Dark Matter Direct Detection of Classical WIMPs

Jodi Cooley1,

1 Department of Physics, Southern Methodist University, Dallas, Texas 75275,
USA

*<EMAIL_ADDRESS>
## Abstract

One of the highest priorities in particle physics today is the identification
of the constituents of dark matter. This manuscript is a supplement to
pedagogical lectures given at the 2021 Les Houches Summer School on Dark
Matter. The lectures cover topics related to the direct detection of Weakly
Interaction Massive Particle (WIMP) dark matter, including the distribution of
dark matter, nuclear scattering, backgrounds, planning and designing of
experiments, and a sampling of planned and ongoing experiments.

###### Contents

  1. 1 Introduction
  2. 2 Dark Matter Distributions
    1. 2.1 A Simplified Model of WIMP Distribution in the Milky Way
  3. 3 Nuclear Scattering of Dark Matter
    1. 3.1 WIMP-nucleus Scattering: Laboratory Frame
    2. 3.2 WIMP-nucleus Scattering: Center-of-Mass Frame
    3. 3.3 Scattering Rates in a Detector - A Simple Picture
    4. 3.4 Scattering Rates in a Detector - A More Complex Picture
    5. 3.5 The Scattering Cross Section
    6. 3.6 Effective Field Theory and the Complexity of Dark Matter Scattering
    7. 3.7 The WIMP Detection Challenge
    8. 3.8 Time-Dependence of the Scattering Rate
  4. 4 Backgrounds to Direct Detection Methodologies
    1. 4.1 Cosmic Rays
    2. 4.2 Radioactivity Primer
    3. 4.3 Event Signatures
  5. 5 Detector Planning and Design
    1. 5.1 Underground Laboratories
    2. 5.2 Activation of Detector Materials
    3. 5.3 Shielding from Environmental Backgrounds
    4. 5.4 Internal Radioactivity in Detector Material
    5. 5.5 Modeling Radioisotope Backgrounds
    6. 5.6 Designing Radiopure Materials
    7. 5.7 Neutrino Interactions in Dark Matter Detectors
  6. 6 A Summary of Needs for Direct Detection Experiments
  7. 7 A Sampling of Ongoing or Planned Direct Detection Experiments
    1. 7.1 Time-Dependent Annual Dark Matter Modulation: DAMA/LIBRA, COSINE, and ANAIS
    2. 7.2 Liquid Noble Detectors
    3. 7.3 Cryogenic Solid-State Detectors
      1. 7.3.1 SuperCDMS
      2. 7.3.2 Edelweiss
      3. 7.3.3 CRESST
    4. 7.4 Bubble Chambers and Superheated Liquids
  8. 8 Conclusion

##  1 Introduction

Detecting dark matter by _direct means_ \- that is, observing the effect of
dark matter as a particle in a controlled experiment that isolates its
interactions - has proven to be a great challenge. Given what we know about it
from astrophysics and cosmology - that it interacts with certainty only
through gravitational influence - poses perhaps the greatest challenge to
experimentalists. We are not assured of its interactions by any other known
force and should it interact by forces _unknown_ , we do not know what
character those forces might possess. As such, there is a vast phase space of
theoretical consideration that lies open to experiment.

To design a dedicated direct dark matter detection experiment, one needs to
factor in a number of considerations:

  * •

What may be the rate of dark matter passing through the experimental
apparatus?

  * •

What may be the rate of interaction between dark matter and baryonic matter?

  * •

What known interactions and particles might confound the observations?

  * •

What will be the potential totality of experimental signatures of dark matter
interacting with the instrumentation?

It’s possible, of course, that we’ve already detected dark matter. Experiments
capable of observing the scattering of a weakly interacting, electrically
neutral particle have existed for decades. The scattering process might happen
off an atomic nucleus or electrons, or both. In principle, however, the
challenge to the experimentalist lies in the lack of constraints on just what
kind of particle we are hunting, perhaps most well-represented by the sheer
lack of constraint on the mass of such a particle. It could be so low in mass
that its effective de Broglie wavelength spans crystal lattices, labs spaces,
or whole planets. On the other hand, it could be so heavy that it ejects heavy
nuclei wholesale from solids, liquids, or gases. A good example of the
challenge, and categories of experiments that have, are, or will rise to meet
that challenge, is shown in Fig. 1.

Figure 1: A logarithmic plot of possible masses for dark matter particles, as
well as general techniques and specific experiments in various ranges that
have, are, or will attempt to constrain dark matter in that region. In these
lectures, the focus is on WIMPs whose masses are typically greater than
$\sim$1 GeV/c2.

##  2 Dark Matter Distributions

The search for dark matter’s particle constituents is for naught if the
distribution of dark matter in our local region of the Milky Way is non-
existent. I will first focus on the distribution of relic Weakly Interacting
Massive Particle (WIMP) dark matter in the Milky Way halo. Understanding the
distribution is a crucial ingredient in considerations for the construction of
experiments and interpretation of their (non-)observations.

###  2.1 A Simplified Model of WIMP Distribution in the Milky Way

WIMPs are assumed to be organized in isothermal spherical halos with a
Gaussian velocity distribution, owing to random fluctuations about some mean
velocity. This is all encapsulated in a Maxwell-Boltzmann model of the halo,
with a“Maxwellian” velocity distribution,

$f(\vec{v})=\frac{1}{\sqrt{2\pi}\sigma}e^{-|\vec{v}|^{2}/(2\sigma^{2})}$ (1)

where $\sigma$ is the WIMP velocity dispersion. The local circular speed,
$v_{c}$, is defined as the motion of our solar system about the center of the
galaxy. The speed dispersion, $\sigma$ is related to the local circular speed
by

$\sigma=\sqrt{\frac{3}{2}}v_{c}.$ (2)

where the local circular velocity is the speed of the our solar system about
the center of the galaxy, taken as $v_{c}=220~{}\mathrm{km/s}$.

The density profile of the sphere is $\rho(r)\propto r^{-2}$ and
$\rho_{0}=0.3~{}\mathrm{GeV/c^{2}\cdot cm^{3}}$, where the local density of
dark matter near Earth is taken to be the average of a small volume within a
few hundred parsecs of the Sun. There are two approaches to measuring the
local dark matter density [1]. The first involves the use of the vertical
kinematics of stars near the Sun [2, 3, 4, 5] and the second involves
extrapolation from galactic rotation curves [6]. In our region of the galaxy,
particles with speeds greater than the escape velocity, $v_{esc}$ are not
gravitationally bound. Hence, the speed distribution needs to be truncated and
will cut off at $v_{esc}=650~{}\mathrm{km/s}$.

It is interesting to then think about what will be the population of WIMPs in
your work area. We have the local dark matter density in our region of the
Milky Way. You need to next select your ”favorite” mass for a WIMP candidate.
I will choose one lighter species, $m_{\chi_{1}}=5~{}\mathrm{GeV/c^{2}}$ and
one heavier species, $m_{\chi_{2}}=100~{}\mathrm{GeV/c^{2}}$. What then will
be their respective number densities? These are

$\displaystyle n_{1}$ $\displaystyle=$
$\displaystyle\rho_{0}/m_{\chi_{1}}=6\times 10^{-2}~{}\mathrm{cm^{-3}},$ (3)
$\displaystyle n_{2}$ $\displaystyle=$
$\displaystyle\rho_{0}/m_{\chi_{2}}=3\times 10^{-3}~{}\mathrm{cm^{-3}}.$ (4)

Using these number densities, we can then take a room with a certain volume
and determine at any moment in time what are the number of particles in the
room. Let’s choose a more familiar volume - that of a large (2 liter) water or
soda bottle. In that case,

$\displaystyle N_{1}$ $\displaystyle=$ $\displaystyle n_{1}V_{bottle}=120,$
(5) $\displaystyle N_{2}$ $\displaystyle=$ $\displaystyle n_{2}V_{bottle}=10.$
(6)

Detector volumes will range from compact (liters in volume) to large-scale
(room-sized or larger in volume). It is clear that, given what we know about
the local dark matter halo, there is a good chance for dark matter to be
passing through such volumes. What will control the rate of actual detection
are two factors: the fundamental strength of the interaction between baryonic
and dark matter (which may be weak-scale, but since that was determined from a
very simplistic hypothesis, perhaps not) and the acceptance and efficiency of
an active detection medium for the energy deposited by these interactions.

##  3 Nuclear Scattering of Dark Matter

Figure 2: A schematic of dark matter (WIMP) elastic scattering off a target
nucleus.

Designing an experiment to directly detect dark matter means establishing
immediately a fundamental hypothesis: _dark matter is not only gravitationally
interacting_. A popular assumption is to consider some kind of weak-
interaction-like framework (even including the actual standard model weak
interaction, involving an exchange of weak bosons or the Higgs boson). This is
couched in the language of ”WIMP searches,” referring to the weakly
interacting massive particles inspired by freeze-out considerations discussed
elsewhere [7]. The term “WIMPs” refer to one candidate class of dark matter
particles that can be heavy and weakly interacting.

The specific nature of the interaction, broadly speaking, doesn’t matter here.
All that matters is that it is stronger than the gravitational interaction
alone which would be so feeble in a subatomic terrestrial experiment as to be
totally undetectable.

Our consideration of galaxy dynamics and ideal gas models has led us to
conclude that a typical dark matter particle velocity in the Milky Way’s dark
matter halo at our radius from the galactic center would be about 220 km/s, or
$\beta=v/c=7.34\times 10^{-4}$. We can begin there - that a single dark matter
particle has an average speed at this level. This certainly qualifies as non-
relativistic, so we can treat a hypothetical scattering process using non-
relativistic quantum mechanics and kinematics. Let us hypothesize a dark
matter particle mass of $100~{}\mathrm{GeV}/c^{2}$ (about 100 times the proton
mass). What kind of kinetic energy, on the scale of nuclear binding energy,
are we expecting here?

$K=\frac{1}{2}mv^{2}=\frac{1}{2}m\beta^{2}c^{2}=27~{}\mathrm{keV}$ (7)

The typical binding energy per nucleon in heavier nuclei is
$6-8~{}\mathrm{MeV}$ …this is no where near enough to fission nuclei.
Therefore, we anticipate that interactions with the nucleus will be elastic in
nature. Even for dark matter with mass of $\mathcal{O}(1TeV)$ we still expect
elastic interactions. Although for this discussion, we will focus on elastic
scattering, it should be noted that inelastic scattering is also possible [8]

The consequences of this kind of nuclear interaction are depicted in Fig. 2.
This scattering process will impart a small amount of energy to the nucleus.
While elastic, spin interactions may nevertheless play a role in the degree
and kind of scatter and will need to be factored into scattering
interpretations. In addition, it will be important to distinguish between
_background scatters_ \- interactions in the nuclei caused by standard model
interactions that have nothing to do with dark matter scattering.

###  3.1 WIMP-nucleus Scattering: Laboratory Frame

Let us approach the scattering process, using Fig. 2, in the _laboratory
frame_ : the frame in which the medium containing the nuclei is at rest. Given
that the typical nuclear mass is going to be
$m_{N}\sim\mathcal{O}(10~{}\mathrm{GeV/c^{2}})$, and that the typical
temperature of experimental material is going to be well below room
temperature (to remove thermal noise from any measurements) at the level of
$T\sim\mathcal{O}(1~{}\mathrm{K})$, the velocity of nuclei executing thermal
motion is going to be

$K=\frac{1}{2}m_{N}v_{N}^{2}\approx k_{B}T\longrightarrow
v_{N}\approx\sqrt{\frac{2k_{B}T}{m_{N}}}\approx 10^{-7}~{}\mathrm{m/s}$ (8)

or a $\beta_{N}=v_{N}/c\approx 10^{-16}$. Since the dark matter is expected to
be traveling at speeds consistent with the solar system’s speed in orbit
around the center of the galaxy
($220~{}\mathrm{km/s}\longrightarrow\beta_{\chi}\approx 10^{-3}$), we can
safely approximate the nuclei to be at rest in the laboratory frame.

We can now develop the mathematics for this scattering. Given the low speeds
involved, I will treat this as a non-relativistic collision subject to
classical mechanics. In an elastic collision, both momentum (always conserved
in closed and isolated systems) and kinetic energy (only conserved in this
case) will be conserved. We can denote initial-state quantities without primes
and final-state quantities with primes. Let $\vec{p}$ ($\vec{k}$) denote the
dark matter (nucleus) momentum. Then:

$\displaystyle\vec{p}+\vec{k}$ $\displaystyle=$
$\displaystyle\vec{p^{\prime}}+\vec{k^{\prime}}$ (9)
$\displaystyle\frac{p^{2}}{2m_{\chi}}+\frac{k^{2}}{2m_{N}}$ $\displaystyle=$
$\displaystyle\frac{(p^{\prime})^{2}}{2m_{\chi}}+\frac{(k^{\prime})^{2}}{2m_{N}}.$
(10)

For the nucleus, $k=0$ (the at-rest approximation). Thus:

$\displaystyle\vec{p}$ $\displaystyle=$
$\displaystyle\vec{p^{\prime}}+\vec{k^{\prime}}$ (11)
$\displaystyle\frac{p^{2}}{2m_{\chi}}$ $\displaystyle=$
$\displaystyle\frac{(p^{\prime})^{2}}{2m_{\chi}}+\frac{(k^{\prime})^{2}}{2m_{N}}.$
(12)

Let us further simplify this by placing the original dark matter particle
entirely along the horizontal axis in Fig. 2. The nucleus will then be
scattered relative to that axis at the indicated angle, $\theta_{R}$, so that
$k^{\prime}_{||}=k^{\prime}\cos\theta_{R}$ and
$k^{\prime}_{\perp}=k^{\prime}\sin\theta_{R}$ , where the parallel ($||$) and
perpendicular ($\perp$) symbols indicate direction relative to horizontal.
Since there was no net momentum in the transverse direction in the initial
state, and since momentum is conserved in each direction separately, it must
be true in the final state that

$\vec{p^{\prime}}_{\perp}+\vec{k^{\prime}}_{\perp}=0\longrightarrow\vec{p^{\prime}}_{\perp}=-\vec{k^{\prime}}_{\perp},$
(13)

which of course tells us that

$p^{\prime}\sin\phi=k^{\prime}\sin\theta_{R}\longrightarrow\sin\theta_{R}=\frac{p^{\prime}}{k^{\prime}}\sin\phi$
(14)

and relates the scattering angles to one another through the ratio of the
final-state momentum magnitudes.

It is useful to define the _momentum transfer_ ,
$\vec{q}=\vec{p^{\prime}}-\vec{p}$. We can then identify that

$q^{2}=p^{2}-2\vec{p^{\prime}}\cdot\vec{p}+(p^{\prime})^{2}.$ (15)

and that

$\vec{q}=-\vec{k^{\prime}}.$ (16)

This let’s use write the kinetic energy of the recoiling nuclear (also known
as the _recoil energy_) as

$E_{R}\equiv\frac{(k^{\prime})^{2}}{2m_{N}}=\frac{q^{2}}{2m_{N}}.$ (17)

Let us try to determine a relationship between the initial dark matter speed
and the final recoiling nucleus speed. To do this, we will need to eliminate
the final dark matter particle speed ($v^{\prime}$) from our equations. From
the conservation of kinetic energy,

$\displaystyle\frac{1}{2}m_{\chi}v^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{\chi}(v^{\prime})^{2}+\frac{1}{2}m_{N}(u^{\prime})^{2}$
(18) $\displaystyle m_{\chi}v^{2}$ $\displaystyle=$ $\displaystyle
m_{\chi}(v^{\prime})^{2}+m_{N}(u^{\prime})^{2}$ (19) $\displaystyle
m_{\chi}^{2}v^{2}$ $\displaystyle=$ $\displaystyle
m_{\chi}^{2}(v^{\prime})^{2}+m_{\chi}m_{N}(u^{\prime})^{2}$ (20)
$\displaystyle m_{\chi}^{2}(v^{\prime})^{2}$ $\displaystyle=$ $\displaystyle
m_{\chi}^{2}v^{2}-m_{\chi}m_{N}(u^{\prime})^{2}$ (21)

where $u$ is the speed of the recoiling nucleus. We can then employ the
conservation of momentum,

$\displaystyle m_{\chi}\vec{v}$ $\displaystyle=$ $\displaystyle
m_{\chi}\vec{v^{\prime}}+m_{N}\vec{u^{\prime}}$ (22) $\displaystyle
m_{\chi}\vec{v^{\prime}}$ $\displaystyle=$ $\displaystyle
m_{\chi}\vec{v}-m_{N}\vec{u^{\prime}}$ (23) $\displaystyle
m_{\chi}^{2}(v^{\prime})^{2}$ $\displaystyle=$ $\displaystyle
m_{\chi}^{2}v^{2}-2m_{\chi}m_{N}\vec{v}\cdot\vec{u^{\prime}}+m_{N}^{2}(u^{\prime})^{2}.$
(24)

Now relate these two initial dark matter speed terms between the conservation
of kinetic energy and momentum,

$\displaystyle m_{\chi}^{2}v^{2}-m_{\chi}m_{N}(u^{\prime})^{2}$
$\displaystyle=$ $\displaystyle
m_{\chi}^{2}v^{2}-2m_{\chi}m_{N}\vec{v}\cdot\vec{u^{\prime}}+m_{N}^{2}(u^{\prime})^{2}$
(25) $\displaystyle-m_{\chi}m_{N}(u^{\prime})^{2}$ $\displaystyle=$
$\displaystyle-2m_{\chi}m_{N}\vec{v}\cdot\vec{u^{\prime}}+m_{N}^{2}(u^{\prime})^{2}$
(26) $\displaystyle 2m_{\chi}m_{N}\vec{v}\cdot\vec{u^{\prime}}$
$\displaystyle=$ $\displaystyle
m_{\chi}m_{N}(u^{\prime})^{2}+m_{N}^{2}(u^{\prime})^{2}.$ (27)

Let us now consider the case of _maximum energy transfer_ , which will happen
when the initial dark matter velocity is entirely collinear with the final
nucleus velocity so that $\theta_{R}=0$. In this case,
$\vec{v}\cdot\vec{u^{\prime}}=vu^{\prime}$ and

$\displaystyle 2m_{\chi}m_{N}\vec{v}\cdot\vec{u^{\prime}}$ $\displaystyle=$
$\displaystyle m_{\chi}m_{N}(u^{\prime})^{2}+m_{N}^{2}(u^{\prime})^{2}$ (28)
$\displaystyle 2m_{\chi}m_{N}vu^{\prime}$ $\displaystyle=$ $\displaystyle
m_{\chi}m_{N}(u^{\prime})^{2}+m_{N}^{2}(u^{\prime})^{2}$ (29) $\displaystyle
2m_{\chi}m_{N}vu^{\prime}$ $\displaystyle=$
$\displaystyle(m_{\chi}m_{N}+m_{N}^{2})(u^{\prime})^{2}$ (30) $\displaystyle
2m_{\chi}m_{N}v$ $\displaystyle=$
$\displaystyle(m_{\chi}m_{N}+m_{N}^{2})u^{\prime}$ (31)
$\displaystyle\frac{v}{u^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{m_{\chi}m_{N}+m_{N}^{2}}{2m_{\chi}m_{N}}.$ (32)

This can be transformed into a statement about the ratios of the kinetic
energies of the initial dark matter particle and the final nucleus. This is a
more relevant physical quantity since most detectors we will consider are
_calorimeters_ , meaning their primary sensitivity is to the amount and kind
of energy deposited in medium:

$\displaystyle\frac{v}{u^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{m_{\chi}m_{N}+m_{N}}{2m_{\chi}}$ (33)
$\displaystyle\left(\frac{v}{u^{\prime}}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{(m_{\chi}+m_{N})^{2}}{4m_{\chi}^{2}}$ (34)
$\displaystyle\frac{\frac{1}{2}m_{\chi}v^{2}}{\frac{1}{2}m_{N}(u^{\prime})^{2}}$
$\displaystyle=$
$\displaystyle\frac{m_{\chi}(m_{\chi}+m_{N})}{4m_{\chi^{2}}m_{N}}$ (35)
$\displaystyle\frac{\frac{1}{2}m_{\chi}v^{2}}{\frac{1}{2}m_{N}(u^{\prime})^{2}}$
$\displaystyle=$ $\displaystyle\frac{(m_{\chi}+m_{N})^{2}}{4m_{\chi}m_{N}}.$
(36)

Since $E_{R}\equiv(k^{\prime})^{2}/2m_{N}=\frac{1}{2}m_{N}(u^{\prime})^{2}$,

$E_{R}^{max}=\frac{1}{2}m_{\chi}v^{2}\frac{4m_{\chi}m_{N}}{(m_{\chi}+m_{N})^{2}}.$
(37)

The quantity $\frac{m_{\chi}m_{N}}{(m_{\chi}+m_{N})}$ is also known as the
_reduced mass_ of the system. If we put ourselves in the center-of-mass frame
of the colliding system (a special point that, by conservation of momentum, is
fixed in space), the reduced mass appears quite naturally. It can be though of
as if a system of that singular mass $\mu$ was experiencing the collision,
taking into account the relative motion of the original two bodies before the
collision. Thus we can write:

$E_{R}^{max}=2m_{\chi}v^{2}\frac{\mu^{2}}{m_{\chi}m_{N}}=2\frac{\mu^{2}v^{2}}{m_{N}}.$
(38)

###  3.2 WIMP-nucleus Scattering: Center-of-Mass Frame

Figure 3: WIMP-nucleus scattering in the center-of-mass frame.

Let’s explore the kinematics of this kind of scattering in more detail, but
moving to the center-of-mass frame. While much of the notation in this section
will look similar to the previous section, it is more of convenience than
invariance. For example, the initial speed of the dark matter in the lab frame
is given by the Earth’s motion through the dark matter halo of the Milky Way,
and the nucleus’s velocity was taken to be zero. In the frame of an observer
at rest with respect to the center of mass of the dark matter-nucleus system,
both will appear to be moving in opposite directions in the initial state.

We can derive the relationship between the lab velocity of the dark matter and
the center-of-mass frame velocity of the dark matter. In the laboratory frame,
the velocity of the dark matter with respect to the nucleus is defined by the
first derivative, with respect to time, of the displacement between the dark
matter particle and the nucleus. Let us denote this as $\Delta(t)$. In the
center-of-mass frame, the velocity of the dark matter particle will be defined
by the time derivative of the displacement between the dark matter particle
and the center of mass position, which I will denote $x(t)$. Let us place the
center of mass at the origin of our coordinate system. In terms of these other
displacements, the position of the center of mass along the x-axis will be
given by:

$0=\frac{m_{\chi}x+m_{N}(\Delta-x)}{m_{\chi}+m_{N}}.$ (39)

Thus

$\displaystyle 0$ $\displaystyle=$ $\displaystyle m_{\chi}x+m_{N}(\Delta-x)$
(40) $\displaystyle m_{N}(\Delta-x)$ $\displaystyle=$ $\displaystyle
m_{\chi}x$ (41) $\displaystyle m_{N}\Delta$ $\displaystyle=$
$\displaystyle(m_{\chi}+m_{N})x.$ (42)

We can now relate the velocities by taking the time derivative of this
equation:

$\displaystyle\frac{d}{dt}m_{N}\Delta$ $\displaystyle=$
$\displaystyle\frac{d}{dt}(m_{\chi}+m_{N})x$ (43) $\displaystyle
m_{N}\dot{\Delta}$ $\displaystyle=$ $\displaystyle(m_{\chi}+m_{N})\dot{x}$
(44) $\displaystyle m_{N}v$ $\displaystyle=$
$\displaystyle(m_{\chi}+m_{N})v_{com}$ (45) $\displaystyle v_{com}$
$\displaystyle=$ $\displaystyle\frac{m_{N}}{m_{\chi}+m_{N}}v$ (46)

where $v$ is the velocity of the dark matter in the lab frame
$V=220~{}\mathrm{km/s}$. You can also show that the velocity of the nucleus
($u$) relative to the center of mass will be:

$u=\frac{m_{\chi}}{m_{\chi}+m_{N}}v.$ (47)

Let us begin by calculating the recoil energy of the nucleus in the center-of-
mass frame (Fig. 3) of the WIMP-nucleus system. Let $p$ label the three-
momentum of the WIMP and $k$ label the three-momentum of the nucleus, now in
the center-of-mass frame and not the laboratory frame. Then

$\displaystyle\vec{p}$ $\displaystyle=$ $\displaystyle-\vec{k}\textrm{
(initial momentum)}$ (48) $\displaystyle\vec{p^{\prime}}$ $\displaystyle=$
$\displaystyle-\vec{k^{\prime}}=\vec{q}+\mu\vec{v}_{\chi}\textrm{ (final
momentum)}$ (49)

where the WIMP-nucleus reduced mass, $\mu$, was defined earlier as:

$\mu=\frac{m_{\chi}m_{N}}{m_{\chi}+m_{N}},$ (50)

$v_{\chi}$ is the mean WIMP velocity _relative_ to the target nucleus and $q$
is the momentum transfer between the WIMP and the nucleus,
$\vec{q}\equiv\vec{p^{\prime}}-\vec{p}$. In the center-of-mass frame and for
elastic scattering, $|\vec{p}|=|\vec{p^{\prime}}|$. We can then write:

$\frac{q^{2}}{2}=\frac{1}{2}(\vec{p^{\prime}}-\vec{p})^{2}=p^{2}-\vec{p}\cdot\vec{p^{\prime}}=p^{2}(1-\cos\theta)=\mu^{2}v^{2}_{\chi}(1-\cos\theta).$
(51)

The angle, $\theta$, is as defined in Fig. 3. Often, this angle is referred to
as the _recoil angle_ and is denoted equivalently by $\theta_{R}$. To remind
us this is about calculating the properties of the recoiling nucleus, which is
what we hope to detect, let us adopt the convention that
$\theta\to\theta_{R}$. Because all the initial-state action happens along the
same x-axis in the laboratory or center-of-mass frames, the angle $\theta$ in
this frame is identical to the angle $\theta_{R}$ in the laboratory frame.

The nuclear recoil (NR) energy will then be given by

$E_{R}=\frac{q^{2}}{2m_{N}}=\frac{\mu^{2}v_{\chi}^{2}}{m_{N}}(1-\cos\theta_{R}).$
(52)

Using this, we can then calculate the minimum dark matter particle velocity
for which we expect a recoil. This will happen under the limiting case
$\cos\theta_{R}=-1$. Then

$\displaystyle E_{R}$ $\displaystyle=$
$\displaystyle\frac{\mu^{2}v^{2}_{\chi}}{m_{N}}(1-\cos\theta)$ (53)
$\displaystyle=$ $\displaystyle\frac{\mu^{2}v^{2}_{\chi}}{m_{N}}(1-(-1))$ (54)
$\displaystyle=$ $\displaystyle 2\frac{\mu^{2}v^{2}_{\chi}}{m_{N}}$ (55)
$\displaystyle\longrightarrow$ $\displaystyle
v_{min}=\sqrt{\frac{m_{N}E_{R}}{2\mu^{2}}}=\frac{q}{2\mu}.$ (56)

This relationship between the minimum speed that generates a recoil and the
other properties of the system has implications:

  * •

Lighter dark matter particles ($m_{\chi}\ll m_{N}$) must have larger minimum
(threshold) velocities to generate a recoil;

  * •

Inelastic scattering, where the nucleus is fissioned or where additional
particles are generated during the scattering process, can further increase
the minimal velocity needed (e.g. by the addition of other mass terms in the
final state that further alters $\mu$).

Consider the average momentum transfer in an elastic scattering between a
WIMP-nucleus. Let’s assume a $10~{}\mathrm{GeV/c^{2}}$ WIMP whose speed is
$\approx 100~{}\mathrm{km/s}$.

$p=m_{\chi}v=(10\times 10^{9}~{}\mathrm{eVc^{-2}})(100\times
10^{3}~{}\mathrm{ms^{-1}})\frac{c}{3\times 10^{8}~{}\mathrm{ms^{-1}}}\approx
3~{}\mathrm{MeV}.$ (57)

If we increase the mass of the dark matter by a factor of ten, we then also
would increase the energy transferred to the nucleus to $\approx
30~{}\mathrm{MeV}$.

We can also consider the de Broglie wavelength that corresponds to the
momentum transfer of $\sim 10$ MeV. This is given by

$\lambda=\frac{hc}{pc}=\frac{1239\times 10^{-6}~{}\mathrm{eV\cdot m}}{10\times
10^{6}!\mathrm{eV}}\sim 12~{}\mathrm{pm}.$ (58)

Typical nuclear size scales go like $r\approx(1.25~{}\mathrm{fm})A^{1/3}$,
where $A$ is the atomic mass number. A Uranium nucleus, for example, would
have a radius of about 8 fm. The de Broglie wavelength of such a momentum
transfer exceeds the size of even large nuclei, which thus implies that any
such elastic scattering will be _coherent scattering_ , scattering off of the
entirety of the nucleus, as if the nucleus were a singular object and not a
collection of nucleons, nor at a deeper level a collection of quarks and
gluons.

###  3.3 Scattering Rates in a Detector - A Simple Picture

Let us now take a simplified picture of elastic scattering rates following the
calculations of Lewin and Smith [9] in an imagined detector composed of atoms,
where the primary target of interest is the nucleus of the atom. The
differential event rate for simplified WIMP interaction (a detector stationary
in the galaxy) is given by:

$\frac{dR}{dE_{R}}=\frac{R_{0}}{E_{0}r}e^{-(E_{R}/E_{0})r}$ (59)

where $R$ is the rate of elastic scattering events, $E_{R}$ is the recoil
energy of the nucleus, $R_{0}$ is the total rate of such events and $E_{0}$ is
the most probable incident energy of a WIMP (e.g. given the Maxwell-Boltzmann
distribution of galactic dark matter). $r$ is a kinematic factor defined by

$r=\frac{4m_{\chi}m_{N}}{(m_{\chi}+m_{N})^{2}}.$ (60)

The total event rate is recovered by

$\int_{0}^{\infty}\frac{dR}{dE_{R}}dE_{R}=R_{0}.$ (61)

and the mean recoiling energy is given by

$\langle E_{R}\rangle=\int_{0}^{\infty}E_{R}\frac{dR}{dE_{R}}dE_{R}=E_{0}r.$
(62)

The spectrum of recoil energy for a given WIMP mass and WIMP velocity
distribution is shown in Fig. 4. Note that this spectrum is featureless. While
changing the hypothetical WIMP mass or kinematics, or the target nucleus mass,
will alter the spectrum, it will always be straight and featureless (albeit
with altered slope, etc.) on a logarithmic scale like this one.

Figure 4: An example of the nucleus recoil energy spectrum from elastic WIMP-
nuclei scattering.

We can now compute the mean nuclear recoil energy deposited in a detector. Let
us assume that we have designed the detector such that $m_{\chi}\approx
m_{N}=100~{}\mathrm{GeV/c^{2}}$. Then

$\langle
E_{R}\rangle=E_{0}r=\left(\frac{1}{2}m_{\chi}v^{2}\right)\left(\frac{4m_{\chi}m_{N}}{(m_{\chi}+m_{N})^{2}}\right)$
(63)

since for our particular case of equivalent or near-equivalent masses, $r=1$.
If we assume that the WIMP halo is stationary and that the Sun-Earth system is
passing through it at our rotation speed around the center of the galaxy, then
$v\approx 220~{}\mathrm{kms^{-1}}=0.75\times 10^{-3}~{}c$. If we substitute
this into our equation, we obtain:

$\langle E_{R}\rangle\approx 30~{}\mathrm{keV}.$ (64)

###  3.4 Scattering Rates in a Detector - A More Complex Picture

Scattering will, of course, be more complex than presented in the previous
section. We need to take into account the following considerations. First, DM
will have a certain velocity distribution $f(v)$. Second, the detector is on
Earth which moves around the Sun while the Sun also moves around the Galactic
Center. Next, the cross-section depends upon the spin-dependence of the
underlying interaction. In the simplest cases, this is either spin-independent
(SI) or spin-dependent (SD).

In addition, the dark matter will scatter on nuclei that have finite size. As
such, we have to consider form-factor corrections which are different for SI
and SD interactions. We must also take into account that the recoil energy is
not necessarily the observed energy. The detection efficiency in real life is
not 100% because real detectors have certain energy resolution and energy
thresholds, as well as physical processes that don’t necessarily translate
into pure conversion of recoil energy to some detectable form.

The total number of dark matter particles, $N$, is a product of the dark
matter flux (particles per unit area per unit time), the effective area of the
target and the exposure time of the experiment. There are two obvious way to
increase the number of dark matter particles on target: increase the effective
target area and/or operate the experiment longer. The formula is

$N=tnvN_{T}\sigma$ (65)

where $n$ is the dark matter number density, $v$ the speed of the particles,
$N_{T}$ the number of target particles, and $\sigma$ the interaction cross
section of dark matter with the target particles. We will need to determine
the spectrum of dark matter recoils which is the energy dependence of the
number of detected dark matter particles. We do this simply by taking the
derivative of Eqn (65) with respect to the recoil energy of the target
particles,

$\frac{dN}{dE_{R}}=tnvN_{T}\frac{d\sigma}{dE_{R}}.$ (66)

We need to consider the dark matter particles are described by their local
velocity distribution, $f(\vec{v})$, where $\vec{v}$ is the velocity of the
dark matter particles in the reference frame of the detector (the so-
called“laboratory frame”). We can insert this in the above equation by use of
an integral that, performed over all velocities would yield the average speed
of the dark matter particles:

$\frac{dN}{dE_{R}}=tnN_{T}\int_{v_{min}}vf(\vec{v})\frac{d\sigma}{dE_{R}}d\vec{v}.$
(67)

where $v_{min}$ is the minimum speed of a dark matter particle that results in
recoil energy in a collision with the nucleus as defined in Eqn. 56.

We have at our disposal some useful relationships:

$\displaystyle n$ $\displaystyle=$ $\displaystyle\frac{\rho}{m_{\chi}}$ (68)
$\displaystyle N_{T}$ $\displaystyle=$
$\displaystyle\frac{M_{T}}{m_{N}}\textrm{ (with }M_{T}\textrm{ the total
target mass)}$ (69) $\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle
tM_{T}\textrm{ (the exposure)}.$ (70)

We can write:

$\frac{dN}{dE_{R}}=\epsilon\frac{\rho}{m_{\chi}m_{N}}\int_{v_{min}}vf(\vec{v})\frac{d\sigma}{dE_{R}}d\vec{v}.$
(71)

The _event rate_ is defined as $R\equiv N/\epsilon$ (number of observable
events per unit target mass per unit time), so the _differential event rate_
(observable events per keV of recoil energy, per kilogram, per day) is then

$\frac{dR}{dE_{R}}=\frac{\rho_{0}}{m_{\chi}m_{N}}\int_{v_{min}}^{\infty}vf(\vec{v})\frac{d\sigma_{\chi
N}}{dE_{R}}d\vec{v}$ (72)

where I solidify the notation so that $\rho_{0}$ is the _local dark matter
density_ (e.g. in our region of the Milky Way), $f(v)$ is the WIMP speed
distribution in the detector frame, and $\sigma_{\chi N}$ is the WIMP-nucleus
scattering cross-section. To convert this latter cross section to a WIMP-
nucleon cross-section requires nuclear physics to map overall nuclear
structure onto individual nucleons in the nucleus.

Dark matter detection, then, is a synthesis of astrophysics (to estimate the
local dark matter density), particle physics (to establish a framework for the
scattering), and nuclear physics (to map the nuclear to the nucleon
properties).

Dark Matter Detection Equations In summary, the dark matter detection
equations of note are
$E_{R}=\left(\frac{m_{\chi}m_{N}}{m_{\chi}+m_{N}}\right)^{2}\frac{v^{2}}{m_{N}}(1-\cos\theta_{R})$
(73) and
$\frac{dR}{dE_{R}}=\frac{\rho_{0}}{m_{\chi}m_{N}}\int_{v_{min}}^{\infty}vf(\vec{v})\frac{d\sigma_{\chi
N}}{dE_{R}}d\vec{v}$ (74)

###  3.5 The Scattering Cross Section

The event rate in a potential detector is determined by integrating over all
recoils.

$R=\int_{E_{threshold}}^{\infty}dE_{R}\frac{\rho_{0}}{m_{\chi}m_{N}}\int_{v_{min}}^{\infty}vf(\vec{v})\frac{d}{dE_{R}}\sigma_{\chi
N}(v,E_{R})\,d\vec{v}$ (75)

In this section, I want to focus on the elements that enter the differential
cross-section, $d\sigma_{\chi N}/dE_{R}$. First, let’s consider that the
scattering process takes place in the non-relativistic limit. It can be
approximated as an isotropic scattering distribution, such that

$\frac{d\sigma}{d\cos\theta}=\mathrm{constant}=\frac{\sigma}{2}.$ (76)

The maximum kinetic energy that can be transferred to the nucleus occurs when
$\cos\theta_{R}=-1$ (100% back-scatter of the nucleus in the center-of-mass
frame), such that $E_{R}^{max}=2\mu^{2}v^{2}/m_{N}$. In terms of this
quantity, Eqn 73 becomes

$E_{R}=\frac{E_{R}^{max}}{2}(1-\cos\theta_{R}).$ (77)

We can obtain the differential of the recoil energy with respect to scattering
angle:

$\frac{dE_{R}}{d\cos\theta}=\frac{E_{R}^{max}}{2}.$ (78)

From this, we can then write the differential of the cross-section with
respect to recoil energy using the chain rule:

$\frac{d\sigma}{dE_{R}}=\frac{d\sigma}{d\cos\theta}\frac{d\cos\theta}{dE_{R}}=\frac{\sigma}{2}\frac{2}{E_{R}^{max}}=\frac{\sigma}{E_{R}^{max}}=\frac{m_{N}}{2\mu^{2}}\frac{\sigma}{v^{2}}.$
(79)

The momentum transfer involved in these collisions corresponds to a de Broglie
wavelength in excess of the nuclear size. Thus, we are coherently scattering
off the entire nucleus. Especially for lighter nuclei, the WIMP doesn’t “see”
the nucleons at all. For light nuclei, one can make a good approximation that
the nucleus is nearly the sum of its parts (nucleons) without significant
corrections to that approximation. For light nuclei, it’s therefore possible
to translate a scattering rate on the nucleus into a scattering rate on
individual nucleons (e.g. to infer the WIMP-proton or WIMP-neutron cross-
section).

However, it’s more typical for heavy nuclei to be used in experiments. Given
our early considerations, this is for obvious reasons: more target cross-
sectional area means more anticipated WIMP interactions per unit mass, per
unit time. Heavy nuclei receive significant corrections from the “sum of
nucleons” approximation when describing the nucleus as a whole. These
corrections are summarized by _nuclear form factors_ , $F$. For example, let
us consider that we are not certain whether the WIMP-nucleus interactions may
have spin-independent or spin-dependent effects. It’s natural, therefore, to
assume that the cross-section in totum will be a sum of both possibilities,
e.g.

$\frac{d\sigma}{dE_{R}}=\left[\left(\frac{d\sigma}{dE_{R}}\right)_{SI}+\left(\frac{d\sigma}{dE_{R}}\right)_{SD}\right],$
(80)

where “SI” (“SD”) denotes the spin-independent (spin-dependent) scattering.
Spin-independent effects would arise from a scalar or vector dark matter
candidate coupling to quarks inside the nucleons, while spin-dependent effects
would arise from an axial-vector coupling to those same quarks. To perform
calculations, it’s straight forward to begin by assuming that these effects
add coherently with corrections from the parton-level up to the nuclear scale:

$\frac{d\sigma}{dE_{R}}=\frac{m_{N}}{2\mu^{2}v^{2}}\left[\sigma_{0}^{SI}F_{SI}^{2}+\sigma_{0}^{SD}F_{SD}^{2}\right].$
(81)

The form factors encode the dependence on the momentum transfer and the
nuclear structure without making explicit the details of these dependencies;
the cross-sections encode the parton-level (e.g. quark-level) particle physics
of the WIMP-parton interaction (e.g. Feynman diagrams describing the leading
interactions possible between fundamental particles).

The Form Factors are an area of rather active research. Nuclear physics, to
say the least, is complicated. The nature of quantum chromodynamics and the
energy domain of the nucleus makes high-precision, direct analytical
calculations of such form factors nearly impossible. Instead, a synthesis of
nuclear scattering data and computational approaches are typically used to
constrain or infer these corrections. It is also possible to make models of
these form factors based on theoretical or data-driven considerations.

One example is the Helm Form Factor [10] [11], which applies to spin-
independent interactions,

$F(q)=\left(\frac{3j_{1}(qR_{1})}{qR_{1}}\right)^{2}\exp(-q^{2}s^{2}/2).$ (82)

Here, $j_{1}$ is the Spherical Bessel Function,

$j_{1}(x)=\frac{\sin(x)}{x^{2}}-\frac{\cos(x)}{x}.$ (83)

$q$ is the momentum transfer, $s$ is the nuclear skin thickness
($s\sim\mathrm{1~{}fm}$), and $R_{1}$ is the effective nuclear radius, which
should be approximated as $R_{1}=1.25~{}\mathrm{fm}\times A^{1/3}$, where $A$
is the atomic mass number.

For spin-dependent interactions, we can write

$F^{2}(E_{R})=\frac{S(E_{R})}{S(0)}$ (84)

where

$S(E_{R})=a_{0}^{2}S_{00}(E_{R})a_{1}^{2}S_{11}(E_{R})+a_{0}a_{1}2S_{01}(E_{R})$
(85)

and

$a_{0}=a_{p}+a_{n}\textrm{, }a_{1}=a_{p}-a_{n}.$ (86)

Here, $S_{ij}$ are the isoscalar (0), isovector (1), and interference form
factors while $a_{i}$ are the isoscalar or isovector coupling constants.

The spin-dependent form factor is the superposition of form-factor components
(isoscalar, etc.) normalized to that superposition at zero recoil energy, the
case that no energy is transferred to the nucleus.

The forms of these SI and SD cross-sections that most often appear in the
literature are

Spin-Independent and Spin-Dependent Cross Sections
$\sigma_{0}^{SI}=\frac{4\mu^{2}}{\pi}\left[Zf_{p}+(A-Z)f_{n}\right]^{2}\propto
A^{2}$ (87)
$\sigma_{0}^{SD}=\frac{32G_{F}^{2}\mu^{2}}{\pi}\frac{J+1}{J}\left[a_{p}\langle
S_{p}\rangle+a_{n}\langle S_{n}\rangle\right]^{2}$ (88)

Eqn. 87 is valid for spin-independent interactions only. Furthermore, we
generally assume a low momentum transfer which corresponds to low $q^{2}$. As
such, the scattering process cannot resolve the difference between the proton
and neutron at the parton level. In that case, the four-fermion coupling
factors satisfy $f_{p}\sim f_{n}$ [12] (This is a model-dependent assumption
that may need to be revisited if WIMP scattering is observed). The scattering
is overall simply proportional to the atomic mass number (larger $A$ means
more interactions) and adds coherently with an $A^{2}$ enhancement.

However, in Eqn. 88 the nuclear angular momentum, $J$, explicitly appears, as
do the individual couplings to the proton and neutron ($a_{i}$) and the
expectation values of the nucleon spins $\langle S_{i}\rangle$. Different
nuclei can lead to very different scattering rates, even for nearly identical
values of $A$; in fact, isotopic variations among elements will potentially
significantly shift the scattering process. An example of how to factor such
considerations into the choice of a particular target element and isotope is
shown in Fig. 5. This table from Ref [13] summarizes efforts that have gone
into modeling the nuclear physics considerations which influence experimental
choices and design.

Figure 5: From Ref. [13], this table provides values for terms that appear in
Eqn. 88. The quantities $C_{A}^{p,n}$ are defined as
$C_{A}^{i}\equiv(8/\pi)\left(a_{i}\langle
S_{i}\rangle\right)^{2}\left[(J+1)/J\right]$.

###  3.6 Effective Field Theory and the Complexity of Dark Matter Scattering

The treatment of scattering so far has been simple and generic, devoid of a
particular underlying theoretical framework except to assume basic non-
relativistic quantum mechanics. In reality, we know that these more
approximate models are rooted in quantum field theories. We do not yet know
what field theory framework provides the proper description of dark matter, so
it is impossible to be explicit about scattering calculations.

However, effective quantum field theories (EFTs) have proven to be a useful
tool for establishing all possible kinds of interactions that can take place
between fermions and bosons. As measurements are made with experiments, it may
be possible to eliminate certain interaction terms from consideration; some
may be ruled out a priori as they provide undesirable or incompatible features
for dark matter particles. Such effective field theories usually begin by
considering leading-order and next-to-leading-order contributions to
interactions, where the ”order” here refers to the cut-off in a perturbation
expansion of the theory that is usually conducted in terms of the coupling
strength(s) of the interaction(s) involved in the theory.

Such an EFT would contain 14 operators that rely on a range of nuclear
properties in addition to the SI and SD cases [14][15][16][17]. They can be
combined and grouped such that the WIMP-nucleon cross section depends on six
independent nuclear response functions: (a) one ”spin-independent, (b) two
”spin-dependent,” (c) and three ”velocity-dependent” functions. Two pairs of
these interfere, resulting in eight independent parameters that can be probed
by comparing EFT predictions with real experimental data (e.g. scattering
observations or non-observations). Each operator connects an initial state and
a final state, and in effect determines the nature and outcome of each kind of
interaction.

Basic matrix element considerations apply here. Consider some generic
operator, $\mathcal{O}$, that describes an interaction that takes some initial
state $\ket{i}$ to some final state $\ket{f}$. The matrix element that
describes the square-root of the probability density for this transition via
this operator is

$\mathcal{M}=\langle i\left|\mathcal{O}\right|f\rangle.$ (89)

The probability of this transition is then given by (or is proportional to)

$|\mathcal{M}|^{2}=|\langle i\left|\mathcal{O}\right|f\rangle|^{2}.$ (90)

The 14 operators in the EFT used to describe WIMP-nucleon interactions are

$\displaystyle\mathcal{O}_{1}$ $\displaystyle=$ $\displaystyle 1_{\chi}1_{N}$
(91) $\displaystyle\mathcal{O}_{3}$ $\displaystyle=$ $\displaystyle
i\vec{S}_{N}\cdot\left[\frac{\vec{q}}{m_{N}}\times\vec{v}^{\perp}\right]$ (92)
$\displaystyle\mathcal{O}_{4}$ $\displaystyle=$
$\displaystyle\vec{S}_{\chi}\cdot\vec{S}_{N}$ (93)
$\displaystyle\mathcal{O}_{5}$ $\displaystyle=$ $\displaystyle
i\vec{S}_{\chi}\cdot\left[\frac{\vec{q}}{m_{N}}\times\vec{v}^{\perp}\right]$
(94) $\displaystyle\mathcal{O}_{6}$ $\displaystyle=$
$\displaystyle\left[\vec{S}_{\chi}\cdot\frac{\vec{q}}{m_{N}}\right]\cdot\left[\vec{S}_{N}\cdot\frac{\vec{q}}{m_{N}}\right]$
(95) $\displaystyle\mathcal{O}_{7}$ $\displaystyle=$
$\displaystyle\vec{S}_{N}\cdot\vec{v}^{\perp}$ (96)
$\displaystyle\mathcal{O}_{8}$ $\displaystyle=$
$\displaystyle\vec{S}_{\chi}\cdot\vec{v}^{\perp}$ (97)
$\displaystyle\mathcal{O}_{9}$ $\displaystyle=$ $\displaystyle
i\vec{S}_{\chi}\cdot\left[\vec{S}_{N}\times\frac{\vec{q}}{m_{N}}\right]$ (98)
$\displaystyle\mathcal{O}_{10}$ $\displaystyle=$ $\displaystyle
i\vec{S}_{N}\cdot\frac{\vec{q}}{m_{N}}$ (99) $\displaystyle\mathcal{O}_{11}$
$\displaystyle=$ $\displaystyle i\vec{S}_{\chi}\cdot\frac{\vec{q}}{m_{N}}$
(100) $\displaystyle\mathcal{O}_{12}$ $\displaystyle=$
$\displaystyle\vec{S}_{\chi}\cdot\left[\vec{S}_{N}\times\vec{v}^{\perp}\right]$
(101) $\displaystyle\mathcal{O}_{13}$ $\displaystyle=$ $\displaystyle
i\left[\vec{S}_{\chi}\cdot\vec{v}^{\perp}\right]\left[\vec{S}_{N}\cdot\frac{\vec{q}}{m_{N}}\right]$
(102) $\displaystyle\mathcal{O}_{14}$ $\displaystyle=$ $\displaystyle
i\left[\vec{S}_{\chi}\cdot\frac{\vec{q}}{m_{N}}\right]\left[\vec{S}_{N}\cdot\vec{v}^{\perp}\right]$
(103) $\displaystyle\mathcal{O}_{15}$ $\displaystyle=$
$\displaystyle-\left[\vec{S}_{\chi}\cdot\frac{\vec{q}}{m_{N}}\right]\left[\left(\vec{S}_{N}\times\vec{v}^{\perp}\right)\cdot\frac{\vec{q}}{m_{N}}\right]$
(104)

where $\vec{v}^{\perp}$ is the relative velocity between the incoming WIMP and
the nucleon, $q$ is the momentum transfer, $\vec{S}_{\chi}$ is the WIMP spin,
and $\vec{S}_{N}$ is the nucleon spin. In addition, each operator can
independently couple to protons and neutrons. You will notice that
$\mathcal{O}_{2}$ is missing from the list; this is because it cannot arise in
the non-relativistic limit, which we are considering here.

(a) Fluorine target

(b) Germanium target

Figure 6: Figures from Ref. [17]. Differential event rate evaluated at the
$\mathcal{O}_{8}/\mathcal{O}_{9}$ constructive interference vector from
Fluorine and Germanium targets.

As a result of this complex web of operators, representing different spin-
independent, spin-dependent, and velocity-dependent interactions, WIMP
scattering can look very different across a range of common or plausible
target materials. For example, nuclear responses for different target elements
vary, some EFT operations have momentum dependence, and EFT operators can
interfere. An example of these effects is shown in Fig. 6. This illustrates
differences evaluating event rates using the $\mathcal{O}_{8}$ and
$\mathcal{O}_{9}$ constructive interference vector. This effect results in
different rates between targets as well as different recoil energy spectral
shapes. Not all targets are equal, nor will two different targets tells you
the same thing about the properties of the dark matter or their interactions
with the nucleus.

A robust dark matter direct detection program with different target materials
will be needed to identify which operators are contributing to any detected
signal. A rich and diverse search program will need multiple targets to map
out the physics of WIMP-nucleon interactions.

###  3.7 The WIMP Detection Challenge

Another challenge that comes from basic or EFT-driven scattering
considerations is the overall low rate at which we can expect nuclear recoils
to occur. For example, event rates in different materials as a function of
energy threshold for a $100~{}\mathrm{GeV/c^{2}}$ WIMP and a WIMP-nucleon
total cross section of $10^{-45}~{}\mathrm{cm^{2}}$ (compatible with upper
bounds placed by leading experiments in the field today), are illustrated in
Fig. 7 [18]. You will notice that we only expect a few events per kg-year of
exposure.

Elastic scattering of a WIMP deposits small amounts of energy into a recoiling
nucleus, amounting to just a few tens of keV. The energy spectrum of the
recoil is featureless and falls exponentially as a function of the recoil
energy. There are not expected to be any resonances, peaks, knees, or breaks
in this spectrum, making it hard to distinguish from other processes with
similar featureless energy spectra. The overall event rate is extremely low,
requiring large targets and long periods of data-taking. This alone won’t be
sufficient. Materials are suffused with naturally occurring radioactive
backgrounds which are, even under fair conditions, capable of producing
nuclear recoils at rates far higher than these.

Figure 7: Event rates in units of counts per 10 kg of material per year of
data-taking vs. energy threshold in keV for different target materials. Taken
from [18]

For lower-mass dark matter candidates, the challenge for such liquid- and
solid-state material approaches is even greater. The event rate vs. energy
threshold for various WIMP candidate masses is shown in Fig. 8. From the
earlier exploration of recoil energy in non-relativistic elastic scattering we
know that as the WIMP mass declines, the challenge of generating a nuclear
recoil grows larger. In addition, different target materials will respond
differently to the same kind of low-mass WIMP candidate (Fig. 9). Lower WIMP
masses will demand, more and more, materials and approaches that can go to
lower and lower recoil energy thresholds.

Figure 8: Event rates in units of counts per kg of material per year of data-
taking vs. energy threshold (in keV) for different low WIMP masses. The target
material is fixed on Germanium. Figure courtesy of Enectali Figueroa-
Feliciano. Figure 9: Event rates in units of counts per kg of material per
year of data-taking vs. energy threshold in keV for different materials. The
WIMP mass is fixed at $5~{}\mathrm{GeV/c^{2}}$. Figure courtesy of Enectali
Figueroa-Feliciano.

###  3.8 Time-Dependence of the Scattering Rate

The focus has been on the event rate integrated over time, but not on the
potential time structure of the rate itself. There is a strong reason to
believe that the interaction rate of dark matter on a target will be time
dependant. The most obvious reason for this is that the average speed of the
dark matter is determined by the estimated speed with which the Sun orbits the
center of the Milky Way. However, the Earth orbits the Sun, so that half the
year our planet is headed “into the WIMP wind” while in the other half of the
year we are headed “out of the WIMP wind.” These tiny expected variations in
the average speed of WIMPs could manifest in a detector as an _annual
modulation_ of the event rate.

Figure 10: A schematic showing the relationship of the Earth’s orbit around
the Sun to the direction of travel of the Sun through the Milky Way. The WIMP
“wind” points opposite that direction of travel. Figure from Ref. [19].

The motion of the Sun through the WIMP halo is in the direction of the
constellation Cygnus (Fig. 10) and is in the plane of the galaxy. However, the
plane of the Earth’s orbit is not aligned with that direction, but rather
tilted about $60^{\circ}$ above that line of motion. We should only consider
the change in the WIMP wind based on the component of the Earth’s orbital
velocity that is parallel to the Sun-Cygnus line.

Earth’s speed with respect to the galactic rest frame, taken also to be the
WIMP halo rest frame, is largest in the summer when the components of Earth’s
orbital velocity in the direction of solar motion is largest. The number of
WIMPs with high (low) speeds in the detector rest frame is largest (smallest)
in the summer. As a result, in the Northern Hemisphere, we would expect any
annual modulation of the event rate to peak during the summer months and be
minimal in the winter months.

We can estimate the degree of this effect [19]. The Earth’s speed around the
Sun is much smaller than the speed of the Sun about the center of the Milky
Way. In numbers, $v_{orbit}/v_{sun}\approx 0.07$. This small value of the
relative speeds provides an opportunity to Taylor expand the differential
event rate to make a leading order approximation of the effect.

$\frac{dR}{dE_{R}}(E_{R},t)\approx\frac{dR}{dE_{R}}\left[1+\Delta(E_{R})\cos\frac{2\pi(t-t_{0})}{T}\right],$
(105)

where $T$ is the period of the modulation and is expected to be one Earth
orbital period, or 365.25 days, and $t_{0}$ is the phase which, due to the
relationship between the plane of orbit and the motion of the Sun is
$t_{0}=150~{}\mathrm{days}$. The factor $\Delta E_{R}$ is the _modulation
amplitude_. To detect the seasonal variation due to the changing recoil energy
spectrum across the year, one needs sufficient detections to tell the
difference between the higher number in the Northern summer and the lower
number in the Northern winter. This requires at least the observation of 1000
interactions to detect what is expected to be a few percent effect. This
effect on an example generic recoil energy spectrum is shown in Fig. 11.

Figure 11: The difference in the recoil energy spectrum (top) comparing the
maximum (solid line) and minimum (dashed line) times of the year for an annual
modulation of WIMP interactions. The bottom graph shows the ratio of the
modulation amplitude for those two cases. Figure excerpted from Ref. [19].

In addition to time dependence, if detectors can be made sensitive to the
direction of the nuclear recoil this can help to understand the modulation
signal. Directionality [20] can also be used for a variety of other purposes,
including the suppression of confounding processes in detector systems. The
Earth’s motion with respect to the Galactic rest frame produces a direction
dependence of the recoil spectrum. The peak WIMP flux will appear to point
from the direction of solar motion, from Cygnus toward us. Assuming a smooth
and isotropic WIMP halo, the recoil rate is peaked in the opposite direction
of our motion toward Cygnus. In the lab frame, this direction varies over the
course of a day due to Earth’s rotation as illustrated in Fig. 12. In
addition, due to the fact that the direction to Cygnus varies by hemisphere,
detectors places on different halves of the Earth should see slightly
different directional effects. This is another example of the importance of
multiple, independent, and complementary dark matter instruments.

Figure 12: Spin axis of Earth with respect to the WIMP “wind” generated by the
motion of the Earth through the Galaxy. Figure excerpted from Ref. [21]

As a result of this directional dependence, we should actually expect the
number of nuclear recoils along a particular direction in the lab frame to
change over the course of a single day. A daily modulation in directionality
could be visible with a sensitive enough detector. Assuming the Standard Halo
model, the dependence is given by

$\frac{dR}{dE_{R}\cos\gamma}=\frac{\rho_{0}\sigma_{\chi
N}}{\sqrt{\pi}\sigma_{v}}\frac{m_{N}}{2m_{\chi}\mu^{2}}\exp\left[-\frac{[(v_{orbit}^{E}v_{sun})\cos\gamma-
v_{min}]^{2}}{\sigma_{v}^{2}}\right]$ (106)

where $v_{orbit}^{E}$ is the component of the Earth’s velocity around the Sun
parallel to the Sun-Cygnus line and $\gamma$ is the angle between the observed
nuclear recoil and the Sun-Cygnus line. The event rate in the forward
direction is expected to be up to an order of magnitude larger than backward
direction [22]. A detector measuring the axis and direction of the recoil with
good angular resolution needs only a few tens of events to distinguish dark
matter interactions from isotropic backgrounds that have no preferred
direction (e.g. not aligned with the Sun-Cygnus line).

##  4 Backgrounds to Direct Detection Methodologies

There are many known and potential sources of confounding detector signatures
- _background processes_ or _backgrounds_ for short. The most prominent of
these is environmental radioactivity. This category includes airborne radon
and its daughters as well as radio-impurities in materials used for the
detector and shielding construction. One of the worst potential backgrounds
connected to environmental radioactivity is radiogenic neutrons with energies
below 10 MeV (neutrons emitted during nuclear decay). These are sourced by
alpha radiation interactions with neutrons and fission reactions. Then there
are the cosmogenic backgrounds, such as cosmic ray muons passing through the
instrumentation or, worse, spallation neutrons produced as a secondary by-
product of cosmic rays which then enter the detector volume. Materials
themselves, later used in construction, become activated due to natural
exposure to cosmic rays or radionuclides near Earth’s surface. Neutrinos from
a wide variety of sources, from terrestrial origins to solar and cosmic
neutrinos, offer a weakly interacting low-mass background that can interact
with electrons or the nucleus. Of course, there is always the worst background
of all: the one no one has thought of yet.

###  4.1 Cosmic Rays

Let’s tackle cosmic rays immediately. The intensity of cosmic rays increases
with altitude and decreases as one goes below the surface of the Earth. For
this reason direct detection dark matter experiments seek laboratories
situated beneath the surface of the Earth, either in specially constructed
shallow underground sites or in deep mines that are also home to laboratory
facilities. An overburden of rock will protect an experiment from the leading
problem, the direct exposure to cosmic rays. However, it will then present the
secondary problem: backgrounds generated by cosmic ray interactions in the
overburden material. As shown in Fig. 13 these secondary particles can be
reduced by increasing further the depth of the laboratory housing the dark
matter experiment.

Figure 13: The cosmic ray primary intensity and the intensity of production of
secondary particles by cosmic ray primaries. This is shown as a function of
depth below the surface of the Earth, where depth is calibrated to meters-
water-equivalent (the equivalent depth if only water composed the overburden).
Figure excerpted from Ref. [23].

Many experiments also employ an _active cosmic ray veto system_ , a detector
subsystem designed to identify moments when a cosmic ray primary or secondary
has entered the volume of the main detector system. The time information about
these events can then be used to veto from consideration any activity in the
primary detector system.

###  4.2 Radioactivity Primer

It is useful to remind ourselves of some basic features of radioactive decay.
Fundamentally, the mathematics of this process is simple. Radioactive decay is
a spontaneous process. The rate of change $-dN/dt$ in the present population
of particles $N$ is proportional to the size of that population:

$-\frac{dN}{dt}=\lambda N.$ (107)

The _activity_ of a sample $A$ is defined as this rate of change of the
population, $A\equiv dN/dt$. Here, $\lambda$ is referred to as the decay
constant (potentially quite different for each radioisotope or new unstable
species). One can find a solution to this equation rather straight-forwardly:
we see a function, $N(t)$, such that the first derivative of this function
with respect to time returns the function itself multiplied by a constant,
$\lambda$. This is a classical hallmark of the exponential function. We guess
that

$N(t)=Ce^{Dt}.$ (108)

We need to solve for $C$ and $D$. To solve for $C$, consider the boundary
condition $t=0$:

$N(0)=Ce^{D}\cdot 0=C.$ (109)

We can identify $C$ as the size of the population at $t=0$, which I denote
$N_{0}$. To identify $D$, take the derivative of the solution:

$\displaystyle\frac{d}{dt}N(t)=\frac{d}{dt}N_{0}d^{Dt}=N_{0}De^{Dt}=DN(t)=-\lambda
N(t).$ (110)

We identify $D=-\lambda$. Thus our solution is of the form:

$N(t)=N_{0}e^{-\lambda t}.$ (111)

$\lambda$ has units of $s^{-1}$. As such, it can be identified with a time
constant, $\lambda=1/\tau$. With a little more work we can see that the decay
time of the population, $\tau$, is the time such that when $t=\tau$, 63.2% of
the original population has decayed. This can be related to the _half life_ of
the population - the time $t_{1/2}$ at which 50% of the population has decayed
- by the equation:

$\lambda=\frac{\ln(2)}{t_{1/2}}\longrightarrow\tau=\frac{t_{1/2}}{\ln(2)}.$
(112)

The initial number of radioisotopes in a population can be determined by

$N_{0}=\frac{N_{A}}{m_{isotope}}\times M_{isotope},$ (113)

where $N_{A}$ is Avogadro’s Number, $m_{isotope}$ is the atomic mass of the
radionuclide, and $M_{isotope}$ is the total mass of the the radionuclide.
There is also the concept of _abundance_ of an isotope, which refers to the
total amount of that isotope present relative to the stable isotopes in a
given sample of material.

###  4.3 Event Signatures

When radiation emitted by an unstable isotope impinges on a detector medium,
what may be the signatures of these interactions? We would expect two kinds of
signature: interactions with the atomic electrons or with the nucleus. For
example, photons (e.g. gamma rays) and electrons (e.g. beta radiation) will
tend to scatter off the atomic electrons. Alpha particles will ionize atomic
electrons, but can also interact with the atomic nucleus. Neutrons will
interact exclusively with the atomic nucleus. This latter fact is what makes
them particularly pernicious: WIMPS and neutrons preferentially are expected
to interact with the same part of an atom.

The interaction with atomic electrons generates what is known as an _electron
recoil_. Gamma rays are the most prevalent source of such recoils, while beta
radiation (fast electrons) tend to cause recoils near the surface of detector
materials and due to their penetrating capabilities recoils in the _bulk_ of
the material. Electron recoils generate significant electron currents in
materials, in addition to other phenomena, and in general can be more easily
differentiated from WIMP interactions.

However, neutrons and WIMPs will both interact with the nucleus and generate
the nuclear recoils discussed earlier in this chapter. Neutron interactions
appear like a 1 GeV WIMP striking a nucleus with a well-defined interaction
cross-section, with the rest of their effects being defined by what speed they
strike the nucleus. In fact, you can apply Eqns. 73 and 74 using the neutron
mass and its nuclear interaction cross-section, which is very well understood.

Electric currents are not the only response of materials to electron and
nuclear recoils. In fact, materials are generally selected because they
provide a range of signatures whose relative rates vary by what is interacting
in the material. These signatures can be summarized broadly as follows:

  * •

Heat

    * –

Phonons: for systems with a regular order (e.g. solid state materials) and
restoring forces to maintain the structure, phonons which are quanta of
vibrational energy, are a key signature. For these to be useful the material
must be kept extremely cold to reduce or remove normal thermal noise from the
system. In a crystal lattice kept at such low temperatures, a single atomic
recoil whether electronic or nuclear can setup phonons in the system. Phonons
can carry about 10 meV per quanta and, if detected at high efficiency, can
represent 100% of the original energy deposited in the material.

    * –

Superheating: for systems kept in a liquid state and under immense pressure,
it is possible to contrive the thermodynamics of the system such that a
nuclear or electronic recoil will cause the material to boil, generating small
bubbles that can be imaged. This is precisely the operating characteristic of
bubble chambers, and new versions of these devices play a role in dark matter
searches.

  * •

Light

    * –

Scintillation: this is a key means to detect interactions in materials.
Radiation absorbed by a material will result in the emission of light. With
appropriate chemistry and detector instrumentation, it is possible to observe
and capture this light and determine a portion of the energy present in the
original interaction. However, in general scintillation only captures a few
percent of the original interaction energy so even with high detection
efficiency one is only sampling a small portion of the original interaction.

  * •

Ionization

    * –

The removal of electrons from parent atoms creates both a population of free
electrons that can then traverse the material. These electrons and absence of
electron (holes) can be read out using modern electronics. Ionization lends
itself to the application of an electric field in the material to amplify the
signals or drift them to desirable locations for read out. Each conduction
electron will carry about 10 eV and with efficient readout one has access to
$\sim$20% of the original interaction energy.

A diagram representing these major signatures as well as experiments that
employ each by itself, or signatures in combination, is shown in Fig. 14.

Figure 14: Signatures for detecting electron and nuclear recoils in materials
and experiments that employ them individually or in combination.

WIMPs and neutrons scatter off of nuclei in nuclear recoils (NR). Most
backgrounds scatter off of electrons in electron recoils (ER). Detectors have
different responses to NR and ER, allowing for potential discrimination of
these different kinds of interations. However, the concept of the _quenching
factor_ (QF) becomes quite important in the discussion of these interactions
and detection of their signatures.The QF describes the difference in the
amount of visible energy in a detector to these two classes of events (NR and
ER). The ER signal will be measured in keVee (_keV electron equivalent_) while
the NR signal will be measured in keVnr (_keV nuclear recoil_). These are two
different energy scales that, in principle, need to be calibrated one to the
other to know, in a material, how to translate between then. For NR events,

$E_{visible}(\mathrm{keVee})=QF\times E_{recoil}(\mathrm{keVnr}).$ (114)

The employment of well-defined radiation sources such as a calibrated gamma or
neutron emitter whose emission energies are known can be used to determine the
quenching factor that relates these two energy scales. Gamma emitters will
preferentially induce electron recoils, while neutron sources will induce
nuclear recoils. The energy spectra of these two sets of recoils, taken in the
same material and taking into account the energies of the gamma and neutron
radiation, can be combined to determined the relationship between the ER and
NR scales.

Experiments can be designed to measure both ER and NR signatures. Their
combination is then employed to develop regions of the ER and NR signatures
rich in potential WIMP signatures (e.g. low ER, high NR) or backgrounds (e.g.
high ER, low NR). One example of this is a nuclear recoil in an ultra-cold
germanium crystal). Such a solid-state device allows for both the production
of electron-hole pairs through ionization and for the production of phonons
through recoils of the lattice atoms resulting from striking the nucleus of
such an atom. In germanium, we find that $E_{visible}\approx(1/3)E_{recoil}$,
which means the quenching factor for germanium is $\sim 30\%$. For more
details and examples of experiments that use these techniques see Sec. 7.3.

Another good example of this is a high-pressure liquid state system such as a
bubble chamber. In this case, we have a volume of superheated fluid at high
pressure that is deliberately placed in a metastable thermodynamic state. A
particle that interacts with the medium and deposits energy above a threshold
within a critical radius will result in the formation of an expanding bubble
in the medium. An interaction below threshold or one that is more diffuse in
the medium, spread beyond the critical radius, will form bubbles that
immediately collapse. These chambers can be tuned so that nuclear recoils
induce the expanding bubbles, but electron recoils do not. For more details
and examples of experiments that use these techniques see Sec. 7.4.

The rich set of signals that result from the variety of direct detection
experiments lends themselves to a variety of analysis approaches. These range
from simple “box cuts” - a set of one-dimensional selection criteria on a set
of variables ${\vec{x}}$ that each have sensitivity to ER, NR, or combinations
of these effects - to multidimensional approaches like profile likelihood
analysis and machine learning.

##  5 Detector Planning and Design

In this section, I will explain the many choices that have been made in the
planning and design of dark matter direct detection experiments in response to
the nature of the WIMP signal and the challenges posed by the many possible
background processes.

###  5.1 Underground Laboratories

As noted earlier, the key means by which substantial cosmic ray backgrounds
are mitigated is to site detectors, as well as construction facilities, deep
underground. There are presently 17 sites world wide that are utilized as, or
designed to be, underground laboratories (Fig. 15). The background intensities
from muons and spallation neutrons are shown in Fig. 16 as a function of
depth, with laboratory sites indicated as labels at their depths.

Figure 15: A map showing the 17 current underground facilities that can house
dark matter experiments, as well as other scientific payloads that benefit
from a location below the Earth’s surface. Figure 16: The intensity of muons
vs. depth, with labels indicating the depth locations of various underground
laboratory facilities. Taken from [24]

A key unit of depth measurement is the meters-water-equivalent (mwe). This
unit is needed to fairly compare two different sites with two different
overburdens. One lab may sit below a mountain of dense rock, while another
laboratory may sit below an equivalent depth of loose rock and soil. These
depths, stated directly, do not fairly compare the shielding differences of
the two overburdens. Instead, geological surveys of the overburden are
conducted and the densities and relative amounts of differnt materials is
accounted. This information is then converted into the equivalent depth of a
body of water, in meters, that would be represented by the combined shielding
capabilies of the lab’s overburden. This is the origin of the mwe unit.

The hadronic component of the cosmic ray flux is negligible after only
$\mathcal{O}(10~{}\mathrm{mwe})$ of overburden. Pions, kaons, and other light
hadrons produced in nuclear interactions by cosmic rays are attenuated quickly
by virtue of their electromagnetic and nuclear interactions. However, cosmic
ray muons can penetrate far deeper and can produce high energy neutrons (_fast
neutrons_). These, in turn, can enter the detector volume and result in keV
nuclear recoils in detector atoms. Neutron production in the vicinity of the
detector is facilitated by a few processes: (a) $\mu^{-}$ capture (an inverse
beta decay reaction that can up-convert a proton to a neutron, which is then
ejected from the nucleus); (b) photo-nuclear reactions initiated by
electromagnetic showers in the surrounding laboratory materials; (c) deep-
inelastic muon-nucleus interactions that “cleave off” neutrons from the
nucleus; (d) hadronic interactions of nucleons with pions and kaons produced
in hadronic showers induced by cosmic ray activities.

The challenge to the experimentalist is to anticipate fast neutron
backgrounds, recognizing that surrounding overburden material as well as well-
intentioned detector shielding materials (e.g. lead) can present production
targets for fast neutrons. The experimentalist must then identify ways to
either predict the rate of such backgrounds or veto them by passive and active
means.

###  5.2 Activation of Detector Materials

Activation of a detector material itself or materials that end up close to the
detector is another concern. This can happen during one or more of phases of
detector design, fabrication, transport, assembly, and operation. For example,
raw materials used in detector construction, even if originally purified to
remove radioisotopes, can be activated by cosmic ray interactions during
transport at the surface of the Earth. In addition, routes and locations on
the surface can matter. For example, cosmic ray spectra vary with geomagnetic
latitude and the flux varies with height above Earth. Choosing a mountainous
shipping route, for example, may lead to much higher activation of materials
than if the same material were transported along a route that is close to sea
level.

Another challenge in the activation of materials is that there is, despite
nearly a century of detailed work on radioisotopes, an incomplete
understanding of the cross-sections of interactions that lead to production of
isotopes. It is simply a fact that not all such production processes and their
rates have been measured. In general, radioisotope production is dominated by
$(n,x)$ reactions (95%) and $(p,x)$ reactions (5%).

###  5.3 Shielding from Environmental Backgrounds

A combination of high-atomic-number (_high- $Z$_) and low-$Z$ materials are
employed to diminish the neutron and gamma fluxes. Commonly employed materials
are lead (prized for its density of target nuclei), polyethelyne (prized for
its high neutron capture cross-section owing to the many hydrogen atoms in its
molecular structure), and copper. In addition, purging volumes with gases like
nitrogen can help to drive out airborne radioisotopes like radon gas.

Active and passive shielding is a key part of dark matter detector design.
Some experiments employ large water shields to passively reduce or moderate
both environmental radioactivity (e.g. gamma rays from surrounding unstable
nuclei) and muon-induced neutrons (e.g. resulting from the presence of
hydrogen in the water whose unpaired nuclear proton can readily capture a
neutron). Current efforts have determined that a water shield with a thickness
of 1-3 m can reduce underground fluxes of gamma radiation or radiogenic fluxes
by a factor of $\sim 106$. An active muon shield, employing materials doped
with scintillators (e.g. boron, gadolinium, etc.) that emit light when
traversed by charged particles, can be used to identify background-triggering
events. For example, a nuclear recoil induced by a neutron rather than a WIMP)
could occur when a muon passes through the detector and spallates a neutron
into the bulk detector material. An active muon shield would “tag” the passage
of that muon with high efficiency, allowing experimenters to veto from
consideration any nuclear recoil activity within a time window of the muon’s
passage.

The active dark matter detector volumes, nestled within a layer of active or
passive shielding, can also provide the own shielding capabilities. This is
known as _self-shielding_. For example, a large volume of liquid xenon
employed for ionization and scintillation observations will generally absorb
exterior backgrounds close to the surface of the xenon volume.
Experimentalists who design such detectors with good three-dimensional
position capabilities can identify these _surface events_ and veto them. This
is typically done by using external calibration sources to bombard the
material, observing where most of those interactions then occur and defining a
veto region in some outer volume of the active material. Of course, this self-
shielding effect comes at a cost: the active mass of the detector is reduced
by discarding a volume from consideration in the actual dark matter search. To
compensate for the loss of target mass, experimentalists may have to run the
experiment much longer (see Eqn. 74).

###  5.4 Internal Radioactivity in Detector Material

No amount of external shielding can protect a dark matter experiment from
itself. For example, any uranium, thorium, potassium, cesium, cobalt, argon,
or krypton (to name just a few) atoms embedded either in the active detector
material or nearby shielding material can result in radioisotope decays that
generate backgrounds. Most of these atoms are naturally unstable in their
abundant isotopes, and some of them have unstable radioisotopes as a small
fraction of their more abundant stable isotope(s).

(a) The XIA Alpha Screening Facility at SMU. Photo provided by the SMU LUMINA
laboratory.

(b) ICMPS at the Gran Sasso Laboratory in Italy [25]

Figure 17: A small sample of the many facilities at universities and
laboratories for screening materials before they are used in final
construction of a dark matter experiment. Taken from [26].

Elimination of these radioisotopes from the construction process requires a
diverse portfolio of techniques. In some cases, purification techniques can be
used in situ, as is the case for experiments using liquid xenon as a target.
Large distillation columns are used to purify the liquid xenon, bringing the
levels of krypton and radon into acceptable ranges. When in situ purification
is not feasible, materials have to be characterized and selected prior use in
the construction. Care needs to be taken to avoid activation after
construction, and detailed handling records (including altitude, duration of
time spent outside of shielding, exposure to atmosphere, etc.) aid in later
recognizing materials that may have become overly contaminated. This challenge
is made particularly large because modern dark matter experiments are now so
sensitive that radioisotope backgrounds have to be reduced below the 1 ppb
level.

Active screening of materials (e.g. by placing materials in drift chambers or
other detector instrumentation) is a part of this process. For example, highly
sensitive alpha counters (Fig. 17(a)) can be used to estimate how much
activity is implanted at or near the surface of small amounts of detector
material. This activity of samples can then be extrapolated to the activity of
all detector volumes constructed from this material. Another highly sensitive
technique is inductively coupled plasma mass spectrometry (ICPMS) (Fig. 17(b))
which can detector very low levels of radioisotopes in samples. A combination
of these approaches is the most common approach to insure compatibility
between independent means of surveying material contamination.

Screening facilities utilize combinations of two major practices to establish
a workflow for material assays. One approach is to augment commercially
available systems. For example, a commercially purchased High-Purity Germanium
(HPGe) detector can be placed in a custom-designed shield consisting of lead,
copper, neutron moderation and capture materials, an active cosmic ray veto,
and/or an underground location. Alternatively, a fully custom system can be
developed by a research group or a multi-institution collaboration. Here, in
addition to a custom shield, a custom cryostat design will be utilized with
attention to design and placement of electronics to minimize background
sources (such as uranium, thorium, or potassium). A comparison of these
approaches is shown in Fig. 18. Many screening options exist (Fig. 19), and no
doubt many more will be needed and developed as detector technology
sensitivity continues to advance to meet the challenge of detecting dark
matter interactions.

Figure 18: A comparison of material screening capabilities for various
isotopes showing the difference between augmented commercial approaches (top
three rows, light red) and custom approaches (bottom three rows, light
yellow). Standard-sized samples are compared between both approaches, while
large-sized samples with long counting times to integrate radiation exposure
to the instrumentation is shown only for an augmented commercial approach.
Figure 19: A comparison of material screening approaches and their current
standard sensitivities.

###  5.5 Modeling Radioisotope Backgrounds

Screening materials affords a chance to estimate radioisotope contamination of
small samples of the detector. Material handling records and tracking affords
a chance to estimate exposure to natural radioisotopes, like radon, or
activation by cosmic rays. However, it is impossible to monitor the whole
instrument in situ while it is operating. Inevitably, a computational model of
the possible backgrounds and their impact on the experiment must be developed.
These models employ fundamental physics, including the details of particle
interactions in material (cross sections); data from material screening and
material handling information; data from the experiment itself; and experience
from past phases of experimentation with earlier instrumentation.

Three software frameworks exist to calculate the spectra of neutrons produced
by ($\alpha-n$) interactions.

  * •

SOURCES - (using the EMPIRE2.19 libraries for cross section inputs)

  * •

USD WebTool (TENDL 2012 libraries which are validated by TALYS for cross
section inputs)

  * •

NeuCBOT (utilizing TALYS for cross section inputs)

TENDL is a validated library and EMPIRE is recommended by the International
Atomic Energy Agency, but neither can properly calculate all resonant behavior
that is experimentally observed. The output energy spectra from these
simulations can be used in a full simulation of the whole experiment to
predict the number of background events from neutrons. Validation of these
approaches [27] is essential before applying them to background estimates in
current or future experiments. For example, SOURCES-4A and USD agree on the
neutron yield (neutrons/s/cm3) from copper metal within about 20% of one
another (including separate predictions of the neutrons from the
${}^{238}\mathrm{U}$ and ${}^{232}\mathrm{Th}$ decay chains). However, for a
material like stainless steel these only agreed within about 50%.

Another example is to estimate the neutron fluxes from radioisotopes in
photomultiplier tube (PMT) glass [28]. PMTs are a common technology for
detecting and amplifying light emissions in detectors. The NeuCBOT
(SOURCES-4C) framework would predict 15 (13) n/year from PMT glass. Within the
ranges of constraints on these calculations and uncertainty on the underlying
processes, these are fair comparisons.

Predicting neutron fluxes from materials is still a place where significant
improvements are needed to keep up with the advances in detector sensitivity.

Simulation frameworks like GEANT4 can be used to develop detector and material
geometry and response models. Those models can then be subjected to the
expected fluxes of radioisotope backgrounds. An example of such a model for
the SuperCDMS dark matter search experiment is shown in Fig. 20. SuperCDMS is
a good example of the layouts I have been describing so far in this text. An
exterior series of shields that include lead, polyethylene, and ancient lead
depleted of radioisotopes encloses the cryostat (shown in red) which houses
the active dark matter detector volume and maintains it at micro-Kelvin
temperatures. The active detector volumes are stacks (”towers”) of hockey-puck
sized single crystals of Germanium. Each crystal is photolithographically
etched with a pattern of sensors for ionization and phonon readout.

Figure 20: A GEANT4 model of the SuperCDMS dark matter detector. Image
provided by the SuperCDMS Collaboration.

All of this complexity can be laid out in GEANT4 including gaps, services, and
other realities of the detector. Each material can be modeled, along with its
expected or measured level of radioactivity. A simulation of what the detector
would see over a period of time can then be generated. From that the energy
recoil spectrum purely from background processes can be produced. This can
include radiogenic backgrounds, cosmic ray-induced backgrounds, and neutrino
backgrounds and others not yet discussed here (Fig. 21). Signal events (dark
matter nuclear recoils) will be defined by experiment- and analysis-specific
thresholds; in the example shown, a dark matter nuclear recoil is counted only
for $E_{R}>0.27\mathrm{keVnr}$. A $10~{}\mathrm{GeV/c^{2}}$ WIMP candidate can
deposit up to $\mathcal{O}(10)~{}\mathrm{keVnr}$; for the highest such recoil
energy regions these models predict just fractions of a background event per
kilogram of material, per $\mathrm{keVnr}$, per year.

(a) Pre-selection

(b) Post-selection

Figure 21: Figures from Ref. [29]. Background spectra, before (left) and after
(right) analysis cuts in Germanium iZIP detectors, shown as a function of
nuclear recoil energy (keVnr) to allow direct comparison of the various
backgrounds to the natural dark matter interaction energy scale. Thick black
lines represent the total background rates. Electron recoils from Compton
gamma-rays, ${}^{3}\mathrm{H}$ and ${}^{32}\mathrm{Si}$ are grouped together
(red). The Ge activation lines (grey) are shown convolved with a 10 eV r.m.s.
resolution. The remaining components are surface betas (green), surface
${}^{206}\mathrm{Pb}$ recoils (orange), neutrons (blue) and neutrino
interactions (cyan).

###  5.6 Designing Radiopure Materials

It is increasingly the case that experiments cannot use commercial-grade
materials in the construction of experiments. For example, let’s consider
copper. This metal is essential in cryostat design and in the construction of
shielding for sensitive electronics. Mined from the Earth and refined, copper
is naturally contaminated by radioisotopes, primarily uranium and thorium. In
addition, radioactive elements such as radon are known to implant on copper
when it is left in open air. Radiopure copper is an essential ingredient in
dark matter detector construction, but sourcing this material has grown more
difficult as the sensitivity of dark matter detectors have been forced to
increase.

The philosophy of physicists in this kind of situation is straight-forward:
_if you cannot buy it, make it yourself._ For example, a facility at the
Pacific Northwest National Laboratory (PNNL) has been constructed to
electroform copper in a controlled manner. This controlled process can prevent
naturally occurring contaminants from playing a significant role during the
formation of the metal from a bath of copper ions (Fig 22). This manufacturing
process results in copper with $\leq 0.1~{}\mathrm{\mu Bq/kg}$ activity levels
from either the uranium or thorium chain.

(a) Copper nuggets prior to electroformation

(b) Copper cylinders from electroformation

Figure 22: (Left) Copper nuggets prior to their use in copper electroforming.
(Right) Copper cylinders resulting from the electroformation process. The blue
solution in the background is copper sulfate from the electroforming bath in
which these copper pieces were grown. Image provided by PNNL researchers Eric
Hoppe, Brian LaFerriere, Jason Merriman and Nicole Overman and made available
courtesy of Pacific Northwest National Laboratory.

Another example is the purification of noble gases and liquids for use in
detector volumes. Xenon is utilized both in dark matter and rare neutrino
search experiments. High purities are necessary in order to suppress the
radioisotope backgrounds that would otherwise render unusable these
instruments. The XENON Collaboration has designed, built, and operated their
own xenon purification facility [30]. This system can both purify and distill
the noble gas. Whereas commercial xenon is contaminated by krypton at the
level of 1 ppm - 10 ppb, the XENON 1-ton detector depends on levels at or
below 0.2 ppt to meet its design goals. This is approximately 100,000 -
500,000 times higher purity than commercial-grade xenon. This is only
achievable with their custom distillation and refinement approach, consisting
of a 5.5 m distillation column that can generate 6.5 kg of refined xenon each
hour.

###  5.7 Neutrino Interactions in Dark Matter Detectors

The neutrino is a low-mass, weakly interacting particle. In principle, it’s a
perfect test of whether or not a dark matter detector is capable of observing
a weak-scale nuclear interaction like those we hope occur between dark matter
and standard model matter. The challenge with neutrinos is that they are
relativistic – they travel at nearly the speed of light – because of their
extremely low masses. The sum of all three neutrino mass-eigenstates is
constrained by the CMB to be less than $0.12~{}\mathrm{eV}$ [31]. While the
individual mass eigenstates’ masses are not known, clearly they are a low-mass
WIMP-like candidate at best.

The most copious nearby source of neutrinos is the Sun. The fusion reactions
in its core, particularly the $pp$ chain of reactions, generate high fluxes of
low-energy neutrinos (Fig. 23). These primarily contribute to the ER
background via $\nu-e$ scattering at a level of 10 - 25 events per tonne of
detector material per year at low energies. The lower-rate ${}^{8}\mathrm{B}$
fusion neutrinos have higher energies, and as a result their NR can not be
distinguished from WIMP signals. Those are expected to lead to 1,000 events
per tonne of detector material per year of operation. This will especially
affect experiments using heavy targets such as germanium.

Figure 23: A few components of the solar neutrino flux spectrum.

Finally, there are atmospheric neutrinos produced by cosmic ray interactions
(e.g. muon decay in the atmosphere). These collisions produce pions which
subsequently decay to muon and electron neutrinos and antineutrinos. Current
and upcoming direct detection dark matter experiments are primarily
susceptible to the component of the atmospheric neutrino spectrum whose energy
is less than 100 MeV. There is also a diffuse background of neutrinos from
supernova explosions. There also exists higher energy neutrinos than even the
${}^{8}\mathrm{B}$ but their flux is even lower. These are expected to
contribute at the level of 0.01 - 0.05 events per tonne per year [32].

As direct detection technologies continue to advance, the neutrinos will
provide a “fog” of electron and nuclear recoils that will have to be modeled
with increasing precision. It will need to be measured using dedicated
instrumentation or in situ, and subtracted from the data to isolate any WIMP
nuclear recoil signatures. They are not, at the time of the writing of this
text, currently a problem. However, they are clearly on the horizon.

Directional detection capability may prove to be a crucial tool in future
experiments for rejecting the solar neutrino background. Neutrinos from the
Sun will, ideally, result in directional interactions, e.g. an ejected
electron or recoiling nucleus, that points back toward the Sun. However, the
weak interactions that, for example, cause a neutrino to scatter an atomic
electron will result in multiple final-state particles and doesn’t guarantee
perfect alignment of the electron with the original neutrino direction.

The degree of directional correlation has been explored at leading order in
the weak interaction [33]. The correlation between the scattering angle of the
final-state electron and the incident angle of the original neutrino, as a
function of the energy of the neutrino and folding in the solar neutrino
spectrum, is illustrated in Fig. 24. This plot is based on the cited work, but
is not published and is provided via an internal communication. There is a
reasonable expectation that, at the 68.3% (95.4%) confidence level, the
scattered electron will be within 1 radian (1.4 radians) of the incident
neutrino direction. This may be sufficient information for future directional
detection experiments to veto or categorize such events as more neutrino-like
than WIMP-like.

Figure 24: (Left) A model of the leading-order scattering of an atomic
electron by an incident neutrino (e.g. a solar neutrino). Image from Ref.
[33]. (Right) The degree of correlation between the ejected electron energy
and the scattering angle with respect to the incident neutrino direction
(private communication).

##  6 A Summary of Needs for Direct Detection Experiments

Direct detection experiments will clearly need the following characteristics
to make progress:

  * •

The ability to see low-energy WIMP induced recoils ($>10~{}\mathrm{keV}$).
This will require

    * –

Radiogenic purity;

    * –

Low energy thresholds.

  * •

Ability to distinguish nuclear recoils:

    * –

Differentiate between electronic recoils and nuclear recoils;

    * –

Differentiate between alphas and nuclear recoils.

  * •

Radiogenic and cosmogenic backgrounds mitigation

    * –

Passive and/or Active shielding from these backgrounds;

    * –

Position reconstruction and fiducialization;

    * –

Characterization of these backgrounds.

  * •

Long exposures with long term stability

    * –

This will be especially true to observe annual and diurnal modulation

The search space for dark matter is, in principle, vast. A variety of
theoretical models provide compelling candidates for dark matter, and the
space of their possible properties (e.g. cross-section vs. mass) is shown in
Fig. 25. A non-exhaustive list of models include NMSSM [34], Asymmetric Dark
Matter [35], MSSM [36] and CMSSM [37]. Existing experimental constraints can
be overlaid on this space to demonstrate what has, and what has not, been
excluded by the current portfolio of projects. I will conclude this chapter
with a look across several ongoing or planned dark matter direct detection
experiments.

Figure 25: An example of the theory-driven search space for dark matter across
a portfolio of plausible theoretical models for dark matter. Superimposed
exclusions of that space based on current experimental efforts. Figure
generated using [38]. Figure 26: Experimental exclusion regions in spin-
independent cross-section vs. mass space. The neutrino ”fog” background
expectation is shown in yellow at the bottom of the plot. The red arrows
indicate directions that forthcoming experiments will push into the parameter
space, toward better sensitivity at low mass and increased sensitivity at high
mass. Image generated using [38].

##  7 A Sampling of Ongoing or Planned Direct Detection Experiments

###  7.1 Time-Dependent Annual Dark Matter Modulation: DAMA/LIBRA, COSINE, and
ANAIS

A very fair question at this stage of the text is: _have we already seen
strong evidence of a dark matter signature in any experiment?_. An
illustration to the complexity of the subject of direct detection, is the
unfolding series of results from the DAMA/LIBRA experiment and two independent
instruments with sensitivity to the same proceses, COSINE-100 and ANAIS.

First DAMA, and then later DAMA/LIBRA, have been reporting positive results in
the search for an annual modulated dark-matter-like signal since 1998. The
DAMA phase of this experiment was a 100 kg NaI (Sodium Iodide) crystal array
operated in Laboratori Nazionali del Gran Sasso from 1996 - 2002. The LIBRA
phase of the experiment, operated by essentially the same collaboration of
institutions, is a 250 kg CsI crystal array operating since 2003 with first
results reported in 2008 and updated periodically since then.

The DAMA/LIBRA instrument is designed to measures scintillation from particle
interactions in detectors. However, this signal is the only one from this
instrumentation at the disposal of the experimentalists. This prevents them
from being able to discriminate between a wide range of background hypotheses
and the possible dark matter annual modulation signature. The primary effort
of their work has been to search for such a time-dependent effect on top of a
large background (e.g. a modulation with the right frequency and phase, see
Section 3.8). This means accounting for time dependence in the possible
backgrounds through simulation. A non-exhaustive list of potential backgrounds
includes a time dependence in the annual cosmic ray rate, or in radioisotope
backgrounds due to freeze/thaw cycles around the mountain laboratory that
trap/release radioisotopes throughout the year. The experiment has no direct
discrimination between nuclear and electron recoils.

They have reported a signal observed over 14 cycles with a significance above
the null (no modulation) hypothesis of $12.9\sigma$ in the
$2-6~{}\mathrm{keV}$ energy range (Fig. 27(a)). The collaboration is also able
to lower their energy threshold with the new LIBRA phase of the experiment and
have reported from that more limited sample alone a $9.5~{}\sigma$ signal for
single-scatter events in the $1-6~{}\mathrm{keV}$ energy range observed over 6
cycles (Fig. 27(b)).

Again, the instrument has no in situ background discrimination capabilities.
The collaboration has interpreted their observations in various ways,
resulting in regions of the plane shown in Fig. 25 where we would expect to
observe, with other independent instruments, those signals. However,
independent experiments using very different technologies have not reported
observing candidates with the expected masses and cross-sections. This has
prompted a robust and extended debate over whether or not the DAMA/LIBRA
background models are comprehensive, or whether or not the dark matter
interpretation is tied to the NaI technology of DAMA/LIBRA and is not
transferrable to other experiments.

(a) DAMA/LIBRA Annual Modulation

(b) LIBRA-only Annual Modulation

Figure 27: Results from the DAMA/LIBRA Experiment. (Top) The full 14-cycle
DAMA/LIBRA annual modulation result, after subtracting flat background
contributions. This analysis uses a higher energy threshold in the range of
$2-6~{}\mathrm{keV}$ of recoil energy. (Bottom) The LIBRA-only results
allowing for a lower-energy threshold of $1-6~{}\mathrm{keV}$ in recoil
energy. Figures extracted from [39].

Over the past two decades, with significant activity in the last five years,
there have been efforts to construct experiments that can directly cross-check
the claims of the DAMA/LIBRA collaboration. While earlier efforts may have
been deflectable with arguments about dark matter models that allow for a
signal in the DAMA/LIBRA experiment but no other running dark matter
experiment, more recently independent collaborations have concentrated on
developing high-purity NaI(Tl) crystals (to the level claimed by the
DAMA/LIBRA collaboration in their own instrumentation) for use in similar
scintillation-based experiments. Although I will focus on just two experiments
with recent results whose methodologies are intentionally similar to
DAMA/LIBRA, it should be noted that their are other collaborations also
working on the cross-check including the SABRE project which plans to deploy
detectors in both the northern and southern hemisphere.

Figure 28: The COSINE-100 detector. Photograph courtesy of the COSINE-100
Collaboration.

The first is the COSINE-100 experiment [40] (Fig. 28). The instrument is
located in Yangyang Laboratory in South Korea. It consists of 8 copper-
encapsulated NaI(Tl) crystals. The total active detector mass is 106 kg. The
detection of scintillation light is accomplished by two 3-inch PMTs per
crystal, with an event trigger set at the level of a 0.2 photoelectron
threshold. Calibration of the instrument is accomplished through insertion of
sources close to the detectors via access tubes. The total background expected
in this instrument is at the level of $2-4$ times the DAMA/LIBRA average, or
aboe 2.7 counts/day/kg/keV on average in the same $2-6~{}\mathrm{keV}$ energy
region. One favorable comparison to DAMA/LIBRA is that the in situ U/Th/K
contamination in these NaI(Tl) crystals is below DAMA, while the
${}^{210}\mathrm{Pb}$ contamination is at nearly the same level. This crystal
detector design permits for a very high light yield during interactions in the
material.

COSINE-100 has performed a similar annual modulation search. The signal is so
obvious in the DAMA/LIBRA data that even with a modest exposure (e.g. just a
couple of years) one should already expect to see this striking signal.
COSINE-100 have published results from just 1.7 years ($97.7~{}\mathrm{kg\cdot
years}$) of exposure. They perform a global maximum likelihood fit using a
cosmogenic background and simple harmonic components fit simultaneously across
all of the crystals, treating each crystal as an independent detector
subsystem. The collaboration have excluded three of the crystals (crystal 1,
5, and 8) in this published analysis due to low light yield and excessive PMT
noise. In regions of the data poor in anticipated WIMP signals (”sideband
regions”), they observe an event yield in energy that decreases exponentially.
This is consistent with known cosmogenic components.

The COSINE-100 Collaboration reported a best-fit result to their data that
prefers an amplitude of $0.0092\pm 0.0067~{}\mathrm{cpd/kg/keV}$ and a period
of $127.2\pm 45.9~{}\mathrm{days}$, which is consistent with both the null
hypothesis (no modulation) and with the DAMA/LIBRA best-fit value. This is
still very early data from this specific experiment, and the collaboration
projects that within 5 years of the start of data-taking they should be able
to cover the DAMA/LIBRA signal region with $3\sigma$ significance from the
null hypothesis. They have projected that future versions of their data
analysis will lower the recoil energy threshold to $1~{}\mathrm{keV}$ and
improve on the selection of recoil events to reduce to total exposure time
required to achieve the $3\sigma$ significance threshold for a DAMA/LIBRA-like
signature.

The second experiment I wish to discuss is the ANAIS instrument [41]. This is
also a NaI(Tl) crystal-based experiment, consisting of 112.5 kg of active
material. The experiment is housed in the Canfranc Underground Laboratory
(LSC), in Spain, and has been operating since 2017. In March 2021 they made
public the results of an analysis of three years of data-taking
($313.95~{}\mathrm{kg\cdot y}$). They found a best fit in the [1-6] keV ([2-6]
keV) energy region a modulation amplitude of $–0.0034\pm
0.0042~{}\mathrm{cpd/kg/keV}$ ($0.0003\pm 0.0037~{}\mathrm{cpd/kg/keV}$), and
determine that this is consistent with the absence of modulation signals in
their data. Their results are compatible with the COSINE-100 and incompatible
with DAMA/LIBRA result at $3.3\sigma$ ($2.6\sigma$) for an expected
sensitivity to the DAMA/LIBRA-like modulation signature of of $2.5\sigma$
($2.7\sigma$) (Fig. 29). While not completely definitive, taken together with
the COSINE-100 results there is a growing body of evidence, using
scintillation-based crystal experiments, that the DAMA/LIBRA result cannot be
replicated as a dark matter signal.

Figure 29: The ANAIS best-fit modulation amplitude result compared with the
DAMA/LIBRA best-fit result for both recoil energy ranges considered by
DAMA/LIBRA. Figure from Ref. [41].

###  7.2 Liquid Noble Detectors

Generally speaking, liquid noble detector technologies use a dual-phase (e.g.
liquid and gas) approach to broaden the range of signatures at their disposal.
This adds background discrimination capabilities, especially for high-mass
dark matter candidates, that are fairly unmatched in the field right now. A
few experiments that use this approach are XENON, LZ, Darkside, and PandaX.
They operate the dual-phase systems as a time projection chamber (TPC),
wherein electric fields are used to drift ions that result from nuclear and
electronic recoils. A grid of electrodes provides two-dimensional information
about the position of the the interaction when ions reach the electrodes. The
third dimension is provided from the known drift speed of ions in the
instrument, combined with information about when the original primary and
secondary interactions occurred.

Figure 30: The operating principle of a liquid Xenon TPC using the XENON-1T
detector as an example. Figure from Ref. [42].

The principle of these instruments is straight-forward and based on
significant experience with TPCs. Figure 30 illustrates the operations
principle of this type of detector [42]). Interactions in the liquid phase
(the largest single volume in the instrument) produce both atomic excitations
and ionization as illustrated in Fig31. Excitation leads to an immediate pulse
of scintillation light, establishing the $t_{0}$ of the interaction. This
first signal is known as $S1$. Ionization electrons are drifted with an
applied electric field and guided from the liquid phase into the gas phase. In
the gas phase, electrons are further accelerated through a stronger electric
field producing proportional scintillation. The pulse of energy from this
effect is known as $S2$. PMTs on the bottom and top of the chamber record
scintillation light. The pulse distribution in the PMTs provides the $x-y$
coordinates of the original interaction, while the drift time of the ions
through the phases of the experiment yields the $z$ coordinate. The ratio of
the energy deposited in $S1$ and $S2$ is a powerful discriminant against
background processes, separating especially electron and nuclear recoils as a
population.

Figure 31: Schematic of scintillation production in two phase xenon detector.
Ref. [43]

For example, a gamma ray that ionizes an atom inside the liquid noble volume
will result in a small pulse of initial energy (S1), and a comparatively
larger pulse of secondary energy (S2) as the avalanche of ionization occurs
when the original ionization electrons drift into the gas phase. A WIMP
striking a nucleus in the liquid phase, however, will cause a large pulse of
S1 energy followed by an equally large pulse of energy in S2. For the WIMP,
$S2/S1$ will be closer to 1.0 than for the gamma ray.

The liquid noble experiments have come to dominate the search sensitivity for
WIMP candidates with large masses $m_{\chi}\gtrsim
10~{}\text{Ge\kern-1.00006ptV/}c^{2}$. This is evident from Fig. 26 where
experiments like XENON1T, LUX, and PandaX define the parameter space
exclusions in that mass region.

###  7.3 Cryogenic Solid-State Detectors

Cryogenic solid state detectors are crystal detectors are generally designed
to readout both ionization and phonon signals from interactions withing the
detectors. Compared to ionization detectors, phonon detectors can provide
better energy resolution and smaller energy deposits in a variety of materials
because many more phonons are generated in an interaction than electron-hole
pairs. As an example, in semiconducting detectors an energy of $\sim 3-5$ eV
is required to create an electron-hole pair. This equates to approximately 300
pairs per keV. As a comparison, scintillation detectors produce 20-40 eV per
scintillation photon. When combined with the quantum efficiency of a
photodetector is equates to to a few photoelectrons per keV.

There are two families of sensors for reading out phonon signals: thermal and
athermal sensors. Thermal sensors measure an increase in temperature. A
thermal sensor must wait for the full thermalization of the phonons within the
bulk of the detector and the sensor itself, a process that occurs on the
timescale of ms. Athermal sensors, on the other hand, measure fast, non-
equilibrium phonons which contain information on the location and type of
recoil that occurred. Schematically, the detector is comprised of an absorber
with a weak thermal link to a refrigerator as illustrated in Fig. 32. The
measured temperature increase in the cryogenic detector is given by

$\Delta T=\frac{E}{C(T)}e^{-\frac{t}{\tau}}$ (115)

where E is the deposited energy, and C(T) is the temperature dependent heat
capacity of the absorber. For pure dielectric crystals and superconductors at
temperatures lower than their critical temperatures, the heat capacity is
given by

$C(T)\sim\frac{n}{M}\biggr{(}\frac{T}{\Theta}_{D}\biggr{)}^{3}$ (116)

where m is the absorber mass, M is the molecular weight of the absorber and
$\Theta_{D}$ is the Debye temperature. From this we see that the lower the
crystal’s temperature, the larger the $\Delta$T per unit of absorbed energy.
For an 100 g detector at 10 mK, a 1 keV energy deposition increases the
temperature by $\sim$1 $\mu$K, a temperature increase that can be measured!

Figure 32: Schematic of a generic cryogenic detector. A thermometer measures
the temperature of an absorber that has a weak thermal link to a refrigerator.
The measured temperature increase is proportional to the deposited energy
divided by the heat capacity of the absorber. Figure courtesy of Matt Pyle.

The two most widely used technologies to measure these signals are
semiconductor thermistors such as neutron doped germanium sensors (NTDs) where
the resistance is a strong function of temperature and transition edge sensors
(TESs). In both NTDs and TESs, an energy deposition produces a change in the
electrical resistance that can be measured.

NTDs are small Ge semiconductor crystals that have been exposed to a neutron
flux to make a large, controlled density of impurity. The resistance in these
devices is strong function of temperature. NTDs measure small temperature
variations relative to a $T_{0}$, which is set to be on the transition from
the superconducting and resistance regime. The resistance is continuously
measured by flowing current through and measuring the resulting voltage. The
sensors are glued to the detectors.

A TES is a thin superconducting film that is operated near its critical
temperature. TESs are fabricated onto crystal substrates using
photolithography. A heater with an electrothemal feedback system maintains the
temperature of the TES at its superconducting edge. When a particle interacts
in the absorber (crystal target), phonons are generated and propogate to the
surface of the absorber where they are absorbed by aluminum collection fins.
When phonons enter the aluminum fins, Cooper pairs within the aluminum fins
are broken, creating quasiparticles. Those quasiparticles diffuse into the
TES, releasing their binding energy and increasing the temperature of the TES.
This increase in temperature increases the resistance of the voltage-biased
TES. Temperature changes are then detected by a change in the feedback current
and collected by a Superconducting Quantum Interference Device (SQUID). This
process is illustrated in Fig. 33.

Figure 33: Schematic of a SuperCDMS phonon sensor on the surface of a Ge
substrate. Phonons produced by particle interactions in the substrate break
Cooper pairs in the aluminum fin producing quasiparticles. The quasiparticles
diffuse into the TES increasing its temperature. Images provided by the
SuperCDMS Collaboration.

####  7.3.1 SuperCDMS

The SuperCDMS collaboration has a long history of using TES technology for
detector readout. Currently, the collaboration is constructing an upgraded
Generation 2 (G2) experiment within SNOLAB in Sudbury, Canada. The cryostat
will be able to accommodate up to seven towers and operate at a temperature of
15 mK. The initial payload will contain 24 detectors amounting to a mass of 30
kg. The detectors configured into four towers with each tower containing six
detectors. Each detector will be a 100 mm diameter, 33.3 mm thick germanium or
silicon crystal with mass 1.39 kg (Ge) or 0.61 kg (Si). Two of the detector
towers will contain detectors operated in high voltage (HV) mode. Each HV
tower will contain four germanium and 2 silicon detectors. The detectors in
the remaining towers will be operated in interdigitated, z-sensitive
ionization and phonon mediated (iZIP) mode. One of those towers will contain
six germanium detectors and other tower will contain four germanium and 2
silicon detectors. The background from gamma particles is expected to be less
than 0.1 dru (differential rate unit = event/kg/day/keV).

Each iZIP detector will have eight phonon channels and two charge channels on
each detector face. The ionization sensors are interleaved between the phonon
sensors on each face, as illustrated in Fig. 34. When a particle interacts in
these detectors, simultaneous measurements of prompt phonon and ionizations
signals allow for the robust discrimination between nuclear recoiling and
electron recoiling events. One issue that was a concern in previous detector
designs was that beta events occurring near the surface of the detector could
mimic a WIMP interaction. The new interdigitated design allows for the
identification of surface events by the distribution of the charge signal for
a given event. If the event has symmetric charge distribution between the top
face and the bottom face, it is a bulk event. If the charge is distributed
primarily on one side of the detector, it is a surface event. As such, all
surface and electron recoil events above a few keV can be removed using
appropriate selection criteria. These detectors have sensitivity in a
“background-free” mode to WIMPs with masses above $\sim$5 GeV and are
sensitive to WIMPs with masses above $\sim$1 GeV in a “limited-discrimination”
mode.

Figure 34: Schematic of a SuperCDMS iZIP detector. Left: Lines of ionization
sensors (red) are interleaved between grounded phonon sensors (blue). Right:
Each phonon sensor consists of an aluminum superconducting fin (grey) and a
TES(blue.) Figure provided by the SuperCDMS Collaboration. Figure 35:
Ionization yield versus phonon recoil energy with the $\pm$2$\sigma$
ionization yield range of neutrons indicated (area within green lines). Taken
from 900 live hours of calibration data from with a ${}^{2}10$Pb source facing
one side of a detector. The hyperbolic black line is the ionization threshold
(2 keVee); the vertical black line is the recoil energy threshold (8 keVr).
Symmetric charge events (large blue dots) in the interior of the crystal and
the events that fail the symmetric charge cut (small red dots) include surface
events from betas, gammas and lead nuclei and are clearly separated from the
nuclear recoil band. Figure provided by the SuperCDMS Collaboration.

The SuperCDMS HV detectors will have only six phonon channels on each face to
take advantage of the Neganov-Trofimov-Luke (NTL) effect. When electrons are
drifted across a potential (V) a large number of phonons are generated. The
total phonon energy (Et) is a combination of the primary recoil energy (Er)
and the NLT phonon energy. The NTL phonon energy is given by

$E_{NTL}=N_{eh}eV_{b}$ (117)

where Vb is the bias voltage across the detector, the number of electron-hole
pairs generated is N${}_{eh}=E_{r}$/$\epsilon_{\gamma}$ and
$\epsilon_{\gamma}$ is the average energy needed to generate an electron-hole
pair. In germanium, ($\epsilon_{\gamma}$) is 3 eV. Thus the total phonon
energy can be written as

$E_{t}=E_{r}+N_{eh}eV_{b}.$ (118)

When Vb is very large, the NTL phonons dominate the signal and allow lower
energy thresholds to be reached.

HV detectors provide ultra-high resolution by indirect charge measurements,
but these detectors do not provide yield or detector face discrimination. The
capabilities of HV detectors was demonstrated using the SuperCDMS Soudan
CDMSlite detectors where thresholds of 75 eVee and 56 eVee were achieved as
illustrated in Figure 36. The HV detectors to be operated in SNOLAB will be
able to search for WIMP dark matter with masses down to $\sim$0.3 GeV.

Figure 36: Illustration of the resolution of a SuperCDMS HV detector using the
WIMP-search spectrum after application of all cuts and correcting for
efficiency (except trigger efficiency) in the CDMSlite detectors. Inset shows
a zoom (with smaller bin size) of the energy range actually used to the set
the limit. 71Ge activation peaks are marked with blue dashed lines. Adapted
from [44].

As noted earlier, nuclear recoils produce electron-hole pairs less efficiently
than electron recoils. As such the energy and interaction type can be defined
through the yield (Y) where $Y\equiv 1$ for electron recoils.

$N_{eh}=Y(E_{r})\frac{E_{r}}{\epsilon_{\gamma}}.$ (119)

Thus, the total recoil energy can be written as

$E_{t}=E_{r}\biggr{(}1+Y(E_{r})\frac{eV_{b}}{\epsilon_{\gamma}}\biggr{)}$
(120)

The energy scale is also dependent upon the interaction type. The detectors
are calibrated using electron recoils and the resulting energy scale is
labeled as keVee. To convert to the nuclear equivalent energy (keVnr) by
equating the equation above for nuclear recoils and electron recoil. Thus

$E_{nr}=E_{ee}\biggr{(}\frac{1+eV_{b}/\epsilon_{gamma}}{1+Y(E_{nr})eV_{b}/\epsilon_{gamma}}\biggr{)}$
(121)

where Y(E${}_{nr})$ must be determined by either direct measurement or with a
model. The most common model used for ionization yield is the Lindhard model
[45, 46, 47]:

$Y(E_{nr})=\frac{k\cdot g(\epsilon)}{1+k\cdot g(\epsilon)}$ (122)

where g($\epsilon)=3\epsilon^{0.15+0.7\epsilon^{0.6}+\epsilon}$,
$\epsilon=11.5E_{nr}$(keV)$Z^{7/3}$, $Z$ is the atomic number of the material
and k = 0.157 in germanium. Although this value of k agrees with experimental
measurements above 1 keVnr, there are few measurements at lower energies. As
such, for the lowest energy events an uncertainty must be taken into account.
This is typically done by varying k uniformly between the majority of
experimentally observed data [48, 49, 50].

Figure 37: Comparison of ionization yield versus recoil energy for various
models and experimental data. Models are Barker and Mei [48], Collar [50] and
Lindhard [45] Data points are from Jones and Kraner [51, 49], Barbeau et al.
[50], TEXONO [52], CDMS[53] Messous et. al [54], Chaseman et al. [55], Shutt
et. al [56], Sattler et al. [57], Baudis et. al [58], and Simon et.a l [59].
Figure adapted from [48].

The SuperCDMS collaboration is also pursuing the development of HVeV
detectors. These detectors have demonstrated single electron/hole pair
resolution and have sensitivity to a variety of sub-GeV dark matter models
using just gram-day exposures [60, 61, 62].

####  7.3.2 Edelweiss

The Edelweiss III collaboration has a long history of using bolometer-style
readout technology. The experiment is located in Laboratoire Souterrain de
Modane (LSM) and consists of 24 cylindrical high purity germanium crystals,
each with mass $\sim$860 g operated at 18 mK [63]. Their fully interdigitated
(FID) detectors collected charge via concentric aluminum ionizatiation sensors
that were interleaved on all absorber surfaces and two NTD Ge sensors that
were glued onto the top and bottom face of the detectors. The configuration of
the ionization sensors allowed for the fiducialization of bulk events in the
inner region of the detectors as illustrated in Fig. 38. These detectors show
strong separation on an event-by-event NR versus ER basis down to $\sim$5 keV
as illustrated in Fig. 39. The rejection of ER events to NR events based on
yield alone is $<2\times 10^{-5}$. The readout of all electrodes provides for
a $<4\times 10^{-5}$ rejection of the especially concerning class of
backgrounds, events occurring near the detector surfaces.

Figure 38: (a): An FID detector showing the interleaved ionization sensors and
NTD sensor. (b): Cross-section of an FID detector illustrating the fiducial
zone for an event generated in the bulk of the detector. (c): Illustration of
charge collection for an event near the surface of the detector. Adapted from
[63]. Figure 39: Ionization yield versus recoil energy for data taken with an
AmBe neutron calibration source illustrating clear separation between electron
recoils with yield $\sim$1 and nuclear recoils with yield $\sim$0.3. Blue
lines indicate the 90% C.L. for electron recoils, red lines indicate the 90%
C.L. for nuclear recoils, and the magenta lines illustrate the inelastic
scattering of neutrons on the first (13.28 keV) and third (68.75 keV) excited
state of 73Ge. Adapted from [64].

There are three issues that are somewhat limiting the Edelweiss experiment
performance. The most concerning for WIMP searches is that the neutron
background is five times what is expected. It is too large for searches of
WIMPs with masses $>\sim$10 GeV/c2. However, this neutron background is not of
concern for searches of lighter mass WIMPs. Similar to SuperCDMS, the
Edelweiss collaboration is developing a new detector called ”RED” to pursue
very light, electron interacting dark matter models. The NTD phonon resolution
and an unknown background source of “heat-only” events (events that deposit
energy in the detector without the creation of electron-hole pairs) are
currently limiting factors in their searches [65]. These issues make it
difficult to disentangle, noise-triggered events, possible events from leakage
currents and other such backgrounds from potential dark matter signal
candidates.

####  7.3.3 CRESST

The CRESST experiment is operated in the Laboratori Nazionali del Gran Sasso.
The experiment uses an array of 10 cryogenic scintillating CaWO4 crystals that
are self-grown. Each crystal has dimensions of 20 mm x 20 mm x 10 mm and is
instrumented with tungsten TESs to readout phonon signals. Each detector
module consists of a scintillating crystal with mass of $\sim$25 g equipped
with a TES next to a silicon on sapphire light absorber which is also equipped
with a TES. The modules have fully scintillating housing and are instrumented
“iStick” holders which provide the ability to veto radiogenic backgrounds from
surrounding surfaces and suppress thermal signals from particle interactions
in surrounding materials [66]. Similar to SuperCDMS and Edelweiss, the CRESST
detectors allow for the discrimination between ER and NR. A schematic of the
detectors is shown in Fig. 40.

Figure 40:  Left: Schematic of the CRESST-III detectors highlighting aspects
of the improved detector design which includes fully scintillating housing and
CaWO4 instrumented holding sticks. Right: Picture of a CRESST-III detector
module. Adapted from [66].

The most recent results [67] from the CRESST detector were obtained with a
single module. The combination of light sensors and the element oxygen in the
detector target allows sensitivity to dark matter particle masses as low as
160 MeV/c2. An AmBe neutron calibration source is used to define the NR band
for each element in the crystal. Selection criteria was used to remove periods
of abnormally high trigger rates due to electronic disturbances. Events that
triggered only the light channel were removed in addition to events of poor
quality. Finally, events triggered in coincidence with the muon veto were
removed. The livetime of the data set after all criteria were applied was 3.64
kg days. The energy and yield of the events passing all selection criteria are
shown in Fig. 41.

Figure 41:  Left: Light yield vs phonon energy for events passing selection
criteria in the dark matter data set. The blue lines enclose the 90% C.L.
electron recoil band, the red and green lines enclose the 90% C.L. nuclear
recoil bands for oxygen and tungsten, respectively. The yellow band encloses
the dark matter candidate acceptance region. All events in this region are
taken as dark matter candidates. Right: Energy spectrum of all events passing
selection criteria in the dark matter data set. Candidate events in red fall
within the dark matter candidate acceptance region. Adapted from [67]

All 441 events in the acceptance region were considered to be dark matter
candidates with no background subtraction when setting the upper-limit on the
WIMP-nucleon scattering cross section using the Yellin Optimum Interval method
[68]. This result excluded new parameter space down to 0.16 GeV/c2 and
improved exclusion limits by a factor of 6 over the previous best limit.

###  7.4 Bubble Chambers and Superheated Liquids

Bubble chambers are sealed vessels filled with a liquid under pressure. To
understand the principles of bubble nucleation [69], consider a bubble in
thermal and chemical equilibrium. In that case the temperature of the liquid
is equal to the temperature of the bubble. Assuming that there is no surface
tension, if the pressure of the bubble, $P_{b}$, is greater than the pressure
of the liquid, $P_{l}$, the bubble will expand. If we include the pressure due
to surface tension, $P_{s}$, the bubble will grow when

$P_{b}>P_{l}+P_{s}$ (123)

and the radius of the bubble is bigger than a critical radius. The pressure
due to surface tension can be written as

$P_{s}=\frac{2\sigma}{r}.$ (124)

where $\sigma$ is the surface tension which is material dependent. Thus, we
write the radius condition for bubble growth as

$r>r_{c}=\frac{2\sigma}{P_{b}-P_{l}}$ (125)

Bubbles that do not meet these conditions collapse. The threshold for bubble
nucleation is given by

$E_{T}=4\pi
r^{2}_{c}\bigg{(}\sigma-T\bigg{[}\frac{d\sigma}{dT}\bigg{]}_{\mu}\bigg{)}+\frac{4\pi}{3}r^{3}_{c}\rho_{b}(h_{b}-h_{l})-\frac{4\pi}{3}r^{3}_{c}(P_{b}-P_{l})$
(126)

where $\rho$ is the bubble density and h is the specific heat of the liquid
($h_{l}$) or bubble ($h_{b}$).

Bubble chambers as dark matter detectors are attractive because of their
unique ability discriminate between different particle interactions [70].
Figure 42 shows the response of superheated C4F10 to various particles.
Electron recoiling events have much lower thresholds compared to events that
produce nuclear recoils. Thus, the detectors can be tuned to trigger only on
nuclear recoiling events.

Figure 42: Detector response to different particles in superheated C4F10. From
left to right: 1.75 MeV $\gamma$-rays and minimum ionizing particles (dot-
dashed); model of 50 GeV/c2 WIMP (red); polyenergetic neutrons from an AcBe
source (dotted); $\alpha$ particles at the Bragg peak from 241Am decays (open
triangles); and 210Pb recoil nuclei from 226Ra spikes(dots) Taken from [70]

.

To mitigate nuclear recoils from neutrons, the experiments can be located
underground and use active and/or passive shielding. Nuclear recoils from
alpha events can be identified using acoustic techniques. Alpha particles
deposit their energy over tens of microns whereas nuclear recoils deposit
their energy over tens of nanometers. The result is that alpha particles are
$\sim$4 times louder than neutrons and dark matter candidates, an effect that
can be measured by piezoelectric sensors as illustrated in Fig 43.

Figure 43: Detector response to alpha particles and nuclear recoils as
characterized by an acoustic parameter (AP) in the COUPP-4kg dark matter
experiment operated in SNOLAB. Adapted from [71].

The PICASSO and COUPP experimental collaborations have joined forces to create
a new collaboration called PICO. The new collaboration has ambitions to
ultimately build and operate a 500L bubble chamber. The collaboration has
already produced results from two runs with PICO-2L [72, 73] and PICO-60 [74].
Operation of these detectors pointed towards problems connected to the water
piston-active fluid interface. As such, a redesign was implimented in the
PICO-40L experiment where the superheated target sits on top of a fused silica
piston.

##  8 Conclusion

The next decade will be very exciting for dark matter direct detection. The
next stages of multiple large target experiments (called Generation 2 or “‘G2”
experiments) with sensitivity to the neutrino fog will be coming online. The
XENON and LUX program two phased xenon experiments are expected to have
sensitivity to or past the neutrino fog. Research and development for even
more massive xenon and argon detectors are taking place within the DARWIN and
GADMC collaborations. The solid state detector program is making fast progress
in pushing their technologies to sensitivities to the lowest mass WIMPs.

Although WIMPs remain a viable and interesting candidate, other dark matter
scenarios are gaining traction in the theoretical community and technological
advances are allowing for the exploration of these ideas. This has opened up a
new window of exploration for solid state cryogenic detectors.

Given the wealth of possibilities, a diverse set of experimental designs and
targets are needed to constrain the theory and couplings of any discovered
signal. As a community, we are well positioned to take on those challenges.

## Acknowledgements

I would like to thank the organizers of the Les Houches 2021 Summer School on
dark matter physics for the opportunity to teach such an engaged bunch of
students. I would also like to thank the students for their lively discussions
and inquisitive questions which led to invaluable contributions to improving
these notes. In particular I would like to thank Dan Salazar-Gallegos for his
extensive editorial suggestions. I would also like to thank Stephen Sekula for
discussions and editorial assistance.

## References

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