Exploring for sub-MeV Boosted Dark Matter from Xenon Electron Direct Detection

Qing-Hong Cao,1, 2, ∗ Ran Ding,3, 1, † and Qian-Fei Xiang1, ‡

1Center for High Energy Physics, Peking University, Beijing 100871, China2Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China

3School of Physics and Materials Science, Anhui University, Hefei 230039, China

Direct detection experiments turn to lose sensitivity of searching for a sub-MeV light dark mattercandidate due to the threshold of recoil energy. However, such light dark matter particles canbe accelerated by energetic cosmic-rays such that they can be detected with existing detectors.We derive the constraints on the scattering of a boosted light dark matter and electron from theXENON100/1T experiment. We illustrate that the energy dependence of the cross section playsa crucial role in improving both the detection sensitivity and also the complementarity of directdetection and other experiments.

Light dark matter (DM) candidate is well motivatedand can be naturally realized when the DM candidatecouples feebly to visible sector [1–5]. In particular,it is difficult for a sub-MeV DM candidate to satisfyobserved relic abundance through the thermal freeze-outmechanism [6–8]; therefore, freeze-in via annihilation ofelectron-positron pairs is a primary mechanism for DMproduction [2, 3, 9]. The traditional direct detectionof DM-nucleus scattering loses sensitivity rapidly for aDM candidate whose mass is below ∼ GeV due to thethreshold of recoil energy. An alternative way to searchfor a light DM candidate is through the scattering offelectrons [3, 10, 11], which is not sensitive to a sub-MeV DM candidate neither. It is crucial to develop newapproach to probe freeze-in DM in such mass range.

A certain fraction of DM candidates in the Galactichalo would be accelerated by energetic Cosmic-Ray(CR) particles as long as the DM candidate interactswith SM particles. The CR-boosted mechanism relaxesthe threshold problem and improves the sensitivity ofdetecting a light DM candidate [12, 13]. It has beenextensively discussed in DM-nucleus direct detections,neutrino experiments and CR observations for variousDM models [14–22]. In this Letter we investigate theCR-boosted effect on the DM-electron direct detectionin the freeze-in scenario and show that the existing datafrom xenon experiments are able to probe a sub-MeVDM candidate.

For illustration, we consider a typical freeze-in DMmodel based on the vector-portal, in which the DMcandidate is a Dirac fermion (χ) that couples to thevisible sector through an additional gauge boson A′µ,named as “dark photon”. The Lagrangian is given by

L ⊃ χ(i∂/−mχ)χ+gχχγµχA′µ+gSMeγ

µeA′µ+1

2m2A′A′µA

′µ ,

(1)where mχ and mA′ denote the mass of DM candidateand the dark photon, respectively. gχ and gSM are thecoupling strength of A′ to the DM candidate and theelectron, respectively. When the DM candidate scattersoff an incident CR electron with a given kinetic energy

(TCR), the distribution of the DM recoil energy Tχ is

dσχedTχ

= σe

(α2m2

e +m2A′

)2µ2χe

× (2)

2mχ (me + TCR)2 − Tχ

((me +mχ)

2+ 2mχTCR

)+mχT

2χ

4 (2meTCR + T 2CR) (2mχTχ +m2

A′)2 ,

where σe denotes the cross section of DM-free electronscattering for a fixed momentum transfer q = αme [3].The maximal recoil energy of the DM candidate is [23]

Tmaxχ =

2mχTCR(TCR + 2me)

(me +mχ)2 + 2TCRmχ. (3)

Convoluting the Tχ distribution in Eq. (2) with theenergy spectrum of incident CR electrons dΦe/dTCR

yields the recoil flux of boosted DM candidate [20]

dΦχdTχ

= Deff

ρlocalχ

mχ

∫ ∞TminCR

dTCRdΦedTCR

dσχedTχ

, (4)

where Deff ≡∫dΩ4π

∫l.o.s

dl is an effective diffusiondistance. See supplement materials for details. Fora homogeneous CR distribution and NFW DM haloprofile [24, 25] (scale radius rs = 20 kpc and local DMdensity ρlocal

χ = 0.4 GeV cm−3), integrating along theline-of-sight to 10 kpc yields Deff = 8.02 kpc [13]. Inorder to produce a recoil energy Tχ after the DM and CR-electron scattering, the minimum kinetic energy (Tmin

CR )of the incident CR electron is given by

TminCR =

(Tχ2−me

)(1±

√1 +

2Tχmχ

(me +mχ)2

(2me − Tχ)2

),

(5)where the plus and minus sign corresponds to Tχ > 2me

and Tχ < 2me, respectively.Figure 1 plots the recoil flux dΦχ/dTχ distributions

as a function of Tχ for various mA′ ’s. Two simplifiedmodels are also plotted for comparison; one is the crosssection σχe being a constant (black-solid curve), the otheris that the the squared matrix element of the DM-electron

arX

iv:2

006.

1276

7v1

[he

p-ph

] 2

3 Ju

n 20

20

2

10-10

10-5

100

105

1010

10-4 10-3 10-2 10-1 100 101 102 103

mχ = 1 keV

dΦχ/

dTχ

(MeV

-1cm

-2s-1

)

Tχ (MeV)

Const |M|2

Const σχe

10-10

10-5

100

105

1010

10-4 10-3 10-2 10-1 100 101 102 103-4

-3

-2

-1

0

1

2

Log 1

0 (m

A’/M

eV)

FIG. 1. Recoil flux distributions of the DM candidate forvarying for mA′ ’s with the choice of mχ = 1 keV andσe = 10−30 cm2. For comparison, the recoil flux distributionsfor the approximation of a constant σχe (black solid) and a

constant |M|2 (black dashed) are also plotted.

scattering (|M|2), averaged over initial and summed overfinal spin states, is a constant (black-dashed curve), i.e.

dσχedTχ

=

σeTmaxχ

, σχe = const,

σeTmaxχ

(mχ +me)2

(mχ +me)2

+ 2mχTCR

, |M|2 = const.

(6)The former case is commonly used in the study of non-relativistic DM candidates, the later one takes the energydependence from phase space into account. However, theboth treatments are not appropriate for an energeticallyboosted DM candidate whose kinetic energy is muchlarger than its mass such that the momentum transferq cannot be neglected. We consider the relativistickinematics throughout this work. As shown in Fig. 1, theflux distribution exhibits a significant enhancement atthe large Tχ range with increasingmA′ . Note that variousrecoil flux curves intersect at Tχ = (αme)

2/(2mχ), and

the recoil flux distribution of the constant |M|2 slightlydeviates from that of the constant σχe when 2mχTCR >(me +mχ)2.

It is worth mentioning that the recoil flux distributionis independent of mA′ when the dark photon is veryheavy (mA′

√2mχTχ) or ultralight (mA′ αme).

See the red and blue boundaries of the contour. Therecoil flux distributions in the above two limits exhibitdistinct dependence on Tχ; for example, the recoil flux ofultralight dark photons drops rapidly with Tχ while therecoil flux of heavy dark photons mildly decreases withTχ. The heavy dark photon represents the so-called Z ′-portal model while the ultralight dark photon the milli-charged DM model [26].

Equipped with the boosted DM flux, we now discuss

the DM direct detection through the DM interaction withthe electron in xenon atoms. For the ionization process ofχ+A→ χ+A++e− with the atom A in the (n, l) atomicshell, the velocity-averaged differential cross section withrespect to the electron recoil energy ER is given by [3, 27]

d〈σnlionv〉d lnER

=σe

8µ2χe

∫qdq |FDM (q)|2

∣∣fnlion(k′, q)∣∣2 η (Emin) ,

(7)where FDM is the DM form factor, η denotes the mean

inverse speed function and∣∣fnlion(k′, q)

∣∣2 represents theionization form factor for an electron with initial state(n, l) and final state with momentum k′ =

√2meER. In

the case of boosted DM, the DM form factor FDM is

|FDM (q)|2 =

(α2m2

e +m2A′

)2(2meER +m2

A′)2

×2me (mχ + Tχ)

2 − ER(

(mχ +me)2

+ 2meTχ

)+meE

2R

2mem2χ

.

(8)

In the non-relativistic limit, Tχ, ER me, it reproducesthe form factor without CR-boost effects, i.e.,

|FDM (q)|2 =

(α2m2

e +m2A′

q2 +m2A′

)2

. (9)

The mean inverse speed function η is replaced by [12]

η (Emin) =

∫Emin

dEχΦ−1halo

m2χ

pEχ

dΦχdTχ

, (10)

where Φhalo = nχvχ is the background DM flux inGalactic halo with vχ being the corresponding averagevelocity. Here, Emin is the minimal DM energy totrigger electron with recoil energy ER. Similarly, Eq. (10)reproduces the conventional expression

η(vmin) =

∫vmin

1

vf(v)d3v (11)

in the non-relativistic limit. The ionization formfactor

∣∣fnlion(k′, q)∣∣2 is calculated by using the Roothaan-

Hartree-Fock radial wavefunction [28] for initial electronstate and applying plane wave approximation for finalstate. For initial electron state, we take into accountcontributions from (5p6, 5s2, 4d10, 4p6, 4s2) xenon elec-tron shells. The differential ionization rate is obtainedby multiplying Eq. (7) with background DM flux Φhalo,the number of target atoms NT , and sum over differentelectron shells,

dRiond lnER

= NTΦhalo

∑nl

d〈σnlionv〉d lnER

. (12)

Figure 2 shows the ionization rate as a function of theelectron recoil energy ER (in unit of tonne−1 year−1)

3

100

105

1010

1015

1020

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

mχ = 1 keV-σe =10-30 cm2

dRio

n/dl

nER

(to

ne-1

yea

r-1)

ER (MeV)

Ultralight A’

Heavy A’

100

105

1010

1015

1020

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100

FIG. 2. Recoil spectra of electrons for benchmark DM massmχ = 1 keV with scattering cross section σe = 10−30 cm2.Here we simply consider recoil electron from 5s state fordemonstration.

for both ultralight (red) and heavy (blue) dark photonswith the choices of mχ = 1 keV and σe = 10−30 cm2.The vertical band represents the order of magnitude ofenergy coverage for current xenon experiments. Theultralight dark photon prefers to produce electrons withsmall recoil energy; however, the heavy dark photon islikely to generate electrons with large recoil energy. Thedistinct difference follows from the energy dependence inthe distribution of dσχe/dTχ and the DM form factorFDM (q). It implies that one might distinguish betweenthe dark photon and the heavy dark photon from therecoil energy spectrum of the ionized electron when thebackground is well understood.

The recoiling electron are then converted into scintil-lation (S1) and ionization (S2) signal in liquid xenonexperiments, and the observable is the number ofphotonelectrons (PE). We consider S2 signal hereafteras the XENON100 and XENON1T collaborations releasethe data sets that are based only on the ionizationsignal [29, 30]. The event spectrum can be schematicallywritten as following:

dN

dS2= Texp · εS2

∑nl

∫dER pdf (S2|∆Ee)

dRnliondlnER

, (13)

where Texp is the exposure of detector and εS2 is theefficiency of triggering and accepting the S2 signal.For a given deposit energy ∆Ee = ER + |EnlB | with|EnlB | the binding energy of (n, l) shell, the conversionprobability of S2 is pdf (S2|∆Ee), which is modeled asfollows [10, 11]. The number of primary quanta produced

at the interaction point is n(1)Q = Floor(ER/W ) with

W = 13.8 eV, and n(1)Q is divided into ne observable

ionized electrons escaping from interaction point and nγunobservable scintillation photons. The fiducial value

of the fraction of primary quanta identified as electronsis chosen as fe ' 0.83. In addition, in the case ofthe DM candidate ionizes an inner shell electron, thesecondary quanta is produced by subsequent electrontransitions from outer to inner shell. The number ofthe secondary quanta is n

(2)Q = Floor((Ei − Ej)/W )

where Ei denotes the binding energy of the ith shell.The production number of secondary electrons follows

a binomial distribution with n(1)Q + n

(2)Q trials and the

success probability fe. Finally, the number of PE

converted from electrons (with total number ne = n(1)e +

n(2)e ) is described by a gaussian distribution with mean

value neµ and width√neσ. The parameters are chosen

as µ = 19.7 (11.4) and σ = 6.9 (2.8) [29, 31].

We derive the limits of σe imposed by the XENON100data [29] (Texp = 30 kg − years) and by the XENON1Tdata [30] (effective Texp = 22 tonne− days), using thesame bin steps. We choose the detection efficiency asεS2 = 1 for simplicity and obtain the limits by demandingthat signal does not exceed 1σ upper bound in each bin.Figure 3 presents benchmark signal spectra versus PE forthe ultralight and heavy mediator cases.

Figure 4(A) shows the exclusion limits in the mχ-σe plane for the case of a ultralight mediator, derived

0

10

20

30

40

50

60

70

0 200 400 600 800 1000

(A)

mχ = 1 keVEve

nts

(PE

-1)

S2 (PE)

Ultralight A’, -σe = 1.5*10-31 cm-2

Heavy A’, -σe = 1.5*10-33 cm-2

Xenon100

0

10

20

30

40

50

60

70

0 200 400 600 800 1000

10-1

10-0

101

102

103

104

90 120 150 200 500 1000 3000

(B)

Eve

nts/

(ton

ne ×

day

× k

eVee

)

S2 (PE)

Ultralight A’, -σe = 1.5*10-31 cm-2

Heavy A’, -σe = 1.5*10-33 cm-2

Xenon1T

10-1

10-0

101

102

103

104

90 120 150 200 500 1000 3000

FIG. 3. Example of expected PE spectra for DM-electronscattering in XENON100 (A) and XENON1T (B) experi-ments, for both ultralight and heavy mediator cases. Signalspectra are shown for mχ = 1 keV with scattering crosssection σe = 1.5× 10−31 cm2 (1.5× 10−33 cm2) in ultralight(heavy) mediator case.

4

10-45

10-40

10-35

10-30

10-25

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

- σ e (cm-2)

mχ (MeV)

const |M|2

Ultralight A'10

-45

10-40

10-35

10-30

10-25

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

XENON100

XENON1T

SN 1987

RG

WDHB

Freeze-in

(A)

10-45

10-40

10-35

10-30

10-25

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

- σ e (cm-2)

mχ (MeV)

const |M|2

Heavy A'10

-45

10-40

10-35

10-30

10-25

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

DDSolar RefectionXE

NON100

XENO

N1T

Super-K

(Cappiello et al.)(B)

FIG. 4. (A): exclusion limits in the mχ-σe plane from the XENON100 data (red-solid) and the XENON1T data (green-

solid) for ultralight mediator scenario. For comparison, corresponding limits for constant |M|2 are presented by dashed lines.Also shown are cooling constraints from supernovae 1987A (“SN 1987”) [32], energy loss of Red-Giant and Horizontal-Branchstars (“RG & HB”), as well as white dwarfs (“WD”) [33]; we also plot parameter region where DM obtains the correct relicabundance via freeze-in mechanism [3, 9]. (B): exclusion limits for heavy mediator scenario. We also plot constraints fromSuper-Kamiokande neutrino experiment (“Super-K”) [17], solar reflection (“Solar Reflection”) [12], as well as previous limitsfrom the XENON10 and the XENON100 direct detections (“DD”) [10, 11].

from the XENON100 data (red) and the XENON1T data(green). The acceleration mechanism greatly enhancesthe discovery potential of direct detection experiments ona light DM candidate. For comparison we also plot theparameter region for the freeze-in DM (brown curve) [3,9]. Even though the parameter space of freeze-in DMis well below the current direct detection sensitivity, itcan be reached when large experimental exposures areachieved. For example, the experimental exposure of 30tonne-years can probe the signal region of freeze-in DMwith mχ ∼ 1 eV when the background is fully controlled.In addition, the DM with a ultralight mediator (orequivalent milli-charged DM) can also be constrainedby astrophysical observations from supernova coolingand stellar energy loss [32, 33]. The bounds from thedirect detection experiments are comparable to thoseastrophysical constraints.

Figure 4(B) displays the exclusion limit of σe forthe case of a heavy mediator. We also plot the limitsfrom Super-Kamiokande neutrino experiment [17], solarreflection [12], and the direct detection without CR-DM scattering effect [11]. After considering the CR-DM effect, the direct detection experiments have a bettersensitivity in the sub-keV mass region.

In summary, we studied the effect of boosted DMon DM-electron direct detections and demonstrate thatthe current data from liquid noble gas experiments issensitive to light DM candidates in the range of sub-MeV. More importantly, the energy dependence in crosssection plays a crucial role in improving the exclusion

limits, e.g., the recoil spectra increase with recoil energyfor heavy mediator case while decrease with recoilenergy for ultralight mediator. Such opposite energydependences imply that the neutrino experiments suchas Super-K are more powerful for heavy mediator due totheir much larger acceptance volume and higher energycoverage [34]. On the other hand, direct detection hasmore advantage on ultralight mediator. Such two kindof experiments are complementary to each other.

The CR boosted DM mechanism has very richphenomenologies. For example, it is interesting toinvestigate boosted DM flux coming from the Galacticcenter which possesses high DM density and CR flux.One also expects that the morphology of signal resultedfrom the Galactic center is different from that originatedfrom local interstellar [35]. Moreover, light DM with sig-nificant CR acceleration and heavy DM (mχ & 10 MeV)with negligible CR acceleration could potentially producedegenerate signal; therefore, discrimination of such twokinds of scenarios in both model independent and modelspecific way is an intriguing issue [36]. The boostedmechanism might explain or be constrained by therecoiled energy spectrum of electrons recently reportedby the XENON1T collaboration [37].

Acknowledgments. We thank Tien-Tien Yu and Su-jie Lin for helpful discussions. The work is supportedin part by the National Science Foundation of Chinaunder Grant Nos. 11725520, 11675002 and 11635001.QFX is also supported by the China Postdoctoral ScienceFoundation under Grant No. 8206300015.

5

∗ qinghongcao@pku.edu.cn† dingran@mail.nankai.edu.cn‡ xiangqf@pku.edu.cn

[1] L. J. Hall, K. Jedamzik, J. March-Russell, and S. M.West, JHEP 03, 080 (2010), arXiv:0911.1120 [hep-ph].

[2] X. Chu, T. Hambye, and M. H. G. Tytgat, JCAP 1205,034 (2012), arXiv:1112.0493 [hep-ph].

[3] R. Essig, J. Mardon, and T. Volansky, Phys. Rev. D85,076007 (2012), arXiv:1108.5383 [hep-ph].

[4] S. Knapen, T. Lin, and K. M. Zurek, Phys. Rev. D96,115021 (2017), arXiv:1709.07882 [hep-ph].

[5] N. Bernal, M. Heikinheimo, T. Tenkanen, K. Tuominen,and V. Vaskonen, Int. J. Mod. Phys. A32, 1730023(2017), arXiv:1706.07442 [hep-ph].

[6] C. Boehm, M. J. Dolan, and C. McCabe, JCAP 1308,041 (2013), arXiv:1303.6270 [hep-ph].

[7] K. M. Nollett and G. Steigman, Phys. Rev. D89, 083508(2014), arXiv:1312.5725 [astro-ph.CO].

[8] Q.-H. Cao, T. Gong, K.-P. Xie, and Z. Zhang, Sci. ChinaPhys. Mech. Astron. 62, 981011 (2019), arXiv:1810.07658[hep-ph].

[9] C. Dvorkin, T. Lin, and K. Schutz, Phys. Rev. D99,115009 (2019), arXiv:1902.08623 [hep-ph].

[10] R. Essig, A. Manalaysay, J. Mardon, P. Sorensen, andT. Volansky, Phys. Rev. Lett. 109, 021301 (2012),arXiv:1206.2644 [astro-ph.CO].

[11] R. Essig, T. Volansky, and T.-T. Yu, Phys. Rev. D96,043017 (2017), arXiv:1703.00910 [hep-ph].

[12] H. An, M. Pospelov, J. Pradler, and A. Ritz, Phys.Rev. Lett. 120, 141801 (2018), [Erratum: Phys. Rev.Lett.121,no.25,259903(2018)], arXiv:1708.03642 [hep-ph].

[13] T. Bringmann and M. Pospelov, Phys. Rev. Lett. 122,171801 (2019), arXiv:1810.10543 [hep-ph].

[14] C. V. Cappiello, K. C. Y. Ng, and J. F. Beacom, Phys.Rev. D99, 063004 (2019), arXiv:1810.07705 [hep-ph].

[15] Y. Ema, F. Sala, and R. Sato, Phys. Rev. Lett. 122,181802 (2019), arXiv:1811.00520 [hep-ph].

[16] J. Alvey, M. Campos, M. Fairbairn, and T. You,Phys. Rev. Lett. 123, 261802 (2020), [Phys. Rev.Lett.123,261802(2019)], arXiv:1905.05776 [hep-ph].

[17] C. Cappiello and J. F. Beacom, Phys. Rev. D100, 103011(2019), arXiv:1906.11283 [hep-ph].

[18] J. B. Dent, B. Dutta, J. L. Newstead, and I. M.Shoemaker, (2019), arXiv:1907.03782 [hep-ph].

[19] G. Krnjaic and S. D. McDermott, (2019),arXiv:1908.00007 [hep-ph].

[20] K. Bondarenko, A. Boyarsky, T. Bringmann, M. Huf-nagel, K. Schmidt-Hoberg, and A. Sokolenko, (2019),arXiv:1909.08632 [hep-ph].

[21] J. Berger, Y. Cui, M. Graham, L. Necib, G. Petrillo,D. Stocks, Y.-T. Tsai, and Y. Zhao, (2019),arXiv:1912.05558 [hep-ph].

[22] W. Wang, L. Wu, J. M. Yang, H. Zhou, and B. Zhu,(2019), arXiv:1912.09904 [hep-ph].

[23] H. Goldstein, C. P. Poole, and J. L. Safko, ClassicalMechanics, 3rd ed. (Addison-Wesley, 2001).

[24] J. F. Navarro, C. S. Frenk, and S. D. M. White,Astrophys. J. 462, 563 (1996), arXiv:astro-ph/9508025[astro-ph].

[25] J. F. Navarro, C. S. Frenk, and S. D. M. White,

Astrophys. J. 490, 493 (1997), arXiv:astro-ph/9611107[astro-ph].

[26] B. Holdom, Phys. Lett. 166B, 196 (1986).[27] R. Essig, M. Fernandez-Serra, J. Mardon, A. Soto,

T. Volansky, and T.-T. Yu, JHEP 05, 046 (2016),arXiv:1509.01598 [hep-ph].

[28] C. F. Bunge, J. A. Barrientos, and A. V. Bunge, Atom.Data Nucl. Data Tabl. 53, 113 (1993).

[29] E. Aprile et al. (XENON), Phys. Rev. D94, 092001(2016), [Erratum: Phys. Rev.D95,no.5,059901(2017)],arXiv:1605.06262 [astro-ph.CO].

[30] E. Aprile et al. (XENON), Phys. Rev. Lett. 123, 251801(2019), arXiv:1907.11485 [hep-ex].

[31] E. Aprile et al. (XENON), Phys. Rev. D 99, 112009(2019), arXiv:1902.11297 [physics.ins-det].

[32] J. H. Chang, R. Essig, and S. D. McDermott, JHEP 09,051 (2018), arXiv:1803.00993 [hep-ph].

[33] H. Vogel and J. Redondo, JCAP 1402, 029 (2014),arXiv:1311.2600 [hep-ph].

[34] K. Bays et al. (Super-Kamiokande), Phys. Rev. D85,052007 (2012), arXiv:1111.5031 [hep-ex].

[35] E. Carlson and S. Profumo, Phys. Rev. D92, 063003(2015), arXiv:1504.04782 [astro-ph.HE].

[36] Q.-H. Cao, R. Ding, and Q.-F. Xiang, in preparation.[37] E. Aprile et al. (XENON), (2020), arXiv:2006.09721

[hep-ex].[38] A. C. Cummings, E. C. Stone, B. C. Heikkila, N. Lal,

W. R. Webber, G. Jahannesson, I. V. Moskalenko,E. Orlando, and T. A. Porter, Astrophys. J. 831, 18(2016).

[39] M. Aguilar et al. (AMS), Phys. Rev. Lett. 113, 121102(2014).

[40] G. Ambrosi et al. (DAMPE), Nature 552, 63 (2017),arXiv:1711.10981 [astro-ph.HE].

[41] A. W. Strong and I. V. Moskalenko, Astrophys. J. 509,212 (1998), arXiv:astro-ph/9807150 [astro-ph].

[42] I. V. Moskalenko and A. W. Strong, Astrophys. J. 493,694 (1998), arXiv:astro-ph/9710124 [astro-ph].

[43] M. S. Potgieter, E. E. Vos, R. Munini, M. Boezio, andV. Di Felice, Astrophys. J. 810, 141 (2015).

[44] S. Abdollahi et al. (Fermi-LAT), Phys. Rev. D95, 082007(2017), arXiv:1704.07195 [astro-ph.HE].

[45] J. Kopp, V. Niro, T. Schwetz, and J. Zupan, Phys. Rev.D80, 083502 (2009), arXiv:0907.3159 [hep-ph].

[46] Z. X. Wang and D. R. Guo, Special Functions (WorldScientific, 1989).

[47] T. Doke, A. Hitachi, J. Kikuchi, K. Masuda, H. Okada,and E. Shibamura, Jap. J. Appl. Phys. 41, 1538 (2002).

[48] J. Thomas and D. A. Imel, Phys. Rev. A36, 614 (1987).[49] P. Sorensen and C. E. Dahl, Phys. Rev. D83, 063501

(2011), arXiv:1101.6080 [astro-ph.IM].

6

SUPPLEMENTAL MATERIALS

The supplemental materials provide additional details to various results presented in the main text. Some of theresults can be applied to other light DM models.

CALCULATION OF CR ELECTRON FLUX

In order to obtain accurate DM recoil flux, the reliable inputs of electron CR flux are in order. The observed CRelectron spectrum at the Earth extend many orders of magnitude energy, ranging from GeV to TeV. Such energetic CRelectrons are easy to accelerate a fraction of DM particles to relativistic speeds. The flux of CR electrons is obtainedby solving the diffusion equation with a widely used galactic CR propagation model. The flux is also modulatedperiodically according to the solar activity due to interactions of CR electrons with the heliosphere magnetic field. Asa result, the CR spectrum observed at the Earth is different from the one in the interstellar. Such solar modulation ismore important for low energy CR electrons and is negligible for energy above several GeV. The unmodulated localinterstellar spectra of CR electrons has been measured by Voyager 1 collaboration which covers energy range with2.7− 74 MeV [38]. For high energy CR electrons, AMS-02 [39] and DAMPE [40] measurements cover energy rangesfrom 1 GeV to 4.6 TeV. We use the GALPROPv54 [41, 42] to obtain the best-fit flux for AMS-02 and DAMPE data sets,and combine the best estimation of Voyager 1 data [43]. Corresponding local interstellar spectrum of CR electrons isshown in Fig. 5 with measurements.

10-15

10-10

10-5

100

105

100 101 102 103 104 105 106 107 108

dΦe/

dTC

R (

m-2

s-1

MeV

-1 s

r-1)

TCR (MeV)

CR flux

Voyager (2016)

AMS-02 (2014)

DAMPE (2017)

Fermi-LAT (2017)

10-15

10-10

10-5

100

105

100 101 102 103 104 105 106 107 108

FIG. 5. Local interstellar flux of CR electrons as a function of electron kinetic energy TCR with the data sets from Voyager1 [38], AMS-02 [39] and DAMPE [40] measurements. For completeness, we also present Fermi-LAT [44] measurement.

DERIVATION OF THE CR-DM DIFFERENTIAL SCATTERING CROSS SECTION

In the CR-DM scattering, the initial DM particles are treated as being at rest since their typical velocities (v ∼ 10−3)are negligible compare to the velocities of incoming CR electrons. The recoil energy of DM for a given CR kineticenergy TCR can be calculated from standard relativistic kinematics of 2-body scattering process [23] and are given as

Tχ = Tmaxχ

(1− cos θCM)

2, Tmax

χ =2mχTCR(TCR + 2me)

(me +mχ)2 + 2TCRmχ, (14)

where θCM is the center-of-mass scattering angle. From above equation, θCM and Tχ are related as

d cos θCM

dTχ= − 2

Tmaxχ

, (15)

which allows us to translate the variable in differential cross section from solid angle dΩ to DM kinetic energy dTχ via

dσχedTχ

=dσχedΩ· dΩ

dTχ=|M|216πs

1

Tmaxχ

, (16)

7

where |M|2 = 14

∑spins |M|2 is the squared DM-electron scattering matrix element, averaged over initial and summed

over final spin states. Using Eq. (16) and expressions of Mandelstam variables s = (mχ +me)2 + 2mχTCR ,

t = −2mχTχ = −q2 ,u = (mχ −me)

2 − 2mχ(TCR − Tχ) ,(17)

one can drive formula of dσχe/dTχ for a given interaction. As below, we list expressions of dσχe/dTχ for some typicalinteractions, which are widely used in light DM model:

• Scalar interaction: L ⊃ gχχχφ+ gSMffφ ,

|M|2 = g2χg

2SM

4mχ (2mχ + Tχ)(2m2

e +mχTχ)(

2mχTχ +m2φ

)2 , (18)

dσχedTχ

= g2χg

2SM

(2mχ + Tχ)(2m2

e +mχTχ)

8π (2meTCR + T 2CR)

(2mχTχ +m2

φ

)2 . (19)

• Vector interaction: L ⊃ gχχγµχA′µ + gSMfγµfA′µ ,

|M|2 = g2χg

2SM

8mχ

(2mχ (me + TCR)

2 − Tχ(

(me +mχ)2

+ 2mχTCR

)+mχT

2χ

)(2mχTχ +m2

A′)2 , (20)

dσχedTχ

= g2χg

2SM

2mχ (me + TCR)2 − Tχ

((me +mχ)

2+ 2mχTCR

)+mχT

2χ

4π (2meTCR + T 2CR) (2mχTχ +m2

A′)2 . (21)

• Axial-vector interaction: L ⊃ gχχγµγ5χA′µ + gSMfγµγ5fA′µ ,

|M|2 = g2χg

2SM

8mχ

(2mχ

((me + TCR)

2+ 2m2

e

)+ Tχ

((me −mχ)

2 − 2mχTCR

)+mχT

2χ

)(2mχTχ +m2

A′)2 , (22)

dσχedTχ

= g2χg

2SM

2mχ

((me + TCR)

2+ 2m2

e

)+ Tχ

((me −mχ)

2 − 2mχTCR

)+mχT

2χ

4π (2meTCR + T 2CR) (2mχTχ +m2

A′)2 . (23)

For the purpose of this paper, we concentrate on vector interaction, while the limits for other interactions canbe obtained in a straightforward way by using our calculation procedures. The DM-electron elastic scattering crosssection is conventionally normalized to σe with following definitions [3]:

|Mfree|2 = |Mfree (αme) |2 × |FDM (q)|2 , (24)

σe =µ2χe|Mfree (αme) |2

16πm2χm

2e

, (25)

where µχe is the DM-electron reduced mass, Mfree (αme) is corresponding matrix element for momentum transfer atreference value q = |q| = αme. The DM form factor, FDM (q), encapsulates all remaining energy dependence of theinteraction. With the notation of Eq. (25), the DM-electron reference cross section for benchmark model in Eq. (1)is given by

|Mfree (αme) |2 =16g2

χg2SMm

2em

2χ

(α2m2e +m2

A′)2 , (26)

σe =g2χg

2SMµ

2χe

π (α2m2e +m2

A′)2 . (27)

8

Combining Eqs. (21) and (27) then gives expression of dσχe/dTχ in Eq. (2)

dσχedTχ

= σe

(α2m2

e +m2A′

)2µ2χe

2mχ (me + TCR)2 − Tχ

((me +mχ)

2+ 2mχTCR

)+mχT

2χ

4 (2meTCR + T 2CR) (2mχTχ +m2

A′)2

' σe

2mχ(me+TCR)2−Tχ((me+mχ)2+2mχTCR)+mχT

2χ

4µ2χe(2meTCR+T 2

CR), heavy A′

α4m4e

16m2χT

2χ

2mχ(me+TCR)2−Tχ((me+mχ)2+2mχTCR)+mχT2χ

µ2χe(2meTCR+T 2

CR), ultralight A′

. (28)

Finally, from Eqs. (16) and (25), one can easily drive dσχe/dTχ corresponding to constant scattering cross section

(|M|2/(16πs) ≡ σe) and constant matrix element. Which are respectively given as

dσχedTχ

= σe

1

Tmaxχ

, constantσχe(mχ+me)

2

(mχ+me)2+2mχTCR

1Tmaxχ

, constant |M|2. (29)

Given differential cross section in Eqs. (28) and (29), we can calculate DM recoil flux as a function of DM kineticenergy according to Eq. (4). In Fig. 6, in additional to the mχ = 1 keV recoil flux in the main text, we also presentDM recoil fluxes for mχ = 1 eV, 10 eV and 0.1 MeV.

10-10

10-5

100

105

1010

10-4 10-3 10-2 10-1 100 101 102 103

mχ = 1 eV

dΦχ/

dTχ

(MeV

-1cm

-2s-1

)

Tχ (MeV)

Const |M|2

Const σχe

10-10

10-5

100

105

1010

10-4 10-3 10-2 10-1 100 101 102 103-4

-3

-2

-1

0

1

2

Log 1

0 (m

V/M

eV)

10-10

10-5

100

105

1010

10-4 10-3 10-2 10-1 100 101 102 103

mχ = 10 eV

dΦχ/

dTχ

(MeV

-1cm

-2s-1

)

Tχ (MeV)

Const |M|2

Const σχe

10-10

10-5

100

105

1010

10-4 10-3 10-2 10-1 100 101 102 103-4

-3

-2

-1

0

1

2

Log 1

0 (m

V/M

eV)

10-10

10-5

100

105

1010

10-4 10-3 10-2 10-1 100 101 102 103

mχ = 1 keV

dΦχ/

dTχ

(MeV

-1cm

-2s-1

)

Tχ (MeV)

Const |M|2

Const σχe

10-10

10-5

100

105

1010

10-4 10-3 10-2 10-1 100 101 102 103-4

-3

-2

-1

0

1

2

Log 1

0 (m

A’/M

eV)

10-10

10-5

100

105

1010

10-4 10-3 10-2 10-1 100 101 102 103

mχ = 0.1 MeV

dΦχ/

dTχ

(MeV

-1cm

-2s-1

)

Tχ (MeV)

Const |M|2

Const σχe

10-10

10-5

100

105

1010

10-4 10-3 10-2 10-1 100 101 102 103-4

-3

-2

-1

0

1

2

Log 1

0 (m

V/M

eV)

FIG. 6. DM recoil fluxes for benchmark DM masses mχ = 1 eV,10 eV, 1 keV and 0.1 MeV with varying mediator mass mA′ .

DERIVATION OF THE DM-ELECTRON SCATTERING CROSS SECTION

The cross section of DM particle scattering with electron in a bound state can be derived in a standard way usingquantum field theory. In the derivation, one conventionally treats the electron is bounded in a static background

9

potential, which means that the recoiling of atoms is neglected. Under such approximation, the cross section forelastic 2→ 2 scattering process χ(p) + e(k)→ χ (p′) + e (k′) is given by

dσ =|Mfree|2vχe

1

2k02p0(2π)4δ4 (k + p− k′ − p′) d3p′

(2π)32p′0

d3k′

(2π)32k′0

=|Mfree|2vχe

1

64π2EχE′χEeE′e

1

(2π)3δ (∆Eχ −∆Ee)

[(2π)3δ3

(k − k′ + q

)]d3qd3k′ , (30)

where vχe is the relative velocity of incoming DM and electron, q = p − p′ is the momentum transfer from DM toelectron. ∆Eχ is the amount of energy lost by DM in the scattering. Notice that for the initial state is boundedelectron, one just need to take replacement (2π)3δ3

(k − k′ + q

)→ |fi→k′(q)|2 in Eq. (30). The atomic form factor,

fi→k′(q) =√V∫d3rψi(r)ψ∗k′(r)eiq·r, accounts for transition from initial to final electron states, and V is the volume

for wavefunction normalization. To understand the consistency of such replacement, notice that for both initialand final states are free electrons, such atomic form factor reduces to fi→k′(q) = (2π)3δ3

(k − k′ + q

). Here we

have included the normalization of the wavefuctions in terms of the volume V , and used the large volume limit(2π)3δ3(0)/V → 1. Then for the ionization process χ+A→ χ+A+ + e− in the (n, l) atomic shell, Eq. (30) is recastas

dσ =|Mfree|2vχe

1

64π2EχE′χEeE′e

1

(2π)3δ (∆Eχ −∆Ee) |fnl(q)|2 d3qd3k′. (31)

Here both initial bounded and recoil electron are non-relativistic, but incoming DM particle could be relativistic ingeneral. The initial bounded electron and recoil electron respectively have energy Ee = me−|EnlB | and E′e = me+ER,with ER, |EnlB | me. One can thus take replacement EeE

′e ' m2

e in Eq. (31). The deposit energy in electron, ∆Ee,is determined by energy conservation ∆Ee = ∆Eχ with

∆Eχ = Eχ − E′χ = mχ

(√1 +

p2

m2χ

−

√1 +

p2 + q2 − 2pq cos θ

m2χ

), (32)

∆Ee = E′e − Ee = |EnlB |+ ER , (33)

where q = |q|, p = |p|. Applying the definitions in Eqs. (24) and (25), one can simplify Eq. (31) to

dσvχe =σe4π

m2χ

µ2χe

|FDM (q)|2

EχE′χδ (∆Eχ −∆Ee)

1

(2π)3|fnl(q)|2 d3qd3k′. (34)

In order to express differential cross section with respect to electron recoil energy ER, using the relation d3k′ =12k′3 d lnER dΩk

′ , and rewrite δ-function as

δ (∆Eχ −∆Ee) =E′χ

pq sin θδ(θ). (35)

Then by taking derivative of ∆Eχ in Eq. (33) with respect to θ, Eq. (34) is recast to the expected form

dσvχed lnER

=σe

8µ2χe

∫qdq |FDM (q)|2

2k′3

(2π)3

∑deg

|fnl(q)|2( m2

χ

pEχ

). (36)

Integrated with the incoming flux of boosted DM dΦχ/dTχ, we finally obtain the velocity averaged differentialionization cross section

d〈σnlionv〉d lnER

=σe

8µ2χe

∫qdq |FDM (q)|2

∣∣fnlion(k′, q)∣∣2 η (Emin) . (37)

Here the DM form factor FDM (q) is evaluated by inverting matrix element in Eq. (20)with applying Eqs. (24) and(26), which reads

|FDM (q)|2 =

(α2m2

e +m2A′

)2(2meER +m2

A′)2

2me (mχ + Tχ)2 − ER

((mχ +me)

2+ 2meTχ

)+meE

2R

2mem2χ

'

2me(mχ+Tχ)2−ER((mχ+me)

2+2meTχ)+meE2R

2mem2χ

, heavy A′

α4m4e

8m2eE

2R

2me(mχ+Tχ)2−ER((mχ+me)2+2meTχ)+meE

2R

mem2χ

, ultralight A′. (38)

10

It is easy to verify that |FDM | is reduced to conventional expression |FDM (q)|2 = ((α2m2e + m2

A′)/(q2 + m2A′))2 in

non-relativistic limit, e.g., Tχ, ER me.The generalized η function is given by

η (Emin) =

∫Emin

dEχΦ−1halo

m2χ

pEχ

dΦχdEχ

, (39)

where Φhalo ≡ nχv is the background DM flux in the Galactic halo. Emin is the minimum incoming DM energy toproduce an electron with recoil energy ER, which is determined by energy conservation ∆Eχ = ∆Ee when p and qare parallel (cos θ = 1) and

pmin =q

2 (1−∆E2e/q

2)

(1− ∆E2

e

q2+

∆Eeq

√(1− ∆E2

e

q2

)(1 +

4m2χ

q2− ∆E2

e

q2

)). (40)

Notice that the flux is related to the velocity distribution f(v) with dΦχ(v) = nχ|v|f(v)d3v. Eq. (39) can be expressedas standard form

η (Emin) =

∫Emin

(1

nχv

)m2χ

vE2χ

nχvf(v)d3v

=

∫Emin

m2χ

vE2χ

f(v)d3v . (41)

Similarly, in the non-relativistic limit, one has

pmin ' q

2

(1 +

∆Eeq

2mχ

q

)=q

2+mχ∆Ee

q, (42)

vmin =pmin

mχ=

q

2mχ+

∆Eeq

. (43)

Equation (41) reduces to standard mean inverse speed function η(vmin) =∫vmin

1vf(v)d3v.

Finally, the atomic ionization form factor∣∣fnlion(k′, q)

∣∣2 is defined as

∣∣fnlion(k′, q)∣∣2 ≡ 2k′3

(2π)3

∑deg

|fnl(q)|2 , (44)

where fnl(q) is the atomic form factor for (n, l) electron shell. For our interested case, the final electron state is alwaysionizaed thus can be taken as a free wavefunction with momentum k′ =

√2meER. In this case, fnl(q) is simplified to∑

deg

|fnl(q)|2 =∑deg

∣∣〈k′|eiq·r|nlm〉∣∣2 =∑deg

∣∣∣∣∫ d3re−ik·rψnlm(r)

∣∣∣∣2=∑deg

∣∣∣χnl(k)Ylm(k)∣∣∣2 , (45)

where we have used the definition of momentum space wavefunction of the initial bounded electron ψnlm(k) =∫d3rψnlm(r)e−ik·r ≡ χnl(k)Ylm(k), with the normalization

∫d3k |ψnlm(k)|2 = (2π)3. χnl(k) is the radial

wavefunction in momentum space, and Ylm(k) is the spherical harmonic function which accounts for angular part ofthe wavefunction. Writing the sum of degenerate states explicitly, we arrive at

∑deg

|fnl(q)|2 = 2

∫dΩk

l∑m=−l

∣∣∣χnl(k)Ylm

(k)∣∣∣2 , (46)

where factor 2 takes account of electron spin. Applying the property of harmonics function

l∑m=−l

∣∣∣Ylm (k)∣∣∣2 =2l + 1

4π, (47)

11

and change the integration variable to initial electron momentum k by using sin θdθ = kdk/(k′q), we obtain theexpression of atomic ionization form factor in the literature [3]∣∣fnlion(k′, q)

∣∣2 =2k′3

(2π)3

(2l + 1

2π

∫dΩk |χnl(k)|2

)=

2k′3(2l + 1)

(2π)3

∫sin θdθ

∣∣∣χnl (√k′2 + q2 − 2k′q cos θ)∣∣∣2

=(2l + 1)k′2

4π3q

∫ |k′+q||k′−q|

kdk |χnl(k)|2 . (48)

CALCULATION OF THE RADIAL ROOTHAAN-HARTREE-FOCK WAVEFUNCTION

We here give the detailed computation of the momentum space radial wave function χnl(p) for DM-electron elasticscattering, which is used to calculate atomic ionization form factor. χnl(p) is obtained by splitting the coordinatespace wavefunction ψnlm(x) into its radial part Rnl(r) and its angular part Ylm(θ, φ), the exact expression is givenby [45]

χnl(p) =4π

2l + 1

∑m

ψnlm(p)Ylm (θp, φp)

= 4πil∫drr2Rnl(r)jl(pr). (49)

Here, p is a momentum space vector with arbitrary orientation (θp, φp), and p = |p|. Pl(cos θ) is a Legendre polynomial.To obtain above result, we have used the orthogonality of the spherical harmonics∫ π

0

∫ 2π

0

Y lm(θ, φ)Y l′m′(θ, φ) sin θdθdϕ = δll′δmm′ , (50)

and the Gegenbauer formula

jl(pr) =(−i)l

2

∫ π

0

−d(cos θ)Pl(cos θ)eipr cos θ , (51)

which expresses the spherical Bessel function jl(x) with Fourier type integration over Legendre polynomial. In theRHF method, the radial wavefunctions Rnl(r) is approximated by a linear combination of Slater-type orbitals [28]:

Rnl(r) =∑k

Cnlk(2Zlk)

nlk+1/2

a3/20

√(2nlk)!

(r/a0)nlk−1

exp

(−Zlkr

a0

), (52)

where a0 is the Bohr radius, and the values of coefficients Cnlk, Zlk and nlk are provided in Ref. [28]. Then χnl(p)can be expressed as

χnl(p) = 4πil∑k

Cnlk(2Zlk)

nlk+1/2√(2nlk)!

a1−nlk−3/20

∫ ∞0

dr rnlk+1 e−Zlk/a0 jl(pr) . (53)

Applying the Hankel transform formula [46]∫ ∞0

e−at Jν(bt) tµ−1dt =Γ(µ+ ν)

aµ+νΓ(ν + 1)

(b

2

)ν2F1

[µ+ ν

2,µ+ ν + 1

2, ν + 1,− b

2

a2

], (54)

with 2F1 (a, b, c, x) being the hypergeometric function, Jν(x) the Bessel function of the first kind and jl(x) =√π2xJν+ 1

2(x), we can evaluate Eq. (49) analytically, which yields

χnl(p) =∑k

Cnlk2nlk−l(

2πa0

Zlk

)3/2(ipa0

Zlk

)lΓ (nlk + l + 2)

Γ(l + 32 )√

(2nlk)!

× 2F1

[1

2(nlk + l + 2) ,

1

2(nlk + l + 3) , l +

3

2,−(pa0

Zlk

)2]. (55)

12

We notice that Eq. (55) has a slightly different expression from Eq. (C3) in Ref. [45], which leads to a small differenceof χnl(p) value especially for high l. As a crosscheck, we have performed full numerical integration to Eq. (53) forsample points and found a good agreement with our analytical result.

MODELING OF THE ELECTRON AND PHOTONELECTRON YIELDS

We provide additional details to convert the recoiling electron’s recoil energy into a specific number of electrons.Our modeling procedure is closely follow Refs. [10, 11]. A primary electron with deposit energy ∆Ee = ER + |EnlB |can produce ne observable electrons, nγ unobservable scintillation photons and heat. The relevant quantities satisfyfollowing relations

ER = (nγ + ne)W ,

nγ = Nex + fRNi ,

ne = (1− fR)Ni . (56)

Here W = 13.8 eV is the average energy required to produce a single quanta (photon or electron), Ni and Nex arecorresponding numbers of ions and excited atoms created by ER and follow Nex/Ni ' 0.2 [47] at energies above akeV. fR is the fraction of ions that can recombine, and we assume fR = 0 at low energy [48]. This then implies thatne = Ni and nγ = Nex, and the fraction of initial quanta observed as electrons is given by [49]

fe =ne

ne + nγ=

1− fR1 +Nex/Ni

' 0.83. (57)

Furthermore, we assume that the photons associated with the de-excitation of the next-to-outer shells can

photoionize to create an additional n(2)Q quanta, which is listed in Table I for full Xenon electron shells. While

in the calculation, we only consider contributions from (5p6, 5s2, 4d10, 4p6, 4s2) shells. The total number of electrons

is given by ne = n(1)e + n

(2)e , where n

(1)e is the primary electron and n(2) are the secondary electrons produced. n(1)

equals to 0 and 1 with probability fR and 1−fR respectively, and n(2)e follows a binomial distribution with n

(1)Q +n

(2)Q

trials and success probability fe. As an example, in Fig. 7 we plot differential rate dN/dne as a function of numberof electrons ne for both ultralight and heavy mediator cases.

Shell 5p6 5s2 4d10 4p6 4s2 3d10 3p6 3s2 2p6 2s2 1s2

|EnlB | [eV] 12.4 25.7 75.6 163.5 213.8 710.7 958.4 1093.2 4837.7 5152.2 33317.6

n(2)Q 0 0 4 6-9 3-14 36-50 17-68 9-78 271-349 22-372 2040-2431

TABLE I. Binding energy and number of additional quanta for full Xenon electron shells.

13

1010

1011

1012

1013

1014

1015

1016

1017

1018

5 10 15 20 25 30 35 40 45 50

mχ = 10 eV

dN/d

n e

ne

5s

4s

5p

4p

4d

total

1010

1011

1012

1013

1014

1015

1016

1017

1018

5 10 15 20 25 30 35 40 45 50

(a)Ultralight mediator, mχ = 10 eV

10-2

10-1

100

101

102

103

104

105

106

107

5 10 15 20 25 30 35 40 45 50

mχ = 1 keV

dN/d

n e

ne

5s

4s

5p

4p

4d

total

10-2

10-1

100

101

102

103

104

105

106

107

5 10 15 20 25 30 35 40 45 50

(b)Ultralight mediator, mχ = 1 keV

106

108

1010

1012

1014

1016

5 10 15 20 25 30 35 40 45 50

mχ = 10 eV

dN/d

n e

ne

5s

4s

5p

4p

4d

total

106

108

1010

1012

1014

1016

5 10 15 20 25 30 35 40 45 50

(c)Heavy mediator, mχ = 10 eV

100

101

102

103

104

105

106

107

108

109

5 10 15 20 25 30 35 40 45 50

mχ = 1 keV

dN/d

n e

ne

5s

4s

5p

4p

4d

total

100

101

102

103

104

105

106

107

108

109

5 10 15 20 25 30 35 40 45 50

(d)Heavy mediator, mχ = 1 keV

FIG. 7. Differential rate dN/dne versus number of electrons ne for mχ = 10 eV and 1 keV, where top (bottom) panelcorresponding to ultralight (heavy) mediator cases. The colored lines present the contributions from various xenon shells andthe black lines show total contributions.