#  DIRECT DETECTION OF DARK MATTER:
A CRITICAL REVIEW

Marcin Misiaszeka and Nicola Rossib
aJagiellonian University, Krakow, Poland
bLaboratori Nazionali del Gran Sasso (INFN), L’Aquila, Italy

###### Abstract

The nature of the dark matter in the Universe is one of the hardest unsolved
problems in modern physics. Indeed, on one hand, the overwhelming indirect
evidence from astrophysics seems to leave no doubt about its existence; on the
other hand, direct search experiments, especially those conducted with low
background detectors in underground laboratories all over the world seem to
deliver only null results, with a few debated exceptions. Furthermore, the
lack of predicted candidates at the LHC energy scale has made this dichotomy
even more puzzling. We will recall the most important phases of this novel
branch of experimental astro-particle physics, analyzing the interconnections
among the main projects involved in this challenging quest, and we will draw
conclusions slightly different from how the problem is commonly understood.

Keywords— dark matter, dark matter direct search, low background experiments,
gravity modification, science philosophy

## Introduction

According to Standard Cosmology, the visible Universe emerged, after a very
short phase of rapid inflation, from a space-time singularity, with infinite
temperature and density, about 13.8 billion years ago, and its evolution from
that moment is determined by the distribution of its energy content among its
distinguished constituents, namely visible (or baryonic) matter (4.9%), dark
(non visible) matter (26.5%) and dark (non visible) energy (68.5%) [2].

Besides dark energy, whose physical nature is still a subject of intense
debate, dark matter offers more specific clues. These clues have led the
scientific community to strongly believe in its existence and even to consider
possible ways to experimentally detect its presence when it passes through the
Earth.

The dark matter hypothesis fills the gap in the explanation of many
astrophysical observations, from gravitational collapse of structures in the
early Universe, patterns in cosmic microwave background, anomalous dynamics
and gravitational lensing of astronomical objects from small-sized galaxies to
large super-clusters, with some exceptional case as the Bullet Cluster,
considered by someone as an incontrovertible smoking gun. These strong
experimental evidences seem to converge towards the existence of a given
amount of invisible massive particles piled up around massive astronomical
objects with some rudimentary properties.

In the dominant paradigm, this matter, indeed, should have had a negligible
velocity at the time of early structure formation, for this reason is called
_cold_ , as stated by the widely accepted _Lambda Cold Dark Matter_ paradigm
($\Uplambda$CDM) [3]. This scenario can be easily actualized with the
existence of heavy and collision-less particles, _e.g._ massive particles with
mass comparable or larger than typical atomic nuclei (tens of GeV/c2 or more).
If massless or much lighter, the dark matter should have some interaction
properties, or some coherent aggregation, sufficient to simulate a non-
relativistic gravitational clustering, capable of speeding up the structure
formation, as observed, after the matter-radiation decoupling epoch in the
early Universe [4].

Alternative theories, as the _Modified Newtonian Dynamics_ (MOND) [5], super-
fluid dark matter [6], or even full General Relativity approach [7], have been
showing waves of interest in the scientific literature. It is worth
mentioning, even though it is not the main topic of this article, that some
empirical evidences, as the Tully-Fisher relation [11] and the Renzo’s rule
[12] for galaxies are hardly explainable by standard cold dark matter galactic
halos, and this issues, very well addressed by the MOND theories, seem
sometimes completely ignored when considering whatever particle explanation of
the missing mass of the Universe. And, along with this, many issues related to
the $\Uplambda$CDM model are still unsolved, such as the _missing satellites_
problem, the _cusp-core_ problem and _too-big-to-fail_ problem (see _e.g._ [2]
and Refs. therein).

Nevertheless, the particle hypothesis for dark matter, even if strongly
weakened by the astonishing absence of the super-symmetry (SUSY) at the Higgs
scale in the LHC accelerator [13], shows still a constant interest in the
scientific community, mostly for its experimental feasibility. In other words,
the costs and technology for the direct dark matter search in underground
laboratories is still so affordable that the biggest part of the experimental
community believes that it is still worth trying, as the main goal in
scientific programs for the forthcoming decades.

Since the early 90s, when the hypothesis of dark matter began to be taken
seriously by scientists, many candidates have been proposed, with mass ranging
from axions with masses starting from $10^{-22}$ eV/c2 to primordial black
holes with masses up to 5 M⊙ $\approx$ ($10^{67}$ eV/c2) [2]. This wide range
of 89 orders of magnitude is filled almost homogeneously with many variants of
the proposed models, and it is basically limited by astrophysical constraints
related to the observed macro-structures. Within this range, it is worth
mentioning the most important candidates with increasing mass: axions, with
sub-eV/c2 mass, detectable in haloscopes or low background detectors [2];
sterile neutrinos [15], with mass of the order 1 keV/c2, evergreen candidates
for many presumed anomalies, never detected; weakly interacting massive
particles (WIMP) [16, 17], with mass in the range 1–$10^{3}$ GeV/c2, predicted
by the SUSY extension of particle Standard Model (SM) and considered the top
reference model almost up to the first null results by LHC [18], with some
monster extensions up to the Grand Unification scale (GUT) of $10^{15}$ GeV/c2
often called WIMPzillas [19], yet never detected; finally, dark objects
(MACHOs) with the size of a planet ($\sim 10^{24}$ kg) or so [20], and
primordial black holes, with huge mass, up to 5 $M_{\odot}$ [22]. Finally, it
is also worth mentioning the Mirror Matter model [23], conceptually different
from WIMPs, but predicting the existence of dark matter candidates with mass
comparable with the mass of visible chemical elements (1–100 GeV/c2).

Another problem is the cross section scale between visible and dark matter. If
one assumes that the typical cross section of the weak interaction
($\sigma\sim 10^{-44}$ cm2, or so), is already experimentally at reach, the
range of possible interaction cross sections is actually really large. If one
takes the squared Planck length as lower bound $\sigma\sim 10^{-66}$ cm2,
there are 20 possible orders of magnitude, even though not all of them are
actually testable.

All the models discussed above, basically agree on the _invisible_
(electrically neutral) nature of the dark matter candidate: astrophysical
observations constrain its possible electric charge and self-interaction, and
require temporal stability with a lifetime comparable to the age of the
Universe [2].

Indirect dark matter searches, such as hidden channels in accelerators or
annihilation/decay in visible diffused particle backgrounds in the Galaxy,
have also yielded no results [2].

Property | Value
---|---
Composition | –
Statistics | –
Family | –
Generation | –
Interaction | –
Symbol | $\chi$
Antiparticle | –
Mass |  $10^{-22}$ eV $\div 5$ M⊙
Half-life |  $\gtrsim 10^{10}$ y
Electric charge |  $\lesssim 10^{-7}$ e [at 1 GeV]
Self interaction |  $<0.5$ cm2 [at 1 GeV]
Magnetic Moment | –
Spin | –
Weak isospin | LH: –, RH:–
Weak hypercharge | LH: –, RH:–
Others | –
Table 1: Properties of the dark matter particle candidate to date. The choice
of the symbol $\chi$ is not universal, but it is assumed in the present
article as a reference for a generic dark matter candidate. From the top:
composition, Bose or Fermi statistics, particle generation or family,
fundamental interaction, symbol, antiparticle, mass, half-life, electric
charge, self interaction, magnetic moment, spin, weak isospin, weak
hypercharge and other characteristics. Symbol “–” stands for _not available_.

Table 1 summarizes the known physical properties of dark matter particle
candidates. The Table is essentially empty, except for some naive properties
inferred by indirect astrophysical observations. The absence of experimental
evidence of dark matter in either direct or indirect search, has sometimes
induced the literature to hide or rename some of the historical candidates,
often with fancy names as “axion-like” (ALPs) or “WIMP-like” [14], just
because the original proposed theory seemed not to hold any more in upgraded
experimental contexts. Furthermore, often there are unconscious biases, like
the belief that, if the dark matter is not found in the range of the WIMPs,
maybe it is really important to search for some _light_ WIMP-like dark matter
particle with mass below 1 GeV/c2, down to a few tens of MeV/c2. This kind of
suggestion is of course much weaker, if one considers that the theoretical
prior on some plausible model, from $10^{-22}$ eV/c2 to $5$ M⊙ is currently
basically uniform.

Figure 1: Schematic view of the expected Keplerian fall. The behavior of the
rotation curve $v(r)$ (top) is shown both inside and outside the galaxy bulge,
with its cross section displayed at the bottom.

The discussion about the details and the status of the direct dark matter
search will proceed as follows: In Sec. 1, the reason, according to the
authors, why the direct dark matter detection is so popular, and increasingly
supported and financed is outlined; in Sec. 2, the basic ideas about the
direct detection of dark matter are reviewed; in Sec. 3, important and often
undervalued aspects of the direct dark matter analysis are highlighted; in
Secs. 4 and 5, the case of both NaI-based and noble gas (xenon and argon)
detectors are largely reviewed, respectively. Finally, in Sec. 6, the
evolution of dark matter search in the new millennium is critically reviewed
and analyzed.

##  1 A simple theory

The direct dark matter search has received an increasing consensus in the
recent decades also thanks to its simplicity, and then to the (apparent)
solidity of its foundations, invoking some principle invented in the context
of theology as the _Occam’s razor._ The possibility of such a detection is
enclosed in simple equations, understandable at the level of an average high
school student. Three basic formulas, concerning the dark matter hypothesis
and its possible detection, are hidden among a list of seemingly simple
equations. Here is the list:

$\displaystyle\nexists A:\aleph_{0}<\\#A<2^{\aleph_{0}}$ (1) $\displaystyle
v=\sqrt{GM/r}$ (2)
$\displaystyle\vec{\nabla}\cdot\vec{\mathcal{F}}=0,\vec{\nabla}\times\vec{\mathcal{F}}=i\frac{\partial\vec{F}}{\partial
t}$ (3) $\displaystyle\mathcal{H}u=-\frac{\partial v}{\partial
t},\mathcal{H}v=\frac{\partial u}{\partial t}$ (4) $\displaystyle
E_{A}=\frac{4M_{A}M_{\chi}}{(M_{A}+M_{\chi})^{2}}E_{\chi}$ (5) $\displaystyle
x^{3}+x=1$ (6) $\displaystyle\mathcal{R}=n\sigma\Phi$ (7)
$\displaystyle\int_{\Omega}\mathrm{d}\omega=\int_{\partial\Omega}\omega$ (8)
$\displaystyle(i\hbar\gamma^{\mu}\partial_{\mu}-mc)\psi=0$ (9) $\displaystyle
e^{i\pi}+1=0$ (10)

An expert reader has immediately recognized that the relevant three equations
are Eq. (2), Eq. (5) and Eq. (7). In the following Subsections, an explanation
for each of the three Equations is discussed111For the sake of completeness,
the explanation why the rest of the listed equations look simple, but they are
not, is hereby reported. The expression (1), both simple and deeply
complicated, represents in symbols the Continuum Hypothesis, partially still
the subject of discussion among modern mathematicians. The Eqs. in (3)
represent the Maxwell equation in vacuum, just taking the electric and the
magnetic field as the real and the imaginary part of $\vec{\mathcal{F}}$.
Whereas, the Eq. (4) displays the Shröedinger equation, but this time with two
real functions $u$ and $v$! Equation (6) is really simple, but it has
“complex” solutions, definitely not easy to be found if one ignores some
algebraic technicalities. The Eq. (8), usually called the Stokes theorem
within the $k$-form formalism, is probably the most beautiful theorem in
advanced calculus, as it encloses the Fundamental Theorem of Calculus, the 3D
Green and Stokes theorems, their 4D version in the Riemann space-time and so
on. The Equation (9) is the famous Dirac equation, whose correct
interpretation in Quantum Field Theory predicts the existence of antiparticles
for Fermions. Finally, the Eq. (10), the Euler identity, often advertised by
Richard Feynman for its beauty, is a complex number identity that is far more
tricky than its apparent simplicity, as it involves the top five numbers in
mathematics..

###  1.1 Existence – Eq. (2)

A spiral galaxy contains about two thirds of its mass ($M$) in the galactic
core (or bulge) [24, 25]. Assuming for the latter a spherical shape with
spherically symmetric density $\rho(r)$, the Gauss theorem predicts, outside
the bulge, a gravitational pull $\propto M/r^{2}$, where $r$ is the distance
from the center of the sphere and $M$ is the enclosed galactic mass (see Fig.
1). More precisely, the profile of the visible matter depends on the galaxy
type, but its specific realization does not invalidate this general argument.
This force provides a centripetal acceleration $\propto v^{2}/r$ for all
objects gravitating around the galaxy center. Here one has implicitly assumed
the Newtonian weak field approximation of General Relativity at the galactic
scale and at the galactic speed (this point has been recently debated, see [7]
already anticipated above).

The model described so far immediately implies that the velocity of stars as a
function of $r$ should follow the so-called “Keplerian fall” described by Eq.
(2). But this is not what is observed [26]: rotation curves always lie largely
above the expected behavior coming from the independent quantification of the
visible galactic mass from mass-to-luminosity ratio of stars and non-luminous
(gaseous or solid) mass. Sometimes it is somewhat inaccurately said that
observed rotation curves are _flat_. Those curves are actually exceeding the
Keplerian fall, but with a family of universal curves that depends on the
visible size of the galaxy, upon which the ratio between visible and dark
matter also depends [24, 25]. The dark matter distributed in a spherically
symmetric halo, present in the interstellar space inside each galaxy, fills
this discrepancy between predictions and observations.

There is considerable debate about the profile of the radial dependence of the
spherically symmetric matter distribution in the hypothetical dark matter
halo, depending on whether it is inferred from numerical simulations of
collision-less particle clustering or from observed rotation curve analysis.
The majority of the proposed profiles can be described by the generic formula:

$\rho_{s}(r)=\frac{\rho_{0}}{\sum_{i=0}^{3}a_{i}\left(\frac{r}{r_{s}}\right)^{i}},$
(11)

where $\rho_{0}$ is a normalization constant, and $r_{s}$ is a characteristic
size scale. According to the choice of the dimensionless coefficients $a_{i}$,
the profile can be more “cuspy” (as in the Navarro-Frenk-White model [8]) or
more “cored” (as in the pseudo-isothermal or Burkert models [9]); or something
different, but anyway traceable by approximation to Eq. (11), as in the
Einasto profile family [10].

To conclude this Section, it is worth mentioning that, interestingly, the
absence of the Keplerian fall has been recently questioned, at least from some
interpretation of accurate observations of the Milky Way by GAIA [27, 28]. The
correctness and the implication of such results have yet to be validated and
fully understood.

###  1.2 Kinematics – Eq. (5)

A typical dark matter candidate $\chi$ of mass, say, $M_{\chi}=50$ GeV/c2,
hitting a target (visible) nucleus $A$ with mass of about $M_{A}=50$ GeV/c2 in
laboratories, has a typical velocity $v_{\chi}$ in the galaxy of the order of
a few hundreds of km/s ($v_{\chi}\ll c$). As a consequence, the kinematics of
the collision $\chi$–$A$ is not different from two billiard balls with masses
$M_{\chi}$ and $M_{A}$ hitting each other (in a non relativistic regime,
actually $\beta=v_{\chi}/c\sim 10^{-3}$). The classical energy and momentum
conservation leads to Eq. (5), in case of linear collision, otherwise the
decrease for the cosine of the scattering angle has to be included. The
kinetic energy of the target recoil $E_{A}$ is a _kinematic_ fraction of the
dark matter kinetic energy $E_{\chi}$. Based on the numbers provided above,
calculations show that the expected recoil energy in a detector on Earth is of
the order of $\lesssim 10$ keV, and so extremely difficult to detect, but not
impossible with the present technologies.

###  1.3 Rate – Eq. (7)

Lastly, Eq. (7) represents the interaction rate $\mathcal{R}$ expected in a
given detector on Earth, exposed to the dark matter wind caused by the Solar
System’s relative motion inside the Milky Way: the rate is given by the target
density $n$ times the $\chi$–$A$ cross section per nucleon $\sigma$, typically
assumed to be spin-independent (SI), and the dark matter flux $\Phi$. In
detail, assuming a Maxwell distribution of the dark matter particle velocities
inside the gravitational potential “box” of the Galaxy, and the experimental
features of the detector, Eq. (7) has to be integrated over the velocity
distribution, taking into account the detector energy threshold. Then, all the
data has to be combined with the experimental resolution, normalized to the
energy quenching for nuclear recoils, and, finally, scaled according to the
detector acceptance [29]. In the SI case, the full formula can be summarized
as

$\frac{d\mathcal{R}}{dE}=\frac{\rho_{\rm\chi}A^{2}\sigma
F^{2}(E)}{2M_{\chi}\mu^{2}}\int\limits_{v_{\rm min}}^{v_{\rm
esc}}\frac{f(v,v_{0})}{v}\mathrm{d}v\otimes\mathcal{G}(E)$ (12)

Where $\sigma$ is the cross section per nucleon of the target $A$, $F(E)$ is
the nuclear form factor, $\mu$ is the reduced mass between the target and the
dark matter mass. Whereas, assuming the standard (galactic) halo model (SHM)
[30, 31], $\rho_{\chi}=0.3$ GeV/cm3 represents the local dark matter density
in the Solar System, $v_{\rm min}$ the minimum delectable velocity (given the
experimental energy threshold) and $v_{0}=220$ km/s the circular rotation
velocity and $v_{\rm esc}=544$ km/s is the Milky Way escape velocity, taken as
the cut-off for the Maxwellian velocity distribution. The Sun velocity is
$v_{\odot}=232$ km/s. Those parameters have been slightly updated recently,
but those small variations go beyond the basic purpose of this article. It is
worth noticing that the notion of SHM could be criticized as non-realistic,
and one can imagine a vast zoology of monster halos with local density
anisotropies and streams: this can change a little the result interpretation
(as the rate normalization can change), but cannot create specific detection
anomalies out of nothing. Finally, the convolution, $\otimes$, with the energy
dependent function $\mathcal{G}(E)$ symbolically includes the experimental
features, such as resolution, nuclear quenching and acceptance in the region
of interest.

Figure 2: The greenish region represents the typical excluded region of dark
matter experiment with null result in the SI $\sigma$–$M_{\chi}$ plot. The
violet region represents the so-called neutrino floor. Finally, the red curve,
with a cusp, represents a limit in case of annual modulation analysis, see
Sec. 3.

The Equation (12) can be integrated over the experimental energy window and
made explicit as $\sigma_{\rm SI}(M_{\chi})$. An experimental limit, _e.g.,_
the absence of events at 90%CL, typically looks like an asymmetric hyperbole
branch, as depicted in Fig. 2 (greenish region): the branch on the left
represents the experimental threshold wall, instead the branch on the right
corresponds to the loss of sensitivity due to the reduced density of targets
for heavier dark matter masses. An experiment has typically the maximal
sensitivity for $M_{\chi}\simeq M_{A}$, corresponding to the minimum of the
green curve. The violet region in the Figure corresponds to the so-called
_neutrino floor_ , _i.e._ the region in which neutrinos coming from Sun,
atmosphere and diffused supernova background gives a nuclear recoil from
coherent scattering, similar to the expected equivalent dark matter
interaction with a given mass. This actual experimental limitation can be
overtaken only by future experiments exploiting the “directionality” of the
dark matter wind, _i.e._ the preferred direction along the galactic plane, due
to the relative motion of the Sun with respect to the galactic center, against
the neutrino background, assumed as uniform and isotropic. This particular
aspect will be further discussed in Sec. 6.

Figure 3: Example of a simple detector made of a scintillating crystals
optically coupled to a PMT. The scintillation signal is converted into a
series of photo-electrons producing an electrical pulse related to the time
distribution of the scintillation light emission.

##  2 Detectors

A grand piano is constructed from durable and raw materials. Indeed, it is
basically made of hard substances, such as wood, cast iron, steel and felt. A
single string holds a tension of about 80 kg and many of its 88 keys own 2 or
3 strings, for a total tension of 15000 kilogram-force. However, only after a
wise assembly and a proper (fine) tuning, this terrific music instrument can
produce a very soft and profound sound, which one can appreciate for example
in the famous Chopin’s Nocturnes. Similarly, a particle detector is made of
hard and raw stuff as well: from metals, crystals or liquids and electronics,
one can get an answer about fundamental question in particle physics and
cosmology, but this process is not that straightforward, and a poor knowledge
of its functioning can deceive the smart experimentalist, and the brilliant
theoretician who wants to jump too quickly to the conclusion.

Consider a simple example. A typical particle detector is made of a
scintillating target (solid or liquid) coupled with a light sensor, for
example a photo-multiplier tube (PMT) [32]. When an ionizing particle hits the
target, a given number of electromagnetic field quanta are excited, and part
of them (depending on detection efficiency) collapse to form what is known as
a “photo-electron” on the light detector. The corresponding electric pulse
output is then shaped and amplified and eventually converted into a binary
number, and finally stored on a computer hard drive. The corresponding data
are retrieved and analyzed by numerical algorithms or passed to some
artificial intelligence black box, see Fig. 3.

A real detector is, of course, sensitive to internal and external radioactive
backgrounds, as well as hypothetical dark matter particles, whose foreseen
amount plays a crucial role in the goodness of the proposed experimental
setup. Furthermore, the electronic (non physical) noise, intrinsic and/or
picked up from the environment, can mimic, to some extent, the signal produced
by particle interactions. As a matter of fact, only a very deep knowledge of
all those effects described above, can enable a good investigation. And
sometimes, as it happens in most of the experiments, part of those effects are
not known _a priori_ and can be addressed properly only after a lot of years
of calibrations, analysis and hardware improvements. Even a seemingly simple
device like a PMT can produce a wide variety of noise pulse signals that might
not be distinguishable by a basic classifier, and requires a deep
characterization, sometimes using novel techniques based on multidimensional
mathematical algorithms such as multivariate likelihood ratios or support
vector machines, or even nonlinear methods based on boosted decision trees or
multi-layer perceptrons, and machine learning in general.

Figure 4: Classifier distribution for NR (gray) from calibration and ER (blue)
from a physics data collection. The acceptance region (red box) for NR
candidates can be defined e.g. as the median of the NR distribution. The
observed three events in the NR band are to be interpreted according to
event/noise understanding and expected physical background.

Finally, the optimal detector has some properties capable of discriminating,
not only between signal and noise, but also between the physical
characteristics of the primary interaction, as between electron and nuclear
recoils (ER–NR, hereafter), especially useful for addressing the nature of a
possible dark matter candidate. In the example mentioned above, the
discrimination could be enabled by the time distribution of the scintillation
light. Typically, a classifier, i.e. a parameter defined through e.g. a
likelihood ratio or an artificial neural network, shows a characteristic
distribution depending in general on the particle energy and exhibiting a
partial overlap (inefficiency) in the region of interest. Usually, after a
deep training with known sources, an acceptance region, in which the dark
matter candidate is expected, is defined (See Fig. 4 as an example). If a
statistically significant group of events emerges over the expected
background, one can reasonably claim evidence of new physics, which should
later be confirmed by other equally sensitive experiments or complementary
techniques.

Dark matter detectors typically exploit various phenomena to develop
classifiers. An ionizing particle hitting a material can, in general, produce
scintillation, ionization or phonon excitation. Many detectors, depending on
state and on the temperature, can utilize one or more signals, offering
broader capabilities in particle discrimination [33].

The high level standard of low background detection requires the choice of
highly radio-pure target materials to be operated in shielded and, possibly,
actively vetoed detectors, located in underground laboratories, far enough
from the atmospheric muons and cosmogenic-induced radiations. These materials
are usually not available on the market and require a long and accurate R&D
program, sometimes with no predictable outcome.

Lastly, an important experimental aspect to consider is nuclear quenching. A
NR indeed releases only a portion of its actual kinetic energy due to non-
radiative excitation. For that reason, the observable energy is less than the
one released by an equivalent ER. This relative ratio is important for the
final interpretation of the experimental result, for this reason results are
usually presented in electron-equivalent energy (_e.g._ keVee) and the
quenching factor is given independently by dedicated calibrations with
neutrons.

Besides the counting of outliers in the expected acceptance region, direct
dark matter detection in underground laboratories can also be pursued through
the detection of annual modulation. This signal arises from the relative
motion of the Earth around the Sun with respect to the center of the Galaxy.
In addition, tracking detectors sensitive to the directionality of the dark
matter can be employed.

With these basic concepts in mind, one can now move to the present
experimental situation.

##  3 Analysis

Imagine that one gets a few (unsuspected) points in the square box of Fig. 4.
Can one claim for the dark matter discovery based on that? A wise response
might be that the number and the distribution (_e.g._ in energy and space) of
the expected background has to be declared in advance. For example, if an
experiment observes 4 events out of 2 (expected), one may argue that the
Poissonian fluctuation of 2 can likely give 4 in a reasonable number of cases.
But if one gets 8 out of 2, the story becomes more intriguing, as the
fluctuation of 2 can hardly return 8.

###  3.1 Which statistics?

The way in which these naive words “likely” or “hardly” are converted into
some quantitative parameters is not universally accepted by a common
procedure, and there are multiple approaches based on different statistical
interpretations, often with heated debates, like the ones between Frequentists
(_e.g._ Feldman-Cousin [35]) or Bayesians [36] statisticians.

Of course this is a fundamental and longstanding controversial debate that
cannot be solved for sure here and sometimes the solution is not unique, but
depends on the specific situation. Therefore, for an experimentalist very
often it is more convenient to quote results (whether it is a measurement or a
limit) in more than one approach, just to delegate some possible controversial
matters to others.

The real challenge might be how to properly combine results coming from
different approaches. However, it is also true that as long as results are
just limits the process itself is not really harmful, and the nuances in
results stemming from different philosophies are basically hidden by the line
width in the $\sigma$–$M_{\chi}$ exclusion plot. For real positive results,
the problem could be more delicate.

###  3.2 How many $\upsigma$’s?

If some statistical procedure is assumed, it becomes important to show how
many sigmas (actually the $p$-value over the background fluctuation) are
necessary to claim for a discovery. Among physicists, there is a common
(questionable) practice to associate a naive meaning to a certain number of
sigmas, such as: _mild indication_ ($1\sigma$), _indication_ ($2\sigma$),
_evidence_ ($3\sigma$), _never quote that, it’s bad luck!_ ($4\sigma$) and
_discovery_ ($5\sigma$). The real problem here is that all of them are only
indicators: forgotten systematic uncertainties and mistakes are always
possible and may unexpectedly arise, and unfortunately there is plenty of such
examples in literature.

Besides these folkloristic topics, to claim a scientific discovery, the way is
much harder, as it is explained in the following Subsection.

###  3.3 No background

Given that background fluctuations can deceive experimentalists, it would
ideally be preferable not to have any background at all. In other words, when
planning a detector, it is ideal for the expected background events to be
significantly fewer than the expected signal to avoid being misled by
fluctuations. This has created the fashion of having high-sounding names as
_background-free_ experiments, or even _zero-background_ experiments.

What is the difference between “Cola Light” and “Cola Zero”? In reality, there
is no difference: both contain chemical sweeteners instead of sugar, but in
marketing names really matter. Some people might avoid a drink labeled as
“Light”, but would choose one labeled “Zero”, as if it is not for everyone
facing the zero with no fear. The same concept applies to the background in
dark matter experiments: it is primarily marketing. The honest name should be
“low background”, where the adjective “low” should go with some objective
quantifier.

In a popular, and absolutely interesting, review on neutrinoless double beta
decay, a footnote reads [37]:

> “For a truly zero background experiment, one event is enough to claim a
> discovery. In a similar fashion, encountering a unicorn is enough to claim
> its existence, provided that we have a template of a unicorn to which to
> compare the observed candidate.”

When the authors asked GPT-4[38] to comment on this statement, its response
was notably insightful. Here is its reply:

> “A single observation of an event can suggest that something exists, but in
> a scientific context, such conclusions are usually considered weak evidence.
> Scientists value repeatability and reliability of data, meaning that
> multiple, consistent, and independent observations are necessary to
> establish existence.
>
> In the case of a unicorn sighting, a single observation would likely not be
> considered sufficient proof of their existence. Scientists would need to
> examine the evidence, conduct additional observations, and demonstrate that
> these observations are indeed consistent and repeatable. For rare or
> extraordinary phenomena like unicorns, scientists may be more cautious and
> require even more evidence to accept their existence.
>
> In summary, a single observation can lead to the belief that something
> exists, but scientists typically rely on repeatable, reliable, and
> independent data. A lone unicorn sighting would likely be insufficient, and
> further research and observations would be needed to confirm their
> existence.”

If one ignores this crispy reasoning, one can be easily fooled and forget the
basic principles of good science. Indeed, continuing with the metaphor, the
unicorn is quite an unfortunate example, not only for our prior about its
existence, but also because there is plenty of alicorns (unicorn’s horns) in
museums, coming from the Middle age [39], sometimes made actually of bone.
Yes, but a lot of people, even today as in the Middle Age, ignore the
existence of narwhals.

To conclude, what one sees in a detector is not a unicorn anyway, if for
unicorn one means something completely weird and different from ordinary
stuff. In a real particle detector, what one records is nothing else than a
pure electronic signal, digitized and analyzed as a discrete waveform, that
resembles either physical events and/or noise artifacts more often than
expected.

###  3.4 Annual modulation

A “model-independent” approach, which in principle can ignore the nature of
the $A$–$\chi$ interaction, involves detecting an annual modulation signal
from a shielded detector over a prolonged (multi-annual) exposure period. In
this case, the ER–NR discrimination is not necessary, as one is interested
only in the typical signature of such a signal. The Earth, indeed, revolves
around the Sun at 30 km/s while the Solar System, tilted of 60° with respect
to the galactic plane, moves altogether around the center of the Galaxy at
about 232 km/s, as depicted in Fig. 5. As a result of this simple geometry,
the expected signal is

$\mathcal{R}(t)=\mathcal{R}_{0}(t)+\mathcal{S}_{m}\cos(\omega(t-t_{0})),$ (13)

Figure 5: Diagram of the Earth and Sun velocity vectors with respect to the
galactic plane. Figure 6: Two limit cases of distortion: June 2nd (orange),
with maximum relative velocity of the Earth with respect to the dark matter
halo, and the opposite case on December 2nd (cyan). Finally, the red lines
mark a possible experimental integration interval. Figure 7: Genealogy of the
NaI-based detector. Besides COSINUS, which is planning to deploy cryogenic
crystal with ER–NR discrimination, all the others are using the model-
independent approach exploiting the annual modulation expected signal.

Where $\mathcal{R}_{0}(t)=\mathcal{S}_{0}+\mathcal{B}(t)$ is the total trend
given by the sum of a constant part $S_{0}$, due to the _unmodulated_ dark
matter component, and the detector background $\mathcal{B}(t)$ that in
principle can depend upon time, as _e.g._ in presence of radioactive decaying
contamination, or time varying cosmogenic background; $\mathcal{S}_{m}$ is the
modulation amplitude of the (expected) dark matter signal, basically given by
the relative speed of the Earth with respect to the local dark matter velocity
distribution in the Galaxy, and therefore of the order
$30/232*\cos(60\textdegree)\simeq 6.5\%$ with respect to $\mathcal{S}_{0}$;
$\omega$ is the annual angular velocity corresponding to $2\pi/(365\,{\rm
d})\simeq 0.0172$ rad-1; finally, $t_{0}$ is the time (phase) corresponding to
the maximum rate, _i.e._ to the date in which Earth and Sun go towards the
same direction (on June 2nd). If one builds up a radio-pure and shielded
setup, acquires data for some years and performs a temporal regression
analysis in which $S_{m}$, $\omega$ and $\phi$ are free parameters returned
simultaneously in the expected ranges, one can in principle claim that such a
signal is compatible with the presence of a diffused dark matter gas in the
interstellar space at the Earth distance from the Galaxy center. Indeed,
Freese et al. [40] clearly states:

> “We argue that a modulation can itself be used as primary means of detecting
> WIMPs [more in general dark matter (A.N.)] (rather than as a confirmation)
> even in presence of a large background.”

This sentence is _for common sense_ correct but actually _theoretical_ and
_practically_ wrong. First of all, it corresponds to the material implication
“IF the dark matter exists THEN a modulation is visible”, _i.e._

$\textsc{dark matter}\rightarrow\textsc{modulation},$ (14)

and so, looking at its truth table, the fact that one can see modulation even
if dark matter does not exist is still a valid possibility: one can argue for
example, that a time-varying background related to some possible seasonal (o
seasonal-induced) signal is still possible [41, 42]; moreover, a time-varying
$\mathcal{B}(t)$ background, if not properly accounted for, can bias the final
results [43], as it is discussed later. Then a robust modulation analysis has
to show the consistency of all terms of Eq. 13, no one excluded, at the very
least. As a further example, it is also worth mentioning that the literature
is full of apparent violations of the exponential law with annual modulation
components in radioactive decays, opportunely criticized, see [44] and Refs.
therein.

Finally, in case of annual modulation the limit in Fig. 2 has a cusp around
the minimum (red curve). This is an artifact of integrating each time bin over
a limited energy window. It is possible, indeed, that the distribution of the
target recoil spectrum is distorted between June 2nd ad December 2nd in such a
way that the shape changes, but the total area not: in this case there is a
specific dark matter mass for which the sensitivity is lost, see Fig. 6. The
same accurate analysis also leads to the so-called _phase inversion_
phenomenon, happening in the very low energy region, that should be present in
the experimental data, but sometimes not visible because of a too high
experimental threshold with respect to the inversion point that depends also
on the specific dark matter mass.

###  3.5 Blind analysis

The blind analysis, consisting for example in closing, completely or in part,
the red box of Fig. 4, is a good practice, adopted by other disciplines such
as medicine, to avoid biases in the analysis, especially in low-rate critical
conditions. A collaboration that decides to apply this practice, usually
closes the box for physical data taking [45, 46], training the event
reconstruction and the selection criteria only on a subset of data, possibly
not used in the final analysis. The collaboration decides to freeze the data-
set, and eventually to open the box. The scenario is significantly different
when the “un-blinding” is done in public with journalists. With a few
exceptions [47], this is usually done behind closed doors instead, and nobody
knows what happens inside. In this case one can only rely on the honesty and
professionalism of colleagues.

###  3.6 Data sharing

It is worth citing what Wikipedia [48] says about the important item of the
“data sharing”:

> “When additional information is needed before a study can be reproduced, the
> author of the study might be asked to provide it. They might provide it, or
> if the author refuses to share data, appeals can be made to the journal
> editors who published the study or to the institution which funded the
> research.”

Raw data are usually not understandable for non-experts. However, all
physically reconstructed events and procedures must be available. The dark
matter community should, sooner or later, converge towards an open data
policy, especially if large amounts of funding are going to be spent on
reproducing some experiments.

##  4 The NaI case

Sodium iodide crystals doped with thallium NaI(Tl) are largely used as
particle detectors exploiting their scintillation properties. The light
produced when an ionizing particle hits the crystal is usually detected by
PMTs, based on a very well known and consolidated technology, which nowadays
can be easily manufactured with highly radio-pure materials [45].

Figure 8: Experimental residual rate of the single-hit scintillation events
measured by DAMA/LIBRA Phase-I and DAMA/LIBRA Phase-II in the (2–6) keV energy
intervals as a function of the time. The superimposed curve in Eq. 13, with
period fixed to one year, phase fixed to 152 days (June 2nd) and amplitude
equal to the central value obtained by best fit on the data [49].

The possibility of using these crystals for dark matter direct detection was
first explored by the DAMA collaboration (see [49] and Refs. therein for a
detailed review). DAMA, since its smaller version, has been detecting a
modulation signal compatible with period and phase with the one expected by
the presence of the dark matter halo in the Milky Way, rejecting the no-
oscillation hypothesis at $12.9\sigma$ (20 annual cycles) in the 2–6 keVee
energy interval. These interesting results, already released in its first and
less massive version at the turn of the new millennium, has become a media
event, and has pushed, to some extent, the direct dark matter search in the
first decade of 2000s, as it will be detailed in Sec. 6. The signal detected
by DAMA is significantly incompatible, in the framework of the SI interaction
with SHM, with the absence of a corresponding signature in detectors with
higher sensitivity, as the ones based on noble gases that will be discussed in
Sec. 5, and with other detectors exploiting a big variety of different
techniques, briefly reviewed in Sec. 5.4. In other words and in practical
terms, the DAMA signal is so intense that the same WIMP-like interactions
should be visible in a cup of liquid xenon. Instead, xenon-based detectors, as
discussed later, are currently operating multi-ton targets with no results.
The remote explanation that the sodium and iodine nuclei have some “special
feature”, not met by other target nuclei, has anyway strongly motivated the
scientific community to reproduce the DAMA experiment using similar NaI
crystals independently, as it will be discussed later.

Figure 7 shows approximately the genealogy of experiments born after DAMA to
accomplish this goal. At the moment, COSINE-100 [50] (originated by the
merging of KIMS-NaI [51] and DM-Ice [52], and ANAIS-112 [53] are the only two
experiments already taking data for dark matter search. SABRE [54] and PICOLON
[55] are taking data in a R&D stage, to basically prove the detection
principle and quantify the intrinsic contamination. COSINUS [56] is the only
detector using NaI crystals with a bolometric technique, and thus exploiting
both scintillation and temperature signals for ER–NR discrimination. After
some preliminary results on small crystal samples [57], the COSINUS
collaboration is building a bigger and complete setup. In the following, the
major experiments in this genealogy are discussed.

Finally, the interpretation of the NaI iodide results in terms of SI solution
on the $\sigma$–$M_{\chi}$ parameter space could depend on the quenching
factor for both Na and I nuclei separately, that could in principle depend on
the specific crystal. There is a lot of debate on this issue and a lot of
controversial measurement of those important parameters [58].

###  4.1 The DAMA experiment

The impressive radiopurity of the DAMA crystals, as low as one count per day
per kg per keV (cpd/kg/keV), with a threshold of $1\div 2$ keVee, made it
possible to reach a sensitivity on the $A$–$\chi$ cross section better than
$10^{-42}$ cm2 for the model independent annual modulation in the energy
region $\lesssim 10$ keV. However, the typical NaI(Tl) light yield of the
order of $\sim 10$ PE/keV does not allow one for statistically sensible ER–NR
discrimination. The DAMA collaboration had been operating a 100 kg detector in
the early phase called DAMA/NaI. The target was replaced with 250 kg of high
purity NaI crystals in the subsequent Phase-I and Phase-II. In the latter, the
energy threshold has been lowered from 2 keV to 1 keV. The DAMA collaboration
is currently operating an empowered Phase-II with threshold as low as 0.5 keV
with the aim of adding an further extreme low energy point in the analysis.
This region is extremely important as it could show the phase inversion
described above, not yet present in the recoil spectrum published so far by
the DAMA collaboration.

The modulation measured by DAMA is $\mathcal{S}_{m}\simeq 0.01$ cpd/kg/keVee
in the 2–6 keVee energy interval, extracted through a time fit of the residual
single-hit (non coincident) rate as reported in Fig. 8 of Ref. [49].

If one considers that this quantity represents only 6.5% of the total rate
detected in the crystals, one can naively assume, by scaling, that in the same
energy window the unmodulated component of the total rate is about
$\mathcal{S}_{0}\simeq 0.15$ cpd/kg/keVee, and then smaller than the total
rate, _i.e._ $\lesssim 1$ cpd/kg/keV.

As from Fig. 8, the modulation reported by DAMA is extracted on the
experimental residual rate of the single scintillation hits, _i.e._
$\mathcal{S}(t)-\hat{\mathcal{S}}$(t) where $\hat{\mathcal{S}}(t)$ is the
_detrend_ function. The detrend function used by DAMA is a piecewise function
made with the average annual cycles of the total rate for each crystal. It is
worth noticing that this method applies without bias, if and only if, the
total rate of single-hit is constant; in all other cases, in which there is an
explicit dependence upon time of the total rate, amplitude and phase are
biased as described in [43]. Basic signal processing theory, indeed, warns
about the fact that injecting a periodical manipulation in time series will
make the injected frequency itself appear in the final periodogram. The total
and explicit rate $\mathcal{S}(t)$ is never reported by the collaborations
(see always [49] and Refs. therein), even if, from the same publications, it
is evident that this rate is different in the two experimental phases, is time
dependent because of the presence of decaying contaminants, and is presumably
affected by many discontinuities because of hardware operations.

One is not saying that the DAMA signal is completely artifact from an
incorrect analysis, but one is only suggesting that the result in phase and
amplitude could be biased, and consequently the interpretation in terms of
dark matter not exactly correct.

###  4.2 Reproducing DAMA

At present, SABRE is the only R&D project that was able to manufacture high-
purity crystals with a counting rate close to 1 keV/kg/keV comparable with
DAMA [61], and plans to deploy two different detectors in both Earth’s
hemispheres, to factor out all possible systematics due to unaccounted
seasonal effects. PICOLON R&D also plans to deploy a target of 250 kg of NaI
crystals in forthcoming years. Finally, the COSINUS collaboration has proved
that the combined scintillation and temperature signals in bolometric usage of
the NaI can feature the ER–NR separation, and it is building the first real
experiment for a counting experiment with a relatively small crystal.
Furthermore, it plans, after the first data-taking, to increase the target and
exploit the annual modulation analysis as well.

In summary, the NaI case is not solved yet. The experiments trying to
reproduce DAMA have not yet definitively clarified this debated result,
therefore the funding agencies should strongly support all the attempts to
independently clarify the question as soon as possible, with substantial
resources and manpower. If there is a physical explanation for the DAMA
signal, this explanation would be anyway highly interesting.

##  5 Noble gases

At present, noble gas based experimental setups are the most promising dark
matter detectors in terms of sensitivity in a wide dark matter mass range,
from 1 GeV/c2 to 1 TeV/c2 [45, 46, 62, 63, 64], and recently even for masses
lower than 1 GeV/c2 [66, 45]. Liquid targets made of these special elements,
can reach high level of radiopurity and, thanks to their scalability, in terms
of target mass, represent a realistic technology and a candidate for the
ultimate experiment in dark matter search, capable of reaching the optimal
sensitivity as will be discussed in Sec. 6.

Among the known noble elements, argon and xenon are the sole gases permitting
a feasible realization in terms of reliable technology and sustainable costs.
Both gases can be exploited in single phase (liquid) detectors or double phase
(liquid and gas) time projection chambers (TPCs).

Table 2 summarizes the main properties of the two noble elements, making the
two technologies complementary in terms of _pros_ and _cons_. All details will
be discussed in the two following subsections dedicated to xenon and argon,
respectively.

Figure 7 sketches the genealogy of both technologies, showing weak (dashed)
and strong (solid) relationships in terms of collaborators and/or merging of
the corresponding experimental groups, while the gray block represents future
projects. Basically the two technologies have a common origin in large noble
gas neutrino detectors, as the ICARUS project [68, 69]. Two distinct branches
originated from some preliminary R&D projects: the argon and xenon
communities, even if this nomenclature is not formally shared among all
collaborators. The genealogy includes neither the DAMA/Xe project [70] , a
single-phase small scintillation not upgraded by the collaboration, nor XMASS
[71], another single-phase xenon-based detector that has anyway set a fair
limit on WIMP-like dark matter. Some of the experimental aspects of the two
approaches will be discussed below.

Figure 9: Genealogy of the noble gas detector. Starting from a common origin,
the xenon (green) and argon (red) community split into two main branches. The
dashed lines point to some weak connection between different projects, while
the solid line represents the natural evolution of different detector scales.
Finally, the gray diagrams represent future projects. The xenon is divided
into three main projects: ZEPLIN, XENON, and PandaX. The first two will
possibly evolve in a common final project. The argon community, started with
three projects (ArDM, WArP and DarkSide), are currently reunited in the common
project DarkSide-20k.

Noble gases can be operated in single phase detectors (liquid) or in double
phase (liquid and gas) TPCs. In the first case, the sole scintillation signal
(S1) can be used for position reconstruction and pulse shape, where possible.
In the second case, for each primary interaction in the liquid target, two
signals are exploited: scintillation in liquid (S1) and electro-luminescence
in the top gas pocket, due to ionization electrons accelerated and extracted
by strong electric fields (S2). Exploiting both S1 and S2 can help in volume
fiducialization, background rejections and particle identification when S1
alone, as in the case of the xenon, is not capable of performing the ER–NR
separation. A single phase detector, indeed, is easier from a technical point
of view, but in general does not allow for a high performance in event
reconstruction. S1 and S2 are usually correlated through the ion recombination
process, therefore, to improve the performances, the energy estimator is
defined as a linear combination of the two, whose parameters are calculated
from accurate energy calibrations. A scheme of the noble gas detectors in
single and double-phase is depicted in Fig. 10.

Recently, the possibility of using an ionizing only signal (S2-only) has been
exploited both by argon and xenon detectors, allowing to set a limit, after an
accurate low energy background modeling, and considering the shape of Eq. (12)
for the expected dark matter signature. These technologies have confirmed a
competitive performance also for the light dark matter detection with mass
below 1 GeV [66, 67].

Figure 10: Scheme of the noble gas signals S1 and S2 in double-phase TCPs. A
collision of an ionizing particle with the noble gas (dashed brown) produces
both scintillation light and ions. In the presence of an electric field, part
of the ions does not recombine producing additional light (S1), but is
extracted and produces the secondary signal (S2). As a consequence, S1 and S2
are partially correlated.

###  5.1 Xenon

The xenon has a high atomic number, and so an optimal self-shield from
external background. Xenon-based detectors can search for both spin-dependent
and spin-independent dark matter interactions. Even if the nuclear form factor
is not that favorable [29], the xenon-based detectors can be a factor 5–7
smaller than argon-based detectors, mostly because of the factor $A^{2}$ in
the cross section.

Because of the relatively higher boiling point (165 K), the xenon does not
present technical issues, typical of electronic components at very low
cryogenic temperatures: PMTs and the electronics chain are usually working
smoothly with high performances. Even though the scintillation wavelength is
in the near UV region (178 nm), the corresponding light can be easily detected
by commercial photo-cathodes. It does not require wavelength shifting,
resulting in high performance in event reconstruction. Since the two decay
components of the scintillation light are very close to each other, S1 alone
is not able to discriminate between ER and NR, while the ratio S2/S1 is
usually used. Typically, the discrimination power of this classifier is one
over 300. For this reason, xenon-based detectors require a high accuracy in
the background control through deep purification systems and high accuracy in
the material screening and selection. If not, there is a real risk of
saturating the detector sensitivity in a very short exposure and creating
puzzling results. Investing in the S1 only discrimination with digitizers with
high sampling rate, of the order of 1 GSa/sec, could be anyway valid: even a
mild preference in the NR–ER discrimination could be enough to improve the
separation when combined with S2/S1 and other observables in a multivariate
approach, but at present no progress has been made in this direction.

In xenon detectors it is easy to remove volatile radioactive contaminants as
85Kr, but it is more difficult to remove radon with comparable atomic mass,
even if a lot of progresses has been recently accomplished in distillation
techniques [45].

Figure 12 reports the most updated comparison between the leading dark matter
detectors LZ, XENONnT (5.9 ton) and PandaX-4t (3.7 ton) with same order
target, as reported in [45]: LZ, originated from the merging or the groups
ZEPLIN and LUX is a double phase TPC, in operation since 2022 with a target of
5.5 ton [64]. XENONnT, coming from a long preceding history of versions with
increasing mass (XENON10, XENON100 and XENON1T), has shown an unprecedented
low background level [72] and is currently taking data with 5.9 ton target;
PandaX-4t, is an independent collaboration that, after different preceding
versions, is currently operating a 3.7 ton version [63].

From the experience of experiments described above, it is evident a gradual
difficulty in the scaling process: a working prototype can prove the detection
principle, but cannot prove the increasing technical complication coming from
the increase of the target mass and the detector volume. Only a step-by-step
scaling, with intermediate stages, can guarantee a solid progression of the
project and a success against a highly probable failure.

The current generation of projects can reach a sensitivity very close (even
though a logarithmic decade above) to the neutrino floor. All collaborations
are planning future projects with targets bigger of a order of magnitude
(30–50 ton) to reach a sensitivity that will basically touch the neutrino
floor, as ultimate experiments on this research field. In particular, PandaX
is moving independently towards a multi-ton detector, while LZ and XENON (with
a possible middle scale DARWIN [73]) are discussing a possible joint venture
in a project called XLZD [74].

###  5.2 Argon

The argon is lighter than xenon, therefore the self-shielding is less
effective. Its boiling point is 87 K, with a corresponding technical
difficulty as the one observed in the PMT electronics at this temperature [75]
(even if this argument is a kind of myth: a failure of a specific PMT batch
does not mean that a possible R&D with the PMT producer would not have solved
the observed issues). The scintillation light of 128 nm is hardly matching the
photo-cathode sensitivity, for this reason typically a wavelength shifter (as
the TPB at 420 nm [75]) is used, with a corresponding degradation of the event
reconstruction, due to the extra diffusion of light. Nevertheless, the argon
shows an excellent ER–NR discrimination using S1 only: a very large separation
(three order of magnitude) in the fast and slow scintillation components
permits to discriminate ER from NR as one over ten billions [62]. It is worth
mentioning that the argon nucleus has spin equal to zero, therefore the spin-
dependent search cannot be performed, and the non-relativistic expansion of
all possible relativistic operators can be done only for a reduced subset
[76].

Contrary to xenon, argon coming from the atmosphere is highly contaminated
with the long-lived beta emitter 39Ar. For this reason, part of the argon
community has moved towards the usage of deep underground argon in which this
contaminant is reduced by a factor better than one over 1000 [77].

For the reasons described above, the argon-based detectors have a different
story. Two independent lines, indeed, have emerged over the years: single
phase as DEAP-3600 [62] and double phase TPCs as DarkSide-50 (and marginally
ArDM [79]). DEAP, operating 3.6 tons of atmospheric argon, has set the
strongest limit to date for this technology, showing its intrinsic
limitations.

The double phase liquid argon TPC has a more complex history. Before
DarkSide-50 with atmospheric argon (A-Ar) [75], a smaller prototype of WArP
had the only result for this technology using 2.3 liters of A-Ar [79]. This
limit was further improved by the first data of underground ultra-pure argon
(U-Ar) distilled from the deep Earth mantle CO2 [77]. After this result,
DEAP-3600 currently holds the best limit for argon-based detectors (in single
phase).

DarkSide-50, DEAP-3600 and ArDM are now joining a common project called GMDMC
(Global argon Dark Matter Community) [80]. The first instance of this joint
venture is DarkSide-20k, a giant TPC containing about 50 ton of U-Ar,
currently under construction [81]. Furthermore, the GADMC is also planning a
futuristic version with 300 tons of active target mass, called ARGO-300.

DarkSide-20k presents a lot of challenging aspects, and many technological
novelties as compared with the existing TPCs for dark matter: first, the use
of SiPM-based photo-detector modules instead of standard PMTs to overcome the
issue of the cold electronics, with quite a few critical issues discussed in
literature [82, 83]; second, extraction and distillation of more than 50 ton
of U-Ar, and preservation of its radio-purity; third, realization of a very
big TPC with many challenges for high voltage, purity and event pile-up
handling; fourth, use of multi-ton acrylic vessels; finally, A-Ar veto with
acrylic doped with gadolinium. Even in the case of DarkSide-20k, it is not
clear why the funding agency panels have not supported intermediate scale
detectors (like _e.g._ 1-ton scale) with the intermediate physics goal of
exploring the light dark matter mass, pushing instead for something bigger,
just to fill the gap in a phantom competition with the xenon-based detectors.
This choice is anyway questionable: even in case the argon technology was left
behind the xenon, the unknown behavior in terms or radiopurity of multi-ton
xenon-based detectors is enough worrisome to justify an alternative, even
smaller but solid, with high background rejection capability, as featured by
the argon.

In conclusion, as a matter of fact, the xenon-based technology has always been
considered full of criticality, but in the real world it continues to be the
most advanced branch of the noble gas based detectors. Liquid argon, on the
contrary, is struggling to keep up and future stages are not completely clear.

Property | Argon | Xenon
---|---|---
$Z$ | 18 | 54
$A_{r}$ | 39.9 | 131.9
$\rho$ | 1.4 g/cm3 | 3 g/cm3
$T_{B}$ | 87 K | 165 K
$\lambda$ | 128 nm | 178 nm
$\tau_{\rm fast}$ | 4 ns | 6 ns
$\tau_{\rm slow}$ | 25 ns | 1600 ns
LY | 40 PE/keV | 46 PE/keV
ER–NR Classifier | S2/S2 | S1(t)
Table 2: Comparison between xenon and argon in dark matter detectors. From the
top: atomic number, atomic weight, density, boiling point, scintillation
wavelength, singlet decay, triplet decay, light yield, ER–NR classifier.
Figure 11: _Left:_ single phase detector in which the light emission from the
scintillation process is collected by PMT’s instrumented all around the
detector. _Right:_ double phase TPC. A primary scintillation signal (S1) is
produced in the liquid. The drift electric field between the cathode (K) and
the grid (G) moves the ionization electrons upwards, finally extracted by
another electric field between the grid and the anode (A). The accelerated
electrons in gas produce a second and stronger light signal (S2). Figure 12:
Comparison of the three leading xenon-based dark matter detectors, as reported
in [45], which is the publication referred to as “this work”.

###  5.3 Solar neutrinos

Before moving ahead, it is worth mentioning that the multi-ton noble gas
detectors have some promising by-product purposes. The very low achievable
background, high target and high scintillation light yield can make them
optimal detectors for precision measurement of solar neutrino fluxes coming
from the proton-proton chain and the CNO cycle [84], the two processes
responsible of the hydrogen fusion in the Sun.

The current precision measurements, produced by detectors such as GALLEX/GNO
[85], Super Kamiokande [86], SNO [87] and especially Borexino [88, 89, 90, 91,
92], have helped to better understand solar physics and neutrino oscillation.
However, further improvement in precision can address plenty of other open
problems, as the solar metallicity abundance [91], the tensions on the solar
$\Delta m^{2}$ with reactor experiments [2], the precision constraint of the
total solar luminosity in the low energy spectrum for extra source of energy
in the Sun (as dark matter decay, indeed) search for solar axions [93], and
non-standard neutrino interaction as smoking gun of new physics beyond SM
[94]. In other words, trying to build large dark matter detectors with some
multi-purpose possibility, such as solar neutrino and also neutrino-less
double beta decay detection, could in principle be reasonably acceptable.

###  5.4 Others

Besides NaI-, xenon- and argon-based detectors, there is plenty of other
experiments and R&D’s using a big variety of target nuclei and techniques.
Some of those, which are less sensitive to the WIMP-like particles (or
sensitive only to light masses) are not discussed for the purpose of present
articles. It is interesting to mention some of them, which are playing an
important role in direct dark matter search: CRESST using a target with CaWO4
[95], CDMSLite and SuperCDMS [96] using germanium, DAMIC using silicon [97],
PICO–60 using C3F8 [98], and NEWS-G using neon [99].

If one may think that DAMA, made of sodium (light nucleus, $A\simeq 23$) and
iodine (heavy nucleus, $A\simeq 127$) cannot be compared to xenon (heavy
nucleus, $A\simeq 131$) because the origin of the annual modulation comes from
the interaction with sodium rather than iodine, one should also think to what
happens with a big variety of other atoms, used by other experiments, with
null results. And they are made of carbon ($A\simeq 12$), oxygen ($A\simeq
16$), fluorine ($A\simeq 19$), calcium ($A\simeq 40$), argon ($A\simeq 40$),
tungsten ($A\simeq 183$), germanium ($A\simeq 73$), silicon ($A\simeq 28$) and
neon ($A\simeq 20$). Results do not easily reconcile even in case of spin-
dependent (SD) interactions, as _e.g._ the xenon contains approximately the
same percentages of isotopes with nuclear spin 0, 1/2 and 3/2. Considering the
full list of nuclei tested so far in the framework of SI (and also SD)
interaction with SHM, it is extremely hard to believe that sodium plays such a
special role among all the elements in the periodic table.

##  6 Evolution of results

A good way to explore the evolution of results on direct detection of dark
matter can be done reading the dark matter review as reported by the Particle
Data Group (PDG) [100]. Downloading old versions from 1996 to the latest
update (2022) (see [101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111,
112, 113, 2] and Refs. therein ), one can see that the Review has increasingly
dedicated a larger and larger number of pages, passing from about 4 to more
than 30 in about 25 years. This fact, of course, is not only related to the
increasing interest in dark matter, but also to the overall increase of the
space dedicated to physics reviews in PDG. A correct comparison should be
somehow normalized.

The first SI $\sigma$–$M_{\chi}$ plot appeared only in 2010 and it was updated
on 2011 [108]. In this paradigm, one can immediately see the strong tension of
the two DAMA solutions (as interpreted in the SI framework by [114]), made of
two islands (one for the sodium solution and one for the iodine solution),
with the other experiments, as XENON-100, EDELWEISS and CMDS-Si. Along with
DAMA, another small-scale detector based on germanium and called CoGeNT [115]
shows a similar solution close to the DAMA sodium regions. In the same plots,
it appears the region in which possible SUSY candidates can be potentially
discovered, as hidden channels, at LHC.

In 2012 nothing was really changed, but in the 2013 update [109] (see Fig.
13), persisting also in 2014 [110], new islands appeared, very close to DAMA:
both CDMS-Si and CRESST had an excess over the predicted background. All these
positive results appeared singularly close to the Higgs boson discovery in
July 2012 at LHC [116, 117]. At the same time, the absence of evidence of SUSY
candidates started to be digested by the scientific community, but still non
widely consolidated [118].

Figure 13: Cross section as a function of the WIMP-like mass in the SI
framework, as reported in PDG 2013. In the same plot, limits (solid curves)
and positive results (islands with a given CL) are reported for various
experiments, see text.

In 2015, CRESST-II observed no excess, and then the previous islands due to
CaWO4 were removed in the updated PDG [110]. In 2016 nothing was really
changing, except for some improvement on previous limits by many experiments
[111]. In the 2017 update [111], from null results by further upgraded
versions of CoGeNT, the corresponding island was removed, and new limits, such
as PICO-60 and DEAP-3600 were added. In the 2018 update [112], coinciding with
important releases from ANAIS-112 and COSINE-100, the DAMA islands were
removed from the PDG. Notice that the DAMA SI islands are almost never
presented in the official DAMA publications, and recently survive mainly in
the publications of the NaI-based experiments trying to reproduce DAMA.
Hereafter, up to the last upgrade in 2022 [2], the PDG SI $\sigma$–$M_{\chi}$
plot reports only upgraded limits (see Fig. 14), closer and closer to the
neutrino floor, for both low and high masses.

Figure 14: Cross section as a function of the WIMP-like mass in the SI
framework, as reported in PDG 2022. Only limits (solid curves) are reported by
various experiments, see text.

Recently, since the statistical fluctuations of the neutrino floor can become
important in present and next generation of dark matter experiments, the name
has been changed to _neutrino fog_ or _mist_ , considering the real impact of
how this expected background grows with the experimental exposure [119].

The common practice of focusing on the SI $\sigma$–$M_{\chi}$ plot has
received some criticisms. One may think that comparing all experimental
results in the same SI $\sigma$–$M_{\chi}$ plot could not be a comprehensive
and accurate way to address the dark matter problem, and would be only a
generally subjective, limited and imprecise action. One can reasonably accept
this criticism, but it remains still unexplained how a unique positive result
(from DAMA) can be compatible, independently of the model, with a tens of
other null results made by experiments of comparable or larger sensitivity and
using even similar nuclei as discussed above. Those experiments are not
detecting any positive signal anyway, regardless of the fact that nature has
chosen a SI interaction, or whatever, for visible and dark matter particles.

As has happened many times in the history of physics, the research is going
through a hard period in which the knowledge is stuck and experiments are
becoming more and more challenging and expensive. The whole scientific
community is moving by inertia, after the thrust of a strong theory that is
now left behind the shoulders, and becomes everyday fuzzier and fainter.
Moreover, this motion proceeds so smoothly that someone could not even realize
how he ended up in it.

Furthermore, the belief is so strong that people are already thinking about
the future made, not only of bigger and bigger detectors capable of reaching
the neutrino floor, but also thinking about how to drive into the neutrino fog
with a huge detector capable of exploiting the directionality of the dark
matter. The latter in particular looks very futuristic, given the present
status of results.

From recent history, it is clear that there was some original enthusiasm,
supported by a very appealing theory as SUSY, and close to Higgs discovery
there was a cluster of experimental “excesses”, leaded by the DAMA result that
has probably amplified the expectation that dark matter particles could be
really detectable.

It is anyway commonly accepted that reaching at least the neutrino floor is a
kind of “moral duty” for the scientific community, before trashing completely
WIMPs and all WIMP-like paradigms all at once, in favor of other particle or
non-particle solutions of the problem of the missing mass of the Universe.

Figure 15: Excess of events above the threshold reported by XENON1T, compared
with their background model (red).

Finally, to conclude with another interesting sociological case, one can
remind when XENON1T published a presumed excess above the threshold as
reported in Fig 15 [121], and later non confirmed by XENONnT with a lower
overall background [72]. The collaboration was also warning that such a
spectrum could even come from an unaccounted tritium beta spectrum. Regardless
of whether or not some trained eyes see these two outlier points above the
event threshold as an excess, it is curious that these two points have
received about 600 citations to date, mostly from theorists and
phenomenologists.

What can one do as a particle physicist? The choice of spending some more time
in this kind of search will for sure pay back, as advances in knowledge and
technological implications are granted by basic research. Anyway, one should
not exaggerate, because if the main purpose of an experiment are its secondary
goals, in this case one can propose to search for whatever non-falsifiable
theory, and this would make the science become a practical paradox.

###  6.1 A Drake equation

The Drake equation is a probabilistic argument used to estimate the number of
active, communicative extraterrestrial civilizations in the Milky Way galaxy
[122]. Of course, this equation is more useful for “understanding” rather than
“quantifying”, as its parameters are affected by large uncertainty and
supported sometimes by naive arguments.

One can imagine to follow the same approach for the probability
$\mathcal{P}_{\chi}$ that dark matter is made of WIMP-like (or light WIMP)
particles and can be detected by experiments on the Earth.

A possible ansatz, containing the main terms, is:

$\mathcal{P}_{\chi}=f_{\rm e}\cdot f_{\rm s}\cdot f_{\rm th}\cdot f_{\rm
exp}\cdot f_{\rm det}\cdot f_{\Omega}$ (15)

where:

$f_{\rm e}=$ fraction of energy in the full mass interval for dark matter
candidates. As there is no actual reason to prefer one candidate to another,
one can estimate this ratio as 5/89, where 5 corresponds in the logarithmic
scale to 0.1–10000 GeV/c2 interval over the full range, from axions to massive
primordial black holes.

$f_{\rm s}=$ fraction of possible cross section available for a WIMP-like
interaction. If one considers the full range, from the already reached
($\log(\sigma)=-46$) down to the squared Plank mass ($\log(\sigma)=-66$) of 20
orders of magnitude, and that this kind of candidates cannot be much lower
than the neutrino floor ($\log(\sigma)=-50$) for experimental reasons, this
factor has to be taken as 4/20.

$f_{\rm th}=$ probability that the missing mass of the Universe is explained
by particles or by some acceleration anomaly, _i.e._ the failure in
extrapolating the Newton law (General Relativity) for the Solar System
($10^{8}$ km) to the galactic scale (kpc), or something else. There is no real
reason why this fraction should not be at least 1/2, or even lower (1/3).

$f_{\rm exp}=$ probability that only one experiment (DAMA) has detected dark
matter over about 10 with comparable sensitivity, that is about 1/10.

$f_{\rm det}=$ probability that the dark sector cannot exchange information
with the visible sector with other (weak) interaction but gravity, as in some
versions of the so-called Mirror Matter models [123]. This probability can be
set to 1/2, as there is not a real reason why the dark matter should share the
same interaction properties of the visible matter.

$f_{\Omega}=$ catastrophic probability that the Big Bang cosmology is all
wrong. Nobody will ever admit that, but also saying that the Big Bang
cosmology is 100% correct, would look weird as well. Besides static or quasi-
static physical cosmological models and the like, one could consider
alternative and radical views of the Universe, such as the Simulation
hypothesis [124] or the Mathematical Universe hypothesis [125]. In this case,
one would reasonably accept a 50% chance from a philosophical point of view,
even though one may also think that dark matter could anyway be included in
this kind of Simulation. Furthermore, this probability could be correlated to
$f_{\rm th}$, if modification of gravity can explain the missing mass. Anyway,
assuming one for this factor is part of the game, even if quite disturbing.

With those basic terms, one gets about 1:3560 for $\mathcal{P}_{\chi}$. If the
equation is empowered with other terms, each of them will be $\lesssim 1$,
then the updated probability will be likely less than this first estimation.
Now it is time to bet.

###  6.2 Mala tempora

Until the early 80’s of the past century, new particles came out of
accelerators every day, and it was relatively easy to understand the hidden
logic among particles and interactions in the frameworks of quantum field
theory. Theorists got carried away and fantasized about many extensions of SM,
which, although constituting an excellent description of the observed
phenomena, left and still leaves indications of a more complete high-energy
theory. And SUSY, with its by-product “WIMP miracle” [126], was the most
awaited guest at the party.

LHC, close to the discovery of the Higgs boson, has shown not to be really
suitable as a discovery machine for the new physics beyond SM. It is therefore
probable that the planned high luminosity stage [127] will end up in
controversial _anomalies_ and _tensions_ , which will only complete an already
existing long list. It would have been probably better to shut down LHC and
speed up the construction of the Future Circular Collider, also known as FCC,
with a center-of-mass collision energy of 100 TeV, that is almost one order of
magnitude higher than LHC. To be honest, one should also admit that, in
principle, there is no indication that a possible new physics emerges just at
100 TeV rather than at 1 PeV or more. Therefore, it is anyway like sailing in
the open sea without knowing if and where the next land will be. If one can
speak of a possible “crisis in modern particle physics”, now is precisely that
moment. The only (weak) hope, in light of the phrase “mala tempora currunt”,
is the memory that in similar situations in the history of physics, a
significant revolution often followed a deep crisis.

The only serious risk is that, in the absence of concrete scientific
objectives, experimental collaborations may become uncontrollable inefficient
organizations, whose main goals can be something different from scientific
research.

###  6.3 Falsifiable and Scientific

Karl Popper in his book _The Logic of Scientific Discovery_ (1934) suggested
that a statement, a hypothesis, or theory, to be considered scientific, should
be “falsifiable”, _i.e._ logically contradicted by an empirical test. The
material implication “IF it is scientific THEN is falsifiable”, _i.e._ ,

$\textsc{scientific}\rightarrow\textsc{falsifiable},$ (16)

leaves a wide room to theories that are falsifiable but not scientific.
Proposing a massive particle detectable in underground experiments is for sure
falsifiable, but not necessarily scientific, especially if there is no theory
behind, and no clear motivation why it should be worth searching for it.
Reading the material implication in the opposite direction (as basic logical
fallacy) has sometimes made a lot of confusion, not only for the dark matter
case, but also for numerous extensions of SM, based on aesthetic argument
instead of real necessity. This misunderstanding becomes even more threatening
when the properties of a given dark matter candidate are updated after the
initial dark matter candidate is not found in the place in which it was
proposed [128].

This is the case of WIMP particles emerging from SUSY. The Minimal
Supersymmetric Standard Model (MSSM), introduced to accommodate the problem of
the hierarchy of the Higgs mass, should have broken at the Higgs mass scale
($\sim 100$ GeV) and should have predicted the existence of stable massive
dark matter candidates. The absence of a SUSY particles discovery at the LHC
has pushed theorists to abandon the “naturalness” concerns [129], add other
parameters and mechanisms, and increase the SUSY breaking scale, creating a
big family of X-MSSM models, where X stands for the acronym of the case, see
[130] and Refs. therein. The PDG has recently removed those families of
allowed regions in SI dark matter parameter space [2], being already halved by
xenon-based dark matter detectors. Furthermore, saying that there is still a
50%, or so, unexplored region is, for what is discussed, definitely pointless.

What one needs is a change of paradigm, and this is very well summarized by F.
Nesti _et al._ [132]:

> “In detail, we advocate for a paradigm according to which, after abandoning
> the failing $\Uplambda$CDM scenario, we must be poised to search for
> scenarios without requiring that: (a) they naturally come from (known)
> “first principles” (b) they obey to the Occam razor idea (c) they have the
> bonus to lead us towards the solution of presently open big issues of
> fundamental Physics. On the other side, the proper search shall: (i) give
> precedence to observations and the experiment results wherever they may lead
> (ii) consider the possibility that the Physics behind the Dark Matter
> phenomenon be disconnected from the Physics we know and and does not comply
> with the usual canons of beauty”.

This strategy is quite reasonable. However, the discovery of some elusive
particle solution in underground laboratories, not supported by any
theoretical framework, will be a big deal from an epistemological point of
view, but anyway better than nothing.

###  6.4 The emperor is naked!

The dark matter in shape of WIMP or WIMP-like or light WIMP particles is
widely accepted as true or professed as being plausible, due to an
unwillingness of the general scientific community to criticize it or be seen
as going against the mainstream opinion.

The _Emperor’s New Clothes_ fairy tale by Hans Christian Andersen is a
metaphor about logical fallacies, that reads in this case: no one believes in
such a dark matter, but everyone believes that everyone else believes in it,
until some child comes out of the crowd and shouts: “the emperor is naked”!
But, in this case, such a child, if ever any, has yet to be born.

###  6.5 A way out

Even though direct searches have not solved the dark matter problem, there are
a few interesting things to do. First, one might need to rethink our
understanding of gravity. MOND, or similar, is one idea, essentially saying
that our current gravitational laws might not apply on larger scales.
Alternatively, one might discover new particles that interact even more weakly
than previously thought, making them incredibly hard to detect.

Another possibility is that one is missing something in our theoretical
framework. Maybe our understanding of particle physics needs an upgrade. Some
theories propose new types of particles, like sterile neutrinos or axions,
which could be potential dark matter candidates. These particles would be
elusive and might interact with normal matter in ways that are not fully
understood yet. An interesting change of paradigm is proposed here [133].

On the observational side, upcoming experiments, like the James Webb Space
Telescope [134] and Euclid [135] might provide more insights into the cosmic
web and the distribution of matter in the Universe. High-energy cosmic-ray
observatories and gravitational wave detectors could also bring new clues.

In essence, solving the dark matter puzzle might require a combination of
refining our theories, developing more sensitive detection methods, and
pushing the boundaries of our observational capabilities.

## Conclusions

We do not want to spoil the party, so we will conclude by reiterating the
conviction that the current direct detection of dark matter in the region of
the WIMP-like and light dark matter is absolutely valid, as the cost/benefit
is still affordable and justifiable. About the next generation of detectors,
the same cannot be said with the same certainty.

However, we want to point out that this quest is currently moved by inertia
after the strong thrust impressed a few years ago by a solid theoretical
framework that now is becoming farther and fainter. It happened many times in
the history of physics that some puzzle has been solved because there was a
clear indication where to search for its solution. Now it looks as though it
is not the case for dark matter any more.

The missing mass of the Universe, explained as a heavy particle forming a halo
all around galaxies is an simple theory, so simple and easy to attract many
scientists who may be not so willing to dwell in complicated calculations and
ideas. But it is also persisting, because we are led to think that easier
solutions are always the ones chosen by nature.

The fact that the presence of the missing mass of the Universe is supported by
so many irrefutable pieces of evidence makes the direct search of dark matter
a kind of moral duty that we seem to pursue at all costs. But this idea is
pointless, and probably dangerous.

In this report, we have seen that the leading role in the dark matter search
has been driven first by the DAMA annual modulation result, followed by other
results singularly close to the discovery of the Higgs boson. This apparent
convergence slowly disappeared in the subsequent years, and was overcome by
the liquid noble gas detectors, which are continuing showing null results, and
increasing their sensitivity with bigger and bigger targets.

Given the very small chance of detecting such dark matter particle candidates,
as inferred by a procedure similar to the Drake equation, we can conclude that
the present stage of the quest, without getting sidetracked by science
fiction-like projects, is anyway necessary to start rethinking the Universe.

## Acknowledgements

First of all, we would like to thank V. Fano and his team from University of
Urbino: the origin of this work can somehow be traced back to the interaction
with him. A lot of pieces of information for Fig. 7 and 9 are coming from
colleagues, as they are not officially reported in collaboration articles. We
would like to thank in particular, G. Di Carlo, L. Grandi, A. Ianni, N. Di
Marco, K. Pelczar and C. Vignoli for useful discussions about the dark matter
problem and for helping with the reconstruction of the experiment genealogies.
Finally, we would like to thank G. Ranucci, F. Nesti, A. Di Giovanni and R.
Biondi, for comments, suggestions and proofreading.

## List of acronyms

LHC = Large Hadronic Collider

$\Uplambda$CMD = Lambda Cold Dark Matter

MOND = Modified Newtonian Dynmics

SUSY = Super Symmetry

WIMP = Weakly Interacting Massive Particle

GUT = Grand Unification Theory

MACHO = Massive Astrophysical Compact Halo Object

ALP = Axion-Like Particle

SI = Spin Independent

SD = Spin Dependent

SM = Standard Model

SHM = Standard Halo Model

CL = Confidence Level

PMT = Photo-Multiplier Tube

ER = Electron Recoil

NR = Nuclear Recoil

R&D = Research and Development

SiPM = Silicon Photo-Multiplier

TPC = Time Projection Chamber

S1 = Primary scintillation light in TPCs

S2 = Secondary scintillation light in TPCs

UV = Ultra Violet

FCC = Future Circular Collider

MSSM = Minimal Supersymmetric Standard Model

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