1. Constraining maverick dark matter through direct
detection experiments
Zachary David Nasipak
Vassar College
Department of Physics and Astronomy
May 2015
3. Chapter 1
Historical background
As scientists we have spent centuries probing the universe. Our investigations have navigated
both the large and the small, from studying the interactions between atoms and molecules
to observing the dynamic evolution of the stars and galaxies that populate the expanse of
the night sky. And yet this matter, known as baryonic matter—the matter we directly see,
measure, and with which we constantly interact; the matter that we have devoted almost
the entire history of science to investigating—only appears to make up a minor fraction of
our universe.
The ΛCDM model currently provides the most accurate picture of our physical universe.
It models the cosmos through four different components: radiation, baryonic matter, cold
dark matter, and dark energy. According to recent data from the Planck satellite, in this
model, baryonic matter constitutes less than 5% of the total mass and energy in this universe.
Cold dark matter constitutes roughly 26%, providing the largest contribution to the matter
content of our universe [1].
Unfortunately we know very little about cold dark matter. In fact, we have never directly
observed it. But as we continue to observe the universe, we repeatedly make unpredicted
observations that are best explained by the existence of dark matter. Evidence supporting
dark matter’s existence dates back to the 1930’s, when Sinclair Smith’s and Fritz Zwicky’s
analyses of galaxy velocities in the Virgo and Coma Clusters led to discrepancies between
predicted and measured galactic masses [2, 3]. Based on the luminosity of the galaxies
in these clusters, astronomers had predicted Virgo and Coma to be much less massive than
what both Smith and Zwicky calculated using the virial theorem. Astronomers hypothesized
that a significant amount of non-luminous or extremely faint matter must exist within the
2
4. individual galaxies or in the intergalactic medium, although researchers were unsure if this
matter was just an overabundance of brown dwarfs, black holes, neutron stars, or something
entirely new and different.
In 1970, Vera Rubin and Kent Ford analyzed the rotation curve of M31—our neighboring
galaxy, Andromeda—and discovered that the rotational velocities of stars, gas, and dust in
the outer disk remain constant beyond 10 kpc. However the distribution of luminous, bary-
onic matter predicted that the velocities would drop off far away from the galactic center
[4]. Morton Roberts and Robert Whitehurst expanded upon Rubin and Ford’s research with
radio observations and demonstrated that these flat rotation curves extended to the farthest
ends of the galaxy, even out at 30 kpc from the galactic center [5]. Ultimately, these measure-
ments indicate that a majority of the galaxy’s matter resides in an extended, spherical halo,
yet when we look at the baryonic matter in M31, it is predominantly distributed in just the
galactic plane. Thus astronomers were further convinced that some halo of non-luminous,
non-baryonic matter constituted a majority of M31’s mass. As astronomers investigated the
rotation curves of other galaxies, they discovered that M31’s flat rotation curve is not a
unique case, but a common feature of spiral galaxies, indicating that this dark matter halo
is an integral component of most galaxies, even the Milky Way [6].
Of course, dark matter has not been the only proposed solution to these unexpected
observations. Many have suggested that these inconsistencies are not due to a fundamen-
tal misunderstanding of the matter content of the universe, but instead due to an incorrect
understanding of gravity at large-scale distances and small accelerations. While general rela-
tivity and the Newtonian approximation may be accurate for solar system-scale interactions,
some believe that gravitational interactions behave differently at small accelerations and that
the force of gravity is actually described by a new theory of Modified Newtonian Dynamics
(MOND) [7].
However recent weak lensing observations—measurements of the bending of light by grav-
ity as predicted by general relativity—of galaxy cluster collisions, such as 1E 0657-558, the
Bullet Cluster, favor theories of dark matter over MOND [8]. When two galaxy clusters
collide, the stellar matter—stars, planets, black holes—from the two galaxies hardly collides
or interacts with itself due to the large distances that separate these objects. On the other
hand, the gas that pervades the interstellar and intergalactic mediums in these clusters col-
lides and is separated from the clusters as they pass through one another. When astronomers
observed 1E 0657-558 by plotting its mass distribution through weak gravitational lensing
they noticed that a majority of the mass resides around the stellar regions [9]. If dark mat-
3
5. ter does not exist, then most of the system’s mass is from the gas and therefore after two
galaxy clusters collide astronomers would expect the mass distribution to peak in-between
the two clusters, centered around the hot, ionized gas. If dark matter does exist, then most
of the cluster’s mass comes from the dark matter, in fact over 80% of it, and therefore, if the
dark matter also hardly collides or interacts just like the stellar matter, then astronomers
would expect most of the mass to reside in the two clusters themselves, not with the gas.
Ultimately the observations of 1E 0657-558 agree with dark matter predictions.
Therefore, over the last century, observations have consistently promoted the existence of
a non-baryonic, non-luminous, massive form of matter. Unfortunately this cold dark matter
cannot be accounted for by the standard model of particle physics: the model that best de-
scribes the particle nature of the universe. The standard model is one of the most predictively
powerful and experimentally tested theories in science. According to the standard model,
all matter in the universe is comprised of fundamental particles either known as fermions or
bosons, which interact with each other through at least three fundamental forces: the strong
force, the weak force, and the electromagnetic force. Fermions (half-spin particles) are made
up of six quarks—the up, down, charm, strange, top, and bottom quarks—and six leptons—
the electron and electron neutrino, the muon and muon neutrino, and the tau lepton and tau
neutrino. Quarks are bound together by the strong force, and systems composed of the three
quarks are known as baryons. Protons and neutrons are the most stable baryons. Protons
are made up of two up quarks and one down quark, while neutrons are made up of two down
quarks and one up. On the other hand, gauge bosons (integer-spin particles)—the photon,
the Z boson, the W boson, the gluon—typically mediate the exchange of forces between
particles in the standard model. Bosons also include the Higgs boson, however the Higgs
does not primarily function as a force carrier.
Again, while the standard model of particle physics is well tested, it still provides a
limited framework for understanding the universe. This limitation in the standard model,
along with its inability to incorporate the force of gravity, indicates the need for a higher
fundamental theory—a beyond the standard model (BSM) theory—which would possess
additional particles and forces, including potential dark matter particle candidates. In fact,
the theoretical models of BSM dark matter particles are limitless since there are few physical
constraints, but our observations do reveal a sliver of information about dark matter. For
instance, if dark matter exists, then it must be cold dark matter rather than hot dark
matter. Hot dark matter would lead to top-down large-scale formation in the universe: after
decoupling from photons and baryonic matter in the early universe, cold dark matter particles
4
6. would move at relativistic speeds, a process known a “free-streaming,” eventually smoothing
out the primordial fluctuations observed in the cosmic microwave background (CMB); only
superclusters would form at first, then eventually galaxy clusters, then galaxies, etc. with
larger gravitational systems developing before smaller ones. However, when we look into
the distant universe, we observe stars and galaxies forming first, then coming together to
form clusters, clusters forming superclusters, and so on—the type of large-scale structure
formation instead predicted by non-relativistic cold dark matter [10].
There are many different models of cold dark matter, however they are motivated by
theory and phenomenology more than by observation. Some models predict dark matter to
be small, primordial black holes, while others predict that dark matter is merely composed
of axions—hypothetical pseudoscalar particles that also suppress charge-parity (CP) viola-
tion in quantum chromodynamic (QCD) interactions [11, 12]. Weakly interacting massive
particles (WIMPs) provide possibly the most popular dark matter candidates due to their
simplicity and experimental falsifiability. They are required to interact with baryonic matter
through a weak-scale interaction (weakly interacting) and the gravitational force (massive
particles) [10]. Due to their weak-scale interactions, WIMPs can be detected not just by
their gravitational effects on massive systems, but also by their interaction with detector
nuclei due to scattering cross-sections. While each model has its own advantages, no the-
ory possesses overwhelming experimental basis over the rest. Therefore, while confidence in
dark matter continues to grow, our understanding remains in the dark. For now we can only
speculate and ponder the properties of dark matter, until a direct observation, beyond grav-
itational interactions, can provide more evidence on the second most important component
of our universe.
5
7. Chapter 2
Introduction
2.1 Motivation
WIMP models provide well-motivated frameworks for studying potential dark matter can-
didates, because they require dark matter to interact with other standard model particles
independent of the gravitational interaction. This provides astrophysicists an avenue for
both predicting and measuring scattering interactions between dark matter particles and
detector nuclei for various WIMP models.
2.1.1 The WIMP miracle
In the ΛCDM model, our universe begins with the big bang. In the first few moments, the
universe is small and dense; it is highly energetic and filled with a primordial soup: a plasma
of standard model particles that are continually interacting and annihilating with each other
in a state of thermal equilibrium. As the universe expands and cools, the particles move
farther apart and are less likely to interact. Eventually particles begin to decouple as their
decay rate, Γ, falls below the expansion rate of the universe, which is defined by the Hubble
parameter, H ≡ ˙a/a, where a is the scale factor of our universe (the current scale factor
a0 = 1). For weakly interacting particles, this moment of decoupling is also known as the
time of freezeout, tf , because after this point it is very unlikely that two weakly interacting
particles will annihilate. Therefore the number of particles for a weakly interacting species is
essentially “frozen in.” By calculating the density of a species at freezeout, it is then possible
to correlate it with the present relic density of the species. Since freezeout depends on the
moment when Γ H and the decay rate of a species depends on its annihilation cross
6
8. section, σX ¯X→ψ ¯ψ, such that Γ ∼ σX ¯X→ψ ¯ψ, we can constrain the annihilation cross section
of WIMPs by using our current understanding of the thermodynamics of the early universe
and the present relic density of dark matter, ΩXh2
.
The evolution of a species’ number density is dictated by the Boltzmann equation
ˆL[f] = C[f] (2.1)
where ˆL is the Liouville operator that describes the evolution of the particle density in phase
space according to the laws of statistical mechanics, C is the collision operator that describes
the loss and gain of particles in phase space due to interactions, and f is the distribution of
particle momenta in phase space [13]. For WIMPs, f is given by the Fermi-Dirac distribution,
f(p) =
1
eE/T + 1
(2.2)
assuming that the chemical potential µ = 0, E is the energy of a WIMP defined as E2
=
|p|2
+ m2
X, p is the momentum of the WIMP and mX is the mass of the WIMP. For an
expanding universe, the Boltzmann equation for the number density of WIMPs, nX, is
derived to be
dnX
dt
+ 3HnX = − σX ¯X→ψ ¯ψ|v| [(nX)2
− (neq
X )2
] (2.3)
where σX ¯X→ψ ¯ψ|v| is the thermally-averaged annihilation cross section of WIMPs and neq
X is
the number density of WIMPs prior to freezeout, when the WIMPs were in thermodynamic
equilibrium [14, 15]. Note however that this is only true if self-annihilation is the dominant
decay route for WIMPs and that coannihilations and resonance annihilations with other BSM
particles do not occur or are heavily suppressed [16]. From thermodynamics, the number
density of a species in thermal equilibrium is
neq
=
g
(2π)3
f(p)d3
p (2.4)
Since, even in the early universe, WIMPs are both cold (meaning non-relativistic) and mas-
sive, we can take the limiting case of mX >> T of equation 2.4, so that
neq
X = g
mXT
2π
3
2
e−
mX
T (2.5)
where g is the number of internal degrees of freedom such that g = 2 for WIMPs. Theoreti-
7
9. cally we can now solve the differential equation, equation 2.3, to get nX as a function of H,
σ|v| , and neq
X . Note that we have simplified notation so that σX ¯X→ψ ¯ψ|v| is re-expressed
as σ|v| . Unfortunately this leaves us with three unconstrained parameters: σ|v| , mX,
and T since neq
X depends on the last two. If we want to constrain the WIMP annihilation
cross section using measurements of the present relic density, ΩXh2
, we first need to find a
way to constrain the other parameters.
Let us define the variables x ≡ mX/T and Y ≡ nX/T3
. This allows us to rewrite the
equations above as
Yeq = g
x
2π
3
2
e−x
(2.6)
dY
dt
= −T3
σ|v| [Y 2
− Y 2
eq] (2.7)
dY
dx
= −
λ
x2
[Y 2
− Y 2
eq] (2.8)
where λ ≡ m3
X σ|v| /H(m) [17]. While equation 2.8 does not have any analytic solutions,
we can make analytical approximations by assuming that λ is a very large constant and that
after freezeout Y Yeq. We will also define freezeout as the point when x = xf ≡ mX/Tf .
These approximations are reasonable because λ will be large with a m3
X dependence and
because after freezeout we expect WIMPs to annihilate less often and therefore not decay
as quickly as they did when in thermal equilibrium. Additionally, as the universe expands,
temperature decreases also contributing to an increase in Y . This allows us to solve for Y
as a function of x. We can solve for xf by comparing Y (x) with Yeq since these functions
begin to diverge at freezeout. Across multiple values of λ 1, numerical analyses lead to
xf ∼ 10–20 [17].
This same process can be repeated more rigorously. Other theorists prefer to define
Y ≡ nX/s, where s is the entropy density given by s = 2π2
g∗T3
/45, and g∗ is the number
of relativistic degrees of freedom. The entropy density is typically preferred because it is a
conserved quantity such that sa3
= constant [14]. Using this method changes equation 2.8
to
dY
dx
= −
x σ|v| s
H(m)
[Y 2
− Y 2
eq] (2.9)
Ultimately even a more mathematically and theoretically rigorous computation for weakly
interacting particles reveals a similar freezeout value to before: xf ∼ 20–30 [14] [18]. This
is great news—so great that this is often referred to as the “WIMP miracle.” By finding a
constraint of xf = 20, σ|v| remains as the only free parameter in equation 2.9. This allows
8
10. us to finally relate the present relic density of dark matter to Y and σ|v| , leading to the
approximate solution [19]
ΩXh2
≈
3 × 10−27
cm3
s−1
σ|v|
(2.10)
The latest measurements from the Planck satellite, combined with analyses of baryonic
acoustic oscillations (BAOs) and measurements from the Atacama Cosmology Telescope
(ACT), the South Pole Telescope (SPT), and the Wilkinson Microwave Anisotropy Probe
(WMAP) [1], reveal
ΩXh2
= 0.1187 ± 0.0017 (2.11)
with a best fit value of
ΩXh2
= 0.11889 (2.12)
allowing us to constrain the thermally averaged annihilation cross section of WIMPs, such
that
σ|v| ≈ 2.5 × 10−26
cm3
s−1
(2.13)
While initially this value may not seem inherently fascinating, it is truly a miraculous
result. As described earlier, we know that there are several problems with the standard
model, which suggests that some BSM theory must exist which would include new particles,
like dark matter. If a BSM theory predicts a stable particle with interactions mediated by
an electroweak scale interactions then its cross section can be approximated by
σ|v| ∼ α2
(100 GeV)−2
(2.14)
where α is the relative strength, also know as the coupling constant, of the mediating force
[19]. At an electroweak scale, α ∼ 10−2
. Therefore
σ|v| ∼ 10−25
cm3
s−1
∼ σX ¯X→ψ ¯ψ|v| (2.15)
This means that BSM theories with particles acting on electroweak scales, and many of
these theories do exist, make perfect dark matter candidates. And there is no fundamental
reason for dark matter to be so easily reconcilable with BSM theories. If the universe had a
different relic density, then the constrained WIMP cross section could be vastly different than
theoretical candidates. If we were unlucky, the cross section could have been constrained
on the order of 100 cm3
s−1
, leaving us with a dark matter model that is far removed from
our current understanding of particle physics. Instead we have derived a cross section that
9
11. corresponds with our need need for BSM theories.
Therefore the WIMP model provides a powerful framework because many theoretical
models have natural WIMP candidates and we can constrain the annihilation cross section
of WIMPs using the present relic density of dark matter, informing us of the relationship
between the coupling strength of WIMPs and their masses.
2.1.2 Comparing with direct detection results
By understanding the interaction properties of the WIMP, such as its interaction type,
coupling strength, and mass, we can predict the ability of WIMP models to scatter off of
detector nuclei, providing a way to experimentally test and constrain WIMP candidates.
Several direct detection experiments have conducted measurements within the last decade,
the major four projects being XENON100, CDMS II, SuperCDMS, and COUPP. Therefore
there is finally experimental data that can test the legitimacy of many WIMP models and
theories.
Unfortunately, to date, direct detection experiments have been fruitless in detecting dark
matter. Of course this does little to inform us of what dark matter is like, but from these
results we can learn a lot about what dark matter is not like. This is important because
each WIMP model is free to choose several parameters, such as its interaction type, its
coupling strength, and the particle mass. This leads to a nearly infinite set of WIMP mod-
els to test. Theorists have devised a multitude of WIMP models, each motivated by their
own BSM theory, however by only studying theory-motivated WIMP models we exclude
ourselves from considering WIMP scenarios that have not yet been theoretically envisioned.
Therefore to investigate a broad, yet finite set of WIMP candidates, it is best to consider
phenomenologically-motivated dark matter models—models that are physically accessible
by experiments such as high-energy particle colliders or dark matter direct detectors. Ulti-
mately we can constrain certain combinations of WIMP interaction properties by comparing
their theoretical scattering properties with detector nuclei with the lack of significant results
reported by the most advanced direct detection projects. By doing so we can exclude specific
mass ranges, interaction types, or model-dependent assumptions for WIMPs.
2.2 Maverick WIMPs
In this paper we will consider the maverick WIMP model: a simple, theory-independent,
phenomenologically-motivated dark matter framework [15, 20, 21]. To suppress possible
10
12. coannihilations or resonance annihilations, this model assumes that the maverick WIMP is
the next heaviest particle beyond the standard model and that it interacts with fermions,
predominantly standard model quarks, through a BSM force. This force is mediated by a
gauge boson whose mass is outside the energy range of the Large Hadron Collider (LHC)
(Mψ > 14 TeV) and is much more massive than maverick WIMPs (Mψ mX). As a theory-
independent model, both scalar (spin-0) and Dirac fermion (spin-1/2) dark matter WIMPs
are considered along with all interaction types—scalar, pseudoscalar, vector, axial vector,
and tensor—though the maverick model assumes that only one set of interactions dominates.
The values for the coupling strengths of these interactions are also left as free parametersm
however two types of coupling will be considered: universal coupling and a Yukawa-like
coupling. In particle physics, Yukawa coupling is the coupling that takes place between
fermions and the Higgs field and is ultimately how the masses of the twelve fermions arise
in the standard model [22]. A similar mechanism could mediate the coupling of maverick
WIMPs with standard quarks, therefore, for Yukawa-like coupling, the interaction strength
between WIMPs and quarks depends on the mass of the quark, such that Fi,q, Gi,q ∼ mq. For
universal coupling, there is no mass dependence. Due to its phenomenological motivation,
the maverick WIMP mass is restricted to 2 GeV mX 104
GeV, with the lower limit set
by the sensitivity of direct detection experiments and the upper limit by the energy reach of
the LHC.
Taking this model of dark matter, we can derive the thermally-averaged cross section of
maverick WIMP annihilations as a function of WIMP mass, particle type, interaction type,
and coupling strength. By examining each interaction and particle type independently, we
can find a relationship between mass and coupling strength, since, as demonstrated above,
the thermally-averaged cross section is constrained by the present relic density. From this
we can then relate the theoretical scattering cross section of maverick WIMPs with detector
nuclei to the WIMP mass and coupling strength, giving us theoretical detection limits for
different types of maverick dark matter.
As a model-independent analysis, we will consider the thermally-averaged cross sections
for all possible combinations of particle and interaction types. The particle and interaction
types are outlined in Table 2.1 and Table 2.2 respectively.
11
13. Particle
Spinor/Scalar
Spin
field operators
Scalar maverick WIMP φ 0
Dirac fermion maverick WIMP χ 1/2
Standard model quark q 1/2
Table 2.1: A table of the particle types considered in the analysis of maverick WIMPs. Scalar
particles are described by solutions to the Klein-Gordon equation: (∂µ
∂µ + m2
)φ = 0. Dirac
fermions are described by solutions to the Dirac equation: (i/∂ − m)ψ = 0 where ψ → χ for
WIMPs and ψ → q for quarks.
Interaction Operator
Scalar 1
Pseudoscalar γ5
Vector
∂µ
γµ
Axial Vector γµ
γ5
Tensor σµν
Table 2.2: A table of the interactions types considered in the analysis of maverick WIMPs.
12
14. Chapter 3
Deriving annihilation cross sections
for maverick WIMPs
3.1 Effective field theory
The standard model and the field of particle physics provides a framework for deriving
particle interactions based on the fundamental qualities of particles and the forces exchanged
between them. This framework can be extended to the maverick dark matter model for
which we can derive a thermally-averaged annihilation cross section that depends only on
the coupling strength and the mass of maverick WIMPs. We do so by first developing an
effective field theory for maverick dark matter. This effective field theory approximates how
maverick dark matter interacts with the rest of the standard model particles and can be
described by the Lagrangian
L = LSM + Lkinetic + Lmass + Lint (3.1)
where LSM is the standard model Lagrangian, Lkinetic + Lmass describes the non-interacting,
free-particle terms, and Lint describes interaction terms for maverick WIMPs. For non-
hermitian scalar maverick WIMPs, the interaction Lagrangian is
Lkinetic + Lmass = ∂µφ†
∂µ
φ −
1
2
m2
φφ†
φ (3.2)
Lint =
q
Fi
√
2
φ†
Γiφ ¯qΓj
q (3.3)
13
15. where φ is a solution to the Klein-Gordon equation, (∂µ
∂µ + m2
)φ = 0, φ†
= (φ∗
)T
, and
Γi is the set of all interaction types possible for scalar maverick WIMPs, such that Γi =
{1, γ5
, ∂µ
, γµ
, γµ
γ5
} [21, 22].
Likewise, for fermionic WIMPs,
Lkinetic + Lmass = ¯χ(i/∂ − m)χ (3.4)
Lint =
q
Gi
√
2
[¯χΓiχ] ¯qΓj
q (3.5)
where χ is a solution to the Dirac equation, (i/∂ − m)ψ = 0, /∂ = γµ
∂µ, ¯χ = χ†
γ0
, and
Γi is the set of all interaction types possible for fermionic maverick WIMPs, such that
Γi = {1, γ5
, γµ
, γµ
γ5
, σµν
} [21, 22].
All possible interactions between maverick WIMPs and standard model quarks are de-
tailed in Table 3.1, along with the labels corresponding to each interaction.
3.2 Representative cross section calculation
As an illustration of the derivation process from an effective field theory to an annihilation
cross section, we will derive in full the thermally-averaged annihilation cross section of vector-
interacting maverick WIMPs (F-V).
3.2.1 Interaction Lagrangians and matrix elements
In the center of mass frame, the annihilation cross section is
σ =
1
64π2s
|p∗
f |
|p∗
i |
|Mfi|2
dΩ∗
(3.6)
where ∗ indicates the center of mass frame, pi is the momentum of an initial state particle,
pf is the momentum of a final state particle, |Mfi|2
is the spin average of the squared
Lorentz-invariant matrix element, and dΩ is the differential solid angle typically defined as
dΩ = dφ d cos θ. Additionally s is one of the Mandelstam variables, defined as s = (p1 +p2)2
,
where p1 and p2 are the four-momenta of the annihilating particles. Therefore for the rest
of this derivation we will assume the center of mass frame.
For vector interactions between Dirac fermions, the Lorentz-invariant matrix element is
14
16. Particle Operator Coupling Cross Section Label
φ
φ†
φ¯qq FS σS-S S-S
φ†
φ¯qq FS,q ∼ mq σS-SQ S-SQ
φ†
φ¯qγ5
q FSP σS-SP S-SP
φ†
φ¯qγ5
q FSP,q ∼ mq σS-SPQ S-SPQ
φ†
∂µφ¯qγµ
q FV σS-V S-V
φ†
∂µφ¯qγµ
γ5
q FV A σS-VA S-VA
χ
¯χχ¯qq GS σF-S F-S
¯χχ¯qq GS,q ∼ mq σF-SQ F-SQ
¯χχ¯qγ5
q GSP σF-SP F-SP
¯χχ¯qγ5
q GSP,q ∼ mq σF-SPQ F-SPQ
¯χγ5
χ¯qγ5
q GP σF-P F-P
¯χγ5
χ¯qγ5
q GP,q ∼ mq σF-PQ F-PQ
¯χγ5
χ¯qq GPS σF-PS F-PS
¯χγ5
χ¯qq GPS,q ∼ mq σF-PSQ F-PSQ
¯χγµχ¯qγµ
q GV σF-V F-V
¯χγµχ¯qγµ
γ5
q GV A σF-VA F-VA
¯χγµγ5
χ¯qγµ
γ5
q GA σF-A F-A
¯χγµγ5
χ¯qq GAV σF-AV F-AV
¯χσµνχ¯qσµν
q GT σF-T F-T
Table 3.1: A table of the possible interactions between maverick WIMPs (both scalar and
fermionic) and standard model quarks. The operator within the interaction Lagrangian is
expressed for each interaction combination in the Operator column, while the labels corre-
sponding to the coupling strength and cross section of each set of interactions are listed in
the far right columns. In the Coupling column, Fi,q, Gi,q ∼ mq denotes Yukawa-like coupling.
In particle physics, Yukawa coupling is the coupling that takes place between fermions and
the Higgs field and is ultimately how the masses of the twelve fermions arise in the standard
model [22]. A similar mechanism could mediate the coupling of maverick WIMPs with stan-
dard quarks, therefore, for Yukawa-like coupling, the interaction strength between WIMPs
and quarks is proportional to the mass of the quark.
15
17. defined as
Mfi =
g2
V
q2 − M2
ψ
[¯v(p2)γµ
u(p1)][¯u(p3)γµv(p4)] (3.7)
where gV is the coupling strength at each interaction vertex, Mψ is the mass of the exchanged
gauge boson, q is its four-momentum, and u and v are the spinor field operators. We are
considering Γ3 = γµ
, therefore equation 3.5 becomes
Lint =
q
GV
√
2
[¯χγµ
χ][¯qγµq] (3.8)
where GV is the coupling constant at each interaction vertex, q is the summation across
interactions with each type of quark, and the
√
2 is introduced according to convention in
Fermi’s effective theory of the weak interaction. To find the spin average of the squared
matrix element, we sum over all the spin states of the spinors and divide by the number of
spin combinations, giving us
|Mfi|2
=
1
4 spin
|Mfi|2
(3.9)
By recollecting terms and introducing the WIMP tensor Lµν
(χ) that sums the spins over all
initial state χ and ¯χ spinors and the quark tensor L
(q)
µν that sums the spins over all initial
state q and ¯q spinors, we can reintroduce equation 3.9 as
|Mfi|2
=
1
4
Lµν
(χ)L(q)
µν (3.10)
Additionally, if we consider the completeness relations, we can take advantage of trace algebra
techniques to solve for the spin average of the squared Lorentz-invariant matrix element as a
function of the Mandelstam variable, s, the mass of the quark, mq, the mass of the maverick
WIMP, mχ, and the angle of the incoming WIMP particle in the center of mass frame, θ.
This gives us
|Mfi|2
=
1
4 q
3
G2
V
2
Tr([/p2
− mχ]γµ
[/p1
+ mχ]γν
) × Tr([/p3
+ mq]γµ[/p4
− mq]γν) (3.11)
where /p = γµ
pµ. We include the summation so that interactions across all quarks are
considered, while the factor of 3 accounts for the three quark colors.
16
18. 3.2.2 Momenta calculations
In order to solve equation 3.11, we first need to solve for the momentum of each of the four
particles involved in maverick WIMP annihilations. Due to energy conservation, the energy
of the initial state, E, will be split equally between the two initial state particles and will
also be equivalent to the energy of the final state, E , which is also split equally between the
two final state particles. This leads to
E = E =
√
s
2
and s = 4E2
(3.12)
We can express the four-momentum of each particle as
p1 = (E, p)
p2 = (E, −p)
p2
1 = E2
− |p|2
= m2
χ
p3 = (E , p )
p4 = (E , −p )
p2
3 = E 2
− |p |2
= m2
q
(3.13)
By combining equations 3.12 and 3.13 we find that
|p|2
=
s
4
− m2
χ
=
1
4
(s − 4m2
χ)
|p| =
1
2
(s − 4m2
χ)1/2
|p |2
=
s
4
− m2
q
=
1
4
(s − 4m2
q)
|p | =
1
2
(s − 4m2
q)1/2
(3.14)
Additionally by taking the dot product between the momenta of the two initial state particles
and the two final state particles described by 3.13
p1 · p2 = E2
+ |p|2
=
s
4
+
s
4
− m2
χ
=
1
2
(s − 2m2
χ)1/2
p3 · p4 = E 2
+ |p |2
=
s
4
+
s
4
− m2
q
=
1
2
(s − 2m2
q)1/2
(3.15)
while if we take the dot product across the combination of initial and final state momenta,
we find
p1 · p3 = p2 · p4 = E2
− |p||p | cos θ
=
s
4
−
1
4
(s − 4m2
χ)1/2
(s − 4m2
q)1/2
cos θ
(3.16)
17
20. Now by considering the momenta relations we derived in Section 3.2.2, we can simplify
equation 3.20 to
Tr([/p2
− mχ]γµ
[/p1
+ mχ]γν
) × Tr([/p3
+ mq]γµ[/p4
− mq]γν) (3.21)
= 32[
1
16
(s + (s − 4m2
q)1/2
(s − 4m2
χ)1/2
cos θ)2
+
1
16
(s − (s − 4m2
q)1/2
(s − 4m2
χ)1/2
cos θ)2
+
1
2
m2
χ(s − 2m2
q) +
1
2
m2
q(s − 2m2
χ) + 2m2
χm2
q]
= 2[s2
+ 2s(s − 4m2
q)1/2
(s − 4m2
χ)1/2
cos θ + (s − 4m2
q)(s − 4m2
χ) cos2
θ
+ s2
− 2s(s − 4m2
q)1/2
(s − 4m2
χ)1/2
cos θ + (s − 4m2
q)(s − 4m2
χ) cos2
θ
+ 8s(m2
χ + m2
q)]
= 2 [2s2
+ 2(s − 4m2
q)(s − 4m2
χ) cos2
θ + 8s(m2
χ + m2
q)]
3.2.4 Annihilation cross section
Taking our analysis in the previous section, we can now derive an expression for the anni-
hilation cross section of fermionic, vector-interacting, maverick WIMPs. First we plug our
simplified trace calculations found in equation 3.21 into our expression for the spin average
of the squared matrix element, equation 3.11. Then we can plug 3.11 into our expression for
the annihilation cross section, equation 3.6, leading to
σF-V =
1
64π2s q
3
4
G2
V
2
|p∗
f |
|p∗
i |
4[s2
+ (s − 4m2
q)(s − 4m2
χ) cos2
θ
+ 4s(m2
χ + m2
q)]dΩ
(3.22)
=
1
64π2s q
3
4
G2
V
2
|p |
|p|
2π
1
−1
4[s2
+ (s − 4m2
q)(s − 4m2
χ) cos2
θ
+ 4s(m2
χ + m2
q)]d cos θ
(3.23)
19
21. =
1
64πs q
3 G2
V
|p |
|p|
[2s2
+
2
3
(s − 4m2
q)(s − 4m2
χ) + 8s(m2
χ + m2
q)] (3.24)
=
1
32π q
3 G2
V
s − 4m2
q
s − 4m2
χ
s + 4m2
χ +
(s − 4m2
q)(s − 4m2
χ)
3s
+ 4m2
q (3.25)
3.2.5 Thermally-averaged cross section
In order to constrain our cross section we need to finally derive its thermal average. The
thermally-averaged cross section is given as
σ|v| =
1
8m4
χTK2
2 (mχ/T)
∞
4m2
χ
σ(s − 4m2
χ)
√
sK1(
√
s/T)ds (3.26)
where K1 is the first order modified Bessel function of the second kind and K2 is the second
order modified Bessel function of the second kind [13].
Substituting in x ≡ mχ/T, y ≡ mq/mχ, and z ≡ s/4m2
χ, we can rewrite equation 3.25 as
σF-V =
1
32π q
3 G2
V
z − y2
z − 1
z + 1 +
1
3
(1 − z−1
)(z − y2
) + y2
4m2
χ (3.27)
This allows us to rewrite the thermally-averaged cross section, equation 3.26 as
σ|v| F-V =
64m6
χ
8m4
χK2
2 (x)
∞
4m2
χ
σ
4m2
χ
(z − 1)
√
s
T
K1(2x
√
z)dz (3.28)
=
8m2
χ
K2
2 (x)
∞
4m2
χ
σ
4m2
χ
(z − 1)2x
√
zK1(2x
√
z)dz (3.29)
=
8m2
χ
K2
2 (x)
∞
4m2
χ
1
32π q
3 G2
V
z − y2
z − 1
× z + 1 +
1
3
(1 − z−1
)(z − y2
) + y2
(z − 1)2x
√
zK1(2x
√
z)dz
(3.30)
20
22. Therefore the thermally-averaged cross section for the annihilation of fermonic maverick
WIMP with vector interactions is
σ|v| F-V =
3
2π q
G2
V m2
χ
x
K2
2 (x)
∞
4m2
χ
z−1/2
(z − 1)1/2
(z − y2
)1/2
× z2
+ z +
1
3
(z − 1)(z − y2
) + zy2
K1(2x
√
z) dz (3.31)
By setting equation 3.31 equal to the relic density constraint derived in Section 2.1.1, 2.5 ×
10−26
cm3
s−1
, we finally have an implicit correlation between the coupling strength, GV and
the mass of a maverick WIMP, mχ.
3.3 Extended results
While this derivation only considered one of the many interaction forms possible between
maverick WIMPs and standard model quarks, it is straightforward to extend this same
derivation process to all other interaction operators outlined in Table 3.1 [21]. The results
for all interactions with scalar maverick WIMPs are
σS-S =
1
16π q
3 F2
S
s − 4m2
q
s − 4m2
φ
s − 4m2
q
s
(3.32)
σS-SQ =
1
16π q
3 F2
S,q
s − 4m2
q
s − 4m2
φ
s − 4m2
q
s
σS-SP =
1
16π q
3 F2
SP
s − 4m2
q
s − 4m2
φ
σS-SPQ =
1
16π q
3 F2
SP,q
s − 4m2
q
s − 4m2
φ
21
23. σS-V =
1
16π q
3 F2
V
s − 4m2
q
s − 4m2
φ
2(s − 4m2
φ)(s + 2m2
q)
3s
σS-VA =
1
16π q
3 F2
V A
s − 4m2
q
s − 4m2
φ
2(s − 4m2
φ)(s − 4m2
q)
3s
Likewise the results for all interactions with fermionic maverick WIMPs are
σF-S =
1
32π q
3 G2
S
s − 4m2
q
s − 4m2
χ
(s − 4m2
χ)(s − 4m2
q)
s
(3.33)
σF-SQ =
1
32π q
3 G2
S,q
s − 4m2
q
s − 4m2
χ
(s − 4m2
χ)(s − 4m2
q)
s
σF-SP =
1
32π q
3 G2
SP
s − 4m2
q
s − 4m2
χ
(s − 4m2
χ)
σF-SPQ =
1
32π q
3 G2
SP,q
s − 4m2
q
s − 4m2
χ
(s − 4m2
χ)
σF-P =
1
32π q
3 G2
P
s − 4m2
q
s − 4m2
χ
2s
σF-PQ =
1
32π q
3 G2
P,q
s − 4m2
q
s − 4m2
χ
2s
σF-PS =
1
32π q
3 G2
PS
s − 4m2
q
s − 4m2
χ
2(s − 4m2
q)
22
24. σF-PSQ =
1
32π q
3 G2
PS,q
s − 4m2
q
s − 4m2
χ
2(s − 4m2
q)
σF-V =
1
32π q
3 G2
V
s − 4m2
q
s − 4m2
χ
s + 4m2
χ +
(s − 4m2
q)(s − 4m2
χ)
3s
+ 4m2
q
σF-VA =
1
32π q
3 G2
V A
s − 4m2
q
s − 4m2
χ
s + 4m2
χ +
(s − 4m2
q)(s − 4m2
χ)
3s
− 4m2
q −
16m2
χm2
q
s
σF-A =
1
32π q
3 G2
A
s − 4m2
q
s − 4m2
χ
s − 4m2
χ +
(s − 4m2
q)(s − 4m2
χ)
3s
− 4m2
q +
32m2
χm2
q
s
σF-AV =
1
32π q
3 G2
AV
s − 4m2
q
s − 4m2
χ
s − 4m2
χ +
(s − 4m2
q)(s − 4m2
χ)
3s
+ 4m2
q −
16m2
χm2
q
s
σF-T =
1
32π q
3 G2
V
s − 4m2
q
s − 4m2
χ
6s + 4m2
χ +
8(s − 4m2
q)(s − 4m2
χ)
3s
+ 4m2
q
Similarly we can derive thermally-averaged cross section for all interactions. For scalar
maverick WIMPs,
σ|v| S-S =
3
4π q
F2
S
x
K2
2 (x)
z−1/2
(z − 1)1/2
(z − y2
)3/2
K1(2x
√
z) dz (3.34)
σ|v| S-SQ =
3
4π q
F2
S,q
x
K2
2 (x)
z−1/2
(z − 1)1/2
(z − y2
)3/2
K1(2x
√
z) dz
σ|v| S-SP =
3
4π q
F2
SP
x
K2
2 (x)
z1/2
(z − 1)1/2
(z − y2
)1/2
K1(2x
√
z) dz
23
25. σ|v| S-SPQ =
3
4π q
F2
SP,q
x
K2
2 (x)
z1/2
(z − 1)3/2
(z − y2
)1/2
K1(2x
√
z) dz
σ|v| S-V =
2
π q
F2
V
x
K2
2 (x)
z−1/2
(z − 1)1/2
(z − y2
)1/2
(z + y2
/2)K1(2x
√
z) dz
σ|v| S-VA =
2
π q
F2
V
x
K2
2 (x)
z−1/2
(z − 1)1/2
(z − y2
)3/2
K1(2x
√
z) dz
For fermionic maverick WIMPs,
σ|v| F-S =
3
2π q
G2
Sm2
χ
x
K2
2 (x)
z−1/2
(z − 1)3/2
(z − y2
)3/2
K1(2x
√
z) dz (3.35)
σ|v| F-SQ =
3
2π q
G2
S,qm2
χ
x
K2
2 (x)
z−1/2
(z − 1)3/2
(z − y2
)3/2
K1(2x
√
z) dz
σ|v| F-SP =
3
2π q
G2
SP m2
χ
x
K2
2 (x)
z1/2
(z − 1)3/2
(z − y2
)1/2
K1(2x
√
z) dz
σ|v| F-SPQ =
3
2π q
G2
SP,qm2
χ
x
K2
2 (x)
z1/2
(z − 1)3/2
(z − y2
)1/2
K1(2x
√
z) dz
σ|v| F-P =
3
2π q
G2
P m2
χ
x
K2
2 (x)
z3/2
(z − 1)1/2
(z − y2
)1/2
K1(2x
√
z) dz
σ|v| F-PQ =
3
2π q
G2
P,qm2
χ
x
K2
2 (x)
z3/2
(z − 1)1/2
(z − y2
)1/2
K1(2x
√
z) dz
σ|v| F-PS =
3
2π q
G2
PSm2
χ
x
K2
2 (x)
z1/2
(z − 1)1/2
(z − y2
)3/2
K1(2x
√
z) dz
24
26. σ|v| F-PSQ =
3
2π q
G2
PS,qm2
χ
x
K2
2 (x)
z1/2
(z − 1)1/2
(z − y2
)3/2
K1(2x
√
z) dz
σ|v| F-V =
3
2π q
G2
V m2
χ
x
K2
2 (x)
z−1/2
(z − 1)1/2
(z − y2
)1/2
× z2
+ z +
1
3
(z − 1)(z − y2
) + zy2
K1(2x
√
z) dz
σ|v| F-VA =
3
2π q
G2
V Am2
χ
x
K2
2 (x)
z−1/2
(z − 1)1/2
(z − y2
)1/2
× z2
+ z +
1
3
(z − 1)(z − y2
) − zy2
− y2
K1(2x
√
z) dz
σ|v| F-A =
3
2π q
G2
Am2
χ
x
K2
2 (x)
z−1/2
(z − 1)1/2
(z − y2
)1/2
× z2
− z +
1
3
(z − 1)(z − y2
) − zy2
+ 2y2
K1(2x
√
z) dz
σ|v| F-AV =
3
2π q
G2
AV m2
χ
x
K2
2 (x)
z−1/2
(z − 1)1/2
(z − y2
)1/2
× z2
− z +
1
3
(z − 1)(z − y2
) + zy2
− y2
K1(2x
√
z) dz
σ|v| F-T =
3
2π q
G2
T m2
χ
x
K2
2 (x)
z−1/2
(z − 1)1/2
(z − y2
)1/2
× 6z2
+ z +
8
3
(z − 1)(z − y2
) + zy2
K1(2x
√
z) dz
As seen in these final results, several interaction forms ultimately share similar constraints,
leading to the following redundant results,
σS-SP σS-S (3.36)
σS-SPQ σS-SQ
25
27. σS-VA σS-V
σF-SP σF-S
σF-SPQ σF-SQ
σF-PS σF-P
σF-PSQ σF-PQ
σF-VA σF-V
σF-AV σF-A
Additionally, the scattering cross sections for S-SP, S-SPQ, S-VA, F-P, F-PQ, F-PS, and
F-PSQ, F-VA and F-AV are suppressed in the low-velocity limit and consequently cannot be
constrained by direct detection experiments [21]. Therefore only S-S, S-SQ, S-V, F-S, F-SQ,
F-V, F-A, and F-T interactions will be considered for the remainder of this analysis.
26
28. Chapter 4
Excluding parameter spaces for
maverick WIMPs
4.1 Scattering cross section
In order to constrain maverick dark matter with the results of direct detection experiments,
we must finally consider how maverick WIMPs will interact with detector nuclei. Thankfully,
the elastic scattering of WIMPs with a target nucleus has been well discussed in the literature
and therefore we will take advantage of the work of previous researchers. Goodman and
Witten first recognized that WIMPs will generally interact with detector nuclei one of two
ways: either by coupling to the spin of the nucleon or to the mass of the nucleon [23, 19]. In
its most general form, the cross section for the elastic scattering of fermionic WIMPs with
a nucleon, assuming that no momentum is transferred between the WIMP and the nucleon,
is given by
σ = 4G2
i µ2
χN C (4.1)
where µXN = mX mN
mX +mN
is the reduced mass of the system, Gi is the coupling strength of the
interaction, and C describes the particle interactions between WIMPs and the target nuclei
[19]. For fermionic WIMPs with scalar interactions,
C =
1
πG2
S
[Zfp + (A − Z)fn]2
(4.2)
where A is the atomic mass of the target nucleus, Z is the atomic number of the target nu-
cleus, and fp and fn describe the coupling of WIMPs with protons and neutrons, respectively
27
29. [19]. This leads to the elastic scattering cross section,
σχ−N =
4
π
µ2
φN [Zfp + (A − Z)fn]2
(4.3)
For scalar interactions (F-S and F-SQ), WIMPs interact quite differently with quarks of
different mass. The coupling of the Higgs bosons with heavy quarks—the charm, bottom,
and top quarks—results in the coupling of WIMPs to gluons through the triangle diagram
of a heavy quark loop [24]. Therefore fp and fn are expressed by
fp,n =
q=u,d,s
GS
√
2
f
(p,n)
T,q
mp,n
mq
+
2
27
f
(p,n)
T,G
q=c,b,t
GS
√
2
mp,n
mq
(4.4)
where GS describes a universal scalar coupling to quarks, mp and mn are the masses of the
proton and neutron, respectively, f
(p,n)
T,q describes the coupling of WIMPs with light quarks—
up, down, and strange—in the protons and neutrons, and f
(p,n)
T,G describes the coupling to
gluons. These last two factors have been determined experimentally and are given as [25]:
f
(p)
Tu = 0.020 ± 0.004, f
(p)
Td = 0.026 ± 0.005, f
(p)
Ts = 0.118 ± 0.062 (4.5)
f
(n)
Tu = 0.014 ± 0.003, f
(n)
Td = 0.036 ± 0.008, f
(n)
Ts = 0.118 ± 0.062
with f
(p,n)
T,G = 1 − u,d,s f
(p,n)
T,q leading to
f
(p)
T,G = 0.84, f
(n)
T,G = 0.83 (4.6)
Note that if we consider a Yukawa-like scalar coupling, then GS,q/mq becomes a constant,
simplifying equation 4.4 to
fp,n =
GS,q
√
2
mp,n
mq
25
27 q=u,d,s
f
(p,n)
T,q +
2
27
(4.7)
We can derive a similar elastic scattering cross section for scalar particles participating in
scalar coupling (S-S and S-SQ), although, due to there being one less fermion in the initial
and final states, a factor of (4mφ)2
is removed from equation 4.3 [21], giving
σφ−N =
1
4π
µ2
φN
m2
φ
[Zfp + (A − Z)fn]2
(4.8)
28
30. where fp and fn are similarly given by, for the S-S case,
fp,n =
q=u,d,s
FS
√
2
f
(p,n)
T,q
mp,n
mq
+
2
27
f
(p,n)
T,G
q=c,b,t
FS
√
2
mp,n
mq
(4.9)
while for Yukawa-like scalar coupling in the S-SQ case, they are given as
fp,n =
FS,q
√
2
mp,n
mq
25
27 q=u,d,s
f
(p,n)
T,q +
2
27
(4.10)
Considering vector interactions is a bit simpler, since the conservation of vector current
ensures that each quark in the detector nucleus adds coherently. Unlike in scalar interactions,
sea quarks and gluons do not contribute to the vector current and therefore there is no need
to consider the strangeness content of the nucleon—the scattering cross section only depends
on the coupling to each quark in the nucleon [19]. Therefore, for fermionic vector interactions
(F-V), the particle interactions are defined by
C =
1
256πG2
V
[2Zbp + (A − Z)bn]2
(4.11)
leading to the elastic scattering cross section
σχ−N =
1
64π
µ2
χN [2Zbp + (A − Z)bn]2
(4.12)
where bp and bn describe the WIMP’s vector coupling to the proton and neutron, respectively.
Since each quark in the nucleon adds coherently, bp = 2bu + bd and bn = bu + 2bd, where
bu and bd describe the vector coupling of maverick WIMPs to the up and down quarks [19].
For vector interactions, maverick WIMPs possess a universal coupling with standard model
quarks, therefore bu = bd = GV /
√
2, giving
bp,n = 3
GV
√
2
(4.13)
Again this derivation can be extended to the scalar maverick case, though this time a
factor of (4mφ)2
is preserved due to the WIMP’s four-momenta contribution in the elastic
scattering cross section [21]. Therefore, the scattering cross section for scalar particle vector
interactions (S-V) is
σφ−N =
1
64π
µ2
φN [2Zbp + (A − Z)bn]2
(4.14)
29
31. where similarly the coupling to protons and neutrons is defined as
bp,n = 3
FV
√
2
(4.15)
The spin-dependent scattering cross section takes a different form, since the coupling
depends on the spin content of the nucleon rather than just its mass. Spin-dependent
scattering only occurs, in terms of this analysis, for axial vector and tensor interactions
(F-A and F-T), and the particle interactions are described by [19]
C =
8
πG2
i
Λ2
J(J + 1) (4.16)
This leads to the spin-dependent scattering cross section
σχ−N =
16
π
µ2
χN Λ2
J(J + 1) (4.17)
where J is the total angular momentum of the nucleus. Λ is given as
Λ =
1
J
Sp
Gi
√
2 q=u,d,s
∆q(p)
+ Sn
Gi
√
2 q=u,d,s
∆q(n)
(4.18)
where Gi → GA for axial vector interactions, Gi → GT for tensor interactions, Sp and Sn
are the expectation values of the spin content of the proton and neutron, respectively, and
∆q(p,n)
is the fraction of the nucleon spin carried by a quark q [24]. Values for Sp , Sn ,
and ∆q(p,n)
have been determined experimentally, such that [25]
∆u(p,n)
= 0.78 ± 0.02, ∆d(p,n)
= −0.48 ± 0.02, ∆s(p,n)
= −0.15 ± 0.02 (4.19)
while values for Sp and Sn depend on the target nucleus of the detector and can be found
in Reference [19].
Now by inserting into these elastic scattering cross sections the experimental values listed
in 4.5, 4.6, and 4.19, along with the mass of the proton, the mass of the neutron, and the
30
32. mass of the quarks, which are given by [26]
mp = 938.3 MeV, mn = 939.6 MeV, (4.20)
mu = 2.15 MeV, md = 4.70 MeV, ms = 93.5 MeV,
mc = 1.27 GeV, mb = 4.18 GeV, mt = 173 GeV,
we can simplify the cross sections so that they depend only on the coupling strength and
mass of the maverick WIMP [21]. For scalar maverick WIMPs, this leads to
σS-S = 9.89 × 10−4
pb
µφp
1 GeV
2
100 GeV
mφ
2
FS
5.88 × 10−5 GeV−1
2
(4.21)
σS-SQ = 9.17 × 10−8
pb
µφp
1 GeV
2
100 GeV
mφ
2
FS,q × (1 GeV/mq)
3.01 × 10−5 GeV−1
2
(4.22)
σS-V = 1.97 × 10−3
pb
µφp
1 GeV
2
FV
1.88 × 10−6 GeV−2
2
(4.23)
For fermionic maverick WIMPs,
σF-S = 1.22 × 10−1
pb
µχp
1 GeV
2
GS
5.88 × 10−5 GeV−2
2
(4.24)
σF-SQ = 2.29 × 10−5
pb
µχp
1 GeV
2
GS,q × (1 GeV/mq)
1.67 × 10−6 GeV−2
2
(4.25)
σF-V = 1.19 × 10−4
pb
µχp
1 GeV
2
GV
4.62 × 10−7 GeV−2
2
(4.26)
σF-A = 3.92 × 10−5
pb
µχp
1 GeV
2
GA
3.26 × 10−6 GeV−2
2
(4.27)
σF-T = 1.82 × 10−7
pb
µχp
1 GeV
2
GT
2.22 × 10−7 GeV−2
2
(4.28)
31
33. Therefore by numerically analyzing the implicit relationships between the coupling con-
stant and WIMP mass, which were constrained in Section 3.2.5, and substituting that rela-
tionship into equations 4.21–4.28, we can plot the scattering cross sections as a function of
just the WIMP mass for all interaction types.
4.2 Direct detection constraints
The XENON100 detector is a 161 kg pure liquid xenon scintillator housed in Laboratori
Nazionali del Gran Sasso in L’Aquila, Italy. During its second run, the detector had an ex-
posure time of 224.9 live days. A 2013 analysis of the data reported that that no statistically
significant signals due to spin-independent or spin-dependent scattering from dark matter
WIMPs had been measured [27]. The detector sensitivity from this 2013 analysis is plotted
in Figures 4.1 and 4.5.
The Cryogenic Dark Matter Search (CDMS) II is a semiconductor array composed of over
5 kg in silicon and germanium detectors at the Soudan Underground Laboratory in Sudan,
Minnesota. CDMS II began an experimental run in 2006, with an exposure of 397.8 kg-
days before and 121.3 kg-days after all cuts, using all 30 available detectors. A 2009 release
reported that no statistically significant measurements had been made of spin-independent
scattering with dark matter WIMPs [28]. The detector sensitivity from this 2009 analysis is
plotted in Figures 4.1 and 4.5.
The SuperCDMS experiment is an extension of CDMS II, consisting of mainly updated
hardware to impressive sensitivity in the low energy limit. SuperCDMS began one of its
latest experimental runs in 2012. A 2014 report analyzed data from a subset of the ongoing
experiment, for which they considered an exposure of 577 kg-days. Again no statistically
significant measurement of spin-independent scattering with dark matter WIMPs had been
made [29]. The detector sensitivity from this 2014 analysis is plotted in Figures 4.1 and 4.5.
The Chicagoland Observatory for Underground Particle Physics (COUPP) uses a 4 kg
CF3I bubble chamber detector at SNOLAB in Sudbury, Canada to measure nucleation rates
due to spin-dependent scattering with dark matter WIMPs. The latest run began in 2010,
with an exposure time of 437.4 kg-days. Like many other analyses, results released in 2012
indicated no statistically significant measurement of spin-dependent scattering with dark
matter WIMPs was made [30]. The detector sensitivity from this 2012 analysis is plotted
in Figure 4.9. Since COUPP is one of the few projects focused on detection spin-dependent
scattering, the detector sensitivity from a 2011 analysis of a previous run at Fermilab is
32
34. included in Figure 4.9 to demonstrate the progress in spin-dependent detector sensitivity.
The previous run began in 2009 and used a 3.5 kg CF3I bubble chamber with an exposure
time of 28.1 kg-days [31].
The second generation of direct detection experiments is currently under development.
The National Science Foundation (NSF) granted the COUPP project over $2 million in
2012 to construct a 500 kg bubble chamber, while a 60 kg bubble chamber is currently
being installed and tested at SNOLAB [32]. However, COUPP, during the development of
COUPP-500, decided to partner with the Project In Canada to Search for Super symmetric
Objects (PICASSO) to develop its next generation bubble chamber detector. This new PICO
collaboration will instead use a new liquid C3F8 target that will avoid background anomalies
encountered with the CF3I target used for COUPP. Thanks to the success of the PICO 2-
liter experiment, the collaboration is using research and development from the COUPP-500
project to construct the next generation PICO 250-liter experiment [33, 34].
The NSF also granted over $6 million in 2012 for the construction of XENON1T—a one
ton liquid xenon scintillator meant to expand upon the success of XENON100 [35]. Addi-
tionally, just within the last year the NSF and the Department of Energy (DOE) approved
funding for the continued development of SuperCDMS at SNOLAB along with the construc-
tion of the LZ 7.2T—a seven tonne liquid xenon scintillator that will operate at the Sanford
Underground Research Facility (SURF) [34, 36]. Therefore the projected sensitivities for the
next generation of experiments are included in Figures 4.2, 4.4, 4.6, and 4.8.
As a side note, the direct detection experiments considered in this analysis are merely
representative of the variety of projects currently under development. Newer measurements,
such as those by the Large Underground Xenon (LUX) detector and PICO 2-liter, have been
recently completed and currently provide the best sensitivity limits for the direct detection
of dark matter [37, 38]. Unfortunately, while results have been released for these projects,
the recency of the measurements has limited their accessibility to the public. Therefore
without access to the latest data from LUX and PICO 2-liter, these experiments could not
be considered when constraining maverick dark matter through direct detection experiments
in this analysis. Future work should look to incorporate these recent measurements as
the data becomes more readily available. Additionally it is important to note that the
funding and development of future projects is always tentative and therefore the projections
considered provide insight into the future parameter spaces that may be probed by the next
generation of experiments rather than claiming that these are the parameter spaces that
will be probed. Therefore, while SuperCDMS at SNOLAB has been projected to probe for
33
35. WIMPs with masses mX < 1 GeV, in our analysis of future experiments, we instead consider
more conservative sensitivity limits that bound the SuperCDMS sensitivity around mX ∼ 2
GeV.
4.3 Exclusion plots
This following section overlays the scattering cross section of relic density-constrained mav-
erick WIMPs with the sensitivity limits of several direct detection experiments. Figures 4.1,
4.5, and 4.9 compare maverick WIMP constraints with recently conducted direct detection
experiments; therefore these plots include highlighted “exclusion regions” along with the
constrained results. These exclusion regions are parameter spaces that have already been
probed by dark matter direct detectors. Since to date no detector has measured a statis-
tically significant dark matter signal we can exclude the aspects of the maverick WIMP
models that fall in these regions. This ultimately allows us to constrain the mass range and
interaction types that maverick WIMPs can theoretically still possess.
Figures 4.2 and 4.6 compare maverick WIMP constraints with the next generation of
direct detection experiments, which are still currently under development. Additionally,
Figures 4.3, 4.4, 4.7 and 4.8 focus on the maverick WIMP parameter spaces that have not yet
been excluded by detectors, but will be probed by these next generation detectors. Figures
4.3 and 4.7 focus on the low-mass range, while Figures 4.4 and 4.8 focus on higher masses.
Altogether, Figures 4.1–4.8 consider constraints on spin-independent scattering, which occurs
for the S-S, S-Q, S-V, F-S, F-SQ, and F-V interactions, while Figure 4.9 considers constraints
on spin-dependent scattering, which occurs for the F-A and F-T interactions [21].
34
36. σφ-nucleon(pb)
mφ (GeV)
10
-16
10
-14
10
-12
10-10
10
-8
10
-6
10
-4
10-2
100
10
2
10
1
10
2
10
3
10
4
S-S
S-V
S-SQ CDMSII (2009)
XENON100 (2012)
SuperCDM
S(2014)
Figure 4.1: A comparison between relic density constraints on scalar maverick WIMPs and
constraints from direct detection experiments for spin-independent elastic scattering with
standard model quarks in the extreme low-velocity limit. Constraints on scalar maverick
WIMPs are represented by dashed lines, while constraints from recent spin-independent
direct detection experiments are illustrated by solid lines. The “exclusion regions”—the
parameter spaces that have been experimentally tested yet in which no dark matter particles
have been detected—are highlighted in yellow. Due to the phenomenologically-motivated
nature of the maverick WIMP models, only WIMP masses within the energy reach of the
LHC, mχ 10 TeV, are plotted. SuperCDMS [29] constrains the low-mass end, however due
to current technological limits, direct detection experiments leave scalar WIMP candidates
with mχ 3 GeV unconstrained. CDMS II [28] and XENON100 [27] constrain the upper-
mass limit of fermionic maverick WIMPs, excluding S-S interactions with mχ 2500 GeV,
S-SQ interactions with mχ 170 GeV, and S-V interactions with mχ 4000 GeV.
35
37. σφ-nucleon(pb)
mφ (GeV)
S-S
S-V
S-SQ
SuperCDMS SNOLAB
XENON1T
LZ 7.2T
10-16
10
-14
10
-12
10
-10
10-8
10
-6
10
-4
10-2
100
10
2
10
1
10
2
10
3
10
4
Figure 4.2: A comparison between relic density constraints on scalar maverick WIMPs and
projected constraints for future direct detection experiments for spin-independent elastic
scattering with standard model quarks in the extreme low-velocity limit. Constraints on
scalar maverick WIMPs are represented by dashed lines, while projected constraints from
future spin-independent direct detection experiments are illustrated by solid lines. Super-
CDMS at SNOLAB [39] will continue to probe the low-mass end, while XENON1T [40] and
LZ 7.2T [36] will further constrain masses around the mass of the top quark. Due to the
phenomenologically-motivated nature of the maverick WIMP models, only WIMP masses
within the energy reach of the LHC, mφ 10 TeV, are plotted.
36
38. σχ-nucleon(pb)
mχ (GeV)
SuperCDMS SNOLAB
10-7
10
-6
10
-5
10-4
10
-3
10
-2
10
-1
100
10
1
2 5 10
S-SQ
S-V
S-S
SuperCDMS (2014)
Figure 4.3: A low-mass comparison of the constraints from direct detection experiments
and of the projected constraints for future direct detection experiments for spin-independent
elastic scattering with standard model quarks in the extreme low-velocity limit. Note that
the next-generation SuperCDMS experiment at SNOLAB [39] will further exclude scalar
interactions in the low-mass range, eliminating low-mass maverick WIMP candidates down
to 2 GeV.
37
39. σφ-nucleon(pb)
mφ (GeV)
XENON1T
LZ 7.2T
10-13
10
-12
10
-11
10-10
10
-9
10
-8
10-7
10
-6
10
2
10
3
S-SQ
XENON100 (2012)
Figure 4.4: A high-mass comparison of the constraints from direct detection experiments
and of the projected constraints for future direct detection experiments for spin-independent
elastic scattering with standard model quarks in the extreme low-velocity limit. Note that
the next-generation XENON1T [40] and LZ 7.2T [36] detectors can further constrain the
existence of a scalar maverick WIMP that couples through scalar interactions proportional
to quark mass by excluding S-SQ candidates in the mass range 100 GeV mφ 300 GeV.
However they will not further constrain other scalar maverick WIMP models.
38
40. σχ-nucleon(pb)
mχ (GeV)
10
-12
10-10
10
-8
10
-6
10-4
10
-2
10
0
102
10
4
10
1
10
2
10
3
10
4
F-S
F-V
F-SQ
CDMSII (2009)
XENON100 (2012)
SuperCDM
S(2014)
Figure 4.5: A comparison between relic density constraints on fermionic maverick WIMPs
and constraints from direct detection experiments for spin-independent elastic scattering with
standard model quarks in the extreme low-velocity limit. Constraints on fermionic maverick
WIMPs are represented by dashed lines, while constraints from recent spin-independent
direct detection experiments are illustrated by solid lines. The “exclusion regions”—the
parameter spaces that have been experimentally tested yet in which no dark matter particles
have been detected—are highlighted in yellow. Due to the phenomenologically-motivated
nature of the maverick WIMP models, only WIMP masses within the energy reach of the
LHC, mχ 10 TeV, are plotted. SuperCDMS [29] constrains the low-mass end, however due
to current technological limits, direct detection experiments leave scalar WIMP candidates
with mχ 3 GeV unconstrained. CDMS II [28] and XENON100 [27] constrain the upper
mass limit of fermionic maverick WIMPs, excluding F-S interactions with mχ 10000 GeV,
F-SQ interactions with mχ 200 GeV, and F-V interactions with mχ 1000 GeV.
39
41. σχ-nucleon(pb)
mχ (GeV)
F-S
F-V
F-SQ
SuperCDMS SNOLAB
XENON1T
LZ 7.2T
10-12
10
-10
10
-8
10-6
10
-4
10
-2
10
0
102
10
4
10
1
10
2
10
3
10
4
Figure 4.6: A comparison between relic density constraints on fermionic maverick WIMPs
and projected constraints for future direct detection experiments for spin-independent elas-
tic scattering with standard model quarks in the extreme low-velocity limit. Constraints
on fermionic maverick WIMPs are represented by dashed lines, while projected constraints
from future spin-independent direct detection experiments are illustrated by solid lines. Su-
perCDMS at SNOLAB [39] will continue to probe the low-mass end, while XENON1T [40]
and LZ 7.2T [36] will further constrain masses around the mass of the top quark. Due to the
phenomenologically-motivated nature of the maverick WIMP models, only WIMP masses
within the energy reach of the LHC, mχ 10 TeV, are plotted.
40
42. σχ-nucleon(pb)
mχ (GeV)
SuperCDMS SNOLAB
10-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
101
10
2
10
3
2 5 10
F-SQ
F-V
F-S
SuperCDMS (2014)
Figure 4.7: A low-mass comparison of the constraints from direct detection experiments
and of the projected constraints for future direct detection experiments for spin-independent
elastic scattering with standard model quarks in the extreme low-velocity limit. Note that
the next-generation SuperCDMS experiment at SNOLAB [39] will further exclude fermionic
interactions in the low-mass range, eliminating low-mass maverick WIMP candidates down
to 2 GeV.
41
43. σχ-nucleon(pb)
mχ (GeV)
XENON1T
LZ 7.2T
10-12
10
-11
10
-10
10-9
10
-8
10
-7
10
-6
10-5
10
-4
10
2
10
3
F-SQ
XENON100 (2012)
Figure 4.8: A high-mass comparison of the constraints from direct detection experiments
and of the projected constraints for future direct detection experiments for spin-independent
elastic scattering with standard model quarks in the extreme low-velocity limit. Note that
the next-generation XENON1T [40] and LZ 7.2T [36] detectors can further constrain the ex-
istence of a fermionic maverick WIMP that couples through scalar interactions proportional
to quark mass by excluding F-SQ candidates in the mass range 100 GeV mχ 300 GeV.
However they will not further constrain other fermionic maverick WIMP models.
42
44. σχ-nucleon(pb)
mχ (GeV)
10-12
10
-10
10
-8
10-6
10-4
10
-2
10
0
102
10
4
10
1
10
2
10
3
10
4
F-A
F-T
COUPP-3.5kg (2011)
COUPP-4kg (2012)
PICO-250
Figure 4.9: A comparison between relic density constraints on fermionic maverick WIMPs
and constraints from direct detection experiments for spin-dependent elastic scattering with
standard model quarks in the extreme low-velocity limit. Projected constraints for future
direct detection experiments for spin-independent elastic scattering with standard model
quarks in the extreme low-velocity limit are also included. Constraints on fermionic mav-
erick WIMPs are represented by dashed lines, while constraints from recent spin-dependent
direct detection experiments led by COUPP [30, 31] and projected constraints from the
next-generation PICO 250-liter spin-dependent detector [33] are illustrated by solid lines.
The “exclusion regions”—the parameter spaces that have been experimentally tested yet in
which no dark matter particles have been detected—are highlighted in yellow. Due to the
phenomenologically-motivated nature of the maverick WIMP models, only WIMP masses
within the energy reach of the LHC, mχ 10 TeV, are plotted. Recent direct detec-
tion experiments have not been sensitive enough to exclude spin-dependent interactions for
fermionic maverick WIMPs. However PICO 250-liter will finally be able to constrain F-A
and F-T maverick WIMPs, though it will only be able to exclude maverick WIMPs with
masses mχ 200.
43
45. Chapter 5
Conclusions
Astronomical observations of our universe provide compelling evidence for the existence of a
non-luminous, non-baryonic form of matter. If this cold dark matter exists and the ΛCDM
model provides an accurate description of our universe, then dark matter not only plays
an important role in the dynamics of galaxies and galaxy clusters, but by constituting a
significant portion of the matter-energy content of our universe, also plays a pivotal role
in the large-scale formation of the cosmos. We recognize that our current physical theories
are nowhere near complete. Even the standard model, despite its predictive power and
experimental success, seems to only be a low-energy approximation of a grander beyond the
standard model (BSM) theory, which could include new forces and particles, including cold
dark matter candidates. Therefore in order to better understand the physics and evolution
of our universe, it is essential for us to further investigate the nature of cold dark matter.
We started this analysis by considering the weakly interacting massive particle (WIMP)
framework for dark matter, since this framework can be examined both theoretically and ex-
perimentally. By considering the cosmic abundance of WIMPs, described by the present relic
density of cold dark matter, and the thermodynamics of the early universe, we constrained
the thermally-averaged annihilation cross section of WIMPs. This constraint provided the
first step in connecting our theoretical framework to cosmological observations.
Next we considered how direct detection experiments could further constrain WIMP
candidates. To do so we examined the various properties that WIMPs can theoretically pos-
sess by adopting the theory-independent and phenomenologically-motivated maverick WIMP
model. With a broad basis established for our investigation, we developed an effective field
theory for maverick WIMPs in order to analytically derive the thermally-averaged annihila-
44
46. tion cross section of maverick WIMPs as a function of WIMP mass and coupling strength
for each particle-interaction set. By connecting these analytical results to the WIMP relic
density constraint, we could constrain the relation between the mass of the WIMP and its
coupling strength.
This additional constraint allowed us to finally derive the elastic scattering cross section
of WIMPs with detector nuclei as a function of the WIMP mass, giving us a set of two-
parameter spaces that we could finally compare and constrain with the sensitivity limits
of dark matter direct detection experiments. Out of all the original 19 particle-interaction
types considered, only eight could be constrained by direct detection experiments, with six
maverick WIMP cases—S-S, S-SQ, S-V, F-S, F-SQ, and F-V—interacting with detectors
through spin-independent elastic scattering and with two maverick WIMP cases—F-A and
F-T—interacting with detectors through spin-dependent elastic scattering.
For spin-independent interactions, maverick WIMPs that interact through universal cou-
plings—S-S, S-V, F-S, and F-V—have been excluded for a majority of the mass range ac-
cessible by the energy limits of current detector and collider experiments. Maverick WIMP
masses mX 3 GeV remain unconstrained, however the next generation of direct detectors
should be able to further constrain WIMP masses down to mX ∼ 1 − 2 GeV. Currently S-S,
S-V, and F-V are unconstrained for higher masses of mX 4000 GeV, and the next genera-
tion of direct detection experiments considered in this analysis have not yet been projected
to further probe this mass range.
Maverick WIMPs that interact through Yukawa-like coupling—S-SQ and F-SQ, where the
coupling strength is proportional to quark mass (Fi,q, Gi,q ∼ mq)—are much less constrained,
with higher-mass exclusions limited by about the mass of the top quark, mX 173 GeV.
This makes sense, since higher mass WIMPs have an additional annihilation channel through
the top quark, and maverick WIMPs with Yukawa-like coupling will “favor” this additional
channel, suppressing the abundance of maverick WIMPs in the early universe.
Therefore, a considerable expanse of the maverick WIMP parameter space is excluded
for spin-independent scattering WIMPs. This may be due to the fact that WIMPs simply
do not have a universal coupling to fermions or are either very heavy or very light. On the
other hand, it may simply mean that the maverick model itself makes incorrect assumptions.
For instance, it is still possible that dark matter is best described by a fermionic WIMP with
mass mX ∼ 10 GeV that universally couples with fermions through vector interactions if
there were additional processes occurring in the early universe that were not considered in the
maverick WIMP derivation, but that could prevent an overabundance of maverick WIMPs
45
47. at freezeout. The existence of additional BSM particles with masses similar to the WIMP
mass, or a gauge boson with a mass that is not significantly greater than the WIMP mass,
could lead to a series of resonance annihilations and coannihilations in the early universe,
that would have further suppressed the density of WIMPs [15, 16].
Additionally, in the maverick WIMP model, WIMPs are assumed to predominantly in-
teract with quarks since they are the most massive of the fermions and therefore interactions
with leptons are treated to have negligible affects on WIMP coupling with fermions. However
it is possible that WIMPs are leptophilic: they preferentially couple with leptons. Though
leptophilic WIMPs would not couple with quarks at tree level, higher order processes would
still allow leptophilic WIMPs to couple with quarks. Therefore, leptophilic WIMPs can still
be measured and constrained through direct detection experiments, however the scattering
cross section with detector nuclei would be significantly smaller and fundamentally different
from the maverick scenario due to the dependence on higher order processes [41]. Ultimately
in this case, collider experiments may provide better constraints on leptophilic models.
We must also consider that dark matter could be a composite particle and that it cannot
be simply expressed by a single BSM candidate. The baryonic matter in our universe is built
upon the complex interactions of quarks and gluons; if cold dark matter makes up an even
more significant portion of our universe than baryonic matter, then it is not unreasonable
to consider that, much like baryonic matter, it is comprised of constituent BSM particles,
bound together by additional BSM forces. Unfortunately the implications of this scenario
are much more difficult to consider in detail.
Thankfully the assumptions of the maverick WIMP model may still hold valid. Spin-
dependent interactions—F-A and F-T—remain completely unconstrained by recent experi-
ments, therefore fermionic, axial vector-interacting or fermionic, tensor-interacting maverick
WIMPs remain as non-falsified candidates for cold dark matter. The accuracy of these
models will be determined by the next-generation PICO 250-liter, which will finally probe
their parameter spaces and provide constraints for the spin-dependent scattering of maverick
WIMPs.
Despite the significant growth of dark matter research at both an experimental and
theoretical level, we still do not know much about this exotic and yet crucial component of
our universe. While this analysis cannot claim to have discovered the particle properties of
dark matter, we have demonstrated that we now can form a better idea of what particle
properties dark matter does not possess. If the assumptions of the maverick model are right,
dark matter most likely does not universally couple with quarks through vector or scalar
46
48. interactions. Yukawa-like coupling still provides a promising property of higher mass WIMP
candidates, yet spin-dependent scattering arising from axial vector and tensor couplings
remains the most unconstrained interaction form for maverick WIMP candidates. The next
generation of direct detectors will continue to probe unconstrained parameter spaces, most
notably in the spin-dependent case.
If next-generation direct detectors also fail to measure dark matter scattering, it may be
time to reconsider our assumptions about WIMP candidates: possibly dark matter is lep-
tophilic; possibly dark matter is a composite particle; or maybe the early universe dynamics
of dark matter is complicated by a variety of other BSM particles. Dark matter particles
might not even be WIMPs after all. They may only interact with other particles through the
gravitational force, making it virtually impossible to directly observe them. Ultimately it is
impossible to say exactly what dark matter is until we finally detect it, but by considering
all of the possible properties of this mysterious matter, we can continue investigating and
probing and testing, hoping that, through these efforts, one day the nature of dark matter
will finally be revealed.
47
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