New Constraints and Prospects for sub-GeV Dark Matter Scattering off Electrons in Xenon
YYITP-SB-17-09, CERN-TH-2017-042
New Constraints and Prospects for sub-GeV Dark MatterScattering off Electrons in Xenon
Rouven Essig, ∗ Tomer Volansky, † and Tien-Tien Yu
1, 3, ‡ C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794 Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland
We study in detail sub-GeV dark matter scattering off electrons in xenon, including the expectedelectron recoil spectra and annual modulation spectra. We derive improved constraints using low-energy XENON10 and XENON100 ionization-only data. For XENON10, in addition to includingelectron-recoil data corresponding to about 1 − (cid:38) (cid:38) Introduction.
Direct-detection experiments play a cru-cial role in our quest to identify the nature of dark matter(DM), and the last few years have seen intense interestand significant progress in expanding their sensitivity toparticles below ∼ ∼ (cid:28) MeV) by electrons [10–13]. For other direct-detectionideas see [1, 14–20]. Direct-detection techniques andcomplementary probes are summarized in [21].Currently, the most stringent direct-detection con-straint on DM as low as a few MeV comes fromXENON10, a two-phase xenon time projection chamber(TPC). When a DM particle scatters off an electron andionizes a xenon atom in the liquid target, the recoilingelectron can ionize other atoms if it has sufficient energy.An electric field accelerates the ionized electrons throughthe liquid, across a liquid-gas interface, and through axenon gas region in which interactions between the elec-trons and xenon atoms create a scintillation (“S2”) signalthat is proportional to the number of extracted electronsand detected by photomultiplier tubes. XENON10 [22]has taken data consisting of events that have an S2 signal corresponding to one or more electrons, without an ob-servable prompt scintillation signal (“S1”). The data cor-responding to events with three electrons or less ( n e (cid:46) n e (cid:38)
4. The rate ofsuch events is lower than for n e (cid:46)
3, leading to sig-nificantly improved constraints for DM masses m χ (cid:38)
50 MeV. We also analyze S2-only data from XENON100,containing n e (cid:38) a r X i v : . [ h e p - ph ] M a r exposure [kg-yrs] fiducial mass [kg]XENON10 [22] 0.041 1.2XENON100 [23] 29.8 48.3LUX [27] 119 145XENON1T [28] 2,000 1,000LZ, XENONnT [28, 29] 15,000 5,600TABLE I. Analyzed (XENON10, XENON100, LUX) and ap-proximate projected (XENON1T, LZ, XENONnT) exposuresand fiducial masses. ity in the near future are likely with SuperCDMS [5, 24],SENSEI [25], and possibly other experiments. Neverthe-less, these experiments will initially have target massesof only O (1 kg), far less than current and future xenonexperiments (Table I). Understanding the S2-only eventsin two-phase TPCs could thus lead to dramatic improve-ments in cross-section sensitivity and, as we will show,probe simple and predictive benchmark models. Thelarge exposures will also allow for an annual modulationanalysis [26], which can significantly improve upon thecurrent limit even if the background rates are high. Theoretical Rates and Recoil Spectra.
To calculatethe DM-electron scattering rate in liquid xenon, we fol-low the procedure in [2] (see appendices for more details).We treat the target electrons as single-particle states ofan isolated atom, described by numerical RHF boundwave functions from [30] [31]. The velocity-averaged dif-ferential ionization cross section for electrons in the ( n, l )shell is d (cid:104) σ nlion (cid:105) d ln E er = σ e µ χe (cid:90) qdq | f nlion ( k (cid:48) , q ) | | F DM ( q ) | η ( v min ) , (1)where η ( v min ) = (cid:104) v θ ( v − v min ) (cid:105) is the inverse meanspeed for a given velocity distribution as a function ofthe minimum velocity, v min , required for scattering. Weassume a standard Maxwell-Boltzmann velocity distri-bution with circular velocity v = 220 km/s and a hardcutoff of v esc = 544 km/s [32, 33]. σ e is the DM-free elec-tron scattering cross section at fixed momentum transfer q = αm e , while the q -dependence of the matrix elementis encoded in the DM form-factor F DM ( q ). | f nlion ( k (cid:48) , q ) | is the ionization form factor of an electron in the ( n, l )shell with final momentum k (cid:48) = √ m e E er . We calcu-late this form factor using the given bound wave func-tions and unbound wave functions that are obtained bysolving the Schr¨odinger equation with a potential thatreproduces the bound wave functions. We consider elec-trons in the following shells (listed with binding energiesin eV): 5 p (12.4), 5 s (25.7), 4 d (75.6), 4 p (163.5),and 4 s (213.8). The differential ionization rate is dR ion d ln E er = N T ρ χ m χ (cid:88) nl d (cid:104) σ nlion v (cid:105) d ln E er , (2)where N T is the number of target atoms and ρ χ =0 . is the local DM density. ��� � �� � �� � �� � �� � �� � �� - � ��� � �� � �� � �� � �� � � � / �� � ( σ � = ���� - �� � � � ) � � / �� � ( σ � = ���� - �� � � � ) � �� = � � χ = ��� ������� ������ ���� �� - ����� � � � � � � � � � �� �� �� �� �� �� �� - � �� - � ��� � �� � �� � �� � �� � �� � �� � �� - � �� - � �� - � �� - � ��� � �� � �� � �� � �� � �� � � � � � / �� � ( σ � = ���� - �� � � � ) � � / �� � ( σ � = ���� - �� � � � ) � �� ~ � / � � � χ = ��� ������� �� �� �� ���� �� - ����� FIG. 1.
Top ( bottom ): Spectrum of expected numberof events for DM-electron scattering in xenon, for m χ =100 MeV and 1000 kg-years for F DM = 1 ( α m e /q ). Forthe left axes, we set σ e to the maximum allowed values bycurrent constraints for two popular benchmark models; forthe right axes, the indicated σ e produces the correct relicabundance. Colored lines show individual contributions fromvarious xenon electron shells, while the gray band encom-passes the spectrum when varying the secondary ionizationmodel. See text for details. We follow [2] to model the conversion from E er to elec-tron yield, n e . The recoiling electron will ionize and ex-cite other atoms, producing n (1) = Floor( E er /W ) addi-tional “primary quanta”, either observable electrons or(unobservable) scintillation photons. For fiducial values,we choose the probability for the initial electron to re-combine with an ion to be f R = 0, W = 13 . f e = 0 .
83. To capture the uncertainty in the fiducial val-ues, we vary these parameters in the range 0 < f R < . . < W <
16 eV, and 0 . < f e < .
91. In addi-tion to primary quanta, if DM ionizes an inner-shell elec-tron, n (2) = Floor(( E i − E j ) /W ) secondary quanta canbe created by photons produced in the subsequent outer-to-inner-shell electron transitions with binding energies E i,j . The number of secondary electrons produced fol-lows a binomial distribution with n (1) + n (2) trials andsuccess probability f e .In Fig. 1, we show the recoil spectra as a function of n e for a hypothetical xenon detector with 1000 kg-years ofexposure for F DM = 1 (top) and F DM = α m e /q (bot-tom). The colored lines show individual contributionsfrom different shells, while the black line shows their sum(for fiducial values). Gray bands show the variation awayfrom the fiducial values discussed above.To emphasize the importance of studying electron re- h n e i �� ��� ��������� � � � � � ��� � � � � � � / � � ������� �� �� - ���� � χ = �� ��� � χ = � ��� �� % �� ����� �� ��� �������� � � � � � � � � � � � � � / �� � � h n e i �� ��� ��� ��� ��� �������� � �� � � � � � ��� � � � � � � / � � �������� �� �� - ����� � χ = �� ��� � χ = � ��� �� % �� ����� �� ��� ��� ��� ��� ������������������� � � � � � � � � � � � / �� � � FIG. 2. Observed number of events versus photoelectrons(PE) in XENON10 ( top ) [22] and XENON100 ( bottom ) [23].DM spectra are shown for m χ = 10 MeV (blue) & 1 GeV (red)with a cross section fixed at our derived 90% C.L. limit (weassume fiducial values for the secondary ionization model).Insets show spectra in bins of 27PE (20PE), the mean numberof PE created by one electron in XENON10 (XENON100). coil events at current and upcoming xenon experiments,we have fixed σ e to specific values that are allowed bysimple and predictive benchmark models [1, 5, 34–40] andfurther below. We consider the DM (a Dirac fermion orcomplex scalar χ ) to be charged under a broken U (1) D gauge force, mediated by a kinetically-mixed dark pho-ton, A (cid:48) , with mass m A (cid:48) . The A (cid:48) mediates DM-electronscattering, and F DM ( q ) = 1 ( α m e /q ) for a heavy (ul-tralight) dark photon. The left axis for top (bottom) plotof Fig. 1 shows the event rate for σ e fixed to the maxi-mum value allowed by current constraints for m A (cid:48) = 3 m χ ( m A (cid:48) (cid:28) keV), while the right axis of the top (bottom)plot fixes σ e so that scalar (fermion) DM obtains the cor-rect relic abundance from thermal freeze-out (freeze-in).Clearly, a large number of DM events could be seen inupcoming detectors. These results are easily rescaled toother DM models that predict DM-electron scattering. New XENON10 and XENON100 bounds.
We nowrecalculate the bounds from XENON10 data [2] (15 kg- �� - �� �� - �� �� - �� σ � [ � � � ] � � � � � � ( � � � � ) � � � � � � ��������������� � �� = � �� �� � �� � �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� � χ [ ��� ] σ � [ � � � ] � � � � � � ( � � � � ) � � � � � � ��������������� � �� ∝ � / � � FIG. 3. 90% C.L. limit on the DM-electron scattering crosssection from XENON10 data (blue) and XENON100 data(red) for F DM = 1 ( top ) & F DM = α m e /q ( bottom ). Dot-ted black lines show XENON10 bounds from [2]. days), including for the first time events with n e (cid:38)
4, aswell as from XENON100 data [23] (30 kg-years). Sincethe experimental observable is the number of photoelec-trons (PE) produced by an event, we convert n e to PE.An event with n e electrons produces a gaussian dis-tributed number of PE with mean n e µ and width √ n e σ ,where µ = 27 (19 .
7) and σ = 6 . .
2) for XENON10(XENON100). We multiply the signal with the triggerand acceptance efficiencies from [2, 23] and then bin boththe signal and data in steps of 27PE (20PE), startingfrom 14PE (80PE) for XENON10 (XENON100). Thefirst bin for the XENON100 analysis is 80-90PE, corre-sponding to roughly half an electron. We require thatthe resulting signal is less than the data at 90% C.L. ineach bin. For XENON10, the 90% C.L. upper boundson the rates (after unfolding the efficiencies) are r < . , r < . , r < . , r < . , r < . , r < . , r < .
35 counts kg − day − , corresponding tobins b = [14 , , b = [41 , . . . , b = [176 − r < . , r < . , r < .
17 counts kg − day − corresponding to bins b =[80 , , b = [90 , , b = [110 , � � � � � � � � � �� �� �������������������������� � � � ��� � � � � �� � χ = ��� ��� � χ = � ��� � �� ∝ � / � � � �� = � FIG. 4. Annual modulation amplitude for F DM = 1 (solid) & F DM = α m e /q (dashed) for m χ = 100 MeV (blue) & 1 GeV(black). comparison with the XENON10 bound derived in [2]. Inthe Appendices, we show cross-section bounds for theindividual PE bins, taking into account the systematicuncertainties from the secondary ionization model. For F DM = 1, the inclusion of the high-PE bins in XENON10significantly improves upon the bound from [2] for m χ (cid:38)
50 MeV (small differences at lower masses are fromthe limit-setting procedure). The new XENON10 andXENON100 bounds are comparable for m χ (cid:38)
50 MeV.For F DM = α m e /q , the low PE bins determine thebound, and XENON100 is therefore not competitive dueto its high analysis threshold. Modulation.
A useful discriminant between signaland background is the annual modulation of the signalrate [26] due to the Sun’s motion through the DM halo.Fig. 4 shows f mod versus n e , where f mod = R max − R min R avg is the modulation amplitude, derived by calculating therates for the average Earth velocity and varying it by ± . f mod spectrum is distinctive, whichshould provide a helpful discriminant between signal andbackground. The significance of a signal S over a flatbackground B is then given by sig = f mod S √ S + B .To demonstrate the power of an annual modulationsearch, we imagine that a future detector with 1000 kg-years of exposure observes the same S2-only event rateand spectrum as observed in XENON10 data, R Xe10 . Re-quiring the signal rate to be less than the observed eventrate yields the same constraints as with XENON10 data, σ e, Xe10 . However, an annual modulation analysis wouldpotentially see a signal of high statistical significance,and in the absence of one a fraction of the observed eventrate must be background. Requiring the significance ofthe annual modulation signal to be less than sig , theexpected sensitivity is σ mod e = sig × σ e, Xe10 f mod √ R Xe10 × exposure . (3) �� �� � �� � �� � �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� � χ [ ��� ] σ � [ � � � ] ����� ����� � ���� ���� ������� ���� �� ��������� �������� � �� � � �� � � � � �� � ����� / ��� � � ⩾ �� ����� / ��� � � ⩾ ���� ���� ���� �������� ���� ����� ����� � �� = � � � � = � � χ � � � � � � �� �� � � � �� � �� �� � �� � �� � �� � �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� � χ [ ��� ] σ � [ � � � ] ����� ����� � ���� ���� ������� ���� �� �� � � � � � � - � � � � � � � � � � � � � � � � � � ����� / ��� � � ⩾ �� ����� / ��� � � ⩾ ���� ���� ������� ���� ����� � �� = α � � � � / � � FIG. 5. Sensitivity reach from an annual modulation analysiswith a hypothetical 1000 kg detector and 1-year exposure, as-suming the observed spectrum and data rate are the same asin XENON10 [22] (solid blue) or XENON100 [23] (solid red).DM-electron scattering event rates assuming a 1-electron (4-electron) threshold are shown in dashed (dotted) green. Blue(red) shaded regions show our XENON10 (XENON100) lim-its. These lines/regions are overlaid on several simple andpredictive benchmark models for DM ( χ ) scattering off elec-trons via a dark photon A (cid:48) . Top: ( F DM = 1) A complexscalar obtains the correct relic density from thermal freeze-out (light orange), while a fermion, which obtains its cor-rect relic abundance from an initial asymmetry, must have σ e above the dark brown line (assuming no additional anni-hilation channels) to avoid indirect-detection constraints [41–43]. Bottom: ( F DM = α m e /q ) Fermion DM coupled toan ultralight mediator A (cid:48) obtains the correct relic densityfrom freeze-in (thick brown line). Gray regions show con-straints as in [5], updated on the top plot with data fromMiniBooNE [44] and BaBar [45]. Due to earth-scatteringeffects [46], no XENON10/100 limit exists in the top rightregion. We calculate σ mod e for bins of n e = 0 . − . , . − . , . . . and show with a blue line the best sensitivity across allbins in Fig. 5 for sig = 1 .
65 (90% CL) (see Appendicesfor sensitivities from each bin). Similarly, a red solid lineshows σ mod e assuming the future observed rates/spectrumcorrespond to the current XENON100 rate/spectrum.We overlay these lines on the DM benchmark modelsdiscussed above. While hypothetical, this analysis em-phasizes the power of an annual modulation analysis. Large Event Rates.
To further emphasize the impor-tance of understanding the electron recoil events in xenonTPCs, we show the expected event rates in Fig. 5 for a1000 kg detector for two thresholds, n e ≥ n e ≥ A (cid:48) that obtains its abundance from an initial asym-metry could produce about one event every two secondsat LZ. This underscores the point that while there areseveral sources of backgrounds that can produce single-or few-electron events, a large event rate can be consis-tent with a DM signal and should not be simply writtenoff as a detector curiosity. Conclusions.
We derived new constraints on DM-electron scattering, improving upon the previous bound,and showed spectra for the expected number of elec-trons and the modulation amplitude. While there areseveral possible detector-specific origins of the observedXENON10/100 events, in principle almost all the ob-served events could originate from DM-electron scatter-ing without coming into conflict with other existing DMconstraints. This is not the case when interpreting theseevents as arising from few-GeV DM recoiling elasticallyoff nuclei [22, 23], which is excluded by existing resultsfrom e.g.
LUX [27] and CDMSlite [47]. Moreover, sim-ple and predictive DM benchmark models predict largeevent rates in current and future xenon TPCs. An ex-panded and dedicated effort by the xenon collaborationsto understand the origin of their low-energy electron re-coil data is thus imperative and well worth the effort.
ACKNOWLEDGMENTS
We would like to thank especially Aaron Manalaysayand Peter Sorensen for many insightful discussions. Wealso thank Ran Budnik, Daniel McKinsey, Matt Pyle,and Jingke Xu for useful discussions. We are also verygrateful to Jeremy Mardon for contributions at the be-ginning of this project as well as many useful discussions.R.E. is supported by the DoE Early Career researchprogram DESC0008061 and through a Sloan Founda-tion Research Fellowship. T.-T.Y. is also supported bygrant DESC0008061. T.V. is supported by the Euro-pean Research Council (ERC) under the EU Horizon2020 Programme (ERC-CoG-2015 - Proposal n. 682676LDMThExp), by the PAZI foundation, by the German-Israeli Foundation (grant No. I-1283- 303.7/2014) and bythe I-CORE Program of the Planning Budgeting Com-mittee and the Israel Science Foundation (grant No.1937/12). T.-T.Y. thanks the hospitality of the AspenCenter for Physics, which is supported by National Sci-ence Foundation grant PHY-1066293, where part of thiswork was completed.
APPENDIX
Here we provide additional details to the calcula-tions described in the main text. We also show spec-tra plots for additional DM masses, as well as theXENON10/XENON100 limits and the prospects for anannual modulation analysis from each PE bin. For com-pleteness, we also show the expected daily modulation ofthe signal rate due to the Earth’s rotation.
Theoretical Rates.
We first quote additional formulasthat are required for the rate calculation (see also [2, 5]).The velocity-averaged differential ionization cross sectionfor electrons in the ( n, l ) shell is given in Eq. (1). Thefull expression for v min is v min = (cid:16) | E nl binding | + E er (cid:17) q + q m χ , (4)where E nl binding is the binding energy of the shell and q is the momentum transfer from the DM to the electron.The form factor for ionization of an electron in the ( n, l )shell with final momentum k (cid:48) = √ m e E er is given by | f nlion ( k (cid:48) , q ) | = 4 k (cid:48) (2 π ) (cid:88) l (cid:48) L (2 l + 1)(2 l (cid:48) + 1)(2 L + 1) × (cid:20) l l (cid:48) L (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) r drR k (cid:48) l (cid:48) ( r ) R nl ( r ) j L ( qr ) (cid:12)(cid:12)(cid:12)(cid:12) , (5)where [ · · · ] is the Wigner 3- j symbol and j L are the spher-ical Bessel functions. We solve for the radial wavefunc-tions R k (cid:48) l (cid:48) ( r ) of the outgoing unbound electrons takingthe radial Schr¨odinger equation with a central poten-tial Z eff ( r ) /r . This central potential is determined fromthe initial electron wavefunction by assuming that it is abound state of the same potential. We include the shellslisted in Table II. Electron and Photoelectron Yields.
We provide ad-ditional details to convert the recoiling electron’s recoilenergy into a specific number of electrons. The relevant
Shell 5 p s d p s Binding Energy [eV] 12.4 25.7 75.6 163.5 213.8Photon Energy [eV] – 13.3 63.2 87.9 201.4Additional Quanta 0 0 4 6-10 3-15TABLE II. Xenon shells and energies. “Photon energy” refersto energy of de-excitation photons for outer-shell electrons de-exciting to lower shells. This photon can subsequently pho-toionize, creating additional quanta. The range of additionalquanta takes into account that the higher energy shell mayhave more than one available lower energy shell to de-exciteinto. For our limits, we take the minimum of this range. � � � � � � � � � �� �� �� �� �� �� �� - � �� - � �� - � �� - � �� - � �� - � �� - � �� - � �� - � � � � � / �� � ( σ � = ���� - �� � � � ) � � / �� � ( σ � = �� - �� � � � ) � �� = � � χ = � ������� ���� �� ���� �� - ����� � � � � � � � � � �� �� �� �� �� �� �� - � ��� � �� � �� � �� � �� � �� � �� � �� - � �� - � �� - � �� - � ��� � �� � �� � �� � �� � � � � � / �� � ( σ � = ���� - �� � � � ) � � / �� � ( σ � = ���� - �� � � � ) � �� ~ � / � � � χ = � ������� �� ���� ���� �� - ����� � � � � � � � � � �� �� �� �� �� �� �� - � �� - � ��� � �� � �� - � �� - � ��� � �� � � � � � / �� � ( σ � = ���� - �� � � � ) � � / �� � ( σ � = ���� - �� � � � ) � �� = � � χ = ��� ������� ���� �� ���� �� - ����� � � � � � � � � � �� �� �� �� �� �� ��� � �� � �� � �� � �� � �� � �� � �� - � �� - � �� - � ��� � �� � �� � �� � �� � � � � � / �� � ( σ � = ���� - �� � � � ) � � / �� � ( σ � = ���� - �� � � � ) � �� ~ � / � � � χ = ��� ������� �� �� �� ���� �� - ����� FIG. 6. Expected number of events as a function of number of electrons observed for 1000 kg-years of xenon. The left-axissets σ e to the maximum allowed value by current constraints while the right-axis sets σ e to the predicted value for a freeze-out(freeze-in) model for F DM = 1( α m e /q ), respectively. The different colored lines show the contributions from the variousxenon shells while the gray band encodes the uncertainties associated with the secondary ionization processes. quantities are E er = ( n γ + n e ) W ,n γ = N ex + f R N i , (6) n e = (1 − f R ) N i .E er is the amount of deposited energy from the primaryelectron, which results in a number of observable elec-trons, n e , unobservable scintillation photons, n γ , andheat. W is the energy needed to produce a single quanta(photon or electron). We take W = 13 . ± . E er can create both a number of ions, N i , and a number of excited atoms N ex , where N ex /N i (cid:39) . f R , is effec-tively zero at low energy. This implies that n e = N i and n γ = N ex . The fraction of initial quanta observed as elec-trons is given by f e = (1 − f R ) / (1 + N ex /N i ) (cid:39) .
83 [51].To capture the uncertainty in f R , W , and N ex /N i , wecalculate the rates and limits varying these parametersover the ranges 0 < f R < . , . < N ex /N i < . . < W <
16 eV. For our fiducial values, we set f e = 0 . , f R = 0 , W = 13 . E er , we assumethat there are additional n (1) =Floor( E er /W ) quantacreated. Furthermore, we assume that the photonsassociated with the de-excitation of the next-to-outershells (5 s, d, p, s ), which have energies (13.3, 63.2,87.9, 201.4) eV, can photoionize to create an additional n (2) =( n s , n d , n p , n s )=(0, 4, 6-10, 3-15) quanta, re-spectively (see Table II). The range in values for the4 p and 4 s shells takes into consideration that there maybe more than one outer-shell electron available that cande-excite down to them. For example, if the 4 d shellde-excites to 4 p , 6 additional quanta are created, while ifthe 5 s shell de-excites to 4 p , it would create 10 additionalquanta. For our fiducial values, we take the lower num-ber of quanta to be conservative. However, the choiceof the number of additional quanta only affects n e > n e = n (cid:48) e + n (cid:48)(cid:48) e ,where n (cid:48) e is the primary electron and n (cid:48)(cid:48) e are the secondaryelectrons produced. n (cid:48) e = 0 or 1 with probability f R or1 − f R , respectively, while n (cid:48)(cid:48) e follows a binomial distri-bution with n (1) + n (2) trials and success probability f e .Given this conversion from E er into n e , we can calcu- �� �� � �� � �� - �� �� - �� �� - �� �� - �� � χ [ ��� ] σ � [ � � � ] ������� � �� = � ������������� ���� �� - �� ���� - �� ���� - �� ���� - ��� ����� - ��� ����� - ��� ����� - ��� �� �� �� � �� � �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� � χ [ ��� ] σ � [ � � � ] ������������� ���� ��� - ��� ����� - ��� ���� - ��� ���� - �� ���� - �� ���� - �� �� ������� � �� ∝ � / � � FIG. 7. New XENON10 limit (black) obtained as described inthe text. The colored bands are from the uncertainty in thesecondary ionization model. The shaded gray region showsthe parameter space previously excluded by the 1, 2, and 3electron XENON10 data. By including the contributions tothe S2 signal from 14PE to 203PE, we see that the limitsimprove considerably for DM masses above ∼
50 MeV for F DM = 1, while there is no improvement due to the momen-tum suppression for F DM = α m e /q . late the differential rate as a function of number of elec-trons. In addition to the m χ = 100 MeV spectra shownin the main text, we show the spectra for m χ = 500 MeVand 1 GeV in Fig. 6. XENON10 and XENON100 constraints for in-dividual photoelectron bins.
In the main text, weshow the cross-section limits from the XENON10 andXENON100 data using the fiducial values above. InFigs. 7, 8, we show the individual limits for each PE binas well as the uncertainty bands due to the secondaryionization model.
Modulation.
In Fig. 5, we showed how an annual mod-ulation analysis of a hypothetical xenon detector with anexposure of 1000 kg-years could significantly improve on �� �� � �� � �� - �� �� - �� �� - �� � χ [ ��� ] σ � [ � � � ] � � � � � � ( � � � � ) � � � � � � �������� ��������� ������ - �� ���� - ��� ����� - ��� �� ���� ���� �������� � �� = � �� �� � �� � �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� � χ [ ��� ] σ � [ � � � ] � � � � � � ( � � � � ) � � � � � � �������� ��������� ������ - �� ���� - ��� ����� - ��� �� ���� ���� �������� � �� ∝ � / � � FIG. 8. New limit obtained using the XENON100 data (red).The XENON100 data starts at 80PE electrons, so we showthe individual limits for the 80-90, 90-110, and 110-130 PEbins. The colored bands are from the uncertainty in the sec-ondary ionization model. The shaded gray region shows theparameter space excluded by our updated XENON10 analy-sis, while the dotted black line shows the XENON10 boundfrom [2]. current constraints even if the background rates are sig-nificant. In Fig. 5, we only showed the best constraintsacross all individual n e bins. In Fig. 9, we show the in-dividual n e bins. Furthermore, for completeness, we alsoshow the daily modulation amplitude due to the Earth’srotation with respect to the DM wind. The daily modu-lation is calculated by modifying the average earth veloc-ity by ± .
23 km/s to obtain the maximum and minimumrates. We show the daily modulation fraction in Fig. 10,where we see that the daily modulation fraction is aboutan order of magnitude smaller than that of the annualmodulation. �� �� � �� � �� - �� �� - �� �� - �� �� - �� �� - �� � χ [ ��� ] σ � ������ ����������� �� % �� � � - � � - � � - � � - � � - � � - ������� � �� = � ���� �� - ����� �� �� � �� � �� - �� �� - �� �� - �� �� - �� �� - �� � χ [ ��� ] σ � ������ ����������� �� % �� � � - � � - � � - � � - ������� � �� ∝ � / � � ���� �� - ����� �� �� � �� � �� - �� �� - �� �� - �� �� - �� �� - �� � χ [ ��� ] σ � ������ ����������� �� % �� � � - � � - � � - �������� � �� = � ���� �� - ����� �� �� � �� � �� - �� �� - �� �� - �� � χ [ ��� ] σ � ������ ����������� �� % �� � � - � � - � � - �������� � �� ∝ � / � � ���� �� - ����� FIG. 9. Individual bin sensitivities to the 90% C.L. annual modulation reach for a 1000 kg-year xenon detector. The backgroundrates and spectra are taken to be the XENON10 (XENON100) rates scaled up to 1000 kg-years for the top (bottom) panels(see also text and Fig. 5). � � � � � � � � � �� �� ��������������������������� � � � � �� � � � �� � χ = ��� ��� � χ = � ��� � �� ∝ � / � � � �� = � FIG. 10. Daily modulation amplitude for F DM = 1 (solid)and F DM = α m e /q (dashed) for m χ = 100 MeV (blue) & 1GeV (black). ∗ [email protected] † [email protected] ‡ [email protected][1] R. Essig, J. Mardon, and T. Volansky, Phys. Rev. D85 ,076007 (2012), arXiv:1108.5383 [hep-ph].[2] R. Essig, A. Manalaysay, J. Mardon, P. Sorensen, andT. Volansky, Phys. Rev. Lett. , 021301 (2012),arXiv:1206.2644 [astro-ph.CO].[3] P. W. Graham, D. E. Kaplan, S. Rajendran, and M. T.Walters, Phys. Dark Univ. , 32 (2012), arXiv:1203.2531[hep-ph].[4] S. K. Lee, M. Lisanti, S. Mishra-Sharma, and B. R.Safdi, Phys. Rev. D92 , 083517 (2015), arXiv:1508.07361[hep-ph].[5] R. Essig, M. Fernandez-Serra, J. Mardon, A. Soto,T. Volansky, and T.-T. Yu, JHEP , 046 (2016),arXiv:1509.01598 [hep-ph].[6] S. Derenzo, R. Essig, A. Massari, A. Soto, and T.-T. Yu,(2016), arXiv:1607.01009 [hep-ph].[7] Y. Hochberg, Y. Kahn, M. Lisanti, C. G. Tully, andK. M. Zurek, (2016), arXiv:1606.08849 [hep-ph]. [8] Y. Hochberg, M. Pyle, Y. Zhao, and K. M. Zurek,(2015), arXiv:1512.04533 [hep-ph].[9] Y. Hochberg, Y. Zhao, and K. M. Zurek, Phys. Rev.Lett. , 011301 (2016), arXiv:1504.07237 [hep-ph].[10] H. An, M. Pospelov, and J. Pradler, Phys. Rev. Lett. , 041302 (2013), arXiv:1304.3461 [hep-ph].[11] H. An, M. Pospelov, J. Pradler, and A. Ritz, Phys. Lett. B747 , 331 (2015), arXiv:1412.8378 [hep-ph].[12] I. M. Bloch, R. Essig, K. Tobioka, T. Volansky, andT.-T. Yu, (2016), arXiv:1608.02123 [hep-ph].[13] Y. Hochberg, T. Lin, and K. M. Zurek, (2016),arXiv:1608.01994 [hep-ph].[14] R. Essig, J. Mardon, O. Slone, and T. Volansky, (2016),arXiv:1608.02940 [hep-ph].[15] R. Budnik, O. Cheshnovsky, O. Slone, and T. Volansky,to be submitted to Nature. (2017), arXiv:1703.xxxxx[hep-ph].[16] K. Schutz and K. M. Zurek, (2016), arXiv:1604.08206[hep-ph].[17] S. Knapen, T. Lin, and K. M. Zurek, (2016),arXiv:1611.06228 [hep-ph].[18] C. Kouvaris and J. Pradler, (2016), arXiv:1607.01789[hep-ph].[19] C. McCabe, (2017), arXiv:1702.04730 [hep-ph].[20] P. C. Bunting, G. Gratta, T. Melia, and S. Rajendran,(2017), arXiv:1701.06566 [hep-ph].[21] R. Essig, J. A. Jaros, W. Wester, P. H. Adrian, S. An-dreas, et al. , (2013), arXiv:1311.0029 [hep-ph].[22] J. Angle et al. (XENON10), Phys. Rev. Lett. , 051301(2011), [Erratum: Phys. Rev. Lett.110,249901(2013)],arXiv:1104.3088 [astro-ph.CO].[23] E. Aprile et al. (XENON100), (2016), arXiv:1605.06262[astro-ph.CO].[24] P. Cushman et al. , in
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