PHYSICAL REVIEW LETTERS 122, 171801 (2019)

Novel Direct Detection Constraints on Light Dark Matter

Torsten Bringmann1 and Maxim Pospelov2,3 1Department of Physics, University of Oslo, Box 1048, N-0371 Oslo, Norway 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J 2W9, Canada 3Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8P 5C2, Canada (Received 5 November 2018; published 1 May 2019)

All attempts to directly detect particle dark matter (DM) scattering on nuclei suffer from the partial or total loss of sensitivity for DM masses in the GeV range or below. We derive novel constraints from the inevitable existence of a subdominant, but highly energetic, component of DM generated through collisions with cosmic rays. Subsequent scattering inside conventional DM detectors, as well as neutrino detectors sensitive to nuclear recoils, limits the DM-nucleon scattering cross section to be below 10−31 cm2 for both spin-independent and spin-dependent scattering of light DM.

DOI: 10.1103/PhysRevLett.122.171801

Introduction.—Attempts to discover nongravitational sensitivity completely disappears, making even cross sec- interactions of dark matter (DM) are a global effort, tions parametrically larger than weak-scale cross sections pursuing many possible avenues—perhaps as many as (e.g., σ ≫ 10−36 cm2) completely undetectable. there are viable microscopic models that link DM with Recently, it has been realized that several physical the rest of fundamental physics [1,2]. The simplicity of the processes allow us to circumvent this limitation. If the early Universe suggests that DM may be realized in the scattering on the nucleus results in the emission of a photon form of some relic particles [3,4], remnants of the big bang, or ejection of an atomic electron, e.g., the electromagnetic which we denote here as χ. fraction of the deposited energy can be larger than for Among the very few things known about the galactic elastic nuclear recoils, improving the sensitivity for mχ in v ∼ 10−3c – component of DM is the scale of its velocity, χ;gal . the few 100 MeV range [8 10]. Further constraints derive E ∼ m v2 The energy carried by DM particles, χ χ χ;gal, can be from multiple collisions of light DM. For example, inter- shared with an atomic nucleus in the process of a collision actions with fast moving nuclei or electrons inside the Sun and therefore, in principle, be detected [5]. The search for can accelerate the DM above threshold for direct detection such DM-nucleus scatterings—commonly referred to as [11–13]. This contribution typically does not exceed a direct DM detection—has seen several generations of fraction of Oð10−5Þ times the total DM flux on Earth, but experiments with ever-improving sensitivity. In the absence nevertheless greatly enhances the mass reach of existing of a credible positive signal, this has translated to conti- detectors, especially for χ − e− scattering [12]. nuously tightening limits. The latest results from the In this Letter, we consider another inevitable component v XENON1T Collaboration [6] bring the sensitivity to the of the DM flux, with velocities much higher than esc.It −46 2 cross section per nucleon below the σχ ¼ 10 cm level originates from energetic galactic cosmic rays (CRs) for the “optimal” DM mass range, mχ ∈ 15–100 GeV. This colliding with cold DM particles in the Milky Way halo, significantly constrains many models of weak-scale DM creating a secondary DM component of CRs with (semi-) (see, e.g., [7]). relativistic momenta. This new component of the DM flux, Below that mass range, and especially below 1 GeV, the called CRDM throughout this Letter, will scatter again in direct sensitivity to DM worsens rapidly. This is because the detectors, but now with much greater energy available. the nuclear recoil energy becomes smaller and cannot The goal of this Letter is to make use of this idea, Emax 2m2 v 2=m v ∼ 540 = exceed recoil ¼ χð escÞ A, where esc km s employing data from the most sensitive current direct is the galactic escape velocity and mA is the nuclear mass. detection and neutrino experiments, to establish new direct Emax E If recoil is below some detector threshold thr, the limits on DM-nucleon scattering that extend to small DM masses (formally even to mχ → 0). We will adopt a simple two parameter model fmχ; σχg without reference to a specific underlying theory. For the Published by the American Physical Society under the terms of DM-nucleon elastic cross section we assume for simplicity the Creative Commons Attribution 4.0 International license. σ σ ≡ σ Further distribution of this work must maintain attribution to the isospin-singlet structure, χn ¼ χp χ, but will the author(s) and the published article’s title, journal citation, consider both spin-dependent and spin-independent 3 and DOI. Funded by SCOAP . scattering. DM models with light mχ often require

0031-9007=19=122(17)=171801(6) 171801-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 122, 171801 (2019) Z Z subelectroweak-scale mediators [14,15] and, therefore, can dΦ dΩ ρ dΦ ρlocal dΦLIS χ dlσ χ i ≡ σ χ i D : be amenable to additional constraints from cosmology, ¼ χi χi eff dTi 4π LOS mχ dTi mχ dTi colliders, neutrino, and beam dump experiments (see, e.g., Ref. [16] for a review). However, all such constraints are ð4Þ necessarily model dependent, while constraints derived Here we introduced an effective distance D out to which in this Letter have greater generality. Despite invoking eff we take into account CRs as the source of a possible high- DM-CR interactions, in particular, they build on the same velocity tail in the DM velocity distribution. The local microscopic picture of DM-nucleon scattering as adopted gradient in the DM density is relatively well constrained in the standard presentation of limits from conventional [19,20], and that in the cosmic-ray density is very small [21]. direct detection experiments. The main uncertainty in D thus derives from the extension Step 1: From CR to DM fluxes.—Compared to CR eff of the diffusion zone, which reaches out to at least a few velocities, DM can be considered effectively at rest. Then, – the kinetic energy transferred to a DM particle in a single kiloparsec from the galactic disk [22 24]. Assuming a collision is Navarro-Frenk-White profile [25,26] for the DM distribu- tion and a homogeneous CR distribution, e.g., performing the full line-of-sight integration out to 1 kpc (10 kpc) results 1 − θ max cos in D ¼ 0.997 kpc (D ¼ 8.02 kpc). Using Eq. (1),we Tχ ¼ Tχ ; eff eff 2 can finally express the DM flux in terms of the DM energy 2 T 2m T by integrating over all CR energies Ti, Tmax i þ i i ; χ ¼ 2 ð1Þ Z Ti þðmi þ mχÞ =ð2mχÞ dΦ ∞ dΦ 1 χ dT χ Θ Tmax T − T : ¼ i max ½ χ ð iÞ χ ð5Þ dTχ 0 dTi Tχ ðTiÞ where θ is the center-of-mass system scattering angle and The flat distribution over recoil energies that follows Ti ≡ Ei − mi is the kinetic energy of the incoming CR particle i. The (spacelike) momentum transfer in the from Eq. (1) for isotropic scattering is an assumption that 2 we modify by the inclusion of the hadronic elastic scatter- collision is given by Q ¼ 2mχTχ.Forisotropic CR-DM 2 ing form factor in the simplest dipole form [27], scattering, both Tχ and Q thus follow a flat distribution, T Tmax with χ ranging from 0 to χ . Inverting Eq. (1) gives the G Q2 1= 1 Q2=Λ2 2: minimal incoming CR energy required to obtain a DM ið Þ¼ ð þ i Þ ð6Þ recoil energy Tχ, Here, Λi scales inversely proportional with the charge radius and is hence smaller for heavier nuclei; for proton sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Λ ≃ Tχ 2Tχ ðmi þ mχÞ (helium) scattering due to a vector current, one has p Tmin − m 1 1 ; 770 Λ ≃ 410 i ¼ i þ 2 ð2Þ MeV ( He MeV) [28]). We thus relate the 2 mχ ð2mi − TχÞ scattering cross section to that in the pointlike limit by

dσ dσ − T > 2m T < 2m χi χi G2 2m T : where the þ ( ) sign applies for χ i ( χ i). dΩ ¼ dΩ i ð χ χÞ ð7Þ The local interstellar (LIS) population of CRs is well Q2¼0 measured and typically described by their differential Putting everything together, we expect the following intensity dI=dR, where R is the particle’s rigidity. We CR-induced DM flux: adopt parametrizations [17,18] for dIi=dRi of protons 4 and He nuclei, the two dominant CR components. The dΦ ρlocal χ D χ differential CR flux (number of particles per area, kinetic ¼ eff dTχ mχ energy, and time) is then obtained as dΦ=dT ¼ Z 4π dR=dT dI=dR X ∞ dΦLIS=dT ð Þð Þ. For an elastic scattering cross section σ0 G2 2m T dT i i : × χi i ð χ χÞ i max ð8Þ σχi i min T T , the collision rate of CR particles with energy in the i Ti χ ð iÞ range ½Ti;Ti þ dTi inside a volume dV thus becomes Here, we only include i ∈ fp; 4Heg in the sum. In Fig. 1, LIS we plot these CRDM fluxes for various DM masses, for ρχ dΦi dΓ →χ ¼ σχi × dTidV: ð3Þ σ σ σ CRi m dT spin-independent χ ¼ n ¼ p. The contribution from χ i helium can be even larger than that from protons, but is form-factor-suppressed at large recoil energies. The flux is The resulting CR-induced DM flux is thus obtained by related to the 1D velocity distribution fðvÞ, more familiar dividing by 4πd2, where d is the distance to the source, in the context of direct DM searches, as fðvÞ¼ 2 local −1 3 implying that the volume integration reduces to an angular mχðρχ Þ γ dΦχ=dTχ. For illustration, we compare this average over a line-of-sight (LOS) integral to the Maxwellian distribution of the standard halo model

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In the last step, we have assumed Tχ ≪ mN in Eq. (1). Integrating this equation, we can relate, very approxi- mately, the differential DM flux at depth z to the one at the top of the atmosphere as dΦ dT dΦ 4m2ez=l dΦ χ χ χ χ χ; z ¼ z ¼ z z z=l 2 ð11Þ dTχ dTχ dTχ ð2mχ þ Tχ − Tχe Þ dTχ

where dΦχ=dTχ needs to be evaluated at

0 z z z=l z z z=l −1 Tχ ¼ TχðTχÞ¼2mχTχe ð2mχ þ Tχ − Tχe Þ : ð12Þ

0 For Tχ ≪ mχ, our treatment of the energy attenuation FIG. 1. Expected flux of CRDM for different DM masses reduces to that previously considered in Ref. [33]. l mχ ¼ 0.001, 0.01, 0.1, 1, 10 GeV (from top to bottom). Dotted For the mean free path of the DM particles , we use lines show the contribution from CR proton scattering alone. The DarkSUSY [34] to calculate the average density nN of Earth’s flux is directly proportional to the effective distance Deff and the 11 most abundant elements between surface and depth z elastic scattering cross section σχ, chosen here as indicated. (based on mass density profiles from Ref. [35]). We also (Inset) Compares the corresponding 1D velocity distributions need to relate the nuclear cross sections to the one on f v c 1 ð Þ, in units where ¼ , to that of the standard halo model nucleons, σχ. For spin-independent scattering, there is the (SHM) (dashed line). usual coherent enhancement, leading to [29], displayed as a dashed line in the inset. As expected, m m m 2 SI 2 Nð χ þ pÞ the CRDM population peaks at (semi-)relativistic velocities σχN ¼ σχ A : ð13Þ mpðmχ þ mNÞ and is highly subdominant at the galactic DM velocities typically considered. We neglect nuclear form factors in obtaining l. Along with — Step 2: Attenuation of CRDM flux. Very large scatter- the energy-loss ansatz (9), as compared to full numerical ing cross sections generally constitute a blind spot for direct simulations [33], this leads to conservative limits. DM detection, because they would lead to a significant Step 3: CRDM scattering in detectors.—Once a CRDM attenuation of the DM flux from the top of the atmosphere particle reaches a detector at depth z, it can transfer (part of – to the location of the detector [30 33]. The degradation in its) energy to a target nucleus inside the detector. Exploiting energy should also occur for the CRDM component, and completely analogous formulas to the case of DM → CR we can estimate the energy loss of DM particles propa- scattering discussed above, in particular, the flat distribu- gating through a medium as T Z tion of the target nucleus recoil energy N for a given DM X max dT Tr dσ energy, we find the differential recoil rate per target nucleus DM − n χN T dT : dx ¼ N dT r r ð9Þ to be N 0 r Z ∞ dT dΦ Here, Tr refers to the energy lost by a CRDM particle in a dΓN 0 2 χ χ ¼ σχNGNð2mNTNÞ z : ð14Þ N dT z;min Tmax T dT collision with nucleus . This process, in analogy with N Tχ ðTχ Þ r;N ð χÞ χ neutrino scattering, can be elastic, quasielastic, or inelastic. 2 The latter two are likely to dominate at high energies Here GNðQ Þ is a nuclear form factor and dΦχ=dTχ is T > 100 max z;min χ few MeV. (In a quasielastic process, one or more given in Eq. (8); the quantities Tr;N and Tχ follow from nucleons are dislodged from N, while in an inelastic Eqs. (1) and (2), by replacing χ → N and i → χ. process, additional hadrons are created in the χ − N The broad energy distribution of CRDM particles allows collision.) In this Letter, we will limit ourselves to elastic us, based on Eq. (14), to use both conventional direct scattering, leaving a more elaborate treatment for future detection and neutrino experiments to set novel limits on considerations. Using the uniform distribution of the σχ. It is clear that for small enough σχ the overburden mass nuclear recoil energy for isotropic scattering, we have above the detectors is transparent to CRDM, and the overall dσ =dT σ =Tmax 2 N r ¼ N r , and hence strength of the signal hence scales as σχ. For large cross sections, on the other hand, the strong attenuation of the dT 1X 1 χ − n σ Tmax ≈− T2 2m T ; CRDM energy as given in Eq. (12) also leads to an dx ¼ 2 N χN r 2m lð χ þ χ χÞ N χ exponential suppression of the signal. X 2m m Resulting limits.—We begin by addressing constraints l−1 ≡ n σ N χ : where N χN m m 2 ð10Þ from conventional direct detection experiments, which we N ð N þ χÞ derive from reported limiting values for heavy DM cross

171801-3 PHYSICAL REVIEW LETTERS 122, 171801 (2019) sections on nucleons as a function of the DM particle mass, σSI;lim m DM ð DMÞ. Assuming a nonrelativistic DM velocity f v distribution NRð Þ, and hence a standard DM flux of dΦ =dT m−2 ρlocalf DM DM ¼ DM DM NR, we can relate the count rate per target nucleus N to the average heavy DM-nucleus DM cross section σχN inside the experimentally accessible window of recoil energies TN ∈ fT1;T2g. In the limit of large DM masses, this gives Z Z T2 ∞ dΦ Θ Tmax T − T ΓDM dT σDM dT DM ½ N ð DMÞ N N ¼ N χN DM dT Tmax T T1 0 DM N ð DMÞ σDM ≃κ χN v¯ρ local m ≫ m : m ð DMÞ for DM N ð15Þ DM FIG. 2. Constraints on spin-independent DM-nucleon scatter- Here v¯ denotes the mean DM velocity and κ is an Oð1Þ ing imposed by the XENON-1T and MiniBooNE experiments. constant that, for a Maxwellian distribution, equals Solid (dashed) lines assume a CR density that equals, on average, κ −2T = πm v¯ 2 − −2T = πm v¯ 2 the local value out to a distance of 1 kpc (10 kpc). We compare ¼ exp½ 1 ð N Þ exp½ 2 ð N Þ. our limits to those deriving from cosmic microwave background In order to constrain the CRDM component, we now (CMB) observations [40], gas cloud cooling [38], the x-ray need to compare Eq. (15) with Eq. (14), taking into account quantum calorimeter experiment (XQC) [39], and a selection of σ0 m ≫ m that χN is evaluated for DM N only in the former direct detection experiments [43–45] after taking into account the case. For spin-independent scattering, we can use Eq. (13) absorption of DM in soil and atmosphere [33]. to compute the ratio of these cross sections. Realizing that the coherence factors for σχN are identical between ordinary −32 −1 Γp Tp > 35 MeV < 1.5 × 10 s : 17 DM and CRDM scattering then allows us to recast conven- ð Þ ð Þ ; tional limits on the scattering rate σSI lim per nucleon to an DM This additional exclusion region is also shown in Fig. 2. equivalent limit resulting from the CRDM component Strong constraints on spin-dependent scattering, finally, 2 SI;lim can be obtained from proton upscattering by CRDM in mχ þ mN σ σSI;lim κ v¯ρ local DM neutrino detectors like Borexino [52]. From a search for χ ¼ ð DMÞ m m m χ þ p DM m →∞ events with higher energy than solar neutrino scattering Z Z DM −1 T2 ∞ dTχ dΦχ [53,54], we deduce that the limiting scattering rate per × dTN : ð16Þ z;min Tmax dT proton is (see Supplemental Material [49]) T1 Tχ ðTχ Þ r;N χ −39 −1 ΓpðTe > 12.5 MeVÞ < 2 × 10 s : ð18Þ For the recent Xenon 1T data ([6], Fig. 5), e.g., one has σSI;lim=m 8 3 10−49 2= m ≳ 100 DM WIMP ¼ . × cm GeV for χ GeV, To apply this limit, we need to convert the proton recoil T ∈ 4 9; 40 9 κ ≃ 0 23 and Xe ½ . . keV implies . . The resulting energy to an apparent electron Te equivalent. For liquid σ limits on χ are shown in Fig. 2 for different assumptions scintillators the recoil energy of the nucleus TN and the about the size of the diffusion zone (with solid lines detected energy Te are related by the empirical law corresponding to an ultraconservative choice). For small Z DM masses, these limits exclude cross sections in the range TN dT T T N ; 10−31 ≲ σSI ≲ 10−28 2 m eð NÞ¼ ð19Þ χ cm , almost independently of χ. 0 1 þ kBhdTN=dxi Clearly, these constraints are highly complementary to existing limits on light DM [33,36–41]. Direct detection where kB is a material-dependent constant. Following the of light energetic dark sector particles was also discussed procedure outlined in Ref. [55], and thus using PSTAR in Ref. [42]. tables from [56] for hdTN=dxi, we numerically tabulate and Due to its shallow location, MiniBooNE [46] gives a invert Eq. (19) for pseudocumene (the scintillator used by particular advantage in limiting CRDM fluxes with large Borexino). The resulting constraint on spin-dependent scattering cross sections that prevent χ from reaching scattering is plotted in Fig. 3. Here the CRDM component deeply placed experiments. We utilize the measurement is produced exclusively by p − χ collisions, since 4He of elastic ν − p scattering [47] and a recent DM run [48] nuclei do not carry spin. For the mean free path in Eq. (10), that allows us to extract the beam-unrelated scattering rate. we assumed exclusively elastic scattering on nuclei as Requiring the scattering rate of CRDM on protons at derived from spin-dependent couplings σχ ¼ σn ¼ σp to MiniBooNE depth not to exceed the beam-unrelated back- nucleons (dashed and solid lines). In reality, quasielastic ground, we obtain (see Supplemental Material [49]) scattering on nucleons would dominate for energy transfers

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Theoretical Physics for generous support and hospitality during the preparation of this manuscript. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science, and Economic Development and by the Province of Ontario through the Ministry of Research, Innovation, and Science.

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