Experiment: Dark-Matter Searches and Their Null Results

Contents

What is being searched for, and what a null result bounds
LUX-ZEPLIN: the largest xenon target (2023)
XENONnT: the same signature in an independent detector (2023)
Cryogenic bolometers: the low-mass frontier
DAMA/LIBRA and the annual-modulation claim (2008–2018)
ADMX: the axion haloscope (2018–2020)
ATLAS: missing transverse momentum at the Large Hadron Collider (2021)
Fermi-LAT: gamma rays from dwarf spheroidal galaxies (2009–)
The state of the search

The Dark Sector: Evidence Without Explanation records the evidence that gravitating matter exceeds luminous matter by about a factor of five on every scale from galaxies to the observable universe, and names the candidates that might account for it. This chapter is the experimental face of that one, and its result is stated at the outset because it is the whole point: there is no laboratory detection of a dark-matter particle. Four decades of dedicated searching — deep underground detectors, resonant microwave cavities, collider missing-energy analyses and gamma-ray telescopes — have returned exclusion bounds and nothing else. One positive claim exists, the annual modulation of DAMA/LIBRA [Bernabei:2008], and it has not been reproduced by any other experiment, including experiments using the same target material [Adhikari:2018]. That discrepancy is reported here as what it is: unresolved.

A null result is evidence, and it is worth being precise about what kind. An exclusion curve is a statement about a two-dimensional parameter space — candidate mass against interaction strength — and it says that if the candidate existed with those parameters, this apparatus, with this exposure and this background, would have seen recoils that it did not see. It is therefore conditional on a halo model, on an assumed interaction, and on a background estimate, and every one of those conditions has to be stated with the bound. The searches reported below are chosen because between them they cover four different couplings of a hypothetical dark sector to the Standard Model of Electroweak Unification and the Higgs Boson: nuclear recoil, photon conversion in a magnetic field, production at a collider, and annihilation in an astrophysical halo. They fail independently, which is what gives the combined null result its weight.

Two boundaries of scope are worth marking. This chapter does not adjudicate between a particle explanation and the modified-dynamics alternative of The Dark Sector: Evidence Without Explanation; it reports what the particle searches found. And it does not treat the neutrino, whose mass bounds from Experiment: Neutrino Oscillations and whose free-streaming length exclude it as the dominant component. What survives unexplained is carried to What We Observe but Do Not Understand.

Derivation pending.

Experiment: Dark-Matter Searches: the standard halo model and the nuclear form factor that together convert a measured recoil spectrum into a bound on a cross-section, and the statistical construction of a 90 per cent confidence exclusion contour in the presence of an estimated background, are pending. The kinematic and rate arguments of this chapter are derived inline.

What is being searched for, and what a null result bounds

The halo the searches assume

Every direct-detection bound is quoted against a common set of astrophysical assumptions, so that different experiments can be compared on one plot. The conventional choice is an isothermal sphere: a local mass density of about \(5\times 10^{-22}\,\mathrm{kg}/\mathrm{m}^{3}\), a Maxwellian speed distribution in the Galactic rest frame with a dispersion corresponding to a solar motion through the halo of roughly \(2.3\times 10^{5}\,\mathrm{m}/\mathrm{s}\), and a truncation at a local escape speed of order \(5\times 10^{5}\,\mathrm{m}/\mathrm{s}\). The density is uncertain by a factor of about two and the velocity structure by more, so a published exclusion curve carries an astrophysical systematic that no detector improvement removes.

[Reserved: the standard halo model written out, with the velocity integral \(\eta(v_{\min})\) that enters the rate; the dependence of a quoted limit on the local density, which is linear and therefore trivially rescalable, against the dependence on the high-speed tail, which is not; substructure — streams and debris flow — and why it matters more for an annual-modulation measurement than for a total-rate measurement; the Galactic rotation curve constraints of The Dark Sector: Evidence Without Explanation as the input.]

Elastic recoil kinematics

The signature sought underground is a single nuclear recoil [Goodman:1985]. A halo particle of mass \(M\) and speed \(v\) scattering elastically off a nucleus of mass \(m_{N}\) deposits, for centre-of-mass scattering angle \(\theta\), the recoil energy

\begin{equation}\tag{136.1} E_{R}=\frac{\mu^{2}v^{2}}{m_{N}}\left(1-\cos\theta\right)\ec \qquad \mu=\frac{M\,m_{N}}{M+m_{N}}\ec \end{equation}

so that \(E_{R}\) ranges from zero to

\begin{equation}\tag{136.2} E_{R}^{\max}=\frac{2\mu^{2}v^{2}}{m_{N}}\ep \end{equation}

Two features of Equation (136.2) govern the entire enterprise. First, \(v/c\) is of order \(10^{-3}\), so the deposited energy is smaller than the rest energy of the projectile by six orders of magnitude: a candidate of mass \(100\,\mathrm{GeV}/c^{2}\) striking a xenon nucleus produces recoils of order tens of kilo-electronvolts, which is why every one of these detectors is built around the lowest energy threshold it can reach. Second, the reduced mass saturates. For \(M\gg m_{N}\) one has \(\mu\to m_{N}\) and \(E_{R}^{\max}\to 2m_{N}v^{2}\), independent of \(M\); for \(M\ll m_{N}\) one has \(\mu\to M\) and \(E_{R}^{\max}\propto M^{2}\), which collapses below any fixed threshold. A heavy target therefore buys signal at high mass and loses it entirely at low mass — the reason the xenon experiments of Section 136.2 and Section 136.3 and the cryogenic detectors of Section 136.4 are complementary rather than competing.

How to read an exclusion curve

The shape of every published spin-independent exclusion curve follows from Equation (136.2) and one rate argument, and it is worth deriving because the shape is often mistaken for a property of the apparatus. Let \(\rho_{\mathrm{loc}}\) be the local mass density of the candidate. Its number density is \(n=\rho_{\mathrm{loc}}/M\), and the event rate per unit target mass is

\begin{equation}\tag{136.3} R\;\sim\;\frac{n\,\sigma\,\avg{v}}{m_{N}} \;=\;\frac{\rho_{\mathrm{loc}}\,\sigma\,\avg{v}}{M\,m_{N}}\ep \end{equation}

An experiment that observes no excess sets an upper limit on \(R\). At high mass the kinematics of Equation (136.2) are saturated and nothing else in Equation (136.3) depends on \(M\), so the excluded cross-section scales as

\begin{equation}\tag{136.4} \sigma_{\text{limit}}\;\propto\;M\ec \end{equation}

which is the straight line of unit slope that terminates every curve on a log–log plot. It is not a loss of sensitivity: it is the statement that at fixed mass density a heavier candidate is proportionately rarer. At low mass the curve instead turns upward almost vertically, because \(E_{R}^{\max}\propto M^{2}\) falls below the detector threshold and the number of detectable recoils, not the number of recoils, goes to zero. The minimum between the two regimes sits where the reduced mass is comparable to the target mass, which is why a xenon experiment is most sensitive near a candidate mass of \(30\,\mathrm{GeV}/c^{2}\) and a light-target experiment is most sensitive far below that.

LUX-ZEPLIN: the largest xenon target (2023)

Apparatus

A two-phase xenon time-projection chamber holding about seven tonnes of active liquid xenon, operated about \(1.5\,\mathrm{km}\) underground at the Sanford Underground Research Facility so that the cosmic-ray muon flux of Cosmic Rays and Astroparticle Physics is suppressed by several orders of magnitude. Two arrays of photomultipliers view the target from above and below; the liquid is overlaid by a gas phase across which an extraction field is applied, and the whole chamber sits inside an instrumented liquid-xenon skin and an outer gadolinium-loaded liquid-scintillator veto for neutrons.

[Reserved: the detector dimensions, the drift and extraction fields, the photomultiplier count and quantum efficiency at the xenon scintillation wavelength of \(175\,\mathrm{nm}\); the radioassay campaign on every construction material, which is what sets the achievable background and is the real content of the apparatus section for an experiment of this class; the rock overburden in metres of water equivalent and the residual muon rate.]

Procedure

A nuclear recoil in the liquid produces prompt scintillation, recorded as a first pulse, and a cloud of ionization electrons that the drift field carries to the surface and the extraction field pulls into the gas, where they produce a second, proportional electroluminescence pulse. The delay between the two pulses gives the depth of the event and the light pattern of the second gives its transverse position, so that every event is located in three dimensions and an inner fiducial volume can be defined in which the outer layers of xenon have absorbed the external radioactivity. The ratio of the two pulse areas discriminates nuclear recoils, produced by neutrons and by the sought signal, from electron recoils produced by gamma rays and beta decays.

[Reserved: the calibration programme — neutron sources for the nuclear-recoil band and injected radioactive sources dissolved in the xenon for the electron-recoil band; the fiducial mass and live time of the first science run; the blinding and salting protocol adopted before unblinding, which is the procedural guarantee that the analysis was not tuned to the data.]

Observations and data

[Reserved: the observed event distribution in the plane of ionization against scintillation, with the nuclear-recoil band drawn; the expected background from accidental coincidences, from residual radon daughters, from atmospheric and solar neutrinos and from detector neutrons, each with its uncertainty; the number of observed events against the number expected, and the profile-likelihood test statistic from which the limit is constructed.]

Phenomenon 136.1 (No nuclear-recoil excess in seven tonnes of xenon).

In an exposure of about \(0.9\,\mathrm{t}\,\mathrm{yr}\), the LUX-ZEPLIN experiment observed a distribution of events in its nuclear-recoil signal region consistent with the estimated background, and no excess attributable to halo particles scattering off xenon nuclei. Converted under the standard halo assumptions of Section 136.1.1 into a bound on a spin-independent cross-section per nucleon, the result excludes cross-sections above about \(9\times 10^{-52}\,\mathrm{m}^{2}\) at a candidate mass near \(36\,\mathrm{GeV}/c^{2}\), with the bound weakening on either side in the manner derived in Section 136.1.3 [Aalbers:2023]. This was, on publication, the most stringent such bound over most of the mass range, and it is the strongest single statement in this chapter of what is not there.

Derivation. The bound quoted is a cross-section per nucleon, not per nucleus, and the conversion is the one place where an assumed interaction enters visibly. If the coupling is scalar and identical for protons and neutrons, the amplitudes of the \(A\) nucleons add coherently in the forward direction, so the nuclear cross-section carries a factor \(A^{2}\) relative to the nucleonic one, modified by the reduced-mass ratio and by a form factor \(F(q)\) that accounts for the loss of coherence once the momentum transfer \(q\) resolves the nuclear radius:

\begin{equation}\tag{136.5} \sigma_{N}(q)=\sigma_{n}\, \frac{\mu_{N}^{2}}{\mu_{n}^{2}}\,A^{2}\,\abs{F(q)}^{2}\ec \end{equation}

with \(\mu_{N}\) and \(\mu_{n}\) the candidate–nucleus and candidate–nucleon reduced masses. The factor \(A^{2}\) is the reason xenon, with \(A\approx131\), is used at all: it buys more than four orders of magnitude in rate over a single nucleon, and it is what allows a bound near \(10^{-51}\,\mathrm{m}^{2}\) — more than twenty orders of magnitude below a typical nuclear cross-section — to be reached with a tonne-scale target in a year. It is also the assumption that makes the bound conditional. A candidate coupling to nuclear spin rather than to mass gets no coherent enhancement, because the paired nucleons cancel and only the odd nucleon contributes; bounds on spin-dependent couplings are accordingly weaker by several orders of magnitude, and are quoted separately.

∎

Derivation pending.

The form factor and the recoil spectrum: derivation of the Helm form factor as the Fourier transform of a uniform nuclear density convolved with a surface term, its first zero in relation to the recoil energies of interest for xenon, and the resulting distortion of the differential rate away from the pure exponential that the maximum-recoil relation alone would give; together with the detector response function that converts a recoil energy into an expected number of scintillation photons and ionization electrons, which is what actually sets the threshold

Interpretation

[Reserved: what the bound excludes and what it does not; the mass range over which the weak-scale thermal relic parameter space of The Dark Sector: Evidence Without Explanation is now closed for the simplest scalar coupling, and the regions — low mass, spin-dependent coupling, inelastic and momentum-suppressed couplings — that survive untouched; the honest statement that a null result excludes a model, never a phenomenon, and that the gravitational evidence of The Dark Sector: Evidence Without Explanation is undisturbed by it.]

Primary references

[Aalbers:2023].

XENONnT: the same signature in an independent detector (2023)

Apparatus

A two-phase xenon time-projection chamber of about six tonnes of active target, operated under the Gran Sasso massif in Italy, with a water Cherenkov muon veto and — the distinguishing feature of this apparatus — a surrounding gadolinium-loaded water neutron veto and an online cryogenic distillation column that continuously removes radon from the circulating xenon. The radon daughters dissolved in the liquid are the dominant electron-recoil background of a xenon experiment, and removing them from the target rather than rejecting them statistically is what allowed the electron-recoil rate to be pushed to a new low.

[Reserved: the active and fiducial masses, the drift field and electron lifetime, the photomultiplier arrays; the radon and krypton distillation columns and the residual specific activities achieved; the neutron-veto tagging efficiency, which converts the dominant radiogenic neutron background into a measurable and subtractable one.]

Procedure

Identical in principle to Section 136.2: prompt scintillation and delayed proportional ionization signals, three-dimensional event reconstruction, fiducialization, and discrimination of nuclear from electron recoils by the ratio of the two signals. The scientific value of running the two experiments in parallel lies precisely in the places where they are not identical — different rock, different veto technology, different radon-mitigation strategy, different analysis teams and different unblinding protocols — so that a common systematic is unlikely to survive in both.

[Reserved: the calibration sources and the neutron-veto efficiency measurement; the exposure and live time; the blinding scheme; the statistical treatment by which a nuclear-recoil signal is tested against a background model with several components.]

Observations and data

[Reserved: the observed events in the signal region against the background expectation, the achieved electron-recoil rate quoted per tonne per year per kilo-electronvolt, and the resulting exclusion curve with its minimum located in mass and cross-section; the comparison plot against Section 136.2 and the earlier generation of xenon experiments, which is where four decades of improvement becomes visible as a stack of curves descending by more than six orders of magnitude.]

Phenomenon 136.2 (Two independent xenon targets, one null result).

In an exposure of about \(1.1\,\mathrm{t}\,\mathrm{yr}\) XENONnT observed no excess of nuclear recoils over its background model, and set a spin-independent exclusion bound of the same order as Phenomenon 136.1 over the same mass range [Aprile:2023]. The two experiments differ in overburden, in veto technology, in radon mitigation and in analysis team, and they agree. The agreement is the observation: a single null result bounds a model, but two null results obtained with independent systematics bound the model and simultaneously test each other's background estimate.

Derivation. The reason the next decade of this measurement is not simply a matter of building a larger tank is that an irreducible background is approaching, and its position in the mass–cross-section plane can be located by kinematics alone. Neutrinos scatter coherently off nuclei through the neutral current, producing exactly the signature sought: a single, unaccompanied nuclear recoil of a few kilo-electronvolts, carrying no feature by which it can be told from the sought signal event by event. For a massless projectile of energy \(E_{\nu}\) on a nucleus at rest, energy and momentum conservation give the maximum recoil

\begin{equation}\tag{136.6} E_{R}^{\max}=\frac{2E_{\nu}^{2}}{m_{N}c^{2}+2E_{\nu}} \;\simeq\;\frac{2E_{\nu}^{2}}{m_{N}c^{2}}\ec \end{equation}

since \(E_{\nu}\ll m_{N}c^{2}\) for every relevant source. Setting Equation (136.6) equal to the halo-particle maximum Equation (136.2), written as \(2\mu^{2}c^{4}\left(v/c\right)^{2}/\left(m_{N}c^{2}\right)\), the nuclear mass cancels and leaves the mimicking relation

\begin{equation}\tag{136.7} \mu c^{2}\;\simeq\;E_{\nu}\,\frac{c}{v}\;\sim\;10^{3}E_{\nu}\ep \end{equation}

Equation (136.7) is the whole structure of the so-called neutrino fog in one line. Solar boron-8 neutrinos, of energies up to about \(16\,\mathrm{MeV}\), imitate candidates of mass of order \(10\,\mathrm{GeV}/c^{2}\); atmospheric neutrinos, of energies from hundreds of \(\mathrm{MeV}\) upwards, imitate candidates one to two orders of magnitude heavier, in precisely the region where Phenomenon 136.1 is strongest. Because this background scales with exposure exactly as the signal does, the statistical advantage of a longer run collapses once it dominates: the reach ceases to improve in proportion to the exposure and improves at best as its square root, and then more slowly still [Billard:2014]. It is a fog rather than a floor for the same reason — it degrades sensitivity progressively rather than forbidding progress [OHare:2021] — and directional detection, which uses the different arrival directions of solar neutrinos and halo particles, is the only known way through it.

∎

Interpretation

[Reserved: the combined status of the two xenon experiments and the parameter space they have jointly closed; the distance in cross-section still separating the current bounds from the atmospheric-neutrino component of Equation (136.7), which is what determines whether a next-generation experiment is worth building; the first measurements of coherent neutrino-nucleus scattering from solar boron-8 neutrinos in xenon detectors, reported in 2024, which demonstrate that the fog is not a theoretical projection but an observed background — to be cited when the primary references enter the bibliography.]

Primary references

[Aprile:2023].

Cryogenic bolometers: the low-mass frontier

The xenon experiments lose sensitivity below a candidate mass of about \(3\,\mathrm{GeV}/c^{2}\) for the kinematic reason derived in Section 136.1.2: a light projectile on a heavy nucleus deposits an energy falling as \(M^{2}\), which passes below any threshold. The complementary technology uses light targets and thresholds several orders of magnitude lower, at the cost of target mass. A cryogenic bolometer is a single crystal held at a few tens of \(\mathrm{mK}\), in which a recoil is measured as a temperature rise of order microkelvin sensed by a superconducting thermometer at its transition edge. Because the phonon signal counts the deposited energy directly, rather than the small fraction of it that appears as light or charge, the threshold is set by thermal noise rather than by a quantum efficiency, and thresholds in the range of tens of electronvolts have been achieved.

Two long-running programmes define this line. The Cryogenic Dark Matter Search operated germanium and silicon crystals reading out ionization and athermal phonons simultaneously, the ratio of the two providing event-by-event discrimination of nuclear from electron recoils in the same way that the ratio of scintillation to ionization does in xenon [Agnese:2013]; its successor operates at deeper overburden with detectors optimized for masses below \(1\,\mathrm{GeV}/c^{2}\). CRESST operates scintillating calcium tungstate crystals with simultaneous phonon and light readout, and its low-threshold detectors currently reach the lowest candidate masses of any direct-detection experiment [Angloher:2016] [Abdelhameed:2019]. Neither has reported a detection.

[Reserved: this section is deliberately not written up as an experiment, because the primary references for the CDMS, SuperCDMS and CRESST results are not yet in the bibliography of this treatise and the house rule forbids an experiment section without its primary source. To be written when they are entered: the detector masses, operating temperatures and thresholds; the transition-edge sensor and neutron-transmutation-doped thermistor readouts; the CDMS silicon result of 2013, in which three candidate events were observed against a smaller expected background and which was correctly reported as not constituting a detection; the CRESST low-mass exclusion curve; and the surface-event and low-energy-excess backgrounds that limit both. The compiled exclusion curves for these experiments are collected in the Particle Data Group review [Navas:2024].]

DAMA/LIBRA and the annual-modulation claim (2008–2018)

Apparatus

An array of highly radiopure thallium-doped sodium iodide scintillator crystals, of total mass of order \(250\,\mathrm{kg}\), viewed by photomultipliers and operated under the Gran Sasso massif with passive shielding and a controlled, continuously monitored environment — temperature, radon concentration and nitrogen flux — since the claimed signal is a modulation of about one per cent with a period of one year, and any seasonal drift of the apparatus itself would imitate it. The replication experiments use the same target compound, at different sites and with active liquid-scintillator vetoes surrounding the crystals.

[Reserved: the crystal growth and radiopurity programme, which is the central technical claim of the experiment; the light yield in photoelectrons per kilo-electronvolt, which sets the analysis threshold; the shielding stack; the corresponding parameters of the replication arrays and the respects in which they differ.]

Procedure

The measured quantity is not a rate but the time dependence of a rate. Single-hit events — energy deposits in one crystal only, multiple hits being rejected as neutron or gamma-ray induced — are binned in energy and in time over many annual cycles, and the low-energy rate is fitted to a constant plus a cosine of period one year. Three parameters are tested against prediction: the amplitude, the period and the phase. The phase is the discriminating one, because the expected date of maximum is fixed by celestial mechanics and is not adjustable.

[Reserved: the single-hit selection and its efficiency; the energy calibration and the stability of the threshold over two decades; the treatment of the known seasonal modulation of the cosmic-ray muon flux, whose phase differs from the expected halo phase by several weeks and which is the leading conventional explanation that has been proposed and contested.]

Observations and data

[Reserved: the residual single-hit rate against time in the lowest energy bins, over the accumulated exposure of DAMA/NaI and DAMA/LIBRA; the fitted amplitude, period and phase with their uncertainties; the energy dependence of the modulation amplitude and its disappearance above a few kilo-electronvolts; the corresponding null fits from the replication arrays, quoted as the amplitude they exclude.]

Phenomenon 136.3 (A modulated rate that no other experiment sees).

The DAMA/LIBRA experiment reports, in the single-hit low-energy event rate of its sodium iodide crystals, a sinusoidal modulation with a period compatible with one year and a phase compatible with a maximum in late May or early June, sustained over many annual cycles and of combined statistical significance in excess of eight standard deviations [Bernabei:2008]. The modulation is a genuine feature of the data and has never been shown to be an artefact. It is nevertheless not accepted as a dark-matter detection, for two independent reasons. Interpreted as elastic spin-independent scattering under the halo model of Section 136.1.1, the implied cross-section is excluded by Phenomenon 136.1 and Phenomenon 136.2 by many orders of magnitude. And experiments using the same target compound, which are therefore not subject to the objection that a different target might respond differently, do not reproduce the modulation [Adhikari:2018] [Amare:2021]. Neither a dark-matter interpretation nor a conventional one is presently established: this is an unresolved experimental discrepancy, and it is reported here as one.

Derivation. The phase is the part of the claim that can be derived with certainty, and it is what makes the measurement interesting at all. The Sun moves through the Galactic halo at a speed \(v_{\odot}\) of about \(2.3\times 10^{5}\,\mathrm{m}/\mathrm{s}\); the Earth orbits the Sun at \(v_{\oplus}=2.98\times 10^{4}\,\mathrm{m}/\mathrm{s}\) in a plane inclined to the direction of solar motion by an angle \(\gamma\) of about \(60\,^\circ\). The Earth's speed through the halo is therefore

\begin{equation}\tag{136.8} v_{E}(t)=v_{\odot} +v_{\oplus}\cos\gamma\, \cos\!\left[\omega\left(t-t_{0}\right)\right]\ec \qquad \omega=\frac{2\pi}{1\,\mathrm{yr}}\ec \end{equation}

with \(t_{0}\) the date on which the orbital velocity is most nearly aligned with the solar motion, which falls in early June. The amplitude of the speed modulation is \(v_{\oplus}\cos\gamma\approx1.5\times 10^{4}\,\mathrm{m}/\mathrm{s}\), about \(6\,\mathrm{\%}\) of \(v_{\odot}\). Because a detector counts only recoils above its threshold, and by Equation (136.2) only particles with \(v\) above a threshold-dependent minimum can produce them, the counted rate samples the steeply falling tail of the speed distribution; a \(6\,\mathrm{\%}\) modulation of the speed therefore produces a rate modulation of order a per cent, with the sign depending on where the threshold sits relative to the distribution. Three predictions follow with no free parameters: the period is one year, the phase is fixed by the Earth's orbit and not by the experiment, and the amplitude is of order a per cent of the rate [Drukier:1986]. That all three are satisfied is why the claim could not simply be dismissed; that the implied cross-section contradicts other experiments is why it could not be accepted.

∎

Derivation pending.

The quantitative incompatibility: the calculation that converts a modulation amplitude measured in sodium iodide into an allowed region in the mass–cross-section plane, including the quenching factors for sodium and for iodine recoils, which are measured quantities with their own uncertainties and which set the recoil-energy scale of the claimed region; and the comparison of that region against the exclusion curves of the xenon experiments, together with an enumeration of the non-standard halo and interaction assumptions — streams, inelastic scattering, isospin-violating couplings — that have been proposed to reconcile them and the reasons each is now disfavoured

Interpretation

[Reserved: how a positive claim is adjudicated when it is not reproduced — the distinction between an experiment being wrong, an experiment being right about an effect that is not dark matter, and a model-dependence in the comparison; the conventional explanations advanced, including the seasonal muon flux and seasonal detector effects, and the status of each; the methodological conclusion, which is that same-target replication is the only decisive test and that it has now been performed [Adhikari:2018].]

Primary references

[Bernabei:2008] [Adhikari:2018].

ADMX: the axion haloscope (2018–2020)

Apparatus

A tunable cylindrical microwave cavity of high quality factor, immersed in the bore of a superconducting solenoid producing a field of order \(8\,\mathrm{T}\), cooled to a few hundred \(\mathrm{mK}\), and read out through a near-quantum-limited amplifier. The resonant frequency is swept by moving dielectric or metallic tuning rods within the cavity, covering a band from roughly \(650\,\mathrm{MHz}\) to \(800\,\mathrm{MHz}\) across the two data runs reported here.

[Reserved: the cavity dimensions and unloaded quality factor; the tuning-rod mechanism and the mode map, including the mode crossings at which sensitivity is lost; the dilution refrigerator and the physical against noise temperature of the receiver; the superconducting quantum interference device and Josephson parametric amplifier readouts that took the system noise near the standard quantum limit, which is the advance that made a Dine–Fischler–Srednicki–Zhitnitsky coupling reachable.]

Procedure

The cavity is tuned to a frequency, power is integrated for a dwell time, the resulting spectrum is searched for a narrow excess above the thermal noise, and the cavity is retuned by a fraction of its bandwidth. A candidate excess is rescanned, and persistence under rescan, together with the expected line shape and the expected disappearance when the magnetic field is lowered, distinguishes a signal from a noise fluctuation or an environmental radio-frequency leak.

[Reserved: the dwell time and scan rate, and the trade between them that fixes the total mass range covered in a run; the candidate rescan protocol and the number of candidates that survived it, which was none; the synthetic-signal injections used to verify the analysis end to end.]

Observations and data

[Reserved: the excluded axion–photon coupling as a function of mass over the scanned band, against the two benchmark model bands; the achieved system noise temperature; the fraction of the band lost to mode crossings.]

Phenomenon 136.4 (No axion signal in the scanned band).

Scanning a tuned microwave cavity in a strong magnetic field, the Axion Dark Matter Experiment found no persistent narrow excess of microwave power over the thermal noise, and excluded halo axions with the coupling of the Dine–Fischler–Srednicki–Zhitnitsky model over the mass range covered by its two reported runs, together spanning roughly \(2.7\text{–}3.3\,\mu\mathrm{eV}\) [Du:2018] [Braine:2020]. This is the first exclusion of that benchmark coupling by a haloscope, and it is worth stating what a narrow band it is: the theoretically motivated mass range for an axion that solves the strong CP problem of Quantum Chromodynamics and also constitutes the dark matter spans several orders of magnitude, and this experiment has closed a fraction of one of them.

Derivation. The design of the apparatus follows from two facts about the sought signal, both of which are certain given the hypothesis. First, a halo axion is non-relativistic, so its total energy is \(E_{a}=m_{a}c^{2}\left(1+v^{2}/2c^{2}+\dots\right)\); converting in a static magnetic field into a single photon [Sikivie:1983], it deposits that energy and no other, so the signal is a line at

\begin{equation}\tag{136.9} \nu=\frac{m_{a}c^{2}}{h}\ec \end{equation}

and a cavity of fixed resonant frequency is sensitive to one axion mass only. The search must therefore be a scan, and the total time required is the time per mass step multiplied by the number of steps — which is why haloscope results cover narrow bands and why the figure of merit of such an experiment is a scan rate rather than an exposure. Second, the same expansion fixes the linewidth. The velocity dispersion of the halo gives \(v/c\sim10^{-3}\), so

\begin{equation}\tag{136.10} \frac{\Delta\nu}{\nu}\simeq\frac{v^{2}}{2c^{2}}\sim5\times10^{-7}\ec \end{equation}

that is, a signal quality factor of order \(10^{6}\). Two consequences follow. The cavity quality factor should be of the same order, since a narrower cavity resonance collects no more signal while a much broader one admits more noise; and the analysis must resolve the line, because a signal spread over \(10^{-6}\) of the carrier frequency is overwhelmingly easier to find in a spectrum binned at that resolution than in the integrated power. Equation (136.10) is thus simultaneously the reason the search is feasible and the reason it is slow.

∎

Interpretation

[Reserved: what a haloscope null result excludes — a coupling at a mass, conditional on the axion constituting the whole of the local halo density, which is an assumption of a different character from those of the nuclear-recoil experiments and rescales the bound linearly if relaxed; the relation to the strong CP problem of Quantum Chromodynamics, and the fact that an axion remains motivated independently of cosmology; the complementary helioscope and light-shining-through-walls searches, which do not assume the axion is the dark matter and are correspondingly weaker.]

Primary references

[Du:2018] [Braine:2020].

ATLAS: missing transverse momentum at the Large Hadron Collider (2021)

Apparatus

The ATLAS detector at the Large Hadron Collider, in the configuration described in Electroweak Unification and the Higgs Boson: an inner tracker in a solenoidal field, electromagnetic and hadronic calorimeters of nearly complete solid-angle coverage, and a muon spectrometer. For this search the calorimeter hermeticity is the critical property, since the observable is a momentum imbalance and any crack through which energy escapes unmeasured produces a false one.

[Reserved: the calorimeter coverage in pseudorapidity and the resulting resolution of the missing transverse momentum; the trigger thresholds; the integrated luminosity of the full Run 2 data set at a proton–proton centre-of-mass energy of \(13\,\mathrm{TeV}\), and its uncertainty.]

Procedure

Events are selected containing at least one energetic jet and a large imbalance of transverse momentum, with events containing charged leptons or photons vetoed. If a pair of invisible particles is produced in association with a jet radiated from the initial state, they carry away momentum that the detector cannot see, and the visible recoil against them is the signature. The dominant irreducible background is the production of a \(Z\) boson decaying to neutrinos in association with a jet, which yields exactly the same final state; it is estimated not from simulation alone but from control regions in the data — the same process with the \(Z\) decaying to charged leptons, and \(W\) production with the lepton lost — which is what makes the background estimate testable.

[Reserved: the signal region definition and its binning in missing transverse momentum; the control regions and the simultaneous fit that transfers their normalization to the signal region; the systematic uncertainties, dominated by the jet energy scale and the control-region transfer factors.]

Observations and data

[Reserved: observed against expected event yields in each bin of missing transverse momentum, with uncertainties; the resulting limits on the parameters of simplified mediator models, translated into the mass–cross-section plane of the direct-detection experiments for comparison; the compiled collider limits, including those on supersymmetric candidates, from the Particle Data Group review [Navas:2024].]

Phenomenon 136.5 (No excess of jets recoiling against nothing).

In the full Run 2 proton–proton data set, ATLAS observed event yields in the energetic-jet plus missing-transverse-momentum signal region consistent with the Standard Model expectation, dominated by \(Z\to\nu\bar{\nu}\) production in association with a jet, and no excess attributable to the pair production of invisible particles [Aad:2021]. Limits on the mediator and candidate masses of simplified dark-matter models follow, and the compiled collider constraints, including those on supersymmetric candidates, are tabulated in the Particle Data Group review [Navas:2024].

Derivation. The observable is missing transverse momentum and not missing energy, and the reason is a kinematic fact about hadron colliders that determines the whole design of the search. The colliding objects are partons carrying unknown fractions \(x_{1}\) and \(x_{2}\) of their protons' longitudinal momenta, so the initial state has longitudinal momentum \(\left(x_{1}-x_{2}\right)p\), which is not measured event by event, and total energy \(\sqrt{x_{1}x_{2}}\,\sqrt{s}\), likewise unmeasured. Neither the longitudinal momentum balance nor the energy balance is therefore available. In the transverse plane, however, the initial-state momentum is zero up to the intrinsic transverse momentum of the partons, which is of order the confinement scale of Quantum Chromodynamics and negligible against the momenta of interest. Momentum conservation transverse to the beam then gives, for the visible particles \(i\),

\begin{equation}\tag{136.11} \vect{p}_{T}^{\text{miss}} =-\sum_{i\ \text{visible}}\vect{p}_{T,i}\ec \end{equation}

which is a measurement, not an inference, provided the calorimeter is hermetic. Equation (136.11) also explains the requirement of a recoiling jet: a pair of invisible particles produced alone would leave a detector with nothing in it at all, which cannot be triggered on and is indistinguishable from no collision, so the search must rely on a jet radiated from the initial state and look for the imbalance against it. The essential limitation of the method is not statistical. Equation (136.11) establishes only that something left the detector without interacting — that is, a particle stable on the time scale of a detector crossing, some tens of nanoseconds. Dark matter must be stable over the age of the universe, a factor of order \(10^{25}\) longer. A collider can therefore never, by itself, detect dark matter; it can produce a candidate, and the identification must come from elsewhere.

∎

Interpretation

[Reserved: the complementarity argument — colliders bound light candidates and light mediators where nuclear recoil is kinematically suppressed, direct detection bounds heavy candidates, and neither statement translates into the other without a model of the mediator; the model-dependence this introduces, which is of a different and stronger kind than in Section 136.2; the invisible-width constraint on candidates lighter than half the \(Z\) or Higgs boson mass from Electroweak Unification and the Higgs Boson; and the honest note that the parameter space of the thermal relic scenario has been substantially reduced from both sides without being closed.]

Primary references

[Aad:2021], the collider search; [Navas:2024], the compiled limits.

Fermi-LAT: gamma rays from dwarf spheroidal galaxies (2009–)

Apparatus

The Large Area Telescope aboard the Fermi Gamma-ray Space Telescope: a pair-conversion instrument in which an incident gamma ray converts in one of a stack of tungsten foils interleaved with silicon-strip tracking planes, the electron–positron pair is tracked to reconstruct the arrival direction, and the energy is measured in a segmented caesium iodide calorimeter below. A segmented anticoincidence detector surrounding the tracker vetoes charged cosmic rays, which outnumber the gamma rays by several orders of magnitude. The instrument covers photon energies from about \(20\,\mathrm{MeV}\) to above \(300\,\mathrm{GeV}\) [Atwood:2009].

[Reserved: the effective area, point-spread function and energy resolution as functions of energy and incidence angle; the survey observing mode, in which the whole sky is covered every two orbits; the anticoincidence segmentation, introduced to avoid the self-veto by backsplash that limited the previous generation of instruments.]

Procedure

Dwarf spheroidal satellites of the Milky Way are chosen as targets because they have the largest known mass-to-light ratios, are nearby, contain little gas and no recent star formation, and are therefore not expected to emit gamma rays by any conventional astrophysical process. The photon counts within a region around each dwarf are fitted jointly with the diffuse and point-source backgrounds, and a likelihood analysis stacking many dwarfs sets a limit on the velocity-averaged annihilation cross-section for an assumed annihilation channel and candidate mass.

[Reserved: the target selection and the kinematically determined astrophysical factor for each dwarf, with its uncertainty, which dominates the error budget; the joint-likelihood construction over targets, energies and channels; the background model for the Galactic and isotropic diffuse emission.]

Observations and data

[Reserved: photon counts against background expectation for the stacked sample; the resulting exclusion curves on the velocity-averaged annihilation cross-section as a function of candidate mass for the quark–antiquark and tau-pair channels, with the thermal relic value drawn for comparison; the primary references for the stacked dwarf analyses to be added when they enter the bibliography of this treatise, the instrument paper alone being cited below. Separately: the excess of gamma rays from the Galactic centre, whose spectrum resembles annihilation but which is equally consistent with a population of unresolved millisecond pulsars, to be reported as an ambiguity and not as a signal.]

Phenomenon 136.6 (No annihilation signal from the darkest galaxies).

Gamma-ray observations of the dwarf spheroidal satellites of the Milky Way with the Large Area Telescope [Atwood:2009] show no emission above the modelled diffuse and point-source backgrounds. Stacked likelihood analyses of these targets bound the velocity-averaged annihilation cross-section below the canonical thermal relic value of about \(3\times 10^{-32}\,\mathrm{m}^{3}/\mathrm{s}\) for candidate masses up to of order \(100\,\mathrm{GeV}/c^{2}\) in the standard annihilation channels [Ackermann:2015]; the compiled limits are collected in the Particle Data Group review [Navas:2024]. The significance of crossing that particular value is that it is the cross-section a thermal relic must have had in the early universe to leave the observed abundance [Lee:1977a], so this is the first class of search to reach the theoretically preferred value rather than merely to approach it.

Derivation. Why dwarf spheroidals, and why the astrophysical factor is the dominant uncertainty, both follow from one line. If the candidate is its own antiparticle, the annihilation rate per unit volume is \(\tfrac{1}{2}n^{2}\avg{\sigma v}\) with \(n=\rho/M\), so the photon flux received from a direction within a solid angle \(\Delta\Omega\) is

\begin{equation}\tag{136.12} \Phi=\frac{\avg{\sigma v}}{8\pi M^{2}}\, \frac{\dd N_{\gamma}}{\dd E}\, \int_{\Delta\Omega}\!\!\int_{\text{l.o.s.}}\rho^{2}\, \dd\ell\,\dd\Omega\ec \end{equation}

the last factor being the astrophysical factor, conventionally called the \(J\)-factor, and the first two the particle-physics factor. The structure of Equation (136.12) dictates the observing strategy. The flux is quadratic in the density, unlike every gravitational probe in The Dark Sector: Evidence Without Explanation, which is linear; so the best targets are those that are dense and nearby rather than merely massive, and so the halo profile enters squared and its uncertainty is amplified. Dwarf spheroidals are selected because their \(J\)-factors can be estimated from stellar kinematics and because, having negligible gas and no star formation, they have no competing gamma-ray emission — the background is instrumental and diffuse, not intrinsic. The Galactic centre has a \(J\)-factor larger by orders of magnitude and is therefore a stronger target by Equation (136.12), but it is crowded with conventional gamma-ray sources, which is exactly why the excess observed there is an ambiguity rather than a detection. Finally, the \(M^{-2}\) prefactor means that a fixed flux sensitivity translates into a bound on \(\avg{\sigma v}\) that weakens as \(M^{2}\), which is why these limits cross the thermal value at low masses and rise above it at high ones.

∎

Derivation pending.

The thermal relic cross-section: solution of the Boltzmann equation for a species annihilating in an expanding universe, freeze-out when the annihilation rate falls below the expansion rate, and the resulting inverse proportionality of the relic abundance to the annihilation cross-section, which is what fixes the benchmark value quoted above from the abundance measured in the dark sector chapter; and the reason the required value falls near a weak-scale cross-section, which is the coincidence that motivated a generation of searches

Interpretation

[Reserved: what an indirect null result excludes and how it differs from a direct one — it constrains the annihilation cross-section now rather than the scattering cross-section, so a candidate that is not its own antiparticle, or that annihilates through a velocity-suppressed channel, escapes it entirely; the antiproton and positron channels and the astrophysical backgrounds that limit them; the Galactic-centre excess and the millisecond-pulsar explanation, presented with the evidence for each side; the neutrino channel from the solar core, bounded by neutrino telescopes.]

Primary references

[Atwood:2009], the instrument; [Navas:2024], the compilation. The stacked dwarf-spheroidal likelihood analyses on which the quoted bound rests are not yet entered in the bibliography of this treatise; only the instrument paper and the compilation are cited, and the bound is stated here at second hand.

The state of the search

Search	Coupling tested	Conditional on
LUX-ZEPLIN and XENONnT	elastic scattering off a nucleus	the local halo density and speed distribution; a coherent spin-independent coupling; the nuclear form factor
Cryogenic bolometers	the same, at low candidate mass	the same, plus the detector response near threshold
DAMA/LIBRA	the same, measured as an annual modulation	the same, plus the quenching factors of sodium and iodine
ADMX	conversion to a photon in a magnetic field	the axion constituting the whole local halo density
ATLAS	production in a proton–proton collision	a mediator model relating production to scattering; stability only over a detector crossing
Fermi-LAT	annihilation to gamma rays in a halo	a self-conjugate candidate; the astrophysical factor of each dwarf; an assumed annihilation channel

The searches reported in this chapter, by the coupling each would need in order to succeed. Every entry is a null result. The last column names the assumption that the bound is conditional on, so that no line of the table can be read as a bound on the phenomenon of The Dark Sector: Evidence Without Explanation rather than on a model of it.

The ledger of Table 136.1 should be read in two directions. Read as a bound on models, it is substantial: the simplest thermal relic scenario, in which a weak-scale particle with a weak-scale annihilation cross-section leaves the observed abundance, has been squeezed from three independent sides — scattering, production and annihilation — over most of the mass range where it was expected, and the region that motivated the whole programme is largely closed. Read as a bound on the phenomenon, it is nothing at all. The gravitational evidence assembled in The Dark Sector: Evidence Without Explanation is untouched by every line of the table, because none of these experiments tests it: they test hypotheses about what the gravitating mass is made of, and the mass remains where it was.

That distinction is the reason this chapter exists, and it is worth stating in the form the treatise uses elsewhere. An experiment excludes a model; it does not exclude an observation. The honest summary of four decades of searching is therefore the conjunction of two statements that are often heard separately and belong together: the gravitational discrepancy of The Dark Sector: Evidence Without Explanation is one of the best-established facts in observational astronomy, and there is no laboratory detection of any particle that would account for it. The searches continue, and the next decade of direct detection will run into the neutrino background of Equation (136.7), at which point the method as presently practised reaches its limit and either directional detection or a different idea is required. What is carried forward to What We Observe but Do Not Understand is not a preferred candidate but the ledger itself.

[Reserved: a consolidated figure of the exclusion curves reported in this chapter, drawn in one mass–cross-section plane with the neutrino fog and the DAMA/LIBRA allowed region marked, to be authored as TikZ from the published tabulations; and a dataset entry in the evidence library holding the digitized curves, keyed to [Aalbers:2023] [Aprile:2023] [Du:2018] [Aad:2021], so that every statement in Table 136.1 can be recomputed from the numbers as published.]