Observational signatures of Dark matter- nucleus bound states.

Maxim Pospelov FTPI, U of Minnesota

In collaboration with A. Berlin, H. Liu, H. Ramani (to appear this month)

1 Plan

• Introduction: a closer look at rare DM species. • Dark photon mediated bound states. • Auger-style capture process: Atom + DM à (Atom-DM) + Energy • Constraints from Xenon1T. • Conclusions

2 Pushing down the sensitivity to energy deposition in direct detection • In the last decades there has been a push to extend the sensitivity of direct detection to very light dark matter, and go below the 1 keV energy deposition scale Ionization, Large direct detection Large neutrino E > few eV, th experiments experiments E >200 102/kg/day/keV th keV counting rates at Eth >1keV counting -2 rates ~ 10-4/kg/day/keV ~ 10 /ton/day/MeV for E ~ few keV for E ~ few MeV

More sensitivity to small energy deposition: superCDMS, CRESST, Damic etc Bigger and cleaner DM detectors: Xenon 1T, LUX, Panda-X 3 Xenon-based dark matter experiments

• Based on two signals: initial scintillation “on impact” (S1) and final scintillation (S2) from drift electrons. • Ratio of S1/S2 is used to discriminate between electron and nuclear recoils • More or less same technology is used in Xenon10, Xenon100, LUX, Panda-X

Motivation for today’s talk: full use of available data in search of new physics: Today we will use Xenon-type experiments to probe DM species with relative abundance ~ 10-14. Search for WIMP-nucleus scattering (latest LUX, XENON 1T and PANDA-X results)

Strong constraints on nuclear recoil

On y-axis: Abundance * cross section scN à fc × scN fc = rc /rDM is the abundance of DM sub-component, fc ≤ 1.

§ Optimum sensitivity, mWIMP ~ mNucleus irrespective of abundance.

§ No sensitivity below mWIMP ~ few GeV, due to exceedingly small recoil that does not give much light or scintillation. § Summer 2020 – interesting hint on excess in electron recoil. 5 Many well-motivated models are constrained DM particles themselves + may be extra extra be may + themselves DM particles Very economical extensions of the SM. of the extensions economical Very predictive. very be Can force. mediator

6 Impressive results by Xenon1T in achieving low backgrounds and high sensitivity

2015 projections, 1512.07501

This is the most sensitive device for rare keV-scale events.

2020 results, 2006.09721 There is a slight excess in low- energy bins à lots of attempts to explain it, including using

rare DM species, �!< or ≪ 1 7 keV could be “intrinsic” scale built into dark matter

• 3 keV dark matter has 105 cm-3 abundance, and 1012/cm2/sec flux. If it is quasi-stable it can get absorbed by atoms, avoiding stellar bounds • Dark photons with mixing in the 10-16 10-15 range, ~ 3keV mass, 2006.11243, .13929, .14521 (Alonso-Alvarez et al, An et al, etc) • Alternatively, ALPs, same mass, coupling to electron axial vector current, 2006.10035, or ~2 keV long-lived excitations of DM (Berlin et al) • keV scale can be the scale of DM- nucleus bound state [today’s talk] 8 Several blind spots for direct detection

• ~MeV scale dark matter: Kin Energy = mv2/2 ~ (10-3)2MeV~eV. Elastic scattering is below the ionization threshold

• Relatively strongly-interacting subdominant component of Dark Matter. Thermalizes before reaching the underground lab, Kin energy ~ kT ~0.03 eV Elastic scattering is below the ionization threshold A blind spot: thermalized DM component • Series of papers with Ramani, Rajendran, Lehnert, et al.

10-24 XQC[Rocket] • 1 per mil - 1ppm dark matter

10-26 DM with strong-ish cross CRESST

] RRS[Balloon] section is invisible. Drowning 2 -28 cm

[ 10 n in backgrounds at the surface, -I CDMS and thermalized deep inside. 10-30 UG Deep -6 fDM=10 • One can use nuclear isomers, 10-32 10-1 1 101 102 103 104 105 106 i.e. extremely long-lived M [GeV] 10-24 nuclei, to search for unusual Current Limit (3a+3b) de-excitations. Usual selection 10-26 rules are avoided because even ] 2 Projection -28 cm [ 10 ( + ) thermalized DM provides

n 3a 3b ) (3a large momentum transfer. 10-30 Projection 180m -6 • New limits from Ta. fDM=10 10-32 10-1 1 101 102 103 104 105 106

M [GeV] Several blind spots for direct detection

• ~MeV scale dark matter: Kin Energy = mv2/2 ~ (10-3)2MeV~eV. Elastic scattering is below the ionization threshold

• Relatively strongly-interacting subdominant component of Dark Matter. Thermalizes before reaching the underground lab, Kin energy ~ kT ~0.03 eV Elastic scattering is below the ionization threshold Does not have to be a blind spot. Can be easily responsible for the e.g. Xenon1T electron excess due to the bound state formation Past work on DM-nucleus bound state in other models • Can occur if there is a “doublet” of DM: neutral state + charged state, separated by ~ 20 MeV or less. (MP, Ritz, 2008; An, MP, Pradler 2012). See also Fornal, Grinstein, Zhao 2020.

• Inside a large nucleus negatively charged “WIMPs” have a binding of up to 20 MeV. Therefore, for smaller D m, e.g. a weak style capture becomes possible Z + c 0 à (Z+1 c -) + Energy. • The model is a bit “tuned” as natural splitting between charged and neutral for an EW-charged particle is ~ 150 MeV.

• First search of this process was reported this year by KamLAND- 12 Zen, 2101.06049 hep-ex, Abe et al. Dark photon induced nucleus-dark matter bound states

1, 2 1, 2 1, 2, 3, Asher Berlin, Hongwan Liu, Maxim Pospelov, ⇤ and Harikrishnan Ramani † 1School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA 2William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA 3Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA (Dated:) Electroweak scale dark matter particles may form bound states with nuclei If there exists an 3 attractive force of sucient strength. In this paper we show that the dark photon (A0)withO(10 ) kinetic mixing and mass in the MeV-to-100-MeV range provides enough attractive strength to generate keV-scale binding with nuclei. The process of DM-nucleus bound state formation liberates energy in the form of electron and gamma radiation, and for direct detection experiments this will be consistent with monno-energetic electron-like events. We show that the small concentrations of 14 such dark matter particles, O(10 ), from the total DM energy density is sucient to generate the observable signal consistent with XENON1T electron recoil excess, provided that the strength of DM-Xe binding is in 2.5 keV range. The recombination signal can have a time structure built to it, with daily and seasonal⇠ modulations present.

I. INTRODUCTION of bound states. Specifically, we are exploring the process of -atom “recombination”,

Over the years, direct detection dark matter exper- A+ (A)b.s. + Q, (1) iments have developed into a precision tool of learn- ! ing about sub-MeV energy deposition by exotic sources. where Q represents electromagnetic energy release coin- While primary focus and motivation for these experi- ciding with the binding energy. For the process (1) to ments is to search for the weakly interacting massive par- occur, the -nucleus coupling have to be sizeable, which ticle (WIMP) elastic scattering in nuclei, the scope of the in turn leads to quick thermalization and drastic over- searches has been extended to include electron scattering, concentration of inside the Earth [11–13] (+ our paper the absorption of dark matter, exo-and endo-thermic in- in prep). Thermal energies at depths corresponding to elasticity in the WIMP-nucleus scattering etc []. locations of underground laboratories housing the direct Among the direct detection dark matter experiments detection experiments means that this component of DM the suit of large scale dual-phase xenon detectors play is invisible in the elastic scattering channels. The forma- especially important role. With the background counts tion of the bound states, however, can release a substan- 5 1 1 1 tial amount of energy, and Q<10 keV is of primary below 10 kg day keV , XENON1T experiment is setting new benchmark sensitivity not only in the WIMP consideration in this paper. nucleus scattering, but also for the electron recoil of The possibility of observing DM bound states with O(keV) and below [1]. Recently, the collaboration re- nuclei has been pointed out several times in the liter- ported O(2 3) excess of events consistent with elec- ature [14–16] with the main focus on MeV-to-10-MeV tron recoil, and centered around the 2 3 keV energy energy release range. Specifically, the charged-neutral deposition [2]. As pointed out in the variety of theoreti- pairs of DM states can undergo charge exchange reaction cal studies, this energy can be consistent with a variety with nuclei and form stable bound states, provided that m m < 20 MeV. The search of such process of models. These models are typically based on a rather charged neutral substantial fluxes of particles (neutrinos, DM, axions etc) has been performed recently by the KamLAND-Zen col- that traverse the detector and have a very small rate of laboration [17]. Other examples include a possibility of interaction with matter due to very small coupling (e.g. DM-neutron transition in the field of the nucleus, with 16 the capture of resulting neutron into a bound state [16]. dark photon dark matter with the 10 coupling to electrons [3, 4], tiny electromagnetic⇠ moments of neutri- While these models require a certain degree of intricate nos and dark radiation [5–9], exothermic dark matter [10] model building, the model considered here is perhaps one etc.) of the most studied in the literature of the last fifteen In this paper, we explore a conceptually di↵erent possi- years [18–20]. bility. A very subdominant flux of dark matter particles, Specifically, we consider a WIMP charged under new 14 DarkU(1) 0photonforce that mediated has kinetic mixing Dark with Matter the SM interaction photon, that can be as small as O(10 ) fraction of galactic dark matter, having a relatively sizeable interaction with mat- which a↵ords bound states with nuclei in a very well de- ter, can induce an electron recoil signal via the formation• Considerfined corner a stable of elementary the parameter particle space. charged Namely, under we considerU(1)’. the dark sector Lagrangian with m m , A0 2 1 2 " mA 2 = (F 0 ) F 0 F + 0 (A0 ) +¯(iD m ), ⇤ [email protected] L 4 µ⌫ 2 µ⌫ µ⌫ 2 µ µ µ † [email protected] (2)

-3 • The choice of parameters of interest: e ~ up to 10 ; mA’~ 10-100 -2 MeV, mc ~ 10 - 1000s GeV or larger, adark ~ 10 – 1.

• Given the choice of parameters abundance can be calculated, assuming the standard cosmological history. However, I am going to

treat fc as a free parameter taking it small. (No E injection limits)

• Thus, the standard visible dark photon constraints apply. 13 Constraints on visibly decaying dark photons −2

ε 10 e (g-2) NA48/2 A1 HL-LHC −3 CMS 10 E774 LHCb APEX BaBar mu3e (phase1) −4 NA64(e) Belle II 10 mu3e (phase2) HPS LHCb: 15 fb-1 (solid), 300 fb-1 (dotted) E141 −5 10 18 20 FASER nu-Cal NA62-dumpDarkQuest: 10 pot (solid), 10 pot (dotted)

−6 10 FASER2CHARM

−7 10 E137 SHiP 10−8 SN1987A 10−9 10−3 10−2 10−1 1 10 102 103 mA' [GeV]

Figure 17: Dark photon into visible final states: Á versus mAÕ . Filled ar- eas are existing limits from searches at experiments at collider/fixed target (A1 [412], LHCbBound [235],CMS state [413],BaBar formation [354], KLOE is [ 256possible, 355, 414, in415 ],this and NA48/2corner [358]) and old beam dump: E774 [352], E141 [353], E137 [346, 416, 417]), ‹-Cal [418, 419], CHARM 14 (from [420]), and BEBC (from [421]).Bounds from supernovae [126] and (g 2) [422] are ≠ e also included. Coloured curves are projections for existing and proposed experiments: Belle- II [423]; LHCb upgrade [424, 425]; NA62 in dump mode [426] and NA64(e)++ [338, 339]; FASER and FASER2 [376]; seaQUEST [194]; HPS [427]; Dark MESA [428], Mu3e [429], and HL-LHC [372]. Figure revised from Ref. [9].

– 70 – 2

where “primed” fields stand for dark photon, Dµ = for a heavy m and small mediator mass, i.e. taking ef-

@µ igdA0 is the covariant derivative w.r.t. dark U(1), fectively mA0 ,RN 0 limit, the approximate Bohr-like and is a stable particle, the sub-component of DM. The expression must be! valid: fermionic nature of is not essential, and all considera- tions in this paper equally apply to scalar as well. The 2 2 "e↵ Z µ self-interaction of ¯ pairs induced by the attractive in- Eb.s. 7.8 keV 3 . (6) ' ⇥ 10 54 100 GeV teraction mediated by A0 has important consequence for ⇣ ⌘ ✓ ◆ ⇣ ⌘ annihilation, as resonances and capture to (¯) bound states can significantly increase the annihilation cross 2 section [19, 21–23]. As a result, the annihilation rate can 2

wheresignificantly “primed” fields exceed stand the for WIMPdark photon, benchmarkDµ = ratefor a 1pbn heavy mc, and small mediator mass, i.e. taking ef- where@ “primed”ig A is the fields covariant stand derivative for dark w.r.t. photon, dark UD(1),µ = fectivelyfor a heavym⇥ ,Rm and0 limit, small the mediator approximate mass, Bohr-likei.e. taking ef- µ possiblyd 0 making a subdominant component of DM.A We0 N @ ig A0 is the covariant derivative w.r.t. dark U(1), fectively m ,R! 0 limit, the approximate Bohr-like µ and dareis a stable going particle, to consider the sub-component modern value of DM. of Thef expression⇢ /⇢ musttoA0 be valid:N ! andfermionic is a stable nature particle, of is the not sub-component essential, and all of considera- DM. The ⌘ expression DM must be valid: fermionictions inbe nature this a small paper of equally freeis not parameter, apply essential, to scalar and noting allas well. considera- that The deviations from 2 2 "e↵ Z µ 2 tionsself-interaction in thisthe paper standard of equally¯ pairs thermal apply induced to cosmological scalar by the attractiveas well. scenario in- The couldEb.s. result7.8 keV 3 2 . (6) ' ⇥ 10 "e↵ 54 Z100 GeV µ teraction mediated by A0 has important consequence for ✓ ◆ self-interactionin tiny off¯.pairs Furthermore, induced by the we attractive assume unbroken in- E chargeb.s. 7.8 keV⇣ ⌘ 3 ⇣ ⌘ . (6) annihilation, as resonances and capture to (¯) bound ' ⇥ 10 54 100 GeV teractionsymmetry mediated by inA the0 has importantsector, ı.e. consequence no mass for splitting among ⇣ ⌘ ✓ ◆ ⇣ ⌘ annihilation,states can significantly as resonances increase and capture the annihilation to (¯) crossbound section [19,states. 21–23]. At As the a result, same the time, annihilation the phenomenology rate can of A0 is states can significantly increase the annihilation cross significantly“standard”, exceed the and WIMP usual benchmark limits on rate dark 1pbn photonc, apply [24], sectionpossibly [19, making 21–23]. Asa subdominant a result, the component annihilation of DM. rate⇥ We can significantlyso that exceed e↵ theectively WIMP for benchmark all . rate 1pbn c, are going to consider modern value of f ⇢/⇢DM to⇥ possiblybe a small making free parameter,a subdominant noting component that deviations⌘ of DM. from We arethe going standard to consider thermal modern cosmological value scenario of f could⇢ /⇢ resultto ⌘ DM be ain small tiny f free.II. Furthermore, parameter, BOUND noting we STATE assume that unbroken PARAMETER deviations charge from SPACE FIG. 1: Critical value of coupling, as function of m symmetry in the sector, ı.e. no mass splitting among the standard thermal cosmological scenario could result that allow binding to di↵erent elements. Mediator mass states. At the same time, the phenomenology of A0 is in tiny f.The Furthermore, Yukawa interaction we assume between unbroken point-like charge and elec- symmetry“standard”, in the and usualsector, limits ı.e. on no dark mass photon splitting apply among [24], is fixed to 15 MeV. so thattrons e↵ectively and protonsfor all . is given by states. At the same time, the phenomenology of A0 is “standard”, and usual limits on dark photon apply [24], exp( mA0 r ri ) so thatII. e↵ectively BOUNDV for(r STATE all)= . PARAMETER"p↵↵d Q SPACEi | | (3) r rFIG. 1: Critical valueIn this of coupling, expression, as functionµ is of them reduced mass of a WIMP- i=e,p i X | that| allow binding tonucleus di↵erent pair,elements. and Mediator normalization mass of Z corresponds to The Yukawa interactionexp( betweenm point-liker r )and elec- is fixed to 15 MeV. tronsII. and BOUND protons STATE is given PARAMETER by A0 e SPACE Xenon atom. In reality, very low mass of mediator "e↵ ↵ | | Z↵"e↵ V (rFIG.,RN ) 1:, Critical value of coupling, as function of m ! r r mA is cut o↵ by particle physics constraints so that e e that allow binding to0 di↵erent elements. Mediator mass | exp( m| A0 r ri ) The YukawaNucleusV (r)= interaction"Xp↵↵-dDM betweenQipotentia point-like | andl | (3) elec- mAis fixed10 to MeV. 15 MeV. Saturating this inequality and equating r r In this expression, µ is0 the reduced mass of a WIMP- i=e,p i trons andwhere protons in is the given second byX line we| take | into accountnucleus that pair, pro- and normalizationRN to that of ofZ Xenon,corresponds and for to the same choice of µ, "e↵ , exp( m r r ) tons are incorporatedA0 e in a single nucleus of chargeXenonZ atom.and In reality,one can very calculate low mass the of mediator binding energy to be 2.58 keV, a "e↵ ↵ | |exp(Z↵"meA↵ Vr(r,RrNi )), ! r r 0 mA is cut o↵ by particle physics constraints so that Vradius(r)=e R"Np.↵↵| Ind thee| Q limiti of small| nuclear| (3) radius,In0 this for expression, a factorµ is of the three reduced less than mass naive of a WIMP- estimate (6). X r ri mA 10 MeV. Saturating this inequality and equating nucleus locatedi= ate,pr , | | nucleus0 pair, and normalization of Z corresponds to where in the second lineX we takeN into account that pro- RN to that of Xenon, andIt is for clear the same that choice the ofbindingµ, "e↵ , energy is very sensitive to tons are incorporatedexp( mA in0 r a singlere ) nucleus of charge Z and Xenon atom. In reality, very low mass of mediator " ↵ | | Z↵" V (r ,R ), one can calculate thethe binding choice energy of µ, toe↵ be,m 2.A58. keV, However, a it is always true that radiuse↵ R . In the limit of small nucleare↵ radius, forN a 0 ! N V (rr, 0)r =e exp( mA0 r rN )/ r factorrmNA.0 ofis three(4) cut less o↵ thanby particlenaive estimate physics (6). constraints so that nucleus locatede at| r , | | | | | would preferentially bind to heavy elements, while not X N Itm isA0 clear10 that MeV. the Saturating binding energy this is inequality very sensitive and to equating where in the second line we take into account that pro- forming bound states with light elements at all. This The ¯ potential has an opposite sign and is ofthe noR choiceN interestto that of µ,e of↵ ,m Xenon,A0 . However, and for it the is always same true choice that of µ, "e↵ , V (r, 0) = exp( mA0 r rN )/ r rN . (4) opens up an opportunity to search for the bound state tons arefor incorporated us in this in paper.a single| nucleus The| parameter| of charge| Z enteringand wouldone these can preferentially for- calculate bind the tobinding heavy elements, energy to while be 2 not.58 keV, a forming bound states with light elements at all. This radiusTheR ¯Npotential. In the has limit an opposite of small sign nuclear and is of radius, no interest for a factor of three lessformation than naive using estimate the direct (6). detection experiments, sensi- mulae, "e↵ (which we define to be positive),opens importantly up an opportunity to search for the bound state nucleusfor us located in this at paper.rN , The parameter entering these for- It is clear thattive the to bindingO(keV) energy energy is very release. sensitive In to Fig. 1, we plot the depends on the kinetic mixing and the darkformation charge, using the direct detection experiments, sensi- • For a pointmulae,-like"e ↵nucleus(which we = defineYukawa to be potential. positive), importantly the choice of µ,ecritical↵ ,mA . binding However, curves, it is always true that V (r , 0) = exp( m r r )/ r r . (4) tive to O(keV) energy release.0 In Fig. 1, we plot the depends on the kinetic mixingA0 andN the dark charge,N would preferentially bind to heavy elements, while not | | | < | critical binding curves, • Since adark can be large, "e↵ " ↵d/↵ O(10)", forming(5) bound states with light elements at all. This The ¯ potential has an opposite⌘ sign<⇥ and is of no interest "e↵ " ↵d/↵ O(10)", ⇠ (5) opens up an opportunity to search for the bound state for us in this paper.⌘ The⇥ parameter⇠ p entering these for- Two important consequenceswhere in the lastofpsizeable inequalitycouplings: we took ↵d < O(1).formation using the direct detection experiments, sensi- mulae,where" in(which the last we inequality define to we be took positive),↵d < O(1). importantly III. PROBABILITY OF RECOMBINATION e↵ It is easy to see that there are two important⇠ tiveIII. conse- to PROBABILITYO(keV) energy OF release. RECOMBINATION In Fig. 1, we plot the 1. Elasticdepends scatteringIt is on easy the cross to kinetic see thatsection mixing there on and are nuclei two the important dark⇠ is large charge, conse- quencesquences of relatively of relatively large "e↵ and large mediator"e↵ and mass mediator giving masscritical giving binding curves, the range of the force comparable or larger than RN : i. IV. RECOMBINATIONIV. RECOMBINATION SIGNAL IN DIRECT SIGNAL IN DIRECT the range" of" the↵ force/↵ < comparableO(10)", or larger(5) than RN : i. 2. Strong enoughThe elasticattractive scatteringe↵ ⌘ ⇥force crossd sectionaffords on bound nuclei are states signif- DETECTION DETECTION icant,Theii. bound elastic states scattering with nuclei⇠ cross may section form. Indeed, on nuclei are signif- where in the last inequalityp we took ↵ < O(1). 15 icant, ii. bound states withd nuclei may form.III. Indeed, PROBABILITY OF RECOMBINATION It is easy to see that there are two important⇠ conse- quences of relatively large "e↵ and mediator mass giving the range of the force comparable or larger than RN : i. IV. RECOMBINATION SIGNAL IN DIRECT The[1] elasticE. Aprile scattering et al. cross Dark Matter section Search on nuclei Results are from signif- a 121(11):111302, 2018. DETECTION icant, ii.Onebound Ton-Year states Exposure with of XENON1T.nuclei mayPhys. form. Rev. Indeed, Lett., [2] E. Aprile et al. Excess electronic recoil events in [1] E. Aprile et al. Dark Matter Search Results from a 121(11):111302, 2018. One Ton-Year Exposure of XENON1T. Phys. Rev. Lett., [2] E. Aprile et al. Excess electronic recoil events in

[1] E. Aprile et al. Dark Matter Search Results from a 121(11):111302, 2018. One Ton-Year Exposure of XENON1T. Phys. Rev. Lett., [2] E. Aprile et al. Excess electronic recoil events in 2

Example of the bound state profile where “primed” fields stand for dark photon, Dµ = for a heavy m and small mediator mass, i.e. taking ef- @µ igdA0 is the covariant derivative w.r.t. dark U(1), fectively mA0 ,RN 0 limit, the approximate Bohr-like ! and is a stable particle, the sub-component of DM.Naïve The Bohrexpression-style mustformula be valid:for the bound state with massless fermionic nature of is not essential, and all considera- mediator: tions in this paper equally apply to scalar as well. The 2 2 "e↵ Z µ self-interaction of ¯ pairs induced by the attractive in- Eb.s. 7.8 keV 3 . (6) ' ⇥ 10 54 100 GeV teraction mediated by A0 has important consequence for 0.100 ⇣ ⌘ ✓ ◆ ⇣ ⌘ annihilation, as resonances and capture to (¯) bound Binding in Actual binding0.050 for m of 10 MeV in Xenon = 2.6 keV. states can significantly increase the annihilation cross A’ Different Elements m =15 MeV section [19, 21–23]. As a result, the annihilation rate can A significantly exceed the WIMP benchmark rate 1pbn c, ⇥ 0.010 possibly making a subdominant component of DM. We Nitrogen 0.005 eff are going to consider modern value of f ⇢ /⇢ to Silicon ⌘ DM be a small free parameter, noting that deviations from 30 fm Iron Xenon the standard thermal cosmological scenario could result 0.001 Germanium * in tiny f. Furthermore, we assume unbroken charge Tungsten 5.×10-4 symmetry in the sector, ı.e. no mass splitting among Thallium states. At the same time, the phenomenology of A0 is 3 Figure 1: Top: binding1 energy5 in10 keV as function50 100 of "eff/50010 1000. Bottom: “standard”, and usual limits on dark photon apply [24], Radial wave function of the bound state Rb.s. multiplied by r. Z =54 m [GeV] so that e↵ectively for all . 3 3 (xenon), Eb.s. =Figure 2 keV, 1: Top:"eff binding=0 energy.85 in keV10 as function, mV of="eff 15/10 MeV, . Bottom:µ = 100GeV. The Radial wave function of the bound⇥ state Rb.s. multiplied by r. Z =54 3 w.f. peaks at 10(xenon), fm.Eb.s. (Barely= 2 keV, "eff consistent=0.85 10 , m withV = 15 MeV, treatingµ = 100GeV. the The potential with a 16 ⇥ w.f. peaks at2 10 fm.2 (Barely consistent with treating the potential with a pointlike nucleus.)pointlike nucleus.)Rb.s.r drR2 =1.r2dr =1. 2 II. BOUND STATE PARAMETER SPACE b.s. FIG. 1: CriticalR value of coupling, as function of m that allowR binding to di↵erent elements. Mediator mass The Yukawa interaction between point-like and elec- is fixed to 15 MeV. trons and protons is given by exp( m r r ) V (r )= "p↵↵ Q A0 | i| (3) d i r r In this expression, µ is the reduced mass of a WIMP- i=e,p i X | | nucleus pair, and normalization of Z corresponds to exp( mA0 r re ) Xenon atom. In reality, very low mass of mediator "e↵ ↵ | | Z↵"e↵ V (r,RN ), ! r r mA is cut o↵ by particle physics constraints so that e e 0 X | | m 10 MeV. Saturating this inequality and equating A0 where in the second line we take into account that pro- RN to that of Xenon, and for the same choice of µ, "e↵ , tons are incorporated in a single nucleus of charge Z and one can calculate the binding energy to be 2.58 keV, a radius RN . In the limit of small nuclear radius, for a factor of three less than naive estimate (6). nucleus located at rN , It is clear that the binding energy is very sensitive to the choice of µ,e↵ ,mA . However, it is always true that V (r , 0) = exp( m r r )/ r r . (4) 0 A0 | N | | N | would preferentially bind to heavy elements, while not forming bound states with light elements at all. This The ¯ potential has an opposite sign and is of no interest opens up an opportunity to search for the bound state for us in this paper. The parameter entering these for- formation using the direct detection experiments, sensi- mulae, " (which we define to be positive), importantly e↵ tive to O(keV) energy release. In Fig. 1, we plot the depends on the kinetic mixing and the dark charge, critical binding curves,

"e↵ " ↵d/↵ < O(10)", (5) ⌘ ⇥ ⇠ where in the last inequalityp we took ↵ < O(1). d III. PROBABILITY OF RECOMBINATION It is easy to see that there are two important⇠ conse- quences of relatively large "e↵ and mediator mass giving the range of the force comparable or larger than RN : i. IV. RECOMBINATION SIGNAL IN DIRECT The elastic scattering cross section on nuclei are signif- DETECTION icant, ii. bound states with nuclei may form. Indeed,

[1] E. Aprile et al. Dark Matter Search Results from a 121(11):111302, 2018. One Ton-Year Exposure of XENON1T. Phys. Rev. Lett., [2] E. Aprile et al. Excess electronic recoil events in Curves of marginal stability

0.100 Binding in 0.050 Different Elements

mA =15 MeV

0.010 Nitrogen 0.005 eff

Silicon Iron Xenon 0.001 Germanium Tungsten 5.×10-4 Thallium

1 5 10 50 100 500 1000

m [GeV] • Binding to heavier elements is much easier • There can be a situation with no binding to light elements but keV-scale binding to heavy elements so that capture process becomes possible, Z + c à (Z c) + Energy 17 1.Exploration of the parameters space

IwillcallthemediatortobeV ,andmV its mass. To avoid pitfalls, mediator =darkphoton.Wewillassumethatthemassofthemediatorissuchthat

Z↵m m 50MeV (1) e ⌧ V ⌧

for simplicity. The atom, matom mnucleus, will have a reduced mass with the DM, µ. As a benchmark I will' take

mV =15MeV,µ Xe =100GeV (2)

Obviously, the binding of an atom and an mCP is possible when mV is very small, as discussed in Hari & Maxim. Does the bound state exist for MeV scale mediator? If so, we can then get to a more ”traditional” particle physics 3 model. The hope is to get to " at 10 level, so that it is not excluded by the collider experiments. Let us calculate the binding energy with Z =54 nucleus (xenon) and with Z = 26 (iron). I use the simple Yukawa potential. For the choice (2) (µ with Xenon is 100 GeV), I need to take mdm 550 GeV, which will translate to a twice smaller value, µ =47.5GeVwithiron.' The choice of the mediator mass, with Compton wave length smaller than K-shell radius and larger than the nuclear size allows us, with some e↵orts, keep the nucleus description point-like, and ignore screening by electrons. In that approximation, the nucleus-DM interaction is given by

Z↵1/2↵1/2" Z↵" V (r)= d exp( m r)= eff exp( m r)(3) r V r V Elastic cross section is large This interaction gives the following elastic cross section on nuclei (not nucle- ons), • Using a perturbative formula 2 2 2 16⇡↵ Z "eff 2 el 4 µ (4) ' mV 3 and itsFor non a-perturbativeZ =26and generalizations,µ =47.5 GeV, one and discovers✏ 0. 9that 10for the, we get el 4 22 2 ' -22⇥ -24 2 ⇠ ⇥ range 10of parameterscm . This where is too bound large, state and exists a non-perturbativeimplies 10 – 10 answercm needs to be 2 24 2 elasticapplied cross sections ( 4⇡RN 5 10 cm ) which is two orders of magnitude smaller. Either⇠ way, this⇠ is⇥ still enough to moderate the flux when it comes -3 2 • Rapidto 1 km thermalization! depth, as the Typical collision energy length drops in thefrom rock (10 maybe) mc/2 = few cm. Given Insidesurface over-densities experiments however,500 keV I believefor TeV thatmass the to fraction0.03 eV of⇠ this dark matter must be small, f 1. ⌧

Galactic escape velocity 1 Galactic DM distribution

DM velocity • Density of DM component shoots up = called “traffic jam”. 18 6

A. The DM TracJam This terminal velocity vterm is lower than the initial (galactic) DM velocity, leading to the DM pile up and a To estimate the density enhancement in the DM traf- resulting density enhancement. From flux conservation, fic jam, we begin by first estimating the terminal velocity the density enhancement is: ⇢ v with which the DM sinks through the ground. The den- ⌘ = lab = vir (15) sity enhancement then follows from flux conservation. ⇢ss vterm We work in the limit where the DM interacts su- where ⇢lab is the DM density at a location of an under- ciently strongly with nuclei so that it thermalizes when ground lab, ⇢ss is the solar system DM density, and vvir it goes underground. This is the range of parameters that is the local virial velocity of DM. is of most interest, since the scattering of DM is otherwise This density enhancement exists as long as the DM constrained by low threshold detectors such as CRESST. thermalizes with the rock. However, for heavy enough Thermalization is of course progressively harder at heav- DM there are two additional e↵ects that need to be taken ier masses since several collisions are necessary for the into account. For large m the thermalization requires DM to thermalize with the rock. To avoid rather strong more scattering, and there will eventually not be enough constraints on anomalous isotopic abundances, we will column depth in the rock to achieve thermal velocity at assume that the strongly interacting DM has repulsive a given laboratory depth. Moreover, when the downward interaction with nuclei. velocity of DM becomes smaller in magnitude than vterm, To perform an estimate of the density enhancement, the thermalization is not complete, as on average the we need a coherent (transport) scattering cross-section t vertical component of the DM velocity is larger than the of DM with nuclei of atomic mass A. We notice that in terminal sinking velocity. Both of these e↵ects cut o↵ the principle, there are two main regimes for such a scattering density enhancement for heavy DM, as shown in Fig. 2 cross section. The first regime can be achieved when and discussed below. the perturbative treatment is possible. Then, given the Many underground labs with developed DD program input cross section on an individual nucleon, the overall are located at depths exceeding 1 km. However, the pre- elastic cross section on the nucleus could be described as cision experiments with metastable tantalum were per- 2 2 2 4 el = A nµ (mA,m)/mp,whichreducestoA n at formed in the Hades observatory, at a more shallow loca- M m . On the hand, if we keep increasing this A n tion. For our estimates, we take the Hades observatory to scaling with A breaks down. Describing the DM-nucleus be 300 m below the surface. In our estimates, we take the potential as a square barrier, we observed that the strong 3gm density of soil/rock to be ⇢ = 3 , ambient temperature interaction limit corresponds to R  1, where  is the cm A T = 300 K, mgas A GeV and take A 30 for rock. virtual momentum inside the barrier [18], and the elastic With these numbers,⇠ we⇥ plot the density enhancement⇠ ⌘ cross section is expected to be 4⇡R2 . For the slow-down A Dark matterfor threetraffic di↵erent massesjam M = 100 GeV, 1 TeV, 10 TeV process, we• needRapid a transport thermalization cross section, and we assume in Fig. 2 (Left). There are three distinct regimes at play. it to be on the same order of magnitude as the elastic one. For small cross-sections,Incoming there particles is an exponential regime Thus, we• chooseFlux the conservation: following ansatz vin fornhal theo = -nucleus where the column density is not enough to slow DM par- transport cross section, Rapid thermalization vterminal nlab. ticles down to the thermal velocity vth. As the downward 4 2 velocity approaches the thermal velocity, the slow down t =Min(A n, 4⇡RA). (13) • Terminal sinking velocity is is enhanced leading to a jump to vth. Next, for cross- sections where vertical velocity drops below v , the ad- After DM isdetermined fully thermalized, by the it iseffective not stationary, but th ditional column densityDiffusion leads biased to furtherby slowing down, continues slowly sinking towards the center of the Earth mobility (~ inverse cross section) leading to a linear regime: the DM density enhancement due to the Earth’s gravitational field. The average ter- gravitational drift is linearly proportional to the size of the elastic cross sec- minal downwardand velocity gravitational in any medium forcing is given by [19] tion. Finally, once vterm is reached, there is no further slow down and a flat regime for the density enhancement 3MgT vterm = 2 3 (14) is achieved. A lab mgasn tvth h i Fig. 2 (Right) shows contours of equal ⌘ in the N vs M plane. ⌘ increases as a function of n till n where M is the DM mass, mgas is the mass of gas par- 30 2 ⇠ • 10 cm which corresponds to the saturated geometric ticle, n is theChange number densityin velocity of gasfrom particles, t is the 7 cross-section in Eqn.(13) and there is no further enhance- transport crossincoming section, v~th 10thecm/s thermalto velocitytypical of gas particles (for solids, velocity due to vibrational motion)1. ment. As mass of DM, M is dialed up, the terminal sinking velocity of 10 cm/s results velocityMP, Rajendran, increases linearly Ramani as 2019 in (MP,14), and as a result ⌘ 6 Ramani 2020, Berlin, Liu, MP, in nlab ~ 10 nhalo ! decreases linearly. However for large enough mass, the relevantRamani, columnin prep depth is not enough to thermalize and hence there is an exponential decrease in ⌘ as a function 1 This e↵ect• wasAt discussedmasses in< [210]. However,GeV upward their estimate flux dif- fers from theis calculations important of [19 and]. Moreover, density [2]didnotaccount goes up. of M. Thus, we conclude that the value of the enhance- for the saturation of the DM nucleon scattering cross-section at ment factor is quite sensitive to particular details of the large A and did not use the correct reduced mass in the collision strongly-interacting DM model (mass, cross section), and between DM and nuclei. can vary in a large range. A possible scenario for direct detection (including Xenon excess)

• Small enough fc so that surface and balloon experiments are not sensitive. • Density enhancement after thermalization (traffic jam). Becomes invisible to elastic scattering. • No bound states with light elements – no efficient capture during the sinking • Efficient capture in an experiment containing heavy enough elements (Xenon, of course. Also, Iodine, Tl etc…). Z + c à (Z c) + Energy • Main feature of the signal: electron-like mono-energetic energy release.

• Possibly non-trivial time structure (i.e. daily and seasonal 20 modulation) More on light vs heavy nuclei bound states

1 500

EB=2.5 keV in Xenon -2 eff=10 0.100 GeV 100 =1 m 50

] -2.5 eff=10 GeV 10 eff 0.010 = MeV m [ A

GeV m =100 10 -3 m eff = 10 0.001 5 =1 TeV m

EB=2.5 keV in Xenon 10-4 1 0.1 1 10 100 1000 1 10 100 1000 104 mA' [MeV] m [GeV]

• Solid part of the curves: binding to Fe and lighter elements not possible while binding to Xe is significant

21 Capture process • Auger-style process with the ejection of an atomic electron. A + c à (A+-ion c) + electron Dominates over photon emission. • Calculable using perturbation theory Unbound electron

Bound electron orbit

R RN bound state

Unbound nucleus-DM Bound nucleus-DM

22 (photon emission is suppressed), A + (A+ ) + e. (6) ! b.s. Extra interactions that our DM has with particles inside the atom look like this: Z↵" ↵" V = eff exp( m r r )+ eff exp( m r r )(7) r r V | N | r r V | i| N i i | | X | | Here N refers to the nucleus, and i refers to electrons. The first term that we have here is responsible for the formation of the bound state, and in that sense is taken into account already, because the ”in” function (modified plane wave of relative N bound state) and ”out” function (bound state w.f. of N) are the solutions of the Hamiltonian 6.Conclusions that contains first term of (7). To makeIdonotthinkthatthiswasfullyappreciatedintheliterature:10MeV this whole calculation simple enough, I will take that there is a 3 single boundto few state. tens Multipleof MeV dark bound photon states mediator, can be treated and ✏ the(g0/e same) way,10 , and which is ⇠ the capturenot excluded, usually is will dominated lead to binding by the capture of heavy onto nuclei out-most with such orbits DM. with Capture largest angularcrossCalculation/estimate sections momentum are small (like but in muonic not negligible. atoms).of the As The capture discussed morphology in rate our of paper the signal with Hari,will the be consistentconditions with of perturbation mono-energetic theory electron are generally recoil. This met, model and we can be can• applyIn theprobed Fermistandard either golden usingFermi rule surface formula to calculate detectors, the exploring cross section: large elastic , or using this recombination signature. In both cases f is required to be small. d3p d =2⇡(Energy) e f V in 2 (8) Given suchc an industry around⇥ (2⇡ Xenon1T)3 |h | pert and| i DAMA,| this seems to be well worth pointing out in print. The tricky part would be to get f to a small Integratingthe mostvalue,interesting over as you the would energy question typically of the is expectoutgoingthe perturbation. a WIMP-ish electron weIt type is end e of independent, up abundance. with the expressionand e enters for thenon cross-perturbatively, section via Eb.s. . pemed⌦e 2 d = V 1 (9) V = V + V + V = V (r )+(2r⇡)2 |h Vi(|r )+ ri rj i j V (r )+... (32) 0 1 2 e N · rre e 2 N N rre rre e This expression should be summed over available electrons2 for this process, 1 1 µ i.e. those that... are+ more(r )2 looselyV (r bound)= (r than)2 N bound V (r states.)(33) In this ! 6 N re e 6 N m re e expression the initial wave function is the relative✓ N ◆ wave of incoming N system times the w.f. of bound electron, normalized on 1. The relative wave of N is to be taken at really small energy (typical thermal energy), and we will take the scattering normalization for this wave that contains 1/pv, the relativeGives transition velocity between of -atom continuum system. of c- ThusGives we willtransition use between s-wave bound state N and bound state E>0 electron E0 1 3 E<0 2 23 N (r) exp(ikr)(larger); d re e =1. ! pv | | Z 4

15 (↵me/mV ). We have the following expression: 3 4 2 3 (4⇡) µ (Z↵me) (↵me) 2 2 s sv = ⇢N ⇢e (16) 9 m m7 ✓ N ◆ V (I am a little puzzled by such a large numerical coecient, to be honest) Dimensionless quantities ⇢ are defined in the following way:

7/2 1 4 ⇢N = mV dr r G(r)Rb.s.(r)(17) 0 ⇥ Z 2 2 3 Rpe0(0) (⇢ ) =2(↵m ) R (0) pv (18) e e n0 2p e n e X ✓ ◆ In these expressions, Rb.s. is the normalized N bound state function, 2 2 r drRb.s. =1.Theelectronicintegralprojectson (0) due to the delta function, and since we take NR wave functions, we ought to take into account R Rp 0(0) only the s waves, as the rest is zero at r =0. e is the wave function 2pe of the ionized electron, and in the limit of neglecting the potential energy in Rpe0(0) the field of the ion, j0(per). Because of the long range Coulomb 2pe ! field there is no vanishing at the threshold as Rpe0(0) v const as v 0. 2pe p e e The sum over the principal quantum numbers occur for! energies that allow! electron ejection, and for Xe, and 2 keV binding this is for n =3, 4, 5. Factor 2 of 2 in ⇢e accounts for the double occupancy of ns1/2 shells. To evaluateCalculation/estimate the cross section numerically, of the I would capture need exact rate wave func- tions, in principle. For the N system I solve for it numerically. For Xenon electrons,S-wave I do (DM not-nucleus) have a corresponding to outgoing electron code (although s-wave capture they exist),rate: so I will (↵m /m ). We have the following expression: be using(↵ somemee/mVV approximations.). We have the following Direct expression: calculation for the parameter spot we (4⇡)33 µ 4 (Z↵m )2(↵m )3 (4⇡) µ (Z↵me) (↵me) 2 22 chose gives s sv = ⇢N ⇢e (16) s sv = 7 ⇢N ⇢e (16) 99 mmN mV ✓✓⇢NN ◆◆ 4.3. V (19) (I(I am am a a little little puzzled puzzled by by such such a a large large' numerical coecient, to to be be honest) honest) The expression under the integral peaks at 40 fm (because of r in high wDimensionlesshereDimensionless radial integrals quantities quantities are⇢⇢ areare given defined defined by in the following⇠ way: power entering ⇢N . 1 77//22 1 4 ⇢N = m dr r4G(r)Rb.s.(r)(17) ⇢N = mVV dr ⇥ r G(r)Rb.s.(r)(17) For the electron w.f. I will use theZ00 following⇥ ansatz: Z 2 2 3 1/2 Rpe0(0) (⇢ )2 =2(↵m )3 3 R (0)Rpe0(0)pv (18) (⇢ e) =2(↵Zm(↵e)m ) Rn0(0) R (0)pve 2⇡Z↵ (18) e e e n0 2ppee0 e R (0) = 2 n ; 2pe = (20) n0 3 Xn ✓✓ ◆ that can be evaluated numerically.n X 2pe ve InIn these these expressions, expressions,✓RRb.s. isis the the normalized normalized◆ N boundr state state function, function, 2 2 b.s. This is veryr2 drR crude,2b.s. =1.Theelectronicintegralprojectson but can be improved a lot with a(0) proper due to code. the delta With this At fiducialr drRb.s. choice=1.Theelectronicintegralprojectsonof parameters, (Xenon, mediator (0) mass due to= the15 MeV, delta m function,function, and and2 since since we we take take NR NR wave wave functions, functions, we ought to take take into into account account we are gettingR ⇢e 100, and the resulting crossRpe section0(0) is = R100only GeV, the s waves,effective as thee giving rest is zero2 keV at rbinding)=0. Rpe 0the(0) isestimate the wave is function only the s waves,⇠ as the rest is zero at r =0. 2pe is the wave function 2 2pe ofof the the ionized ionized electron, electron, and and in in the the limit limit33 of2 neglecting⇢e the potential energy energy in in Rpe0(0)v 10 cm ( c)(21) the field of the ion, Rspe0s(0) j0(per). Because of the long range Coulomb the field of the ion, 2pe j (p r). Because of the long range Coulomb 2pe '! 0 e ⇥ 100 ⇥ ! Rpe0(0) field there is no vanishing6 at the threshold as Rpe0(0)pve const-27 as 2ve 0. 24 • fieldSince there actual is no c/v vanishing ~ 10 at, the the actual threshold cross as section2pe pv e~! 10constcmas v. eNot! 0. tiny The sum over the principal quantum numbers occur2pe for energies that allow The sum over the principal quantum8 numbers occur for! energies that allow! electron ejection, and for Xe, and 2 keV binding this is for n =3, 4, 5. Factor electron ejection,2 and for Xe, and 2 keV binding this is for n =3, 4, 5. Factor of 2 in ⇢2e accounts for the double occupancy of ns1/2 shells. of 2 in ⇢e accounts for the double occupancy of ns1/2 shells. To evaluate the cross section numerically, I would need exact wave func- To evaluate the cross section numerically, I would need exact wave func- tions, in principle. For the N system I solve for it numerically. For Xenon tions, in principle. For the N system I solve for it numerically. For Xenon electrons, I do not have a corresponding code (although they exist), so I will electrons, I do not have a corresponding code (although they exist), so I will be using some approximations. Direct calculation for the parameter spot we be using some approximations. Direct calculation for the parameter spot we chose gives chose gives ⇢N 4.3. (19) ⇢N ' 4.3. (19) The expression under the integral peaks' at 40 fm (because of r in high The expression under the integral peaks at ⇠ 40 fm (because of r in high power entering ⇢N . ⇠ power entering ⇢N . For the electron w.f. I will use the following ansatz: For the electron w.f. I will use the following ansatz: Z(↵m )3 1/2 R (0) 2⇡Z↵ R (0) = 2 e 3 1/2 ; pe0 = (20) n0 Z(↵m3 e) Rpe0(0) 2⇡Z↵ R (0) = 2 n ; 2pe = ve (20) n0 ✓ n3 ◆ 2p r v This is very crude, but can✓ be improved◆ a lot withe a properr e code. With this Thiswe are is very getting crude,⇢2 but100, can and be the improved resulting a lot cross with section a proper is code. With this 2e ⇠ we are getting ⇢e 100, and the resulting cross2 section is ⇠ 33 2 ⇢e s sv 10 cm 2 ( c)(21) 33 2 ⇢e s sv ' 10 cm ⇥ 100(⇥c)(21) ' ⇥ 100 ⇥ 8 8 The smallness is related to the fact that one needs ”to take the energy out” of the small volume (30fm)3, and there is no easy way to do it. Notice that this it looks much much⇠ smaller than the elastic cross section if we consider c =1,butthisisv,not. In fact, if we compare elastic with capture we 33 2 27 2 find that capture 10 cm (c/vthermal)andwegetto10 cm level, as c/v is 106. / ⇥ The estimate of the p-wave to s-bound state transition is actually simpler. It turns out that it can be related to the total cross section of the photoelectric e↵ect on Xe, because V1 can be reduced to dipole dipole interaction on e.o.m. To make the long story short, I quote the final⇥ result.

3 2 2 2 ! me(2µEN ) 4 p sv = photoelc drr F (r)Rb.s.(r) (22) ⇥ 9↵ m7 ⇥ V ✓Z ◆

As already mentioned, the result is suppressed by kinetic energy EN T and is kind of small. ! is the binding energy, and we take it to be 2 keV,⇠ as before. 18 2 Photoelectric cross section at 2 keV is well known, and it is 10 cm . Putting this all together we have the following estimate,

34 2 p sv 10 cm ( c). (23) ' ⇥ Estimate of the event rate for Xenon1T 3.Estimates of the signal rate • If the bound states are formed [i.e. dark photon/dark matter To estimateparameters the signal are right] rate we Xenon1T need the electron sinking recoil velocity translates that determines into the density enhancement. The sinking velocity is given by our previous papers extreme sensitivity to fc . with Hari. Taking m =500GeVorso,andtheelasticcrosssectionof1bn, we get• Taking the thermal~ 500 and GeV sinkingmass crossc, (µ sectionwith Xe to = be100 roughlyGeV) in one this gets, range. and

v 104cm/s; v 101cm/s(24) thermal ' sinking ' 3 -3 We can immediatelytraffic-jam deduce-enhanced several density important becomes consequences, ~ 10 cm iffc in. addition, following previous section we take the capture cross section-33 in2 the range of • Taking33 2the capture cross section to be (s v) ~ 10 cm × c we get v =10 cm c. the estimate⇥ of the counting rate in the detector as

15 -1 -1 6 • ”Tracjam”enhancementfactorisR ~ 5 × 10 × f⌘c tonvhaloyear/vsinking. 10 . The 3⇠ 3 3 ⇠3 enhanced density of DM is then ⌘ 10 cm f =10cm f. Where f ⇥ -14 • isThis someimplies fractionsensitivity< 1. to abundance as good as fc ~ 10 . Next best probes are ~ 10 orders of magnitude away. • The border-line value of abundance9 can be consistent with 25 anomalous events. Zooming in onto target parameter space

10-2 EB=2-3 keV in Xenon m =5 TeV

10-3

=0.1 D

-4 =1 10 D

10-5 1 101 102 103 mA' [MeV]

§ Because of the unknown adark, we do not know “exact” e parameter that can explain the excess. 26 Zooming in onto target parameter space

10-2 EB=2-3 keV in Xenon m =5 TeV

10-3

=0.1 D

-4 =1 10 D

10-5 1 101 102 103 mA' [MeV] § A roughly triangular shape of the parameter space, ~ one decade

long on each side can explain the Xenon1T excess at small fc .

§ This parameter space is [hopefully] going to be explored by the 27 LHCb and HPS experiments. What if the formation of bound states is not possible?

§ Density enhancement may still exists if bound states do not.

§ Thermalized but very slow EW relics contain a reservoir of “momentum transfer” for the excited nuclei – so called isomers – to recoil at and de-excite. For example, exothermic reaction appear 180mTa + c à 180Ta(g.s.àdecay) + c.

DJ DJ § Usual suppression: Amplitude ~ (RN / lg) =(RN DE) is removed, 1/2 DJ (and DJ is e.g. 9) becoming (RN (µ DE ) ) can be O(1).

§ The idea is published in MP, Ramani, Rajendran, 2019, and the first

search is done in Lehnert et al, 2020, 1911.07865. Sensitivity to fc down to ~ 10-5. 28 Conclusions • Direct detection experiments are sensitive not only to main components of DM, but can also be sensitive to rare DM species.

• Usual blind spot – thermalized DM component – may be detected via the bound state formation.

• Bound states between DM and nuclei are possible within one of the most studied models of dark photon mediation, with specific corner of

{mA’, e } parameter space.

• Binding to heavier elements with O(keV) is far more likely, opening up a possibility to study capture into bound states. Capture will look as a mono-energetic electron recoil.

-14 • Xenon1T anomaly can be explained this way or else fc < 10 29 constraint can be imposed. A rarity frontier.