The Neutrino-Floor in the Presence of Dark Radation

The neutrino-ﬂoor in the presence of dark radation

Marco Nikolic,1 Suchita Kulkarni,1 and Josef Pradler1 1Institute of High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18, 1050 Vienna, Austria In this work we analyse the ultimate sensitivity of dark matter direct detection experiments, the “neutrino-floor”, in the presence of anomalous sources of dark radiation in form of SM or semi-sterile neutrinos. This flux-component is assumed to be produced from dark matter decay. Since dark radiation may mimic dark matter signals, we perform our analysis based on likelihood statistics that allows to test the distinguishability between signals and backgrounds. We show that the neutrino floor for xenon-based experiments may be lifted in the presence of extra dark radiation. In addition, we explore the testability of neutrino dark radiation from dark matter decay in direct detection experiments. Given the previous bounds from neutrino experiments, we find that xenon- based dark matter searches will not be able to probe new regions of the dark matter progenitor mass and lifetime parameter space when the decay products are SM neutrinos. In turn, if the decay instead happens to a fourth neutrino species with enhanced interactions to baryons, DR can either constitute the dominant background or a discoverable signal in direct detection experiments.

1. INTRODUCTION ation in these experiments, XENONnT [15] and LZ [16] is already under construction and/or commissioning and they will be sensitive enough to see a small number of The neutral current-induced coherent neutrino-nucleus neutrino background events. Finally, exposures of sev- scattering process [1, 2] once inspired the conception eral hundred ton-yr may be achieved with the respec- of dark matter (DM) direct detection experiments [3– tive liquid xenon and argon detectors DARWIN [17] and 5]. Neutrinos with energies up to several hundred MeV DarkSide-20k [18], and their reach in (mχ, σn) will be elastically scatter on atomic nuclei with a cross section limited by the neutrino floor. that is approximately enhanced by the squared number 2 More than 40 years after its prediction, coherent of neutrons, N . Similarly, the spin-independent scat- neutrino-nucleus scattering has finally been observed by tering of weakly interacting massive particles (WIMPs) 2 the COHERENT collaboration using accelerator-based is enhanced by the square of the atomic number, A , neutrino beams [19, 20]; efforts to detect the process boosting the prospects of observing an atomic recoil sig- using reactors are underway [21, 22]. These measure- nal from DM in ultra-low background detectors with keV ments provide valuable new insight into the interactions energy thresholds. Whereas DM has yet to be directly of Standard Model (SM) neutrinos with the constituents observed in the laboratory, the very process of coherent of atomic nuclei, thereby constraining non-standard in- neutrino-nucleus scattering that once started the field, teractions to quarks and the presence of new forces. In may also be the defining process in closing the window of the context of DM direct detection, the most impor- opportunity in our direct searches for electroweak-scale tant neutrino source is the sun and beyond-SM neutrino DM. Neutrinos produced in the sun, in the atmosphere physics utilizing these fluxes has been explored in [23, 24]; or in supernova explosions, among other sources, con- for more recent works see [25–33] and the review [34]. A stitute a steady flux that cannot be shielded and, given common theme in many of these studies is that new in- enough observation time, detector volume, and detection teractions of neutrinos will modify, and typically elevate sensitivity will eventually be seen as an irreducible back- the standard neutrino floor by within a factor of a few, ground in DM direct detection experiments. This limits when the new physics is subjected to complementary con- the ultimate sensitivity to discover DM of mass mχ and straints. nucleon cross section σn, and the combination of param-

arXiv:2008.13557v1 [hep-ph] 31 Aug 2020 In this work, we consider a principal alternative option. eters where this occurs is conventionally referred to as Rather than modifying neutrino interactions per se, we the “neutrino floor” [6–8]. shall primarily study the presence of new neutrino fluxes The previous years have seen steady advances in in- and their influence on DM detectability. Concretely, we creasing the sensitivity of direct detection experiments. consider DM decay as a source of SM neutrinos; only Besides a tremendous effort that is underway and aims at in a second step we shall also consider the possibility of developing and operating ultra-low threshold detectors— new interactions in an extended neutrino sector. Sub- see [9] and references therein—the classical WIMP de- stantial fluxes of these neutrinos will originate from DM tectors have now gone beyond the ton-yr mark in expo- decay within our own galaxy as well as globally, by the 47 2 sure, reaching a sensitivity of σ 10− cm and better cosmological decay of DM. Indeed, it is entirely possible n ' at a DM mass mχ of several tens of GeV. Full expo- that the Universe is filled in significant number with rela- sure results have been reported from liquid xenon exper- tivistic particles, dark radiation (DR), that may have es- iments LUX [10], PANDAX-II [11], and XENON1T [12], caped detection to date. Taking the present energy den- followed by results from current liquid argon detectors sity in dark matter, ρDM, as a calibration point, DR may DEAP-3600 [13] and DarkSide-50 [14]. The next gener- contribute as much as several per cent, ρDR . 0.1ρDM, 2 while still being allowed by gravitational cosmological or of a new type, that originate from the decay of an probes [35]. When compared to the present number (av- unstable DM progenitor X. Due to cosmological con- erage energy) of cosmic microwave background (CMB) straints [35], we restrict ourselves to DR scenarios, where photons, nCMB ( ECMB ), this implies that only a mass-fraction fX = 10 % of the total DM abun- h i dance2 injects monochromatic neutrinos via two body EDR nCMB decays of a lepton-number carrying Majoron-type scalar h i . 500 . (1) ECMB nDR relic X νν; a description of such asymmetric model h i is provided→ in the original paper [44]. As we are prin- Hence, at the expense of having much less DR quanta cipally interested in neutral current processes, the flavor than CMB photons nDR nCMB, their typical energy evolution of injected neutrinos is of no relevance. may be significantly larger, E E . If the en- h DRi h CMBi ergy is in the several tens of MeV ballpark, DR neutrinos The DR flux arriving at the Earth is by and large a induce keV-scale nuclear recoils in direct detection exper- combination of two components, the galactic flux Φ iments, altering the predictions of the neutrino floor. ν,gal and the extra-galactic flux Φν,e.gal. Assuming no direc- In fact, the neutrino energy range between 15-100 MeV tional sensitivity, for a DM particle X with lifetime τ is of particular interest because it is a window of X and mass mX decaying within the Milky Way, the differ- opportunity—framed by solar and atmospheric neutrino ential galactic flux is given by, fluxes at the respective low- and high-energy ends—to search for the diffuse supernova neutrino background (DSNB) [36, 37]. Its non-observation to-date puts a limit on the flux of electron antineutrinos φ(¯νe) < 3 /cm/s [38], t0 dΦν,gal Nν fX τ and an upper limit on a cosmological νe flux from DM = e− X r ρ Jdec δ(Eν Ein), (2) decay in this window has been established with super- dEν τX mX h i − Kamiokande data in [39]; see also [40].1 In this work, we shall consider that (a component of) DM decays into neutrinos ν but not anti-neutrinosν ¯. This possibility has been analyzed in some generality in a previous work by where t0 = 13.787 0.020 Gyr is the age of the uni- ± some of us [44], where constrains on the combination of verse, Nν = 2 is the number of neutrinos in final state, lifetime and progenitor mass from neutrino and direct de- Ein = mX /2 is the injection energy, r = 8.33 kpc is tection experiments have been derived. In this work we the distance between the observer at the Earth and the explore in detail the consequences on the DM neutrino galactic center, ρ = 0.3 GeV/cm3 is the local DM den-

floor in the allowed regions of parameter space. sity and Jdec 2.19 is the angular averaged J-factor h i ≈ The paper is organized as follows: in Sec. 2 we estab- obtained from an NFW profile [45]. Compared to t0, lish the principal DR fluxes from DM decay, in Sec. 3 we Eq. (2) probes the amount of DM that is decaying “to- introduce the neutrino-induced event rates at direct de- day.” tection experiments and list the standard neutrino fluxes, in Sec. 4 we establish the statistical tools for quantifying In contrast, DR that arrives from cosmological dis- the principal reach for DM or DR discovery. The main tances probes the decaying DM fraction at a time t dec ≤ results are then presented in Sec. 5 before concluding in t0, or, in terms of redshift, at z 0. Hence, the extra- Sec. 6. galactic flux is assembled by contributions≥ from all red- shifts. To estimate this flux, remember that neutrinos arriving at the Earth with relativistic energy Eν = ~pν 2. DARK RADIATION FROM DM DECAY em | | were emitted with higher energy Eν = Eν (1 + z). The energy-differential extra-galactic flux is then obtained via We consider the possibility that non-thermal DR is a redshift integral, which for monochromatic injection made of neutrinos, either from the Standard Model (SM) can be resolved [44],

dΦν,e.gal fX Ωdmρcrit 1 1 t(α−1) = N e− τX Θ(α 1), (3) ν 3 dEλ H0mX τX ~pν √α Ω + Ω − | | M Λ

1 Limits on the ν +ν ¯ ﬂux from MeV-mass DM annihilation have −3 that fX /τX < 6.3 × 10 Gyr for lifetimes larger than the age been derived in [41]. Fluxes of new stable decay products, of the Universe. For simplicity we take fX = 0.1 and arbitrary “boosted DM” from DM decay or annihilation have e.g. been lifetime [35] even if it implies that we occasionally slightly slip considered in [42, 43]. into the disfavored region. 2 The precise statement, at 95% C.L., is that either 3.8% of all of DM could have decayed between recombination and today, or 3

3 where α = Ein/Eν 1. We set the density parame- with h = 0.674 [46]. Considering DM lifetimes such ters Ω h2 = 0.12, Ω≥ = 0.315, Ω = 1 Ω consis- that its decay proceeds after matter-radiation equality, dm M Λ − M tent with the ΛCDM model of a ﬂat Universe. H0 = we are allowed neglect the radiation content of the Uni- 2 100h km/s/Mpc and ρcrit = 3H /(8πG) are the Hub- verse and may ﬁnd an analytic expression for the look- ble parameter and critical density at the present time back time t(z),

q Z 1 + (1 + z)3 ΩM + 1 ∞ dz0 1 ΩΛ t(z) = = ln q , (4) (1 + z )H(z ) 3H √Ω 3 ΩM z 0 0 0 Λ 1 + (1 + z) 1 ΩΛ −

p 3 where H(z) = H0 (1 + z) ΩM + ΩΛ is the Hubble the galactic and extra-galactic component, is obtained by rate at redshift z. In this way, a connection be- integration over Eν . The galactic flux reads, tween the elapsed time and the redshift is established. f t0 Equations (2) and (3) are also applicable for any X τ Φν,gal = Nν e− X r ρ Jdec(θ) . (5) two body massive final states with replacement E τX mX h i q ν → m2 + ~p 2, where m , p are the mass and momenta ν | ν | ν ν One may in fact also obtain a useful analytic expression of the massive species [44]. for the extra-galactic flux for the two-body decay with The total DR flux ΦX2ν = Φν,gal. + Φν,e.gal, a sum of relativistic final states (see App. A),

 ! 1  p − 3H0√ΩΛτX fX Ωdmρcrit ΩM /ΩΛ + 1 + 1 Φν,e.gal = Nν 1 p  , (6) mX − Ω /Ω + 1 1 M Λ −

where ρcrit is the Universe’s critical energy density to- redshift z corresponds to a mere transferal from one par- day. The extra-galactic flux asymptotically approaches ticle species (X) to another (2ν) with the comoving num- maximum for lifetimes τX . t0, and can be parametrised ber of their sum being conserved. “Early DR” originates as from a cosmological DM density that is larger by a factor (1 + z)3, hence compensating for the flux dilution max 5 fX 50 MeV 2 1 3 Φν, e.gal 1.5 10 cm− s− . (7) by a factor (1 + z)− from area expansion and increase ' × 0.1 mX in the time-interval of subsequent particle arrivals. For τ t , the total extragalactic flux becomes hence in- In Fig. 1, we exemplify the DR fluxes originating from X 0 galactic (dashed lines) and extra-galactic (solid lines) dependent of progenitor lifetime. Nevertheless, given the finite energy thresholds of any experiment, the number components for various progenitor masses mX assuming of “active” neutrinos in a detector diminishes for ν orig- a 10% mass-fraction of decaying DM, i.e. fX = 0.1. As both fluxes are inversely proportional to progenitor mass, inating at high redshift, because of their redshifting in for a fixed progenitor lifetime the flux decreases as pro- energy. The maximum flux (as the sum of galactic and extragalactic contributions) is attained when τ t . genitor mass increases. For large lifetime, both galactic X ∼ 0 and extra-galactic fluxes fall exponentially, for small life- Finally, it is interesting to note that the galactic flux is time, the galactic flux diminishes exponentially, whereas a double-valued function in mX : for a fixed progenitor the extra-galactic flux asymptotes to a constant value mass, the same flux is attained for two different progenitor lifetimes. As we will see later this feature leads to Nν fX ΩDM ρcrit/mX . This is manifest from (6) and is easily understood: a complete decay of X at some early interesting consequences for the experimental sensitivity to DR.

3 There is a well-known current discrepancy between the CMB- 3. SIGNATURES OF DR AT DIRECT inferred value of H0 and the distance ladder estimates from DETECTION EXPERIMENTS SNIa [47] (among other, more recent local measurements). Our results only depend mildly on the adopted value of H0, but we note that decaying DM scenarios such as the one considered here Dark radiation in the form of neutrinos produced by can alleviate this tension [48, 49]. unstable progenitors as described in the previous section 4

1011 takes place during the Kelvin-Helmholtz cooling phase. m = 1 MeV The prediction depends crucially on the star formation t0 X mX = 10 MeV rate [54] as a function of redshift and in this work we use the analytical ﬁt provided in [55]. The emission spectrum 109 mX = 100 MeV mX = 1 GeV is expected to be thermal [56], with each neutrino compo-

fX Ωdmρcrit mX = 10 GeV nent being assigned a speciﬁc temperature, Tνe 4 MeV, Nν ≈ mX Tν¯ 5 MeV, Tx 8 MeV (x = νµ, ν¯µ, ντ , ν¯τ ). Finally, 7 ∼ e

/s] 10 ≈ ≈

2 we note in passing that reactor- and geo-neutrinos are location dependent and relatively small, and we neglect fX = 0.1 them in this work. [1/cm 105 The second relevant aspect is the diﬀerential recoil ν 2 cross section of neutrinos on atomic nuclei. Within SM, X

Φ the process is mediated by neutral current interactions 103 for spin-independent scattering and can be written as 2 2 2 d σNν (Eν ,ER) QW GF mN F ( ~q ) ERmN = | | 1 2 , Φν,gal dE 4π − 2E 101 R ν Φν,e.gal (8) 5 2 3 1 1 3 5 where GF = 1.1663787(6) 10− GeV− is the Fermi 10− 10− 10 10 10 constant, Q = (4 sin2 θ ×1)Z + N is the weak charge W W − τX [Gyr] of the nucleus and Eν is the neutrino energy. The differential xsec is primarily a function of number of Neutrons (N) because of a near cancellation in the charge (Z) de- FIG. 1. The integrated neutrino flux originating from galactic 2 (dashed lines) and extra-galactic (solid lines) components for pendent part of QW ; sin θW 0.23 is the weak angle. The degree of coherence is given≈ by the Helm form factor different DM masses mX as a function of DM lifetime. The F ( ~q ) [60] where ~q is the three-momentum transfer to flux has the general 1/mX scaling, is inversely proportional | | to progenitor lifetime for τX . t0 and the extragalactic com- the nucleus. ponent asymptotes to a constant in the converse limit. The As a second possibility we shall consider the case that maximum DR flux as the sum of solid and dashed lines is X decays into a pair of new neutrinos νB which interact obtained for τ ∼ t0. with baryon number through a new vector particle of mass mV and gauge coupling gB [61]. In the following we shall only be concerned with relativistic states and we can potentially be detected at direct detection experi- may take the mass of the new neutrino to zero, mν 0, ments at the Earth via neutrino-nucleus coherent scatter- → so that ~pν = Eν . The nuclear recoil cross section is ing. Even in absence of such DR neutrinos, the neutrino- coherently| | enhanced with atomic number A2 and then nucleus scattering at direct detection takes place due to reads [44, 61], neutrino fluxes arising from standard ambient neutrino sources. These include solar, atmospheric and supernova- dσ (E ,E ) A2q2g4 m F 2( ~q ) m E Nν ν R = ν B N | | 1 N R . generated neutrinos. We shall refer to those fluxes as the 2 2 2 dER 2π (mV + 2mN ER) − 2Eν “standard” ones. (9) The standard neutrino fluxes adopted in this work together with their standard errors are listed in Tab. I. For For the sake of presentation, we shall only consider the solar neutrinos we use the ones based on the chemical case of a mediator that is heavy compared to the typi- 4 composition determination of [50]. While solar neutri- cal momentum transfer ~q = √2ERmN , implying mV | | & nos dominate for Eν 20 MeV, atmospheric and super- 100 MeV in practice. In that case the strength of the new . 2 2 nova neutrinos are otherwise the most important fluxes. interaction can parameterized by GB qν g /m allow- ≡ B V There are no measurements of the atmospheric flux below ing for a convenient comparison to GF of the SM sector. 100 MeV and we use the results from a FLUKA simula- The phenomenology of this model has been explored in tion [53]. The DSNB neutrinos originate from Type II [62–66], and in this work we shall consider allowed values supernovae and their largest emission in all flavours GB > GF as a possibility to boost the direct detection phenomenology of DR. The differential recoil rate introduced by neutrinos of source i is therefore given by 4 The results of this work are only mildly dependent on the solar Emax opacity problem: the more recent determination [51] primarily dR (E ) Z ν dΦ (E ) dσ (E ,E ) i R = N dE ν,i ν Nν ν R . affects the O, N, and F fluxes in addition to a 20% downward dE T ν dE dE shift of the 8B flux. We note in passing that for the last flux, R Eν,min ν R (10) being the most relevant in this context, the measurement [52] p lies in between both low and metallicity determinations of [51] where Eν,min = ERmN /2 is the minimum neutrino en- and [50], respectively. max ergy to produce a recoil on a target of mass mN ; Eν 5

max max Source Flux Φν Eν ER,ν [cm−2 s−1] [MeV] [keV] solar pp 5.98 (1 ± 0.006) × 1010 0.42 2.9 × 10−3 hep 8.04 (1 ± 0.3) × 103 1.88 5.8 17F 5.52 (1 ± 0.17) × 106 1.74 5.0 × 10−2 15O 2.23 (1 ± 0.15) × 108 1.73 4.9 × 10−2 13N 2.96 (1 ± 0.14) × 108 1.20 2.4 × 10−2 8B 5.58 (1 ± 0.14) × 106 16.6 4.5 7Be (line 1) 4.50 (1 ± 0.07) × 108 0.384 2.4 × 10−3 7Be (line 2) 5.00 (1 ± 0.07) × 109 0.861 1.2 × 10−2 pep (line) 1.44 (1 ± 0.012) × 108 1.44 3.4 × 10−2 3 atmospheric νe 1.27 (1 ± 0.2) 10 > 100 3 ν¯e 1.17 (1 ± 0.2) 10 > 100 3 νµ 2.46 (1 ± 0.2) 10 > 100 3 ν¯µ 2.45 (1 ± 0.2) 10 > 100

DSNB νe (fid) 22.12 (1 ± 0.5) 100 > 100 νe (fid) 17.69 (1 ± 0.5) 100 > 100 νx (fid) 11.06 (1 ± 0.5) 100 > 100

TABLE I. Adopted neutrino fluxes in this work; we follow [57–59] in their compilation of the fluxes (original works are referened in the main text). Beside the fluxes, the endpoint or the maximally used energy together with the corresponding maximal nuclear recoil energies on a xenon target are given. is the maximum energy neutrinos of source i can have. that the differential rate is bounded from above by max In case of DR, Eν is given by endpoint energy of the ( 1 1 1 source Ein = mX /2. The number of target nuclei per dRX2ν 100 MeV 11 keV− ton− yr− (gal.) . 1 1 1 . unit mass of detector material is denoted by NT . The to- dER mX 18 keV− ton− yr− (e.g.) tal rate in a detector will then be the sum over all isotopic compositions of relevant elements and over all neutrino This implies for the total detectable rate, sources i. We shall denote by dRν (ER)/dER the total ( 1 1 rate over all “standard” sources and by dRX2ν (ER)/dER 100 MeV 80 ton− yr− (gal.) µX2ν . 1 1 , the DR-induced non-standard recoil rate. The latter car- mX 124 ton− yr− (e.g.) ries two contributions, from galactic and extragalactic fluxes, respectively. An analytic expression for the dif- and demonstrates that for a given progenitor mass, there ferential recoil rate for the extragalactic flux can be ob- is a natural ceiling on the event rate when considering tained and is given in App. A. DR in form of SM neutrinos. The maximum nuclear recoil energy from DR is de- In Fig. 2, we show exemplary differential recoil rates rived from kinematics and depends on the mass of the in xenon introduced by DR in form of SM neutrinos for progenitor, a progenitor mass of 60 MeV in comparison with the

max 2 2 differential recoil rate due to standard neutrino fluxes. max 2(Eν ) mX The progenitor lifetime takes on values from 10 3 Gyr to E = (mX mN ), (11) − R,X2ν max 4 2Eν + mN ' 2mN 10 Gyr. Heavier progenitor particles introduce larger recoil energies at direct detection experiments, as the recoil where Emax = E is the neutrino energy at injection. ν in endpoint energy is controlled by m . Generally speak- It is worth noticing that the end point of recoil energy X ing, m 30 MeV is required to exceed solar neutrino- is directly proportional to the square of progenitor mass. X & induced events—in particular the ones induced by 8B The expected recoil rate is then given by the integral over neutrinos—because of the principal limitations in the the recoil energy, magnitude of incident DR flux. max Z ER,X2ν X dRi The latter point is further exacerbated by the fact that µX2ν = dER , (12) for a percent-level decaying fraction of DM we find that ER dER i=gal,e.gal thr there is no combination of (τX , mX ) such that the total DR-induced event rate in xenon is larger than the one where E is the threshold of the detector. Rthr from 8B neutrinos, As an example, consider the recoil rate induced by a progenitor of 100 MeV mass decaying to SM neutrinos 8 R(DR) < R( B) (ER 1 keV, mX > 1 MeV). (13) in a liquid xenon detector with negligible nominal 1 eV thr threshold. Notwithstanding a general dependence on re- While it is well known that approximately 6 GeV mass coil energy, saturating the possible DR fluxes, we find WIMP recoils mimic 8B neutrino recoils, we find that 6

5 mX = 60 MeV 30 MeV. 109 We now proceed and introduce the formalism to make Total SM ν τX = 10 Gyr quantitative statements on sensitivity of direct detection τX = 10 Myr τX = 100 Gyr 7 10 τX = 100 Myr τX = 1000 Gyr experiments to a WIMP signal in presence of DR and/or τX = 1 Gyr τX = 10 Tyr to the detecability of DR itself. 105

fX = 0.1 4. THE PRINCIPLE REACH OF DIRECT 103 DETECTION EXPERIMENTS

[1/keV/ton/yr] 101 To forecast sensitivity of an experiment, there are two R seemingly similar, yet distinct questions to address: the 1 10− first regards the ability to exclude the presence of a signal and the second regards the ability to discover the dR/dE 3 signal. The exclusion and discovery exercises can in turn 10− be performed either assuming no backgrounds or in the presence thereof. 10 5 −0.001 0.01 0.1 1 10 100 The purpose of this work is to quantify the reach of ER [keV] direct detection experiments to a DM signal in presence of DR in addition to the standard neutrino background. Furthermore, we like to see to what extent direct de- FIG. 2. Energy differential recoil rate on xenon induced by tection experiments can discover DR. We therefore deal standard fluxes (gray line) and induced by DR originating with two different signal and background hypotheses, one from a progenitor with mX = 60 MeV and various lifetimes which involves a DM signal and DR plus standard neu- as labeled (colored lines). trino background, and one which involves DR recoils as signal and standard neutrino sources as background. In each case, we are interested in understanding the exclu- DR induced neutrino recoils from 2-body DM decay can sion potential and discovery reach. neither mimic nor exceed the latter. This puts a principal Before discussing the statistical criteria, we first review limitation on the detectability/influence of DR in direct the DM (χ) signal hypothesis, which has not been dis- detection experiments: any set of prospective parameters cussed so far. The total event rate of DM recoils at the must be such that the DR flux has a component that direct detection experiments, integrated over the differ- 8 exceeds the B flux in energy, hence pointing to mX & ential recoil spectrum dRχ/dER reads

Z Z Z dRχ ρχ mN 2 2 3 f(~v ) µχ = dER = NT σn 2 A dER F ( ~q ) d ~v (14) Ethr dER mχ 2mχ,n Ethr | | ~v vmin v | |≥

In the second equality, we make the assumption of a stan- 100% efficiency of detection for ER Ethr, ignoring finite dard spin-independent DM-nucleus interaction of contact energy resolution effects. Furthermore,≥ for our purposes type. The local DM mass density is ρ 0.3 GeV/cm3 it is sufficient to integrate the differential recoil rate up χ ' and mχ (µχ,n) is the DM (reduced DM-nucleon) mass to a recoil energy of 100 keV. It should be noted that and σn is the DM-nucleon scattering cross-section with µχ, µν , and µX2ν represent recoil rates per detector mass equal coupling to protons and neutrons. At last, f(~v ) is and live-time and need to be multiplied by detector ex- the normalized galactic Maxwell-Boltzmann velocity dis- posure ε to get the total number of observed events. tribution with v0 = 220 km/s, truncated at the escape velocity vesc = 544 km/s and boosted into the detector frame with vlab = 232 km/s. Detector specifics are only entering via the recoil threshold energy E and assume A. Exclusion potential of direct detection thr experiments

In this part, we set up the formalism to obtain the 5 More precisely, if the statistical error bar on the 8B-induced events becomes smaller than the DR-induced event rate, one exclusion potential of direct detection experiments, for may actually subtract the 8B background and DR becomes de- WIMP signal or for DR signal-only hypothesis. For this tectable. This is seen by the extra island at mX ' 10 MeV in we assume the limiting case of zero background and zero left Fig. 8 for the futuristic exposure of 100 ton-yr (see below). observed signal events. It is clear, however, that for arbi- 7 trary large exposures, neutrino-induced events unavoid- of (10), and the contour εµX2ν = 2.3 in the (mX , τX ) ably lead to backgrounds and this limit will no longer be plane needs to be extracted numerically. valid. For a single channel counting experiment it is possible to take the number of events n as the test statistic. The B. Discovery potential of direct detection probability to observe n events, when λ events are ex- experiments pected is given by the Poisson distribution Pois(n λ) = n λ | λ e− /n!. Here λ can be the expected number of signal While the procedure described above works well in events εµχ or εµX2ν , background events, or their sum. the zero background scenario, the upcoming/ongoing ton For zero observed events, n = 0, and zero expected back- scale direct detection experiments with large exposures ground, the 90% C.L. upper limit on λ is obtained from will not remain background free. These backgrounds a p-value pλ = 0.1, corresponding to a 10% probability which originate from standard neutrino recoils suffer from that the outcome of an experiment is at least as extreme uncertainties in the measured neutrino fluxes (see Tab. I) as observed. It is then related to the confidence level via and therefore need to be modelled as distributions rather p 1 CL. Solving p = Pois(0 λ) for λ yields λ = 2.3 λ ≤ − λ | than fixed numbers. Hence, the statistical procedure as usual. needs to take these background distributions as nuisance In a background-free experiment with threshold energy parameters into account. This is accomplished by means Ethr and exposure ε, the 90% C.L. exclusion contour in of a profile likelihood analysis. the (σn, mχ)-plane is found by evaluating (14) under the We now proceed and estimate the discovery potentials condition εµχ = 2.3. The best exclusion that can be ob- for the upcoming direct detection experiments and be- tained in such background-free scenario, i.e. the smallest gin by introducing the likelihood functions necessary for value of σn that can be excluded for a given DM mass the profile likelihood analysis. Any observed spectrum mχ, is the one where the exposure grows to the level that of events is composed of signal and background sources. the first irreducible neutrino background event is seen, They all enter the generalized Poisson likelihood, Z ! dRν P ~ Nobs 1 = ε(Ethr) µν = ε(Ethr) dER . (15) ε α µα(θ) ~ dE e− Y X dRα(ERi , θ) Ethr R (θ~ H) = ε , Levents | N ! dE obs i=1 α R Here we have highlighted the threshold-dependence of the (17) required exposure, ε(Ethr); the neutrino-induced recoil rate is given by the sum over all “standard sources” in where the sum over sources α depends on the hypoth- Eq. (10). Using (15) to find the minimum value of σn esis under question. In this work we consider DM, that can be probed in a background-free experiment is DR, and standard neutrino-induced events, and α = at times colloquially referred to as the “neutrino-floor”; DM,X2ν, and νj together with their associated recoil we will, however, reserve the term for the discovery reach spectra dR (E , θ~)/dE are possible. Model parame- introduced later and not for the exclusion boundary. The α Ri R ~ presence of DR induced recoils leads to an additional ters are part of the vector θ and for DM they are mχ and σn, for DR they are mX and τX and for νj they are source of background and hence needs to be accounted j for while scaling the exposure. Hence the condition now the assumed standard fluxes φν . In our simulations, the reads individual recoil events ERi are random numbers. A total of Nα of them for each source α are drawn from the ! probability density function (PDF) µ 1dR /dE . Note 1 = ε(Ethr)[µν + µX2ν ] , (16) α− α R that Nα itself is a Poisson random number with mean P in an obvious modification to Eq. (15); the DR neutrino value µα. Finally, Nobs = α Nα is the total number of induced recoil rate is denoted by µX2ν . The DR rate observed events. µX2ν is a function of the progenitor mass and lifetime, When the number of observed events becomes large, we will take combinations of (mX , τX ) to evaluate the one may also switch to a binned version of Eq. (17), exclusion potential. N i Finally, it is also possible to consider DR as a signal Nbin ε P µi (θ~) ! obs Y e− α α X instead of background and we can estimate the poten- binned(θ~ H) = µi (θ~) . Levents | N i ! α tial of direct detection experiments to set constraints i=1 obs α on the DR progenitor parameters (mX , τX ). Assuming (18) zero standard neutrino backgrounds, a signal is excluded i at 90% C.L. once εµX2ν 2.3 where µX2ν is the DR- Here Nbin is the number of considered bins, µα is the ≥ i induced total event rate given in Eq. (12). In exact anal- expected number of events for α = X2ν, νj and Nobs = P i ogy to the WIMP case, the exclusion potential for DR α Nα is the total number of observed events in each i is found fixing the exposure to the value for which stan- bin i; in practice, Nobs is the random number that needs dard neutrino sources start being seen, Eq. (15). In this to be drawn. Unless otherwise stated, we shall use the case, however, τX and mX appear inside of the integral unbinned version (17) below. 8

When a source of events is declared a background un- For a discovery with 3σ significance the corresponding der the considered hypothesis, we additionally subject p0-value is given by p0 = 0.00135. If data is observed in their fluxes to appropriate uncertainties. Concretely, the critical region Z Z the hypothesis H is rejected. ≥ obs 0 we model them as Gaussians with mean φβ and vari- For the 3σ significance that we are interested in, Zobs = 2 h i ance σβ, 3. We have verified, that our generated background data 2 satisfies f(q H0) χ . Y 2 | ∼ (θ~) = gauss(φ φ , σ ). (19) In turn, the probability β for H1 being rejected is de- Lbg-flux β|h βi β β fined by,

Here, the product is over all background sources β and Z Z90 the random variables φβ are part of θ~. For the stan- β = P(Z Z90 H1) = dZ f(Z H1), (24) ≤ | 0 | dard neutrino sources the mean values φβ together with their standard errors are listed in Tab.h i I. When where f is now the PDF of Z under H1 and Z90 repre- DR is considered as a background, we take φ = h X2ν i sents the significance that can be at least obtained 90% Φν,e.gal + Φν,gal and assume σX2ν = 0.3. In addi- of the time in an experiment with data generated un- tion, we rewrite the differential rate as dRX2ν /dEν = 1 der H1. Conversely, H1 is accepted at a confidence level φ φ − dR /dE . Here . denotes quanti- X2ν × h X2ν i h X2ν ν i h i 1 β. Assuming a confidence level of 90% the alterna- ties evaluated at fixed values (τX , mX ). The assignment − tive hypothesis H1 is excluded for β = 0.1. It is thus of a 30% uncertainty is somewhat arbitrary, but meant to possible to find a value of signal with 1 β = 0.9, which be conservatively reflective of overall astrophysical uncer- − leads to Z90 3. For this case, an experiment has a 90% tainties concerning DM-induced DR fluxes. Finally, DM probability to≥ detect at least a 3σ signal. is never considered as a background in this work and its Concretely, we generate between 250 and 500 Monte- flux is not part of (19). Taken together, the total likeli- Carlo data sets for the signal plus background hypothesis hood becomes, H1 for each value of DM mass mχ or progenitor mass mX (θ~ H) = (θ~ H) (θ~) (20) and for each cross section σn or lifetime τX . These data L | Levents | × Lbg-flux sets determine the significance distribution. The proce- For a given threshold energy Ethr and exposure ε and for dure is repeated for a range of cross sections or lifetimes every point θ~ in the parameter space, we then generate until we obtain Z90 = 3 for β = 0.1 which corresponds several hundred mock realizations to build a statistical to a 3σ discovery potential at 90% C.L.. We repeat this sample. Each of these realisations we refer to as an “ex- procedure until the entire parameter space is mapped periment”. out. The task at hand is now to define appropriate sig- For each experiment one may then use the negative nal and background hypothesis which can be tested using log-likelihood as test statistic [67], the profile likelihood ratio formalism. WIMP signal/no DR background: In order to dis- ˆ cover a WIMP signal, the background only hypothesis (θ~ H0) q = 2 ln L | , (21) H0: σn = 0 must rejected; the alternative hypothesis is − ˆ (θ~ H1) H1: σn > 0. The first scenario to consider here is the L | complete absence of any DR. This allows us to recover ˆ the standard result on the neutrino floor in the (σ , m )- where θ~ maximizes the likelihood under the background- n χ ˆ plane [7]. In other words we take α = DM, νj in Eq. (17) only hypothesis H and θ~ maximizes for signal plus 0 L and β = νj in Eq. (19) to minimze the likelihood ra- background, H1. The distribution of q under H0 asymp- tio (21). 2 totically follows a χ -distribution with one degree of free- WIMP signal with DR background: The next step is dom as per Wilk’s theorem [68]. The p0-value then char- to switch on DR as an additional source of background acterizes the probability that the background-only hy- and study the influence on the WIMP discovery region. pothesis H is excluded if the q-value is greater than the 0 The background only hypothesis remains H0: σn = 0 and “observed value” q , obs the alternative hypothesis is again H1: σn > 0, but now we take the sums in Eqs. (17) and (19) to include DR, Z ∞ p = P(q q H ) = dq f(q H ), (22) i.e. α = DM,X2ν, ν and β = X2ν, ν , respectively. We 0 ≥ obs| 0 | 0 j j qobs fix progenitor mass and lifetime for this procedure. DR signal with standard neutrino backgrounds: The where f(q H0) is the PDF of q under H0. We obtain the latter| by Monte Carlo generation of mock data as final alternative is to explore the DR discovery poten- described in the next section. In terms of significance tial of direct detection experiments. Hence, we treat the DR recoils as signal in the presence of standard neutrino Z = √q, given in units of standard deviations, the p0- value is obtained by the cumulative standard normal dis- backgrounds but in absence of DM-induced events. The null hypothesis is H : σ = 0 and τ and the al- tribution Φ(x), 0 n X → ∞ ternative hypothesis becomes H1: σn = 0 and finite τ. p = P(Z Z H ) 1 Φ(Z ). (23) We therefore take the sums in Eqs. (17) and (19) with 0 ≥ obs| 0 ' − obs 9

40 40 10− 10− X2νSM Standard neutrino ﬂoor fX = 0.1 3 τX = 5 10 Gyr, mX = 66 MeV × 4 42 42 τX = 4 10 Gyr, mX = 88 MeV 10− 10− × τX = 0.6 Gyr, mX = 144 MeV

] 44 ] 44

2 10− 2 10− fX = 0.1 [cm [cm Ethr n n

σ 46 σ 46 10− 0.001 keV 10− 0.01 keV 0.1 keV 48 1 keV 48 10− 5 keV 10− 10 keV 50 keV 10 50 10 50 − 1 10 100 1000 − 1 10 100 1000 mχ [GeV] mχ [GeV]

FIG. 3. The DM exclusion potential in the presence of DR; the gray region shows the standard result without DR Left panel: the various dashed lines show the limits for varying threshold energy for DR induced by a progenitor of mX = 60 MeV and τX = 10 Gyr. The thick blue curve as the minimum of all curves is the modified neutrino floor. Right panel: dependence of the neutrino floor on progenitor mass and lifetime for exemplary combinations.

40 from DM decay can compete with standard solar neu- 10− 8 X2ν trino ﬂuxes only above the B endpoint energy. Dark fX = 0.1 B radiation introduces changes to the standard picture in GB = 10 GF 10 42 the recoil energy region where the large WIMP detectors − have ample sensitivity and ultra-low thresholds are of little beneﬁt. In the following, we will hence explore the formalism on the basis of ton-scale xenon experiments

] 44

2 10 − like the ones mentioned in the introduction. [cm Ethr n

σ 46 0.001 keV 10− 5. RESULTS 0.01 keV 0.1 keV 48 1 keV A. Exclusion limits 10− 10 keV 20 keV To begin with, we establish the exclusion potential for 50 keV 10 50 DM when DR neutrinos become a source of additional − 1 10 100 1000 background. In the remainder of the paper we shall focus mχ [GeV] on the example of a liquid xenon detector. The opera- tional procedure is then as follows: given the presence of anomalous neutrino flux, we first compute the best FIG. 4. The DM exclusion potential in the presence of DR DM limit at a fixed threshold by finding the exposure in form of baryonic neutrinos νB with GB = 10GF ; the gray for which one background event as explained in Sec. 4 A region shows the standard result without DR. The various dashed lines show the limits for varying threshold energy for is seen. In a second step, the threshold is varied and the minimum of all the exclusion curves, i.e. the mini- DR induced by a progenitor of mX = 60 MeV and τX = 10 Gyr. The red blue curve as the minimum of all curves is mum value of σn for each mχ, is obtained. The resulting the modified neutrino floor. contour in the (σn, mχ) plane then represents the optimal DM exclusion potential of the experiment when irreducible backgrounds are not dealt with. i.e. α = X2ν, νj and β = νj, respectively. In the bary- Figure 3, shows the results of the procedure. In the left onic neutrino scenario, we fix GB to an exemplary value. panel, the results for a fixed progenitor mass of 60 MeV, The formalism described above is not specific to any and lifetime of 10 Gyr decaying into SM neutrinos are particular detector. As was shown above, DR fluxes shown (X2νSM case). The various dashed curves show 10

107 107 SK(sol.) fX = 0.1 GB = 10 GF SK(DSNB) fX = 0.1 5 SK(atm.) 5 10 10 ERthr = 0.01 keV ERthr = 1.2 keV ERthr = 0.1 keV ERthr = 2 keV ERthr = 1 keV ERthr = 3 keV 3 3 ERthr = 10 keV 10 ERthr = 10 keV 10 0 0 /t /t X X τ τ 101 101

Borexino 1 1 10− 10−

10 3 10 3 − 1 10 100 1000 − 1 10 100 1000 mX [MeV] mX [MeV]

FIG. 5. Obtainable exclusion limits, for DR in form of SM neutrinos (baryonic neutrinos) in the left (right) panel, assuming fX = 0.1. The various dashed curves assume diﬀerent nuclear recoil energies as labeled. The thick blue line is the maximum of all curves. The gray areas in the left (right) panel show excluded regions from SK (Borexino) and are taken from [44].

the best limit on σn for various assumed thresholds, from The sizable new coupling boosts the DR-induced nuclear 1 eV to 50 keV, as labeled. The blue solid line is the recoil rate above the 8B neutrino recoil rate, weakening minimum of all curves and represents the optimal exclu- the minimal excludable DM-nucleon cross section by few sion potential. The curves are to be compared with the orders of magnitude once mχ < 50 GeV. gray line and shaded region which represents the stan- The potential modifications entertained in Fig. 3 and dard exclusion potential. As can be seen, modifications Fig. 4 have to be put in perspective with the current sen- to the standard result are obtained in the regions be- sitivity of multi-ton scale neutrino experiments. This has tween 6 50 GeV WIMP mass, which can be several been explored previously in Ref. [44] and the gray shaded − orders of magnitude. The right panel of Fig. 3 shows regions in Fig. 5 show the excluded regions in the DR the modified exclusion potential in the presence of SM progenitor plane (mX , τX ) from Super-Kamiokande and DR neutrinos for currently allowed (mX , τX ) combina- Borexino. The constraints included in the left panel of tions: the parameter point in the left panel is already the Fig. 5 were derived from measurements of solar- and challenged from measurements of atmospheric fluxes and atmospheric fluxes [69, 70], and from searches of DSNB searches for DSNB neutrinos by Super-Kamiokande and neutrinos [38]. In the right panel of Fig. 5 the con- the parameter points in the right panel give the largest, straint was obtained by re-purposing a Borexino solar- albeit modest modification to the exclusion potential (see axion search [71] to νB-induced elastic proton recoils; for below). further details on both cases see [44]. The inside of the The general trend in both panels of Fig. 3 is that the curves labeled according to the assumed threshold cor- exclusion potential is shifted to the right off the stan- respond to a combination of parameters where we find dard 8B shoulder. How far that shift goes, depends on that DR sourced by a DM progenitor is discoverable in kinematics. For a progenitor mass of mX = 60 MeV, a direct detection experiment. As before, we assume a DR neutrinos induce recoils that compete with DM only 10% fraction of decaying DM of type X, fX = 0.1. below mχ . 50 GeV. The exclusion potential for heav- Also shown in Fig. 5 are the exclusion potentials for ier DM is then only challenged by the standard, more DR in form of SM neutrinos (left) and baryonic neutrinos energetic atmospheric fluxes. If we are to entertain the (right) for threshold energies ranging from 0.01 keV up to possibility of mX 100 MeV into the GeV-regime, the 50 keV. In the region τ > t they are determined by the X 0 floor would be lifted by several orders of magnitude in galactic and extra-galactic components whereas in the the electroweak scale DM mass regime. region τX < t0 only the extra-galactic flux contributes. Figure 4 shows the modification to the standard exclu- The prospective regions are hence centered around t0: for sion limit (gray) from DR in form of baryonic neutrinos τ t the flux is too small and for τ t the extra- X 0 X 0 (X2νB case). The red solid line is the optimal exclu- galactic flux becomes too soft. It is interesting to note sion potential with an effective coupling GB = 10 GF . that in the right panel of Fig. 5 below the minimum as- 11 sumed threshold of 1.2 keV, DR in from of SM neutrinos can be seen. Compared to Fig. 6, owing to the increased cannot be excluded, as their induced events are super- strength of interaction, the neutrino floor is also raised in 8 seded by solar neutrinos, especially the ones originating the low mass region left of the B shoulder (mχ . 6 GeV). from the 8B reaction. On the flip side, as the thresh- In both Figs. 6 and 7, the modification of discovery limits old increases, the experiment becomes more sensitive to are seen up to the largest shown DM masses of 1 TeV. higher energy recoils generated by larger mX . Hence, in This is in contrast to the exclusion potentials of the pre- contrast to the WIMP case, and because of the general vious section. For the latter, only the number of events limitations in the DR flux in form of SM neutrinos, ever for varying threshold (up to 50 keV) enter, but not their lower thresholds as they are sought in many direct de- detailed spectral shape. In this section, we fix the thresh- tection experiments are not a beneficiary factor. In sum- old to Ethr = 0.1 keV so that DR spills into the standard mary, as can be seen, except for relatively small regions induced events at any rate, also affecting the discovery located in the quadrant mX . 100 MeV and τX > t0, potential for mχ & 50, GeV. neutrino experiments carry stronger current constrain- Finally, we also present the prospects of discovering ing power to SM DR than DM direct detection experi- DR at direct detection experiments in Fig. 8. We con- ments will every have. On the other hand, when we are sider exposures up to 100 ton-yr for a threshold of 0.1 keV to entertain new physics in the DR itself (in form of νB), for DR in form of SM neutrinos (left panel) and baryonic direct detection sensitive to lower masses mX . 20 GeV neutrinos (right panel). For the extreme case of 100 ton- is gained as the threshold drops below 1 keV. But even yr exposure, reflective of the sensitivity of future multi- with higher threshold values, and expanded region in τX ton xenon detectors such as DARWIN [17], we switch at a given mX can be covered. to a binned likelihood approach (18) with Nbin = 50. The limiting factors for the reach can be understood by analysing the most dominant background fluxes. In the B. Discovery potential bulk of the (mX , τX ) parameter space, hep neutrinos constitute the dominant background to DR. This is because We now turn our attention to the prospects of making hep neutrinos generate a large range of recoil energies with appreciable flux. Around 20 MeV progenitor mass, a discovery of either DM or DR in the presence of irre- 8 ducible backgrounds which are to be faced in the coming the B neutrinos become the limiting factor. As can be generation of direct detection experiments. Charting out seen in the left panel of Fig. 8, it is not possible to make the discovery regions will be achieved using the likeli- a 3σ discovery of DR in form of SM neutrinos, unless a hood approach outlined in Sec. 4 B. Similarly to the pre- 100 ton-yr exposure is assumed (and paralleling progress vious section, we attempt at discovering WIMP recoils with neutrino detectors is neglected.) When νB are in- in presence of standard plus DR neutrino sources and voked (right panel), they can still be discovered in a wide we also estimate the discovery potential for DR itself in mass range with only 1 ton-yr exposure. presence of standard neutrino backgrounds. Importantly, such exercise requires specification of an experiment’s exposure and threshold and is hence specific to these as- 6. CONCLUSIONS sumptions. In the following we choose ε = 1 ton-yr and Ethr = 0.1 keV to normalize on current collected data In the foreseeable future, DM direct detection exper- sets combined with an aggressive assumption on nuclear iments will face irreducible background in the form of recoil threshold. neutrino-nucleus elastic scattering. The responsible neu- In the left panel of Fig. 6, we show the DM discovery trino fluxes originate from the sun and to a smaller de- potential for X2νSM for a progenitor mass of 60 MeV and gree from the atmosphere or from the cosmological SN for progenitor lifetimes of 10 Gyr (blue line) and 100 Gyr history. In this work, we first explore the impact on the (green line). As in the previous sections, modifications to DM discovery and exclusion potential at next generation the standard result (dashed line) appear for mχ & 6 GeV, direct detection searches in the presence of new, anoma- when DR fluxes come to dominate the event rates. The lous neutrino fluxes. The latter are sourced by a per-cent modification of discovery limits are seen up to DM masses fraction of DM that may decay with arbitrary lifetime of 1 TeV. In the right panel of Fig. 6 we show the discov- after recombination. We consider either SM or baryonic ery limits for X2νSM for the same allowed combinations neutrinos νB as DR. The latter designate a semi-sterile (τX , mX ) as chosen in Fig. 3 (right). As can be seen, fourth species that interacts through gauged baryon num- the maximal allowed DR signal is at most comparable to ber. In a next step, we also address the question on the the SM neutrino backgrounds and only mildly alters the detectability of DR itself. standard neutrino floor (gray dashed). The DR fluxes principally fall into two categories, ei- Figure 7 shows the modified DM discovery potential ther constituting neutrinos that travel cosmological dis- for X2νB for a progenitor mass of 60 MeV, for a lifetime tances or neutrinos that originate from DM decay in the of 10 Gyr and GB = 10GF . For this case, we switch to galaxy. Whereas the latter requires τX & t0 as otherwise a binned likelihood approach (18) with Nbin = 50. As the DR-generating sub-component of DM would have al- already shown in Fig. 4, the same behaviour of the curve ready decayed, the former contributes for any τX and we 12

41 41 10− 10− Standard neutrino ﬂoor Standard neutrino ﬂoor 3 τX = 10 Gyr, mX = 60 MeV τ = 5 10 Gyr, m = 66 MeV 42 42 X × X 10 10 4 − τX = 100 Gyr, mX = 60 MeV − τ = 4 10 Gyr, m = 88 MeV X × X 43 43 10− 10− ] ]

2 44 2 44 10− fX = 0.1 10− fX = 0.1 [cm [cm 45 45 n 10− n 10− σ σ

46 46 10− 10−

47 47 10− 10−

10 48 10 48 − 1 10 100 1000 − 1 10 100 1000 mχ [GeV] mχ [GeV]

FIG. 6. The modified neutrino floor at 90% C.L. in case of X2νSM for a threshold ERthr = 0.1 keV and exposure of 1 ton-yr Left panel: for progenitor X mass of 60 MeV and lifetime of 10 Gyr Right panel: for exemplary allowed combinations (τX ,X ). For reference, we also show the ‘standard’ neutrino floor for the DM-only signal hypothesis.

41 tance of the DR signal. 10− Standard neutrino ﬂoor We then establish the DM exclusion potential of future τX = 10 Gyr, mX = 60 MeV WIMP searches. We do so on the example of a ton-scale 10 42 − liquid xenon detector, varying its threshold and assum-

43 ing an exposure such that the ﬁrst neutrino events will 10− be seen. In principle, modiﬁcations to the standard re- fX = 0.1

] sult, i.e. without DR, can be several orders of magnitude.

2 44 10− GB = 10 GF For DR in form of SM neutrinos, we ﬁnd that the best

[cm sensitivity to the DM-nucleon cross section is at most 45 n 10− weakened by about an order of magnitude once the sce- σ nario is subjected to existing constraints from SK data. 46 10− We find that the limitations to exclude values of σn originating from the solar 8B flux remain unaltered. This is 47 owed to the principal cap on the size of the DR flux. In 10− turn, for mχ & 6 GeV, the region in the DM parameter space that is subject to background from atmospheric, 10 48 − 1 10 100 1000 hep, and DSNB neutrinos is affected substantially, reach- mχ [GeV] ing its largest modification for mχ 20 30 GeV. This is owed to the fact that neutrinos of' 30 MeV− have similar recoil characteristics as DM of that mass range, while at FIG. 7. The modified neutrino floor at 90% C.L. in case of the same time the allowance on a non-standard DR flux is X2ν for a threshold E = 0.1 keV and exposure of 1 ton- B Rthr largest (DSNB search window). The region mχ & 50 GeV yr for progenitor X mass of 60 MeV and lifetime of 10 Gyr. becomes again unaltered, as any modification to it would require a progenitor mass mX 100 MeV; the DR then becomes amply visible in the standard neutrino detec- provide a new closed-form expression for its total flux. tors, excluding fluxes in excess of the standard ones. Fi- In the detector, the neutrinos then interact coherently nally, considering νB as DR with GB > GF we find that with the nucleus, with a cross section that approximately the entire region mχ . 50, GeV can be strongly affected, scales as (A Z)2 and A2 for SM and baryonic neutrinos, and the DM exclusion potential is modified by many or- respectively.− The benefit of DR in form of SM neutrinos ders of magnitude. These general conclusions carry over is that its interactions are known. In turn, DR in form of from exemplary parameter points to the (mX , τX ) plane, νB allows for effective interactions that can be stronger shown in Fig. 5. than in the SM, GB > GF , boosting the principal impor- In a second part, we then study the discovery potential 13

107 107 SK(sol.) ε = 1 ton yr fX = 0.1 fX = 0.1 SK(DSNB) Borexino GB = 10 GF SK(atm.) 105 105 ε = 100 ton yr ε = 1 ton yr ε = 0.5 ton yr 3 3 10 ε = 0.1 ton yr 10 0 0 /t /t X X τ τ 101 101 discovery discovery

1 1 10− 10−

10 3 10 3 − 1 10 100 1000 − 1 10 100 1000 mX [MeV] mX [MeV]

FIG. 8. Discovery limits at a 90% C.L. for a threshold ERthr = 0.1 keV Left panel: for DR in form of SM neutrinos Right panel: for DR in form of baryonic neutrinos. for DM in the additional presence of DR and for DR in trast, direct detection experiments bear better potential the presence of SM neutrino backgrounds. For this we to discover non-standard neutrino interactions of DR, as embark into the profile likelihood method which allows exemplified in the νB case, see Fig. 8. Here progenitor for a concise test of the various hypotheses. For vari- masses down to mX = 10 MeV and lifetimes as large as 5 ous combinations of detection threshold and exposure, 10 t0 can be discovered. we build a statistical sample by Monte-Carlo generat- With the advance of multi-ton scale direct detection ing a large set of mock recoil spectra that are subse- experiments, we are entering the “end-game” of WIMP quently used in the maximization of the likelihood func- detection. Neutrinos from standard sources will become tion. Through this numerically expensive procedure we an irreducible background and ultimately limit our abil- are able to obtain a DR-modified “neutrino floor”, i.e. the ity to push for ever smaller event rates and better DM boundary in the (mχ, σn) parameter space for the ability sensitivity. It is hence only timely to ask, what other to detect DM with 3σ significance when neutrino back- irreducible background there could be. Here we inves- grounds cannot be rejected. We find that DR in form tigated the perfect possibility that our Universe is filled of SM neutrinos only marginally affect our prospect for with MeV-scale DR that traces back to the instability of DM discovery (by less than a factor of two), once comple- (a component of) DM. We find that standard expecta- mentary constraints on the progenitor parameter space tions can be altered, and—in the presence of new par- are taken into account. For the νB case, we find that the ticles and interactions—in a significant manner. In the discovery potential for DM can be significantly weakened. latter case, DM direct detection experiments may then In a final step, we then ask the question on the de- be turned into DR detectors instead. tectability of DR itself. Here, standard neutrino sources Acknowledgments We thank Y. Perez-Gonzalez, and are the irreducible background, and we find that SM DR W. Waltenberger for useful discussions. MN is supported cannot be discovered in direct detection experiments un- by the FWF Research Group grant FG1. JP is sup- less a futuristic exposure of 100 ton-yr is assumed. The ported through the New Frontiers Program by the Aus- coverage in new parameter space is modest (and neglects trian Academy of Sciences. SK is supported by Elise- the paralleling advances in neutrino detectors). In con- Richter grant project number V592-N27.

Appendix A: Analytic form of dRX2ν /dER

For 2-body decays into massless neutrino DR, the energy-diﬀerential recoil rate given in Eq. (10) can be integrated analytically. For the extragalactic signal with the ﬂux given in (3) we split the integral over neutrino energy into the 14 two terms that comprise the cross section (8), ! e.gal 2 2 2 Z Ein Z Ein dRX2ν QW GF mN F (q) dΦν,e.g. ERmN dΦν,e.g. 1 = NT dEν dEν 2 , (1) dER 4π Emin dEν − 2 Emin dEν Eν q mX ERmN where Ein = 2 is the injection energy and Emin = 2 is the minimum energy to produce a recoil ER. The integrals are solved by substitution, s E 3 Ω u + 1 u = in M + 1, w = , (2) E Ω u 1 min Λ − and we obtain the expression,

 2 x  e.gal 2 2 2 ! 3 1 1 2 max dR Q G m F ( ~q ) f Ω ρ √2 Ω E m 2F1 , d, ; 1 x X2ν = N W F N | | N X dm crit xd xd 3d Λ R N − 3 3 − 3 − . T ν  min max 2 3  dER 4π mX − − ΩM Ein √ 1 + x − xmin (3) where

r 3 Ein ΩM q 3 + 1 + 1 ΩM E ΩΛ + 1 + 1 1 min ΩΛ d = , xmin = r , xmax = q . (4) −3H √Ω τ 3 ΩM 0 Λ X Ein ΩM + 1 1 3 + 1 1 ΩΛ E ΩΛ min − −

The total flux ΦX2ν in Eq. 6 is obtained from the first two terms in the bracket of (3) by setting xmin = 1. For the galactic component the integral with the flux given in 2 can be evaluated directly,

t0 gal. 2 2 2 dR − τX X2ν QW GF mN F ( ~q ) fX e 2ERmN (5) = NT | | Nν r ρ Jdec(θ) 1 2 . dER 4π τX mX h i − mX The DR rate is then the sum of both, galactic and extragalactic contributions. The baryonic neutrino rate is calculated 2 2 2 2 in the same way replacing QW GF /4π by A GB/2π.

Appendix B: Details on discovery and exclusion above)—the DM and DR spectra are well distinguishable, potentials with DR-induced events exhibiting a harder spectrum. For GeV-scale progenitors, the spectrum is cut off by the Here we show some examples of exclusion and discov- nuclear form factor F (q) 2 explaining thee similarity in | | ery limits, where SM neutrinos may mimic potential DR the rate with an electroweak-scale mass DM particle. We and DM signals in direct detection experiments. The note in passing, that in the latter case, additional inelas- reach of an experiment for a signal depends on the shape tic nuclear channels are possible as the CM energy is in of the recoil rate and the number of events in relation the tens of MeV regime. to backgrounds. In this appendix we study cases where In the right panel of Fig. 9 we follow our usual pro- DR neutrinos exhibit some degree of similarity with DM- cedure and simulate several thousand Monte-Carlo ex- induced events. periments for each neutrino source νj and calculate the In the left panel of Fig. 9 we show two examples where distribution of best parameters (mX , τX ) by minimizing DM and DR yield similar differential recoil spectra for the likelihood from Eq. (17) for α = X2ν. The discovery two different progenitor masses 50 MeV and 1 GeV and limit to SM DR neutrinos for an exposure of 1 ton-yr

DM masses as labeled such that the recoil endpoint en- and threshold ERthr = 0.1 keV is shown. In addition, the ergies match. For the lighter progenitor—and which is dots as labeled are indicative of the standard neutrino most prospective with respect to existing constraints (see ﬂuxes that limit a further extended reach.

[1] D. Z. Freedman, Phys. Rev. D 9, 1389 (1974). [2] V. Kopeliovich and L. Frankfurt, JETP Lett. 19, 145 15

109 107 Total SM ν ε = 1 ton yr fX = 0.1 τX = 10 Gyr, mX = 1 GeV hep 7 10 τX = 10 Gyr, mX = 50 MeV 8B 46 2 5 σ = 2 10− cm , m = 100 GeV n × χ 10 atm. ν 46 2 σ = 3 10− cm , m = 11 GeV 105 n × χ DSNB

fX = 0.1 3 3 10

10 0 /t X [1/keV/ton/yr] 1 τ 10 1 R 10

1 10−

dR/dE 1 10− 3 10−

10 5 10 3 −0.001 0.01 0.1 1 10 100 − 1 10 100 1000 ER [keV] mX [MeV]

FIG. 9. Left: Best ﬁt WIMP recoil spectra (dashed lines) which mimic the DR recoils (cyan solid lined) with progenitor lifetime of 10 Gyr and progenitor mass of 50 MeV (left) and 1 GeV. The total standard neutrino recoil rates are also overlaid (solid grey). Right: Contributions from each neutrino source (best ﬁts) shown by the points as labeled and determining the shape of the discovery limit for threshold ERthr = 0.1 keV and exposure of = 1 ton-yr.

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