Fog on the horizon: a new definition of the neutrino floor for direct dark matter searches

Ciaran A. J. O’Hare1, a

1ARC Centre of Excellence for Dark Matter Particle Physics,The University of Sydney, NSW 2006, Camperdown, Sydney, Australia

The neutrino floor is a theoretical lower limit on WIMP-like dark matter models that are discoverable indirect detection experiments. It is commonly interpreted as the point at which dark matter signals becomehidden underneath a remarkably similar-looking background from neutrinos. However, it has been known forsome time that the neutrino floor is not a hard limit, but can be pushed past with sufficient statistics. As aconsequence, some have recently advocated for calling it the “neutrino fog” instead. The downside of currentmethods of deriving the neutrino floor are that they rely on arbitrary choices of experimental exposure andenergy threshold. Here we propose to define the neutrino floor as the boundary of the neutrino fog, and develop acalculation free from these assumptions. The technique is based on the derivative of a hypothetical experimentaldiscovery limit as a function of exposure, and leads to a neutrino floor that is only influenced by the systematicuncertainties on the neutrino flux normalisations. Our floor is broadly similar to those found in the literature,but differs by almost an order of magnitude in the sub-GeV range, and above 20 GeV. �

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FIG. 1. Present exclusion limits on the spin-independent DM-nucleon cross section (assuming equal proton/neutron couplings) [1–15]. Beneath these limits we show three definitions of the neutrinofloor for a xenon target. The previous discovery-limit-based neutrinofloor calculation shown by the dashed line is taken from the recentAPPEC report [16] (based on the technique of Ref. [17]). The enve-lope of 90% C.L. exclusion limits seeing one expected neutrino eventis shown as a dotted line. The result of our work is the solid orangeline. We define this notion of the neutrino floor to be the bound-ary of the neutrino fog, i.e. the cross section at which any experi-ment sensitive to a given value of mχ leaves the standard Poissonianregime σ ∝ N−1/2, and begins to be saturated by the background:σ ∝ N−1/n, with n > 2. The floor is thus a contour for n = 2.

Introduction.—Modern experiments searching for darkmatter (DM) in the form of weakly interacting massive par-ticles (WIMPs) have become rather large [18, 19]. It has beenanticipated for some time [20–23] that these underground de-tectors might one day become large enough to detect not justDM, but astrophysical neutrinos as well. In fact, it appears as

though the first detection of solar neutrinos in a xenon-baseddetector is just around the corner [15]. Fittingly, the commu-nity has begun to collate a rich catalogue of novel physics tobe done with our expanding multi-purpose network of largeunderground detectors [24–43].

Unfortunately for WIMP enthusiasts, the impending arrivalof neutrinos in DM detectors is somewhat bittersweet—being,as they are, essentially the harbingers of the end of conven-tional searches. These experiments usually look for signals ofDM using nuclear recoils—a channel through which neutri-nos also generate events via coherent elastic neutrino-nucleusscattering (CEνNS) [44–46]. It turns out that the recoil signa-tures of DM and neutrinos look remarkably alike, with differ-ent sources of neutrino each masquerading as DM of varyingmasses and cross sections [47].

Even an irreducible background like neutrinos may not beso problematic were it not for the—sometimes sizeable—systematic uncertainties on their fluxes. The cross section be-low which the potential discovery of a DM signal is prohibiteddue to this uncertainty is what is usually, but not always, la-belled the “neutrino floor” [17]: a limit that has since beenthe subject of many detailed studies [38, 39, 48–58]. Since2013 some form of neutrino floor has been shown underneathall experimental results, often billed as an ultimate sensitivitylimit [59]. Methods of circumventing the neutrino floor havebeen proposed [60–62]. However, only directional detectionseems to be a realistic strategy for doing so with compara-tively low statistics [63–70].

One potentially misleading aspect of the name “neutrinofloor” is the fact that while it does pose an existential threatto DM searches, the floor itself is not solid. Firstly, the sever-ity of the neutrino background—and hence the height of thefloor in terms of cross section—is dependent crucially on neu-trino flux uncertainties, which are anticipated to improve overtime. Secondly, the DM and neutrino signals are never per-fect matches. Even for DM masses and neutrino fluxes withvery closely aligned nuclear recoil spectra—like xenon scat-tering with 8B neutrinos and a 6 GeV WIMP—they are notprecisely the same. This means that with a large enough num-

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FIG. 2. A graphical description of the technique we adopt to map the neutrino fog and plot its boundary. In the main panel we show thespin-independent DM parameter space, colouring the section below the neutrino floor by the value of n, defined as the index with which adiscovery limit scales with the number of background events, i.e. σ ∝ N−1/n. The neutrino fog is defined to be the regime for which n > 2,with the neutrino floor being the cross section for a given mass where this transition occurs. The top right panel shows the evolution of σ withN at mχ = 5.5 GeV between the two cross sections labelled “a” and “b” on the main panel. The lower right panel shows the value n, foundfrom derivative of the curve in the top right panel.

ber of events, the spectra should be distinguishable [60]. Thisfact implies that neutrinos present not a floor, but perhaps a‘fog’: a region of the parameter space where a clear distinc-tion between signal and background is challenging, but notimpossible.

The fogginess of the neutrino floor has become somewhatbetter-appreciated recently [51, 62, 71]. However it is some-thing that is rarely visualised: usually just a single neutrinofloor limit is plotted. The most common version shown re-lies on an interpolation of several discovery limits for a set ofsomewhat arbitrary thresholds and exposures. It is true thatgiven the softness of the neutrino fog, insisting upon a hardboundary will always be slightly arbitrary, however it shouldbe possible to devise a simpler and self-consistent definition.

Since direct detection experiments will venture into theneutrino fog imminently, it is timely to update and refine ourdefinitions. In this article we propose a new definition of theneutrino floor that situates it at the edge of the neutrino fog.

We aim for this definition to 1) not depend upon arbitrarychoices for experimental thresholds, or absolute numbers ofobserved neutrino events, 2) have a single consistent statisti-cal meaning, and 3) be flexible to future improvements to neu-trino flux measurements. The result of this effort can be foundin Fig. 1, contrasted against previously used definitions. Thetechnique for calculating this limit is explained graphically inFig. 2. The rest of this paper is devoted to summarising thebasic ingredients of the calculation and evaluating several il-lustrative examples to compare against previous versions.

Neutrinos versus DM.—To begin, we need to define a DMmodel to frame our discussion around. We adopt the followingDM-nucleus scattering rate,

dRχdEr

= ρ0Cσ

2mχµ2F 2(Er) g(vmin) , (1)

where ρ0 is the local DM density, mχ is the DM mass, σis some DM-nucleon cross section, µ is the DM-nucleon re-

3

duced mass, C is a nucleus-dependent constant that coherentlyenhances the rate, and F (Er) is the form factor that sup-presses it at high energies. Finally, g(vmin) is the mean in-verse DM speed above the minimal speed required to producea recoil with energy Er. The latter is found by integratingthe lab-frame DM velocity distribution. We assume the time-averaged Standard Halo Model, with parameters summarisedin Ref. [72]. For this initial explanation we will frame the dis-cussion around the spin-independent (SI) isospin-conservingDM-nucleon cross section σ ≡ σSI

p , which is the canonicalcross section that experimental collaborations most frequentlyset exclusion limits on. The scattering rate for this model isenhanced by C = A2, for a target with A nucleons.

Neutrinos can scatter elastically off nuclei and produce re-coils with very similar spectra to the ones found by evaluatingEq.(1). Currently, the only measurement of CEνNS is by CO-HERENT [73, 74], but it is well-understood in the StandardModel [44, 45]. Theoretical uncertainties, for example fromthe running of the Weinberg angle [75], or the nuclear formfactor [53], are subdominant to those on the neutrino fluxes.We assume a Weinberg angle of sin2 θW = 0.2387, and forboth neutrino and SI DM scattering we use the standard Helmform factor [76]. Summaries of the calculation of the CEνNScross section dσνN (Eν)/dEr as a function of neutrino andrecoil energy, as well the resulting spectra of recoil energiesfor the various targets we will consider here can be found ine.g. Refs. [17, 60, 68, 77].

The recoil energy spectra are found by integrating the dif-ferential CEνNS cross section multiplied by the neutrino flux,

dRνdEr

=1

mN

∫

Eminν

dΦ

dEν

dσνN (Eν)

dErdEν . (2)

We cut off the integral at the minimum neutrino energy thatcan cause a recoil with Er: Emin

ν =√mNEr/2. We adopt

the same neutrino flux model as in Ref. [68] (Table I), so wewill only briefly summarise some pertinent details. Furtherinformation about neutrino fluxes can be found in Ref. [78].

Solar neutrinos generated in nuclear fusion reactions in theSun form the largest flux at Earth for Eν . 11 MeV. Thesewill be the primary source of CEνNS events for most DM de-tectors and will limit discovery around mχ ∼ 10 GeV. TheSun’s nuclear energy generation is well-understood, and inthe case of the most important flux of neutrino from 8B de-cay, the corresponding flux normalisation, is also measuredprecisely [79–85]. For the less well-measured components,several theoretical calculations of the solar neutrino fluxes areon the market (see e.g. Ref. [86] for a recent review). Herewe adopt the Barcelona 2016 calculation of the GS98 high-metallicity Standard Solar Model [87]. We adopt the quoteduncertainty on each flux normalisation, with the exception of8B which we give a 2% uncertainty in line with global fitsof neutrino data [88]. After 8B neutrinos, the most importantsolar fluxes are the two neutrino lines from electron captureby 7Be, which mimic the signal for sub-GeV masses—thesecome with 6% uncertainties.

Geoneutrinos are a constant flux of antineutrinos producedin radioactive decays of mainly uranium, thorium and potas-sium in the Earth. As might be expected, the ratios andnormalisations of these fluxes are location-dependent. Forconcreteness, we use spectra from Ref. [89] and normaliseour geoneutrino fluxes to Gran Sasso, with correspondinguncertainties ranging from 17–25% [90]. Geoneutrinos im-pede the discovery at mχ ∼ GeV, but only for cross sectionsσSIp . 10−47 cm2.Nuclear reactors generate another source of antineutrinos

and influence the floor at slightly higher masses. We as-sume the fission fractions and average energy releases fromRef. [91] combined with the spectra from Ref. [92] beforesumming over all nearby nuclear reactors to Gran Sasso [93].

The diffuse supernova neutrino background (DSNB) is thecumulative flux of neutrinos from the cosmological historyof core-collapse supernovae, and is relevant for the neutrinofloor in a small mass window around 20 GeV. We adopt thefluxes parameterised in terms of three effective neutrino tem-peratures for the different flavour contributions, and place a50% uncertainty on the all-flavour flux [94].

Atmospheric neutrinos originate from the scattering ofhigh-energy cosmic rays. The low-energy tail of the flux issmall Φ ∼ 10 cm−2 s−1, but is the dominant background athigh recoil energies. We use the theoretical flux model for13 MeV–1 GeV atmospheric neutrinos from FLUKA simula-tions [95], placing the recommended 20% theoretical uncer-tainty. The final exposures of experiments like DARWIN [96]and Argo [97], may reach the high mass section of the neu-trino floor set by this background. Since many WIMP-likemodels with viable cosmologies (see e.g. Refs. [98–106] fora truncated sample) populate the neutrino fog in this regime,it is a critical part to try to reach.

Statistics.—We quantify the impact of the neutrino back-ground in terms of discovery limits. We compute these usingthe standard profile likelihood ratio test [107], which we willnow briefly discuss.

Our parameters of interest are the DM mass and cross sec-tion, as well some nuisance parameters in the form of theneutrino flux normalisations Φ = {Φ1, ...,Φnν}. Since thebackground-only model should be insensitive to mχ, we cantake it as a fixed parameter and scan over a range of values.We use a binned likelihood written as the product of the Pois-son probability in each bin, multiplied by Gaussian likelihoodfunctions for the uncertainties on each neutrino flux normali-sation:

L (σ,Φ) =

Nbins∏

i=1

P

N i

obs

∣∣∣∣N iχ +

nν∑

j=1

N iν(Φj)

nν∏

j=1

G (Φj) .

(3)The Gaussian distributions have standard deviations given bythe systematic uncertainties that we just discussed. The Pois-son probabilities at the ith bin are taken for an observed num-ber of events N i

obs, given an expected number of signal eventsN iχ and the sum of the expected number of neutrino events

4

for each flux N iν(Φj). We bin events logarithmically between

10−4 and 150 keV. The former threshold is clearly not in anyway realistic, however the main advantage of our definition ofthe neutrino floor will be that it is not based on absolute num-bers of events. Our choice of threshold is simply to allow us tomap the neutrino floor down to mχ = 0.1 GeV, but cruciallyit does not impact the height of the limit at other masses.

If we have two models, a background-only modelMσ=0,and a signal+background model M, we can test for σ > 0using the following statistic,

q0 =

−2 ln

[L (0,

ˆΦ|Mσ=0)

L (σ,Φ|M)

]σ > 0

0 σ ≤ 0,(4)

where L is maximised at ˆΦ when σ is set to 0, and (σ, Φ)

when σ is a free parameter. The model Mσ=0 is a specialcase of M, obtained by fixing one parameter to the bound-ary of its allowed space. Therefore Chernoff’s theorem [108]holds, and q0 should be asymptotically distributed accordingto 1

2χ21 + 1

2δ(0) whenM is true [109].Evaluating q0 while avoiding the need to collect many

Monte Carlo realisations for every point in the parameterspace, we exploit the Asimov dataset [107]. This is a hypo-thetical scenario in which the observation exactly matches theexpectation for a given model, i.e. N i

obs = N iexp for all i. The

test statistic computed assuming this dataset asymptotes to-wards the median of the chosen model’s q0 distribution [107].In high-statistics analyses such as ours, this turns out to bean extremely good approximation, and has been demonstratedmultiple times in similar calculations [48, 68, 71]. We fixNobs to be the expected number of neutrino and DM events,and require a threshold test statistic of q0 > 9. Therefore ourlimits are defined as the expected 3σ discovery limits.

The neutrino fog.—We can now explain how the neutrinobackground impacts the discovery of DM. The critical factorto understand is the systematic uncertainty on the background.The way to think about this is to imagine a very feeble DMsignal that closely matches one of the background compo-nents. Such a signal is saturated not just when the numberof signal events is simply less than the background, but whenthat excess of events is smaller than the statistical fluctuationsof the background. The regime of parameter space where thisoccurs is what we define as the neutrino fog.

We can quantify the neutrino fog by considering how somediscovery limit, σ, decreases as the exposure/the number ofobserved background events, N , increases. The limit evolvesthrough three distinct scalings. Initially, when the experi-ment is essentially background free, N < 1, the limit evolvesas σ ∝ N−1. Then, as N increases, the limit transitionsinto Poissonian background subtraction: ∝ 1/

√N . Even-

tually, as the number of events increases further, any would-be detectable DM signal disappears beneath the scale of po-tential background fluctuations, and the limit is stalled atσ ∝

√(1 +NδΦ2)/N [17].

If the DM signal and CEνNS background were identical,

the saturation regime would persist for arbitrarily large N .However there is rarely a value of mχ for which the back-ground perfectly matches the signal. In theory one could al-ways collect enough statistics to distinguish the two via theirtails or some other broad spectral features. So eventually thelimit will emerge from saturation, and the σ ∝ N−1/2 scalingreturns, but only by the time N has grown very large.

The “opacity” of the neutrino fog can therefore be visu-alised by plotting some gradient of this discovery limit. Letus define the index n,

n = −(

d lnσ

d lnN

)−1, (5)

so that n = 2 under normal Poissonian subtraction, and n > 2when there is saturation. The value of this index for each pointin the neutrino fog is shown by the colourscale in Fig. 2. Weadopt xenon in this example as it is the most popular target andits corresponding neutrino floor is the most familiar. Seeingas for every value of mχ there is a value of σ where n crosses2, we can join these points together in a contour to form theboundary of the neutrino fog, or neutrino floor.

The result of this procedure is the solid line in Fig. 1.There, we compared our result alongside two previous def-initions quoted frequently in the literature. The dashed lineshows the computation which follows the technique describedin Refs. [17, 60], with this specific limit taken from Ref. [16].The technique invoked there involves the interpolation of twolimits: a low-threshold/low-exposure one that captures solarneutrinos, and a high-threshold/high-exposure one that ex-cludes solar neutrinos and captures atmospheric and super-novae neutrinos. The two discovery limits are for 400 ob-served neutrino events so as to put them somewhere aroundthe systematics dominated part of the fog for certain masses.

We also showed a “one-neutrino” contour, which is formedfrom the envelope of a series of background-free 90% C.L. ex-clusion limits (2.3 events) with increasing thresholds that haveexposures large enough to see one expected neutrino event.These limits have a similar shape but are higher in cross sec-tion. One-neutrino contours have an advantage in that theyare easy to calculate, but come with the downside that they donot encode any information about the DM/neutrino spectraldegeneracy, and do not incorporate systematic uncertainties.

We compare our new neutrino floors for six different tar-gets in Fig. 3. In the SI space, they are largely similar, withthe general trend that the kinematics of scattering off lightertarget nuclei means their floors are pushed to higher masses.Helium is the extreme case, because Eq.(1) enters the asymp-totic Rχ ∝ ρ0/mχ scaling for much lighter masses than othertargets, hence why the neutrino floor above 10 GeV is sig-nificantly higher, and is set by solar neutrinos rather than at-mospheric neutrinos. Molecular targets like NaI and CaWO4,also have some distinctive shape differences due to their mul-tiple nuclei. We show complete maps of the neutrino fog forthese targets in Fig. S1. The broad features observed in thesefigures largely persist for other interactions, such as spin-dependent scattering. We show examples of these in Fig. S2.

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FIG. 3. Left: The neutrino floor for various popular direct detection targets. The general trend is for lighter targets to have floors shiftedtowards higher masses. The full topology of the neutrino fog for these same targets is shown in Fig. S1. Right: Xenon’s neutrino floor,showing the effect of improvements to neutrino flux uncertainties. The solid line shows our baseline calculation, whereas the dot-dashed anddashed lines show the result after reducing the systematic uncertainties on the solar and atmospheric fluxes by a factor of 10.

Finally, in the right-hand panel of Fig. 3 we show how ourdefinition of the neutrino floor is adaptable to improvementsin the flux estimates. The dashed line imagines that we haveobtained a factor of 10 improvement in the atmospheric neu-trino uncertainty, i.e. down to ∼2%, whereas the dot-dashedline imagines that all solar flux estimates are improved by afactor of 10. These scenarios are intended to be illustrative,rather than in anticipation of any specific improvements. Nev-ertheless, with a flock of large-scale neutrino observatories onthe horizon [110–114], it is not unreasonable to expect someimprovement to our knowledge of the neutrino fluxes, espe-cially the poorer measured solar fluxes or the low-energy at-mospheric flux [68, 115–119].

Discussion.—Given the imminent arrival of the neutrinobackground in underground WIMP-like DM searches, wehave decided to revisit and refine the neutrino floor, or perhapsmore appropriately, neutrino fog. Our floor can be interpretedas the boundary of the neutrino fog in a statistically meaning-ful way. It marks the point at which any experiment will startto be limited by the background: a cross section that is influ-enced only by overlap between the DM and neutrino spectra,and the background’s systematic uncertainties.

In contrast to prior calculations, our definition of the neu-trino floor is not based on arbitrary absolute numbers ofevents, but on the derivative of the discovery limit. As suchwe arrive at a limit that does not depend upon the recoil en-ergy threshold, so long as one does not attempt to calculate thelimit for masses that only scatter below the chosen threshold.This also means that we do not need to interpolate multiplelimits together to map the floor across a wider mass range,

and can do so with a single calculation.

The main quantitative differences between our result andthose found in the literature are the following. Firstly, wecan see for instance in Fig. 1, that our neutrino floor is no-ticeably higher in the sub-GeV region where 7Be neutrinosmimic the signal, as well as the high mass region where thesame is true of atmospheric neutrinos. Previous calculationsthat fixed a number of neutrino events ended up placing theneutrino floor deeper into the systematics dominated regimefor those masses compared to others. On the other hand, theshoulder in the neutrino floor around 6 GeV is lower in ourcase. This is because we have adopted a systematic uncer-tainty on the 8B flux of 2% which is based on a global fit ofneutrino data [88]. The previous calculation of the neutrinofloor assumed a 15% systematic uncertainty coming from thesolar model estimation.

We have applied our new technique to map the neutrino fogfor a range of targets (Fig. S1) as well as a few different DMinteractions (Fig. S2). In the right-hand panel of Fig. 3 wealso showed how our neutrino floor can be updated in the fu-ture as neutrino observatories make continued refinements tomeasured fluxes. Our technique is such that these improve-ments can be incorporated without changing the definition ofthe neutrino floor. To accompany this paper, we have supplieda public code [120] with the tools needed to compute the neu-trino floor and fog in the manner detailed here.

Acknowledgements.—This work was supported by TheUniversity of Sydney

6

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10

Supplementary Figures

Ciaran A. J. O’Hare

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FIG. S1. Detailed maps of the neutrino fog for six popular direct detection targets: Xe, Ge, Ar, NaI, CaWO4 and He, chosen to provide awide range of nuclei masses. Notable differences between the impact of the neutrino background on different targets can be noted using the“opacity” of the neutrino fog encoded by the value of n. In other words, the colourscale displays how similar the neutrino and DM signals arefor certain targets, and thus how strongly DM discovery is impacted by the neutrino background. The bright blue colour of the atmosphericregion for xenon for instance can be contrasted against the same region in the Ar, CaWO4, and He panels. This shows that the mχ =100GeV–10 TeV signal in those targets is notably less well mimicked by atmospheric neutrinos than in Xe or Ge.

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FIG. S2. Spin-dependent (SD) neutrino floors for various targets with nuclear spin: Si, NaI, Ge, Xe, and F. Exclusion limits are taken fromRefs. [12, 13, 121–123] (proton, left) and Refs. [3, 123–129] (neutron, right). The height of the neutrino floor varies much more widely in crosssection due to the range of values of the nucleon spin expectation value 〈Sp,n〉 for each target, as well as the fraction of the target comprisedof a spin-possessing isotope. For instance, for SD-proton interactions, the fluorine floor is the lowest because the proton spin expectationvalue of 19F is high ∼ 0.48 [130]. On the other hand, for SD-neutron interactions, xenon has the highest combination of 〈Sn〉 on its twospin-possessing isotopes 129Xe and 131Xe, which make up around half of natural xenon. We show only a few examples here to reduce clutter,plots for all target nuclei considered can be found at [120].