Cosmic-Ray Upscattered Inelastic Dark Matter

MI-HET-758

Cosmic-ray Upscattered Inelastic Dark Matter

Nicole F. Bell,1, ∗ James B. Dent,2, † Bhaskar Dutta,3, ‡ Sumit Ghosh,3, § Jason Kumar,4, ¶ Jayden L. Newstead,1, ∗∗ and Ian M. Shoemaker5 1ARC Centre of Excellence for Dark Matter Particle Physics, School of Physics, The University of Melbourne, Victoria 3010, Australia 2Department of Physics, Sam Houston State University, Huntsville, Texas 77341, USA 3Mitchell Institute for Fundamental Physics and Astronomy, Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843,USA 4Department of Physics and Astronomy, University of Hawaii, Honolulu, Hawaii 96822, USA 5Center for Neutrino Physics, Department of Physics, Virginia Tech University, Blacksburg, Virginia 24601, USA Light non-relativistic components of the galactic dark matter halo elude direct detection constraints because they lack the kinetic energy to create an observable recoil. However, cosmic-rays can upscatter dark matter to significant energies, giving direct detection experiments access to previously unreachable regions of parameter-space at very low dark matter mass. In this work we extend the cosmic-ray dark matter formalism to models of inelastic dark matter and show that previously inaccessible regions of the mass-splitting parameter space can be probed. Conventional direct detection of non-relativistic halo dark matter is limited to mass splittings of δ ∼ 10 keV and is highly mass dependent. We find that including the effect of cosmic-ray upscattering can extend the reach to mass splittings of δ ∼ 100 MeV and maintain that reach at much lower dark matter mass.

I. INTRODUCTION threshold. This relativistic population can thus provide the leading channel at direct detection experiments. Until recent years low-mass dark matter (DM) was rel- Although non-relativistic inelastic DM scattering has atively unconstrained by direct detection experiments. been studied in-depth in the context of the DAMA ex- The difficulty low-mass DM presents is that the recoil cess, inelastic scattering is in fact a generic feature of energy deposited is proportional to the DM mass, typi- some classes of DM models. As an illustrative exam- cally falling below the detector threshold for masses less ple, we can consider DM which couples to the Standard than a few GeV. While low-threshold detector technolo- Model (SM) by exchange of a dark photon. The DM gies have made advances in recent years, new analysis vector current can only be non-vanishing if the DM is a strategies have lead the field in constraining low-mass complex degree of freedom. But if the continuous sym- DM [1–32]. Two particularly useful strategies, which metries under which the DM is charged are all sponta- have been the subject of several recent studies, are the neously broken, then the DM generically splits into two Migdal effect [18, 23, 32–40] and cosmic-ray boosted dark real degrees of freedom, and the vector current is neces- matter (CRDM) [19, 41, 42]. These studies have all fo- sarily off-diagonal, mediating inelastic scattering. cused on elastic nuclear scattering. However, inelastic Previous model building efforts of inelastic DM have DM scattering is a generic feature of many classes of DM focused on small mass splittings, motivated by a desire to models [43–61]. Here we explore the prospects for inelas- either explain an experimental anomaly or to stay in con- tic DM detection within the CRDM paradigm. tact with experimentally accessible signals. More gener- In the CRDM paradigm, rather than finding a channel ally, there is no reason to presuppose that the mass split- through which small energy depositions can be detected arXiv:2108.00583v1 [hep-ph] 2 Aug 2021 ting be O(keV). In the example given above, the mass (e.g. Migdal electrons), one instead finds a population splitting need only be small relative to the symmetry of fast moving DM which can yield larger energy deposi- breaking scale and could easily be O(MeV-GeV). Such tion. When energetic cosmic rays (mostly protons) scat- large mass splittings are inaccessible to non-relativistic ter off non-relativistic DM particles in the halo, they can direct detection experiments and have only been probed produce a small population of relativistic DM. If these in collider experiments [59, 62]. relativistic DM particles scatter at a direct detection experiment, then the deposited energy can be well above For CRDM, the initial inelastic upscattering process can have a much larger center-of-mass energy, dictated by the cosmic-ray energies available in the interstellar medium. As a result, much larger mass splittings are ac- ∗ [email protected] cessible in this scenario as compared to the standard nu- † [email protected] ‡ [email protected] clear recoil case. Given the long path-length from cosmic- § [email protected] ray upscatter to the detector, we consider two cases: one ¶ [email protected] in which all upscattered particles reach the Earth before ∗∗ [email protected] decaying where they exothermically scatter in a detector, and one in which all upscattered particles decay before 10-6 reaching the detector, where they endothermically scat- elastic mχ1 = 100 MeV δ=0.1 MeV ter. mA = 1 GeV

) δ=1 MeV The plan of this paper is as follows: in SectionII, we -7 gχgN = 0.5 - 1 10 δ=10 MeV derive the energy spectrum of CR-upscattered inelastic s - 2 δ=100 MeV DM (CRiDM). In SectionIII we present the recoil spec- ( cm trum arising from the inelastic scattering of CRiDM, and χ 10-8 comment on the distinguishability of the scenarios under consideration. In SectionIV, we describe the bound on d ϕ / dT χ CRiDM which are placed by XENON1T. Lastly, in Sec- 10-9 tionV, we conclude with a discussion of our results and future avenues. 10-10 T flux, CRDM

II. COSMIC-RAY UPSCATTERING OF INELASTIC DM 10-11 10-5 10-4 10-3 10-2 10-1 100 101 CRDM kinetic energy,T (GeV) The direct detection of DM relies on a non-zero cross χ section for the DM scattering on nucleons or electrons. 10-2 m = 1 MeV elastic Consequently, there is also the possibility that DM can χ1 δ=0.1 MeV ﬁrst be upscattered by cosmic-rays before it reaches the -3 mA = 1 MeV 10 -3 δ=1 MeV ) gχgN = 10

detector [19]. Light DM candidates (below a GeV) can - 1 δ=10 MeV s δ=100 MeV be upscattered to relativistic energies, making their re- - 2 10-4

coils visible to experiments that were previously insensi- ( cm tive to them. Previous analyses have explored CRDM in χ 10-5 the context of simpliﬁed models [41], scattering on elec- d ϕ / dT

trons [22], and inelastic hadronic scattering [42]. In this χ -6 work we consider the effect of inelastic scattering due to 10 the DM candidate which couples to nucleons. Since we will consider processes in which the center- 10-7 of-mass energy may be much larger than the mass of the T flux, CRDM mediating particle, it will be necessary to provide a model 10-8 for DM-SM interactions beyond the contact approxima- tion. For simplicity, we assume that dark sector particles 10-9 10-5 10-4 10-3 10-2 10-1 100 101 are two Majorana fermions, χ1,2 (mχ2 − mχ1 ≡ δ > 0), which couple to a spin-1 particle (A0) through an inter- CRDM kinetic energy,T χ (GeV) 0 µ µ 0 action gχAµ(¯χ2γ χ1 − χ¯1γ χ2). A also couples to nu- 0 µ FIG. 1. Sample spectra of dark matter after upscattering by cleons through an interaction gN Aµnγ¯ n. In particular, we consider the case in which A0 couples to protons and cosmic rays (χ2, dashed) and after subsequently decaying (χ1, solid). The approximate non-relativistic total cross sections neutrons with equal strength. −31 2 these couplings correspond to is:σ ˜0 = 10 cm andσ ˜0 = Note that there are some important consistency con- 5 × 10−30cm2, for the top and bottom respectively. ditions associated with this effective interaction, in order to ensure that it arises from a consistent theory. For example, if the coupling gN remains fixed, then in the limit mA0 → 0 the gauge symmetry is unbroken, and one must have δ → 0 as a result of gauge-invariance. More gen- erally, in order for our tree-level calculation of the cross section to be consistent, one should require g . 1, and δ mχ/g. The latter condition ensures that the Yukawa . 3 couplings which generate the mass splitting are also per- where ρχ = 0.3 GeV/cm is the local DM density, and LIS turbative. In our subsequent analysis, we will focus on dΦi /dTi is the local interstellar flux of the ith species regions of parameter space where these constraints are of incident cosmic-rays (here we include contributions satisfied. from protons and helium only, with the spectra taken

The double-differential rate of cosmic-rays scattering from [63]). Ti is the incoming CR kinetic energy and Tχ2 on DM within an infinitesimal volume element is is the outgoing DM kinetic energy. σχi(Ti,Tχ2 ) is the cross section for scattering of DM with the ith cosmic 2 LIS d Γ ρχ dσχi dΦ ray species. The total upscattered DM flux at Earth is = i dV, (1) obtained by integrating this over the relevant volume and dTidTχ2 mχ1 dTχ2 dTi

2 cosmic-ray spectrum, which are the maximum and minimum kinetic energy of

max the incoming cosmic ray, such that it is kinematically Z Z Ti 3 dΦχ2 dV d Γ possible for the outgoing χ2 to have kinetic energy Tχ2 . = 2 dTi , (2) dTχ 4πd min dTidTχ dV 2 V Ti 2 max Z Ti LIS To account for the variation in the DM density ρχ dσχi dΦi = Deff dTi , (3) throughout the diffusion zone, within which the cosmic- min mχ1 T dTχ2 dTi i ray flux is assumed to be constant, an effective diffusion zone parameter D is found by integrating over the where Deff is an effective diffusion zone parameter. The eff limits of the energy integral are given by: NFW profile. These assumptions give rise to some un- certainty in the value of Deff , which can be minimized by max/min Tχ2 − 2mi + δ following Ref. [19] and conservatively considering a dif- Ti = 2 fusion zone of only 1 kpc, corresponding to Deff = 0.997 1 " # 2 kpc. This choice only modestly reduces the sensitivity of 1 T (2δ + 2m + T ) 4m2 + 2m T − δ2 χ2 χ1 χ2 i χ1 χ2 the analysis. ± 2 , 2 (2mχ1 Tχ2 − δ ) (4) The differential cross section is:

dσχi 2 2 2 2 2 = gχgN Ai F (q ) dTχ2 4m (m + T )2 − 2((m + m )2 + 2m T )T + 2m T 2 − 4m (m + T )δ + (m − T )δ2 χ i i i χ1 χ1 i χ χ1 χ2 χ1 i i χ1 χ2 × 2 2 2 , (5) 2πTi(2mi + Ti)(mA + 2mχ1 Tχ2 − δ )

where Ai is the atomic number of the ith cosmic-ray A. Decay of excited DM species and F 2(q2) is the form factor. For cosmic- ray−DM scattering we follow [19] and take the hadronic The heavier dark sector particle can decay to the form factor to be of the dipole form with Λp = 770 MeV lighter particle by the emission of a photon. This pro- and ΛHe = 410 MeV. cess will affect the energy spectrum of the relativistic dark matter component, and can potentially lead to an We assume that the lighter particle, χ1, is the domi- observable gamma-ray signature. nant constituent of the galactic DM halo, while the heavier exited state, χ , only makes up a negligible fraction. 2 -5 To be satisfied, this assumption requires a sufficiently 10 δ= 0.1 MeV mχ = 1 MeV small upscattering rate and/or a sufficiently short life- δ= 1 MeV mA = 1 MeV time of the χ . However, if the lifetime of the χ is -3 2 2 δ= 10 MeV gχgN = 10 ) -6 10 δ= 100 MeV greater than a few years, the flux of CRiDM could be a - 1 sr mixture of these two states. We therefore consider two Fermi IGRB - 1 scenarios: one in which the DM decays before reaching s the Earth and then endothermically scatters on a tar- - 2 10-7 get nucleus in the detector, and a second in which the χ reaches the Earth and exothermically scatters on a

2 cm ( GeV nucleus in the detector target. These two scenarios rep- γ 10-8 / dE

resent the extremal limits, all ﬁnite lifetime scenarios will γ

lie between these two cases. d ϕ 2 γ -9 E 10 In Figure1, we plot the differential flux of χ2 (dashed lines) produced by cosmic ray up-scattering of the -10 ambient dark matter distribution, for either mχ1 = 10 100 MeV, 1 MeV (upper and lower panels, respectively), 10-3 10-2 10-1 100 101 and δ = 1, 10, 100 MeV (orange, green, and red lines, Eγ (GeV) respectively). Note that, for δ > 0, there is always a minimum kinetic energy for upscattered χ2. This re- FIG. 2. Sample spectra of photons resulting for the decay flects the fact that, in center-of-mass frame, we must have χ2 → χ1 for various values of the mass splitting.

|~pχ2 |< |~pχ1 |, implying that in the frame of the Earth, χ2 must be forward-moving. Let us ﬁrst consider the process χ2 → χ1γ, where we

3 assume that the angular distribution is isotropic in the In the frame of the detector, the photon spectrum is rest frame of the parent particle. This process cannot then given by [64] proceed through a vector current interaction, as a result   of gauge-invariance, but may proceed through a magnetic dΦ Z ∞ dΦ m µν γ χ2 χ2 = dEχ ,(9) dipole interaction ((1/Λ)¯χ2σ χ1Fµν ). We can then find m 2 q  dE χ2 1 dE γ (x+ ) χ2 2E E2 − m2 the CRiDM and photon spectra using the results of [64], 2 x ∗ χ2 χ2 which considered this decay process in the context of in- direct detection. where x ≡ Eγ /E∗, Eγ is the photon energy in the In the rest frame of χ2, the energies of the lighter DM Earth frame. The resulting gamma ray flux is shown in Fig.2, and is generically subdominant to the intergalactic particle (Eχ1 ) and photon (E∗) are given by gamma-ray background for the relevant couplings. m2 + m2 χ2 χ1 Eχ1 = , (6) 2mχ2 m2 − m2 III. DETECTION OF CRIDM χ2 χ1 E∗ = , (7) 2mχ2 The upscattered χ1 (χ2) flux arriving at the Earth can then endothermically (exothermically) scatter in a detec- respectively. tor. The differential event rate (per unit detector mass) The post-decay CRiDM flux in the detector frame is from the incoming DM flux that can be measured by a then given by, terrestrial detector is given by max Z Tχ2 dΦ dΦ T max χ1 χ2 dR 1 Z χ dΦ dσ = dTχ2 χ χT min = dT , (10) dTχ1 T dTχ2 χ χ2 dE m min dT dE T T Tχ χ T m2 × χ2 . (8) p where ET is the recoil energy of the target nuclei, with (mχ1 + mχ2 )δ Tχ2 (2mχ1 + 2δ + Tχ2) mass mT . In contrast to the elastic case, the integral Assuming that all of the χ2 decay produced by CR up- over the DM energies now includes an upper limit due scattering decay back to χ1, the differential spectrum of to the inelastic nature of the scattering. The kinematic this χ1 population is also shown in Fig.1 (solid lines). limits are given by

min/max 1 δ T = ET mT − (2mχ + δ) − mχ χj 2 4 1 j 1/2 E (E + 2m )(2E m + δ2)(2E m + (2m + δ)2) ± T T T T T T T χ1 (11) 4ET mT where j = 1, 2 corresponds to endothermic and exothermic scattering respectively. The diﬀerential cross sections for scattering on a nuclear target with A nucleons are given by

dσχ1T 2 2 2 2 = gχgN A F (ET ) dET 2m E2 − E 2(m + m )2 + 4m T + 2m δ + δ2 + m 4(m + T )2 − δ2 × T T T T χ1 T χ1 χ T χ1 χ1 (12) 2π (m2 + 2m E )2 (T 2 + 2m T ) A T T χ1 χ1 χ1 and

dσχ2T 2 2 2 2 = gχgN A F (ET ) dET 2m E2 − E 2(m + m )2 + 4m T + 4m δ − δ2 + m 4(m + T )2 + 6(m + T )δ + δ2 × T T T T χ1 T χ2 T T χ1 χ2 χ1 χ2 (13) 2π (m2 + 2m E )2 T 2 + 2(m + δ)T A T T χ2 χ1 χ2

for endothermic and exothermic scattering, respectively. cross section at zero momentum transfer. This quantity For DM-nuclear scattering we take the form factor to cannot capture the physics of the high-energy scattering be of the Helm form [65]. It is conventional to express taking place. However, to make contact with the bounds the DM-proton interaction strength in terms of the total on the non-relativistic cross section we deﬁne a reference

4 2 2 2 4 cross-sectionσ ˜0 ≡ 4gχgN µχN /πmA. 103 An upper bound of T = 100 GeV is placed on the DM χ mχ = 100 MeV elastic kinetic energy, as higher energies do not have a signiﬁcant δ=1 MeV 102 mA = 1 GeV

) δ=10 MeV enough ﬂux for the XENON1T detector. Higher energies gχgp = 0.5 - 1 δ=100 MeV and potentially larger mass splittings may be accessible 1 with large volume neutrino detectors, and we leave such keV 10 - 1 explorations to a future work. yr

- 1 0 The resulting differential rates for endo- and exother- 10 mic scattering are shown in Fig.3. The high-energy of ( tonne -1 the incoming CRiDM has two main effects: large mass R 10 splittings of up to 0.1 GeV are accessible, while small -2 mass splittings δ . 10 MeV appear degenerate with the 10 elastic case, especially above the detector threshold. In ae dR / dE Rate, the case of exothermic scattering, sufficiently small mass 10-3 splittings can slightly enhance the observable rate, but this is not as pronounced as in the non-relativistic case. -4 10 0 1 2 3 To highlight the degeneracy in the event rate we plot 10 10 10 10 Recoil energy,E (keV) the spectra normalized to the elastic rate at ET = 5 keV R in Fig.4. We chose to plot this on a linear spectrum because it better represents the likelihood of distinguish- 103 mχ = 1 MeV elastic ablity with the low count rates expected of a dark mat- mA = 1 MeV δ=1 MeV

) g g = 10-3 δ=10 MeV ter signal. With the exception of the mχ1 = 100 MeV χ p - 1 δ=100 MeV and δ = 100 MeV case, the observable event rate spec- 101 tra exhibit an approximate degeneracy between variation keV - 1 of the mass splitting and the coupling strength. More- yr - 1 over, these scenarios are also approximately degenerate 10-1 with elastic scattering of non-relativistic heavy dark mat- ( tonne ter. The degeneracy between mass splitting and cou- R pling strength is present whether there is endothermic or 10-3 exothermic scattering.

To explain this similarity, we can consider inelastic -5 ae dR / dE Rate, 10 DM-nucleus scattering (χiA → χjA) in the center-of- mass frame. This scattering process describes both the initial CR-upscatter of DM, as well as inelastic scatter- -7 10 0 1 2 3 ing (endothermic or exothermic) of relativistic DM at 10 10 10 10 the detector. For the model we consider, DM-nucleus Recoil energy,E R (keV) scattering is isotropic in the center-of-mass√ frame. Thus, given a center-of-mass energy s, assuming a target at FIG. 3. Rates of cosmic-ray dark matter scattering in xenon rest, the energy spectrum of the outgoing products is with a 1 GeV mediator (top) and a 1 MeV mediator (bot- determined by the magnitude of the momentum of the tom). Endothermic and exothermic scattering are shown in solid and dashed respectively, for various values of the mass outgoing products in the center-of-mass√ frame, which is splitting (the δ = 0.1 MeV is omitted since it is essentially de- given by p = [s − (m + m )2]/2 s. We thus see that if √ χj A generate with the elastic case). The vertical line denotes the δ s − (mA + mχ), then the momentum of the outgo- approximate threshold of XENON1T. The approximate non- ing particle will be largely independent of δ. In this case, relativistic total cross sections these couplings correspond to −31 2 −30 2 although the normalization of the event rate may depend is:σ ˜0 = 10 cm andσ ˜0 = 5 × 10 cm , for the top and on δ, the energy spectrum of the outgoing particles will bottom respectively. not. Thus, if δ mχ, we see that the recoil spectrum due to CRiDM is essentially independent of δ, once the energy if δ/mT 1. is well above the threshold for upscattering. Similarly, For the case with mχ1 = 1 MeV, the CRiDM spectra since the target detector nucleus is much heavier than can be very different at low-energies if δ > mχ1 . But the the DM, we will have δ mT for all scenarios which we kinematics of light DM scattering at the detector requires consider. The event rate at the detector will be domi- Tχ & 0.1 GeV and so only the region where the fluxes are nated by events well above threshold, for which again we equivalent is probed. find that the energy spectrum of the outgoing particles There are several avenues to pursue in order to break is independent of δ. We can confirm this from Eq. 12 the degeneracy in the recoil spectrum. Firstly, at low and Eq. 13, which show that the differential scattering recoil energies the rate starts to differ and so lower cross section at the detector is largely independent of δ detector thresholds could aid in discrimination - espe-

5 2.0 DM at the detector is only possible if the lifetime of the m Non-rel. χ = 100 MeV excited state is large enough for the particle to travel from mχ=100 GeV mA = 1 GeV the location of cosmic-ray upscattering to the detector elastic δ=1 MeV without decaying. Given the size of the effective diffusion 1.5 δ=10 MeV zone and the typical energies of the upscattered DM, one δ=100 MeV finds that the lifetime must be & O(10 yr) in order for

( a.u. ) exothermic scattering to occur at the detector. We have T considered the case in which decay of the excited state 1.0 occurs by the process χ2 → χ1γ, mediated by an inelastic magnetic dipole operator. Although this is a simple two- body final state, there is no natural scale for the magnetic ae dR / dE Rate, dipole operator, since the DM is electrically neutral. As 0.5 such, there is no reason why the partial decay width to this channel cannot be sufficiently small. The decay process χ2 → χ1γγγ can be mediated di- rectly by the vector current interaction, with the A0 10 20 30 40 50 60 70 and three photons coupling to a box diagram. For Recoil energy,E R (keV) small masses and mass splittings, the size of this de- 2.0 cay rate can be estimated from chiral perturbation the- mχ = 1 MeV Non-rel. 3 13 4 4 4 mχ=15 GeV ory [66], yielding Γχ2→χ1γγγ ∝ αemδ /mπfπΛ , where mA = 1 MeV elastic Λ ∼ max[δ, mA]. This partial decay width depends very δ=1 MeV strongly on δ; for large enough δ, χ2 will decay before 1.5 δ=10 MeV reaching the detector. In particular, we find that for δ=100 MeV mA ∼ 1 GeV, exothermic scattering at the detector is only possible for δ 25 MeV. Similarly, if mA δ, then ( a.u. ) . . R exothermic scattering at the detector is only possible for 1.0 δ . 5 MeV. ae dR / dE Rate, IV. EXPERIMENTAL BOUNDS ON CRIDM 0.5 At present, the most stringent constraints on CRDM come from the XENON1T experiment. We place bounds 0.0 on the product of the couplings, gχgN , for two bench- 5 10 15 20 mark values for the mediator mass, mA = 1 MeV and Recoil energy,E R (keV) 1 GeV. The bounds are calculated for a variety of accessible mass splittings across a wide mass range. We FIG. 4. Rates of cosmic-ray dark matter scattering in xenon compute the upper limits by finding the coupling value with a 1 GeV mediator (top) and a 1 MeV mediator (bottom) that would produce a total of 12 events in a 1 tonne-year normalized to the scattering rate at threshold (ER = 5 keV). exposure (the 90%CL for additional events given the ex- Endothermic and exothermic scattering are shown in solid pected background of 7.36 and observation of 14 events). and dashed respectively, for various values of the mass split- The total number of events is obtained by integrating the ting. For comparison a sample non-relativistic scattering rate with similar energy dependence is shown in black dashed. The differential rate between recoil energies of 2 to 60 keV, vertical line denotes the approximate threshold of XENON1T. and folding in the energy dependent nuclear-recoil detection efficiency from [67]. The resulting bounds are shown in Fig.5. One feature which is immediately apparent is that CR cially for heavier DM. A potential way to discriminate upscattering allows for the direct detection of a relativis- the endo/exothermic scenarios would be to observe the tic component of dark matter, even if the mass splitting gamma-ray flux arising from the decay process. However is so large that direct detection of the non-relativistic the flux is very small and would be subject to astro- component would not be kinematically allowed. Even physical backgrounds. The degeneracy with the heavy if inelastic scattering of the non-relativistic component non-relativistic DM rate could be broken by exploring is not allowed on Earth, gravitational infall may allow the kinematic endpoint, which would be much larger for scattering to proceed in the Sun, leading to gravitational CRDM. While the flux and rate fall dramatically with capture [68–70]. In that case, neutrino detectors which energy, large volume neutrino detectors may still be sen- look for the neutrinos produced by DM annihilation may sitive to a CRDM signal. be sensitive to these models (see, for example, [71]). But Note that the exothermic scattering (χ2N → χ1N) of CR upscattering allows direct detection of the relativistic

6 beyond the elastic scattering case. This eﬀect is due to the kinematics of the exothermic scattering, for a given -1 10 value of the mass splitting (up to a maximum), there is -18 10 a mass at which T min (Eq. 11) vanishes. This mass is χ2 −3 2 approximately given by ET mT /δ ∼ 10 [GeV] /δ, for 10-2 a representative recoil energy of ET = 10 keV. Around 10-20 this mass upscattered particles of a wider range of en- N g x ) ergies can produce a signal above threshold, enhancing 2 10-3 the rate. For a ﬁxed δ, this enhancement in sensitivity -22 ( cm disappears for small enough m ; in this regime, most 10 0 χ2  σ of the kinetic energy released by exothermic scattering is

g coupling, elastic -4 transferred to the outgoing DM particle, not the nucleus. 10 δ= 0.1 MeV -24 For the large interaction cross sections under consid- δ= 1 MeV 10 δ= 10 MeV eration it becomes important to consider the eﬀects of -5 δ= 100 MeV attenuation of the dark matter in the overburden. If 10 the dark matter interacts too strongly it will not be able 10-3 10-2 10-1 100 101 102 to penetrate to the depth of the XENON1T detector at LNGS. While a detailed treatment of attenuation due DM mass,m (GeV) χ1 to inelastic scattering is beyond the scope of this work, we estimate the couplings at which the eﬀects of atten- 101 uation become important following [41]. We solve the energy loss equation for elastic scattering and determine 10-26 the coupling whereby a dark matter particle arriving at the Earth with Tχ = 10 GeV would be attenuated to an energy where the maximum recoil energy it can im- 0

N 10 part is below the XENON1T threshold (approximately

g −3 x ) Tχ = 10 mχ). Using the elastic scattering cross sec- -28 2 10 tion to estimate the attenuation should be conservative ( cm

0 as inelastic scattering reduces the total cross section.  σ In the non-relativistic case, attenuation causes g coupling, 10-1 elastic δ= 0.1 MeV XENON1T to be blind to DM with nucleon cross sec- −31 2 δ= 1 MeV 10-30 tions & 10 cm . Therefore, for elastic scattering, δ= 10 MeV the XENON1T CRDM signal can provide stronger con- δ= 100 MeV straints than the non-relativistic signal for DM masses even above 100 GeV. However, DM masses larger than 10-2 10-3 10-2 10-1 100 101 102 & 0.1 GeV will have stronger non-relativistic constraints from small-volume shallow detectors (e.g. the CRESST DM mass,m χ (GeV) 1 surface run [72]), rocket-based detectors (e.g. XQC [73]) FIG. 5. Bounds on inelastic CRDM from XENON1T for me- or cosmologically derived bounds (e.g. [74]). For the diator mass of 1 MeV (top) and 1 GeV (bottom) for various inelastic model considered here these constraints are values of the mass splitting where endothermic (exothermic) greatly weakened since non-relativistic DM cannot ac- scattering in the detector is shown in solid (dashed). The light cess mass splittings & 0.1 MeV for mχ . 40 GeV. We grey contours denote the equivalent non-relativistic cross sec- therefore do not include any non-relativistic constraints tion for the given mass and coupling (denoted on right axes). as they are not competitive beyond the elastic scattering scenario. A further consideration for the large coupling region component even for mass splittings so large that scatter- is that the decay photons produced in the endothermic ing in the Sun is not allowed. scenario may be observable by the FERMI satellite. We For the most part, this enhanced sensitivity does not set an upper limit where the decay photon flux would ex- significantly depend on whether or not the upscattered ceed 20% of the FERMI intergalactic gamma-ray back- DM decays before reaching the detector, because the ki- ground [75]. The resulting bound is shown in Fig.6 for netic energy of the dark particle incident at the detector the light mediator case only, since in the heavy media- is in any case well above the kinematic threshold for in- tor case we are not able to constrain perturbative val- elastic scattering. In general we find that with increased ues of the coupling. As expected from the gamma-ray mass splittings the bounds become less stringent. How- flux derived in sectionIIA, the constraints derived from ever, for some regions of the parameter space, particu- the decay process are subdominant compared to those larly in the low-mass mediator case, we find that exother- from direct detection experiments. However, the gamma- mic scattering can significantly increase the sensitivity ray flux is not subject to the effects of attenuation and

7 thus the FERMI bounds provide a complementary search V. CONCLUSION channel in this large-coupling region. Future direct detection experiments with larger expo- sures and/or lower thresholds such as LZ or SuperCDMS will not greatly improve on the bounds derived in this In this paper we have extended the CRDM paradigm work. This is because CRDM is not very sensitive to to include a model of inelastic DM. We have derived con- threshold and requires two scattering events, meaning straints on two scenarios, where the DM decays en-route that the bounds on the product of the couplings will scale to a terrestrial detector and where the DM exothermi- as ∝ (exposure)−1/4. cally scatters inside the detector. The strongest constraints were found to come from XENON1T, however, in the region of large coupling, attenuation eﬀects become important and constraints on the DM decay from FERMI will be competitive. 101 δ= 0.1 MeV δ= 1 MeV 10-14 δ= 10 MeV The large center-of-mass energy available in CR colli- δ= 100 MeV sions has allowed us to probe halo DM with much larger 0 10 mass splittings and much smaller DM masses than can be N -16 g 10 x ) probed with traditional non-relativistic direct detection. 2

-1 ( cm

10 0 

σ Interestingly, we ﬁnd that the nuclear recoil spectrum 10-18

g coupling, observed at the detector exhibits degeneracies between the dark matter mass, mass splitting, and scattering cross -2 10 section. To break this degeneracy, one could use lower 10-20 threshold detectors. ACKNOWLEDGMENTS 10-3 10-3 10-2 10-1 100 101

DM mass,m χ1 (GeV) The work of BD and SG are supported in part by the FIG. 6. Constraints on inelastic CRDM due to endothermic DOE Grant No. DE-SC0010813. The work of JK is scattering in XENON1T (solid) and the gamma-ray ﬂux from supported in part by DOE grant DE-SC0010504. JLN FERMI (dot-dashed), assuming mA0 = 1 MeV. Additionally and NFB are supported in part by the Australian Re- we show the region where the eﬀect of attenuation can not search Council. The work of IMS is supported by the be ignored and thus XENON1T would be blind to CRDM U.S. Department of Energy under the award number DE- (gray). SC0020250

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