcw
2
(1)
(1) 2
2
MW
MW
g M (1)
= −i
2cw MZ(1)
!
2
m2W
g M (1)
1−2
= ∓
(1)
(1)
(1) 2
2 MW
MZ
M
=
(A.31)
(A.32)
(A.33)
W
gW (0) a∓(1) h(1)
±
gA3 (1) a(1) h(0)
0
gA3 (1) G(1) h(0)
0
gB (1) a(1) h(0)
0
gW (1) a(1) h(0)
±
∓
g M (1)
(1)
2 MW
g M (1)
= i
2 MZ(1)
g mZ
= i
2 MZ(1)
= i
gY M (1)
2 MZ(1)
g M (1)
= i
(1)
2 MW
= −i
(A.34)
(A.35)
(A.36)
(A.37)
(A.38)
In unitarity gauge (ξ → ∞) unphysical scalar fields are not present. However, in a general gauge there are also additional vertices including unphysical
scalar fields. I here choose to include only those vertex rules that were explicitly used in Paper III to calculate the one-loop process B (1) B (1) → γγ in the
Feynamn-’t tHooft gauge (ξ → 1):
gA(0) χ+ (0)χ− (0)
=
gZ (0) χ+ (0) χ− (0)
=
gZ (0) χ3 (0) h(0)
=
gW (0) χ∓ (0) h(0)
=
gW (0) χ∓ (0) χ3 (0)
=
gA(0) G(1) G(1)
=
gB (1) χ±(0) a(1)
=
gB (1) χ±(0) G(1)
=
gB (1) χ3 (0) h(1)
=
±
±
+
−
∓
∓
e
cw
sw
g
− gY
2
2
g
i
2cw
g
i
2
g
∓
2
e
gY M (1)
(1)
2 MW
g Y mW
±
(1)
2 MW
gY
−i
2
±
(A.39)
(A.40)
(A.41)
(A.42)
(A.43)
(A.44)
(A.45)
(A.46)
(A.47)
Section A.2.
Vertex Rules
gB (1) G(1) h(0)
=
gW (1) G(1) h(0)
=
0
±
∓
g Y mZ
2 MZ(1)
g mW
i
(1)
2 MW
−i
149
(A.48)
(A.49)
Scalar-Scalar-Scalar Vertices
Couplings between tree scalar fields appear in the kinetic and potential in
the Higgs Lagrangian (5.17). The following four-dimensional Feynman vertex
rules are derived:
S1
S3
igS1 S2 S3
S2
where
gh(0) h(0,1) h(0,1)
gh(0) a(1) a(1)
+
−
gh(0) a(1) a(1)
0
0
=
3 m2
− g h
2 mW
=
−g mW
=
g
− mZ
cw
(A.50)
1+
!
2
1 m2h M (1)
2 m2W M (1) 2
W
2
1 m2h M (1)
1+
2 m2Z M (1) 2
Z
!
(A.51)
(A.52)
In unitarity gauge (ξ → ∞) unphysical scalar fields are not present. However, in a general gauge there are also additional vertices including unphysical
scalar fields. I here choose to include only those vertex rules that were explicitly used in Paper III to calculate the one-loop B (1) B (1) → γγ process
(which was done numerically in the Feynamn-’t Hooft gauge (ξ → 1):
gh(0) χ+ (0) χ− (0)
=
gh(0) χ3 (0) χ3 (0)
=
gh(0) a(1) G(1)
=
gh(0) G(1) G(1)
=
±
±
∓
∓
g
2
g
−
2
−
m2h
mW
m2h
mW
g (1)
M
2
(A.53)
(A.54)
1−
m2h
g
m2h
− mW
(1) 2
2
M
W
(1) 2
MW
!
(A.55)
(A.56)
150
Feynman Rules: The UED model
Appendix A
Fermion-Fermion-Vector Vertices
Couplings between two fermions and the gauge fields appear in the covariant
derivative of the fermion fields, see the Lagrangian in (5.44). The notation
below is that indices i and j indicate which SM generation a fermion belongs
to, and Vij is the Cabibbo-Kobayahi-Maskawa matrix. In the case of leptons
Vij simply reduces to δij (and remember that there are no singlet neutrinos;
QU = Ys,U = 0 for neutrinos.) Furthermore, ξ = U, D denotes the mass eigenstates for up (T3 = +1/2) and down (T3 = −1/2) type quarks, respectively.
Electric charge is defined as usual as Q ≡ T3 + Yd = Ys . With these additional
notations, the following four-dimensional Feynman vertex rules are derived:
ξ2
Vµ
iγ µ gV ξ̄1 ξ2
ξ¯1
where
gA(0) ξ̄(0) ξ(0)
=
Qe
(A.57)
gA(0) ξ̄(1) ξ(1)
=
Qe
(A.58)
gZ (0) ξ̄(0) ξ(0)
=
(A.59)
gZ (0) ξ̄(1) ξ(1)
=
gZ (0) ξ̄(1) ξ(1)
=
gZ (0) ξ̄(1) ξ(1)
=
gA(1) ξ̄(0) ξ(1)
=
(T3 gcw − Yd gY sw ) PL − Ys gY sw PR
g T3 cos2 α(1) − Ys s2w
cw
g T3 sin2 α(1) − Ys s2w
cw
g
T3 sin α(1) cos α(1) γ 5
cw
T3 g cos α(1) PL
gA(1) ξ̄(0) ξ(1)
=
−T3 g sin α(1) PL
(A.64)
gB (1) ξ̄(0) ξ(1)
=
Ys gY sin α(1) PR + Yd gY cos α(1) PL
(A.65)
gB (1) ξ̄(0) ξ(1)
=
(A.66)
gW (0) Ū (0) D(0)
=
gW (0) Ū (1) D(1)
=
gW (0) Ū (1) D(1)
=
gW (0) Ū (1) D(1)
=
−Ys gY cos α(1) PR − Yd gY sin α(1) PL
g
√ Vij PL
2
g
(1)
√ Vij cos α(1)
Ui cos αDj
2
g
(1)
√ Vij sin α(1)
Ui sin αDj
2
g
(1) 5
√ Vij cos α(1)
Ui sin αDj γ
2
s,d s,d
d
s
s
3
d
s
d
d
s
3
d
s
+
+
+
+
i
d,i
s,i
d,i
j
d,j
s,j
s,j
(A.60)
(A.61)
(A.62)
(A.63)
(A.67)
(A.68)
(A.69)
(A.70)
Section A.2.
Vertex Rules
gW (0) Ū (1) D(1)
=
gW (1) Ū (0) D(1)
=
gW (1) Ū (0) D(1)
=
gW (1) Ū (1) D(0)
=
gW (1) Ū (1) D(0)
=
+
+
+
+
+
s,i
i
i
d,i
s,i
g
(1) 5
√ Vij sin α(1)
Ui cos αDj γ
2
g
√ Vij cos α(1)
Dj PL
2
g
(1)
− √ Vij sin αDj PL
2
g
√ Vij cos α(1)
Ui PL
2
g
(1)
− √ Vij sin αUi PL
2
d,j
d,j
s,j
j
j
151
(A.71)
(A.72)
(A.73)
(A.74)
(A.75)
The remaining vertex rules are given by gV ξ̄1 ξ2 = gV∗ † ξ̄2 ξ1 .
Fermion-Fermion-Scalar Vertices
Couplings between two fermions and a scalar originate from the covariant
derivative in Eq. (5.44) and the Yukawa couplings (5.46). The same notation
as for the ‘fermion-fermion-vector’ couplings are used. The following fourdimensional Feynman vertex rules are derived:
ξ2
S
igS ξ̄1 ξ2
ξ¯1
where
gh(0) ξ̄(0) ξ(0)
=
gh(0) ξ̄(1) ξ(1)
=
gh(0) ξ̄(1) ξ(1)
=
gh(0) ξ̄(1) ξ(1)
=
gh(1) ξ̄(0) ξ(1)
=
gh(1) ξ̄(0) ξ(1)
=
ga(1) ξ̄(0) ξ(1)
=
d
s
d
d
s
s
d
s
0
d
mξ
(A.76)
2mW
mξ
−g
sin α(1) cos α(1)
(A.77)
mW
mξ
−g
sin α(1) cos α(1)
(A.78)
mW
mξ −g
1 − 2 cos2 α(1) γ 5
(A.79)
2mW
mξ
−g
sin α(1) PR + cos α(1) PL
(A.80)
2mW
mξ
g
cos α(1) PR − sin α(1) PL
(A.81)
2mW
n
o
mZ
i (1) (T3 gcw − Yd gY sw ) cos α(1) PR + Yd gY sw sin α(1) PL
MZ
mξ M (1) (1)
(1)
−igT3
sin
α
P
−
cos
α
P
(A.82)
R
L
mW MZ(1)
−g
152
Feynman Rules: The UED model
ga(1) ξ̄(0) ξ(1)
s
0
=
i
mZ n
(1)
MZ
(T3 gcw − Yd gY sw ) sin α(1) PR + Yd gY sw cos α(1) PL
mξ M (1) (1)
(1)
cos
α
P
−
sin
α
P
R
L
mW MZ(1)
mDj
g M (1)
mUi
(1)
(1)
δ
sin
α
P
−
cos
α
P
−i √
Dj R
Dj L
(1) ij
mW
mW
2 MW
g mW
(1)
+i √
V cos αDj PR
(1) ij
2 mW
(1)
mDj
mUi
g M
(1)
(1)
i√
δ
cos
α
P
−
sin
α
P
Dj R
Dj L
(1) ij
mW
mW
2 MW
g mW
(1)
+i √
V sin αDj PR
(1) ij
2 mW
(1)
mDj
g M
mUi
(1)
(1)
δ
−i √
cos
α
P
−
sin
α
P
Ui R
Ui L
(1) ij
mW
mW
2 MW
g mW
(1)
−i √
Vij cos αUi PL
2 m(1)
W
mDj
g M (1)
mUi
(1)
(1)
i√
δ
sin
α
P
−
cos
α
P
Ui R
Ui L
(1) ij
mW
mW
2 MW
g mW
(1)
Vij sin αUi PL
−i √
2 m(1)
W
+igT3
ga(1) Ū (0) D(1)
+
i
ga(1) Ū (0) D(1)
+
i
d,i
s,i
=
j
ga(1) Ū (1) D(0)
+
=
s,j
ga(1) Ū (1) D(0)
+
=
d,j
j
=
Appendix A
o
(A.83)
(A.84)
(A.85)
(A.86)
(A.87)
The remaining symmetry related fermion-fermion-scalar vertex rules are
found using gS ξ̄1 ξ2 = gS∗ † ξ̄2 ξ1 for scalar couplings, whereas for all pseudo-scalar
coupling parts (i.e., the part of gS ξ̄1 ξ2 that include a γ 5 ) pick up an additional
minus sign gS ξ̄1 ξ2 = −gS∗ † ξ̄2 ξ1 (this follows from the relation (ψ1 γ 5 ψ2 )† =
−ψ̄2 γ 5 ψ1 for these interaction terms in the Lagrangian).
In unitarity gauge (ξ → ∞) unphysical scalar fields are not present. However, in a general gauge there are also additional vertices including unphysical
scalar fields. Although we in Paper III calculated the one-loop B (1) B (1) → γγ
process in the Feynamn-’t Hooft gauge (ξ → 1) no fermion-fermion-scalar vertexes come in at loop order for this process.
Ghost-Ghost-Vector Vertices
In unitarity gauge (ξ → ∞), all ghosts disappear from the theory. For other
gauges one can derive the vertex rules from the ghost Lagrangian in Eq. (5.34).
I here choose to include only the (one) vertex rules that was explicitly used in
Paper III to calculate the one-loop process B (1) B (1) → γγ in the Feynamn’t Hooft gauge (ξ → 1):
Section A.2.
c∓
(1)
= ±ie q µ
A(1) µ
c̄∓
153
Vertex Rules
q
(1)
Ghost-Ghost-Scalar Vertices
As for the above ghost-ghost-vector couplings, only the (one) ghost-ghostscalar Feynman rule explicitly needed in our calculation in Paper III of the
process B (1) B (1) → γγ is listed:
c∓
(1)
g
= −i mW ξ
2
h(1)
c̄∓
(1)
Vector-Vector-Vector-Vector Vertices
Couplings between four vector fields originate from the gauge field strength
part in the Lagrangian (5.9). The following four-dimensional Feynman vertex
rules are derived:
V1µ
V3ρ
igV1 V2 V3 V4 · 2η µν η ρσ − η µρ η νσ − η µσ η νρ
V4σ
V2ν
where
g2
gW (0) W (0) W (0) W (0)
=
gW (1) W (1) W (1) W (1)
=
gW (1,0) W (1,0) W (0,1) W (0,1)
=
3 2
g
2
g2
gW (1,0) W (0,1) W (1,0) W (0,1)
=
g2
(A.91)
gZ (0) Z (0) W (0) W (0)
=
−g 2 c2w
(A.92)
−
−
−
−
−
−
−
+
−
+
+
+
+
(A.88)
(A.89)
(A.90)
+
+
+
−
+
154
Feynman Rules: The UED model
Appendix A
=
−g 2 c2w
(A.93)
gA(0) A(0) W (0) W (0)
=
−e
2
(A.94)
gA(0) A(0) W (1) W (1)
=
−e2
(A.95)
gZ (0) A(0) W (0) W (0)
=
−egcw
(A.96)
gZ (0) A(0) W (1) W (1)
=
−egcw
(A.97)
gA(1) A(1) W (1) W (1)
=
gA(1) A(1) W (0) W (0)
=
gZ (0) Z (0) W (1) W (1)
−
+
−
+
−
+
−
+
−
+
3
3
3
−
+
3
−
+
gA(1) Z (0) W (0,1) W (1,0)
3
=
−
+
3
− g2
2
−g 2
(A.98)
(A.99)
2
−g cw
(A.100)
Vector-Vector-Scalar-Scalar Vertices
Couplings between two vector and two scalar fields originate both from the
gauge field strength term (5.9) and the kinetic Higgs term (5.17) of the Lagrangian. The following four-dimensional Feynman vertex rules are derived:
V1µ
S1
iη µν gV1 V2 S1 S2
V2ν
S1
where
gZ (0) Z (0) h(0) h(0)
=
gW (0) W (0) h(0) h(0)
=
gB (1) B (1) h(1) h(1)
=
gB (1) B (1) a(1) a(1)
=
gA(1) A(1) h(1) h(1)
=
gA(1) A(1) a(1) a(1)
=
±
∓
0
3
3
0
3
3
0
0
g2
2c2w
g2
2
3gY2
4
2
3gY2 M (1)
4 M (1) 2
Z
3g 2
4
2
3g 2 M (1)
4 M (1) 2
Z
(A.101)
(A.102)
(A.103)
(A.104)
(A.105)
(A.106)
Section A.2.
Vertex Rules
gW (1) W (1) a(1) a(1)
±
±
∓
∓
= −g 2
m2W
155
(A.107)
(1) 2
MW
2
gB (1) B (1) a(1) a(1)
=
gA(1) A(1) a(1) a(1)
=
gW (1) W (1) h(1) h(1)
=
+
−
3gY2 M (1)
4 M (1) 2
W
(A.108)
(1) 2
3
±
3
±
∓
∓
3g 2 MW + 1/3m2W
(1) 2
4
MW
3g 2
4
(A.109)
(A.110)
(1) 2
gW (1) W (1) a(1) a(1)
±
∓
0
0
=
3g 2 MW + 1/3m2W
(1) 2
4
MZ
(A.111)
(1) 2
gW (1) W (1) a(1) a(1)
∓
±
±
∓
gB (1) A(1) h(1) h(1)
3
gB (1) A(1) a(1) a(1)
3
0
0
3g 2 MW − 1/3m2W
=
(1) 2
4
MW
3gY g
= −
4
2
3gY g M (1)
= −
4 M (1) 2
(A.112)
(A.113)
(A.114)
Z
2
gB (1) A(1) a(1) a(1)
3
±
∓
=
3gY g M (1)
4 M (1) 2
(A.115)
W
3gY g M (1)
(1)
4 MW
gB (1) W (1) h(1) a(1)
= ∓i
gB (1) W (1) a(1) a(1)
= −
3gY g M (1)
(1)
(1)
4 MW
MZ
(A.117)
gA(1) W (1) a(1) a(1)
= −
2
g 2 MW
(1)
(1)
2 MW
MZ
(A.118)
gB (1) B (1) h(0) h(0)
=
gZ (0) Z (0) h(1) h(1)
=
gA(1) A(1) h(0) h(0)
=
gZ (0) A(1) h(0) h(1)
=
±
∓
(A.116)
2
±
3
3
±
0
0
∓
∓
3
3
gY2
2
g2
2c2w
g2
2
g2
2cw
(A.119)
(A.120)
(A.121)
(A.122)
2
gZ (0) Z (0) a(1) a(1)
0
0
=
g 2 M (1)
2c2w M (1) 2
Z
(A.123)
156
Feynman Rules: The UED model
gW (0) W (0) a(1) a(1)
= −g 2
gA(0) A(0) a(1) a(1)
= 2e2
gZ (0) Z (0) a(1) a(1)
=
gW (0,1) W (0,1) h(1,0) h(1,0)
=
gW (0,1) W (1,0) h(0,1) h(1,0)
=
gW (0) W (0) a(1) a(1)
=
±
±
∓
±
±
+
+
∓
∓
∓
−
−
Appendix A
m2W
(A.124)
(1) 2
MW
(A.125)
g 2 4c4w m2W + (c2w − s2w )2 M (1)
(1) 2
c2w
2MW
g2
2
g2
2
2
(A.126)
(A.127)
(A.128)
(1) 2
±
∓
0
0
g 2 MW + 3m2W
(1) 2
2
M
(A.129)
Z
2
g MW + m2W
(1) 2
2
MW
gY g
= −
2
egY
=
2cw
=
gW (0) W (0) a(1) a(1)
±
∓
±
∓
gB (1) A(1) h(0) h(0)
3
gB (1) Z (0) h(1) h(0)
∓
∓
gB (1) W (0) h(0) a(1)
= ∓i
gA(0) W (0) a(1) a(1)
= −
±
(A.131)
(A.132)
eg 2c2w m2W + (c2w − s2w )M (1)
(1) 2
cw
MW
e2 M (1)
= ∓i
(1)
2sw MW
gA(0) W (0,1) h(1,0) a(1)
±
(A.130)
=
gA(0) Z (0) a(1) a(1)
±
(1) 2
∓
e2 M (1)
(1)
2cw MW
2
(A.133)
(A.134)
(A.135)
(1)
±
0
e 2 mW + M W
(1)
(1)
2sw MW
MZ
e2 M (1)
= ±i
(1)
2cw MW
∓
gZ (0) W (0,1) h(1,0) a(1)
±
∓
(A.136)
(A.137)
2
gZ (0) W (0) a(1) a(1)
±
0
=
∓
e2 s2w M (1) − 2c2w m2W
(1)
(1)
2s2w cw
MW MZ
(A.138)
In unitarity gauge (ξ → ∞), unphysical scalar fields are not present. However, in a general gauge there are also additional vertices including unphysical
scalar fields. I here choose to include only those vertex rules that were explicitly used in Paper III to calculate the one-loop process B (1) B (1) → γγ in the
Feynamn-’t Hooft gauge (ξ → 1):
gA(0) A(0) G(1) G(1)
+
−
= 2e2
(A.139)
Section A.2.
Vertex Rules
gB (1) B (1) χ(0) χ(0) =
=
gB (1) B (1) G(1) G(1)
=
gB (1) B (1) G(1) a(1)
=
−
+
−
+
∓
±
gY2
2
3gY2 m2W
4 M (1) 2
W
2
3gY mW M (1)
4 M (1) 2
157
(A.140)
(A.141)
(A.142)
W
gB (1) B (1) χ3 (0) χ3 (0)
=
gB (1) B (1) G(1) G(1)
=
gB (1) B (1) G(1) a(1)
=
gB (1) A(0) G(1) χ(0)
=
gB (1) A(0) a(1) χ(0)
=
gW (1) A(0) h(0) G(1)
=
0
0
0
0
±
±
±
∓
∓
∓
gY2
2
3gY2 m2Z
4 M (1) 2
Z
3gY2 mZ M (1)
4 M (1) 2
Z
mW
±iegY (1)
MW
M (1)
±iegY (1)
MW
eg mW
∓i
(1)
2 MW
(A.143)
(A.144)
(A.145)
(A.146)
(A.147)
(A.148)
Scalar-Scalar-Scalar-Scalar Vertices
Couplings between four scalar fields originate from the kinetic Higgs term
(5.17) of the Lagrangian. The following four-dimensional Feynman vertex
rules are derived:
S1
S3
igS1 S2 S3 S4
S4
S2
where
gh(0) h(0) h(0) h(0)
gh(1) h(1) h(1) h(1)
ga(1) a(1) a(1) a(1)
0
0
0
0
= −6λ
= −9λ
= −
(A.149)
(A.150)
3g 2 /c2w m2Z M
(1) 2
(1) 4
MZ
+ 9λM
(1) 4
(A.151)
158
Feynman Rules: The UED model
gh(1) h(1) a(1) a(1)
= −
gh(1) h(1) a(1) a(1)
= −
0
+
0
−
Appendix A
g 2 m2Z /c2w + 6λM (1)
2
(A.152)
(1) 2
2MZ
g 2 m2W + 6λM (1)
2
(A.153)
(1) 2
2MW
2
g 2 m2W M (1) (1 + c4w )/c4w + 6λM (1)
ga(1) a(1) a(1) a(1)
= −
ga(1) a(1) a(1) a(1)
= −
gh(1) h(1) h(0) h(0)
= −λ
0
0
+
−
(1) 2
+
−
−
gh(0) h(0) a(1) a(1)
= −
gh(0) h(0) a(1) a(1)
= −
0
+
0
−
(A.154)
2MW MZ
2
+
(1) 2
4
2g 2 m2W M (1) + 6λM (1)
(1) 2
4
(A.155)
MW
(A.156)
g 2 m2Z /c2w + 4λM
(1) 2
(1) 2
(A.157)
2MZ
g 2 m2W + 4λM (1)
(1) 2
2
(A.158)
2MW
In unitarity gauge (ξ → ∞), unphysical scalar fields are not present. However, in a general gauge there are also additional vertices including unphysical
scalar fields. Although we in Paper III calculated the one-loop B (1) B (1) → γγ
process in the Feynamn-’t Hooft gauge (ξ → 1), no scalar-scalar-scalar-scalar
vertexes come in at loop order for this process.
A Short Note on Conventions in FeynArts
Unfortunately, there is no consensus in sign conventions in the field theory
literature. In our actual implementation of vertex rules into the FeynArts
package [255], which we used in the numerical calculations in Paper III, we
adopted the same convention as for the preimplemented vertex rules for the
SM particles in FeynArts. This requires a minor change for the zero mode
Goldstone bosons vertex rules compared to those given in this Appendix.
Feynman rules in the FeynArts convention would be obtained from this Ap(0)
pendix if we changes the overall sign for each χ30 and multiply by ±i for
(0)
each χ± that appears in the vertex rule.
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VII
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