FERMILAB-PUB-20-065-A

A Dark Matter Interpretation of Excesses in Multiple Direct Detection Experiments

Noah Kurinsky,1, 2, ∗ Daniel Baxter,2, 3, † Yonatan Kahn,4, ‡ and Gordan Krnjaic1, 2, §

1Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA2Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA

3Enrico Fermi Institute, University of Chicago, Chicago, Illinois 60637, USA4University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

(Dated: February 18, 2020)

We present a novel unifying interpretation of excess event rates observed in several dark matterdirect-detection experiments that utilize single-electron threshold semiconductor detectors. Despitetheir different locations, exposures, readout techniques, detector composition, and operating depths,these experiments all observe statistically significant excess event rates of ∼ 10 Hz/kg. However,none of these persistent excesses has yet been reported as a dark matter signal because their commonspectral shapes are inconsistent with dark matter particles scattering elastically off detector nucleior electrons. We show that these results can be reconciled if the semiconductor detectors are seeing acollective inelastic process known as a plasmon. We further show that plasmon excitation could arisein two compelling dark matter scenarios, both of which can explain rates of existing signal excessesin germanium and, at least at the order of magnitude level, across several traditional WIMP searchesand single-electron threshold detectors. Both dark matter scenarios motivate a radical rethinkingof the standard interpretations of dark matter-electron scattering from recent experiments.

I. INTRODUCTION

Searches for particle dark matter (DM) with massesbelow 1 GeV have proliferated in the last decade, drivenby advances in detector technologies which have pushedheat detection thresholds below 100 eV [1, 2] and chargedetection thresholds to the single electron-hole pairlevel [3–5]. While high-mass searches have continuedto advance to larger background-free exposures, sev-eral low-mass searches, including EDELWEISS [1, 6],CDMS HVeV [3], SENSEI [4], DAMIC [7], CRESST-III [2], νCLEUS [8], XENON10 [5, 9], XENON100 [5],XENON1T [10], and Darkside50 [11] – see Table I andFig. 1 – have observed events at low energy superfi-cially consistent with either dark rate or unmodeled back-grounds. As more experiments approach these low-massregions, it is pertinent to ask whether these excess eventsall have different origins, or if a single mechanism canprovide a unifying explanation.

The standard signal interpretation of an excess in adetector with order 100 eV threshold is that of elastic nu-clear recoils from DM (as described by Lewin and Smith[12]), whereas a detector with a single-electron thresh-old is considered primarily sensitive to DM scattering onelectrons (as described in detail by Essig et al. [13] forsemiconductors, see also [14–16] for earlier work). Ashas been recently shown in Refs. [17, 18], the lines be-tween these interpretations blur in the case of inelas-tic below-threshold nuclear recoils with accompanying

∗ kurinsky@fnal.gov† dbaxter@kicp.uchicago.edu‡ yfkahn@illinois.edu§ krnjaicg@fnal.gov

above-threshold ionization, which, for a liquid noble de-tector, is actually the dominant signal component for DMmasses between approximately 100–1000 MeV. Becausethe term “inelastic” has different meanings in the theoret-ical and experimental communities, we emphasize that inthis paper, “inelastic” refers to an energy and momentumtransfer to the detector which differs from the relationsfrom two-to-two scattering. In particular, it refers to ex-citing internal modes of the detector, not internal modesof the DM.

In this paper, we postulate that existing excesses insilicon (Si), germanium (Ge), and sapphire (Al2O3) de-tectors, as well as the observation of excesses at muchsmaller rates in detectors based on disordered materials,can be persuasively interpreted as the excitation of a plas-mon resonance, a ubiquitous feature of nearly every well-ordered solid-state material [19]. The strong plasmon res-onance in highly ordered crystals, and the absence of sucha resonance in less ordered materials, provides a naturalexplanation for the large rate differences observed. In-deed, plasmon excitation is the quintessential many-bodyeffect, and provides an important example of an inelasticprocess that dominates at low momentum transfers andwhich cannot be understood in terms of two-body scat-tering and non-interacting single-particle states, as hasbeen the standard treatment of DM-electron interactions[13].

We argue that a compelling explanation of the com-mon ∼ 10 Hz/kg event rate seen in numerous chargedetection experiments, in widely varying background en-vironments, is lacking if the plasmon excitation is sourcedby Standard Model (SM) backgrounds. In contrast, theserates can be explained by a common DM origin, al-beit one with interactions that primarily excite collectivecharge modes in well-ordered crystals. We will argue that

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these interactions are easily accommodated by the mostwidely-studied DM benchmark models and have simplybeen neglected in previous studies in favor of the morefamiliar electron and nuclear recoils.

This paper is organized as follows. In Sec. II, we re-view the various excesses in low-threshold experimentsand propose a yield model which reconciles the observedionization (Ee) [6] and calorimetric (Edet) [1] spectrain EDELWEISS germanium data, the only material forwhich excesses are currently observed in both Ee andEdet data. In Sec. III, we show that interpreting theyield model as a plasmon is consistent with the similarevent rates measured in silicon detectors as well as withthe comparably lower rates measured in amorphous ma-terials like CaWO4 and liquid Xe. Additionally, we ar-gue that SM sources cannot produce plasmon excitationrates consistent with the observed excesses. In Sec. IV wepresent two illustrative DM scenarios which can explainthe observed rate in Ge, and demonstrate their consis-tency with the other excesses at the order-of-magnitudelevel. We conclude in Sec. V with a number of predic-tions and suggestions for future studies. Further detailson dark counts in semiconductor detectors, yield curves,and plasmons are provided in the Appendices.

II. REVIEW OF RECENT LOW-THRESHOLDRESULTS

We begin by considering the standard interpretationof existing excesses in roughly chronological order of ap-pearance to illustrate the difficulty in explaining themthrough conventional backgrounds. We restrict our dis-cussion to experiments running detectors with source-independent energy resolution (Edet) below 100 eV,where excesses are observed directly.

A. Nuclear Recoil Searches

In a typical nuclear recoil (NR) search, an excess showsup as a rate rising with decreasing energy down to thresh-old on top of an approximately flat background rate.Calorimetric detectors are sensitive to total energy in theform of phonons, which are the longest-lived excitationsafter the relaxation of all charge processes.

CRESST-III: In the nuclear recoil channel (detectorssensitive to Edet), the first hint of an unexpected signalat low energy came from the CRESST-III experiment [2].With a heat threshold of 30 eV in a CaWO4 detector,this was the first result to achieve significant exposure(3.6 kg ·days) below 100 eV, and their initial hypothesisfor the excess of 440 events near threshold was crystalcracking.1 This relatively low rate (3× 10−3 Hz/kg; see

1 This is based on the collaboration’s earlier result, which showedthat low-yield events were due to this phenomenon [20]. That

Table I) and the lack of other measured excesses at thetime suggested this as the least controversial explanation.

νCLEUS: Shortly thereafter, the νCLEUS experi-ment [8], an off-shoot of the CRESST collaboration tar-geting coherent neutrino scattering, published a surfaceNR search in which an excess of 30 Hz/kg above ex-pected background was observed in an Al2O3 detector.This rate, drastically higher than the excess observed inCaWO4, could also be interpreted as crystal cracking[20], though there is no a priori reason to expect sucha drastic difference in the rate of these events, nor a spe-cific model for their energy distribution.

EDELWEISS: Most recently, EDELWEISS publishedsurface results from a Ge detector with a 60 eV thresh-old [1] showing a very large low-energy excess above theexpected background. This excess steeply rises below500 eV and reaches over 100 times the measured flatbackground rate at the detector threshold of 60 eV, leav-ing no argument about its statistical significance; theyobserve on the order of 105 events. The observed rateabove threshold is orders of magnitude larger than theCRESST-III excess and extends to higher energy, makingit inconsistent with a simultaneous elastic nuclear recoilinterpretation of the two. Independently of the CRESST-III excess, the EDELWEISS excess has eluded interpreta-tion as a DM signal, since the sharp rise matches neitherthe expected spectrum of an elastic DM recoil (using thestandard velocity distribution [12]), nor secondary ion-ization induced by DM-nuclear scattering, the so-calledMigdal effect (using Ibe et al. [21] to calculate the crosssection for this process).2

The evidence from the Edet spectra is thus inconclu-sive at this point in the story: multiple experiments ob-serve excesses, none consistent with each other, withouta unifying explanation (apart from crystal cracking, anexplanation which is more a diagnosis of exclusion).

B. Electron Recoil Searches

In a typical electron recoil (ER) search, dark countsare expected to contribute a significant quantity of single-electron events, and for a given detector should producea calculable number of pile-up events with two electrons

interpretation of this excess comes largely by word of mouth inthe community; there is no official explanation that we are awareof beyond this informal speculation.

2 It should be noted that one other experiment running a calori-metric detector has also noted a low energy excess. SuperCDMS,running an 11 g Si detector, has measured an excess above athreshold of around 20 eV [22], but the rate was not published atthat time. The rate rises above background around 30 eV, andalso appears to be sharply rising. When information becomesavailable about the spectrum of this excess, it can be incorpo-rated into this analysis, but at this time it remains a qualitativelyinteresting result which we cannot interpret further.

3

Readout Type Target Resolution Exposure Threshold Excess Rate (Hz/kg) Depth Reference

Charge (Ee)

Ge 1.6 e− 80 g · d 0.5 eVee (∼1e−)a [20, 100] 1.7 km EDELWEISS [6]Si ∼0.2 e− 0.18 g ·d 1.2 eVee (<1 e−) [6, 400] 100 m SENSEI [4]Si 0.1 e− 0.5 g · d 1.2 eVee (<1 e−) [10, 2000] ∼1 m CDMS HVeV [3]Si 1.6 e− 200 g ·d 1.2 eVee (∼1e−) [1 ×10−3, 7] 2 km DAMIC [7]

Energy (Edet)Ge 18 eV 200 g ·d 60 eV > 2 ∼1 m EDELWEISS [1]

CaWO4 4.6 eV 3600 g ·d 30 eV > 3 ×10−3 1.4 km CRESST-III [2]Al2O3 3.8 eV 0.046 g ·d 20 eV > 30 ∼1 m νCLEUS [8]

Photo e−

Xe 6.7 PE (∼ 0.25 e−) 15 kg ·d 12.1 eVee (∼14 PE) [0.5, 3] ×10−4 1.4 km XENON10 [5, 9]Xe 6.2 PE (∼ 0.31 e−) 30 kg · yr ∼70 eVee (∼80 PE) > 2.2 ×10−5 1.4 km XENON100 [5]Xe < 10 PE 60 kg · yr ∼140 eVee (∼90 PE) > 1.7 ×10−6 1.4 km XENON1T [10]Ar ∼15 PE (∼ 0.5 e−) 6780 kg · d 50 eVee > 6 ×10−4 1.4 km Darkside50 [11]

a There is a very small but non-zero sensitivity to single electrons that, when the large exposure is taken into account, becomescomparable in sensitivity to the other electron recoil experiments.

TABLE I. Rates of observed low-energy excesses in experiments with single electron (<100 eV) charge (energy) resolutions.Lower bounds on the rate are given by integrating the rate above 2e− (or above threshold) whereas upper bounds are givenby assuming that the entire 1e− rate is of the same origin, despite likely containing large experiment-specific backgrounds (seeAppendix A for a discussion). Experiments sensitive to charge energy Ee are in the top section of the table, while experimentssensitive to total detector energy Edet are in the middle section. The bottom section lists experiments sensitive to secondaryradiation produced by charge interactions. The main coincidence reported here is that the excesses for ne ≥ 2 across the firstthree charge detectors (∼10 Hz/kg) demonstrate nearly identical rates for their ne ≥ 2 bins – 20, 6, and 10 Hz/kg – despitespanning ∼ 2 km of variation in overburden and almost three orders of magnitude in exposure. The total rate observed in theDAMIC detector is much lower, but the upper bound (7 Hz/kg) is intriguingly of the same order of magnitude.

(see Appendix A). An excess rate can either be inter-preted as the number of events with two electrons exceed-ing this prediction, or the overall dark rate, interpretedas a limit on a putative signal rate.

CDMS HVeV/SENSEI: The successful demonstra-tion of single-electron thresholds in Si detectors byCDMS HVeV [3] and SENSEI [4] led to a leap forward inelectron recoil sensitivity to low-mass DM. Both experi-ments observed a roughly Hz/g dark rate in the singleelectron bin, and only ran for less than a gram-day ofexposure. The relative similarity of the event rates wasstriking, but was considered to be a temporary coinci-dence that would soon be resolved as one of the exper-iments improved on their single electron dark rates. Itis notable that neither experiment has demonstrated animproved dark rate as of this writing, which may point toa dark rate which is independent of detector environmentand is not reduced with additional overburden.

EDELWEISS: Subsequently, the first electron recoilanalysis in Ge was released by EDELWEISS [6]; intrigu-ingly, the observed event rate is within an order of mag-nitude of the Si rates, despite exposures differing by afactor of 400 among the three experiments, and the factthat the EDELWEISS search was conducted with signif-icantly greater overburden. Further investigation revealsthat the event rate per unit mass in the 2–3 electron binsis remarkably similar between the three experiments, theGe rate being only roughly twice the Si rate.

DAMIC: Finally, the latest DAMIC [7] limit is strongerthan the other ER limits, as explained by the significantlyreduced dark rate in the single electron bin comparedto other silicon detectors. The ER analysis presentedby DAMIC does not have single-electron resolution and

instead assumes Poisson-distributed dark counts, fromwhich we extract a robust upper bound on the 1e− binand an inferred lower bound on the 2e− bin. The DAMICdata is most in tension with the narrative presented here,indicating a source of events in CDMS HVeV and SEN-SEI that is absent in the DAMIC detector. Regardless,we would like to emphasize that the origin of the darkcurrent in DAMIC remains unknown and could still beconsistent with some realizations of the interpretationpresented here.

XENON 10/100/1T: At face value, a DM-electronscattering interpretation in the semiconductor detectorsis strongly inconsistent with results from XENON10 [9],which sees a far smaller event rate, even accounting forthe higher threshold. We also list observed event rates atthe bottom of Table I for several noble liquid experimentswith phototube readout; of these, only XENON10 is ableto sample the single charge rate because its threshold isbelow the average energy (13.7 eV) needed to produceone quantum of charge in xenon [23]. The observed eventrates in these experiments are much lower than that ob-served by semiconductor experiments, and thus any con-sistent interpretation of those signals must explain thisdiscrepancy. A significant observation, however, is thatall of these experiments observe unexplained excesses atlow energy, as shown in Tab. I.3 Intriguingly, the ob-served excess rates per unit detector mass in XENON10,

3 We note that a recent result using phototube readout of EJ-301scintillator reports a total rate of about 14 Hz/kg [24], muchlarger than the noble liquid rates and comparable to the semi-conductor rates. However, since this experiment was the firstdemonstration of a new technique for light DM searches and was

4

FIG. 1. Integrated rate of each excess versus approximatedepth (shifted for clarity), separated by detector medium.Ranges are given according to the same criteria in Table I withthe shaded bands indicating regions most consistent with allobserved excess rates for Ge (red), Si (blue), and Xe (green),along with the muon flux from [25] in dashed black to high-light the lack of dependence on depth. For the measurementswhich only give a lower bound on excess rate, the range isdrawn off the top of the plot. Given some reasonable model forthe spectrum of the excess below threshold, an upper boundshould apply to these measurements, but such a bound is out-side of the scope of this paper. We also note that there existssome tension among the silicon measurements shown here.

XENON100, and XENON1T match to within an order ofmagnitude when the same threshold is applied. This im-plies at the very least that some event rate is scaling withmass rather than liquid-gas interface area, supporting aninterpretation as a real signal in the detector rather thana detector readout effect.

C. Determining Signal Origin

The significance of these apparent coincidences is thatthese detectors acquired data in very different environ-ments (both near surface and deep underground), eachwith distinct technologies, at dramatically different tem-peratures and electric fields, with greatly varying degreesof shielding. There is no detector effect or known back-ground that should conspire to produce the same eventrate in these detectors. Furthermore, in all four semicon-ductor detectors, a charge produced with arbitrarily lowenergy above the band edge may be detected: there isno threshold for charge detection. By contrast, the NRsearches have a nonzero energy threshold, below whichevents can be hidden depending on the energy spectrumof the signal.

run with minimal overburden, we regard this result as qualita-tively interesting and await further data from an undergroundrun.

At this point in our discussion we therefore make a boldassumption: that all the excesses in Tab. I are caused bya common source.4 We justify this assumption basedon the electron recoil results, arguing at the very least,that interesting new detector physics is being probed bythese experiments. If this is the case, then it stands toreason that any other detector should be sensitive to thesame rate of these events, and an excess above a modeledbackground can be interpreted as arising from the samesource. The measurement of a statistically significantexcess in Ge in both the Edet and Ee channels allowsus to characterize the nature of these events under thatassumption.

For the last decade, DM experiments have been reject-ing irreducible electron recoil backgrounds using the dif-fering yield between nuclear and electronic recoils, oftencalled the quenching factor, utilizing simultaneous mea-surements of energy in complementary detection channels(see e.g. Refs. [26, 27] and Appendix B). For solid-stateexperiments, the readout typically comprises both a heat(Edet) and charge or light (Ee) signal. The charge (orlight) yield for an event of energy Edet is then computedas y(Edet) = Ee/Edet, where y = 1 is characteristic ofan electron recoil event, and y < 1, following a measuredyield curve [27], can be used to select the expected nu-clear recoil band.

Taking the example of a charge detector, Ee is a de-rived parameter based on the empirical fact that, on av-erage, one electron-hole pair is produced per εeh of Edetenergy.5 In other words, an average of neh = Edet/εehelectron-hole pairs is produced for such an event, giv-ing the relation Ee = Edet = nehεeh for electron recoil.While this relation is usually used to convert measuredcharge to an equivalent energy spectrum, it can also beused to compare measured Ee and Edet spectra from thesame source of events to determine whether they are con-sistent with expectations for electron recoils, nuclear re-coils, or neither. For further details, see Appendix B.

The recent release of the high-voltage EDELWEISSDM search [6] is thus the most significant developmentto date because, taken with the previously published Edetspectrum from a similar detector, it is the first datasetin which we can compare the two spectra directly to de-termine a likely origin. This type of detector actuallymeasures a combination of Ee and Edet as we have de-fined them, producing an Ee measurement according to

Ee = Edet

[y(Edet) +

εehe ·Vdet

], (1)

4 Note here that we do not, at this stage, argue that the commonsource is the same population of dark matter scattering in eachdetector. Even if dark matter turns out not to be the explanationfor these events, the conclusions made here stand independentlyof the particular source of events.

5 εeh is a measured material property and varies material to ma-terial, and is measured such that Ee in different materials for agiven calibration source can be plotted on a consistent energyaxis.

5

where Vdet is the detector operating voltage and e is theelectron charge. This reduces to our definition of Ee onlyin the limit Vdet → ∞; the data considered here weretaken at 78 V. This gives an additional correction termof εeh/qeV ∼ 3.8× 10−2 for εeh = 3.0 eV [27].

Figure 2 shows these spectra under three scenarios forthe origin of the Edet spectrum, assuming it originatesfrom a single type of event:

1. The events are electron recoils, with y = 1 (Fig-ure 2, top left). This is clearly inconsistent becausethe black and orange curves are markedly different,and electron recoils are strongly ruled out.

2. The events are nuclear recoils, and the yield fol-lows the measured Ge nuclear recoil yield model[27] with varying low-energy behavior (Figure 2,bottom left). We consider an extrapolation of mea-sured yield to the bandgap energy (Nominal), ayield constant below 100 eV (High), or a yield thatdrops discontinuously to 0 at 100 eV (Low). Allare clearly also inconsistent with the measured Eespectrum.

3. Finally, we consider a maximally inelastic yield, inwhich every event produces a charge yield indepen-dent of the recoil energy, such that

〈Ee〉 = εeh

(λeh +

Edete ·Vdet

), (2)

where λeh is the mean number of electron-hole pairsproduced by the event (Figure 2, right). Unlikethe previous two cases, this matches the observedspectrum remarkably well both in signal shape andevent rate for λeh < 0.25 (no relative scaling is doneto force the rate to match), suggesting an inelasticinterpretation is correct for the measured Ee spec-trum rather than the two standard scenarios.

Based on this simple analysis, we conclude that aninterpretation of the EDELWEISS events based on stan-dard elastic NR or ER models is misleading, and that themost likely interpretation of these events is an inelasticinteraction, with a yield curve that increases with lowerevent energy. If this is the case, it also helps reconcile theevent rates in well-ordered crystals (which see a rate ofO(10 Hz/kg)) compared to liquids or amorphous solids,which observe much smaller event rates. An inelasticinteraction will be largely driven by condensed matterproperties unrelated to the nuclear mass or electron den-sity we use to relate different targets to each other understandard elastic assumptions.

This observation therefore rules out backgroundscaused by known low-energy interactions of photons,charged particles, and neutrons. It does not preclude theaforementioned crystal cracking events, which would notintrinsically produce light or charge. However, we noteat this point that, if crystal cracking events were trulycausing the Ee background, one would expect there to

be a dependence on applied pressure, temperature, andoperating history, which is in direct conflict with the mea-surements we have presented. We therefore either haveto accept a crystal cracking rate determined only by ma-terial, or ask what other physical process might lead toa consistent rate with an inelastic-like charge yield.

III. PLASMON INTERPRETATION

A. Plasmon Properties

Without committing to a particular source of signalevents yet, we postulate that the nature of the observedexcitations in low-threshold silicon, germanium, and sap-phire detectors is the plasmon.6 In this section we brieflyreview the properties of plasmons relevant for our anal-ysis; see Appendix C for more details.

A plasmon is a long-wavelength collective excitation ofcharges in a lattice which carries energy near the classicalplasma frequency,

Ep '√

4παneme

, (3)

where α is the fine-structure constant and ne is the elec-tron number density; in a semiconductor, ne is to beinterpreted as the density of valence electrons. Sincemost solid-state systems have roughly the same numberdensity, with interatomic spacing of a few Angstroms,Ep ∼ O(10− 100) eV across essentially all materials (seeTab. II). In particular, bulk plasmons exist and have beenobserved in silicon, germanium, and sapphire. The long-wavelength nature of the plasmon is reflected in a mo-mentum cutoff

qc ∼2π

a∼ 5 keV , (4)

where a is the lattice spacing. If a plasmon carries q > qc,it represents a charge oscillation localized to within asingle lattice site, and the plasmon will decay very rapidlyinto a single electron-hole pair in a process known asLandau damping [19]. Note that the creation of such ashort-range plasmon is inconsistent with the analysis ofthe Ge spectra in Sec. II C above, which suggests thatthe plasmon should have a dominant decay channel intophonons only. Thus we will focus on excitation of long-range plasmons only.

The plasmon is most easily observed in electron energy-loss spectroscopy (EELS), where fast (∼ 100 keV) elec-trons impinging on a material have a high probability ofdepositing energy Ep. This probability is only weaklydependent on the incident electron energy E0, scaling as

6 In this work “plasmon” will only refer to a bulk plasmon, incontrast with surface plasmons which are qualitatively differentphenomena.

6

FIG. 2. Comparison of measured Ee [6] and Edet [1] spectra from EDELWEISS in Ge with yield models for converting Edetto Ee, with error bars shown in shaded grey. For each model, the Edet spectrum is shown is converted to Ee space according tothe yield model (see Appendix B for further details). The Edet measurement is a lower bound on the differential rateonce converted to Ee by a yield model. For a model to be viable, it must predict an Ee spectrum that is lessthan or equal to the measured spectrum; the rate will increase as the Edet threshold is lowered. If the modeldoes not fit inside of the black curve, then it is an inconsistent interpretation. Only the plasmon model is ableto fit inside of the measured Ee spectrum. Top Left: Excesses observed in the Edet and Ee spectra, showing inconsistencywith an electron recoil (Ee = Edet) interpretation. Grey shading indicated 1σ uncertainty on the measured Ee spectrum.Bottom Left: Interpretation of the spectra as nuclear recoil (NR) according to the models described in the text. The blueband around the NR curves illustrates the possible shape variation in the NR models due to any possible NR yield between the3 benchmark yield models. None of the yield models match either the shape or the rate of the observed Ee spectrum. Note thatthe dot-dashed curve that lies below the Ee spectrum is the zero-yield portion of the “Low” model; the nonzero charge portionof this yield model still overpredicts the Ee spectrum. Right: Example of a yield model in which an average of λeh ∼ 0.25charges are produced per event regardless of the energy of the event (solid red, with dashed red showing the contribution fromeach integer number of charges), representative of a signal which produces very few charges independent of event energy. Alsoshown is a simpler yield model in which exactly 1 charge is produced for every event regardless of event energy (green). Asshown in the top left plot, the measured Edet spectrum has a fairly high threshold, but the events above threshold can easily fitthe rate and shape of the observed Ee spectrum simultaneously. This is a significant observation, given that the data for the Eespectrum was taken in an ultra-low background environment at 1.7 km of overburden, while the Edet spectrum was acquiredin a relatively high-background surface lab. Note that we are only able to perform this analysis for EDELWEISS because noother experiment has yet released both Ee and Edet spectral information. In order for a DM interpretation to succeed, it mustultimately be shown that common DM model parameters can fit future spectral information in other materials.

log(E0)/√E0 (see Appendix C), and is independent of

the target material except for the core electron contri-bution to the dielectric constant. At the same time, theprobe must be fast in order to deposit a small amount ofmomentum for a given energy Ep. In other words, probeswith sufficient energy E0 � Ep and sufficient velocity willstrongly prefer to deposit energy Ep, regardless of theirinitial energy, at similar rates across diverse materials.This behavior is typical of other resonances encounteredin nuclear physics or electrical engineering; in a sense,the plasmon acts as a band-pass filter for Edet.

The lineshape of the plasmon near the peak is welldescribed by a Lorentzian [30], where the finite widthΓ parameterizes the decay of the plasmon into phononsand/or electron/hole pairs, which are the long-lived ex-citations in the detector. We note that the plasmon isinherently a many-body excitation, and cannot be de-

scribed in terms of non-interacting single-particle states,such as band structure wavefunctions derived using den-sity functional theory. Moreover, typical values of Γ/Epfor semiconductors are 0.2 [30], which is larger than Γ/Mfor most strongly-decaying hadronic resonances and sug-gests that the plasmon couplings which govern its de-cay are large or even nonperturbative. The simple yieldmodel for the Ge spectra suggests that the plasmon musthave a ∼75% branching fraction to phonons only. Toour knowledge, the branching fractions of the plasmon tophonons or electron/hole pairs is unknown, but in prin-ciple these could be determined from a suitably modi-fied EELS experiment with both calorimetric and chargereadout.

Based on this interpretation, assuming some incidentflux of particles is dominantly exciting the plasmon overother elastic or inelastic excitations, detectors with Edet

7

thresholds approaching Ep from above should see a sharprise in events as the threshold is lowered; this qualita-tively explains the results from the silicon, germanium,and sapphire experiments, as well as the null resultsfrom previous experiments with thresholds well aboveEp. Moreover, the plasmon in germanium has a signif-icant high-energy tail and double-peaked structure re-sulting from contributions from the 3d shell [30], furtherexplaining the onset of events in EDELWEISS despite athreshold of 60 eV ∼ 4Ep. By contrast, the plasmon insilicon lacks a corresponding tail, explaining the lack ofa signal excess in higher-threshold analyses of DAMIC[31] and CDMSlite [32] data. Furthermore, materialswithout long-range order such as liquid xenon and, to alesser extent, CaWO4 do not have a pronounced plasmonpeak, explaining the lower event rates from XENON10and CRESST.

B. Plasmons from Known Particles?

An interpretation of the plasmon excitation as sourcedby SM particles or fields is extremely difficult.

• Photons and electromagnetic fields: Trans-verse UV and soft X-ray photons cannot source thelongitudinal plasmon oscillation, and static electricfields cannot source oscillating charges.

• Charged SM Particles: The inelastic mean freepath for charged particles such as electrons ormuons, or for x-rays, is on the order of tens ofnm, so these particles would be expected to un-dergo multiple scattering and deposit many multi-ples of Ep as they traversed a detector (all of whichare much thicker than nm for the experiments weconsider), which would lead to many events abovethreshold contrary to what was observed. A singleenergy deposit under 100 eV is only consistent witha particle of mean free path much larger than thedetector thickness; if charged, this particle wouldhave to have electric charge much less than e.

Material Plasmon Energy Ep (eV) Width Γ (eV)Si 16.6 3.25Ge 16.1 3.65

Al2O3 24.0 [28] ∼ 5GaAs 16.0 4.0

Xe (Solid) 14–15 [29] ∼ 4Ar (Solid) 19–21 [29] ∼ 5CaWO4 Unknown

TABLE II. Plasmon energies in various materials. Crystal val-ues taken from Ref [30] unless otherwise referenced. We wereunable to find measurements of plasmon features in CaWO4,and expect that it has a much weaker plasmon resonance thanthe other crystals considered here. It is significant to note thatthe solid forms of the noble elements show strong resonancefeatures; the liquid forms do not.

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phonons (momentum)<latexit sha1_base64="rxcP5g9JiluFaLF30bt+DUdL0vM=">AAACBHicdVBLS0JBGJ1rL7OX1dLNkAQGIXckUndSm5YG+QAVmTuOOjiPy8zcQC4u2vRX2rQoom0/ol3/prlqUFEHBg7nfIdvvhOEnBnr+x9eamV1bX0jvZnZ2t7Z3cvuHzSNijShDaK40u0AG8qZpA3LLKftUFMsAk5bweQy8Vu3VBum5I2dhrQn8EiyISPYOqmfzcVdLWA4VlJJA7unsCCUoNJG4mTWz+b9ou/7CCGYEFQ+9x2pVislVIEosRzyYIl6P/veHSgSJXnCsTEd5Ie2F2NtGeF0lulGhoaYTPCIdhyVWFDTi+dHzOCxUwZwqLR70sK5+j0RY2HMVARuUmA7Nr+9RPzL60R2WOnFTIaRpZIsFg0jDq2CSSNwwDQllk8dwUQz91dIxlhjYl1vGVfC16Xwf9IsFZHj12f52sWyjjTIgSNQAAiUQQ1cgTpoAALuwAN4As/evffovXivi9GUt8wcgh/w3j4BNsiX0g==</latexit><latexit sha1_base64="rxcP5g9JiluFaLF30bt+DUdL0vM=">AAACBHicdVBLS0JBGJ1rL7OX1dLNkAQGIXckUndSm5YG+QAVmTuOOjiPy8zcQC4u2vRX2rQoom0/ol3/prlqUFEHBg7nfIdvvhOEnBnr+x9eamV1bX0jvZnZ2t7Z3cvuHzSNijShDaK40u0AG8qZpA3LLKftUFMsAk5bweQy8Vu3VBum5I2dhrQn8EiyISPYOqmfzcVdLWA4VlJJA7unsCCUoNJG4mTWz+b9ou/7CCGYEFQ+9x2pVislVIEosRzyYIl6P/veHSgSJXnCsTEd5Ie2F2NtGeF0lulGhoaYTPCIdhyVWFDTi+dHzOCxUwZwqLR70sK5+j0RY2HMVARuUmA7Nr+9RPzL60R2WOnFTIaRpZIsFg0jDq2CSSNwwDQllk8dwUQz91dIxlhjYl1vGVfC16Xwf9IsFZHj12f52sWyjjTIgSNQAAiUQQ1cgTpoAALuwAN4As/evffovXivi9GUt8wcgh/w3j4BNsiX0g==</latexit><latexit sha1_base64="rxcP5g9JiluFaLF30bt+DUdL0vM=">AAACBHicdVBLS0JBGJ1rL7OX1dLNkAQGIXckUndSm5YG+QAVmTuOOjiPy8zcQC4u2vRX2rQoom0/ol3/prlqUFEHBg7nfIdvvhOEnBnr+x9eamV1bX0jvZnZ2t7Z3cvuHzSNijShDaK40u0AG8qZpA3LLKftUFMsAk5bweQy8Vu3VBum5I2dhrQn8EiyISPYOqmfzcVdLWA4VlJJA7unsCCUoNJG4mTWz+b9ou/7CCGYEFQ+9x2pVislVIEosRzyYIl6P/veHSgSJXnCsTEd5Ie2F2NtGeF0lulGhoaYTPCIdhyVWFDTi+dHzOCxUwZwqLR70sK5+j0RY2HMVARuUmA7Nr+9RPzL60R2WOnFTIaRpZIsFg0jDq2CSSNwwDQllk8dwUQz91dIxlhjYl1vGVfC16Xwf9IsFZHj12f52sWyjjTIgSNQAAiUQQ1cgTpoAALuwAN4As/evffovXivi9GUt8wcgh/w3j4BNsiX0g==</latexit><latexit sha1_base64="rxcP5g9JiluFaLF30bt+DUdL0vM=">AAACBHicdVBLS0JBGJ1rL7OX1dLNkAQGIXckUndSm5YG+QAVmTuOOjiPy8zcQC4u2vRX2rQoom0/ol3/prlqUFEHBg7nfIdvvhOEnBnr+x9eamV1bX0jvZnZ2t7Z3cvuHzSNijShDaK40u0AG8qZpA3LLKftUFMsAk5bweQy8Vu3VBum5I2dhrQn8EiyISPYOqmfzcVdLWA4VlJJA7unsCCUoNJG4mTWz+b9ou/7CCGYEFQ+9x2pVislVIEosRzyYIl6P/veHSgSJXnCsTEd5Ie2F2NtGeF0lulGhoaYTPCIdhyVWFDTi+dHzOCxUwZwqLR70sK5+j0RY2HMVARuUmA7Nr+9RPzL60R2WOnFTIaRpZIsFg0jDq2CSSNwwDQllk8dwUQz91dIxlhjYl1vGVfC16Xwf9IsFZHj12f52sWyjjTIgSNQAAiUQQ1cgTpoAALuwAN4As/evffovXivi9GUt8wcgh/w3j4BNsiX0g==</latexit>

FIG. 3. Cartoon of indirect plasmon excitation through ahard scattering event, where the imparted momentum is dom-inantly carried by multiple phonons, while the imparted en-ergy is carried by the low-momentum plasmon.

• Neutrons: In principle, it is possible that hardscattering events induced by neutrons may createsecondary plasmon excitations; indeed, we specu-late on this possibility in Sec. IV A below in thecontext of hard DM-nucleus scattering. However,one would have to explain why the neutron flux isthe same at all the relevant experiments listed inTable I regardless of the shielding, detector envi-ronment, detector construction, and exposure.

• Neutrinos: Astrophysical neutrinos can, in prin-ciple, undergo neutral-current scattering with aseminconductor nucleus whose recoil excites a plas-mon independently of detector overburden. How-ever, the known solar and atmospheric fluxes (as-suming SM weak interactions) cannot account forrates of the observed magnitude [33]. Althoughit may be possible for an unknown populationof very low-energy neutrinos to excite plasmonsthrough non-standard (larger than electroweak) in-teractions, exploring this scenario is beyond thescope of the present work.

We conclude that none of these options offers a satis-factory explanation for the observed excesses.

IV. DARK MATTER SCENARIOS FORPLASMON EXCITATION

Having excluded the possibility that the plasmon couldarise from SM particles, we now make a further leap andconsider the hypothesis that DM could account for theseplasmon excitations. If a DM particle with mass mχ andincident velocity v deposits energy E and momentum qin a detector, energy conservation requires

E = q ·v − q2

2mχ, (5)

which implies

q ≥ E

v, (6)

8

FIG. 4. Left: If DM-SM scattering is assumed to involve elastic interactions off detector particles, there are many reportedbounds on σp assuming equal DM couplings to electrons and protons. Note that all of the bounds in this plot are based ontranslating electron recoil searches according to σp = σeµ

2χp/µ

2χe where µij is the ij reduced mass. However every experiment

shown in dashed contours observes an excess of events as shown in Table I. These excesses are not currently reported as DMsignals because the spectral shape for elastic DM scattering does not provide a good fit to these data (see the left panel ofFig. 2 for examples of such shape mismatch). Consequently, these results are reported as limits, not as evidence of a DMsignal. Right: Parameter space for which an DM-proton interaction with an a secondary plasmon excitation probability Pcan accommodate a 20 Hz/kg rate in a Ge target with Edet up to 100 eV (shaded pink). Note that the CRESST-II [34] boundsare based on a nuclear recoil search, so these constraints still apply; the additional parameter space covered by CRESST-III isalso the region in which an excess was seen. Although we are arguing that, in this scenario, the constraints from the left paneldo not apply straightforwardly due to their observed excesses, these bounds are also model-dependent and inapplicable if theDM interacts only with nucleons since the low-threshold bounds assume electron recoil signals. Furthermore, a leptobhobicDM-nucleus interaction could excite plasmons without ever directly inducing electron recoils, so many of the dashed regions inthe left panel would not even apply in principle. The one exception is the limit from EJ-301 [35] which does observe an excessrate but sets conservative limits based only on the total single-photoelectron rate rather than a fit to an expected spectrum.Also shown are exclusions from CMB scattering [36] and Milky Way satellites [37].

which is saturated in the limit of forward scattering andm→∞. Taking E = Ep = 16 eV for the typical plasmonenergy in Ge, we find that to excite the plasmon directly(i.e. q < qc, see Eq. (4)) we must have

v & 10−2 (direct plasmon excitation). (7)

Since this exceeds Galactic escape velocity in the Earthframe [45], gravitationally-bound DM with v ∼ 10−3 can-not directly excite the plasmon. However, we identify twoqualitatively distinct mechanisms by which DM (or a sub-component) can still account for the observed excesses:

• Scenario 1, Secondary Plasmon: In analogywith the Migdal effect [21, 46], if halo DM with thestandard Maxwellian velocity distribution peakedat v ∼ 10−3 first scatters off a target nucleus, theinteraction can transfer a majority of the momen-tum to phonons, while imparting most of the de-posited energy to the plasmon which carries q < qc

(see Fig. 3).7 The plasmon can then decay tophonons and electron/hole pairs. In this scenario,the signal rates scale as Z2/A where Z and A arerespectively the atomic and mass numbers of thetarget material.

• Scenario 2, Fast DM Sub-Component: Al-though the majority of halo DM in our Galaxy mustsatisfy v . 10−3 to account for observed rotationcurves, it is possible that a small fraction f � 1of the local DM density is accelerated to speedsv & 10−2 above Galactic escape velocity (e.g by

7 Note that the scale of the momentum transfers we will consider,O(15 keV) from Eq. (6), is precisely in the regime between single-phonon excitation and direct nuclear scattering, where directmulti-phonon production is expected to dominate [47]. Indeed,the defect plus vacancy energy of bulk Ge is at least 15 eV [48], sobelow this energy, an elastic nuclear recoil is not even an on-shellstate, and the non-electronic energy must appear in the form ofphonons – see Appendix B.

9

FIG. 5. Same parameter space as Fig. 4 (right panel) butinterpreted in the context of dark photon mediated scattering(see Scenario 1, Sec. IV A). The blue curve represents theparameter space for which direct χχ → SM SM annihilationaccounts for the full DM abundance; parameter space abovethis curve can be viably accommodated if the DM populationis particle-antiparticle asymmetric. The gray shaded regionbelow the bounds shown in Fig. 5 represents accelerator basedconstraints for a representative choice of parameters mA′ =3mχ and αD = 0.5. This envelope contains bounds fromthe LSND [38], MiniBooNE [39], and E137 beam dumps [40],BaBar [41], and NA64 [42]. Also shown are projections for theLDMX missing momentum experiment [43] and the B-factoryBelle-II [44]. As in Fig 4 (right), the pink shaded region iscompatible with a 20 Hz/kg plasmon excitation rate in Gefor a virialized DM halo population; the region to the left canalso accommodate this rate if a DM subcomponent is fasterthan the Galactic escape velocity.

solar reflection [49]).8 Unlike in Scenario 1 above,here the rate scales inversely with the target’s massdensity since the plasmon is excited directly with-out the DM having to first undergo nuclear scat-tering.

These scenarios are complementary: Scenario 1 requiresno non-standard DM ingredients but features large the-oretical uncertainty in the plasmon-phonon coupling; bycontrast, Scenario 2 has no theoretical uncertainty in thedirect plasmon excitation probability, which is in one-to-one correspondence with an EELS measurement, butrequires an explanation for the fast DM sub-component.In both scenarios, a plasmon with a large branching ra-tio to phonons only can accommodate the excess spectral

8 Other possibilities for achieving a fast sub-component of darksector particles include boosted DM [50–52], cosmic ray up-scattering [53–56], direct production in supernovae [57, 58], andacceleration from supernova remnants [59].

shape and match the total observed rate in the EDEL-WEISS 78 V run for 2 or more charges, ∼ 20 Hz/kg [6].

Theoretically, both of these scenarios can be realizedwithin a popular framework for DM below the GeV scale.Let χ be a DM candidate particle of mass mχ coupledto a new spin-1 U(1) gauge boson A′, which kineticallymixes with the SM photon. Here χ can be a scalar ora fermion and such an interaction has long been a stan-dard benchmark for sub-GeV DM studies [60–62]. In themass eigenbasis, the Lagrangian for this scenario can bewritten

L ⊃ −m2A′

2A′µA

′µ +A′µ(κeJµEM + gDJµD), (8)

where JEM is the SM electromagnetic current, κ � 1is a small kinetic mixing parameter, gD is the DM-A′

coupling constant and JµD is the DM current

JµD =

{i(χ∗∂µχ− χ∂µχ∗) Scalar

χγµχ Fermion,(9)

which are analogous to scalar and fermionic versions of“dark electromagnetism” with a massive dark photon.

In the limit where the dark photon is massless, mA′ →0, the DM effectively acquires an electric millicharge κgD;this interpretation holds as long as mA′ � q where qis the typical momentum transfer in the process underconsideration. In the opposite limit, where mA′ � q,the DM effectively has contact interactions with chargedparticles, including electrons and nuclei.

A. Scenario 1: Secondary Plasmon Excitationthrough Hard Inelastic Scattering

One way to interpret the origin of this plasmon res-onance signal is through the inelastic nuclear scatter-ing of 100 MeV-scale DM through a contact interaction(mA′ � q). This is similar to recent calculations of theMigdal effect [21], except that the existing literature pre-senting the formalism for the Migdal effect relies on anisolated atom approximation and cannot be reliably ex-tended to semiconductors.

In light of this uncertainty, we factorize the DM-induced ionization rate R in a semiconductor into a spin-averaged single-proton cross section

σp =16πκ2ααDµ

2χp

m4A′

, (10)

where αD ≡ g2D/4π, times an energy/momentum-averaged plasmon excitation probability P ≤ 1 per in-dividual nuclear scatter, such that the total plasmon ex-citation rate is

R = NAρχmχ

Z2

AσpP, (11)

where ρχ = 0.4 GeV cm−3 is the local DM density [63],NA is Avogadro’s number, A is the atomic mass of Ge,

10

and Z is its atomic number. Note that P is a propertyof the detector material, and would be expected to varysomewhat between Si and Ge, for example, but shouldbe similar across all single-crystal detectors of the samematerial. Our assumption in this scenario is that, despitethe fact that the DM transfers momentum q � qc tothe material, a long-wavelength plasmon with q < qc isexcited, with phonons absorbing the remainder of themomentum. The factor of Z2 in the rate arises becausethe maximum momentum transfer for sub-GeV DM isqmax = 2mχv ∼ 2 MeV, which is not large enough toprobe the nuclear structure, so the interaction is coherentover all the protons in the nucleus (i.e. the nuclear formfactor is unity). For Ge, we find

R ∼ 20 Hz

kg

( P0.1

)(σp

6× 10−34 cm2

)(100 MeV

mχ

). (12)

In Fig. 4 (left), we show the constraints on the relevantparameter space when limits from the various experi-ments shown in Tab. I are interpreted in terms of tra-ditional electron recoil or nuclear recoil models. Thesebounds do not apply in the indirect plasmon excitationmodel we a considering here; in Fig. 4 (right) we showonly the direct nuclear recoil bounds from CRESST-IIand EJ-301 which survive. Indeed, we are proposing thatmany of the “bounds,” which correspond to actual low-energy excesses in the data, are in fact signals when inter-preted in the plasmon model. We shade in red the regionwhere DM at the Galactic escape velocity has kinetic en-ergy 1

2mχv2 > 100 eV in order for gravitationally-bound

DM to explain the observed Edet spectrum in Ge. InFig. ?? we show the same parameter space including avariety of accelerator-based bounds and projections forfuture searches. If this scenario is correct, various fixed-target and B-factory follow-up measurements will be sen-sitive to the full parameter space responsible for the lowthreshold direct detection excesses.

For each value of mχ, there is a specific value ofσp which would generate the observed cosmological DMabundance through thermal freeze-out, or in the case ofasymmetric dark matter, provides a lower bound on thecross section required to annihilate away the symmetriccomponent. This line is shown in blue in Fig. 5 [43, 64].We see that for P of order 1, and a DM mass of 30 MeV,the multiple excesses described in Table I can be con-sistently explained with a dark photon interaction thatalso sets the relic density to SM particles in the earlyuniverse.

Although a large value of P ∼ O(1) is somewhat sur-prising, it is not unreasonable given that the consistencyof this explanation requires a large branching fraction ofthe plasmon to phonons. As we describe in Appendix C,the plasmon is best understood as a nonperturbative ef-fect with relative width even exceeding that of QCD res-onances, so large couplings are expected. Intriguingly,the interaction we are proposing is maximally coherentin the sense that it benefits from coherence over the nu-cleus in the hard scattering event, followed by coherence

over the the lattice sites in the excitation of the plasmon,such that there is no suppression by factors of momen-tum anywhere in the rate. The conventional wisdom isthat such coherent events are always suppressed by ei-ther a small momentum or small phase space [65], butsuch arguments typically involve a single-particle pic-ture, and it is plausible that this intuition is modifiedby many-body effects in condensed matter systems. Toour knowledge, such a secondary excitation process hasnot been previously considered in the condensed matterliterature, though the plasmon-phonon coupling has beencomputed for polar semiconductors [66]. We make somesuggestions in Sec. V for neutron experiments which mayconfirm this effect. Regardless, the near-perfect matchbetween the observed rate and the expected cross sec-tion for thermally-produced DM suggests that this pro-cess should be taken seriously as a signal candidate.

Another example of an inelastic detector signal wasrecently explored by Ibe et al. [21] in the context ofDM scattering from isolated atoms, known as the Migdaleffect. In the standard Migdal effect, orthogonality ofinitial- and final-state wavefunctions makes the rate pro-portional to (me/mN )2q2, as coherence over the final-state wavefunctions is lost [17, 18, 21]. This could explainwhy the event rate per unit mass in noble liquid detec-tors is smaller than in semiconductors, as those amor-phous materials lack a pronounced long-range plasmonmode. Furthermore, the parameter space which lies nearthe thermal relic target is precisely the DM mass rangein which the Migdal and direct electron scattering ratesare comparable when scattering through a heavy medi-ator [17, 18]; we propose that the apparent strength ofthe recent XENON1T limit in this mass range is due tothe fact that electron recoil and Migdal analyses wereperformed separately. In fact, in the dark photon model,both processes will be present, giving a markedly differ-ent spectral shape to the signal. We emphasize, however,that the isolated atom approximations made in the stan-dard treatment of the Migdal effect fail to take into ac-count long-range interactions in the valence shell that areknown to lead to nontrivial collective behavior in solid-state materials; we argue that the dominant signal inthe 10–100 MeV mass range in semiconductors is not theMigdal effect, but instead plasmon excitation.9

B. Scenario 2: Direct Plasmon Excitation Througha Light Mediator

An alternative model is the limit where mA′ � q, suchthat DM is effectively millicharged. The EELS experi-ments which characterized plasmons with electron probes

9 Ref. [18] made strides toward addressing this problem, thoughtheir analysis was restricted to non-interacting single-particlewavefunctions, which cannot describe the plasmon.

11

FIG. 6. Left: Lineshape S(E) of the germanium plasmon from EELS measurements [30] (blue), normalized to match the peakof the best-fit Frohlich model (red) described in Appendix C. Right: Parameter space for a subcomponent of semi-relativisticDM interacting through a light mediator, expressed in terms of the effective millicharge κgD/e. The shaded pink region showsthe preferred parameter space for the 20 Hz/kg plasmon signal in EDELWEISS, assuming a fraction f of the local DM densityhas velocity v = 0.1, for f ranging from 10−5 to 10−1. Exclusions from SN1987A [57], CMB [67], SLAC [68], XENON1T [10],and stellar cooling [69] are shown in grey.

can thus be used to determine the event rate for DM-induced plasmons. As mentioned above, DM cannot ex-cite the plasmon directly unless v & 10−2. Given a veloc-ity distribution with support for v & 10−2, the plasmonexcitation rate per unit detector mass can be derivedfrom the analogous results for EELS. The plasmon exci-tation probability per incident particle per unit time is[30]

P (ω) =α

π2

∫d3q

1

q2Im

{ −1

ε(ω,q)

}

× δ(ω − q ·v +

q2

2mχ

), (13)

where ε(q, ω) is the frequency- and momentum-dependent dielectric function of the target. The singledelta function enforces energy conservation, but there isno corresponding delta function for momentum conser-vation; it is in this sense that we refer to the plasmonas an inelastic excitation. The plasmon contribution isextracted by considering the region of small momentumtransfer and approximating ε(q, ω) ≈ ε(ω), the imagi-nary part of which gives the plasmon lineshape S(ω) (seeAppendix C for details). By taking α → κ2αD, relabel-ing ω → E, multiplying by the number of DM particlesin the detector volume, and integrating over the veloc-ity distribution and momentum transfer, we obtain theDM-plasmon spectrum per unit detector mass:

dR

dE=

fρχmχρT

2κ2αDπ

S(E)

∫ qc

0

dq

qη(vmin(q, E)) , (14)

where ρχ is the DM mass density, ρT is the target massdensity, η(v) is the mean inverse DM speed, and

vmin(q, E) =E

q+

q

2mχ, (15)

is the minimum χ speed required to deposit energy E.Note that we have cut off the q integral at the maximumvalue of qc ∼ 2π/a ∼ 5 keV compatible with sourcing along-range plasmon.

The plasmon lineshape S(E) is taken from Ref. [30]and shown in Fig. 6 (left). Following the analysis ofRef. [30] for silicon, we normalize S(E) to the Frohlichmodel of a single damped harmonic oscillator [70] withcore electron dielectric constant εc = 1 (see Appendix Cfor further details). To understand the order of mag-nitude of the rate, we can use the fact that if η(vmin) isapproximately independent of E, and that in the Frohlichmodel, S(E) is Lorentzian so

∫dR

dEdE ∝

∫S(E) dE ≈ 3

2Ep (16)

(see Appendix C). This underestimates the true rateslightly because it neglects the long high-energy tail ofthe germanium plasmon. For a monochromatic velocitydistribution at velocity v such that mχv

2 > Ep, this givesan approximate total rate

R ≈ 3

π

fρχmχρT v

κ2αDEp log

(mχv

2

Ep

). (17)

In Fig. 6 (right) the gray shaded pink region marksparameter space for which the χ-induced direct plas-mon excitation yields a 20 Hz/kg event rate at EDEL-WEISS, for abundance fractions with v = 0.1 ranging

12

from f = 10−5 to 10−1 . The shaded regions of this figurerepresent astrophysical bounds on millicharged particles,including constraints on χχ emission in red giants [71]and supernovae [57, 58]. The curve labeled “Freeze-In”represents the parameter space for which the dominant,slower 1 − f fraction of the χ population can be pro-duced out of equilibrium through the kinetic mixing in-teraction [14, 67, 72]. The effective DM millicharges κgDwhich match the observed rate for f < 1 are larger thanthe millicharges required to generate the observed relicabundance from freeze-in, so for those parameters, someinteraction within the dark sector would be required todeplete the DM relic abundance [73, 74].

Finally, as in Scenario 1, we would expect to see anonzero rate from the Migdal effect or electron scatteringin noble liquid detectors, but one which is smaller than insemiconductors. The excitation rate for a generic systemis proportional to |〈Ψf |eiq ·x|Ψi〉|2, where Ψi and Ψf arethe full many-body electronic wavefunctions. In a solid-state system, these many-body contributions are incorpo-rated in the dielectric function ε(q, ω) (see Appendix C),and the plasmon represents a many-body state with avery large dipole matrix element |〈|Ψf |x|Ψi〉|2, becausethe wavefunctions have support over many lattice spac-ings. By contrast, in noble liquids the final-state wave-function contains an ionized electron, which will not havelarge overlap with the initial state except in the vicinity ofthe nucleus. However, the large exposure and the persis-tent low-energy excesses in xenon and argon experimentslisted in Tab. I may still be consistent with a combina-tion of DM-electron scattering and the Migdal effect, asin Model 1 above. A more quantitative analysis wouldalso require including the fast DM fraction in the velocitydistribution, which we leave for future work.

Regardless of the particular model for the DM veloc-ity distribution with which we choose to compute therate, we note that plasmon excitation is a striking coun-terexample to the conventional wisdom that inelastic pro-cesses like the Migdal effect dominate at large momen-tum transfers [17, 18]. While this may be true for iso-lated atoms, the long-range Coulomb force creates collec-tive excitations which are enhanced at small momentumtransfer in semiconductors. This example also illustratesthe importance of many-body processes which accountfor electron-electron interactions, as opposed to scatter-ing rates computed using non-interacting single-particlestates.

V. CONCLUSIONS

In this paper we have argued that multiple excesses inlow-threshold dark matter experiments may be explainedby an inelastic excitation, which can be consistently in-terpreted as a plasmon. We thus predict the following:

1. The ratio of Edet to Ee on an event-by-event basismeasures the branching fraction of the plasmon tophonons and electron/hole pairs, respectively. The

statistical moments of this ratio will be a functionof energy, but they should be the same for all eventswith the same Ee in a given detector material. Toour knowledge this branching fraction has not beencalculated in the literature; if our interpretation iscorrect, such a computation would be highly rele-vant to DM experiments.

2. With sufficient resolution (on the order of 1 eV,less than the typical width of the plasmon peakin Si and Ge) and a threshold below the expectedplasmon energy, the Edet spectrum should show arelative maximum at Edet = Ep. The Ee spectrummay not show a peak above the 1e− bin for chargeonly detectors. For the CDMS and EDELWEISSdetectors run in Ee mode, a significant shift awayfrom the one and two electron peaks due to thelarge excess phonon energy should be observed.

3. A similar spectrum should be seen in sapphire,where Ep = 24 eV, once an energy resolution be-low 5 eV (approximately the width of the plasmonpeak) is achieved.

The striking independence of these excesses with re-spect to detector composition, location, and environmentsuggests a common origin. Given the difficulty of explain-ing plasmon excitation in terms of SM backgrounds, wesuggest an interpretation in terms of DM-induced exci-tations. We have proposed two scenarios, where DM in-teracts through a heavy or light mediator, producing sec-ondary and primary plasmon excitations, respectively. Inthe context of these models, we make the following pre-dictions:

1. Explaining these large signal rates with the dom-inant halo DM population (as in Scenario 1 ofSec.IV A) suggests a single-nucleon contact inter-action satisfying σp & 10−35 cm2 and a DM massscale 30 − 200 MeV. Such large SM couplings andlight DM masses imply large DM production ratesat terrestrial accelerator searches. In particular, ifthe underlying interaction is due to an invisibly-decaying dark photon (mA′ > 2mχ), some combi-nation of Belle-II [44], BDX [75], SHiP [76], NA62[77], NA64 [42], DUNE [78], and LDMX [79] amongothers will discover or falsify this scenario (see also[62] for a broad list of follow-up searches at accel-erators).

2. There should be a reduced annual modulation sig-nal in both Scenarios 1 and 2 (Secs. IV A and IV B)compared to the standard expectation from WIMPDM, since DM dominantly deposits energy of or-der Ep inside the detector, regardless of its initialenergy. In particular, the spectrum itself shouldnot show a significant annual modulation, althoughthe overall rate should change due to the modu-lating DM flux. However, this prediction shouldbe interpreted with great care. For instance, in

13

Scenario 2, the signal arises from a boosted sub-population of the cosmic DM, which could arisefrom solar reflection [49] and would not exhibit theexpected annual modulation signature at all. Fur-thermore, even if the source population has a con-ventional Maxwellian velocity distribution, thereare subtleties in interpreting modulation results onsub-annual timescales as the phase of this modula-tion is sensitive to solar gravitational focusing ef-fects [16, 80]. Analyzing this effect is beyond thescope of the present work, but may become im-portant in follow-up studies. That said, given theenormous total event rates which have so far beenobserved, some annual modulation signal should bevisible at high statistical significance with enoughexposure.

3. In an anisotropic material where plasmon-phononinteractions or the dielectric function are direc-tional, a daily modulation may be seen. In thedirect excitation model, there may also be strongdirectional signals in low-threshold experimentssearching for the dominant 1− f cold DM fraction[47, 81–84].

4. The indirect plasmon hypothesis implies a largeplasmon-phonon coupling, which may be seen incondensed matter experiments involving neutronenergy-loss spectroscopy, for example.

5. GaAs, a polar material with a direct gap andEp ∼ 15 eV, should have even larger plasmon-phonon couplings than Si or Ge and a markedlydifferent branching ratio of the plasmon to phononscompared to Si and Ge, which have indirect gaps.This would result in a larger total signal rate inScenario 1, with a different relationship betweenEe and Edet in both Scenarios 1 and 2. Consider-able attention has already been devoted to GaAsas a candidate for sub-MeV DM detection [85, 86],and this signals discussed here further motivate in-vestigation of this material.

A pressing question arising from this analysis is howone might discover or falsify a DM signal which domi-nantly produces plasmons. In the near future, we be-lieve the most promising line of inquiry would be to op-erate a detector similar to the EDELWEISS or CDMSHVeV detectors in both calorimetric or charge mode ina low-background environment. To date, no experimenthas published results from a detector operating in bothmodes in an experimental site with known backgrounds;such an experiment could significantly strengthen thecase for an inelastic interaction. Furthermore, a signalof this magnitude presents the unique challenge that it issignificantly higher than ambient backgrounds. It is thusimportant to expose the detector to an elevated back-ground to verify that the observed excess remains un-changed and does not correlate with photon or neutronrates. The gold standard of proof beyond these tests,

likely at least a few years down the road, is a calori-metric measurement with sufficient resolution and a lowenough threshold to detect and resolve the plasmon peak.In addition, it should be demonstrated that rates in dif-ferent materials should scale according to the strength ofthe plasmon interaction, and more detailed calculationsare needed to reinforce our assertion that the spectrumshould closely resemble the plasmon lineshape (which, westress, can be measured directly with EELS).

We emphasize that it is imperative to try to under-stand the precise rate and spectra of expected signals inthe context of liquid noble and scintillation detectors, andto continue to operate these detectors at lower thresholdsto gain better insight into the shape of the observed spec-tra. Upcoming experiments will continue to shed lighton the source of dark counts observed by xenon and ar-gon experiments, for example, which will contribute toa better understanding of whether a similarly suggestiverate exists in these experiments. In parallel, more workto measure dynamic structure factors for these materialswould elucidate the nature of low-energy inelastic inter-actions that, at this point, are still not well character-ized. An intriguing possibility suggested by Table II isthat solid xenon or argon detectors may allow one to testthe theory that the event rate is strongly enhanced bythe presence of a plasmonic resonance at low energy.

A number of our predictions are nontrivial and repre-sent qualitatively new effects and interpretations of darkmatter interactions in condensed matter detectors. Weeagerly look forward to the results of upcoming experi-ments to either support or refute these conclusions. Ineither case, we expect that the dark matter communitywill benefit greatly with the increased interactions withthe condensed matter community which may be stimu-lated by this work.

ACKNOWLEDGMENTS

We gratefully acknowledge Peter Abbamonte for dis-cussions and for pointing out that plasmons are the likelymechanism by which light dark matter couples to chargedparticles in condensed matter systems, and Lucas Wag-ner for many enlightening conversations which helped tobridge the gap between the languages of condensed mat-ter physics and high-energy physics. In parallel, we wantto acknowledge Alan Robinson for pointing out that plas-mon interactions should impact low-energy reconstruc-tion of electron recoils. None of the observations in thispaper would be possible without the results we cite, butalso without private conversations with the collabora-tions responsible which led to early discussions on thevarious possible background origins of observed excesses.We thus want to acknowledge (in alphabetical order) DanBauer, Karl Berggren, Julien Billard, Alvaro Chavarria,Juan Collar, Rouven Essig, Enectali Figueroa-Feliciano,Jules Gascon, Yonit Hochberg, Ziqing Hong, Sam McDer-mott, Paolo Privitera, Matt Pyle, Karthik Ramanathan,

14

Wolfgang Rau, Florian Reindl, Javier Tiffenberg, Belinavon Krosigk, and Tien-Tien Yu. We are especially grate-ful to Nikita Blinov, Torben Ferber, Jeff Filippini, PaddyFox, Roni Harnik, Dan Hooper, Lauren Hsu, Sam McDer-mott, Harikrishnan Ramani, Albert Stebbins, and Belinavon Krosigk for their feedback on early drafts of this pa-per. We thank the Gordon and Betty Moore Foundationand the American Physical Society for the support of the“New Directions in Light Dark Matter” workshop wherethe key idea for this work was conceived. Fermilab is op-erated by Fermi Research Alliance, LLC, under ContractNo. DE-AC02-07CH11359 with the US Department ofEnergy. This work was supported in part by the KavliInstitute for Cosmological Physics at the University ofChicago through an endowment from the Kavli Founda-tion and its founder Fred Kavli.

Appendix A: Differentiating Dark Counts fromSignal

Part of the difficulty in understanding excesses in sin-gle electron experiments is differentiating dark counts (adistinct type of detector background) from a putative sig-nal, given that the rate of dark counts cannot easily beincreased as a calibration step in the same way as nu-clear or electronic recoil backgrounds. For this reason, itis very hard to interpret the single electron rate as signal.

We can, however, use the single electron rate to deter-mine whether the higher energy bins are consistent withdark count pileup; if not, we can conclude they arise froma distinct source. Given a single electron dark rate Γd andan integration window T , we find a mean number of darkevents λd = ΓdT . For measurements which integrate afixed window of time, such as CCDs, we should there-fore get a Poisson distribution with mean λd of events,as in [7].

CCDs actually sample all pixels, even empty ones, sothose pixels without dark counts show up in the eventhistogram. We thus can determine a Poisson mean fromthe 0 and 1 electron bins, and predict the dark rate inthe second bin; an excess above this rate is from a dif-ferent source. For example, for a dark rate of 400 Hz/kgas seen in the SENSEI data 1-electron bin, a mass ofabout 4 × 10−10 kg (the mass of a single pixel), and anexposure of an hour, we find that the mean λd is roughly5 × 10−4. This means that we expect the second bin tohave a rate of roughly 0.1 Hz/kg, which is more than anorder of magnitude lower than the measured rate. Wecan therefore conclude that the excess in the second binis likely from a different source than the single electronbin, or that these two bins are not entirely dominated bydark counts.

For experiments which take a time-stream of data, wefind that the probability of a dark event being irreduciblepileup on top of another dark event is ppu = Γdδt, whereδt is the minimum separation between events that canbe distinguished. For example, in CDMS HVeV with a

minimum time discrimination of 10µs [3] and a dark rateof 1 Hz in a 1g detector, we thus expect that p = 10−5.For a measured dark rate of 103 Hz/kg, we expect thepileup rate to be 10−2 Hz/kg. This is 103 times lowerthan the measured rate in any of the 2-6 electron bins; wecan therefore conclude that if the first bin is truly all darkcounts, none of the other bins can possibly be dark countsunless the experiment is observing a correlated leakageprocess. Because models of dark counts only produce asingle charge at a time, we conclude that the event ratesin the charge bins above the single electron bin are notdue to dark counts.

This analysis allows us to state the following: while thesingle electron bin in both SENSEI and CDMS HVeVmay be dominated by leakage, we can exclude simpleleakage as being the source of events in the >1 elec-tron bins. There must be, at the least, another unknownsource of events. In the main text, we follow this logicto exclude SM particles as the source, motivating a DMinterpretation.

Appendix B: Using Electron Yield to DetermineRecoil Type

Conventional DM searches, probing physics at energyscales above ∼1 keV, classify events in a binary fash-ion as electron recoil (ER) or nuclear recoil (NR). Thisclean separation is due to the very different microphysicsinvolved in each, and the large number of additional in-teractions which occur during the relaxation process as ahigh-energy particle deposits its momentum and energyin the detector.

In the case of a semiconductor, an electron recoilproduces a pair of high-energy charges, which inter-act primarily with other electrons, efficiently generatingelectron-hole pairs at a mean rate of neh = Ee/εeh, whereneh is the number of generated electron-hole pairs, Eeis the initial energy in the electron system, and εeh isthe mean energy per electron-hole pair produced. A re-markably generic property of semiconductors is that εehis constant across energy scales from eV to MeV (seee.g. Ref [87]); εeh ∼ 3.6 eV for Si and 3.0 eV for Ge,for example. This provides a convenient energy scale forcharge detectors measuring neh, under the assumptionthat all events measured are interaction with electrons.The electron-equivalent energy scale Ee, measure in eVee,is just nehεeh.

Nuclear recoils, on the other hand, involve a more com-plex picture. At very high energies (well above 15 eV inSi and Ge [48]), a nuclear recoil produces a crystal de-fect: a nucleus is physically removed from its lattice siteand bounces through the lattice, displacing some nuclei,ionizing others, and producing phonons. The relativescattering rate for ionization versus other loss processesis thus much more complex than electron scattering; inparticular, screening effects, which depend on the mo-mentum of the nucleus, become important. This means

15

FIG. 7. Top: EDELWEISS excess in Edet [1] used for the var-ious Ee spectrum models in Figure 2. The grey band showsthe uncertainty in the measurement; bins near threshold havevery small uncertainties due to the large number of events ob-served below 100 eV. Bottom: Yield curves used to generatesimulated Ee spectra in Figure 2. The electron recoil (ER)yield curve is flat because energy is initially imparted to theEe system, and thus Ee = Edet. The NR curves show a de-creased yield at lower energy. We have extended the NR yieldmodel from Ref [27] down to the lowest calibrated energy; wethen consider the full range of options from an extrapolation,to constant yield, to yield which discontinuously goes to 0. Wealso show the model from Ref [26] for comparison. Finally, weshow two inelastic yield curves which assume a fixed chargemean produced, one with a Poisson distribution (red), theother with identically one charge produced per event (green).

the relative energy given to the electron system, Ee, ver-sus the phonon system, Er, is momentum-dependent. Atenergies above 1 keV, this distribution of energy is well-modeled by a charge screening model, the so-called Lind-hard model [26, 27], but at low energies experiments havebegun to see sharp departures from this smooth energydependence (see e.g. Ref [88]).

The key point is that, if two of the three quantities(Ee, Er, Edet) can be measured (usually one proportionalto Ee, the other either Er or Edet = Er + Ee), then therecoil type of a given event can generally be determinedusing the ratio of these quantities by comparing to cal-ibrated yield curves. This is generally applicable at en-ergies above ∼1 keV, and has been used successfully byall recent DM experiments searching for DM with massesabove a few GeV. Below these energies, however, betterresolution is required in both measurements to success-

fully discriminate by recoil type, and statistical fluctua-tions begin to wash out discrimination ability on an eventby event level. For this reason, many of the very lowthreshold experiments have resorted back to only mea-suring one quantity. In our analysis, we are primarilycomparing experiments which only measure charge, andthus report spectra in Ee, with experiments which mea-sure heat, and are thus sensitive to Edet = Ee + Er. Allelectron recoil DM searches, by construction, are onlymeasuring Ee.

The lack of a clear distinction between ER and NRat sub-keV energies becomes important when we try tocompare two experiments in this regime in the presenceof some unknown signal. In this paper, we consider theexcesses observed by EDELWEISS [1, 6] because, havingmeasured both Ee and Edet with similar detectors, weare able to statistically determine the mean yield as afunction of energy for a population of events, and therebyshed light on the origin of those events. Consider theEe spectrum shown in the different panels of Figure 2compared with the Edet spectrum plotted in Figure 7.Ideally, both measurements would be made with infiniteprecision, and without a finite threshold. The remarkabledevelopment of single charge detectors over the past fewyears allows the Ee measurement to have effectively nothreshold, but the Edet measurement still has a thresholdEthresh = 60 eV.

We can still, however, try to determine whether eventsare electron recoil, nuclear recoil, or an entirely differentclass of events by applying the yield model and ensur-ing that, when we convert the spectrum from Edet toEe, the result is not in direct conflict with the measuredspectrum. Put another way, we must have

Rtotal =

∫ ∞

0

dR

dEedEe >

∫ ∞

Ethresh

dR

dEdetdEdet. (B1)

We thus convert the Edet spectrum to a modeled Eespectrum as

dRmodeldEe

= y(Edet)−1 dR

dEdet(B2)

where for this paper, this is a convolution done by MonteCarlo. We then check that

∫ Ehigh

Elow

dRmodeldEe

<

∫ Ehigh

Elow

dRmeasdEe

(B3)

for any choice of Elow and Ehigh. If we could sampleall of the Edet spectrum, this would be an equality, butbecause we know we’re only sampling the Edet spectrumabove Ethresh, the model should always either match orundershoot the measured Ee spectrum, which is com-plete.

The yield models we consider in this paper, shown inFigure 7, constitute a constant yield (ER), falling yield(NR), and fixed Ee independent of Edet. Because theyield is a ratio of Ee to Edet, these last models appearto rise in yield space at low energy. One of the primary

16

conclusions of this paper is that the only yield modelswhich satisfy Eq. (B3) for the measured Ee spectrum inFigure 2 are the rising models; in other words, the spec-tra are inconsistent with either a NR or ER interaction,and suggest a novel inelastic interaction. This also illus-trates that interpreting either the Ee spectrum or Edetspectrum independently, without knowledge of the typeof recoil, leads to erroneous exclusion curves. For exam-ple, the NR limit implied by converting the Ee spectrumto an effective nuclear recoil energy scale would be overlyaggressive.

Appendix C: Plasmon Review

Since plasmons in solid-state systems are likely unfa-miliar to many high-energy physics, in this appendix wereview some basic properties of plasmons and their mea-surement using EELS. To facilitate comparison with theliterature, we will use as much as possible the notation ofRef. [30], in contrast to the typical high-energy physicsnotation of the main text (i.e. ω instead of E, and einstead of α).

1. Plasmon measurements with EELS

Plasmons are the quantized longitudinal oscillations ofvalence electrons in a condensed matter system, carryingenergy on the order of the classical plasma frequency. InEELS experiments, plasmons are excited by the electricfield of an electron traversing the material, which hasa longitudinal component (i.e. there is a component ofthe field along the electron’s direction of motion). Theresponse of a material to electromagnetic fields of mo-mentum q and frequency ω can be characterized by acomplex dielectric function ε(ω,q). For an electron withcharge e, mass me, and velocity v traversing a materialwith dielectric function ε, the differential probability perunit time of depositing energy ω is [30]

dP

dtdω=

e2

4π3

∫d3q

1

q2Im

{ −1

ε(ω,q)

}

× δ(ω − q ·v +

q2

2me

), (C1)

where we have converted to the Heaviside-Lorentz unitsconventional in high-energy physics (in contrast to theformulas from [30] which use Gaussian units common incondensed matter physics and contain additional factorsof 4π).10 Note that we can rewrite the energy conserva-

10 Note that in the small-q limit, the q2/(2me) term is negligible, sothis term is typically neglected in the condensed matter literaturewhen considering electron probes.

tion condition enforced by the delta function as

q =E

v cos θ+

q2

2mev cos θ(C2)

where θ is the angle between q and v. We are inter-ested in the forward-scattering region where cos θ > 0where momentum transfer is the smallest. In that partof phase space, both terms on the right-hand side arepositive-definite, so we obtain the inequality q ≥ E/v bydropping the second term. Assuming the plasmon hastypical energy ω = Ep, we find the important relation

q ≥ Epv, (C3)

which is saturated in the forward scattering limit where

q is parallel to v and when the finite-mass term q2

2meis

negligible (which typically holds for the kinematics rel-evant to EELS). This condition is simply an expressionof energy conservation, which must be satisfied to excitethe plasmon at the peak energy. Define qp = Ep/v asthe typical scale of momentum transfer for plasmon ex-citations. As mentioned in the main text, the plasmonhas a cutoff frequency qc ∼ 2π/a. Plasmon resonanceshave been observed with q about a factor of 2 above thiscutoff [19], but for parametric estimates, it will suffice torequire that qp < qc. To satisfy Eq. (C3), we must have

v ≥ Ep/qp = 6.5× 10−3(

Ep16 eV

)(C4)

Note that this condition is independent of the mass of theincident particle; it could be an electron, proton, or mil-licharged DM. In the main text we conservatively tightenthis bound on v to 10−2. This constraint on the velocityexplains why EELS experiments to probe the plasmonare performed with semi-relativistic electrons.

Assuming the plasmon kinematic condition (C4) issatisfied, the presence of the 1/q2 in the integrand ofEq. (C1) implies that the smallest allowed momentumtransfers, q ∼ qp, will dominate. This is just the typicalbehavior of the long-range Coulomb force. In that case,we can approximate ε(ω,q) ≈ ε(ω, 0) and pull it out ofthe q integral, giving

dP

dtdω=

e2

2π2vIm

{ −1

ε(ω)

}log

(2E0

ω

). (C5)

where E0 is the incident electron energy. The logarith-mic dependence on E0 is a manifestation of the uni-versal Coulomb logarithm, which here is cut off by theenergy transfer ω. This formula can be modified in astraightforward way for relativistic probes. Note thatsince v =

√2E0/me, the plasmon excitation probability

scales as log(E0)/√E0, a relation which holds for any

nonrelativistic charged particle (in particular, for mil-licharged DM as well as electrons).

Note that by dividing by v and integrating over ω, wecan convert this expression into a probability per unitlength for the probe to undergo some nonzero energy

17

loss. Setting this to unity gives an inelastic mean freepath, which for electrons (Q = e) with v = 0.1 is about58 nm for 50 keV electrons in silicon [19]. For electrons,then, multiple scattering is an important consideration,and in thick samples this will give rise to energy depositsin integer multiples of Ep. On the other hand, for DMwith a small millicharge, the mean free path scales withκ2gD and is orders of magnitude larger than the detectorsize for the millicharge values we consider. Thus, mul-tiple scattering of DM will typically not occur in small

semiconductor detectors.

2. Role of the dielectric function

In the non-interacting electron approximation, thecomplex dielectric function of a material is given by theLindhard formula [89]:

ε(ω,q) = 1− limη→0

e2

V

1

q2

∑

k

∑

l,l′

fFD(Ek+q,l′)− fFD(Ek,l)

Ek+q,l′ − Ek,l − ω − iη× |〈|k + q, l′|eiq ·x|k, l〉|2. (C6)

Here, V is the volume of the material, l and l′ are bandindices, Ek,l is the energy of the lth band at lattice mo-mentum k, and fFD are Fermi-Dirac factors. The imag-inary part of ε reflects “on-shell” transitions when twostates have an energy difference ω. The dielectric func-tion is closely related to the 1-loop vacuum polarizationin quantum field theory, which has similar properties (i.e.its imaginary part reflects on-shell final states). Notethat Im{−1/ε} = Im ε/|ε|2 selects out these transitionsand weights them by the corresponding squared matrixelement, which is why this expression appears in the for-mula Eq. (C1).

It is important to emphasize that Eq. (C6), whichforms the basis for much of the recent literature on DMinteractions in condensed matter systems [13, 81, 90], as-sumes no electron-electron interactions. In this approxi-mation, the plasmon does not appear. In the language ofhigh-energy physics, the plasmon is analogous to a polein a matrix element which does not correspond to thefields in the Lagrangian. This is by no means unusual,and indeed is behavior characteristic of strongly-coupledfield theories. Moreover, much like the Sudakov factorin QCD, the plasmon is a result of a resummation of aninfinite series of diagrams (known in the condensed mat-ter literature as the random phase approximation [91])and does not appear at any finite order in perturbationtheory. Thus is is best to treat the plasmon phenomeno-logically as a nonperturbative effect which nonethelesscontributes to the imaginary part of the dielectric func-tion.

Finally, we note that the factor of e2 in the dielectricfunction represents the response of a material to (ordi-nary, electromagnetic, possibly virtual) photons. TheDM-induced direct plasmon excitation rate is propor-tional to κ2αD, which roughly speaking represents theprobability of millicharged DM emitting a virtual darkphoton which converts to a virtual photon. The dielectricfunction then parameterizes the response of the materialto this photon, which is proportional to e2 in perturba-

tion theory.11

3. Plasmon lineshape

The Frohlich damped-harmonic-oscillator model [70]posits the following form for ε(ω):

ε(ω) = εc +E2p

(E2g − ω2)− iΓω , (C7)

where εc is the contribution to the dielectric constantfrom core electrons (assumed independent of ω), Ep is theplasma energy of the valence electrons, Eg is an averageband gap, and Γ is the plasmon damping parameter. Asin the analogous Breit-Wigner formulae in high-energyphysics, Γ represents the sum of the partial widths of allthe plasmon decay modes, including to phonons and elec-tron/hole pairs. In high-energy physics, if the decay ofa resonance of energy E can be described perturbatively,the narrow-width approximation Γ/E � 1 applies. Forplasmons, typical values are Γ = 3 eV and E = 16 eV[19, 30], so Γ/E ≈ 0.2 is not particularly small. For com-parison, Γ/E for the top quark is 0.008, and Γ/E forthe lowest-lying ∆ resonance is 0.09, so in this sense theplasmon is even more nonperturbative than QCD reso-nances. Thus it is not unreasonable that the plasmon-phonon coupling, which controls both plasmon produc-tion and plasmon decay, is nonperturbatively large, con-sistent with Scenario 1 in the main text.

11 We note in passing that if DM-SM interactions were mediatedby a light scalar rather than a light vector, a completely differentobject (essentially a scalar response function) would control thematerial response. This function has been calculated in pertur-bation theory for the case of isolated atoms [92], but it wouldbe very interesting to understand the nonperturbative plasmoncontribution in solid-state systems in light of our work, whichcannot be measured with any SM probe because no long-rangescalar forces exist in the SM.

18

Substituting into Eq. (C5), we find

dP

dtdω=

e2

2π2v

ΓE2pω ln(2E0/ω)

ε2c(E2g + E2

p/ε2c − ω2)2 + Γ2ω2

. (C8)

This model is an excellent fit to the observed plas-mon in silicon, but the germanium plasmon has a longerhigh-energy tail due to contributions from the core 3delectrons. However, the region around the peak is well-modeled by a single Lorentzian, so following [30] wetake εc = 1 and use Eq. (C8) to normalize the ger-manium plasmon, taking the best-fit values in the two-parameter model of [30] with an effective plasmon energyE′2p ≡ E2

g + E2p/ε

2c . Integrating over ω (or equivalently

E) for millicharged DM gives Eq. (17). On the otherhand, to obtain the spectrum for a general velocity dis-tribution f(v), we weight Eq. (C1) by f(v) and integrateover v. Solving the delta function by performing the ve-

locity integral, as is standard in DM-electron scatteringtreatments, and performing the q integral up to qc givesEq. (14) in the main text. In that equation, the (dimen-sionless) plasmon lineshape is

S(ω) ≡ Im

{ −1

ε(ω)

}, (C9)

which is

SF (ω) ≡ ΓE2pω

ε2c(E2g + E2

p/ε2c − ω2)2 + Γ2ω2

(C10)

in the Frohlich model. Integrating Eq. (C10) gives∫SF (ω) dω ≈ 3

2εcEp (C11)

for Eg = 0.

[1] E. Armengaud, C. Augier, A. Benot, A. Benoit, L. Berg,J. Billard, A. Broniatowski, P. Camus, A. Cazes,M. Chapellier, and et al., Physical Review D 99 (2019),10.1103/physrevd.99.082003.

[2] A. Abdelhameed, G. Angloher, P. Bauer, A. Bento,E. Bertoldo, C. Bucci, L. Canonica, A. DAddabbo,X. Defay, S. Di Lorenzo, and et al., Physical ReviewD 100 (2019), 10.1103/physrevd.100.102002.

[3] R. Agnese, T. Aralis, T. Aramaki, I. Arnquist, E. Azad-bakht, W. Baker, S. Banik, D. Barker, D. Bauer,T. Binder, and et al., Physical Review Letters 121(2018), 10.1103/physrevlett.121.051301.

[4] O. Abramoff, L. Barak, I. M. Bloch, L. Chaplinsky,M. Crisler, Dawa, A. Drlica-Wagner, R. Essig, J. Estrada,E. Etzion, and et al., Physical Review Letters 122(2019), 10.1103/physrevlett.122.161801.

[5] R. Essig, T. Volansky, and T.-T. Yu, Physical Review D96 (2017), 10.1103/physrevd.96.043017.

[6] J. Gascon, “The EDELWEISS SubGeV program,”(2020), theory Meeting Experiments: Particle Astro-physics and Cosmology.

[7] A. Aguilar-Arevalo, D. Amidei, D. Baxter, G. Cancelo,B. A. Cervantes Vergara, A. E. Chavarria, E. Darragh-Ford, J. R. T. de Mello Neto, J. C. D’Olivo, J. Estrada,R. Gaıor, Y. Guardincerri, T. W. Hossbach, B. Kilmin-ster, I. Lawson, S. J. Lee, A. Letessier-Selvon, A. Mat-alon, V. B. B. Mello, P. Mitra, J. Molina, S. Paul,A. Piers, P. Privitera, K. Ramanathan, J. Da Rocha,Y. Sarkis, M. Settimo, R. Smida, R. Thomas, J. Tiff-enberg, D. Torres Machado, R. Vilar, and A. L. Virto(DAMIC Collaboration), Phys. Rev. Lett. 123, 181802(2019).

[8] G. Angloher, P. Bauer, A. Bento, C. Bucci, L. Canon-ica, X. Defay, A. Erb, F. v. Feilitzsch, N. F. Iachellini,P. Gorla, and et al., The European Physical Journal C77 (2017), 10.1140/epjc/s10052-017-5223-9.

[9] R. Essig, A. Manalaysay, J. Mardon, P. Sorensen,and T. Volansky, Physical Review Letters 109 (2012),10.1103/physrevlett.109.021301.

[10] E. Aprile et al. (XENON Collaboration), Phys. Rev. Lett.

123, 251801 (2019).[11] P. Agnes et al. (DarkSide), Phys. Rev. Lett. 121, 111303

(2018), arXiv:1802.06998 [astro-ph.CO].[12] J. Lewin and P. Smith, Astroparticle Physics 6, 87

(1996).[13] R. Essig, M. Fernandez-Serra, J. Mardon, A. Soto,

T. Volansky, and T.-T. Yu, JHEP 05, 046 (2016),arXiv:1509.01598 [hep-ph].

[14] R. Essig, J. Mardon, and T. Volansky, Phys. Rev. D85,076007 (2012), arXiv:1108.5383 [hep-ph].

[15] P. W. Graham, D. E. Kaplan, S. Rajendran, and M. T.Walters, Phys. Dark Univ. 1, 32 (2012), arXiv:1203.2531[hep-ph].

[16] S. K. Lee, M. Lisanti, S. Mishra-Sharma, and B. R.Safdi, Phys. Rev. D92, 083517 (2015), arXiv:1508.07361[hep-ph].

[17] D. Baxter, Y. Kahn, and G. Krnjaic, (2019),arXiv:1908.00012 [hep-ph].

[18] R. Essig, J. Pradler, M. Sholapurkar, and T.-T. Yu,(2019), arXiv:1908.10881 [hep-ph].

[19] H. Raether, Excitation of plasmons and interband tran-sitions by electrons, Vol. 88 (Springer, 2006).

[20] J. strm, P. Di Stefano, F. Prbst, L. Stodolsky, J. Ti-monen, C. Bucci, S. Cooper, C. Cozzini, F. Feilitzsch,H. Kraus, and et al., Physics Letters A 356, 262266(2006).

[21] M. Ibe, W. Nakano, Y. Shoji, and K. Suzuki, JHEP 03,194 (2018), arXiv:1707.07258 [hep-ph].

[22] S. Watkins, “Performance of a Large Area Photon Detec-tor and Applications,” (2019), 18th International Work-shop on Low Temperature Detectors (LTD-18).

[23] C. E. Dahl, Ph.D. thesis, Princeton University (2009).[24] C. Blanco, J. I. Collar, Y. Kahn, and B. Lillard, (2019),

arXiv:1912.02822 [hep-ph].[25] E. V. Bugaev, A. Misaki, V. A. Naumov, T. S. Sinegov-

skaya, S. I. Sinegovsky, and N. Takahashi, Phys. Rev. D58, 054001 (1998).

[26] D. Barker, W.-Z. Wei, D.-M. Mei, and C. Zhang, As-troparticle Physics 48, 815 (2013).

[27] R. Agnese, A. Anderson, T. Aramaki, I. Arnquist,

19

W. Baker, D. Barker, R. Basu Thakur, D. Bauer, A. Bor-gland, M. Bowles, and et al., Physical Review D 95(2017), 10.1103/physrevd.95.082002.

[28] W. J. Gignac, R. S. Williams, and S. P. Kowalczyk,Phys. Rev. B 32, 1237 (1985).

[29] J. D. Nuttall, T. E. Gallon, M. G. Devey, and J. A. D.Matthew, Journal of Physics C: Solid State Physics 8,445 (1975).

[30] M. K. Kundmann, Study of semiconductor valence plas-mon line shapes via electron energy-loss spectroscopyin the transmission electron microscope, Tech. Rep.(Lawrence Berkeley Lab., CA (USA), 1988).

[31] A. Aguilar-Arevalo, D. Amidei, X. Bertou, M. But-ner, G. Cancelo, A. Castaneda Vazquez, B. A. Cer-vantes Vergara, A. E. Chavarria, C. R. Chavez, J. R. T.de Mello Neto, J. C. D’Olivo, J. Estrada, G. Fer-nandez Moroni, R. Gaıor, Y. Guardincerri, K. P.Hernandez Torres, F. Izraelevitch, A. Kavner, B. Kilmin-ster, I. Lawson, A. Letessier-Selvon, J. Liao, V. B. B.Mello, J. Molina, J. R. Pena, P. Privitera, K. Ra-manathan, Y. Sarkis, T. Schwarz, C. Sengul, M. Settimo,M. Sofo Haro, R. Thomas, J. Tiffenberg, E. Tiouchi-chine, D. Torres Machado, F. Trillaud, X. You, andJ. Zhou (DAMIC Collaboration), Phys. Rev. D 94,082006 (2016).

[32] R. Agnese, T. Aralis, T. Aramaki, I. Arnquist, E. Azad-bakht, W. Baker, S. Banik, D. Barker, D. Bauer,T. Binder, and et al., Physical Review D 99 (2019),10.1103/physrevd.99.062001.

[33] R. Harnik, J. Kopp, and P. A. N. Machado, JCAP 1207,026 (2012), arXiv:1202.6073 [hep-ph].

[34] G. Angloher, A. Bento, C. Bucci, L. Canonica, X. De-fay, A. Erb, F. von Feilitzsch, N. F. Iachellini, P. Gorla,A. Gtlein, and et al., The European Physical Journal C76 (2016), 10.1140/epjc/s10052-016-3877-3.

[35] J. I. Collar, Phys. Rev. D 98, 023005 (2018).[36] W. L. Xu, C. Dvorkin, and A. Chael, Phys. Rev. D97,

103530 (2018), arXiv:1802.06788 [astro-ph.CO].[37] E. O. Nadler, V. Gluscevic, K. K. Boddy, and

R. H. Wechsler, Astrophys. J. Lett. 878, 32 (2019),arXiv:1904.10000 [astro-ph.CO].

[38] P. deNiverville, M. Pospelov, and A. Ritz, Phys. Rev.D84, 075020 (2011), arXiv:1107.4580 [hep-ph].

[39] A. A. Aguilar-Arevalo et al. (MiniBooNE DM), Phys.Rev. D98, 112004 (2018), arXiv:1807.06137 [hep-ex].

[40] B. Batell, R. Essig, and Z. Surujon, Phys. Rev. Lett.113, 171802 (2014), arXiv:1406.2698 [hep-ph].

[41] R. Essig, J. Mardon, M. Papucci, T. Volansky, and Y.-M.Zhong, JHEP 11, 167 (2013), arXiv:1309.5084 [hep-ph].

[42] D. Banerjee et al., Phys. Rev. Lett. 123, 121801 (2019),arXiv:1906.00176 [hep-ex].

[43] A. Berlin, N. Blinov, G. Krnjaic, P. Schuster,and N. Toro, Phys. Rev. D99, 075001 (2019),arXiv:1807.01730 [hep-ph].

[44] W. Altmannshofer et al. (Belle-II), PTEP 2019, 123C01(2019), arXiv:1808.10567 [hep-ex].

[45] N. W. Evans, C. A. J. O’Hare, and C. McCabe,Phys. Rev. D99, 023012 (2019), arXiv:1810.11468 [astro-ph.GA].

[46] A. Migdal, Sov. Phys. JETP 9, 1163 (1939).[47] T. Trickle, Z. Zhang, K. M. Zurek, K. Inzani, and S. Grif-

fin, (2019), arXiv:1910.08092 [hep-ph].[48] Y. Chen and J. MacKay, Philosophical Magazine 19, 357

(1969).

[49] H. An, M. Pospelov, J. Pradler, and A. Ritz, Phys.Rev. Lett. 120, 141801 (2018), [Erratum: Phys. Rev.Lett.121,no.25,259903(2018)], arXiv:1708.03642 [hep-ph].

[50] K. Agashe, Y. Cui, L. Necib, and J. Thaler, JCAP 1410,062 (2014), arXiv:1405.7370 [hep-ph].

[51] L. Necib, J. Moon, T. Wongjirad, and J. M. Conrad,Phys. Rev. D95, 075018 (2017), arXiv:1610.03486 [hep-ph].

[52] J. Berger, Y. Cui, M. Graham, L. Necib, G. Petrillo,D. Stocks, Y.-T. Tsai, and Y. Zhao, (2019),arXiv:1912.05558 [hep-ph].

[53] T. Bringmann and M. Pospelov, Phys. Rev. Lett. 122,171801 (2019), arXiv:1810.10543 [hep-ph].

[54] C. Cappiello and J. F. Beacom, Phys. Rev. D100, 103011(2019), arXiv:1906.11283 [hep-ph].

[55] J. B. Dent, B. Dutta, J. L. Newstead, and I. M. Shoe-maker, (2019), arXiv:1907.03782 [hep-ph].

[56] G. Krnjaic and S. D. McDermott, (2019),arXiv:1908.00007 [hep-ph].

[57] J. H. Chang, R. Essig, and S. D. McDermott, JHEP 09,051 (2018), arXiv:1803.00993 [hep-ph].

[58] W. DeRocco, P. W. Graham, D. Kasen, G. Marques-Tavares, and S. Rajendran, Phys. Rev. D100, 075018(2019), arXiv:1905.09284 [hep-ph].

[59] J.-T. Li and T. Lin, (2020), arXiv:2002.04625 [astro-ph.CO].

[60] R. Essig et al., in Proceedings, 2013 Community SummerStudy on the Future of U.S. Particle Physics: Snowmasson the Mississippi (CSS2013): Minneapolis, MN, USA,July 29-August 6, 2013 (2013) arXiv:1311.0029 [hep-ph].

[61] J. Alexander et al. (2016) arXiv:1608.08632 [hep-ph].[62] M. Battaglieri et al., in U.S. Cosmic Visions: New Ideas

in Dark Matter College Park, MD, USA, March 23-25,2017 (2017) arXiv:1707.04591 [hep-ph].

[63] J. Bovy and S. Tremaine, Astrophys. J. 756, 89 (2012),arXiv:1205.4033 [astro-ph.GA].

[64] E. Izaguirre, G. Krnjaic, P. Schuster, and N. Toro, Phys.Rev. Lett. 115, 251301 (2015), arXiv:1505.00011 [hep-ph].

[65] E. Akhmedov, G. Arcadi, M. Lindner, and S. Vogl,JHEP 10, 045 (2018), arXiv:1806.10962 [hep-ph].

[66] I. Yokota, Progress of Theoretical Physics Supplement57, 97 (1975), https://academic.oup.com/ptps/article-pdf/doi/10.1143/PTP.57.97/5385903/57-97.pdf.

[67] C. Dvorkin, T. Lin, and K. Schutz, Phys. Rev. D99,115009 (2019), arXiv:1902.08623 [hep-ph].

[68] M. D. Diamond and P. Schuster, Phys. Rev. Lett. 111,221803 (2013), arXiv:1307.6861 [hep-ph].

[69] S. Davidson, S. Hannestad, and G. Raffelt, JHEP 05,003 (2000), arXiv:hep-ph/0001179 [hep-ph].

[70] H. Frohlich, Phenomenological theory of the energy loss offast particles in solids (VEB Deutscher Verlag der Wis-senschaften, 1959).

[71] H. Vogel and J. Redondo, JCAP 1402, 029 (2014),arXiv:1311.2600 [hep-ph].

[72] L. J. Hall, K. Jedamzik, J. March-Russell, and S. M.West, JHEP 03, 080 (2010), arXiv:0911.1120 [hep-ph].

[73] G. Krnjaic, JHEP 10, 136 (2018), arXiv:1711.11038 [hep-ph].

[74] J. A. Evans, C. Gaidau, and J. Shelton, JHEP 01, 032(2020), arXiv:1909.04671 [hep-ph].

[75] M. Battaglieri et al. (BDX), (2016), arXiv:1607.01390[hep-ex].

20

[76] S. Alekhin et al., Rept. Prog. Phys. 79, 124201 (2016),arXiv:1504.04855 [hep-ph].

[77] P. Mermod (SHiP), Proceedings, 2017 InternationalWorkshop on Neutrinos from Accelerators (NuFact17):Uppsala University Main Building, Uppsala, Sweden,September 25-30, 2017, PoS NuFact2017, 139 (2017),arXiv:1712.01768 [hep-ex].

[78] V. De Romeri, K. J. Kelly, and P. A. N. Machado, Phys.Rev. D100, 095010 (2019), arXiv:1903.10505 [hep-ph].

[79] T. kesson et al. (LDMX), (2018), arXiv:1808.05219 [hep-ex].

[80] S. K. Lee, M. Lisanti, A. H. G. Peter, and B. R. Safdi,Phys. Rev. Lett. 112, 011301 (2014), arXiv:1308.1953[astro-ph.CO].

[81] Y. Hochberg, Y. Kahn, M. Lisanti, K. M. Zurek, A. G.Grushin, R. Ilan, S. M. Griffin, Z.-F. Liu, S. F. We-ber, and J. B. Neaton, Phys. Rev. D97, 015004 (2018),arXiv:1708.08929 [hep-ph].

[82] A. Coskuner, A. Mitridate, A. Olivares, and K. M.Zurek, (2019), arXiv:1909.09170 [hep-ph].

[83] R. M. Geilhufe, F. Kahlhoefer, and M. W. Winkler,(2019), arXiv:1910.02091 [hep-ph].

[84] S. M. Griffin, K. Inzani, T. Trickle, Z. Zhang, and K. M.

Zurek, (2019), arXiv:1910.10716 [hep-ph].[85] S. Derenzo, R. Essig, A. Massari, A. Soto, and T.-T.

Yu, Phys. Rev. D96, 016026 (2017), arXiv:1607.01009[hep-ph].

[86] S. Knapen, T. Lin, M. Pyle, and K. M. Zurek, Phys.Lett. B785, 386 (2018), arXiv:1712.06598 [hep-ph].

[87] C. Canali, M. Martini, G. Ottaviani, and A. A. Quar-anta, IEEE Transactions on Nuclear Science 19, 9 (1972).

[88] B. Scholz, A. Chavarria, J. Collar, P. Privitera,and A. Robinson, Physical Review D 94 (2016),10.1103/physrevd.94.122003.

[89] M. Dressel and G. Gruner, Electrodynamics of Solids,Optical Properties of Electrons in Matter (CambridgeUniversity Press, 2002).

[90] Y. Hochberg, M. Pyle, Y. Zhao, and K. M. Zurek, JHEP08, 057 (2016), arXiv:1512.04533 [hep-ph].

[91] A. Fetter and J. Walecka, Quantum Theory of Many-Particle Systems, Dover Books on Physics (Dover Publi-cations, 2012).

[92] R. Catena, T. Emken, N. Spaldin, and W. Tarantino,(2019), arXiv:1912.08204 [hep-ph].