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Dark Matter Experiments
A bachelor thesis by Niklas Grønlund Nielsen
Supervised by Professor Francesco Sannino, PhD
August 2013
Department of Physics, Chemistry and Pharmacy
University of Southern Denmark
Dark Matter Experiments
Abstract
Dark matter is an explanation for one of the most important problems in modern physics.
It is a well established scientific paradigm that excess gravity is caused by a new and
unobserved particle.
Some of the most important experiments trying to detect dark matter is direct detection
experiments, a method where one tries to measure recoils from collisions between dark
matter particles and atomic nuclei. These collisions are very rare and hard to measure.
In the first part of this bachelor thesis we will look at the theory behind direct detection,
and we will test some of the astrophysical assumptions that must be made prior to any
experiment.
In the second part we will construct a model, where dark matter is a scalar field that
interacts with a detector nucleus via two channels: through an exchange of the Higgs
boson and through a small dipole interaction that allows a photon exchange. The dipole
allows dark matter to feel the electric charge of the proton very weakly. As shown in e.g.
[11] differentiating how dark matter interacts with protons and neutrons, can alleviate
tension between prominent direct detection experiments. We will see that the correct
differentiation can be achieved via the dipole and Higgs interactions, and find how the
coupling parameters of our model must be tuned.
In the end we will extend the possibilities of our model by considering which possible
interactions can be included in a generic theory of scalar dark matter acting as a singlet
under the symmetries of the Standard Model.
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Dark Matter Experiments
Contents
1 Introduction
2 Dark Matter Phenomenology and Appeal
2.1 Zwicky and the Coma Cluster . . . . . . .
2.2 Rubin, Ford and the Andromeda Galaxy .
2.3 The WIMP and Genesis of Dark Matter .
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of the WIMP
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3 Dark Matter Detection Methods
3.1 Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Detector Kinematics . . . . . . . . . . . . . . . . . . .
3.1.2 The Event Rate . . . . . . . . . . . . . . . . . . . . .
3.1.3 Example of Sensitivity to Astrophysical Assumptions .
3.1.4 Annual Modulation . . . . . . . . . . . . . . . . . . .
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4 Modelling Scalar Dark Matter
4.1 Real Scalar Dark Matter under a Z2 Symmetry . . . . . . . . . . . . . .
4.1.1 Dark Matter-Nucleus Scattering Cross Section . . . . . . . . . .
4.1.2 Investigating whether Real Scalar DM is a good Candidate . . .
4.2 Upgrading to a Complex Field that Differentiates Protons and Neutrons
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5 An
5.1
5.2
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5.5
Effective Dark Matter Theory for Scalar SM Singlets
Self-interactions . . . . . . . . . . . . . . . . . . . . . . . . . .
Interactions with the Higgs . . . . . . . . . . . . . . . . . . .
Interactions with Fermions of the Standard Model . . . . . .
Interactions with Gauge Bosons of the Standard Model . . .
The General Scalar Dark Matter Model up to Dimension 6
Prospectives . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Summary and Conclusions
A Appendices
A.1 Notation . . . . . . . . . . . . . . . . . . . . . . .
A.2 Flat Friedmann Universe . . . . . . . . . . . . . .
A.3 Spontaneous Electroweak Symmetry Breaking . .
A.4 Scattering Cross Section for Scalar Higgs-coupled
A.5 Scalar DM with Dipole Interference:
Derivation with Photon Propagator . . . . . . . .
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1
Introduction
In 1933 astronomer Fritz Zwicky noted that the apparent gravitational mass of the Coma
cluster far exceeded what the visible matter could provide [1]. This discovery is one of the
greatest in the history of cosmology and arguably all of physics. The scientific significance
was not fully appreciated, until Vera Rubin and Kent Ford in the 1970’s observed, that
the radial velocity distribution in the Andromeda galaxy could only be explained by
Newtonian dynamics if the bulk of gravitating mass was of a non-baryonic nature [2].
Since the 1970’s many other observations of large scale astrophysical phenomena have
suggested the existence of dark matter distributed in halos within and around galaxies.
These observations provide some of the strongest empirical evidence for physics beyond
the Standard Model (SM) of particle physics. Today, some 80 years after Zwicky’s initial
observations, the nature of dark matter is still undetermined. In fact the level of ignorance
concerning dark matter is remarkable, it truly is one of the greatest problems in physics
today.
There has been large variety of credible explanations, ranging from new invisible
particles to Modified Newtonian Dynamics (MoND) to dim baryonic matter. In this thesis
we will look at the favoured paradigm: that dark matter consists of massive and nonbaryonic particles, which for some reason remain invisible to science. Usually a dark matter
candidate known as the Weakly Interacting Massive Particle (the WIMP) is assumed,
which as the name suggests mainly interacts via gravity (massive) and only weakly to the
particles of the Standard Model through the other forces.
The latest high precision cosmological survey from the Planck satellite (ref. [4]) estimate
the energy content of our universe to be distributed as follows: 5 % ordinary matter, 26
% dark matter and 69 % dark energy (dark energy is an even deeper mystery than dark
matter!), making dark matter much more abundant than ordinary matter.
Detection of dark matter has naturally been a focus of a large experimental community.
In the first part of this thesis we will focus on the theoretical background for direct detection,
which is one of three major experimental types, that currently tries to detect dark matter
(an perhaps already has).
In the second part we will construct a dark matter model, where our WIMP is a scalar
field that behaves as a singlet under the the Standard Model. This is arguably the simplest
approach to dark matter modelling. First, we will construct the absolutely simplest model,
which is only connected to the standard model via exchange of the Higgs boson. Secondly
we will improve key features of the model ad hoc; such that experimental results can agree.
The formulation will be done in the language of quantum field theory and treatment will
be in the context of direct detection.
In the last chapter an effective field theory will briefly be introduced, to illustrate
a different model building approach and explore the possibilities to extend the already
constructed model.
Mathematical notation and lengthy derivations can be found in the appendices.
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2
Dark Matter Phenomenology and Appeal of the WIMP
In cosmology the Big Bang nucleosynthesis (BBN) is the formation of all the lightest
elements up to 7 Li. In the moments after Big Bang the universe underwent a series of
chemical equilibria as expansion caused cooling. These processes are well understood
and predictions of especially hydrogen and helium abundances (by far the most abundant
constituents) are very close to observations. BBN’s predictions for the mass fraction of
the less abundant materials turns out to be heavily dependent on the total baryonic mass
density, this is especially true for deuterium (see [3] for ref.). Therefore the total baryonic
mass density can be estimated from measuring the abundance of deuterium (and other
light elements). It turns out that ΩB = 0.049, where Ω is the density in units of the critical
density.
Independently cosmologists have determined our universe to be very close to being flat,
i.e. the curvature of space time k = 0. Assuming the Friedmann Equations (implicitly
assuming the Robertson-Walker metric), k = 0 is equivalent to setting the total density
parameter Ω = ΩB + ΩDark ≡ 1 (see Appendix A.2 for argument).
Since the estimates from e.g. Planck sets dark energy at ΩDE = 0.696, there is a large
dark matter component at ΩDM = 1 − ΩDE − ΩB = 0.265. So in terms of mass, dark
matter is by far the largest component in our universe.
2.1
Zwicky and the Coma Cluster
Fritz Zwicky (1898-1974; Bulgarian born and of Swiss nationality) is arguably one of the
greatest astronomers of the 20th century, among his many original ideas and observations
is his work on the Coma cluster of galaxies. Zwicky used the steady state virial theorem
demanding for the cluster’s moment of inertia I that I¨ = 0. From this is obtained that
hV i = −2hKi,
(2.1)
where hV i = −αGM 2 /rh is the average potential and α is a numerical factor depending
on the average distance between galaxies; rh is the half radius of the cluster; and hKi =
M hv 2 i/2 is the average kinetic energy. Zwicky’s expression for the gravitating mass of the
Coma cluster was thus
rh hv 2 i
.
(2.2)
αG
Amazingly the luminous matter only accounted for about 1/50 of the gravitational
mass. Therefore Zwicky predicted the existence of a large amount of dunkle materie or
dark matter. In reality most of the baryonic matter in galaxy clusters are interstellar gasses
and other dim constituents, however ∼ 80% of the total mass is still unaccounted for by
baryonic matter.
Today we know Zwicky discovered one of our universe’s biggest mysteries; sadly in his
time, Zwicky was not duly appreciated for the weight of his work.
M=
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2.2
Rubin, Ford and the Andromeda Galaxy
In 1978 Astronomers Vera Rubin and Kent Ford made a shocking discovery. They observed
the rotational velocities of the Andromeda galaxy via redshifts and found a very unexpected
result. On the galactic scale, the laws of Newtonian dynamics was thought to be firmly
established, therefore rotational velocities should be found using the Newtonian gravity
and the centripetal force
GmM (r)
mv 2 (r)
=
,
r2
r
s
v(r) =
GM (r)
.
r
(2.3)
(2.4)
Approximating galaxies to be spherical one would expect M (r) ∝ r3 until the end of
the luminous disk (r ≈ 15kpc for the Milky Way), so in this region one expects v(r) ∝ r.
If M (r) roughly ended at the edge of the luminous disk, objects beyond this limit would
behave as v(r) ∝ r−1/2 . However, this is not the behaviour observed!
Figure 1: Rubin and Ford data from a 1978 article [2], clearly there is no Keplerian r−1/2
decrease.
The discrepancy between expected and observed results can be explained in one of
two ways: either there is a dark component such that M (r) = Mluminous (r) + Mdark (r),
or dynamics at galactic scales need to be modified such that FG = FN ewton + Fgalactic or
even a correction on FN ewton = GmM (r)/r2+ε . These explanations are called dark matter
(DM) and Modified Newtonian Dynamics (MoND) respectively.
Today MoND is almost totally abandoned for multiple reasons, for example the Bullet
Cluster strongly suggests the existence of weakly interacting and gravitating dark matter.
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It is also a problem that these galactic forces are not seen at even larger scales, so to be
consistent with observations one must add new corrections for every length scale. MoND
also requires corrections to General Relativity, while keeping intact the principle of general
covariance, this proves difficult. Lastly, everybody agrees that neutrinos exist, and they
are in fact weakly interacting massive particles and therefore dark matter. However, for
reasons related to when and how galaxies formed and clustered, neutrinos cannot be more
than a small fraction of the entire dark matter abundance, as they correspond to what
has been coined hot dark matter. In ref. [9] the paradigm of hot vs. cold dark matter is
treated. The point is, that matter invisible to electromagnetic and strong interactions is
not inconceivable or exotic, in fact it exists all around us.
Figure 2:
The Bullet Cluster as seen by NASA’s Hubble Space Telescope
(can be found on NASA’s web page, at the time of writing in the link
http://apod.nasa.gov/apod/ap060824.html). Two galaxy clusters have crashed in a titanic
collision, separating weakly interacting DM from luminous matter; the blue highlighted
regions suggests weakly interacting dark regions that exhibit gravitational lensing of far
away star light. As would be expected for weakly interacting particles, they are less likely
to be captured in the collision compared to the purple highlighted baryonic matter.
2.3
The WIMP and Genesis of Dark Matter
If we make the single assumption, that the missing gravity we observe in our universe is in
fact a result of an invisible particle, what features would such a particle have?
We would at least assume the particle to be electrically neutral and stable. That is, given
that dark matter has been around since the early universe, we would require for the mean
lifetime to be greater than the age of the universe.
Furthermore, the fact that the mysterious dark matter particle has alluded attempts of
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observation for 80 years, heavily limit how strongly DM can interact with atoms, hence
the name Weakly Interacting Massive Particle.
It should however be noted, that actual dark matter doesn’t care about scientific
paradigms, and one could easily think of the scary but plausible scenario: that dark matter
only interacts via gravity. Such a scenario would make all current observational attempts
redundant. Hopefully this is not the case, and if one of the many shades of WIMP actually
exists, an experimental group may soon claim its discovery.
An important feature of the dark matter problem is the question of the origin. The
non-exotic explanation is that dark matter is a thermal relic, frozen out as the universe
cooled. This mechanism assumes some early equilibrium where the temperature of the
universe exceeded the dark matter mass, as expansion caused the number density nχ to
drop (and thus the annihilation cross section), the annihilation rate became smaller than
the expansion rate of the universe (the Gamov condition)
H ≥ nhσvi,
at which point a large amount of sterile particles froze out. H being Hubble’s constant.
This mechanism is well known from other thermal freeze outs such as the photons of the
Cosmic Microwave Background.
Another possible genesis is an early WIMP/anti-WIMP asymmetry. One can imagine
that a large abundance of particles would still remain after the initial annihilation left an
excess. This mechanism mimics the origin of baryonic matter and is termed asymmetric
dark matter. For the possibility of an asymmetric origin, the dark matter particle should
of course be different from its anti-particle, which is not true for all WIMP candidates.
3
Dark Matter Detection Methods
Experiments for detection of a dark matter are very diverse and span a huge energy scale.
The main experiments can be divided in the following methods:
Direct Detection (DD) is a straight forward but technically difficult approach. Atomic
nuclei with well known physical properties are shielded from cosmic background
radiation and other contaminants (typically DD experiments are located deep under
ground). Recoil energies are measured from collisions with possible WIMPs. The
observed recoils can be expressed as a statistical region in the cross section/mass
parameter space where a dark matter particle could possibly be found. The rate of
measuring a recoil for a single nucleus is extremely low, therefore a large number of
targets is used. The total exposure of a given experiment is measured in the exposure
time times the size/mass of the target (kg days).
This method probes the lowest energy region, which is in the 10’s of keV, corresponding
to the energy of the recoil. This is a very low energy indeed and provides considerable
challenges for construction of highly sensitive apparatus. Furthermore background
radiation and other internal/external effects are difficult to exclude and could provide
fake DM signals.
There are many large DD experiments, among the most important is DAMA, CoGeNT,
Xenon, CRESST and others.
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Indirect Detection (ID) observes Standard Model products of dark matter decay or
annihilation. The energy scale of ID is thus wide, from the smallest possible products
up to the unknown mass of the possible dark matter candidate. Indirect experiments
lacks many of the DD challenges, but for a specific indirect experiment, one could
imagine other sources than dark matter annihilation if results are achieved.
A current example of ID is the ongoing AMS-02 experiment on the International Space
Station, which recently released data showing a e+ /e− -fraction larger than expected
from a universe with no dark matter. This ratio could come from an astrophysical
source such as a pulsar or possibly from dark matter annihilation (recent results and
press releases are available on http://www.ams02.org/ at the time of writing).
Collider Detection (CD) is in some sense the opposite of indirect detection. Particle
accelerators attempt to create dark matter by colliding Standard Model particles. If
successful such a detection, would be a detection of missing energy. The drawback is
that energy could go missing in many ways! However if a theoretical dark matter
candidate is suggested and its parameter space is constrained by direct detection,
then observables such as the cross section for hadrons into dark matter can be tested
at colliders if they are energetic enough to kinetically allow the process. This is the
true strength of collider detection.
Direct Detection (DD)
Collider Detection (CD)
Ideally the different methods should not
be seen as competing, but complementary.
Since results in each separate method could
DM
SM
be observations of some unforeseen effect, a
connection between different kinds of experiments operating at largely different energies
is necessary to claim a discovery of a WIMP.
For this thesis we will only focus on the
theory of direct detection. Direct detection
has the nice feature of being thoroughly
non-relativistic, therefore all kinematics can
be treated in this limit. Given a model,
the observables we want to determine is
therefore those related to the upward arrow
in figure 3.
In principle, dark matter can be treated
DM
SM
independent of microscopic models by taking the approach of effective theories. This
Indirect Detection (ID)
can be done by writing down all allowed
terms that can exist in a Lagrangian deFigure 3: Here is shown an unknown interscribing dark matter. For direct detection
action between dark matter and a standard
this is a particularly viable approach, bemodel particle. Different directions in the diacause DD is in the low energy regime, and
gram correspond to specific search strategies.
we always deal with two particles in the
initial and final states, which limits how complicated the effective interactions can become.
An effective DM model will be treated further in chapter 5, where we expand the number
of possible interactions for the model build in chapter 4.
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Dark Matter Experiments
3.1
Direct Detection
Currently a large and varied effort is being made in the field of direct detection. Multiple
experiments at many locations using different techniques have tried to limit the region of
the cross section/mass parameter space where a WIMP could reside. However, considerable
tension has risen between high profile experiments, as seen in figure 4.
Figure 4: This plot is from reference [10] and shows the proton scattering cross section vs.
the dark matter mass. There is assumed a standard contact interaction, i.e. equal proton
and neutron interaction. The DM mass density is taken to be ρχ = 0.3/GeV/cm3 and
the distribution is in an isothermal sphere with a velocity dispersion of 220 km/s. Lines
from XENON and CDMS are upper bounds and contours are allowed regions. There is no
common allowed region agreed upon by all experiments, so either some experiments are
wrong or some assumptions must be changed.
3.1.1
Detector Kinematics
In direct detection methods one can examine elastic and inelastic collisions. Elastic
collisions preserve particle states and only exchanges momentum (p and k) between the
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Dark Matter Experiments
dark matter particle and the target nucleus. A general inelastic scattering is
χ(p) + N (k) → χ0 (p0 ) + N 0 (k0 ).
(3.1)
In the following we will treat the kinematics of elastic scattering, because this is somewhat simpler. One should however keep in mind, that inelastic scattering is an interesting
case to study, if one tries to make experiments agree (in chapter 4 we will go about this in
a different way).
Given a detector of perfect sensitivity and knowledge about the distribution of WIMP
velocities in our galaxy, one could perform the integral
dR
=
dQ
Z ∞
dR
dQ
0
f (v) dv,
(3.2)
where R is the averaged recoil count, Q is the recoil energy deposited in the nucleus
and f (v) is the probability distribution in WIMP velocities. Unfortunately detectors are
not perfect and the lowest detectable energy must be considered. This corresponds to a
minimal velocity that a WIMP must have w.r.t. the detector before a scattering is observed.
To find the minimal velocity with Qth as the threshold energy of the detector, we take
the velocity of dark matter to be firmly non-relativistic; this is a fair assumption since the
actual velocity is at least bounded from above by the galactic escape velocity vesc ∼ 10−3 c
i.
Furthermore the velocity in the laboratory frame is of equal magnitude to the relative
velocity v in the center of mass frame. Take q = p − p0 = k0 − k to be the transferred
momentum and p = µv in the center of mass frame. We have the c.o.m. condition p = −k
and µ = mN mχ /(mN + mχ ) is the reduced mass of the WIMP-nucleus system. The recoil
energy of the nucleus is
Q=
q2
.
2mN
(3.3)
Now demanding conservation of energy |p|2 = |p0 |2 ; (3.3) becomes
Q=
|p|2 +|p0 |2 −2|p||p0 |cos θ
µ2 v 2 (1 − cos θ)
=
.
2mN
mN
(3.4)
Taking a perfect head on collision (θ = π), one obtains the minimal velocity in the nonrelativistic limit. (If the collision is inelastic, where outgoing and incoming masses are
unequal, then there is a small correction to the velocity)
s
vmin '
mN Qth
.
2µ2
(3.5)
The minimal velocity should of course be met by the actual WIMP velocity. This puts a
direct requirement on how sensitive the detector must be. If we assume dark matter to be
distributed in giant halos around our galaxies, and that there is no dark matter wind, then
the average relative velocity would be thermally distributed around the Earth’s velocity in
i
vesc =
1012 M .
p
2GM/R ≈ 500km/s ∼ 10−3 c with: G = 4.3 · 10−3 pc/M (km/s)2 , R = 100 pc, M =
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the galactic frame. If very unlucky there could be a DM wind such that WIMP’s average
velocity in the galactic frame matches the Earth’s, this case would require extremely
sensitive detectors.
3.1.2
The Event Rate
The most important observable in any direct detection experiments is the differential recoil
rate with respect to the recoil energy. This is measured in units of [ #/kg/yr/energy] and
is related to the differential cross section. The recoil for some specific velocity is
R = nχ vσ,
(3.6)
where nχ = ρχ /mχ is the number density of DM, v is the local velocity and σ is the
scattering cross section. The differential recoil rate is
dR
dσ
= nχ v
.
dQ
dQ
This expression needs to be averaged over all velocities from vmin to vesc w.r.t. the velocity
distribution f (v), (R = hRi)
dR
dσ
= nχ v
dQ
dQ
Z vesc
= nχ
f (v)
d3 v f (v)v
vmin
dσ
.
dQ
(3.7)
Here a single dark matter candidate is assumed, but one could easily sum over different
species that could be distributed differently. Here we consider only one dark matter
candidate at a time, since we know of no a priori reason to do anything more complicated.
Scattering Cross Section. The scattering cross section σ is an effective area. A
large cross section corresponds to a large scattering probability and vice versa. Direct
detection depends heavily on the differential cross section of the scattering. The differential
cross section has separate contributions: spin dependent and independent
dσSI
dσSD
dσ
=
+
.
dQ
dQ
dQ
(3.8)
In the model that we will build in chapter 4, we will take
dσ
dσSI
≈
.
dQ
dQ
(3.9)
Usually the differential cross section is written in terms of the scattering cross section
and a function FN (Q) called the nuclear form factor containing information related to the
fact, that the scattering is not point like
dσ
mN
= FN2 (Q) 2 σ.
dQ
µv
(3.10)
Depending on the specific DM model and interaction type the scattering cross section
behaves qualitatively different.
Velocity Distribution. The local velocity distribution must be assumed so the
integral in equation 3.7 can be performed. A reasonable assumption is that the distribution
is Maxwell-Boltzmann, if self-interactions are taken to happen through elastic collisions.
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Dark Matter Experiments
f (v) =
mχ
2πkT
3/2
!
mχ v 2
4πv exp −
.
kT
2
(3.11)
The temperature of the distribution is bounded from above by the galactic escape velocity;
it can however be lower depending on the mechanism for the synthesis of dark matter.
Here is assumed no time dependence (that is neglecting annual modulation effects, discussed
in section 3.1.4). If the nature of DM turns out to be different than expected, one can
imagine completely different velocity distributions with the sole requirement that f (v) is
normalized
Z ∞
1=
d3 vf (v) ≈
Z vesc
0
d3 vf (v).
(3.12)
0
One could for example play the game of having many components of dark matter, with
distributions at different temperatures. Such possibilities makes the necessary assumptions
highly tunable.
In turn this makes the differential recoil rate deceptively simple. In fact it is a product
of two large assumptions. From particle physics we must assume a cross section and from
astrophysics we must assume the velocity distribution. So basically we have
(Differential Recoil Rate) = (Astrophysical Assumptions)×(Particle Physics Assumptions)
(3.13)
Thus when two DD experiments have conflicting results, one cannot conclude that
the paradigm of dark matter is wrong; because one of the assumptions could easily be
incorrect (or even both!). In light of this one would require from a good dark matter model,
that it at least explains the discrepancy between various DD experiments with reasonable
assumptions.
3.1.3
Example of Sensitivity to Astrophysical Assumptions
As a simple exercise in direct detection responses, here we consider a specific detector
and assume a scattering cross section that behaves nicely; specifically we will assume it is
constant w.r.t. velocities, the recoil energy and the dark matter mass
σ = σ0 ,
(3.14)
this is likely an oversimplification, but we take it to be correct to first order. The recoil
rate becomes
dR
MN
= nχ σ0 2 FN2 (Q)
dQ
µ
Z vesc
dv f (v)v.
(3.15)
vmin
The form factor in this example includes all recoil energy dependence and is determined
through nuclear physics. Here we assume a pure 70 Ge detector with an explicit spin
independent form factor from ref. [8] p. 38 (here is only used a single of the many responses
for illustrative purposes):
FN = e−2y (1000 − 2800y + 2900y 2 − 1400y 3 + 350y 4 − 42y 5 + 1.9y 6 − 0.0027y 7 ), (3.16)
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with y ≈ 105 GeV−1 × Q such that the minimal recoil around 10 keV corresponds to the
dimensionless y to be minimal around 1.
In the following figures on the right hand is shown the base 10 logarithm of the
differential recoil rate. The differential recoil rate is in units of ρχ · σ0 ; so we are not looking
for actual counts per year, but differences in qualitative behaviour from one assumption
to another. The left figures are velocity distributions shown as contours in the DM
velocity/mχ phase space, such that for a given DM mass, condition 3.12 holds.
In the first plots we chose the thermal energy kT of the Maxwell velocity distributions,
such that the lightest candidates < 10 GeV are distributed within the galactic escape
velocity. In all plots we have assumed a good detector with a threshold of 10 keV recoil
energy.
Velocity distribution with kT= 10−6 GeV
2
dR/dQ
10
100
0
−2
90
−4
80
−6
mχ [GeV]
mχ [GeV]
70
60
50
40
−8
−10
1
10
−12
−14
−16
30
−18
20
−20
10
0
−22
0
0
0.2
0.4
0.6
0.8
v [c]
1
1.2
1.4
1.6
−3
x 10
10
−5
10
−4
10
Q [GeV]
Figure 5: The velocity distribution plot stretches to vesc ∼ 1.6 · 10−3 c and the recoil rate is
shown for recoil energies between 10keV and 100keV. Every contour line is one power of
10 different from its neighbour with yellow being a high detection rate and green a low
one. Clearly light candidates as close as possible to the energy threshold is most easily
detectable. Interestingly this theoretical detector has a blind spot just above the mχ = 10
GeV line, which is not far from the (e.g.) DAMA and CoGeNT favoured region for scalar
dark matter
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Velocity distribution with kT= 10−5 GeV
2
0
90
−2
80
−4
70
−6
mχ [GeV]
mχ [GeV]
dR/dQ
10
100
60
50
40
−8
1
10
−10
−12
30
−14
20
−16
10
0
−18
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
v [c]
10
−5
10
1.6
−4
10
Q [GeV]
−3
x 10
Figure 6: Here we have allowed the velocity distribution to stretch out from our estimate of
the galactic escape velocity for dark matter of mass mχ ≤ 30 − 40 GeV. This is reasonable
if dark matter actually has a higher mass. The main difference from before is, that an
island of easier detectable candidates emerged beyond the blind spot. This is definitely a
nice feature, although the region is rather small.
Velocity distribution with kT=2 ⋅ 10−4 GeV
3
dR/dQ
10
1000
0
−2
900
−4
800
mχ [GeV]
700
mχ [GeV]
−6
2
600
500
400
10
−8
−10
−12
1
10
300
−14
200
−16
100
0
−18
0
0
0.2
0.4
0.6
0.8
v [c]
1
1.2
1.4
1.6
−3
x 10
10
−5
10
−4
10
Q [GeV]
Figure 7: For this last plot we increase the thermal energy kT all the way to 2 · 10−4 GeV
and consider dark matter up to 1000 GeV. Here only distributions for very heavy DM of
mass mχ ∼ 700 GeV or larger are contained in the allowed velocities within our galaxy. It
seems qualitatively that such heavy candidates are difficult to detect, but there is clearly a
stretching effect of the island beyond the mχ = 10GeV line, so this kind of recoil would
not be completely suppressed.
This is of course a theoretical detector, but it provides some illustrative points. First
of all, if dark matter is Maxwell-Boltzmann distributed (especially if it’s cold) then light
candidates are orders of magnitude more detectable. Secondly, in all cases low energy
thresholds are very important for a detector. Say we had a detector with a threshold of
100 keV and dark matter turned out to weigh 15 GeV, then from our plots the recoil rate
would be suppressed by anywhere between ∼ 10 − 20 orders of magnitude compared to a
10 keV detector.
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Dark Matter Experiments
3.1.4
Annual Modulation
Assuming the local DM velocity is constant
in time is in many cases a fair approximation. However, dark matter is thought to
be distributed as an aether in and around
galaxies through which star systems move.
The average velocity in Earth’s frame thus
depends on whether Earth is moving parallel or anti-parallel to the the Sun’s galactic
orbit.
Earth’s velocity in the galactic frame is
(values from ref. [10])
vE (t) = vG + vS + vmod (t)
(3.17)
30 km/s
∼ 230 km/s
−30 km/s
Here vG and vS are our local system’s
galactic velocity and the sun’s proper ve- Figure 8: Sun moving in the galactic frame.
locity respectively, these are close to being
aligned. vmod (t) is the modulation stemming from Earth’s revolution around the Sun and
thus has a period of one year. The modulation term is of special interest, because increased
dark matter signals should be observed when vmod (t) is aligned with vG (on the 2nd of
June). the velocities are approximately vG = 220 ± 50 km/s, vS = 12 km/s and vmod = 30
km/s. One can think of the modulation effect as giving the velocity distribution a time
dependence f (v(t)), the quantity measured is the integral from vmin to vesc of f (v(t)).
When orbits are aligned more of the distribution lies within the integration borders and
vice versa.
This technique is used by the DAMA collaboration (among others) to observe a signal at a
high confidence level.
Figure 9: Partial results from the 2003 article Dark Matter Search [7] clearly showing
an annual modulation of signals. 10 years ago DAMA already claimed a WIMP at 6.3σ
and today the claim is even stronger statistically, unfortunately it is not entirely certain
whether the signal is DM or something else.
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4
Modelling Scalar Dark Matter
The Standard Model of particle physics is a very good (albeit finely tuned) quantum field
theory for a small part of the energy content in our universe. One of the Standard Model’s
shortcomings is the lack of a dark matter particle, therefore one must go beyond the
Standard Model (BSM).
Sticking to Occam’s Razor we will assume DM to be a scalar field protected under some
global symmetry that ensures stability, this is certainly the most basic starting point we
can imagine. Our goal in this chapter is to construct a model, and examine a general direct
detection of our WIMP candidate.
4.1
Real Scalar Dark Matter under a Z2 Symmetry
The simplest thing one can think of is a real scalar field χ which is a singlet under the
symmetries of the Standard Model, we let χ be protected under a simple symmetry Z2 ,
such that the Lagrangian is invariant under χ → −χ. We want our dark matter candidate
to have some weak interaction with the Standard Model. One place such a connection
could still hide is in a coupling to the Higgs. The Z2 symmetry eliminates all odd power
terms and we can write an extension to the SM Lagrangian density as
1
1
1
L = LSM + ∂µ χ∂ µ χ − m20 χ2 − gχ4 − λH † Hχ2 ,
(4.1)
2
2
4
where m0 is the mass of the DM candidate before spontaneous electroweak symmetry
breaking, g and λ are the DM self coupling and Higgs coupling respectively, these are both
assumed to be real and positive. H is the Higgs doublet which in unitary gauge is (see
appendix A.3 for discussion)
!
1
0
H=√
,
2 h+v
(4.2)
here h is the Higgs scalar field and v ≈ 246 GeV is the vacuum expectation value (vev)
of the mexican hat potential. After the spontaneous symmetry breaking the dark matter
mass thus acquires an extra term, namely m2χ = m20 + λv 2 .
One can easily verify that the vev for the potential of χ is 0 with m20 , g ≥ 0, if this was not
the case the DM candidate would not be stable, so this is a requirement for the model.
Equation 4.1 is arguably the simplest model assuming only one connection to the Standard
Model. Other simple interactions include a mechanism where a hidden U(1) symmetry
gives rise to a new dark photon that kinetically mixes with the SM photon.
Writing out H † H, the Lagrangian reads
1
1
1
λ
L = LSM + ∂µ χ∂ µ χ − m2χ χ2 − gχ4 − χ2 h2 + λvχ2 h .
2
2
4
|2
{z
}
(4.3)
interaction terms
The last two terms give rise to the following vertexes
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χ
χ
h
χ
h
h
χ
In this model, there are no other connections to the SM than through these two Higgs
couplings. The three legged vertex is most important, as it allows a simple scattering
with the Higgs as a propagator. We will not go further than three level in any of our
calculations.
Depending on the experiment employed to detect DM; different calculable quantities
are of interest. For indirect detection cross sections for various annihilation channels to
SM particles are naturally important. For direct detection the important quantity is the
elastic scattering cross section between the DM scalar and a direct detector nucleus.
4.1.1
Dark Matter-Nucleus Scattering Cross Section
In this section we calculate σ(N χ → N χ), the cross section for the DD direction of fig. 3
assuming spin independent scattering. At the tree level we have
N
χ
χ
n
h
h
≈ A×
χ
N
χ
n
where time is upward and the approximation is, that the nucleus scattering is proportional
to a scattering off a single nucleon times the the atomic number. n being the nucleons.
Loop diagrams like
χ
h
N
χ
h
N
are allowed but heavily suppressed. The general elastic scattering cross section for two
initial and final states is (ref. [5])
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1
σ=
4Eχ EN
Z
d3 pχ0 d3 pN 0
|M|2 (2π)4 δ (4) (pχ0 + pN 0 − pχ − pN ).
(2π)6 4Eχ0 EN 0
(4.4)
Working out the Feynman amplitude (see appendix A.4 for details in cross section derivation)
one obtains
|M|2 = 4m2χ m2p A2 fn2 .
(4.5)
where fn is given by
fn = λf
mp
.
m2H mχ
(4.6)
Solving the integral in equation 4.4 yields a simple cross section, which in the limit of zero
momentum transfer is
µ2 2
σ=
ζ ,
(4.7)
4π
where µ is the reduced mass of the DM-nucleus system, ζ = Zfp + (A − Z)fn = Afn , fn/p
describes the strength of coupling to neutrons and protons, A is the number of protons
and neutrons and Z is the number of protons in the nucleus. In this model protons and
neutrons are indistinguishable in interactions i.e. fn = fp .
Before we move on to explore how good this model fits to experiments, we take a closer
look at the actual behaviour of this scattering cross section. This is interesting to us, since
the scattering cross section corresponds to the assumption from particle physics, discussed
in equation 3.13. Taking Amp ≈ mN and Ω to be a constant of energy dimension −4, the
cross section is
!2
mN µ
σ=
Ω.
(4.8)
mχ
As of now we don’t know λ, but we guess the electroweak scale for m2H /λ = v 2 =
(246GeV)2 ∼ 6.0 × 104 GeV2 so that we can plot the behaviour. We have in natural units
1
Ω=
4π
λf
m2H
!2
≈ 2.0 × 10−12 GeV−4 ≈ 3.0 × 10−67 cm4 ,
(4.9)
and we can plot the scattering cross section in the parameter space of dark matter mass
and nucleus mass.
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log10 σ
3
2
[cm ]
10
−34
−35
[GeV]
−36
2
−37
10
−38
m
χ
−39
1
−40
10
−41
−42
0
10
−43
0
20
40
60
mN [GeV]
80
100
Figure 10: Here contour lines are powers of tens of cm2 . Light nuclei have a strong
advantage in this model, especially for detecting light DM.
In this model a light dark matter particle clearly has a much smaller scattering cross
section, and will consequently be harder to detect (if the assumption of the velocity
distribution is not dominating). In the next section we will see that, assuming this model,
a light candidate ∼ 8GeV is in fact favoured by some experiments; although they still
disagree. Note here, that this this cross section behaviour is different from the constant
cross section assumed in the example from section 3.1.3.
4.1.2
Investigating whether Real Scalar DM is a good Candidate
The real scalar dark matter with a Higgs exchange is a nice model because it is very simple,
but there are some problems; chiefly that the important direct detection experiments
disagree under assumption of this model. Furthermore a real scalar cancels the possibility
of an asymmetric genesis, which would be a nice possibility to include.
To see disagreement between experiments under assumption of this model we introduce
the DM-proton cross section by setting A = Z = 1 in equation 4.7
µ2p
|fp |2 .
(4.10)
4π
For the moment we forget fn = fp and we factor equation 4.10 out of the recoil rate and
sum over different isotopes. Now we can write
σp =
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Dark Matter Experiments
X
dR
= σp
κi
dQ
isotopes
µi
µp
!2
2
fn IAi Z + (Ai − Z) ,
fp (4.11)
where κi is the fraction of each isotope in the detector and Ai is the corresponding atomic
number. IAi is the remaining integral from 3.15
IAi
mi
= nχ 2 FN2 (Q)
µi
Z vesc
dv f (v)v.
(4.12)
vmin
We will assume these integrals to be similar for different isotopes (as checked by ref. [11]).
Setting equal interactions with protons and neutrons, the experimental recoil rate would be
X
dR
= σp,exp
κi
dQ
isotopes
µi
µp
!2
IAi A2i .
(4.13)
Equations 4.10 and 4.13 now let us write the actual cross section in terms of the experimental
one (letting IAi cancel)
κi µ2i A2i
.
κi µ2i |Z + (Ai − Z)fn /fp |2
P
σp = σp,exp P
(4.14)
Obviously σp = σp,exp if fn = fp . However it turns out, that by choosing fn /fp 6= 1,
one can force agreement between experiments, although such a fraction cannot be obtained
having only a Higgs exchange.
10-39
10-37
Σ p in cm2
Σ p in cm2
10-40
10-41
10-38
10-42
7.5
8.0
MΦ in GeV
8.5
9.0
7.5
8.0
8.5
9.0
MΦ in GeV
Figure 11: Here is shown the DM/proton cross section vs. the dark matter mass. The
blue contour is CoGent-favoured region at 90% confidence interval, the green region is the
DAMA/LIBRA 3σ and the dashed line is one above which all is excluded by CDMSII. In
the left figure fn /fp = 1 and the right fn /fp = −0.71. The right features an allowed region
shown in red, for a dark matter candidate around 8 GeV and proton cross section around
σp = 2 · 10−38 cm2 . This figure is recreated from plots in reference [11].
The model we have build in the previous section corresponds to the left plot of figure
11. To construct the effect from the right figure, we need an interaction term allowing
fn /fp = −0.71. It should be noted that the allowed overlap at fn /fp = −0.71 is very
sensitive, it could be a coincidence, but it seems like a big one. In the rest of this chapter
we will examine a mechanism that can accommodate the allowed region.
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4.2
Upgrading to a Complex Field that Differentiates
Protons and Neutrons
A few motivated extensions to the previous model are in order. First of all, if the real
scalar field is upgraded to a complex one and we simultaneously extend the Z2 symmetry
to a general complex phase, i.e. a U(1) symmetry, then our candidate could have had an
asymmetric origin, which is a possibility that we gladly include.
We saw empirically that fn /fp = −0.71 would let experiments agree. An interaction we
could ad to accommodate this, is one that feels the positive electric charge of the proton,
since this is the main feature that distinguishes protons and neutrons. A weak interaction
←
→
that couples the dark matter current χ∗ ∂µ χ to the electromagnetic current ∂ν F µν can be
imagined, we call this the dipole interaction. In the end of this chapter we will discuss how
an otherwise electrically neutral scalar could interact via the dipole.
If we only have the dipole, the WIMP would not feel the neutrons, so in order to have
fn 6= 0 we keep the Higgs interaction. The Lagrangian for this upgraded model is thus
1
λ
L = LSM + ∂µ χ∗ ∂ µ χ − m2χ χ∗ χ − g(χ∗ χ)2 − χ∗ χh2 + λvχ∗ χh + Ldipole ,
2
|2
{z
}
(4.15)
interaction terms
→
βe ∗ ←
χ ∂µ χ∂ν F µν ,
(4.16)
2
Λ
where e is the electromagnetic coupling, β contains the unknown coupling between the
WIMP and the photon (in 4.1 β = 0), Λ is some relevant energy scale and is squared since
the dipole is of dimension 6 and the Lagrangian is of dimension 4. F µν = ∂ µ Aν − ∂ ν Aµ is
the standard electromagnetic field strength tensor, Aµ being the photon. Since the field
is now complex the mass term is without a 1/2 pre-factor, so the acquired mass after
spontaneous symmetry breaking is m2χ = m20 + λv 2 /2 instead.
Ldipole =
We make a similar approximation to earlier: that the dipole interaction with the whole
nucleus is roughly the number of protons Z, times proton scattering:
χ
χ
N
χ
γ
≈ Z×
N
p
γ
χ
p
Again time is upward.
This photon exchange interferes with the coupling to the proton. If we call the
interference to the coupling δ, we can write it as a function of the dipole coupling β. We
write
fp (λ, β) = fn (λ) + δ(β).
(4.17)
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Dark Matter Experiments
Now we want to figure out how to tune λ and β to achieve fn /fp = −0.71, this condition
can be written as
1
fn (λ0 ).
0.71
δ(β0 ) = − 1 +
(4.18)
Before we can figure out how β0 and λ0 must be tuned, we must evaluate the diagram.
The evaluation of the Feynman diagram with the photon exchange can be done straight
forwardly by reading off the Feynman rule in the Lagrangian for the vertex
χ
p0
χ
p
γ
pγ
and using the well known rules from quantum electrodynamics for the remainder (see this
derivation in appendix A.5). A less tedious way is to treat the photon exchange as an
effective contact interaction, corresponding to the diagram
χ
p
γ
χ
p
χ
p
−→
χ
p
This is usually only possible for heavy mediators where the transferred momentum is negligible, but in this case there are two derivatives in ∂ν F µν cancelling the 1/q 2 from the
photon propagator, which makes it possible to consider the contact interaction. We use
the non-homogeneous Maxwell equations to couple to the electromagnetic current, which
in the units µ0 = ε0 = 1 are
∂ν F µν = J µ .
(4.19)
The conserved current can be written in terms of the charged protons. For fermionic
conserved current we have (ref. [5])
J µ = eZ p̄γ µ p.
(4.20)
Our dipole interaction thus reads
Zβe2 ∗
(χ ∂µ χ − χ∂µ χ∗ )p̄γ µ p.
(4.21)
Λ2
And now we can evaluate the diagram, we denote dark matter momenta p, p0 and proton
momenta k, k 0 . Writing in momentum Fourier space
Ldipole =
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Dark Matter Experiments
χ ∼ e−ip·x ,
χ∗ ∼ eip·x
(4.22)
we have
χ
p
χ
p
Zβe2
(−ipµ − ip0µ )us γ µ ūr
Λ2
2iZβe2
'−
pµ us γ µ ūr = iM̃dipole ,
Λ2
=
(4.23)
where u and ū are spinors for initial and final protons and r, s is initial and final spins, and
p − p0 = k 0 − k = q with −ipµ − ip0µ = −2ipµ + iqµ → −2ipµ in the limit of zero momentum
transfer. We use g µν and gµν to raise and lower indices, and the identities for momenta
pµ pµ = m2 . Summing over final and averaging over initial spins we get the amplitude
2βe2
Λ2
!2
1
= Z2
2
2βe2
Λ2
!2
2
2βe2
Λ2
1X
1
|Mdipole | =
|M̃dipole |2 = Z 2
2 r,s
2
2
= 4Z
X
(pν us γ µ ūr ) × (ur γµ ūs pν )
r,s
0
m2χ tr[(k/ + mp )γ µ (k/ + mp )γµ ]
!2
m2χ m2p .
(4.24)
We now have the total Feynman amplitude to have two contributions
|MHiggs + Mdipole |2 = 4m2χ m2p (Afn + Zδ)2 ,
δ=−
2βe2
8πβα
=− 2 ,
2
Λ
Λ
(4.25)
(4.26)
where α = e2 /4π is the fine structure constant. Taking δ < 0 as the non-spin summed
M̃dipole < 0 and δ is defined from Mdipole = 2mχ mp Zδ.
If experiments are reconciled at fn /fp = −0.71 we get a constraint,
√ assuming
√ mχ = 8GeV
as suggested by figure 11. Incidentally 1/0.71 is very close to 2 as 1/ 2 = 0.7071, it
further adds to how incredible the coincidence would be if a famous irrational number just
happens to be the correct fraction; even though we have no micro-physical explanation for
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Dark Matter Experiments
this, it seems likely that there is one. We also take mp = 0.938GeV, mH = 125 GeV and
α ≈ 1/137
√
1
δ
=−
≈ − 2,
1+
fn
0.71
√
8παβ0 mχ m2H
= 2 + 1,
2
Λ λ0 f mp
β0
=
Λ2
√
2 + 1 f mp
λ0 ≈ 2.97 · 10−5 GeV−2 × λ0 .
8π αmχ m2H
(4.27)
If we fix σp = 2 · 10−38 cm2 we also get a constraint from the proton cross section in equation
4.10 which is
λ0 =
!
√
2 σp π
β0 mχ m2H
+ 8πα 2
.
µp
Λ
f mp
(4.28)
In the end we find (up to a common sign) by plugging in all values
β0
∼ 2.8 · 10−4 GeV−2 ,
Λ2
λ0 ∼ 9.5.
(4.29)
This is how we fine tune the coupling parameters β/Λ2 and λ if experiments has to agree.
Remember, while discussing the event rate, experimental reconciliation was decided to be a
criterion for a good dark matter model. Luckily reconciliation can be achieved in our model.
Now, if we tune the model as needed, is this dark matter? Well, probably not. The
biggest reason being, that we for simplicity assumed dark matter to be a scalar particle,
and there was no physical reason for doing so. In nature only the Higgs boson has ever been
observed as a scalar, and the Higgs may even be composite and therefore not fundamental.
However, if lucky, a scalar description of dark matter could be effectively correct. If
there are no fundamental scalars in nature, but DM in some energy regime can be described
by a composite scalar, then this model could be true. Although it does not describe any of
the micro-physics composing the scalar.
Another motivation for this scalar to be composite is to explain how the dipole interaction
arises on the electrically neutral particle. A dipole can only exist on the scalar if it has
an electrically charged substructure. In analogy, the substructure of the neutron (1 up
and 2 down quarks) was discovered, by the existence of a small magnetic moment on the
otherwise electrically neutral neutron indicating the existence of quarks.
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5
An Effective Dark Matter Theory for Scalar SM Singlets
Building the dark matter model in the previous chapter we assumed many things. Besides
assuming a scalar singlet protected under a U(1) symmetry, we also picked the Higgs
interaction term and a dipole interaction term. The first was picked as a simple interaction,
and the latter was introduced completely ad hoc to get fn /fp to equal −0.71 and thus
make experiments agree as seen in figure 11.
In this chapter we wish to relax the previous assumptions and explore the correspondingly more general model. This will illustrate the possible expansions and modifications
that we can think of for future scalar DM modelling. The only assumption from now on is
that dark matter can be described by a scalar field acting as a singlet under the symmetries
of the Standard Model. In reference [12] the whole array of operators has been classified,
not only for singlets w.r.t. the SM, but also for doublets and triplets. It would also be
interesting to do a similar classification assuming dark matter to be fermionic, but we do
not consider that option here.
Now we look at all the relevant interaction terms for the general DM Lagrangian, this
is the DM self-interaction and interactions with the Standard Model fields.
To this end we (re-)introduce a cut-off energy scale Λ, in chapter 4 this was just a
dimensionfull constant, here we set it to be some unknown (possibly large) energy scale
above which a more fundamental theory of dark matter emerges, and the effective description
(probably) breaks down.
We will write operators in reciprocal powers of this cut-off and satisfy ourself with
(energy) dimension = 6 interactions, such that possible higher dimensional terms are
suppressed by Λ−k with k > 2. Remembering the energy dimensions [L] = 4, a scalar field
[φ] = 1, [∂φ] = 2, a fermion [ψ] = 3/2 and the field strength tensor [F µν ] = 2. In this
←
→
chapter whenever Jµ appears, it is the DM current χ∗ ∂µ χ, not the EM current as in the
previous chapter.
5.1
Self-interactions
In chapter 4 the included self-interactions was the kinetic, mass and quartic terms
g
(5.1)
∂µ χ∗ ∂ µ χ − m2χ χ∗ χ − (χ∗ χ)2 .
2
If our only limitation is 6 energy dimensions we can write many more terms, including
derivative interactions with non-derivatives, as long as all indices are properly summed
over. Here all coefficients are real and made dimensionless by proper powers of Λ
∂µ χ∗ ∂ µ χ − m2χ χ∗ χ +
3
X
i=2
ai
(χ∗ χ)i
+
Λ2i−4
1
c
(b1 ∂µ (χ∗ χ)∂ µ (χ∗ χ)+b2 ∂µ (χ∗ χ)J µ + b3 Jµ J µ ) + 2 (∂µ ∂ µ χ∗ )(∂ν ∂ ν χ).
2
Λ
Λ
(5.2)
Of the 6 coefficient besides the kinetic and mass term, we only have a2 = − g2 6= 0. It is
clear that many possibilities are left to consider, and some could influence how the velocity
distribution behaves.
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Dark Matter Experiments
5.2
Interactions with the Higgs
In chapter 4 we included two terms with the Higgs scalar
λ ∗ 2
χ χh .
(5.3)
2
Similarly we can imagine many more terms up to dimension 6 if we simply count the
powers and write all terms having both χ and h with or without derivatives
λvχ∗ χh −
χ∗ χ
4
X
ai
i=1
2
X
(∂µ χ∗ )(∂ µ χ)
2
X
hi
hi
∗ 2
+
(χ
χ)
b
+
i
Λi−2
Λi
i=1
1
X
hi
hi
∗
µ
+
∂
(χ
χ)(∂
h)
d
+
µ
i
Λi
Λi+1
i=0
ci
i=1
Jµ (∂ µ h)
1
X
i=0
ei
hi
(∂µ h)(∂ µ h)
∗
+
f
(χ
χ)
.
Λi+1
Λ2
(5.4)
In our microscopic model only coefficients a1 = λv and a2 = −λ/v are different from zero.
5.3
Interactions with Fermions of the Standard Model
We take the interaction with the fermions of the standard model (ψ) to be of the forms
χ∗ Oχ χψ̄Oψ ψ,
(5.5)
Oχ (χ∗ χ)ψ̄Oψ ψ,
(5.6)
where O are operators such that the interaction is a Lorentz scalar, and ψ̄, ψ are any two
SM fermions that in combination are electrically neutral. Our operators are
←
→
Oχ ∈ {1, ∂µ , ∂ µ },
←
→µ
Oψ ∈ {1, γ µ , D },
(5.7)
→µ
←
→µ ←
where D ≡ ∂ − ieQAµ is the covariant derivative, where Aµ is the photon coming
from the SM U(1). The possible interactions are thus
1 ∗
χ χψ̄ψ,
Λ
1
∂µ (χ∗ χ)ψ̄γ µ ψ,
Λ2
1
Jµ ψ̄γ µ ψ,
Λ2
←
→
1 ∗
χ χψ̄i D ψ;
2
Λ
(5.8)
(5.9)
(5.10)
(5.11)
←
→
in the last interaction Dµ is sandwiched between two spinors, therefore the index is
suppressed and need not be summed over like γ matrices. Beyond these terms are 4 other,
because letting Oψ −→ Oψ γ 5 still leaves the interaction as a Lorentz scalar. γ 5 = γ 0 γ 1 γ 2 γ 3
is the chirality matrix.
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In the chapter 4 model we have no explicit fermion interactions. Although interestingly,
the interaction 5.10 arose from the dipole when using the equation of motion from electromagnetism (Maxwell inhomogeneous: ∂ν F µν = p̄γ µ p) in the context of direct detection. In
fact, this is also to be expected, since we effectively treated it as a contact scattering.
5.4
Interactions with Gauge Bosons of the Standard Model
In the Standard Model gauge bosons are carriers of forces. The photon carries the
electromagnetic force, the W and Z bosons carry the weak nuclear force and the gluons
carry the strong force.
In our model we only had the dipole interaction that included the photon. Gauge
invariance under the electromagnetic U(1) part of the standard model only allows the
photon to enter terms via the field strength tensor. In fact no terms of dimension lower
than six appears, but we do find two new ones
1 ∗
χ χFµν F µν ,
Λ2
1 ∗
χ χFµν F̃ µν ,
Λ2
1
Jµ ∂ν F µν ,
Λ2
(5.12)
(5.13)
(5.14)
where F̃ µν = µνρσ Fρσ and µνρσ is the Levi-Civita symbol. The interaction
1
∂µ (χ∗ χ)∂ν F µν ,
(5.15)
Λ2
vanishes identically, since one can perform an integration by parts in the Lagrangian
density and use that ∂ν F µν is the conserved electromagnetic current.
There are many more interactions with the Z and W bosons, the first of which emerge
at dimension 4. The first interactions with the gluons are at dimension 6. Instead of listing
them all here, we refer once again to [12] where all can be found.
Beyond the possibilities listed and referred to in the sections of this chapter, one can
imagine mixed type interactions for example including both χ, ψ and h through
1 ∗
χ χψ̄ψh,
Λ2
(5.16)
or one of many other possibilities.
5.5
The General Scalar Dark Matter Model up to
Dimension 6 and Future Prospectives
From the partial list of possible interactions in this chapter, it is clear that there is a
wide variety of potential parameters to tune in a general theory of complex scalar dark
matter. If we call the set of interactions that has been found I, then we can write the
general Lagrangian density for complex scalar dark matter up to dimension 6 as a linear
combination of i ∈ I
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LSDM =
#I
X
cn in .
(5.17)
n=1
In principle we can write all these terms and tune all the cn coefficients. However,
it is more difficult to tune many coefficient simultaneously. The model in chapter 4 is
obviously included in this general Lagrangian, where most of the coefficients are identically
zero. In fact we only had two free parameters, and the fn /fp = −0.71 requirement gave
us two conditions: a favoured proton cross section and favoured dark matter mass. If we
include more interactions than two, we still only get the two constraints. Therefore our set
of couplings will be under determined and we must scan the parameter space for values
that gives us the desired experimental reconciliation (or find more constraints). In the
future this could be very interesting to do with some of the other interactions that has
been listed in this chapter. With the work from this thesis in mind, it would be especially
interesting to examine some of the other terms that allow differentiation between protons
and neutrons of direct detectors.
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6
Summary and Conclusions
In this thesis we have investigated aspects of dark matter experiments, in particular we
have focused on direct detection, and the theoretical background behind this type of dark
matter experiment. We have constructed a quantum field theory to treat in this context.
In chapter 2 key points of the dark matter problem was presented, and the dark matter
paradigm was introduced.
In chapter 3 the main strategies for detection of dark matter was introduced, and
the theory behind direct detection was treated in greater depth. The only observables in
direct detection are the recoils, these depend on both astrophysical and particle physics
assumptions. Variations in these assumptions have been tested in section 3.1.3, finding
that the assumed velocity distribution of dark matter and detector properties can change
the observed recoil rate significantly. We conclude that detectors with a threshold energy
as low as possible are preferable. Lastly we discuss exploitation of annual modulation
effects, that originates from Earth’s revolution around the Sun, as this is a technique that
yields a strong argument for dark matter.
In chapter 4 we constructed the simplest dark matter extension to the Standard Model
that we could think of. From our model we got a dark matter candidate that interacts
with the nuclei in direct detectors. In this model, dark matter is a real scalar field
interacting through a Higgs exchange. We saw in section 4.1.2, that experiments disagree
when assuming the Higgs as the only mediator. This situation was remedied by treating
protons and neutrons in detector nuclei differently, specifically having the ratio between
the couplings fn /fp = −0.71 (this fraction was obtained from reference [11]). In section 4.2
we introduced a dipole interaction and tuned the couplings to the Higgs and dipole, such
that this feature was accommodated. In the end we saw that a scalar model of dark matter
can in fact alleviate tensions between experiments; if the free parameters are properly
tuned. It is quite remarkable that experimental disagreement can
√ be fixed in a relatively
simple way. Furthermore we noted that the fraction 0.71 ≈ 1/ 2, which is probably a
coincidence, although it is curious that the fraction√alleviating experimental disagreement,
just happens to be the famous irrational number 2.
In the last chapter we introduce an effective theory, where we try to think of all
interactions for scalar dark matter up to energy dimension 6 in the low energy regime.
This encompasses the model from chapter 4 and illustrates possible model extensions, and
provides many interaction that could be interesting to investigate in the future.
Acknowledgements
First of all I thank my supervisor Francesco Sannino for providing great ideas, references
and counselling for this thesis. I also thank Ole Svendsen and Mads Frandsen for help and
for answering a lot of questions, and of course everybody who read and critiqued the thesis.
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A
Appendices
A.1
Notation
We use some notation from quantum field theory, here are some definitions. Covariant
vectors are written with lower index and contravariant with upper index, the derivatives
are
∂
,
∂xµ
∂
,
∂xµ
(A.1)
←
→
φ† ∂ φ ≡ φ† ∂φ − (∂φ† )φ,
(A.2)
Ā = A† γ 0 ,
(A.3)
∂µ =
/ = γ µ Aµ ,
A
∂µ =
where γ µ are the 4 × 4 γ-matrices satisfying the anti-commuting relation (Clifford algebra)
{γ µ , γ ν } = 2g µν ,
(A.4)
where g is the Minkowski metric with sign convention (+, −, −, −) and µ, ν ∈ {0, 1, 2, 3}.
The chirality matrix is defined as (not called γ 4 for strange historical reasons)
γ 5 = iγ 0 γ 1 γ 2 γ 3 .
(A.5)
We write Feynman amplitudes as M̃ and call them M after spin summation has been
performed.
Explicit values in this thesis are derived using the natural unit convention of k = c =
h̄ = 1
A.2
Flat Friedmann Universe
The Friedmann Equations come from Einsteins Field Equations in GR, assuming the
Friedmann-Lemaître-Robertson-Walker metric, i.e. the cosmological principle and an ideal
fluid universe. The first equation is
H 2 (t) +
kc2
8πGρ(x)
=
,
2
a (t)
3
(A.6)
where H(t) = ȧ(t)/a(t) is Hubble’s constant, a(t) is the scale factor, and the critical density
is ρc = 3H 2 (t)/8πG. Rewritten and taking the density fraction to be Ω = ρ/ρc we get
kc2 = ȧ2 (t)(Ω − 1).
(A.7)
Since k ≈ 0 and we live in an expanding universe we have Ω ≈ 1.
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A.3
Spontaneous Electroweak Symmetry Breaking
The electroweak symmetry is SU(2)L × U(1)y where an SU(N ) means N × N matrices that
are unitary U means unitary i.e. group elements u ∈ U(N ) have the property u† u = 1 and
S means special and is the requirement that the determinant det(u) = 1. Since elements
from SU(2) are 2 × 2 matrices, the Higgs is written as a doublet
!
H=
φ1 + iφ2
,
φ3 + iφ4
(A.8)
where φi ’s are real scalar fields. The potential is
V = −m2H H † H − λH (H † H)2 ,
(A.9)
where λH is the Higgs self coupling taken here to be negative. The minimal potential is at
H † H = 0 if m2H < 0, if however m2H > 0 it is at
0=
dV
= −m2H − 2λH (H † H),
d(H † H)
−m2H
.
2λH
φ21 + φ22 + φ23 + φ24 =
(A.10)
Choosing some φi ’s correspond to picking a ground
state and breaking the symmetry. In
q
√
2
particular we take φ1 = φ2 = φ4 = 0 and φ3 = −m2H /(2λH ) = v/ 2 such that
!
1 0
H0 = √
.
(A.11)
2 v
√
√
Plugging in and rewriting φ1 + iφ2 = 1/ 2(ψ1 + iψ2 ) and φ3 + iφ4 = 1/ 2(v + h + iψ3 )
!
1
ψ1 + iψ2
H=√
.
2 v + h + iψ3
(A.12)
By a gauge transformation some fields can be transformed away, in particular choosing
unitary gauge is
!
1
0
.
H=√
v
+
h
2
A.4
(A.13)
Scattering Cross Section for Scalar Higgs-coupled DM
In this section we denote p, p0 as incoming and outgoing momentum for the DM scalar and
k, k 0 for the nucleons.
The dominant Feynman diagram is at the tree level. If the model only couples to the
Standard Model through the Higgs exchange, we have for neutrons n and protons p
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N
χ
χ
n
h
h
≈ Z×
χ
N
χ
χ
n
p
h
+ (A − Z)×
χ
p
The Higgs couples almost equally to protons and neutrons with f mp /v. f is the
dimensionless parameter containing the physics coupling the Higgs to the nucleon, the
value is approximately f ∼ 0.3 (ref. [11]). So with a Higgs exchange here is really only one
diagram with the Feynman amplitude
iM̃Higgs
"
#
f mp r
i
= A × ū (pN 0 ) i
u (pN )
[iλv],
v
(pN 0 − pN )2 − m2H
s
(A.14)
where mp is the nucleon mass and v ' 246GeV. We can take the limit of zero transferred
momentum pN − p0N = q → 0 since it in any case is much smaller than the Higgs mass.
Summing over final spins s and averaging over initial spins r at the amplitude level gives
1X
1
|M| =
|M̃Higgs |2 =
2 s,r
2
2
=
1
2
Af mp λ
m2H
!2
A2 f mp λ
m2H
Af mp λ
=2
m2H
Af mp λ
=4
m2H
X
ūs ur ūr us
s,r
!2
0
tr[(k/ + mp )(k/ + mp )]
!2
(k · k 0 + m2p )
!2
m2p ,
(A.15)
P
where uū = k/ + mp , and taking that the trace of any odd number of γ matrices is zero.
Again in the zero momentum transfer limit k · k 0 → k 2 = m2p .
The cross section σ(N χ → N χ) is spin independent and can be calculated in the center
of mass frame, where the center of mass energy is ECM = mχ + mN and p = −k. In
the non-relativistic limit Eχ and EN is not much greater than the rest energy. We use
A × M as the amplitude, because the way M is defined, we obtain 2 powers of mp and
only multiply once with A, so we must multiply with A once more to take into account
the mass of the whole nucleus.
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d3 p0 d3 k0
1
|A · M|2 (2π)4 δ (4) (p0 + k 0 − p − k)
σ=
4Eχ EN vr
(2π)6 4Eχ0 EN 0
1
|p|
=
A2 |M|2
16π Eχ EN ECM |vχ − vN |
Z
m2χ m2N
µ
=
4π Eχ EN (mχ + mN )
'
Af mp λ
m2H mχ
!2
µ2 2 2
A fn ,
4π
(A.16)
where fn = f mp λ/(m2H mχ ) is a factor that determines the coupling to the nucleons.
A.5 Scalar DM with Dipole Interference:
Derivation with Photon Propagator
If the scattering could feel the electric charge of proton very weakly we would have
fp = fn + δ.
(A.17)
A possible interaction could be:
βe
Jµ ∂µ F µν .
Λ2
Writing out the interaction one immediately finds
Ldipole =
(A.18)
→
βe ∗ ←
χ ∂µ χ [∂ µ ∂ ν + g µν ] Aν .
(A.19)
Λ2
Writing the dark matter fields in momentum Fourier space, we can choose the convention
for positive and negative modes
Ldipole =
χ ∼ e−ip·x ,
0
χ∗ ∼ eip ·x .
(A.20)
The Feynman diagram of interest is
χ
χ
γ
p
p
We treat the vertex by reading of Ldipole and using conservation of momentum p = p0 +pγ
and defining the transferred momentum as p − p0 = k 0 − k = pγ ≡ q
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χ
p0
χ
p
γ
pγ
i
h
βe µ ν
2 µν
0
×
−p
p
+
p
g
−ip
−
ip
γ γ
γ
µ
Λ2
i
h
iβe
= − 2 [2p − q]µ × −q µ q ν + q 2 g µν
Λ
i
2iβe h µ ν
= − 2 pµ −q q + q 2 g µν ,
Λ
=
(A.21)
using the pretty identity −qµ −q µ q ν + q 2 g µν = 0 in the last equality, courtesy of the
photon.
γ
pγ
k0
p
k
p
= us [eZγ µ ]ūr ,
(A.22)
where u, ū are spinors for incoming and outgoing protons respectively with spin indices
s, r.
m2γ
Now we can write the entire diagram (γ is virtual and thus off shell meaning q 2 6=
= 0)
χ
χ
γ
p
p
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i g
2iZβe2 h µ ν
νρ
2 µν
× 2 × ur γ ρ ūs
−q
q
+
q
g
p
µ
Λ2
q
2iZβe2
p·q
r ρ s
=−
p
−
q
ρ
ρ ×u γ ū = iM̃dipole
Λ2
q2
=−
|
{z
Pρ
(A.23)
}
1X
1
|Mdipole |2 =
|M̃dipole |2 = Z 2
2 r,s
2
2βe2
Λ2
!2
1
= Z2
2
2βe2
Λ2
!2
1
= Z2
2
2βe2
Λ2
!2
1
= Z2
2
2βe2
Λ2
!2
1
= Z2
2
2βe2
Λ2
!2
= 4Z
2βe2
Λ2
!2
2
= 4Z
2βe2
Λ2
!2
2
= 4Z
2βe2
Λ2
!2
2
X
tr[P ρ ur γ µ ūs us γµ ūr Pρ ]
r,s
P2
X
tr[ūr ur γ µ ūs us γµ ]
r,s
0
P 2 tr[(k/ + mp )γ µ (k/ + mp )γµ ]
0
P 2 tr[(−2k/ + 4mp )(k/ + mp )]
0
P 2 tr[−2k/ k/ + 4m2p ]
P 2 m2p
m2p
(p · q)2
p −
q2
!
2
m2p m2χ 1 − cos2 θ .
(A.24)
We could have rewritten P in terms of the proton momentum k, which should yield the
same result. Since p and k are oriented anti-parallel in the center of mass frame, q is taken
to be at a right angle to p that is cos2 θ = 0. This condition is the same as saying the
second part of the P operator in equation A.23 is identically zero, which is also true when
q is acting on the free field (see ref. [6] page 134). We now obtain the same result as
treating the scattering as a contact interaction in section 4.2.
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[3] Barbara Ryden Introduction to Cosmology 2006.
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Bachelor Thesis: Hot vs. Cold Dark Matter 2013
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Dark Matter Effective Theory 2012
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