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JHEP01(2021)178 Springer -threshold January 27, 2021 December 7, 2020 (keV) : November 19, 2020 : September 10, 2020 O : : d,e,f Revised Accepted Published Received , , Diego Redigolo, Published for SISSA by https://doi.org/10.1007/JHEP01(2021)178 c a tomerv@post.tau.ac.il , [email protected] , [email protected] Rouven Essig, , b . 3 and Tomer Volansky c Andrea Caputo, The Authors. a c

Motivated by the recent XENON1T results, we explore various new physics , [email protected] [email protected] INFN Sezione di Firenze, Via G. Sansone 1, I-50019Department Sesto of Fiorentino, Physics Italy andSesto Astronomy, University Fiorentino Florence, of Italy Florence, E-mail: [email protected] Edificio Institutos Investigacion, Catedratico JoseC.N. Beltran Yang 2, Institute Paterna, for 46980Stony Theoretical Spain Physics, Brook, Stony NY Brook 11794, University, U.S.A. CERN, Theory Division, CH-1211 Geneva 23, Switzerland School of Physics andTel-Aviv Astronomy, 69978, Tel-Aviv University, Israel Instituto de Fisica Corpuscular, Universidad de Valencia and CSIC, b c e d a f Open Access Article funded by SCOAP case, photophobic axion couplingsanalyze models are of necessary dark matter-electron to scatteringthe avoid to excess. determine X-ray which Standard constraints. models scattering might ofdata explain Second, dark from matter we lower-threshold with experiments. electronsmediator is Momentum-dependent can generically in fit interactions conflict the with with data a with heavy dark matter mass heavier than a GeV but are generically in photons, and scalars, eitherIn as the dark latter case, matter wefit relics find or to that being keV the mass produced databy bosons directly but produced introducing in in are a the the excluded novel Sunevading Sun. the Chameleon-like by provide stellar an bounds. axion stellar adequate model, We cooling findexplanation that which constraints. for absorption of the can bosonic excess We explain dark only matter address if the provides the a this dark excess viable matter tension while is a dark photon or an axion. In the latter Abstract: models that can bedirect-detection experiments. discovered First, through we searches consider the for absorption electron of axion-like recoils particles, in dark Itay M. Bloch, Mukul Sholapurkar Exploring new physics withdirect O(keV) electron detection recoils experiments in JHEP01(2021)178 (100 keV)-mass mediator can fit the data. The O 2006.14521 Beyond Standard Model, Cosmology of Theories beyond the SM probes of the lightCarlo mediator. analysis and Throughout use an our improved study, energy we reconstruction implement ofKeywords: an the XENON1T unbinned events. Monte ArXiv ePrint: states that have astate small to mass the splitting. lightera The state subcomponent exothermic of can (down)scattering dark fit ofthat matter the dark the that matter heavier data is interacting for acceleratedcross though sections keV by an required mass scattering in this splittings. off scenario cosmic are, however, Finally, rays, typically challenged we finding by complementary consider tension with collider constraints. Next, we consider dark matter consisting of two (or more) JHEP01(2021)178 26 37 – 1 – 30 24 30 7 20 25 14 46 42 3 12 21 17 9 22 38 19 1 7 11 regions crete models 6.2.2 Exothermic dark matter-electron scattering: relic abundance for con- 6.2.1 Exothermic dark matter-electron scattering: kinematics and best-fit 4.2.2 Solar scalar 4.3.1 Dark4.3.2 photon dark matter Solar dark photon 4.1.1 ALP4.1.2 dark matter Solar ALPs 4.2.1 Scalar dark matter 6.1 Standard DM-electron6.2 scattering Exothermic dark matter and electron recoils 4.3 The dark photon 4.1 Axion-like particles 4.2 The scalar 3.1 Energy reconstruction3.2 method Statistical method experiments has been ongoing fordirect-detection, more indirect-detection, than and collider three experiments, decades.has no been Despite convincing found numerous signal to searches for date. at particle’s DM Given properties the if profound implications wesignal for were in our to one understanding find of of these the it DM experiments in deserves the to laboratory, be any studied claim carefully. for a possible DM 1 Introduction The quest to identify the particle nature of dark matter (DM) by detecting DM in terrestrial 7 Accelerated dark matter 6 Dark matter-electron scattering 5 Chameleon-like ALPs: circumventing the stellar cooling bounds 3 XENON1T 4 Absorption Contents 1 Introduction 2 Models and summary JHEP01(2021)178 low ] and 29 , 7 keV , we focus ], and DM 4 28 − 14 1 – energy range. 10 ]). These models 35 implying a total of 30 keV − 1 , we discuss DM-electron 6 ]; see also [ in the 34 – tonne year keV 31 day / ]). Here we present a model in which − 27 , we describe the requirements that new – kg (keV) bosonic DM that is absorbed by an 2 20 5 excess), with the excess mainly located in O 2 events 10 σ ± 3 × – 2 – investigates how a density-dependent potential 76 ]. While the most likely explanation is a neglected 36 . 5 1 ]. For the absorption of bosonic DM, we show that 2 ]), and cosmic-ray accelerated DM that here interacts 19 – 30 15 , we describe important features of the XENON1T data, (keV) [ 3 ], bosonic DM that is emitted from the Sun [ O 9 – 2 considers a subdominant DM component that is accelerated by 7 background events. An excess of 53 events has been observed at the This paper is organized as follows. In section We also discuss, DM-electron scattering with different form factors, “exothermic” DM In this paper, we explore several possibilities for the origin of this signal. We will focus The XENON1T collaboration has recently observed an unexplained excess of electronic 1476 can be used to circumventscattering, the reviewing stellar the cooling “standard” bound. casesmall and mass In then splittings. section focusing We on willphenomenology. multi-component see Section DM that with “exothermic” DM scatteringscattering off off electrons cosmic has rays. a rich physics needs to satisfy in orderthat to explain we the will XENON1T excess, discuss. andour also In method detail section for the reconstructing models the energy,on and our the statistical absorption analysis. of In bosonichalo section particles or that are emitted either from (non-relativistic) the DM particles Sun. in our Section focused on electron scattering seewith [ electrons through an intermediate-mass mediatormediators or (for light previous mediators work interacting focused with ondeserve nuclei see heavy further [ studyfocusing in on future the dedicated XENON1T excess. papers, but we provide their salient features dense environments (for a similara effect see pseudo-scalar e.g. is [ produced independent the coupling Sun between the and pseudo-scalar explains and the electrons XENON1T avoids excess, stellar but coolingscattering a bounds. off density- electrons (for previous work focused on nuclear scattering see [ scattering off electrons in xenononly [ the dark photonFor or light bosons a produced “photophobic” in axion-likethe the Sun, particle XENON1T bosons data can with than fit abe massless mass the ameliorated near bosons. XENON1T in 1 models keV hint. The provide where a tension the shape better with of fit star the to scalar cooling potential constraints is substantially can modified in mostly on the possibility that theparticles origin is (pseudo-scalar, attributable scalar to DM, and but vector)have will also produced to consider in be bosonic the a Sun,models DM which previously component. do considered in We not the discuss necessarily electron literature: in in the the xenon context atom [ of the XENON1T excess several XENON1T search has an exposureThe of background rate is∼ reported to be energy region (corresponding tothe roughly 2-keV and a 3-keV energy bins. recoil events with an energy of background source or a statisticalfirst fluctuation, sign the of possibility new thatevents physics does the (not not excess appear necessarily could in even bescattering. the a traditional the search sign Rather, for of nuclear it DM) recoils from is appears elastic intriguing. DM-nucleus as The an excess excess of in a search for electron recoils (ER). The JHEP01(2021)178 , ) ]. 5 18 , 59 ], Red 16 54 – 51 and section 4 ). We consider sev- 7 ], and Supernovae (SN) [ 58 and section ], White Dwarfs (WD) [ 6 50 , 49 – 3 – 4 keV are approximately consistent with background ≥ ] for a compilation), as well as from astrophysical obser- left. 48 4 ], Horizontal Branch (HB) stars [ ]). Models that predict such a signal must evade these bounds. 57 47 – ]. 55 44 We will consider the case that an electron absorbs a bosonic particle: ) ER searches. We first summarize the relevant features of the excess and keV ]. For energies of order 100’s of eV, the XENON1T S2-only analysis is especially & 46 – a temperature of aroundthe 2 production keV. mechanism A could non-zero alsothe cut mass best the fit around solar to 1.5–2.5 emission keV thearise kinematically, depending data. providing from on However, stellar for cooling, bosonsXENON1T produced excess strongly in for disfavoring the the the Sun vanilla couplings strong axion constraints needed and dark to photon explain models. the or relativistic. The formerDM; may in occur this if case the thedata particle only ER due constitutes spectrum to a is the component peakeda experiment’s at of pseudo-scalar finite the the energy can mass resolution. explain ofstellar the We cooling find the XENON1T constraints. that DM, excess, a Next, and while vector light can and bosons a fit may scalar the be is produced in in the conflict Sun, with which has Absorption. pseudo-scalar (axion), a scalar, or a vector. The boson may be either non-relativistic New physics that couplesdump to experiments electrons (see is [ constrainedvations, such by various as collider theGiants and cooling (RG) beam- [ of the Sun [ constraining [ Numerous direct-detection experiments place stringentnying nuclear constraints recoil on signal any (for accompa- ainteractions recent see compilation [ of low-mass DM limits on nuclear and the binsexpectations. with See energies figure Low-threshold direct-detection experiments searching forditional electron constraints recoils on provide any ad- signal36 that also produces sub-keV electron recoils [ The excess events have ana energy potential of signal 2–3 keV. shouldenergy The contribute measured resolution to spectrum of more suggests the thansharp, that experiment allowing a makes for single this rather bin. (statistically narrow weak) spectra However, observation to the less provide finite a reasonable fit. The 1 keV bin Prospective models that could produce the observed excess and satisfy its features The following considerations are important when studying a prospective new • • • • 1. can be separated intoand models those that that predict aneral predict scenarios: absorption a signal scattering (section signal (section physics signal: The XENON1T excess motivates us toscenarios, consider focused various mostly, known but as not wellthreshold solely, as ( on novel DM new physics models thatthen can identify be possible discovered mechanisms via that a may high- explain it. 2 Models and summary JHEP01(2021)178 ]. While we find is the fine structure 33 for a wide DM mass – | δ 31 EM | α keV and coupling to elec- 5 . (keV). The spectrum can be = 2 | ∼ O X δ | m is the Bohr radius, 0 Models that exhibit velocity- or momentum- a – 4 – , where ]. ] or with Cosmic Rays (CRs) [ e m 61 43 , , EM 60 42 α = 0 is the electron’s mass. As a consequence, DM scattering through An unsuppressed high-energy spectrum from DM-electron scatter- A small subcomponent of DM may be accelerated through its in- e /a The DM-electron scattering rate depends on the momentum-transfer- 1 m The stellar cooling constraints on light bosons may be evaded if the cou- . The atomic form factor, together with the scattering kinematics, imply q δ , we summarize the goodness-of-fit of the various absorption scenarios dis- 1 , if there are three or more DM states whose mass is split by different amount (keV), or if the DM mass is well below the GeV-scale. q O the scenario we considerthis here, explanation. direct constraints on the mediator exclude robustly teractions in the Sunthat [ the component acceleratedwithout from being the in Sun conflict cannot withCR explain lower-threshold scattering the direct-detection of XENON1T searches, DM we excess with findthe non-trivial that XENON1T momentum-dependent form excess factor while can evading address other direct-detection constraints. However, in range and can explain thebroadened XENON1T if excess the for DM-electron interactionfer increases with increasing momentum trans- of Accelerated DM. Exothermic DM. ing may stem fromconsisting an of exothermic two scattering or ofnoted more DM as states off electrons, whosea the masses rather result are narrow of electron slightly DM recoil split spectrum by that an is peaked amount near de- Velocity-suppressed DM scattering. dependent heavy-particle-mediated DM-electronmental scattering data are at lower allowed energiesHowever, by such and models provide experi- are an likely adequate ingenerate fit tension this to with operator collider the [ bounds XENON1T on excess. new particles that momenta constant and a light mediator orspectrum a at sub-keV velocity-independent energies heavy and mediator are predict thus disfavored. a steeply rising plings of SM particlesor to high-temperature the stellar corresponding objects. bosonsnomenology are and Such can screened chameleon-like revive inside particles the high-density have Solar a explanationDM of rich scattering. the phe- XENON1T hint. dependent atomic form-factor. This steeply-falling function is highly suppressed for Chameleons. In all the solar cases, the addition of a non zero mass ameliorates the fit by cutting off In figure 6. 5. 4. 3. 2. trons. Among these, thephoton scalar and DM the case axion isthe are excluded anomalous good axion by coupling stellar explanation to constraints of photon while the should the be XENON1T dark set excess.the to spectrum zero In kinematically, to the in avoid X-rays latter better constraints. agreement case with the 1 keV bin being consistent with cussed above to the XENON1Tcurve) measurement. can We fit see the that data the bosonic well with DM scenarios a (red predicted mass of JHEP01(2021)178 , 4.1.1 and in � 4.2.2 � � � ooi DM Bosonic Photon Dark Solar -philic γ Scalar Solar -phobic γ Scalar Solar γ-philic Axion Solar γ-phobic Axion Solar � �� light scalars in section . We also discuss all the solar scenarios: blue � 4.3.1 , in [��] � 4.1.2 � �� ] the scalar provides a very good fit of the data – 5 – 58 � the spectrum falls fast enough towards lower energies , since the electron recoil spectrum rises at low energy, 0 ] are not observed in the data. If the solar production 0 . 50 all the cases of bosonic DM: axion-like particles in section ≤ n > n keV, which cuts the sharp rise towards low energies and hence 9 4.3.2 . and dark photon in section with = 1 dark red n φ q 4.2.1 m ∝ ) q � ( , we summarize the goodness-of-fit of the different scattering scenarios pre- � -� -� -� -� F

�� - form factors with n . Summary of the absorption scenarios considered here, with their p-value as a function axion-like particles discussed in section q dark photon in section ∝ Second, we show that exothermic scattering can fit well the data when the heavy and In figure ) q green ( However, for and provides an adequate fit to the XENON1T data. light DM states arethe split best in fit mass value, by the a p-value is few essentially keV. independent Once of the the splitting DM is mass marginalized as to long as it provide a good fit toto the a excess, massive independently axion of consistently the tends mass to of prefer the asented axion. very above. Therefore, light, First, fitting even we notice massless, thatF axion. elastic scattering cannot explain thein XENON1T hint tension for with complementary direct-detection experiments at lower energy thresholds. to the different energy dependencesscalar of rate the grows axion fast and scalar atfor absorption a low rates scalar energies in mass for of xenon. verygenerates The light a masses bump but between asuppressed 2 good at and low fit 3 energies can keV. and be the On obtained resulting the spectrum other is hand, too flat the at axion energies absorption above rate 3 keV is to the background prediction. For purecompared electron to coupling the the scalar axion orof explanation the the is dark axion disfavored photon. ABChappens The production through reason the [ is Primakoff that processwhile the [ the peaks axion explanation in is the disfavored. spectrum As we will discuss, the reason can be traced back of the mass. Welight show scalars in in section in orange Figure 1 JHEP01(2021)178 . 4 / 6.2 = 1 dashed χ /m �� φ �� m . We also include , heavy mediator =1/4 =1/15 χ χ �� 7 /m -2 -2 /m ϕ discussed section �� ϕ q ∝ q ∝ . We show accelerated DM DM (purple) �� 6.1 2 �� (cyan) m acc.: CR m acc: CR /q q 1 � ∝ ∝ ) 2 . e,F keV, |δ|<4.9 F keV, |δ|=2.5 ) �� q q ( q ∝ q ∝ ( F F DM DM � q ∝ q ∝ �� DM DM :F DM-e: F DM-e: � discussed in section [��] �� can be a scalar or a vector with scalar interactions χ φ � – 6 – � . e,F keV, |δ|<4.9 F keV, |δ|=2.5 �� -� �� -�� ), where 1 ∝ 1 ∝ � (light green) 2 DM DM �� q keV (as discussed in the text, this range of splittings may not 9 ∝ . � momentum suppressed DM-electron scattering with form factors 4 ) light red q ( �� ( < F 15 | . e,F keV, |δ|<4.9 F keV, |δ|=2.5 / δ | green � keV, which implies a lower bound on the DM mass. 5 and = 1 �� . χ , and momentum suppressed form factors = 2 � | /m between the heavy state and the light state in the dark sector is marginalized to δ φ | � �� | -� -� -� -� m δ

�� - (dark green) . We show in (dark blue) ) and q 1 ∝ have fixed ∝ We now present our data analysis framework, before discussing each of these model Third, we discuss accelerated DM by scattering with cosmic rays. In such a case, ) ) q q ( ( dark red around 100 keV.challenged Just by as other other observationfor probes. models future of We work. leave accelerated a DM, more in this depth scenario study isscenarios of in likely this detail. scenario to be the nature of the form factor. the challenge is againrapidly to at energies find lower than a 2 keV. scenariobetween We achieve where the this by the accelerated considering axial-scalar accelerated DM interactions spectrum and falls the sufficiently SM, mediated by a light new mediator with mass is heavier than thedata, splitting without being itself. trivially excluded Theconcrete models, by spectrum the complementary is rich direct phenomenology detection peaked ofof experiments. these near testing DM In 2 them scenario keV could at and providefixed beam other fits splitting, handles dump well a experiments lower the or bound in on the nuclear DM recoil. mass We can also be show derived, that which for varies depending a on The splitting minimize the p-value for capture the entire possible parameterlines space, and should thus be treated with caution). The F DM by cosmic rays scattering for( fixed mass ratios between theon DM the and SM the side mediator and axialdifferent interactions scenarios on for the DM exothermic side.F scattering: This light is mediator discussed in section Figure 2 JHEP01(2021)178 ]), (3.1) 74 , 73 ] at their ], to search 71 [ 71 – 62 , 3 keV 44 . 0 ∼ W, b ], where they simply use 2 2 1 g cS + 1 ] (which uses detector modeling techniques 1 g cS 69 – 7 – = ], and with additional data taken from [ ) plane, and use a maximum likelihood estimator to 72 b are the probabilities for one photon to be detected as a 4 . XENON1T reconstructed (the ‘b’ subscript signifies that only the PMTs at the bottom E ) . In order to reconstruct the energies, we use the procedures = 11 b 2 g cS2 keV , ∼ and (cS1 142 . = 0 1 g In their analysis of the Science Run 1 (SR1) data, the XENON1T collaboration provides The ratio of S2/S1 provides a handle that enables one to differentiate between Nuclear XENON1T collaboration in their ER analysis paper [ where photo-electron in the PMT and the charge amplification factor, respectively, and the mean signal. We useenergy a is Monte distributed on Carlo thefind (MC) (cS1,cS2 the simulation energy to of“our determine the method”, how event. an even Below, though ERpapers; we refer it with we to is a do this based given so way on of to information reconstructing differentiate provided the it energy in from as previous the XENON1T way the energy was reconstructed by the information, as theanalysis XENON1T detector threshold resolution islaid out as by low the as XENON1Tcreated collaboration by in the [ NESTwhich collaboration allows [ us to simulate the detector response and the effects of reconstructing the S1 (cS1) and corrected S2 (cS2), which takes intoa account scatter this plot additional of information. of the detector were usedkeV-binned for the energy S2 spectrum, reconstruction). we Ratherthe than will energies using use their for reconstructed the each data event. from this We do scatter this, plot to since reconstruct the keV-binned data results in a loss of Recoil (NR) andby ER its events. location Further insideS2 information signals, the about and PMTs, a the theinto account given S1 time in and event difference the can S2 betweenavailable. analysis signal be the When by shapes. arrival inferred the the of XENON1T XENON1T This the collaboration collaboration, complementary S1 reports however, information it their and is is data, taken not they publicly use the corrected the detector due to anlayer external (GXe) electric field. at the Whencollide the top electrons with of reach xenon the the atoms, detector, Gaseoussignal, and Xenon they which produce are is a extracted also proportional across measured scintillation the by light, the liquid-gas known PMTs. interface, as the S2 The experiment utilizes a dual-phase xenonfor Time Projection weakly Chamber interacting [ particles. Whenphase one recoils of or the xenon istomultiplier atoms ionized tubes in due (PMTs). the Liquid to Thisaddition Xenon a to signal (LXe) the collision, is photons photons emitted called close are the to emitted prompt the interaction scintillation and point, signal detected ionized electrons (S1). by drift pho- inside In In this section, weand review the the relevant electron aspects recoilreconstruction of analysis, and the statistical with XENON1T analysis a experimental that focus apparatus is used on3.1 throughout describing this our work. treatment Energy of reconstruction the method energy 3 XENON1T JHEP01(2021)178 ] to have 1 ). ] E 1.0 0.8 0.6 0.4 0.2 0 ). 3.1 ]. 3.1 ΔE[σ 75 ) plane, for events b 70 is the expectation value keV 9 keV 8 lines show constant energy away from this expectation keV 7 )). The energy resolution is blue σ keV so the binning leads to 2 60 3.1 ∼ ). In C.L.) for all energies. To avoid keV 6 ] in the (cS1,cS2 1 3.1 . For additional discussion of the 50 5% gray dashed ≤ keV 5 8 eV . are more than 40 = 13 keV 4 ) scatter plot for events tagged in [ cS1 b W – 8 – 30 . The colors of the points correspond to the difference keV 3 black points ]. 1 9 keV 20 ≤ keV 2 . The oprsno reconstructions E of Comparison ). The colored points on the plot are in units of the energy ) to be the best estimator for the energy, the variables cS1 and 10 9keV] 3.1 , (left) our calculation of the keV-wide binned energy spectrum 3.1 keV 1 (above this energy, the resolution is 4 [keV 0 ], eq. ( 1 9 keV 500

2000 1000 , we reproduce the (cS1,cS2 b cS2 3 . Observed events by the XENON1T collaboration [ should be anti-correlated. While this has been validated for high energies, preliminary For the formula eq. ( We show in figure In figure b unbinned energy information forplot our the new background physics model analyses from below. [ We also includecS2 in this measurements appear to suggest that there is only a weak anti-correlation for low energies. sample, large numerical errors maythe occur reconstructed in energy rare of casessuch a errors, where given in the event those calculated is cases likelihood small we for ( use the simplifiedand reconstruction compare method, it eq. with ( theprovides XENON1T confidence spectrum. in The our two spectra energy are reconstruction nearly identical. method, This and allows us to use the full only marginal information loss;so moreover, we the will excessshows not the is be difference concentrated concerned between belowformula the with this used energy events energy, in reconstructed at [ byresolution, our higher calculated method energies). with and the our The simplified method color (see of below). the points Due to the finite size of our MC energy to produce apossible detectable problems quanta with is this simplified energy reconstruction formula, seean [ energy below estimated using our energy reconstructionlines calculation. using The the simplifiedfor energy the reconstruction energy given interval invalue, eq. and we ( did not sampletheir the energy; parameter for space finely these enough points, with we our assume MC to simply reliably that reconstruct their energy is given by eq. ( Figure 3 they tagged as havingin an the energy reconstructed energycalculation (in and units the of simplified the equation energy resolution) used between by our XENON1T energy (eq. reconstruction ( JHEP01(2021)178 , ]. 1 [keV] 8 E ) agrees 1 z 7 0015 . + 0 ]: our agreement 6 E 73 √ h Crslto estimate resolution MC The resolution XENON1T The (MLE) resolution MC Sampled 3171 5 ] (blue line). As can be [keV] E . , red line), and the energy 1 E √ nryResolution Energy 4 Ar line at 2.83 keV presented 39 . 37 [keV] = 0 ]. While the biggest disagreement, at Energy resolution estimated from our E 1 3 σ 21 + 0 Ar data found in [ . ]( 0 37 1 − Right: 2

0.4 0.3 1.0 0.9 0.8 0.7 0.6 0.5

E ] [ σ keV (keV) energies relevant for the excess events. [keV] = – 9 – O E , we find the likelihood of the signal+background σ . This provides further confidence in our energy s b 8 θ disagreement, we note that this is misleading, as many of ) plane for the b σ 1 keV energy bins. ≥ Cestimate MC estimate XENON1T model Background 10 6 − − 5 ∼ [keV] E for the central value, and we can find even better agreement if we 4 indSpectrum Binned ]. 3% 72 Comparison of the naive spectrum reconstructed, and the one by the MC, for keV. In blue is the background model from [ . 9 ]), however, this does not appear to change of the results significantly. Left: 2 75 . , that depends on parameters (right) shows the energy resolution estimated from our MC (black points), a fit to s 4 ] within their error margins. Our simulation also agrees well with the observed ]. Our MC simulation of 2.8 keV events (assuming a uniform distribution in

60 40 20 80 We thank Matthew Szydagis, for helping us verify our results with the more detailed calculation done

[ / ]

keV dE

dN 1 - 1 For our analyses, we usemodel, a likelihood ratio test, with unbinned likelihoods. For each signal by the NEST code [ As the actual smearing ofmight appears provide to an be even notalso more entirely ref. accurate symmetric, [ an description asymmetric than resolution the symmetric one used3.2 here (see Statistical method in [ weak correlation between cS1reconstruction and method, cS2 especially atFigure the these MC data (red line),seen, and the the energy energy resolution resolution estimated estimated from in the [ MC is slightly better than that used in [ In particular, this canin be [ seen from measurementswell with of the contours the in the (cS1,is cS2 better than change the simulation parameters slightly from those given by the XENON1T collaboration decreasing spectrum in the MC (black points), aresolution fit estimated to by these the MC XENON1Tblue data collaboration line). ( in [ energies below the lowest-energy bin, seems tothe be points a were reconstructed onoverall good the agreement edge between of our theis MC bin method observed, causing and enabling small the differences us onean to used to aside, by be use the we magnified. XENON1T the also An collaboration full note unbinned that energy our information binned throughout energy this spectrum paper. does As not have the same monotonically Figure 4 JHEP01(2021)178 = s /dE (3.4) (3.3) b (3.2) µ ( dN left, for an /dE b 6 would correspond dN R 5% = b , , µ ) s θ | . i ) b E ) away from the best fit point ) ( b σ + ) ( b s − bands (see e.g. figure value) L (ˆ N 2 σ L − ( + − σ dE (p s 2 1 distribution). − N log − ( ]; we therefore assume that twice the log- . , 1 . (right), and then calculate the likelihood. F d ]). We will also ignore the look-elsewhere ] by a factor of 2. A p-value of . and 76 F is the number of observed events, O . distribution, with the number of degrees of 4 76 . 75 σ n =1 Y i 2 erfc O 2 n 2 . − b 2 #D – 10 – χ √ 1 µ χ of an excess, we also present the more commonly − ! #D s n µ − Q e value Ar (see e.g. [ − ) = b p 37 is the number of degrees of freedom for the signal hypothesis, + value = Significance = s F − . ( p L O ]) and . ) is the likelihood of the best fit for the signal+background (background- 77 ) , #D b 1 ( 2 L ( is the inverse function to the complementary error function. ) 1 ) band presents the points that are b − σ are the reconstructed energies, + ) is the background spectrum (signal spectrum), and s i − ) are the total expected background (signal) events. We maximize the likelihood (ˆ is the regularized incomplete gamma function (one minus the p-value gives the erfc E 2 L ( significance in our notation. /dE Q , of the xenon detector. We then smear the resulting spectrum by a gaussian with the In our analysis, we will ignore any contribution to the background from, e.g., tritium In each of the following sections, we describe how to derive the spectrum of events. When presenting later 2D plots with To ease interpreting the /dE s σ ) Our p-value definition differs from the one of ref. [ σ s 2 2 ω − ( dN is not flat, and even for such models, thedecays effect (see is e.g. not significant. [ to resolution presented by theThe red effective line exposure in models figure thethe non-flat MC efficiency, stage, and should assimplicity, in we it fact have directly be applied applied relatesyield it during changes to as only the for described signal S1 in models with signal, the a text. and large rate not at Small the the variations 1–2 keV energy. bin on where our the However, efficiency methods for 1 on that graph (i.e. not necessarily the best fitThe point measured in spectrum general). willa be given modified theoretically predicted byE signal, detector we response modify effects. the In spectrum particular, by the for effective exposure, Where example parameter space for theas ALP an DM independent hypothesis), hypothesis each (i.e. point with on a the given graph coupling, is treated mass, etc.). At such graphs, the only) hypothesis. and cumulative distribution function for the used significance freedom set equal to the number of model parameters, where dN to find the bestwe assume fit the points. asymptotic formulaslikelihood-ratio In found of order in [ to thehypothesis estimate signal+background the is hypothesis significance distributed compared and according to quality to the of a background-only our fits, where ( hypothesis for the data as a function of the model parameters, JHEP01(2021)178 ]. 78 (4.1) (4.2) , while σ 4 . ), (ii) scalar ab- 3 − v 4.1 10 I m ; section keV , ), and (iii) dark photon DM ) ). I AE 4.2 m 3 4.3 = − − I ω ( χ δ ρ ) ; section ω ; section ( . The total number of events in the relevant 4 GeVcm . 3 SE 0 I abs PE σ – 11 – = = 1 DM − I I and integrating over energy. sec 3 2 = Φ , depends, however, on the interaction of a given light − ]. For the case of exothermic-DM, the multi-dimensional I abs DM abs cm 1 σ dω [ 13 dR σ 3 10 × , is the same for all bosons, and depends only on the DM relic 2 . ] for a thorough discussion), we briefly discuss here what we expect DM , is kinematically constrained to equal the DM mass (ignoring small 78 Φ ω = 1 DM reproduced in section Φ ) ω ( , and the mass of the light boson E χ with the bounded electrons in the liquid xenon. Here we consider three cases: ρ I The DM flux, In the case of bosonic DM, the rate of events in the XENON1T detector per unit For both standard DM-e scattering models considered here, as well as for the CR- The absorption cross section, boson (i) ALP DM absorptionsorption via via the the axioelectric scalar-electricabsorption effect via effect ( the ( photoelectric effect ( density, where the energy, non-relativistic corrections of orderthe the detector resolution DM as energy), describedXENON1T in and section energy is window then is then smearedexposure obtained to by convolving account the for above rate with the effective relativistic case, for which the boson is produced inenergy the is Sun. 4 Absorption We consider first models ofment. bosonic Three DM, cases confronting areeach them we with considered: explore the pseudo-scalar the XENON1T non-relativistic (axion), measure- case, scalar, in and which vector the bosons. boson constitutes For the DM, and the signal. Indeed, the reportedthe local significance reported by global the one XENON1Tparameter collaboration is space is can greatly affectit both is the location thus and expected width for of the the look-elsewhere signal to spectrum, have and the most drastic effect for these models. accelerated DM presented, the look-elsewhere-effectthe case is of expected particles to producedsince be changing in non-important. the the mass For Sun, of the theenergies, look-elsewhere particle and can will lead yet have to a thethe peaks mild in range solar importance, the of core. signal spectrum possibleto For at masses different the possibly is case be limited of important, by the since roughly DM it the absorption, corresponds temperature the to look-elsewhere of the effect classical is case expected of a highly-localized While a formal calculation ofthis the work global significances (see for ref. eachthe [ model importance is beyond of thethis scope the of paper. look elsewhere effect to be for each of the models considered in effect which is important for determining the global significance of a particular model [ JHEP01(2021)178 �� (4.3) (4.4) (4.5) ) ] 8

=�� ) and Primakoﬀ �� ( �� /�� blue ]( �� = � , 50 [��] ω �� ), and for dark photons � and solar dark photons - that couples to photons 3 / we included the kinematical a 2 e . a ] can be written as [ green 5 v m 4.2.2 γ 1 3 ]( �� = 79 µ � , � right − 5 14 ¯ eγ , �� – �� = /s. 1 , a � is its velocity. We take the photo- 3 2 � ) µ 11 a ∂ ω � 2 e v 2 a ( e � -� -� ω m

aee

�� abs a g �� ] [ ω / Φ 2 σ v + EM 2 Sun I aee g µν dω Φ 3 depends on the production mechanisms of the ˜ πα d F – 12 – �� 16 µν = ], which agrees reasonably well with experimental ) equals 1 event/cm /dω a , solar scalars in section aF 80 ω ( Sun abs Sun I 4 dω aγγ �� Φ PE g dR d 4.1.2 σ �� 10] keV , = , from [ [1 ) ) = a , we show the relevant solar ﬂuxes that are important for the �� ∈ a ω ALP - ( ω 5 ω ( �� L PE is the energy of the ALP and AE [��] ω σ ), for scalars from bremsstrahlung [ 2 a �� σ k

�� + , we assume a massless boson, while on the �� 2 a yellow . In ﬁgure inside the Sun’s environment and needs to be treated case by case. Below m left ]( I p �� 58 = 4.3.2 . Solar ﬂux spectra for the axion production from ABC processes [ a � � ω -�

�� For light bosons produced in the Sun, the differential event rate per unit energy can �� ] [ ω / Φ ). On the red where electric cross section, We consider an axion-likeand particle electrons, (ALP) of arbitrary mass The ALP can be absorbedcross inside section the for detector this material so leading called to axio-electric a (AE) ioniziation effect signal. [ The we discuss solar axions in section in section derivation of the predicted signal’s spectrum for the4.1 different cases. Axion-like particles be written as where the differential solarlight flux bosons threshold due to aspectral features, finite the boson plots mass, areof normalized interest fixed such for to that XENON1T the the , total best integrated flux fit in point the for energy window each case. To highlight the Figure 5 production [ ( JHEP01(2021)178 2 e . 1 /m (4.9) (4.6) (4.7) (4.8) 2 a → . The a m v aγγ g . and as a function of e QCD aee m aee g g Λ and for the ALP DM is generated below the log . is the UV anomaly with ˜ F aγγ u 5 parametrizes the electron d g which decouples as , UV m ) . m aF x 2 E i eff ( + + ∂ e C − d A a u e 2 a f a m m . m f m 4 ) /m 2 3 x 2 e ( = , and in the relativistic limit, log − m A 1 aee eff UV ]. At low energies, one finds UV = 4 C E C a 82 can be introduced as a soft breaking of the x ], v is often related to 2 + a parametrizes the effective coupling to photons, 2 EM π 83 , g −→ QCD α m 4 – 13 – eff aee UV 3 eff with g eff E E E 1 + a = − EM UV 1 πf eff α 2 − C 1 ,C x E √ = = is non-zero, the electron coupling is modified by the running 92 2 . eff 1 aγγ UV C ) can be mapped to concrete models where the pseudo-Nambu- g − E arctan 4.4 eff x E ]. If 81 ) = −→ QCD x ( eff A E . This feature can be traced back to the fact that in the presence of a purely e is the UV coupling of the axion to electrons, while m is the ALP decay constant and a UV f typically implies a non-anomalous global symmetry with respect to QED. motivating chameleon-like ALPs to be discussed in section ultraviolet (UV) couplings to electronsmagnetism. and the UV anomaly with respect to electro- a C More general ALP DM requires suppressed couplings to photonsStandard in solar the ALPs UV, could be which the QCD axion but are excluded by stellar constraints, Fitting the data with QCD axion DM requires a high degree of fine tuning of its In what follows, we will derive the XENON1T best-fit regions for m 2. 3. 1. For the QCD axion, thethe coupling effective to photon the and gluon electronQCD field couplings mesons strength generated below gives by the further the confinement contributions mixing scale to of [ the axion with the derivative coupling to electrons, onlyelectron the threshold [ effective operator contribution induced by the photon coupling [ Here respect to electromagnetism, whichloop is function, model dependent. for where which is related to the UV parameters through To understand these statements,parametrization let of us eq. ( brieflyGoldstone discuss boson the (pNGb) of origin a spontaneously for brokenand global electrons. symmetry couples to An the photons arbitrarilypNGb shift small symmetry. mass More explicitly, we can write which will allow us to identifycan viable be ALP drawn: models. As we shall see below, three conclusions results obtained in the non-relativistic limit, the ALP mass. Theoretically,case, however, X-rays measurements canis then therefore be interesting used to to understand exclude the part theoretical of relation the between parameter the space. two couplings, It data above 30 eV. The above formula is approximate, and chosen to correctly reproduce the JHEP01(2021)178 ) E ( E (4.13) (4.10) (4.11) (4.12) , , ] (darker green). , 7 (left), where the . 44 6 ! 14 − E )) = 15 10 aee tonne-day ]. B g ( × . L 85 200 , 4 / (right). )

84 6 [ B 2 contribution from QCD explains ! / 3 + 14 (1) (1) − S ( . O 10 2 aee 3 a a L keV ( ∼ O g × m m 5 . . π log 2 4 UV 2 ]. 2 aγγ E

16 g ] (light blue) cooling as well as terrestrial limits and the electron coupling dominates the phe- 87 , 23 ∼ – 14 – , = 56 , 14 . 86 a γγ = 0 UV keV − gram and the effective XENON1T exposure, 55 / Γ m 5 C eff eff . 2 10 , and the electron coupling is only generated from the 2 E C UV × cm E together with the = 0 keV. The predicted spectrum is a narrow peak around the 3 1 eff = 4 UV − local significance. The number of signal events is given by, E ∼ C ] for a general discussion of photophobic ALPs and the Ma- σ aee where a = 1133 5 . 88 m DM 3 ρ PE , g σ ] (light green) and the XENON1T S2-only analysis [ 4 GeVcm GeV shown on the right y-axis. We further show constraints from white bands of our likelihood fit is shown in red in figure . , where , where naturally 0 ] as a particularly motivated example of this coupling structure. keV σ 10 91 , 2 5 ALP stability: . 10 90 , 44 , and thus the energy absorbed by the bounded electron in the detector is ' a = 2 and 89 = 33 m a eff σ ] (dark blue) and red giants [ 1 m AE ' 52 /C R a KSVZ models photon coupling via the running [ Photophobic models nomenology. See [ joron [ DFSZ models f E If the coupling to photons is non-vanishing, the ALP DM with the desired range of The Various axion models have been studied, where the different hierarchies between the • • • Imposing that the ALP is stable on timescales of our Universe we get to be dwarfs [ from PandaX [ masses and decay constants is severely challenged by its large decay rate into di-photons, where we used that evaluated at the best fit mass. The predicted coupling to electrons fixes the decay constant best fit point is which corresponds to a with equal to the axion mass.masses Consequently, must in be order around toALP explain the mass, XENON1T with signal the theeffects observed ALP that signal smear spreading the predicted into signal, several as bins shown from in detector figure resolution Of the above,XENON1T and hint without in being the excluded, if absence they4.1.1 of are DM. tuning, only ALP dark theIf matter Photophobic the ALP ALPs is can DM, fit the axio-electric the effect should be treated in the non-relativistic limit why QCD axion DM must be tuned to addresselectron the and anomaly. photon couplings are realized: The strong X-ray limits on JHEP01(2021)178 �� ± 3 . ) and . For region �� (4.14) (4.15) (2 �� contours ' 4.10 = 1 a ν Signal shape eff I �� C a ��  ν � . -- plane. The ALP ) gray shaded ! Right: aee 14 [��] � ]. − , g Dashed brown a 53 aee 10 � �� ]. m g ( × 94 4 -��

2 / =�×�� . � �� =�� 3 � �� σ �� is our best ﬁt point in eq. ( � � � the current direct detection constraints

��

, we show the bounds from star cooling keV keV �� ] and could be substantially improved ] [ /

� - m m 5

contours show the X-rays constraints from

� �� ] [ / 2 2 blue band adapted from [ green red star σ 3 are the XENON1T data, the line is the signal shape after/before smearing and 2 -�� -�� -�� -� − ]. The best fit ALP is predicted to produce Hz 10 94 17 – 15 – ] using the X-ray microcalorimeters in the XQC × �� × �� × �� × �� × �� 10 gray dotted ], and in 1 regions. In × 93 52 , black dots . � σ , the shaded region on the top right is excluded by XQC even CXB =0 2 92 UV = 3 eff eff UV E ]. The -4 a E C E =10 XQC =0 ν RG and the cooling hint UV

= Athena0 proj. 91 ). The UV E blue solid/dashed ,

E UV � E σ 44 1 4.10 . Using this procedure, we ﬁnd 1 yellow − =1 0 [��] PandaX-II θ � � � =π rad 0 θ 2 ] and white dwarfs [

−

� ��

σ oln it2 hint Cooling 56 Allowed parameter space for ALP dark matter in the , region is the resulting signal plus background distribution. CXB bound: � 55 W m -2 regions are the ✶✶ . We also show the weaker bound obtained from CXB [ ] for diﬀerent values of =10 Left: 11 UV S2only E WD − . XENON1T 93 XENON1T = 0 , 10 � 92 UV × -�� -�� -�� -�� dark red blue shaded E

�� 2) . This bound is very similarby to looking the at one individual obtained sourcesconsider in [ and the performing bounds background obtained subtraction. in For instance [ we A very conservative bound canto be be less extracted than by0 the requiring measured the CXB intensity background of at the that photon frequency, line which is point to be stable. Evenof stronger constraints the on the cosmic diphoton width X-raymonochromatic come from background photon-lines observations at (CXB) frequency [ for the best fitis point the in expected eq. background, ( the the which gives already an upper bound on the coupling to photons in order for our best fit of red giants [ from Xenon1T and PandaXXQC [ [ for show the initial misalignmentcompleteness we necessary show to in get the right DM relic abundance for Figure 6 decay constant is plottedthe on the right y-axis. The JHEP01(2021)178 eff eγ C ], as → . We (4.17) (4.16) 96 µ ]. = 1 95 , eff 81 . , non-minimal C 2 ! π 14 . 16 − / 2 0 1 θ aee 10 g ∼ 2 eff × C 4 eff 4 values for the misalignment / C

1 2 / (1) 3 ∗ ], will further improve the X-ray g 90 O ] , is larger than the temperature at which 98 a 2 rh , keV T ! m 101 5 97 . – 14 2 − value is needed to explain the XENON1T 99 , 4 aee 10 − g 95 UV × 10 E 4 – 16 – left. Future X-ray missions like Athena [ ×

2 6 6 . / , due to the irreducible one-loop contribution to the that could be seen at future high intensity muon fa- 1 1 ] for further details). Depending on the actual seesaw ]. and the ALP thermal production is suppressed [ . = 0 59 . A large reheating temperature enhances the thermal production a γa a + osc m eff eff keV m e 103 T . UV , , the correct relic abundance can be generated in the region E C 3 ' a E & → ) = 0 m 01 102 we show that at the moment, even the more optimistic Athena rh . ). The bound from CXB and the one from the XQC rocket are + osc T 6 µ T ( UV 4.7 ] are not enough to test the region of parameter space explaining = 0 E H , we illustrate this limit with dashed gray lines for different values 2 97 , we show in dotted brown lines two and h 6 a 6 a ) and could become important to test the ALP DM interpretation of the Ω + e 4.16 ]. Another interesting consequence is that since → ) we assumed that the reheating temperature, XQC bound: + 90 µ 4.17 , for which the observed DM relic abundance is obtained with 0 . Interestingly, the X-ray bounds discussed so far can exclude a portion of the θ UV All in all, we showed that a very small It is interesting to ask what are the conditions for an ALP DM addressing the anomaly In eq. ( E 3 misalignment contribution [ the ALP starts oscillating, of hot ALPs fromcompared the to SM the thermal cold bath.a one. large This However, parameter hot it space DM is where component easy to could check become that problematic for if an dominant ALP coupled to electrons only there is signals in cilities like MEGII and Mu3escale (see [ one could furtherat explore MEGII the [ parameterproduction space mechanisms of are this required model to by enhance looking the at Majoron relic abundance beyond the anomaly, disfavoring most existinghinting ALP towards models, photophobic and ALPs.ample in of Last, particular photophobic we the ALP: commentloop the QCD level on Majoron. axion, together a with and In particularly LFVfirst this couplings, interesting consequence case after ex- of right-handed electron neutrinos this coupling are framework are integrated is out. generated that at A the XENON1T signal is correlated with future angle, conclude that the standardcondition misalignment can mechanism, address with thecan no ALP be tuning DM made of relic sufficiently the large. density ALP in initial the region of interest as long as of interest via the misalignment mechanism [ On the left of figure prospects derived in [ the Xenon1T excess if to have the observedwith DM a non-dynamical relic mass abundance. If one considers a generic axion-like particle ALP parameter space even if photon coupling in eq. ( shown as shaded graywell regions as in new figure techniques likebound line in intensity eq. mapping ( [ Xenon1T excess. In figure On the left ofof figure rocket. Using these bounds we find for the best fit value JHEP01(2021)178 , . , aγγ (4.19) (4.20) (4.21) (4.18) 9 . /g , 5 . aee . E g )) = 6 tonne-day B ( )) = 11 L 200 / B ) ( 2 E B L , for which the energy /

+ tonne-day 1 ) T − B S ( 200 + L ( GeV a S 4 9 ( m log aγγ − L g 2 ( 12 10 takes on very large values). − log × , . We also consider photophilic ALP 2 3 1 10 eff . aee − 4 E g × , ], which is the conversion of photons into 4 3 keV 2 . 12 . GeV 3 58 − 22 13 10 = 1 − − aee – 17 – 1 a g × 10 10 − 4 m . This is satisfied in many standard QCD axion . × eff sec 3 . The peaked structure of these signals is due to the for the different spectra with a massless (relativistic) 2 . C 1 = 3 − 7 5 − ), the ABC productions are subdominant for 27 = 4 aee local significance respectively. The spectrum for the two cm sec & 2 keV 12 σ ABC, Xe aee − g 1 , g Φ . 10 eff & 2 cm E , in which the spectrum is significantly modified, improving the fit P, Xe aγγ × ω

14 keV Φ 5 . . T 3 and 2 . 10 ] for an explicit model where , g & ). Here we study solar production, not making any assumptions on the aγγ × σ g a 6 = 1 3 . 104 4.4 = 0 1 m a ]. (ii) The Primakoff process [ is the integrated ABC flux in the energy window that is relevant for the a = 61 m is the integrated Primakoff flux, once again, in the energy window that is 50 [ m : = 48 aee : g phobic P, Xe , or equivalently, ABC, Xe − , so that the total signal rate scales as Φ γ solar Φ philic R aee First, we discuss the case of a strong prior on a massless ALP. This prior can be The best fit points in these scenarios are We discuss photophobic ALPs where both production and absorption are controlled Two production mechanisms are of interest: (i) the “ABC” processes: atomic recom- − g γ solar -phobic -philic γ γ R convolution of the solar fluxesprinciple, with the one detector may smearing be andwith efficiency, able suggesting more to that data. differentiate in between the two solar productionjustified mechanisms as a theory bias, given that QCD axion models will typically predict an axion corresponding to cases is shown on the top of figure relevant for the XENON1Tenergy experiment, range and of interest with ( a16 MeV massless ALP. Wemodels find (see that also [ for the while the ALP coupling to the electrons controls the absorption rate. Here one finds where Here XENON1T experiment, calculated for models, where the ALP coupling to photons contributes substantially to the production, axions in the electromagnetic fieldsThis of is the the electrons dominant andwhich production depends ions mechanism on making in up the the energy solar range plasma. relevant forby XENON1T, to the XENON1T data.and See massive figure (non-relativistic) ALPs. bination and de-excitation, bremsstrahlung, andvalue Compton of scattering, all depending on the ALPs can also be producedcouplings in of the eq. Sun ( through processesALPs involving the relic electron density. and We photon considerabsorbed both by the the relativistic bounded case, relativistic electrons case, is independent of the ALP’s mass, as well as the non- 4.1.2 Solar ALPs JHEP01(2021)178 σ ). �� 7 2 region right �� and �� σ 1 �� ] are shown in ��  �� -- gray-shaded ) where the ALP γ- 56 , -� 4.21 55 [��] � , �� 52 , -�� , while the 10 =��×�� γγ � keV shutting of kinematically the �� =� , the middle plot shows constraints � �� 3 �� σ �� . black dots 1 � = 0 region is the predicted binned signal. � �

��

�� ] [ /

� m - ], unless non-trivial dynamics modifies the 83 best fit regions are shown in red, white dwarf plane. As mentioned above, the best-fit value [ �� – 18 – a ) blue-shaded σ a 2 m �� /f �� and GeV σ versus . Stellar cooling constraints [ 1 12 �� , obviously larger than the one in eq. ( ��  � aee -- = 0 (10 �� σ . On the left plot, we show in red the photophilic g 6 a . γ- × 8 2 aγγ m g eV [��] � lines show the signal spectrum before and after detector smearing effects, µ ) we conclude that ABC production provides a slightly better fit to the ) and a photophilic axion with a massless axion as the best fit model ( plane. The 70 . �� left solid -�� 4.21 aee ( g = 5 − and a =��×�� keV, leaving an excess signal in the lowest energy bin. On the other hand, the � �� =�� 3 keV � . m �� . Predicted spectrum for the solar production of a photophobic axion with a best-fit value ) and ( 5 �� σ � � � aγγ . where one can clearly see that the ABC production does not shut off fast enough g 2 � = 1 � 5 �

�� dashed

4.20

The parameter space for the solar production of the photophilic and photophobic Second, we comment on the case of a massive ALP. The ABC production ﬁt can be a

��

�� ] [ /

� m - lies outside the plotblue. at For the same modelin with the the best-fit value (WD), horizontal branch (HB), andand sun limits cooling form limits CAST are are marked shown with dashed-blue in lines, orange. We also show the predicted model lines for eq. ( data than Primakoff after a mass for the ALPaxions is is introduced. shown inbest-fit figure regions in the the one parameter fitmass is was left as a free parameter. sensibly improved by introducingsolar an flux to ALP ameliorate mass theleft. of agreement We with checked the that introducing 1 keV a bin. mass This does not is ameliorate clearly the shown Primakoff in fit. figure Comparing behavior of QCD atdoes high not energies. reproduce The well solarfigure the production spectral of shape a of masslessbelow the photophobic data. axion The reasonmassless can photophilic be model traced provides back the to best-fit one parameter model. The significance of The respectively. The measured XENON1Tis data the is expected shown binned background, as and the mass of Figure 7 of JHEP01(2021)178 ; 1 ], σ 10 2

52 . WD 6 − (4.22) (4.23) orange and RG is again 10 from HB ) σ ' φ 1 ✶✶ ω aγγ XENON1T ( 1 g [keV] a 2 DM oa basin Solar ) for the best-ﬁt m v PE σ red γ-phobic Axion Solar plane. Stellar cooling ] are shaded in HB Sun aee 107 -1 g ] stellar cooling constraints 10 -10 -11 -12 -13 -14 , − 10

10 10 10 10 10 aee g 2 in combination with HB stars. , we discuss brieﬂy a possible ! aγγ ¯ ee . g φ φ -10 5 φ k ω 10 its velocity, and green

1 ). Notice that in the case of scalar

φee φ ⇐ 0 = = Sun

⇓

g a UV v φ

] and Sun [ E = -11

β , 4.5 ,

II 10

, 58

I φ - EM 2 φee µν

g DFSZ -12 F aee XENON1T πα g ) for the photophilic solar axion scenario. The 10 4

µν

CAST ) WD φ red ] is shown in ⇐ – 19 – φF ω -13 ( HB 10 best-ﬁt regions as in the left plot ( 106 [ 4 ⇐ φγγ lines, while limits from CAST [ case and lies outside the plot. White dwarfs (WD) [ PE m γ-philic, Axion Solar σ g ⇓ σ 2 ) can be mapped to concrete models. Two particularly -14 regions in combination with the bound on aee = g -7 -8 -9 10 = 0 -12 -13 -10 -11 ) = 10 10 10

and a

10 10 10 10

4.22

φ γγ a ] [ g GeV

- 1 m σ ω ( scalar 1 1 L :

dashed blue best-ﬁt regions (

10 SE

HB σ

+ HB

: Same as the left plot, but for the photophobic solar axion model. The σ WD + , that couples to photons and electrons blue-shaded 2 is the energy of the scalar φ RG 2 φ Middle ], horizontal branch stars (HB) [ k Right and 56 + , σ 1 1 [keV] 2 φ a 55 oa basin+HB Solar ] m m XENON1T q only Left: γ-philic Axion Solar HB 108 HB = , + . ], the Sun basin bound on φ indicates the best ﬁt point in this case. 14 Sun are shown by the ω -1 105 10 The parametrization of eq. ( ee -22 -23 -24 -19 -20 -21

10 10 10 10 10 10

γγ aee a ] [ GeV g g

- 1 where the photoelectric cross sectionDM, already the used expression in above eq. leads ( to a suppression ofmotivated the scenarios absorption are rate (i) of a light SM singlet mixing with the SM Higgs doublet, and (ii) The cross section for scalar-electricone (SE) as effect [ can be written in terms of the photoelectric 4.2 The scalar Consider now a scalar, the DFSZ and KSVZbest-fit axion regions models. for theconclude Finally, photophobic that on for case the all insevere right cases, red tension plot the with and we solar stellar themechanism show axion cooling stellar to the explanation constraints. circumvent constraints these to in In bounds. the blue. section XENON1T anomaly We is in constraints are indicated with arrows point to regions that areright allowed. part The of theoretical the axion plot. modelred lines star are shown in the bottom- red giants (RG) [ on stars [ Different assumptions about the gravitationalthe ejection dashed timescale green distinguish line. themassless shaded photophilic region solar and axion model, but here shown in the Figure 8 best fit point corresponds to the JHEP01(2021)178 ) φee g 4.24 (4.25) (4.26) (4.24) best-fit σ 2 . and the elec- . e and 7 . ) σ /m 1 φ m keV 2 . )) = 15 . tonne-day B (2 e ( v E m L ). Regions excluded by RG 200 / θ ) B ! 4.24 . Thus, analogously to the ALP ], and the prediction of eq. ( e + 13 = sin − S 112 ( /m is suppressed as φee φ 10 L 2 φee ( m g × , and we fixed the coupling of the Higgs , we show in red the φγγ e plane. On the right y-axis, we map 7 log g . ] 2 9 m φ 1 , g m

111 - , – – 20 – SM φγγ ) is a dilaton, its coupling to the SM are more φ 13 φee κ − g m 109 a keV 4.22 10 πv EM m 5 . 2 α × 2 is the asymptotic value of the SM loop functions from θ 7 . 2 ) e , ignoring possible deviations from its predicted SM value. 3 = 1 = sin , and a large coupling to nucleons is also generated from the /v /m 3 cm e φ / − φee m φγγ m DM g 10 ( . On the left of figure ρ · 7 = GeV O , g − 2 e 4 . , and the final couplings of the singlet to photons and electrons are . y 10 0 3 + keV = 4 / × v/f 5 . 1 11 . = 2 φee . The mixing can be written in terms of the ratio of the Higgs and the singlet = 2 ' = 33 θ /g , with values, a ' H e † 9 m AE sin θ m SM φγγ H R The Higgs-mixing scenario, whereelectron the couplings is ratio fixed. of the relativeThe strength photophobic of dilaton scenario, photon where tron and coupling dominates the phenomenology. κ 2 Conversely, if the scalar in eq. ( A singlet obtaining a VEV would generically mix with the Higgs through the quartic sin φ φγγ • • g ’s and Standard Model fermions for φH The predicted coupling toder electrons corresponds toregions a for mixing the angle with scalarto the DM the Higgs case mixing of in angle or- the for the doublet-singlet model, eq. ( The number of signal events is given by 4.2.1 Scalar darkAs matter in the ALPmass, case, as can the be absorption seenof spectrum in figure the of spectrum the plotted scalar for is the best-fit sharply scalar peaked DM around model its on the right by direct UV couplingsmixing with between the the Higgs CFT canare be and changed. arbitrarily the suppressed SM. [ suppressed In photon particular, In coupling, for this whichcase, framework, a decouples we the as consider dilaton, two dilaton one possibilities: can entertain the possibility of a loop- In this simple framework, theto ratio between the photoncouplings and of the the electron Higgs to coupling gluons. is fixed model dependent and controlled by the infrared (IR) trace anomaly contributions induced Here W to electrons to be these models, pointing to the distinct nature of theirλ photon and electron couplings. VEVs, generated once the mixing is resolved [ the dilaton from a spontaneously broken conformal-invariance. Below we briefly review JHEP01(2021)178 ] �� red 108 �� Right: (4.29) (4.28) (4.27) �� region is ]. , 7 , . gray shaded 91 , ��  44 � plane. The -- ) )) = 13 )) = 18 aee B B [��] � blue shaded ( ( , g a L L �� / / m ) ) ( -�� ] are shown in dark and and 2 sigma regions around B B σ 91 + + 1 . S S =��×�� =�� ( ( ϕ ϕ�� σ �� L L EM ( ( � � 2 � φee

��

]. Similarly, a photophilic scalar is

��

πα

log log

�� ] [ / the scalar DM cannot explain the

� - 4 2 2 are the Xenon1T data. The 50 9 =

, ,

) ) �� θ 14 24 regions are the γ φ − − + + is the signal shape and the 10 10 e e -� -�� -� -� -� black dots × × + + 1 2 – 21 – ). . The rate can be obtained through the rescaling . . dark red φ N N = 2 = 4 + 4.25 �� × �� × �� × �� × �� × �� → → blue line e ] and PandaX-II analysis [ e e φee φee + 44 g � + + ) and the N φγγ N N → 4.10 Γ( Γ( . e ] are shown in light blue, while the exclusion regions due to � ,, g g 5 + 56 , N keV keV ��-�� 9 1 [��] 55 . . � � � = 1 = 1 Allowed parameter space for scalar dark matter in the φ φ �� m m � the present direct detection constraints from Xenon1T and PandaX [ ✶✶ �� : : Left: we show the bounds from star cooling of red giants and horizontal branch stars [ �� . � green blue -�� -�� -�� -�� -�� We fit both the photophilic and photophobic scalar to the XENON1T data. We find is our best fit point in eq. ( -philic

-phobic γ

�� ϕ � γ the best-fit points The above agrees numerically withproduced the via one the Primakoff given process, influxes with [ are a shown rate in similar figure to that of the ALP. The predicted Much like ALPs, light scalarsDM. can For be a produced photophobic scalar, inscalar-bremsstrahlung the the production Sun, in whether the or Sunof not is the they dominated regular by constitute electron-nucleus photon-bremsstrahlungso by we the find ratio of the matrix elements squared. Doing light green, respectively. AsXENON1T one excess can because of see theALP from large case. figure suppression of its absorption rate compared4.2.2 to the Solar scalar Signal shape for the bestregion fit is point the in expected eq.the background, ( resulting the signal plus background distribution. cooling constraints [ the XENON1T S2-only analysis [ Figure 9 star it. In and in JHEP01(2021)178 �� , a 0 A �� (4.30) ). The �� right ( ��  �� -- γ- 1 keV . ]. Much as in the e , -� region is the expected µ = 1 [��] � 113 �� φ ¯ eγ µ -�� m �� eA + =��×�� 0 µ ϕγγ . The dark photon may couple gray-shaded 0 � � �� =�� ). This implies a soft spectrum, A ϕ ϕ�� µ �� σ �� 0 � A � U(1) � 4.23 0

��

�� 2 A

�� ] [ /

� - m while the 1 2 best-fit regions of the solar production of a + σ 2 0 µν F – 22 – black dots . As before, the gray region shows the expected µν �� and F σ 10 2 1 �� − �� region is the predicted binned signal. 0 µν F �� µν 0 ��  � -- �� F γ- 1 4 ) and photophilic scalar with a best fit value blue-shaded − left [��] � ( = L �� keV , we show in red the lines show the signal spectrum before and after detector smearing effects respective. . -�� 11 = 2 solid . Predicted spectrum for the solar production of a photophobic scalar with a best-fit =��×�� φ � �� =�� ϕ ϕ�� m �� σ �� and � � �

The relevant interactions are In ﬁgure

��

�� ] [ /

� - previous sections, we consider thea absorption dark of photon a in dark the photon Sun DM as and explanations the to production the of XENON1T anomaly. 4.3 The dark photon As the finalmassive absorption gauge scenario boson of ofto a this broken ordinary section, (dark) matter gauge let via group us its consider kinetic mixing the with dark the photon visible photon [ the massless ALP providedrapidly the falling best absorption rate fit.which at must be high The cut energies, reason off eq.the at for photophilic ( production this case, through combined can kinematic stellar effects beshown cooling from separately constraints traced a are for massive back shown the particle. in to photophobic blue For scalar. the while those are binned background while the blueXENON1T fillings data show the are binned shown contribution in of black. the signals. The photophilic (left) and photophobic (right)explain scalars. the In XENON1T both anomaly. cases, This only is a in massive contrast scalar to the can photophilic ALP case for which The measured XENON1T data isbinned shown background as and for which we showafter with smearing dashed respectively and in solid figure blue lines the predicted spectrum before and Figure 10 value of dashed JHEP01(2021)178 ] �� 108 , (4.33) (4.31) (4.32) 10 keV. In ∼ ) and photo- ω left indicate the best [��] keV, the production ϕ 3 , � . 0 T red stars Γ �� γ-�� 2 ' ✶✶ 0 pl ,

1 2 A ω �� are the photon and dark pho- Lν − m − µν 0 Lµ . 2 ], and horizontal branch (HB) [ ) ω/T F ω L � regions. The E . 56 -�� -�� -�� -�� , ( q ]

exp

�� + Π 55 and �� ϕ � PE 11 2 10% 1 σ π T ν i · µν 2 F dr ) for the photophilic solar scalar ( 2 T µ i – 23 – red blue-shaded ]. For dark photon DM with mass near 1 keV, we πr 2 , �� ) = 4 , which can be decomposed into longitudinal and E is the kinetic-mixing parameter. After the kinetic X =1 i

i ( 114 R , T 0 ν EM DP Z 11 σ 2 ,J = Π 1 πR µ EM µν 4 J Π h best-fit regions ( = 2 e σ T 2 [��] ) scenarios. Red giants (RG) [ = Φ ϕ dω d � and µν right σ Π γ-�� , and the dark-photon absorption cross-section can be related to the SM 1 e �� larger than the typical solar plasma frequency �� 0 is the mass of the dark photon, Left: + are the polarization vectors. In general, in-medium effects should be accounted A 0 . �� A m ✶✶ L,T m � -�� -�� -�� -�� For Inside a medium, the propagation of electromagnetic fields is determined by the po-

��

�� γγ

�� ] [

� - of dark photons in thesuch Sun a is case dominated the by flux the at transverse the modes Earth at is energies found to be [ where for in order toeffects correctly following the compute discussion the infind dark [ photon that absorption the rate. absorptionlongitudinal is ones We dominated modifies implement by the these rate the by transverse less modes, than and the inclusion of the larization tensor transverse components as, terms are diagonalized, the darkpling photon strength couples to thephotelectric electron cross vector section current by with a a cou- simple rescaling, fit points in bothexplain cases. the data, In due contrast to to its a sharply photophilic rising ALP, absorption thewhere rate scalar at must low be energy. massiveton in order field to strength respectively, and Figure 11 phobic solar scalar ( stellar cooling constraints are shown by the JHEP01(2021)178 . �� light : An 5 keV region (4.34) �� . ]. This �� 4 = 2 Right 0 line delimits is the Sun’s A ]. �� limit [ T m ��  � , -- 0 116 7 , gray-shaded . → 0 115 [��] � A , where dashed blue 4 m ) )) = 15 B /T �� 0 ( , while the ]. The A -�� L / m 44 ) ( . As a consequence, the rate of B � �� =�� 0 � �� σ �� ϵ=��×�� A + � black dots � S �

(

��

) for dark photon DM with a Stuckelberg

�� ] [

/ L

� - ( region is the predicted binned signal. red log 2 , we show an example for the best-ﬁt model, , – 24 – �� 12 16 − blue-shaded . Conversely, if the dark photon’s mass is generated 10 5 the production/absorption of a dynamical dark photon limit for an on-shell × 0 best-ﬁt regions ( �� 8 0 A . σ m �� 2 is dominated by free-free absorption and Compton scattering. → = 5 )- ∼ 0 T represent the RG and HB cooling limits, respectively, and the ]. For a non-dynamical Stuckelberg mass, the dark and visible A Γ h and m m 114 , σ [��] ��

��

�� lines show the signal spectrum before and after detector smearing eﬀects, � ✶✶ �� keV ( . For 5 0 as shown in ﬁgure . 0 : The solid A λ/e = 2 �� m Left 0 √ and A . ' �� ∼ �� m 0 �� ω region which can also explain the anomalous cooling of HB stars [ A Light and darker blue � region is excluded due to the XENON1T S2-only analysis [ σ dashed -�� -�� -�� -�� -�� -�� 2 /m

�� h ϵ If the darkXENON1T detector photon is very plays similar tosubsections. the the On other bosonic role the DM right cases of panel discussed of in DM, figure the previous its predicted absorption spectrum in the m therefore goes predominantly through thecase radial shares component many in the featuresdiscuss with it the here absorption for scenarios the discussed sake so of far, brevity. 4.3.1 and we will not Dark photon dark matter production/absorption of the transverse modestemperature. falls Adding off a as Stuckelberg massaround to the dark photon willthrough then the cut VEV off of thedark solar a Higgs flux dark mass Higgs, is controlled then by the the ratio ratio of between the the Higgs dark quartic and photon the mass dark and gauge coupling the where the interaction rate At lower masses, the behaviorthe of dark the photon flux mass fromsectors [ the decouple Sun in depends crucially the on the nature of the example of the predictedThe spectrum for darkrespectively. photon The DM measured using XENON1Tis the data the is expected best-fit shown binned value background. as The Figure 12 mass. green JHEP01(2021)178 ]. 117 (4.36) (4.37) (4.35) , between 95 , and the 0 8 ˜ ) F 0 e . . 3 . (right). In the φF /m 0 A ) 13 m day ( / 2 keV )) = 17 ] can accommodate the ] explains the DM relic 2 5 . B best-fit regions for dark ton α ( (2 119 L 118 σ ∼ E / 2 200 ) 4 B / 1 and + 16 σ S − ( 1 0 A L keV 10 ( 5 m . × 2 log 7 2 . 5 , GeV 13 – 25 – − 11 a 10 keV 10 m 5 × . ]. The only decay channel allowed kinematically is × 2 4 3 . , we show in red the = 8 = 2 3 I ]. In particular, the mechanism in [ 12 H cm / 122 , – DM ρ GeV keV 119 4 2 . . 0 = 2 and the dark photon. In principle the different inflationary production 0 A φ = 33 m ]. We learn that the explanation of the XENON1T anomaly with dark photon 44 PE , which is induced by dimension eight operators generated at one loop from the γ R 3 Finally, two remarks are in order. First, a major advantage of dark photon DM com- On the left plot of figure → 0 the signal yieldtogether in with the the binned lower background, XENON1Tleft signal, plot, and bins. data we is show The showncooling the in best unsmeared bounds. figure fit and We region smeared learnexcluded for that by spectrum, the the as model, astrophysical for together bounds. the with scalar the and HB ALP, and the RG best-fit stellar regime is robustly 4.3.2 Solar darkFor photon a Stuckelberg dark photon produced in the Sun, the best fitAs point for is the case of the scalar, the presence of a mass cuts off the low-energy flux to reduce minimal mechanisms [ correct DM abundance forthe a inflaton, keV dark photonmechanisms of by dark postulating photon a DMof could coupling the be matter distinguished power by spectrum looking at at short the scales. detailed features The contribution from inflationary fluctuationsabundance explored relating in directly ref. [ the scale of inflation with the dark photon mass Lower scales of inflation can be achieved by producing the dark photon with other non- electron coupling. The widthdark of photon this explanation to process thecaying is DM. XENON1T suppressed anomaly Second, by is the safelyand misalignment outside axion-DM mechanism, relic any which densities, bound comfortably fails explains froma to the de- non-minimal generate scalar- coupling the of observed the dark dark photon field-strength abundance to unless gravity is taken into account [ analysis [ DM is viable. pared to the ALPparticles and is scalar extremely casesA is suppressed that [ the decay rate of a keV dark photon into SM expected binned background is shown inin the blue. figure in The gray, XENON1T while the data binned is signal presented is with shown photon black dots. DM with acooling Stuckelberg limits mass. respectively In and light in and light darker green, blue, the we constraint show from the the RG XENON1T and S2-only HB Dashed and solid linessee represent that, the as unsmeared with the andmass, axion smeared and and spectrum, scalar, detector the respectively. resolution spectral allow shape We for is peaked a around reasonable the fit dark photon to data. As in previous plots, the for which the number of signal events is given by JHEP01(2021)178 �� ]), �� 123 , �� 20 lines show the �� ] cooling limits. ��  � -- 56 , solid 55 [��] � and �� regions are the expected binned dashed -�� ],RG and HB [ � �� =�� σ �� ϵ=��×�� . The 116 , � � �

��

blue-shaded

��

) for a dark photon with a Stuckelberg mass

��

] [ / 3 keV

� - . ]. red and : An example of the predicted spectrum for the 27 = 2 – 0 �� – 26 – A 21 , four stellar cooling bounds may need to be ad- m Right aee �� g . Among the four, the solar cooling bound is the least ) gray-shaded 1 represent the Sun [ best-fit regions ( σ �� ( while the [��] �� Blue regions ✶✶ black dots : The 1- and 2- �� indicates the best fit point. ). Below we entertain a simple novel model of this kind, leaving a more general Left 8 . � red star For the axion-electron coupling, Here we focus on the specific case of chameleon-like ALPs (cALPs). While most -�� -�� -�� -�� -��

�� ϵ framework as well as possible generalizations for future work. dressed: RG, WD, HBcouplings stars, are and summarized Sun in cooling. table constraining The and resulting does not bounds exclude on the the ALP ALP explanation of electron the XENON1T anomaly (see for previous work has focusedto on study suppressing the the suppression axion-photon ofparameter couplings space the in for axion-electron solar stars, coupling, ALP we which(see models figure choose is that sufficient predict to either open only the up latter the or both couplings constraints arise from theenvironment. energy In loss principle, induced the by constraintsdepend can the on be emission evaded the of if environment, the lightChameleon-like thereby properties bosons particles allowing of in have these for been the particles ain star studied suppressed order extensively production to in evade in a fifth-forcebut stars. broader constraints also context, or Such for for play the the example particular role case of of dark ALPs energy [ (see e.g. [ 5 Chameleon-like ALPs: circumventing theAs stellar discussed cooling in bounds theexplanation for previous the XENON1T section, anomaly particles due to produced stringent in stellar cooling the constraints. Sun These are excluded as an The solar dark photon usingsignal the spectrum best-fit before and value data is after shown detector as smearingbackground effects and respective. signal respectively. The measured XENON1T Figure 13 produced in the sun. JHEP01(2021)178 , s is 2 Z s (5.2) (5.1) , where , allows the . a that rotates the c.c. ia/f e ) ] ] ] s PQ ) + ] , roughly four orders 54 57 S + 50 4 , we can neglect the ] + 1 = 0 – – , ( , 58 ). Therefore, a model a R Ref. X ) V f U(1) 51 55 e 49 1 ( 2 X m MeV 2 + 1 m √ 4 2 ) ] + [ ∼ ]. For simplicity, we consider X L S from star cooling with the rough = M e , we assume that the following X [ keV , charged under a Peccei-Quinn m RG S , λ ( + aee S 1.3 [ 8.6 [ 0.8 [ 8.6 [ M 127 1 4 g ρ , WD core − + ρ T 126 2 Θ( is the ALP, which is massless up to the X ) ! a 4 2 4 2 X − − . The two fields are odd under the same 2 X 4 m 10 10 7 3 m X 2 and . . MeV explicitly. For M 7 4 ( × × a – 27 – − M X f 7 3 . . 2 s λ develops a VEV, PQ core − 6 4 ρ λ ρ M ρ S √

U(1) = 1 2 2 11 13 13 13 2 e s − − − − m + M m 10 10 10 10 R 2 ee bound c e × × × × L 4 5 8 3 e = aee . . . . e is such that g 2 9 2 4 m ) 2 aee 2 g S ( ] and a real SM-singlet M XS V . Ensuring that under that symmetry Sun HB WD RG Star ee 124 S ). The HB bound is in marginal tension with the XENON1T explanation c 8 L ⊃ ], where the SM electrons carry charges under the same . Summary of the bounds on the electron coupling couples to density. Below a given cutoff scale, 125 X The potential Consider a complex Standard Model (SM) singlet, For this reason, we focus here mostly on evading the RG and WD bounds. The energy =1). The cut-off scale in such a construction would correspond to the scale of the vector -invariant interactions are generated ] 2 S the massive singlet with mass addition of operators breaking the dynamics and write the effective coupling of the ALP to the electrons [ like-fermions required to generatethe this theory interaction below [ theabove it. Higgs mass scale, ignoring further complications that might arise The interaction term withtion the [ electrons cancomplex be singlet induced inoperator a above Froggatt-Nielsen while construc- forbidding unwanted others (we normalize the singlet charge to be (PQ) symmetry [ and Z that suppresses production onlydensity ones in may high evade densityproduction RG in stars and the while WD Sun keeping unchanged.for constraints it which To and, illustrate production unaltered at this in in the point, high-densityconstraints, we low objects same as now well is time, discuss as suppressed. leave a a A the simple UV-completion more model ALP of thorough this study model, of is the left for future work. instance figure if one accounts for the potentially large systematical uncertainties. losses in RGdegenerate and core, WD where the are centralof density dominated is magnitude of by larger order the than production the core of density light of bosons the in Sun the (see table highly Table 1 value of the densitythe at stellar the modelling core. and should Thecooling be bounds constraints. taken reported as benchmark here values ignore exemplifying the the power systematic of uncertainties the on stellar JHEP01(2021)178 ) ) 1 X line one 10 ] for = 1 = 1 (5.3)

S 2 X WD S regions basin Solar 106 λ m will allow 2 -2 X RG M plane. In the λ =10 p ) ϵ dotted magenta & dark red ρ ,M ✶✶✶✶ ✶✶ X XENON1T 1 m [keV] ( a ). The ]: at low densities, ), for m are circumvented by the dashed dark magenta 5.6 1 5.2 128 =1 p ϵ plane is shown in the white 1 = -phobic γ Axion Solar =1 HB ) p , p . The smallness of the chameleon ϵ ϵ -2 X RG HB λ ) for unsuppressed Sun ﬂux ( ,M , as in eq. ( =10 p X Sun ϵ and 1 otherwise. The second term -1 X WD 4.28 m 0 λ ρ ( 10 -10 -11 -12 -13 .

2 10 10 10 10

x < , to electrons vanishes, shutting down its we show the bounds from Sun basins [ aee allowed by star cooling bounds (we fix g a if M 2 X max X λ green m – 28 – ) � �� . ) = 0 - �� x �� ) ). In is fixed at its best fit point. A lower quartic = � - � �� 2 for different choices of λ core Θ( ( , �� − ) � = �

aee � �� -

λ �� g ( �� ρ � �� max = �� = 10 M λ � ( Parameter space of the chameleon ALP produced in the Sun. The

�� S �� ) RG, WD and HB are shut off while the Sun production is unchanged. � ) �� 5.3 symmetry is restored. As shown in eq. ( Right: . [��] -� either one or the other requirement is not satisfied. The maximal cutoff (λ=λ 2 � �� Z � �� 14 . As we can see, suppressing the sun flux extend the parameter space of the � 2 blue band around the best fit point in eq. ( − -� Allowed parameter space for chameleon-like ALPs in the σ �� - �� 2 , and the coupling of the ALP, and

�� = 10 evaluated at the maximal quartic Left: defined in eq. ( depends on the chameleon quartic coupling -� = 0 S ) expresses nothing more than the idea discussed in [ and . �� is the matter density and �� i σ max 5.1 max 1 ρ X orange and M if we want topressed. avoid WD, The RG, allowed or parameterband HB space of constraints figure in while the keeping the Sun flux unsup- First, in accordance with the discussion above h M -� Several conditions limit the parameter space of the example above: � � � � � � • ��

�� = 1 �� ] [ is positive, and the finds production in stars. model as discussed in the text. The shaded bluewhere band is excluded by Sunin cooling. eq. ( has a negative mass, obtaining a VEV. Conversely, at high densities, its squared mass for a higher cutoff scale. star cooling bounds fromchameleon HB, mechanism WD for and allare HB the the stars parameter summarized space inand shown table for in a the suppressed leftS Sun panel. flux ( The In the scale contours show different values of quartic can be takenshows as a measure ofonce the the fine ALP tuning coupling of to electrons the model. The Figure 14 white region JHEP01(2021)178 is to 2 (5.7) (5.8) (5.4) (5.5) (5.6) as in , and min Xee right, S /M g λ controls M ee 14 c e X . m i . S S . In the limit 1 h 2 λ / X 1 = 2 . Xee 6 core 16 g / ρ MeV 1 − . allowed by stellar cooling . To avoid this, we require to the XENON1T best fit ) even though it is allowed 10 3 X . Xee 1 X keV the RG bounds are the λ × g 5.1 4 2 aee / max X 7 g 1 10 . λ 6 6 M / core MeV 1 . MeV ρ S X to electrons 1 2 λ λ X ! X m 12 12 core MeV − aee − ρ ee 1 . 1 ]. Setting g c 10 10 ), we get aee aee ≡ g g 108 × × keV 6 6 4.20 4 . . – 29 – min GeV Xee 2 2 − MeV g 6 6

. 10 & 8 2 was omitted from eq. ( ≡ / − 24 1 X 2 | λ 10 Xee obtains a VEV, such a quartic induces a new mass term . S max | g 12 , needed in order to make this model phenomenologically a 2 × S − M f X 7 X λ 10 . ≡ SX × , we get the maximal value of λ M ). max X to comply with perturbativity, we get an upper bound on the cutoff = 5 λ 5.3 1 = 1 ee . S . c . Independently of its bare value, this quartic will be generated at one λ X 2 X ee λ the coupling of the chameleon field c m X . m 2 M a f & s SX chameleon masses weaken the phenomenologicalarchy bounds, between the allowing for couplings,dictated a but by milder at eq. hier- the ( price of lowering the cut-off scale The above reveals a hierarchythat between of the the quartic chameleon, ofviable. the PQ-breaking This field hierarchystance, might three be loop difficult to contributionstheir realize to electron quantum the couplings, mechanically. singlet will and For act in- chameleon to quartics induced make by them both of the same order. Higher constraints. In themost mass stringent, and range we find, A conservative bound onsatisfy the parameter the space stellar can coolingand be constraints obtained setting [ by requiring Finally, we needm to avoid theenhanced phenomenological compared to constraints the on one of the ALP and is bounded from below by Requiring scale Third we want to fitthe still solar the ALP XENON1T best-fit hint with model the in cALP. eq. Using ( as a benchmark by all symmetries. When for X that could destroyλ the density-dependent VEV of loop via the electrons. Putting all together we get an upper bound on the VEV of S, Second, the quartic In summary, cALPs could avoid stellar cooling bounds. As shown in figure • • • the stellar cooling from dense stars can be circumvented if a new light scalar JHEP01(2021)178 (6.1) (6.2) , while the . Achieving 2 2 aee / 1 p g p left. 14 lies in the mass vs cut-off , is the mass of the nucleus. 2 , which scatters off a bound X v N v χ m m 1 2 . = 2 χN N , while keeping the detection rate fixed, q µ 2 2 q m aee 2 g − + v · 2 | – 30 – q q and initial velocity χ = − χ m e v to the electron. Energy conservation of the DM-atom 2 χ E m ]. Due to the distinctive kinematics of this process, the q . Increasing ∆ m | 15 4 aee + g p e E ∆ left, its potential is modified by density dependent effects. In the could take any value. The maximum energy that can be deposited 14 q ], before discussing momentum-dependent and exothermic interactions. 17 , , in the solar production of ALPs, the solar flux scales as 15 1 is the energy transferred to the electron and ] for a first discussion of such a possibility). Indeed, for a given suppression e 23 E p ∆ ]. A possibility to relax this constraint, which we do not pursue here, is to suppress Our cALP construction is still challenged by the Sun basins constraint pointed out 106 This can be written as As the initial electron ismomentum in transfer a bound state, it can have arbitrary momentum, and hence the electron, transferring a momentum system gives, where 6.1 Standard DM-electron scattering We begin by reviewingdiscussed in the [ standardConsider DM-electron a scattering DM kinematics particle and with mass formalism here whether other scenarioselectrons can as explain well as the DM-electron XENON1Ttransfer interactions data: (up that increase to exothermic as some scatteringwork a cutoff off function well, of and scale). the momentum-dependent momentum interactionsthe We also XENON1T will excess. provide find a that potential exothermic explanation scattering of off electrons If DM interacts withproduce electrons, an it electron can recoil signal scatterelectron [ off recoil the signal electrons forbelow in “standard” the the DM-electron target keV scattering material energiesthus peaks and needed in conflict at to with recoil explain lower threshold energies the direct-detection well searches. XENON1T However, data; we will this investigate standard process is this suppression requires extraimportant fine-tuning role in in the generalizing model cALPs presented to here the but case could of play light6 an scalars and dark photons. Dark matter-electron scattering in [ the solar production in order to(see relax ref. stellar cooling [ bounds withfactor, respect to direct detection solar detection rate scales as implies a relative suppression in the solar cooling bound, which scales as range shown in figure simplest construction, the chameleon-like scalarcan can be be arranged to light be andmode sufficiently the and high cut-off the if of quartic a the of hierarchy theory the between chameleon the is quartic arranged of as the shown PQ in radial figure the coupling of the ALP to matter. If chameleon-like scalar JHEP01(2021)178 , , 2 ) is | 3 ) q − ( q eff (6.3) (6.4) ( 2 10 ] and Z nl keV. → 1 ∼ E f ], calling 129 − v e 100 E 130 ∆ ≥ . We use the q q → electron recoil. nl f keV, where | 4 (keV) × O , and we get eff q momentum-transfer scale Z , ∼ x e · q i typical e αm ) x eff ) (neglecting the second term, which ( Z 1 . 2 ψ 6.2 ) ∼ v x ( χN ∗ 2 typ µ q 1 2 x ψ few eV. While higher momentum transfers are 3 – 31 – . d × e (keV), since the outgoing wave functions are not ; our ‘Plane Wave’ form factors will therefore not Z E eff x O · Z ∆ q i ) = ∼ e . q ( q 2 , we plot the non-relativistic form factors typ q → 3 1 , and almost the entire kinetic energy of the incoming DM 15 f − χ (keV). Also, within this scheme, the form factors are multiplied ], taking the initial bound-state wave functions from [ m 10 O 16 ∼ ' . is the initial bound-state (final-state) electron wavefunction. There e q E )) χN ∆ x µ ( , 2 ψ N m )( x ( 1 ψ χ Finally, it is important toavailable atomic include form the factors relativistic with correctionsthem relativistic the for ‘Relativistic’ corrections high form computed factors. in [ These form factors are given for orthogonal to the boundresults state by wave much, functions. since This DM-electronform factor issue scattering at ends does up notby not typically a sample affecting Fermi the our factor atomic (see below). Another simple approximation foraccount calculating the relativistic the corrections form iswaves. to factors consider We without the also taking outgoing consider wavefunctions into this asthe plane ‘Plane approach Wave’ here form and factors. calloperator In the this from form approach, the factors we also operator sobe do obtained not correctly as subtract behaved the identity for numerically solving the Schrödinger equation withthe a bound central potential state that wavefunctions reproduces forscheme the of outgoing form wave factors functions. asaccount the We the will ‘Non-relativistic’ relativistic refer form corrections to factors important this as at these do high not momenta. take into First, we follow [ m In the left panel of figure We can see this behavior in more detail by calculating the atomic form factor • • • for different initial electron shells {n,l} of the xenon atom and for two different values of where are various methodsapproaches to for calculate calculating the the form wave factors: functions. We here consider three different momentum that is much higher than the typical momentum. which captures the transition from state 1 to state 2, However, for DM with massesis above set the by MeV the scale,the electron’s effective the momentum, charge seen given by by theis electron. usually From small), eq. ( possible, they are dramatically suppressed, since it is unlikely for the electron to have a For particle can be transferred toa the DM electron. particle Since with the mass typical of DM halo a velocity few is GeV can in principle produce a is then found by maximizing the above equation with respect to JHEP01(2021)178 nl E keV, keV, (6.5) (6.6) (6.7) is the e �� 500 200 �� = E �� -�� ∆ Δ� & )). This is & , where q q nl , 6.2 . In order for E e 2 | �� ) E − ] q ∆ e ( 2 [��] � E 17 is the reference DM- keV. , → ∆ 1 e (see eq. ( 15 f �� σ | = 3 2 2 | e | ) ) E q q ( MeV ( for higher ∆ �� q & , DM �� DM | F q F | 2 keV (dashed lines), where | ) keV. We see that for ) v e × | -� -� -� -� -� -� -� -� · -�� -�� 2 e = 8 2

��

q �� - →Δ

� � = 3 ) (

| e ) m αm � e ( 2 χ E e − ∆ E free αm πm 2 ∆ χN ( q µ 16 |M 2 keV, we need | free – 32 – 1 2 χe + � µ |M e �� & �� E e ≡ ≡ , we compare the three different form factor schemes e E 2 | (∆ σ ) δ ∆ 15 q q Non-relativistic (solid lines), Plane Wave without Fermi factor π ( 3 4 d free � keV (solid lines) and Z 5 �� . |M e Right: 2 χe σ µ is the absolute value squared of the matrix element describing the = 1 [��] � e 2 = | E ) 2 e is the DM-electron interaction form factor and ∆ → Non-relativistic form factors for DM-electron scattering in xenon for the indicated 1 2 | � αm ) �� ( σv q ( �� = �� = Left: � � free . Δ� Δ� . For every shell, the peak also shifts to higher DM e F |M | � � � αm � . This corresponds to different final outgoing electron energies We can now write the cross section for the scattering rate as [ In the right panel of figure -� -� -� -� e ��

��

� �� & � � �� - →Δ E � �� | ) - ( ) ( � where elastic scattering between DM and a free electron. While using the plane wave form factors, where electron cross section defined as the plane wave calculationFermi underestimates factor the in form the factor, calculation justifying of the the inclusion scattering rates of (see the below). an electron in anypossible, shell but to highly give suppressed. considered in this paperthe at relativistic a corrections fixed start value becoming of important. We also see that for (dashed lines) and Relativistic form factors (dot-dashed lines) for ∆ is the binding energyq of the shell {n,l}. We see that the form factor drops sharply for Figure 15 electron shells for entire deposited energy. JHEP01(2021)178 . is E 16 (6.8) (6.9) v km/s. (6.12) (6.15) (6.16) (6.10) (6.11) km/s, an = 544 , , , = 220 esc )) v 0 e ]. The differential v E ∆ 132 , q, ( 131 , min . ) | 2 v , ( E ) η v , cm 2 | + ) with respective binding energies min 40 πζ v q 2 − χ } ( , − ) v − 10 πζ e χ nl 2 v n, l = E q − | − m { e − 2 1 σ e χ )Θ( esc χ E v v + ρ m ( -dependent “heavy” mediator (∆ e 1 ) = χ for -dependent “heavy” mediator 2 Θ( g → − E “heavy” mediator“light” mediatorq q (6.13) (6.14) q 2 eff ) | v nl 16 is the DM velocity in the Earth frame, and ∆ E – 33 – 3 v nl f v 2 0 ,Z | d χ v + e E 2 = {12.4, 14.2, 21.9, 25.0, 26.2, 39.9, 35.7, 35.6, 49.8, | χ v E ) Z v − | q = ( e − min (∆ km/s, and a galactic escape velocity of e v E ) = eff DM ∝ Z F (∆ fermi | min ) ) where v F χ 2 2 n,l ( X = 240 v η qdq ( e e e E χ 2 χe e . ) = 1 v Z g 3 q q q µ σ v ( 8 αm αm αm × χ . We take = e e v g e = = 1 = = E 3 GeV are shown in figure m is defined by, 2 E d GeV/cm ) dR R ∆ q DM DM DM DM 4 . d α F F F F min = 10 v is given by, eff χ ( = 0 Z η m χ min ρ = v ζ For standard DM-electron scattering, we calculate the rates using plane wave atomic The DM form factor depends on the precise DM-electron interaction, but we will . The nl or below the typical momentumrates transfer. for The resulting differential DM-electron scattering form factors and also usingWe the relativistic see form that factors. the We show relativistic both spectra corrections in (dotted figure lines) predict a larger signal rate in the where “heavy” and “light” refer to the mass of the mediator, which is respectively above consider (normalized as the Earth’s velocity in the galacticaverage rest Earth frame. velocity of We take aWe peak set velocity of where and where we sum overE all the occupied initial shells scattering rate will then be given by, with 39.8, 52.9} for the shells {5p, 5s, 4d, 4p, 4s, 3d, 3p, 3s, 2p, 2s, 1s} [ we also include a Fermi factor in the scattering rate given by, JHEP01(2021)178 and keV. σ (6.17) 1) 1 , GeV and 5 does pro- . , e -dependent 3 (0 . q = 10 χ q/αm ) = 7 m ] (light mediators). B = L keV and 45 / DM 5) B . keV bin, as well as less left we show the F + 0 S , 5) �� . 2 L 17 . 0 � � ) , � � /�) (0 � 2 . �� 2 log( =�/α� =� (0 α� =( �/α� =( �� , �� keV to ensure that the dataset is com- 1 χ right. In figure GeV m [��] �� 1 17 are unable to explain the XENON1T signal Δ� 2 -dependent form factor – 34 – × q �� 2 /q � Due to the steep rise at low energy, the spectra 1 A �� cm -�� ∝ ]. We consider two bins: 45 �� =�� =�� − � χ 44 ] (for heavy mediators) and SENSEI [ σ � DM 10 44 F parameter space. We include also a rough estimate of the sig- � �� × �� -� χ keV bin. We see that the best-fit regions for the -�� 6 ��

��

�� m

�

��

[ Δ / ] 1)

� � � - - - , 5 = 2 . e (0 σ versus e , σ for four different DM form factors. Solid lines show results calculated with plane ]. See text for details. and especially 2 1 GeV 130 cm . Electron recoil spectra for standard DM-electron scattering for 90 ∝ 40 & − χ DM We now briefly describe the different DM form factors, focusing first on their ability to F m regions in the = 10 -dependent “heavy” mediator. e σ We avoid considering the S2-onlypletely analysis independent above from theby one requiring used a to fit signalthan the yield 5 signal. of events less in We impose the heavy than mediators a 22 are conservative events bound not in constrained the from the lower-threshold S2-only analysis. and the resulting spectrum is2 shown in figure nal yield of the S2-only analysis [ without being in dramatic conflict withXENON1T lower-threshold (S2-only direct-detection analysis) searches [ from, e.g., q vide a reasonable fit to the XENON1T excess. The best-fit point is given by and then commenting onencoded possible in complementary the probes cutoff related of to the the operators new generating physics“Heavy” the scale DM-electron a interaction. andfor light mediator. region relevant for explaining theform XENON1T factors (solid excess lines). than that predicted with plane wave fit the XENON1T excess without being in conflict with other direct detection experiments Figure 16 σ waves as outgoing electronresults wave from functions [ while the dotted lines include relativistic effects using JHEP01(2021)178 ) �� of F 6.18 ) and Λ �� (6.18) (6.19) e �� y � -dependent q ∝ region is the ��  � -- regions fitting the σ 2 ]. A possible way to [��] � region is the expected 60 � �� blue shaded � and - �� σ 1 �� -�� Signal shape for the best fit point �� =� =��×�� χ gray shaded � interacting with electron through �� σ �� σ � �� φ � � , �

��

to be at its perturbativity bound and

��

Right: �� ] [ / ,

¯ ee � - D e χ y 5 regions are the 5 2 F ¯ eγ ¯ χγ Λ φ i S ∗ D Λ φ y – 35 – e D 3 y dark red y e 10 y lines show the scales at which the operator in eq. ( -2 10 [GeV] . The

) implies new physics below the GeV scale. This is likely XENON1T e 12 ) D 2 y 6.18 e are the XENON1T data, the shaded region shows the current bound from the XENON1T only 10 q/αm /(y F = Dashed cyan Λ is the signal shape after smearing, and the green [GeV] F SCALES CUT-OFF DM χ

F m S2only 1 black dots Allowed parameter space for DM-electron scattering through a

q ∝ F(q) DM-e: ) couplings. Even by assuming XENON1T -dependent cross section, where the scalar DM interacts with the electron 10 q D blue line y Left: ). The 0.1 . is the fermionic DM and the scalar-pseudoscalar interaction is induced by a heavy -dependent form factor is predicted, for example, by the dimension six operator, 1 6.17 q χ -46 -43 -44 -45 -42 A

10 10 10 10 10

e σ ] [ cm 2 spin. The reducedhundreds dimensionality of of GeVs thereby thisand allowing operator electron a could EDMs. UV help completion However,the pushing a consistent bounded the correct with electrons cut-off treatment collider inside ofcross up the constraints section. the xenon to spin-dependent atom We leave interactions is this for necessary interesting to issue correctly for compute future the investigations. DM-e to be excluded by colliderraise bounds the cut-off from scale electron-positron would machines bethe [ to dimension consider five a operator, scalar DM which leads to a scalar which admits a spin dependentcontours interaction of with the the DM. cutoff Inmediator-DM scale the ( same divided plot by we present thetaking the square-root coupling of of the thethe mediator-electron mediator effective to ( operator electrons in or eq. be ( order one, the required cutoff where analysis based on S2is only. generated to obtainin the eq. corresponding ( crossbackground, section. the resulting signal plus background distribution. Figure 17 heavy mediator, with XENON1T excess. The JHEP01(2021)178 2 �� ) e after ) can �� (6.20) (6.21) (6.22) 2 ) could �� Λ � q/αm 6.22 � / ) 6.21 ∝ ) H = ( � �� , �� ( † 3 ��  . � �� -- H -dependent heavy ( DM , is then required q are the XENON1T µ S F ∂ ) Λ bands of our fit to the [��] � φ ) = 12 ∗ is the signal shape after B σ ]. The best fit point is φ the allowed parameter � 2 ( L � �� 44 µ / �� - ∂ 18 B black dots -�� and + �� S σ 1 L blue line �� =�� =��×�� χ � ). The �� σ � σ � �� , � � � 2 log(

)

�� 6.20

��

�� ] [

/ ee � - right. As it is evident by comparing e , (¯ 5 2 µ ∂ ¯ 18 eγ ) cm χ φ 5 regions are the 3 S 2 F . A very low cut-oﬀ scale, ∗ 50 h Λ φ Λ ¯ − χγ ( -dependent form factor, a stronger momentum – 36 – m µ q 3 D 10 Last, we show in ﬁgure ∂ y ∼ 10 e is a fermionic DM. The operator in eq. ( D × y S y dark red 0 e χ . Λ y ∼ = 6 Λ e . The 2

2 2 XENON1T σ ) region is the resulting signal plus background distribution. 10 ] if e shaded region shows the current XENON1T bound from the S2 only , 133 [ [GeV] q/αm h χ region is the expected background, the green GeV m = ( 1 ✶✶ /m e q ∝ F(q) DM-e: 10 blue shaded Allowed parameter space for DM-electron scattering through a = 11 m DM Signal shape for the best ﬁt point in eq. ( is a scalar DM while F χ φ m

Left: S2only

gray shaded XENON1T -dependent form factor could be generated by operators such as Right: 2 1 q -51 -48 -49 -50 -47 A

10 10 10 10 10

-dependent “heavy” mediator.

e σ ] [

cm 2 2 be obtained from thethe dimension Higgs is six integrated “derivative” out.proportional Higgs to This portal will lead toto an get extra a suppression cross of sectionmaking the in this wilson the example coefficient ballpark not of viable the phenomenologically. one The required second by operator our in fit eq. of ( the XENON1T data, and where again and the resulting spectrumthis result is with shown the independence improves previous figure the one XENON1T fit for substantially. a and best fittogether regions with to the the boundgiven XENON1T from by the data XENON1T for S2-only the analysis DM [ form factor analysis. data, the smearing, and the q Figure 18 mediator, with XENON1T data. The JHEP01(2021)178 – 134 , ), while ); in the 0 29 0 , 28 δ > , respectively, δ < δ ]( + 1 134 χ gauge symmetry; if m = 2 U(1) χ symmetry, it is possible to m ), the lifetime of the heavier e ]. m U(1) and 2 30 1 χ . m ]. For sub-GeV DM, the abundance of could be two Majorana fermions that . 2 138 χ , to be the incoming state, which then scatters 6.2.2 1 137 χ and – 37 – is often called “exothermic” DM ( is often called “inelastic” DM [ 1 2 1 , with masses χ χ (which in our notation is always the outgoing state). 2 χ χ 2 χ ’s of eV. and 10 1 χ sufficiently small (typically | δ can be entirely converted into kinetic energy of the electron when | | δ | | δ , we discuss the kinematics and also provide best-fit regions to the is heavier than is heavier than | . For example, 2 1 2 ] and for DM-electron scattering in [ χ χ χ ]. However, the fractional abundance of the heavier state after freeze- 6.2.1 29 , m , 1 χ 28 138 , . Similarly, one can consider two real scalars that originated from a complex m 2 χ 137 | δ and | We focus here on exothermic scattering, since it is able to explain the XENON1T The relic abundance of the two states depends on the precise model. In the minimal We now turn our attention to exothermic DM, which can provide an even better fit to 1 ]. We consider two states, χ be highly suppressed for excess. In section XENON1T excess that are independentbefore of considering the concrete precise relic models abundance in of section the heavier state, the heavier state will typically beheavier small. state However, can even leave asee, dramatic small the fractional signals mass abundance in splitting of direct-detection the scattering experiments, off since, of as it wein in will the a halo target to material. scatter, while The the exothermic inelastic scenario up-scatter allows of all the relic lighter particles to the heavier state will state for decays via thedecays (off-shell) into mediator the into lighter the state lighteruniverse state plus [ plus three two photons, neutrinos, isout or easily for in much the longer early thanstrength universe the and will the age DM depend of and the sensitively mediator on masses [ the precise DM-mediator interaction the scenario where context of direct-detection experiments, the latterscattering was in previously [ discussed for DM-nuclear scenario above and for split them into theto two Majorana fermions, withscalar. the In gauge what boson follows, we couplingoff will ordinary always off-diagonally take matter and converts to The scenario where DM could consist of two139 or more approximately degeneratewith particles, see e.g. [ originated from a Diracthere are fermion mass that terms for is the charged Dirac fermion under that a break new the the XENON1T excess, and also has several interesting features that deserve further study. 6.2 Exothermic dark matter and electron recoils expected cut-off for thethan range 1 GeV of and cross hence in sectiondepth tension and analysis with colliders DM should constraints. masses be Ason of performed mentioned the interest in earlier, electronic a order is side. more to always in correctly lower account for the spin dependence be obtained by integrating out a heavy axion coupled to fermionic DM and electrons. The JHEP01(2021)178 . q . In , and (6.25) (6.26) (6.27) (6.28) (6.23) (6.24) ) = 0 ) (and in keV δ ( O = 0 ), the energy- , ], the scattering δ 2 )) 6.2 ) , e , e , E /q 1 E ! e to the target electron χ ∆ ) can be well above the ∆ δ q m q, e αm ( 2 max + + , since requiring a sizable E v e 2 χ = ( min ∆ v E v = 0 m 1 , we will often simply denote ( , we can simplify this as, 1 2 2 χ 2 η δ (∆ . DM χ 2 χ m . |

F few eV applicable for δ. ) m − 1 2 m and a ﬁxed q ! ,N 1 × − ( ∼ ) 2 χ ∼ = q δ χ nl s 1 1 2 eﬀ m ,N µ E χ 2 max χ χ can be obtained even for very small + 2 − 1 2 Z 2 − v 1 . In contrast to eq. ( χ m e χ e q m m χ 1 χ 2 E E + ρ E m ∼ m χ

on the DM halo velocity, we get upper and m + 2 ∆ ∆ 1 δ 1 2 , transfers momentum (∆ | E → − N 1 − typ ) δ + v 2 max | nl q χ q q − m e v v nl f – 38 – 3 + | 2 χ · 1 E for a given − E 2 | m q ) ∆ q s + esc 10 ) −

q v δ 2 ( e = | = ∼ = e E q + 1 + DM 2 e E e

− χ F (∆ max E E min | ∆ v v v m ∆ 1 n,l 2 max X χ , the electron recoil spectrum will be peaked at to be sizable. However, as we will discuss, this is not true qdq v ) χ m 2 χe e q | Z µ σ m 8 keV × + = = sign(∆ − e ; this was not a problem for the case = ( . Also, for the calculation of exothermic DM-electron scattering, we q E e χ min max ∆ E q m q ∼ O dR ∆ δ also forces d E is again the energy transferred to the electron. Assuming a small mass-splitting e ∆ E gions ∆ The differential scattering rate is given by For DM scattering off electrons throughrate a diverges light at mediator low [ value for anymore for exothermic scattering:To remove a the sizable resulting divergence in this case, we will consider that the light mediator is As there is an upperlower bounds bound on of the allowed values of where the minimum velocity to scatter is given by can explain the XENON1Tthe excess. DM Below, mass since as consider non-relativistic atomic form factors. In contrast tocontrast the also “standard” with DM-electron exothermic“typical” scattering nuclear energy discussed scattering, transfers see above of below), particular, ( for where compared to the mass scale of the DM i.e. We assume that the incoming DM particle, and converts toconservation the equation lighter now reads (outgoing) state, 6.2.1 Exothermic dark matter-electron scattering: kinematics and best-fit re- JHEP01(2021)178 , ) ) | 2 ∼ 2 δ χ | max max keV /q δ n q q 1 4 ( , there − 1 ∝ , where ]. χ χ and n = m , which is smaller or | 19 DM 142 δ δ e | F min E keV (left) and q ), we see that, plane (middle). misses the peak ∆ 5 . , with 1 q 2 2 χ 6.27 . For χ − | ]) from [ n 1 m δ χ - + , both = e n . For 1 ) provides a very good | | and for three different | 141 χ σ δ ∼ | δ δ δ | n | | 2 e . Moreover, the spectrum (MeV) as follows. If , Thomas-Fermi screening keV), we get a sharp peak in [ | < E , the peak (for a fixed cross the peak of the form = = keV (left) and q δ cm 0 − q | 5 ∆ e ( 1 e a q . χ E 40 2 . Thus, the recoil spectrum in E ∝ . From eq. ( ∼ O f e ), we see that the value of − − ∆ GeV for ∆ . In order to have the spectrum χ ∼ O typ E q DM below δ = m , is ∆ typ 6.28 F 1 e = 1 q = 10 δ χ E χ when e ∼ is now only a few keV, and we see from ∆ σ m and 1 χ 1 (due to the enhancement of the integrand f = 0 plane (left) and the max | (keV) q 1 ∝ δ (GeV) and and for three different form factors. Here the MeV for χ min and therefore the integral over – 39 – 2 q m ∼ | ∼ O - DM best-fit regions that explain the XENON1T excess δ ∼ O = 1 F q cm typ as a function of E σ q χ , the allowed values of ). We see that the spectrum is sharply peaked at χ (keV) is higher. Note that for such small 2 | 2 ∆ 40 keV, the 2s- and 2p-shells can also be excited, leading δ m (GeV). From eq. ( m | χ − min 9 ( . O q . However, and 4 ∼ O | 1 (MeV), showing the resulting spectra in figure . The reason is that for ) in the δ | min e σ | χ & q δ = 10 1 . | E | | below e = = 1 δ δ ∆ ∼ O | σ e e 1 χ E E χ ∼ | DM for f (keV), e keV), we see that m ∆ ∆ F ). However, for ], using a Thomas-Fermi radius (called 0 E − increases and the available phase space decreases again, thus giving a (MeV), which is much higher than ( at typ ∆ q ∼ O → 141 e , min q ∼ O , the solid lines show spectra for ∼ O E q , we show the = 0 , δ ∆ | 140 19 20 δ . However, for | max min ) at | v q δ χ | and that the keV (right) for m 6.25 4 15 depends on the behavior of = 0 keV), − In figure We can understand the shape of the spectra for We next consider In figure Consider first DM masses of e δ and is reminiscent of a DM absorption signal, which provides an adequate fit to the − = E ( | δ to additional peaks in theDM spectrum. mass The and precise spectrum splitting.entire thus (sub-)MeV-scale depends In DM sensitively parameter our on space. the parameter scans below, wefor do a not heavy mediator attempt ( to cover the increase, and at some factor. Hence, we seeis a not peak as in sharply theof peaked spectrum the as for momentum it transfer ispeak are for near heavier always 2.5 DM, keV (which sincevalues gives for of a heavier good DM fit the allowed to values the XENON1T data), one then needs larger is barely any kineticalready around energy keV. in So thethe the spectrum DM spectrum sharply to is cuts peaked giveof off roughly recoil eq. above at ( energies largeroccurs than at energies less than wider than a DM absorptionfit signal, to which the (for the XENON1T larger data. value of O figure XENON1T excess. the dashed lines show(right), spectra both for for incomingform DM factors. mass The fraction spectrum has a wide peak, wider than for heavier exothermic DM and δ fractional abundance of thebeing incoming the DM particle, number| density of for of the form-factor, leadingHowever, for to strongly suppressedlarger than scattering rates (assuppression discussed in the above). rate.in Hence, the for spectrum at will then naturally regulatediscussed in the [ divergence. We implement Thomas-Fermiis screening near as ∆ a light dark photon, which couples to electric charge. At low JHEP01(2021)178 ) � � 2 � �� /�) χ � n 1 MeV =� χ �� + � =�/α� ), and �� =� �� =� dashed n 1 = �� α� =( � regions � � χ χ χ �� n = 1 � � �� 1 = ��  � -- χ orange 1 ). The ( χ m f 2 ) [��] � plot, we show an , which is lighter 6.29 � � 2 /q blue-shaded �� χ e [��] -�� ) and MeV ( right and Δ� αm � �� =��×�� =�� σ � χ solid χ �� |δ|=�� σ �� = ( = 30 � � � keV in eq. (

�

��

�� [ /

] 9

�� - . DM m ). In the 4 -�� F 6 =�� ), � gray-shaded �� δ=- 10 =1 GeV ( | ≤ σ 1 1 � δ χ | � ∝ χ ) middle blue q m � � � � ( � �� -� -� -� F plane for 5 ��

��

� δ

��

] [ Δ

/ 10

� � � - - - =1 ( keV ( while the 5 . [keV] DM χ F m 4 = 2 versus 10 | xtemcDM Exothermic e – 40 – δ | σ ). We show 1 � �� black dots χ XENON1T . keV |δ|=2.5 � f /�) 3 � right 10 =� -44 -45 -46 -47 α� =( =�/α� �� =� �� =� ��

�

�� 10 10 10 10

1 χ � e ] [ σ cm � f � plane for � 2 keV ( χ ) in the �� 4 m 4.0 − 1 = 1 ∝ = best-fit regions that explain the XENON1T excess for exothermic DM ) δ q �� 3.5 ( σ DM F 2 [��] versus F � ). 3.0 e Δ� �� XENON1T σ and ) and ✶✶ 1 χ 2.5 σ green f left 1 [keV] |δ| ( � �� ) 2.0 xtemcDM Exothermic �� e -�� keV ( 1.5 . The . Differential recoil spectra for “exothermic” DM, in which a heavy incoming DM state, 0MeV =30 lines show the signal spectrum before and after detector smearing effects, respectively. �� =�� 5 χ � � �� δ=- . σ m q/αm 2 � χ ). We consider three DM form factors, � 1.0 − -47 -46 -45 = ( � � � � �� = solid ��

-� -� -� ), and in the

10 10 10

�� 1 ��

χ e

] [ σ

�� f

, scatters oﬀ an electron and converts to a lighter (outgoing) DM state, ��

2 �

� � ��

Δ / ]

[ DM

1 � � � - - - left dashed example of the predicted spectrumand for the best-ﬁt value with The measured XENON1T data is shownare as the expected binned background and signal respectively. Figure 20 and a heavy mediator( ( χ by ( F Figure 19 JHEP01(2021)178 �� . . . 7 8 MeV blue- � . 7 . � �� / . (6.29) (6.30) (6.31) � �� 55 16 ∝ 15 . 15 �� ' ' �� ' ) = 0 ) ��  � -- ) ). The B B χ B L L L m / / [��] � / B � B B middle �� + below which rela- + + S -�� S S q L L L =��×�� keV ( � � �� =�� σ � χ plane for 5 χ �� |δ|=�� σ �� . � 2 log( δ � � 2 log(

�� 2 log(

��

-dependent heavy mediator

�� [ /

] lines show the signal spectrum � - = 2 q , , | , 2 2 δ 6 2 | versus 10 cm 2 solid cm q cm e / 1 is not forced to be large to obtain 43 σ 47 46 ∝ 1 − − ) and q χ − q 5 ( f 10 F 10 10 10 plane for × × × χ [keV] 3 2 . χ 3 . dashed keV) is given by m regions are the expected binned background and . 1 m 1 4 2 9 , we show the corresponding plots for a light . 10 ) in the xtemcDM Exothermic ' ' ) – 41 – 4 ' 2 e e ) e 21 σ σ versus σ ). The 1 1 /q . keV |δ|=2.5 | ≤ 1 χ 3 e e χ δ χ | σ 10 1 6.30 -40 -41 -42 -43 blue-shaded αm χ

, f

f 10 10 10 10

, f

1 χ e ] [ σ cm f , f 2 plot, we show an example of the predicted spectrum for the best- = ( and keV keV keV 2 9 best-ﬁt regions that explain the XENON1T excess for exothermic 6.0 5 . q 6 DM . /

4 1 = shell 2 n 2 right F σ 2 ∝ 2 5.5 ) q ( ). Here the best-fit point is given by | ' | ' F keV from eq. ( | ' 2 δ δ ✶✶ | 5.0 δ | and ), and in the GeV provides an adequate fit to the XENON1T excess, one can obtain , we show the corresponding plots for a | 9 /q and the form factors typically peak at values of . gray-shaded 1 4 1 σ , e left 4.5 , 1 22 , ∼ ∝ XENON1T E [keV] |δ| )( | ≤ χ ∆ δ 4.0 2 xtemcDM Exothermic | χ MeV m MeV DM n MeV 1 F χ . The + 3.5 55 while the .5MeV =0.55 ); here the best-fit point is given by n 1 . χ 30 q χ 780 m 0 n region has not been included in our scan because the second xenon shells would be excited 3.0 ' ∝ ' a large tivistic corrections become important. Weour therefore calculations. neglect relativistic corrections in While an even better fit for heavy DM by imagining that DM consists of three or more states. The inclusion of relativisticnot corrections essential when for calculating the exothermic atomic scattering, form since factors is ' -44 -42 -43 = χ We now make a few comments: χ

10 10 10 • •

χ e 1 ] [ σ cm

f m DM 2 m χ m F f We see that exothermic DM can explain well the observed XENON1T ER spectrum. Finally, in figure ( In the right plot, we showdata how the and signal at background the best-fit model.mediator point ( compares with In the figure XENON1T signal respectively. We see that the best-fit point (with ( shaded (see text for details).fit In the value with before and after detector smearingblack effects, dots respectively. The measured XENON1T data is shown as Figure 21 DM and a light mediator ( JHEP01(2021)178 . , , 4 . ¯ 1 ∼ f χ ,N f 1 N R GeV χ m . µ m (6.32) 1 E | ↔ 6.2.2 δ − | and the ∆ 2 ∼ 2 eV. This 1 χ χ 1 6.2.2 1 χ 21 χ m i ∼ χ R m ∼ ≡ E h R 21 E δ ∆ , the relative fraction is | δ 3 | − all negative, the electron 10 32 δ ∼ , v 31 δ , , ∗ is the reduced mass of χ 21 /T | δ ,N δ 1 , driving the relative abundance to the , with mass splitting χ −| 1 3 e µ χ χ 1 ' χ , with 2 1 χ χ ↔ , and scattering off a xenon atom, with – 42 – m /n 2 2 in two explicit simple models of exothermic DM. 2 1 χ χ − χ 2 ∗ χ , 3 n χ χ 1 χ T χ ≡ m ∗ f ≡ ]. For eV, while the typical spread in energy around the mean recoil 32 δ 29 8 , is much lower than the mass splitting 28 ∗ [ i ∼ χ 2 T R , and v 1 E χ h ,N 1 m ]. If χ is the mass of the nucleus and µ − | keV, 3 δ N | 138 1 χ , 8 m m ∼ is the temperature of the dark sector at which the DM-DM scattering decou- q δ 137 ≡ ∗ ,N , N χ crete models 1 χ m T 31 µ in the early universe willstrength depend and sensitively the on the DMwill precise and typically DM-mediator mediator also interaction be masses.experiments, and other Moreover, colliders. constraints in We from investigate a searches two concrete concrete at models model beam in there dumps, section fixed-target It is possibleexothermically to off obtain nuclei; electron thiscareful could recoils study lead from that to we the additional leave Migdal constraints, to which future effectAs requires work. when a mentioned DM above, scatters the fractional abundance of the heavier state after freeze-out and energy for the same parametersis and below a the DM XENON1T velocity and of manythe other threshold experimental achieved thresholds, by although not CRESST-III; below we will discuss this further in section could scatter exothermically off nuclei. WeDM can contrast scattering the kinematics off for exothermic electronsnuclei. For with exothermic scattering the off kinematics nuclei,where the for mean recoil exothermic energy DM is nucleus; scattering the off spread in energy around the mean recoil energy is given by δ recoil spectrum wouldvarious show peaks up will depend to sensitively threeand on hence the peaks. depend relic abundances sensitively of Of on the the course, three model DMIf the parameters. states, the actual DM couples size also of to nuclei the (for example, if the mediator is a dark photon), DM For example, for three states 134 For the cases of interest here, the DM-DM scattering will always decouple after the scattering of DM • • • 4 ples [ of the excited statestemperature will of chemical be decoupling exponentially suppressed. In what follows,with electrons, we and compute hence the the DM-DM scattering will set the relative abundance in the dark sector. After the dark stateslibrium freeze and out scatter from with theequilibrium each SM value other, bath, they continue to be in chemical equi- where 6.2.2 Exothermic dark matter-electron scattering:In relic the abundance case of for exothermicof con- DM, the it heavier is state ofthe crucial after standard importance freeze-out freeze-out to is in calculate dominated the the by relic early the abundance Universe. annihilations into We SM consider fermions the case where JHEP01(2021)178 . ) = �� † 1 keV φ while 5 plane , χ �� . (6.33) (6.34) (6.35) 2 ∝ �� δ χ �� = 2 �� | = ( ��  � δ -- | ¯ ψ , versus black dots e [��] � 1 σ � 1 χ �� χ µ f -�� , ¯ σ 1 † 2 plane for χ χ � �� =��×�� µ χ � �� =�� σ � χ χ ∂ �� − � �� |δ|=�� σ �� m 2 � � 2 χ

��

) in the

��

χ �� [ / ] is the dark photon coupling, )

� - e µ − ¯ σ D 2 † 1 6 , versus χ χ 10 πα µ µν e 0 q/αm 4 q ∂ σ i ∝ 1 1 F √ ) − χ χ q = ( ( f µν 5 F ≡ = F 10 D ) = 2 DM g ∗ F ψ lines show the signal spectrum before and after [keV] φ + µ χ µ m 4 ¯ ψγ 10 φ∂ DM µ xtemcDM Exothermic solid ( – 43 – J i i µ 0 − − ), and in the . keV |δ|=2.5 and XENON1T A 3 ]. The ﬁnal allowed parameter space compatible with = = D left 10 g -47 -48 -49 145

)(

DM DM 10 10 10

µ µ

χ e ] [ σ cm f 2 2 J J dashed χ L ⊃ n 1 regions are the expected binned background and signal respectively. χ + 5 n 1 best-fit regions that explain the XENON1T excess for exothermic χ q n ∝ σ ). The ) 2 q ( = 4 ]. Our treatment of the cosmology here agrees with the one first F 1 6.31 For the cosmology of the scalar case, we follow a similar treatment, χ and plot, we show an example of the predicted spectrum for the best-fit value with f 144 5 blue-shaded , σ 3 ]. 1 scalar DM: XENON1T [keV] |δ| ✶✶ right 143 and and the Dirac fermionic current in terms of its Weyl components 143 MeV ( xtemcDM Exothermic 2 fermionic DM: 2 is the field strength of the dark U(1), √ . The . GeV =0.8 the dark matter current given by χ / m ) 0 keV from eq. ( µν = 780 2 1 ). In the 9 F . DM µ χ iχ -48 -46 -47 4 J The interaction Lagrangian between the SM and the dark sector reads We study models where the coupling between the SM and the DM sector arises via

We thank Hongwan Liu for correspondence and a thorough comparison of our results. 10 10 10 m

gray-shaded

1 χ e ] [ σ cm

f 5

1 2 | ≤ χ middle δ where we write the( complex scalar current in terms of the real scalar components where and atomic form factors. the pseudo-Dirac DM andconsidered its in implications [ forpresented the in XENON1T [ excessobtaining have results been that recently agree withthe [ XENON1T excess willour be improved statistical somewhat analysis different of than the the XENON1T previous data analyses and because a of different treatment of the the kinetic mixing ofof the a dark complex photon scalarmetrical DM with dependence and the on a the SM physical pseudo-Dirac photon. quantities fermion up DM, We to discuss numerical which factors. have both very The the cosmology similar cases of para- for ( | detector smearing effects, respectively. Thethe measured XENON1T data is shown as Figure 22 DM and a momentum-dependent heavy mediator ( JHEP01(2021)178 , , e 2 / 7 m − χ (6.40) (6.41) (6.42) (6.36) (6.37) (6.38) (6.39) & . The m 1 is small χ ∼ D T/π m ), needs to 2 y 2 χ & 1 6.5 & χ . . 0 A Γ | ) i δ ∗ χ m | ] in the fermionic D T , ( H , h 2 145 H / , D 1 y h.c ' . . As long as h.c ) for the fermion (scalar) ) , , ' + ∗ i 2 χ m + δ 2 2 v ] and [ T /π, δ 1 2 φ ( = 2 χ χ 2 to distinguish the scalar and 2 ψ D 1 T χ , defined in eq. ( 143 D 1 χ 1 D 2 e H 1 in the dark sector, generating a χ α χ MeV 4) ¯ σ χ ↔ ( MeV 2 H D Γ / 2 m y χ m 1 D 5 2 . , y 100 , we need to consider the rate of the arises from the threshold velocity for + χ max 100 1 U(1) σ 2 χ + δ φ h / = 0 ∗ 3 χ 1 10 8 /n φ = (1 χ ¯ − − 2 D fermion DM: m 2 ψψ best-fit regions for the case of a heavy dark n χ α 2 D χ κ 0 10 10 m m σ n 4 A 2 /T | ∼ ∼ = = πα δ m – 44 – , 2 and y y m m −| breaks the 2 √ on the DM mass needs to be extracted numeri- and e χ i L L interactions fixes the total number density of dark κ D D σ y is then defined as m ¯ = f 1 16 H H m ∗ 1 f χ h χ T = 3 1 2 D 2 χ ↔ y 0 i ' χ v A ↔ 2 . This can be chosen to match the DM abundance today if 1 . 2 ' χ 1 2 χ m χ 1 / ]). χ 1 scalar DM: δ 2 7 χ χ and χ χ n Γ m 1 ↔ fermionic DM: 2 ' 147 χ scalar DM: , χ ∼ 2 2 is fixed as a function of the DM mass. In the limit χ χ ∗ 17 1 fermionic DM: 4 n σ f ) χ h 0 1 + A χ 1 , we show in red the χ scalar DM: /m ↔ n 23 χ 2 m χ ( ] (see also [ 2 2 χ 146 D In figure The compute the relative abundance of The freeze out of the α = photon mediator with mass case. The requiredbe electronic larger interaction for cross lowerdecreases section DM for masses lower DM becausewhich mass. the then fraction Indeed, implies of the rate primordial of excited de-excitation states scales rapidly as fermionic cases. Thisand formula scalar agrees case, with respectively. thethe The results scattering dependence of process, on [ andchemical decoupling in temperature most of the parameter space of interest where the thermally averaged scattering cross section reads and we introduce the numerical coefficient cally [ process The precise relation dependence of neglected in the rest of our discussion. sector states y we find roughly The splitting can beenough, easily the suppressed mass compared of to the the mass dark Higgs can be made arbitrarily heavy and, hence, will be where the VEV ofmass the splitting between dark Higgs The mass terms in the two models can be written as JHEP01(2021)178 . ] � ]. D α dark 152 149 ]. Of , 61

- �� [

�� 161

�� · =

� - the BABAR -� �� and coupling γA �� χ → m − e [��] = 3 χ +

e �

* ��

· =

� - A �� dark orange

�� -�

m �� the bound from E137 [ �� ⇒ ⇒ ]. Alternatively, one could �� χ ], in ]. Conversely, in the scalar 160 cyan lines indicate the fractional relic �� = 148 � � = � �� α

�� 144 � ]. We also show the limit from self , ], in �� = |δ| ) that are split in mass by an amount -� �� 143 ], while the dotted dark-red line shows -�� -�� -�� -�� -�� -�� -�� 156 151 , ] (blue solid line), and the projections

right

��

� ��

] [

σ 153

� ). The XENON1T signal is produced from - 155 150 148 best-ﬁt regions for exothermic DM in which the σ right shaded region is the bound from the heavy state ( 2 5 – 45 – . � and Dotted black vertical ]. We show the bound on a DM beam produced in σ = 0 ] and from CMB distortion due to energy injection ( 1 ]. D 154 α , 157 ) or two fermions ( light green

�� 159 ,

��-�� *

�� ·

= left � - and -� 158 line) [

χ �� ⇒ ⇒ �� the bound from LSND [ ]. The m

�� 154 [��] = 3 , the bound from the NA64 experiments [

�� χ 0

purple

� 61

* �� · = A

line shows the reach of LDMX [ �� - regions indicate the

�� -� m blue ] and Belle-II [ , we show the accelerator constraints on the dark photon decaying that are at the boundary of perturbativity [ ], in �� red

�� D χ 153 23 , of the excited state. α 149 , ∗ ) and dotted red vertical � =

�� f � � � = 61 . The α �� left � ( �� = |δ| keV, and interact through a dark photon mediator with mass -� 5 ) for the fermionic case [ . �� dotted light-red In ﬁgure In the fermionic model, for values of the cross section that explain the XENON1T -�� -�� -�� -�� -�� -�� -�� = 2

��

�

�� [ σ ] = 2

D � - “invisibly” to DM: the BABAR constraint(dark from a orange monophoton solid search line),dark from photon bremsstrahlung an (to electron-beam-dumpfrom DM) missing-energy LDMX at search [ NA64 induced [ by explore a region ofdark parameter photon space mass, where andcourse, resonant the this effects DM is enhance mass aannihilation the is specific channels annihilation within almost feature the cross degenerate of dark sectionat with the sector the [ could simple the price open models of the a presented parameter less here space minimal substantially and model. adding further excess it isThis easy result to is get incase, the fitting agreement correct the with DM XENON1T previousto relic excess values analysis abundance for pushes [ the for parameter perturbative space values with of the relic abundance scattering exothermically off nucleiinteraction ( in CRESST-III [ yellow abundance, α the heavy state (down)scatteringbased to searches: the in lighterconstraint [ state. WeThe show also the boundsthe on reach of accelerator- Belle-II [ Figure 23 DM states consist ofδ two scalars ( JHEP01(2021)178 ] ]. × ]. 157 156 pb (6.43) , 167 . 155 σv ∗ f ], which is the ]. 164 162 , 146 [ 2 . ] or neutrino experiments such as ]. However, for standard DM with MeV 43 152 10 , χ / 0 GeV 42 m A 60 m 2 ]. Such an accelerated component may then 2 CMB power spectrum. Planck observations ], we calculate only the bound from [ ]. Since the exposure is on the order of gram- / 1 ` χ χe as 166 – 46 – χ m , µ 156 ]. In fact, residual co-annihilations into SM states MeV threshold), about five orders of magnitude 156 e , , ¯ σ /m ∗ 43 , 100 f 159 155 pb 155 , 42 ∼ . , MeV 158 e 33 ¯ σ – 06 . 31 0 . ]. This was used to derive world-leading limits on DM-electron D 33 ), the predicted accelerated DM spectrum does not vary by more – α ] and 30.1 eV in [ 31 = 1 g; in this case, the relevant scattering cross section is the elastic scatter- 163 / 2 ]. We display the self interaction constraints (dotted red vertical line) [ DM ]. , which gives a bound on F cm ] and of order kg-days in [ 151 , 165 10 163 , the resulting flux ends at around 2 keV, thereby naively disfavoring a simple fit GeV . 150 χ 60 = 1 / Naively, DM acceleration from CRs cannot address the XENON1T anomaly either, Last, we include direct detection bounds on DM inelastic scattering off nuclei. The In the fermionic case, important constraints can be derived from CMB distortion due χ /m DM SI m form factor ( than an order ofuses magnitude 176 between kt-years keV oflarger and data than 100 the (with MeV. 0.65 a tonne-year However, exposureat the available XENON1T in would Super-K XENON1T. be analysis Consequently, naively any signal excluded by Super-K. F to the XENON1T data. Acosmic second rays energetic DM (CRs) flux [ iscouplings generated for through interactions DM with in the eV to few keVfor mass the range following using reason. the Super-K For experiment the [ previously studied DM-electron interactions with trivial that impinges on the Earthbe [ detected with experiments such aswhich XENON1T, otherwise allowing for cannot sensitivity be totwo very probed distinct light DM, mechanisms without have sub-keVinterior been threshold to produce suggested. experiments. a significantly harder In Specifically, spectrum the [ first, DM interacts with the solar elsewhere [ 7 Accelerated dark matter A fraction of DM could be accelerated to high velocities, producing an energetic DM flux old of 19.7 eV in [ hours in [ To derive this bound,method we reproduce used Yellin’s by optimumthe the interval recasting method CRESST of [ collaboration low-threshold nuclear for recoil their bounds for own different bounds. DM models More will be details given about For the scalar case instead,leptonic the final CMB states bounds is are weakened, p-wave because suppressed. the co-annihilation to most relevant searches here are from CRESST-III, which obtained a very low energy thresh- to energy injection (dark yellow) [ can reionize hydrogen andlimit distort the the cross high- section for annihilation to electromagnetic final state to be LSND [ σ ing of the light statelow with masses itself, and which reads is loop-suppressed. This bound is relevant only at high-intensity beam dump experiments such as E137 [ JHEP01(2021)178 , the (7.2) (7.3) (7.1) χ m is the DM density , we show the pre- kpc, a local density χ 24 ρ , corresponds to a scalar- . i = 20 χ e 2 e s ) E and for two different form Φ r 2 φ d d − E φ m χ ) χ , e m ρ . In figure + ]. e m E e ( E 2 2 energies. However, measurements ) ¯ σ | 35 χ q e ) ( E q 150 eV) S2-only XENON1T analysis, m / max e ( χ ] and therefore an extrapolation must 2 E ∼ q m h (keV) + 2 DM 170 ∝ Θ F 2 O | + 2 ) ) 2 : e | χ 2 2 e χe ) E E ¯ q σ ( m ( ( 1 cm χ dl + – 47 – max χ 40 e m DM E ] with a scale radius of 2 − F m | l.o.s. χ e ( Z E Φ 169 is the CR-electron flux. d d = = 10 π Ω e e 4 d e E E with ¯ σ d max χ Z /d e E 24 Z = Φ d = χ e E χ Φ χ d d E Φ ] for a summary). We expect this feature to be quite generic in all MeV and d d , and 3 168 , and the DM-electron cross-section, = 1 φ left) and evade the constraints from the S2-only analysis of XENON1T cm are the DM’s and CR-electrons’ kinetic energy, χ / m e 4 m E 4 GeV and is the DM-electron (momentum-dependent) cross-section, . χ χe E = 0 The left panels of figure pseudoscalar or vector-pseudovector interaction,tion where is the on the spin-dependent DM interac- side so that the mediator has scalar or vector coupling to the SM. σ The model discussed here has three independent parameters: the DM mass, In order to derive limits from the low threshold ( The DM flux obtained from interactions with CRs is given by, We consider DM that interacts solely with electrons via a light mediator. In order to The above argument does not hold for a DM interacting with electrons via a light me-

• ]. In this mass range, and for the range of electron couplings we consider, the mediator ρ 44 DM flux in xenonfactors, fixing (bottom) for three different values of of using the S2-analysisthus to simply exclude assume the that CR-accelerated the DM CR solution. flux drops In tomediator what zero mass, follows, below we MeV energies. dicted accelerated DM flux (top) and expected electron spectra induced by the accelerated of one crucially needs to know theonly provide CR the flux spectrum down to down tobe MeV used. energies [ Strictly speaking, this implies that systematic uncertainties hinder the possibility and Here profile, taken to be an NFW profile [ where mediator (see e.g. [ the models of DMwith accelerated significantly lighter by mediator cosmic masses rays. is upcoming A [ complementary study to this scenario fit the XENON1T anomaly, thethe benefit mediator of mass having must a beis low-threshold experiment i) lost in lighter comparison ii) than tobins a heavier the (see few Super-K than experiment MeV figure roughly or else [ 1 keV in ordercoupling to to the not SM overshoot model ends the up XENON1T being excluded energy by complementary searches for the light diator. Indeed, in such afalling case, spectrum both towards the higher produced energies,exposure flux thereby of and easily scattering XENON1T compensating rate with predict forlower its a thresholds the significantly steeply may relative lower then low threshold. be However, more experiments constraining. with We now study this possibility in detail. JHEP01(2021)178 ) � ∝ ) � 2 | left black ) � ( =1 eV q �� 2 ( φ ) Bottom:

2 φ m � �� - �� (⨯ �� =� � �� =�� =�� DM ϕ ϕ ϕ � . The m � � F � 2 | �� + 2 cm q � ( 30 / ) and �� − � 2 ). In these plots the q [��] [��] left � ( � ∝ �� = 10 2 Δ� � �� 2 green � ) e | ) � ϕ . As discussed in the text, ) ) 2 φ ¯ σ � ϕ 2 q � � ( m � +� � +� � cm �� + � � DM

2 30 �� F �� /( ∝ q � � �� =�� | �� =� � �� =�� /( ∝ − ( ϕ ϕ ϕ � -�� -�� / � � � (�) 10 2 =100 GeV ( (�) � �� =� �� =� �� =�� q =�� φ χ χ � � �� |� σ � σ � |� m ∝ -� ) are shown. The DM mass is ﬁxed as 2 � | �� -� -� -� -� -� -� -� -� -� -� -� -� -� -� -� -� -� ) -�� -�� -�� -�� -�� -�� -�� q

��

�� (

��

�

��

] [

��

] [ Δ

/ ), and

green

� � � - - - � � � - - - DM F | GeV ( � orange – 48 – ) � � �� = 100

). Three diﬀerent values of the mediator mass,

� �� - with an intermediate mediator mass can viably address the φ �� (⨯ �� =� � �� =�� =�� ϕ ϕ ϕ � 2 m � � � =100 keV ( ) �� φ right 2 φ ( m m 2 � ) ), + 2 φ �� ), and � 2 m q [��] blue [��] ( � � / + �� 2 � �� 2 � Δ� q ) orange q ϕ � � ) ( ϕ ∝ / � ). The three different solid colored lines show the flux for varying values of the � +� � =1 eV ( � 1 Accelerated dark matter flux due to interactions with cosmic ray electrons. The 2 +� φ | �� /( ) � �

� ∝ m � /(

q � �� right �� ( � ∝ �� =�� ( �� =� � �� =�� 2 ϕ ϕ ϕ � ∝ | Top: -�� 2 ) -�� (�) DM ) . q =100 keV ( �� =� (�) =�� φ 2 �� ( F χ �� =� � =�� φ | χ � |� σ � �� σ � m MeV, and the DM-electron cross section is taken to be lines indicate the energy thresholds for the Super-K and XENON1T experiments. m |� -� DM � � � + � F � � � � �� -� -� -� -� -� -� -� -� -� � ), | -�� -� -� -� -� 2 �� = 1 ��

��

( �

��

[ ] ��

�� χ

[ Δ / ]

� � � - - - � � � / - - - blue ( m only the XENON1T data. 1 mediator mass, DM mass is set to 1dashed MeV and the DM-electron cross-section isElectron taken to recoil be spectra fromand cosmic ray accelerated DM flux for Figure 24 flux is shown for two different DM form factors: JHEP01(2021)178 �� ) � blue � �� + (7.4) � Φ �� , /( � magenta 1 ∝ . �� . The ��  � φ -- : The signal region is the m . The [��] � ) = 12 eff Right B � N ], and the region at ]. �� L / -�� plot we show the best B 172 173 , , + � �� =�� =�� =��×�� blue shaded χ ϕ � S corresponds to a scalar- � �� σ �� σ � �� 61 171 L � 2 � �

)

��

are the XENON1T data, the

��

�� ] [ / middle 2 φ

� - m 2 log( + �� 2 line is the signal shape after detector π 4 , q 2 = region the mediator thermalizes before ( D y / black dots dec e . Exclusion regions due to the XENON1T 1 T cm �� 2 ) region, while the limits from the Super-K ∝ 32 2 φ before − e ) for cosmic-ray accelerated DM that interacts 2 orange ) m | enhances the suppression from the atomic 2 blue solid ) - [��] �� + g 10 ( q 2 ϕ red 2 ( � q as a function of the mediator mass, × /q ( e / 1 6 – 49 – DM Thermalization ]. . y 2 - , we see that when choosing appropriate media- HB . Various complementary constraints strongly bound �� q ϕ F ∝ | 44 colliders ∝ = 6 2 - 24 | e 2 + ) measurement of the electron [ �� e ) XENON1T | e 2 ) q σ ], while in the �� ( q , and at higher energies compared to the mediator mass, -� -� -� -� -� -� -� -� with − ( light green-shaded 4 φ

�� g � � , 108 DM best-ﬁt regions ( DM 24 /m F � F | line is the signal before smearing. The 2 σ | �� = 4 2 q χ ∝ and � 2 | /m �� darker shaded green ) σ φ Super-K q 1 ( � blue dotted DM [��] ��

, m (right) we show its expected signal. The dashed and solid blue lines show χ F : The

| � 25 �� Left region is the expected background, the /4 MeV χ .

4 . is excluded by collider constraints from monophoton searches [

=m ϕ ] are shown in e

XENON1T-S2only XENON1T m y -� tophilic scalar or aspectrum does dark not photon flatten mixingthe out with XENON1T at S2-only low the energies, analysis SM and [ photon. thus we find Here it the to be predicted excluded by mass for where the suppression form factor. The right panel of figure scalar or vector-vector interaction, which can be obtained by the exchange of a lep- As expected, for lighter mediatorthe masses, bottom the DM left flux panel peakstor at masses, of lower we figure energies. can From get a decreasing spectrum at lower energies below the mediator 33 = 0 �� We perform a wide scan of the accelerated DM parameter space for different DM -�� -�� -�� -�� -�� χ •

��

� �� ] [ σ m � masses and mediator masses. The best fit point we find is and in figure the unsmeared and detector-smeared spectra. The gray region is the expected binned resulting signal plus background distribution. electron decoupling, and may thereforeshaded suffer regions from is (model-dependent) excluded limits byhigher on the ( spectral shape for thegray best shaded fit pointsmearing, for and this the model. The with electrons via a form factor S2-only analysis aredata shown [ by the the light-mediator coupling tofit electrons. region for To the illustrateregion mediator-electron is this, coupling excluded in by the HB cooling [ Figure 25 JHEP01(2021)178 ] ], . 44 , as q keV. e 172 , y 30 171 ) as a red [ & 2 q 7.4 to thermalize − φ g (left), we show the 25 , one finds keV , this justifies neglecting the q ' ], the electron e E 108 to the maximal value allowed by unitarity. D ) and since – 50 – y 6.11 ]. To understand why it is justified to neglect rela- . We plot the mediator coupling to electrons, 130 25 ] gives an upper bound on the DM mass (which is related 33 , and we explore the parameter space as a function of the DM 4 / best-fit regions in red and the best fit point in eq. ( χ . Using eq. ( c σ m 2 03 = . 0 ]. In summary, it seems that this explanation of the XENON1T excess φ and m ∼ σ ]. At higher mediator masses, the required coupling to electrons is so high 173 , 1 61 174 [ γφ → Before closing, a remark is in order. In deriving the limits above, we used non- Since the mediator mass must be small enough for the spectrum to be suppressed − e + RE and MS are supportedin Physics in Award part 623940. by TV DoENo. is Grant 2522/17), supported DE-SC0017938 by by and the the Simonsropean Binational Israel Investigator Research Science Science Council Foundation-NSFC Foundation (ERC) (grant (grant under— No. the Proposal 2016153) EU n. and Horizon 682676 by 2020 LDMThExp). the Programme (ERC-CoG-2015 Eu- thank G. Alonso-Alvarez,G. L. Marquez-Tavares, P. Panci, Calibbi, E. Salvioni, M.for F. Szydagis, many useful L.J. Ertas, Thormaehlen, feedbacks on K. J. the VanScholarship, Tilburg draft. Huang, The IB Buchmann is J. grateful Scholarship,support for Jaeckel, and the from support the F. of the Azrieli the Kahlhoefer, (CIDEGENT/2018/019), Alexander “Generalitat Foundation. Zaks as Valenciana” well AC as (Spain) acknowledges national grants through FPA2014-57816-P, FPA2017- the 85985-P. “plan GenT” program Acknowledgments We thank T. Bringmann,G. R. Rossi, Budnik, F. H. Sala, Kim, Y. H. Soreq, Liu, L. P. Ubaldi, Meade, and G. N. Perez, Weiner J. for Pradler, useful discussions. We also tivistic corrections, we notevelocities that above a DMSince with the mass atomic around and 1 DMrelativistic MeV form must corrections, factors which be both become dominate accelerated important at to only low at significantly higher values of e is robustly excluded. relativistic form factors forrelativistic limit the can DM-electron be interactions. found in Corrections [ that arise in the We include stellar coolingconstraints, constraints and from a HB linebefore that stars electron shows [ decoupling. the coupling Thisconstraints required region [ for is the likely mediator toto be be subject robustly to (model-dependent) excluded BBN by direct production of the light mediator at colliders through and the Super-K experiment [ to the mediator mass). at high energies,illustrate the in relevant the parameter middlea space of function is figure of subject the to mediator severe mass, constraints that fixing we simplicity, we fix mass. Our conclusions do notcorresponding depend much on thisstar. choice. In In agreement figure with thelies previous close discussion, we to see the how boundary in this of setup the the region best-fit excluded region by the XENON1T S2-only analysis [ background, while the blue shaded regions show the contribution of the binned signal. For JHEP01(2021)178 , , , ]. (2013) 102 Phys. , . 08 Phys. Rev. SPIRE , IN ][ JCAP Astropart. Phys. , ]. , Phys. Rev. D (2008) 008 , ]. ] appeared. A number 07 SPIRE ]. ]. IN Axio-electric effect 194 – ][ SPIRE ]. Searching for Dark Absorption (2019) 095021 ]. IN JCAP SPIRE [ SPIRE , Chicago University Press, , 175 arXiv:1608.02123 , IN IN [ 100 [ SPIRE ][ Searching for a solar relaxion or ]. SPIRE IN 145 IN , ][ ][ ]. Atomic Enhancements in the Detection of (1986) 145 115 SPIRE IN (2017) 087 Direct Detection Constraints on Dark Photon ]. arXiv:1412.8378 ][ (1987) 2752 168 [ SPIRE 06 ]. Phys. Rev. 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