Cosmic ray e+e− spectrum excess and peak feature observed by the DAMPEexperiment from dark matter

Hong-Bo Jin,1, 2 Bin Yue∗,1, 2 Xin Zhang,1, 2 and Xuelei Chen†1, 2, 3

1Key Laboratory of Computational Astrophysics, National Astronomical Observatories,Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, 100012, China

2School of Astronomy and Space Science, University of Chinese Academy of Sciences3Center of High Energy Physics, Peking University, Beijing 100871, China

(Dated: January 12, 2018)

The Chinese satellite Wukong, also known as the DArk Matter Particle Explorer (DAMPE)experiment, has released its observation data of the cosmic ray (CR) electrons and positrons. Thedata shows an excess in the energy spectrum up to TeV energy, and possibly also a peak-like finestructure at ∼1.4 TeV. We investigate the scenario that the source of the excess comes from theannihilation or decay of dark matter (DM) particles. We consider the W+W− channel and directe+e− channel (model A), and the double τ+τ− channel and direct e+e− channel (model B). Wefind that the annihilation or decay of dark matter particles in the galactic halo can give excellentfits to the broad excess. However, the annihilation cross section is of the order of 10−23 cm3 s−1,larger than required for obtaining the correct relic abundance when dark matter freeze out. Wethen study whether the narrow peak at ∼1.4 TeV could be explained by a nearby subhalo, whichthanks to the smaller distance, could supply e+e− within a narrow energy range. We find that inorder to produce a peak width less than the DAMPE energy bin width (0.2 TeV), the source mustbe located within r <∼ 0.6 kpc. Our global fit models do not produce the peak-like feature, insteadat ∼1.4 TeV the spectrum show either a slope or a cliff-like feature. However, if less than optimalfit to the data is allowed, the peak-like feature could be generated. Furthermore, an excellent fitwith peak could be obtained with model B if the background is rescaled. If the dark matter decayand annihilation rates are determined using the broad excess, the required subhalo mass could be∼ 105 M� for decay model with DM particle lifetime 7.3 × 1025 s, or ∼ 104.5 M� for annihilationmodel with cross-section 10−23cm3s−1 and a shallower density profile slope α = 1.2, or ∼ 102.5 M�if the density profile slope is a steep as α = 1.7.

PACS numbers:

I. INTRODUCTION

According to recent astronomical observations, about26% of the total cosmic density is made up by non-baryonic dark matter (DM), much more than the 4.7%baryonic matter we know of [1]. The DM plays a crucialrole in large scale structure growth and galaxy formation,but its presence is only inferred from its gravitational ef-fects, while its nature is still unknown. A simple andplausible conjecture is that the DM are made up of un-known particles, which are electromagnetically neutralbut participate in weak interactions. Such DM particlesmay annihilate or decay into standard model particles.In the indirect search of weakly massive interacting par-ticle (WIMP) dark matter, one looks for the signatureof dark matter by observing γ-ray, energetic neutrino orcharged cosmic ray (CR) particles as annihilation or de-cay products (for a review see e.g. [2]). As these are alsoproduced by other astrophysical processes, which are notalways well understood, it is necessary to look for distinctsignatures from the dark matter annihilation/decay pro-cesses. In the case of cosmic ray, most CR particles follow

∗Co-first author†Corresponding author

power law distributions, which could be nicely explainedas the result of diffusive shock acceleration by supernovaeremnants (SNR), followed by a complicated transporta-tion process in the Galaxy [3]. On the other hand, thedark matter particle has a specific mass, so the energyof particles produced in dark matter annihilation or de-cay would be distributed in a particular energy range,which may produce an excess or deviation from the sim-ple power law in the CR energy spectrum. In particular,if the dark matter particle could annihilate or decay di-rectly to a pair of standard model particles, this mayproduce a narrow line feature, which would be a smok-ing gun signature of dark matter, as there is no otherknown mechanism to produce such narrow line featurein the multi-GeV energy range.

The cosmic ray electrons and positrons have been mea-sured by a number of balloon or space-borne experi-ments. The HEAT[4], ATIC [5], PAMELA [6], Fermi[7, 8], and AMS-02 [9] have measured the electron andpositron spectrum up to 2 TeV. An intriguing excessabove a few tens GeV were found by these experiments,though the uncertainty remains large at the high energyend where the CR flux drops. In addition, the groundbased HESS experiment observes the e+e− spectrum in-directly at higher energies [10]. From joint analysis ofthese experiment, a possible break in the energy spec-trum was found around TeV scale, though in such analy-

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sis the systematic uncertainty is sizable [11, 12]. Such anexcess could be generated by dark matter decay or anni-hilations (e.g. [13–15]), but to obtain a large excess, thedark matter annihilation rate must be higher than usu-ally assumed for achieving the correct abundance duringits freeze out in the Early Universe, or are from a rela-tively nearby source such as a subhalo [10, 16–22]. Alter-natively, the excess could be produced by some energeticastrophysical processes such as pulsars or SNRs whichinject energetic positrons or electrons in the CR energyspectrum [23–29].

The DAMPE satellite has made a new measurementof the electron and positron spectrum (the electrons andpositrons are not distinguishable in its observation). Thissatellite is designed to have low background contamina-tion from the much greater proton component of the cos-mic ray, and have high energy resolution so that sharpfeature in the cosmic ray electron and positron spectrumwould not be erased by energy resolution issue [30]. Re-cently, the DAMPE experiment has released their mea-surement from 25 GeV to 4.6 TeV [31]. Their energyspectrum is consistent but higher than those from theAMS-02. A break in the energy spectrum is indeed foundat about 1 TeV, confirming the earlier result from HESSexperiment, and provides a very precise measurement ofCR electron positron energy spectrum at the TeV energyrange. These results raises new interest on the dark mat-ter contribution to the cosmic ray electron and positronspectrum.

In addition to the break, there is also a single peak fea-ture at 1.4 TeV, with a statistical significance of ∼ 3σ.We should take a cautionary note here: the number ofactual events detected in this energy bin is only 93, whilefor the two adjacent bins the numbers are 74 and 33 re-spectively, so it is possible that the large peak is due toa statistical fluctuation. This would become more clearin the future as more data is accumulated. Nevertheless,the peak is very intriguing, because in sources of astro-physical origin, such as SNR or pulsar, it is not easy togenerate particles with a single energy. So if the peak isreal, it would be of great importance to the indirect darkmatter search. Indeed, as we shall see below, even inthe dark matter scenario it is not easy to produce a nar-row peak in the charged particle spectrum, because theelectrons and positrons produced in dark matter decayor annihilation will diffuse into broad energy distribu-tions during their propagation. So to generate a narrowpeak in the spectrum, the source must be located at avery small distance where the diffusion does not take longtime. One such possibility is a dark matter subhalo whichis accidentally located near the Solar system.

In this paper, we shall investigate the scenario that thedark matter annihilation or decay serve as possible sourceof the excess cosmic ray electrons and positrons in light ofthe new DAMPE data. The paper is organized as follows:In Sec. II we describe our method for computing theCR electron positron spectrum. In Sec. III we presentthe results from fitting the broad spectral excess with

the galactic halo dark matter contribution. In Sec. IV,we study whether the peak feature at 1.4 TeV could begenerated by a nearby subhalo. In Sec. V we investigatewhat is the required mass for such a subhalo, and theconstraint on the model from inverse Compton scatteredphotons and CR anisotropy. We conclude in Sec. VI.

II. METHODS

The propagation of charged CR particles in the Galaxycan be described as a diffusion process, with interactionwith other particles, and energy loss by radiation andcollision, as well as re-acceleration in the turbulent mag-netic field [32]. To quantitatively model this process,the GALPROP software package has been developed, whichsolves the propagation equation with a Crank-Nicholsonimplicit second-order scheme [33]. The primary electronsare mostly produced in SNRs. Their injection spectrumis modeled as a simple power-law form, as is expectedfrom the diffuse shock acceleration mechanism [34]. Inthe propagation models, the source item of the primaryelectrons is also often described as a broken power lawspectrum multiplied by the assumed spatial distributiondescribed in the cylindrical coordinate [33]. The densityof CR electron source is modeled as an exponential disc.The secondary electrons and positrons are produced dur-ing collisions of CR nucleons (protons dominant) with theinterstellar gas [35]. The distribution of the interstellargas, which collide with CR particles and generate sec-ondary particles, is derived from the observational dataof H and CO, etc. [36, 37]. The CR propagation timecan be constrained observationally by the ratio betweenprimary particles and the secondary particles, such asthe Be10/Be9 and B/C ratios in the cosmic ray. Alter-natively, with the advent of high quality data from theAMS-02 experiment, the CR model parameters may alsobe determined solely from the proton[38] and B/C[39]data of AMS-02, as is demonstrated first in Ref. [40],which yield tighter constraints. In this paper, we shallalso follow this approach.

If some CR particles are produced by dark matter anni-hilation or decay, their propagation process are the same,only the injection source distribution and energy spec-trum differ. For DM annihilations, we model the sourceterm as

q(r, p) =ρ(r)2

2m2χ

〈σv〉∑X

ηXdN (X)

dp, (1)

where 〈σv〉 is the velocity-averaged DM annihilationcross section multiplied by DM relative velocity (referredto as cross-section), ρ(r) is the DM energy density dis-tribution function, and dN (X)/dp is the injection energyspectrum of antiprotons from DM annihilating into stan-dard model (SM) final states through all possible inter-mediate states X with ηX the corresponding branchingfractions. Similarly, the source term for decay can be

3

modeled as

q(r, p) =ρ(r)

mχΓχ

∑X

ηXdN (X)

dp, (2)

where Γχ is the decay rate. The density profile ρ(r) ofthe dark matter halos were derived in N-body simula-tion, such as NFW [41], Isothermal [42], and Moore [43].In this paper we use the Einasto [44] profile to modelthe dark matter distribution. The energy spectrum ofelectrons and positrons of each channel is calculated byPYTHIA v8.175 [45]. The hadronic model interactionpart of the code is also modified to improve the treat-ment of secondary particles [46].

In the dark matter particle annihilation or decay, theelectrons and positrons can be produced directly, herewe shall call this the e+e− channel. Note that the an-nihilation or decay may produce an e+e− pair, or eitheran e+ or an e− plus other particles, but on the wholeproduce equal numbers of e+ and e−. For our inves-tigation these differences do not matter, though in theformer case the energy of the electron or positron wouldbe exactly mχ(annihilation) or mχ/2 (decay), and theprimary energy spectrum of electrons and positrons is avery narrow peak around E0 = mχ/2 (decay) or E0 = mχ

(annihilation) for the pair production. The electrons andpositrons may also be produced by decay or annihilationinto intermediate particles, such as the W+W− pairs orτ+τ− paris, which further decays to produce electronsand positrons. In these cases the energy spectrum is gen-erally more broadly distributed. In this paper we shallconsider 1) model A: electrons and positrons are pro-duced through the e+e− channel and the W+W− chan-nel, and 2) model B: through the e+e− channel and thedouble τ+τ− channel. In previous analysis, it has beenfound that in fitting the broad spectrum excess measuredby the AMS-02 experiment, the other channels usuallydo not yield as good fit as the ones listed above [46].The branching ratio of e+e− channel to W+W− channel(double τ+τ− channel), f , is treated a free parameter toexplore the favored final state.

According to their origins, we divide the CR electronsand positrons into three components : 1) an astrophys-ical background, this is the primary electrons from dis-tant astrophysical sources and the secondary electronsand positrons generated when these primary electronspropagate in the space between sources and observers; 2)a diffuse dark matter component, including the electronsand positrons produced by the annihilation or decay ofthe dark matter particles in the galactic halo and pos-sible distant subhalos, and the secondary electrons andpositrons generated during their propagation; 3) a com-ponent from a nearby subhalo, which may accidentally tobe located at a very close distance so that it significantlyalter the CR energy spectrum.

A. The broad spectrum

Precise measurements of CR electrons and positronswere reported by the AMS-02 experiment [47]. Theyfound that in the measured flux of electrons pluspositrons, the positron fraction reaches the maximumof 15.9% at 305 GeV [48]. There is a positron excessagainst the astrophysical background, which is usuallyinterpreted as produced by either pulsars or dark matterannihilation or decay [40, 49–52].

If equal amount of positrons and electrons at the sameenergy are produced by the additional sources such aspulsars, dark matter annihilation or decay, then by sub-tracting the same amount of electrons as positrons fromthe total spectrum, one can obtain the contribution ofthe astrophysical primary electrons. In Ref. [53], the pri-mary electron background is extracted from the AMS-02experiment, and it indeed follows a power law spectrumas is expected for electrons of the astrophysical origin.This knowledge of primary electrons in turn helps us toimprove the understanding of the additional sources.

In a recent work [46], it is found that the CR elec-trons and positrons background as measured by AMS-02 agrees well with the prediction by a convection andre-acceleration diffusion (DCR) model using the CR pro-tons, positrons and B/C data of the same experiment.Based on the two-peaks feature of the CR electrons andpositrons background spectrum, they further pointed outthat the positron excess in AMS-02 experiment could beinterpreted by diffuse dark matter annihilation, and themost relevant channels are the W+W−( in the bosonsand quarks) and double τ+τ− (in the leptons) annihila-tion channels.

The dark matter profile is chosen as Einasto [44], whichis described approximately by a power-law density distri-bution:

ρ(r) = ρ� exp

[−(

2

αE

)(rαE − rαE�

rαEs

)], (3)

with αE ≈ 0.17 and rs ≈ 20 kpc. The local DM en-ergy density is fixed at ρ� = 0.43 GeV cm−3[54]. Inthis model there is no nearby sources such as dark mat-ter subhalos, and the CR electrons and positrons comingfrom the dark matter annihilation is a diffuse distribu-tion in the Milky Way. The energy spectrum of electronsand positrons of each channel is calculated by PYTHIAv8.175 [45]. Then based on the above density profile, thefinal energy spectrum after propagation is obtained viathe GALPROP [55].

In this paper, we use the spectrum template generateas [46] for the W+W−, double τ+τ− and e+e− channelsrespectively, as the theoretical models for the broad spec-trum excess. We leave the normalizations and branchingratio as free parameters.

4

B. Nearby source

Even for the dark matter originated scenario, typi-cally the observed cosmic ray electron and positron spec-trum would still be a broad bump rather than a narrowpeak. Charged particles move in curved trajectories inthe stochastic interstellar magnetic field, so their trans-portation in the galactic scale can be described as a dif-fusion process. The TeV electrons and positrons alsolose their energy rapidly by synchrotron radiation andinverse Compton scattering, with a life time of 105 ∼ 106

years. Even if the electrons and positrons are producedat a single energy by dark matter decay/annihilation,the peak would still be broadened quickly as they prop-agate through the space. For the electron and positronspectrum to retain a narrow peak, the source must befairly close by, so that the peak is not greatly dampedby energy loss or momentum space diffusion during thepropagation.

The N-body simulations show that in the galactic darkmatter halo there could be a large number of subhaloswhich were produced during the hierarchical structureformation process [56, 57]. Many of such halos did notform stars so they can not be observed directly (see Ref.[58] for a recent review). So below we compute the thecontribution to the electron and positron spectrum froma nearby subhalo, and see if the peak in the energy spec-trum could be reproduced.

The CR spectrum from a nearby subhalo can be sim-ulated with the GALPROP code. However, as the locationand mass of such a subhalo is unknown, searching theparameter space with such simulation would be compu-tationally expensive. Here, we consider an analytical so-lution of the CR diffusion equation from a nearby source.While this is not as accurate and comprehensive as theGALPROP code, for the nearby source it could provide afast approximation.

We first assume the source produce electrons andpositrons with a single energy E0, from a point source.The propagation equation of primary cosmic ray elec-trons (ignoring secondary ones produced in particle col-lision) can be written as

∂ψ

∂t= ∇ · (Dxx∇ψ − Vcψ) +

∂

∂p[p2Dpp

∂

∂p

ψ

p2]

− ∂

∂p[pψ − p

3(∇ · Vc)ψ] + S (4)

where ψ(r, p, t) is the number density of electrons perunit energy, Dxx and Dpp are the spatial and momentumdiffusion coefficients respectively, Vc the bulk velocity,and S the source distribution. For the present problem,p ≈ E. This is a linear equation, so we may solve thesubhalo contribution separately, and just adding it to thesolution for the whole galaxy.

The diffusion coefficient and other parameters in thecosmic ray propagation model were fitted with other ob-servations [59], Dxx = D0(ρ/ρ0)δ, whereD0 = 6.59×1028

cm2 s−1, ρ = pc/Ze and ρ0 = 4× 103 MV, here we takeδ = 1/3 which is for Kolmogorov spectrum of interstel-lar turbulence. In the TeV energy range, the electronsand positrons lose their energy primarily by synchrotronradiation and inverse Compton scattering,

b =

(dEedt

)syn

+

(dEedt

)ICS

. (5)

For example, at 1.5 TeV, b ≈ 6.8 × 10−13 erg s−1 =4.3 × 10−10GeV s−1. Also, if the source is nearby, thediffusion in momentum space which is due re-acceleration

can be neglected, asDppE ∼ 2.9× 10−17 erg s−1 � b.

The dynamical time scale of the Milky Way is of or-der 108 year, so the time scale for the subhalo to passby is much longer than the life time of the electron andpositrons. We may then assume a steady state for thecosmic ray spectrum is reached, ∂ψ/∂t = 0. We ignorethe bulk velocity Vc. For a point source of monochro-matic electron/positron injection, S = Qδ(r)δ(E − E0),then the propagation equation could be simplified as

Dxx∇2ψ − b ∂∂E

ψ +Qδ(r)δ(E − E0) = 0. (6)

This equation can be solved by making a 3D Fouriertransform over the spatial dimensions and a 1D Laplacetransform over time. If the source is very nearby, wedo not need to consider the global confinement to thegalactic disk by the magnetic field. The solution in thespherically symmetric case is given by

ψ(E0, E, r) =Q

b

exp[− r2

(4πDxx/b)(E0−E) ]

[(4πDxx/b)(E0 − E)]3/2. (7)

Then the observed specific intensity of particles numberflux is J ≈ ψc/(4π).

If the injected electrons and positrons are not of a sin-gle energy but has an energy distribution of dN/dE (e.g.for W+W− channel or double τ+τ− channel), the corre-sponding spectrum could be obtained by

J =cf

4π(1 + f)

∫dE′

dN

dE′G(E′, E, r) (8)

where the Green function G(E′, E, r) is given by Eq. (7).If the annihilation or decay has both direct e+e− channelsand other channels, the result is given by the sum of allthe channels.

We show the electrons and positrons spectrum pro-duced via the different channels in Fig. 1, for model A(top panel) and model B (bottom panel). Here we fixE0 = 1.5 TeV and f = 0.01. In each case we show theresults for r = 0.1 and 1 kpc. The total injection rate Qare adjusted (Q = 1033 and 1036 s−1 for r = 0.1 and 1 kpcrespectively) so that the curves can be show on the sameplot. As we expect, the closer the source, the narrowerthe peak. For r = 0.1 kpc (blue) the peak is very sharp,while for r = 1 kpc (red) the peak is a much broader one.

5

Compared with the direct production case, the electronsand positrons produced via the W+W− or double τ+τ−

channel have broader distributions, as the electrons pro-duced in this channel is not monochromatic. Moreover,the peak is located at higher energy for the r = 0.1 kpc(blue) case than the r = 1 kpc (red) case. If we want togenerate a peak in the spectrum, the direct e+e− chan-nel have to be used, and to achieve the narrow peak withwidth less than ∼0.2 TeV, the source must be locatedwithin r <∼ 0.6 kpc.

200 400 600 800 1000 1200 1400 1600

E [GeV]

10−1

100

101

102

E3J

[GeV

2m−

2s−

1sr−

1]

r = 0.1 kpc

200 400 600 800 1000 1200 1400 1600

E [GeV]

10−1

100

101

102

E3J

[GeV

2m−

2s−

1sr−

1]

r = 0.1 kpc

r = 0.1, Q = 1033, double τ+τ−

r = 0.1, Q = 1033, e+e−

r = 1.0, Q = 1036, double τ+τ−

r = 1.0, Q = 1036, e+e−

FIG. 1: The specific intensity of the electrons and positronsnumber flux from a nearby subhalo with different distances.Top panel is for model A while bottom panel is for model B.

We fit the normalization A of the broad spectrum elec-trons and positrons due to W+W− or double τ+τ− chan-nel and the branching ratio f of the e+e− channel to theW+W− or double τ+τ− channel, the distance r, and theamount of injection Q from the statistics:

χ2(A, f, r,Q) =∑ [〈Jnr

i (f, r,Q)〉+ 〈Jci (A, f)〉 − J iobs]2

σ2stat,i + σ2

sys,i

,

(9)where 〈Jnr

i (f, r,Q)〉 is the mean intensity from the nearbysource in the i-th energy bin; while

Jci (A, f) = AJbr

W/τ + fAJbre + Jbg. (10)

The astrophysical background Jbg, the the broad spec-trum template from the annihilation/decay of Galactic

diffuse DM through the W+W− or double τ+τ− channel,JbrW/τ , and through the e+e− channel, Jbr

e , are all gener-

ated using the algorithm presented in [46]. For simplicitywe fix E0 = 1.5 TeV. Adopting a different value within∼ 0.1 TeV does not change our result.

III. FITTING WITH THE BROAD SPECTRUM

The CR electron spectrum has been measured by anumber of experiments, such as the Veritas [60], HESS[61, 62] and Fermi-LAT [63] before the DAMPE data re-lease. These observations indicate a remarkable break inthe spectrum at TeV scale. However, the electron datafrom the ground based atmospheric Cherenkov telescopessuch as HESS have large uncertainties from the subtrac-tion of hadronic background and discrimination againstgamma rays events [61]. In detail, the very-high-energyflux of HESS electrons is described by an exponentiallycutoff power law with an index of 3.05± 0.02 and a cutoffat 2.1± 0.3 TeV in the range of 700 GeV to 5 TeV [61].The low-energy extension of the HESS electron measure-ment are from 340 GeV to 1.7 TeV with a break energyat about 1 TeV [62].

The TeV break of CR electrons are now confirmed bythe highly precise DAMPE observation. If we assume theprimary electron is described by a power law, while theexcess electrons and positrons have the same spectral dis-tribution, the degeneracy between the background andexcess can be broken, and the origin of the TeV breakis connected to the positron excess [64]. This is con-firmed in [53], where the features of the primary electronspectrum is carefully studied, which showed a power lawprimary electron spectrum without TeV break.

Here, we derive the excess of electrons under the sameassumption that the excess electrons and positrons arethe same, and use the AMS-02 experiment data onpositron fraction for subtraction [47]. We then computethe excess produced by dark matter particle annihilationsin the galactic halo. We consider the annihilation to elec-trons and positrons via the e+e− channel and W+W−

channel (model A) or double τ+τ− channel (model B).In Fig. 2, we show dark matter annihilation model fit

to the DAMPE measured spectrum data. We plot theprimary electrons derived from AMS-02 e− − e+ data,the dark matter contribution, and the dark matter plusbackground total. The top two panels show the model Aresults, while the bottom two show the model B results.We see all these models provide good fit to the data. Themodel A fit is better than model B, at the lower energypart, its spectral shape agrees with the data very well.

In the left panels of Fig.2, we try to fit all data pointsincluding the peak point in the DAMPE data, while inthe right panels we excluded the peak point. For modelA, by excluding the peak point, the fit to the other pointsis improved. For model B there is steep drop of the spec-trum at E0, this cliff-like edge fits the peak point better,however the fitting is worse for the energy just below E0.

6

1 0 1 1 0 2 1 0 3 1 0 4

1 0

1 0 0

E3 Flux(G

eV2 m-2 sr-1 s-1 )

E n e r g y ( G e V )

D A M P E A M S - 0 2 ( e - - e + ) D M D M + b a c k g r o u n d b a c k g r o u n d P r i m a r y e l e c t r o n s

D M - > W + W - ( mχ= 1 . 7 T e V , < σν> = 5 . 8 x 1 0 - 2 3 m 3 / s )

1 0 1 1 0 2 1 0 3 1 0 4

1 0

1 0 0

E3 Flux(G

eV2 m-2 sr-1 s-1 )

E n e r g y ( G e V )

D A M P E A M S - 0 2 ( e - - e + ) p r i m a r y e l e c t r o n s D M b a c k g r o u n d D M + b a c k g o u r n d

D M - > W + W - ( mχ= 1 . 3 T e V , < σν> = 4 . 5 8 x 1 0 - 2 3 m 3 / s )

1 0 1 1 0 2 1 0 3 1 0 4

1 0

1 0 0

E3 Flux(G

eV2 m-2 sr-1 s-1 )

E n e r g y ( G e V )

D A M P E A M S - 0 2 ( e - - e + ) b a c k g r o u n d D M D M + b a c k g r o u n d

D M - > 4 t a u , e + e - ( mχ= 1 . 4 T e V , < σν> = 3 . 3 x 1 0 - 2 3 m 3 / s )

1 0 1 1 0 2 1 0 3 1 0 4

1 0

1 0 0

E3 Flux(G

eV2 m-2 sr-1 s-1 )

E n e r g y ( G e V )

D A M P E A M S - 0 2 ( e - - e + ) D M D M + b a c k g r o u n d b a c k g r o u n d

D M - > 4 t a u , e + e - ( mχ= 1 . 1 7 T e V , < σν> = 2 . 6 7 x 1 0 - 2 3 m 3 / s )

FIG. 2: The primary electrons background derived from (e− − e+) of the AMS-02 data [48] and the total flux of CR electronsand positrons background measured by DAMPE. The curves that fit the data including and excluding the peak point near 1.4TeV are shown in the left and right panel respectively. We write the best-fit DM particle mass mχ and the velocity weightedcross-section 〈σv〉 in each panel.

For model A the mass of the dark matter particle are1.7 TeV and 1.3 TeV respectively for the left and rightpanel, and 1.4 TeV and 1.2 TeV respectively for modelB. The required annihilation cross sections are typically〈σv〉 ∼ 10−23 cm3 s−1, within the constraints derivedfrom the AMS-02 electron data [49]. The cross-sectionvalues are however larger than the one required to getcorrect abundance of WIMPs during the thermal de-coupling. To obtain large abundance, the dark mattermust be produced non-thermally in the early Universe,or the cross-section of its annihilation in the current timemust be somehow enhanced [16, 17]. Moreover, while thecross-section is derived for electrons and positrons pro-duction, we note that these are larger than the obser-vational limits on cross-section of annihilation into pho-tons derived from observations of the dwarf spheroidalgalaxies [65, 66]. The required cross-sections here areone order of magnitude greater than those of VERITAS[65], and three order larger than the one from Fermi-LAT[66]. Although the cross-sections of γ-ray and electronsand positrons could be different, we expect them to be of

the same order, and special mechanism may be needed toachieve the large cross-section to electrons and positronswhile not violating the γ-ray bound.

1 0 1 1 0 2 1 0 3 1 0 4

1 0

1 0 0

E3 Flux(G

eV-2 m-2 sr-1 s-1 )

E n e r g y ( G e V )

D A M P E A M S - 0 2 ( e - - e + ) P r i m a r y e l e c t r o n s b a c k g r o u n d D M D M + b a c k g r o u n d

D M - > W + W - ( mχ= 2 . 9 T e V , τ= 7 . 3 x 1 0 2 5 s )

FIG. 3: Spectrum for dark matter decay via the W+W− case.

7

In the above we have discussed the case of annihila-tions. In the case of decay, except for the mass which isdoubled and the rate is determined from decay life timeinstead of the annihilation cross-section, others are simi-lar. In principle, the dark matter annihilation and decayresults may differ, because the annihilation rate ∝ n2,while the decay rate ∝ n, where n is the number densityof the dark matter particle, so the source distribution isin principle different. Nevertheless, in the end we findthat the results are quite similar in our cases.

In Fig.3 we show the spectrum of the decay in modelA. The spectrum is very close to that of the annihilations,though the energy is doubled with mχ = 2.9 TeV. Thelife time of the decaying particle is τ = 7.3 × 1025 s.As the spectrum for the annihilation and decay are quitesimilar, we shall mainly discuss the annihilations, thoughmost of the results are also applicable to decays.

IV. THE PEAK AND NEARBY SOURCE

In the last section we see that the Galactic halo darkmatter annihilation could provide a reasonably good fitto the DAMPE e+e− spectrum, though it does requirea very large annihilation cross-section. However, there isno peak produced in the spectrum. Here, we tentativelytreat the peak as real, and consider whether such a peakcould be produced in dark matter annihilation or decayfrom a nearby subhalo. To do this, we use the GALPROPcode to compute the primary background and dark mat-ter contribution from the galactic halo, then adding thecontribution from the nearby subhalo. The dark matterparameters and subhalo parameters are varied to fit theDAMPE spectrum.

A. Global Fit Models

First we consider the two models introduced in Sec. II:model A, in which the e+e− are produced by the directchannel and the W+W− channel; and model B, in whichthe e+e− production are mainly from the direct channeland the double τ+τ− channels. The branching ratio ofthe direct channel to other channels are taken as freeparameters.

Model A. In Fig.4, we plot the best fit to the DAMPEspectrum. The CR background, the various contribu-tions from dark matter annihilations, and the total ofthese are all plotted in the figure. As we we can seefrom the figure, there is no peak in the total spectrumcurve, which behaves as a gentle break, despite the factthat a subhalo is introduced. Looking into the details,we see the W+W− channel broad spectrum (i.e. the onefrom Galactic halo annihilations) gives an excellent fitto the broad excess, which are the majority of the dataand also have much smaller measurement error bars thanthe few data points near the 1.4 TeV peak. As a result,the W+W− channel is the dominant contribution to the

DAMPE spectrum. The broad e−e+ spectrum (i.e. thedirect channel contribution from the Galactic halo), onthe other hand, does not have the correct shape to fit thebroad excess, so its contribution must be suppressed. Weplot the logf − logA distribution in Fig. 5, with otherparameters marginalized. We find that the branchingratio is limit to be f <∼ 1.1 × 10−4 at 3σ, meaning thatthe contribution from e+e− channel is quite limited com-pared with the W+W− channel. Thus, the whole spec-trum shape is determined by the W+W− channel. Thebranching ratio of the direct channel to the W+W− chan-nel f is however a single parameter, so the direct e+e−

from the subhalo which has the desirable spectral shapeof a peak is also suppressed. The final spectrum fits mostof the data, but does not have a peak, and is not signif-icantly better than the broad spectrum fit given in Sec.II.

Is it possible for the direct e+e− channel to dominateover the W+W− channel? Unfortunately, for realisticparameters for CR propagation and dark matter halo,the Galactic halo annihilation via e+e− channel producefairly hard spectrum, even after the diffusion process, asshown by the cyan colored curve in Fig. 4. The curveis broader than that of the nearby subhalo contribution,but still too narrow to produce the broad excess.

101 102 103

E [GeV]

100

101

102

E3J

[GeV

2m−

2s−

1sr−

1]

r = 0.1 kpc

backgroundallnearbybroad, W+W−

broad, e+e−

DAMPE data

FIG. 4: The fit to DAMPE spectrum with model A, withvarious contributions shown and marked. The envelope cor-responds to 1σ in parameter space .

Model B. In Figs. 6 and 7 we show the correspond-ing results for model B. In this case, the fitting curvepasses closer to the peak point, but the spectral shapeis more like a cliff than a peak, the flux on the left sideof the peak is over-predicted. In fact, in this model themain contribution to the 1.4 TeV spectrum is not fromthe nearby subhalo, but instead from the direct e+e−

channel component in the broad spectrum, i.e. from theGalactic halo dark matter annihilation or decay. An-other problem is that the flux at energy around severaltens GeVs were under predicted. Though the deviationis not very obvious on this plot, the measurement errorin that energy range is much smaller. In this model, weobtain 5.6× 10−4 <∼ f <∼ 1.2× 10−3 at 3σ.

8

−6.0 −5.5 −5.0 −4.5 −4.0 −3.5log(f )

3.50

3.52

3.54

3.56

3.58

3.60

log(A

)

1σ

2σ

3σ

FIG. 5: The constraints on f and A for model A.

101 102 103

E [GeV]

100

101

102

E3J

[GeV

2m−

2s−

1sr−

1]

r = 0.1 kpc

backgroundallnearbybroad, double τ+τ−

broad, e+e−

DAMPE data

FIG. 6: Same as Fig. 4, but for model B.

−3.4 −3.2 −3.0 −2.8 −2.6log(f )

3.25

3.30

3.35

3.40

3.45

log(A

)

1σ

2σ

3σ

FIG. 7: Same as Fig. 5, but for model B.

In all these global fit models, the contribution from apossible nearby subhalo is limited to <∼ 10 - 20%. Thissmall contribution is not apparent in the flux, and failsto produce the peak-like feature.

B. Eye Ball Fit Models

In the above we have seen that even with nearby sub-halo, the global fit models do not produce a sharp peak inthe spectrum. Here we consider how such a peak mightbe generated if we relax the requirement on the fit.

The different components have different spectralshapes. The background electrons have a simple powerlaw shape and its normalization is determined by usingthe positron spectrum measured in the AMS-02 experi-ment. The direct e+e−, W+W− and double τ+τ− chan-nels from the Galactic halo annihilation also have es-sentially fixed spectral shape, though it depends on theGalactic halo model and the CR propagation parame-ters. The normalization of these depends on the darkmatter annihilation cross-section or decay rate. In fact,the shapes are slightly different for the case of annihila-tion and the case of decay, as the radial profile of pro-duction in the Galactic halo are different, though we findthat the difference is not large. On the other hand, forthe nearby components the shape depend on the distanceof the subhalo. This change is most obvious for the di-rect e+e− channel. By adjusting the branching ratio,one could change the relative contributions of the direct,W+W− and double τ+τ− channels, and by adjusting thesubhalo distance and mass, one could change the relativecontribution from the subhalo and the Galactic halo. Thevarious different components are summed up to producethe total spectrum.

We note that the measurement error on the spectrumis smaller at lower energy scales. When performing aglobal fit with natural weight according to Eq. (9), thelow energy part of the spectrum would largely determinethe fit, that is the reason why the contribution of the di-rect e+e− component is strongly suppressed in the above,because the shape of broad (i.e. Galactic) direct e+e−

component does not fit the lower energy part of the spec-trum. This is also why the W+W− component is stronglyfavored, because its shape agrees extremely well with theobserved excess. Here, to study whether a peak featurecan be generated, we will give up the minimal variancefit based on the χ2, but use an “eye ball fit” to producethe spectrum. Note that even though the spectrum mayappear to the eye to fit the data well, the actual χ2 valuewould be much larger than the global fits we obtainedabove.

In performing the “eye ball fit”, we shall ignore thedata above 2 TeV, where a dip appeared in the E3Fluxplot. In this part the error is too large to draw definiteconclusions.Model A. Since the broad (Galactic halo) W+W−

component has a shape very well suited for fitting the

9

broad excess, we try to produce a spectrum with thebroad and nearby (subhalo) W+W− components as prin-cipal contributions in addition to the background. Wehave tried various combination of the two, and show oneexample of such “eye ball fit” in Fig.8. We see the broadW+W− has a relatively flat shape up to 0.7 TeV, abovewhich it drops rapidly. By contrast, the nearby W+W−

source produce a one sided peak at ∼ 1.4TeV, it drops tozero above 1.4 TeV, while declines rapidly at smaller en-ergy. If we adjust the relative contribution of the two, wecould create a slight hill in the spectrum at the desiredposition. However, we see that the hill has a gentle slopeon the left side, though on the right side it could be verysteep. Also, on the low energy part the fit is not good: itis obviously below the data point. To get a better fit onthe low energy part, one has to increase the amplitude ofthe broad W+W− component. This can be achieved byincreasing the annihilate cross-section or decay rate, butthen the nearby source component would also increase,increasing the peak height while keep the same shape. Ifone wants to maintain the height by reducing the nearbysource component (this can be done by adjusting the sub-halo mass), the relative amplitude of the two componentswill be changed, and the slope below 1.4 TeV would beeven more gentle, making the peak disappear into a onesided slope, which is essentially the result of global bestfit of model A above.

101 102 103

E [GeV]

100

101

102

E3J

[GeV

2m−

2s−

1sr−

1]

r = 0.1 kpc

backgroundDAMPE datanearby W+W− templatebroad W+W− templaterescaled nearby W+W−+ backgroundrescaled nearby W+W−+ rescaled broad W+W− + background

FIG. 8: An eye ball fit to the DAMPE spectrum with principalcontribution from the broad and nearby W+W− components.

Model B. Next we consider the fit with the directe+e− channels. The advantage of this channel is that thenearby source of this channel may generate very sharppeak, though its broad component has the wrong shapeto fit the broad excess. If we relax the fit to the broadexcess though, we could produce a peaked spectrum inthis case. The double τ+τ− components which have flat-ter shapes may also be employed to help improve thefit to the low energy broad excess part. An “eye ballfit” is shown in Fig.9. Here we see the total spectrum(marked as ”adjusted spectrum”) can well reproduce thepeak feature at 1.4 TeV. This peak feature is dominatedby contribution from the nearby direct e+e− source. In-

terestingly, the broad e+e− source, the nearby and broaddouble τ+τ− source together make up the contributionto the broad excess at a few hundred GeVs. Neverthe-less, the total spectrum at below the few hundred GeVsare still significantly below that of the observation. If onewants to improve the fit to that part from these compo-nents, however, one would have to increase the doubleτ+τ− contribution, which would again make the peakchange into a slope or cliff, as the case of global fit modelB.

101 102 103

E [GeV]

100

101

102

E3J

[GeV

2m−

2s−

1sr−

1]

r = 0.1 kpc

backgroundDAMPE datanearby double τ+τ− templatenearby e+e− templatebroad double τ+τ− templatebroad e+e− templateadjusted spectrum

FIG. 9: A model B fit with adjusted background.

C. Rescaled Background Model

In our analysis of the DAMPE data, we distinguishthe “background”, which is the part of the spectrumcontributed by normal astrophysical sources, and theexcess, which we attribute to the contribute of darkmatter. In the above we have used the experimen-tally derived background from the AMS-02 experiment.However, there are uncertainties in such decomposition.Here we consider the some other way of decomposition.To do this, we rescale the background as follows: werescale the experimentally derived background by a fac-tor 1.13× exp(−E/5TeV).

With this change, the fit to the broad excess can besignificantly improved for the model B. We show the fitto the model with the rescaled background in Fig. 10.Here, compared with the global best fit model of theoriginal background, the reduced χ2 of the best fit modeldecreased from 2.2 to 0.7. This is because with the ad-justed background, the shape of the double τ+τ− com-ponents have much better match with the broad excess.The best fit model still does not show the narrow peak,but a model with the narrow peak is only slightly worsefit to the model than the best fit one. See the greendashed line in Fig. 10.

10

This example shows that we have to view the variousfits with some caution. Systematic uncertainty such asthe definition of the background could be very importantin our interpretation of the data.

101 102 103

E [GeV]

100

101

102

E3J

[GeV

2m−

2s−

1sr−

1]

r = 0.1 kpc

rescaled backgroundallnearbybroad, double τ+τ−

broad, e+e−

best fita model with narrow peakDAMPE data

FIG. 10: A fit to the DAMPE spectrum with adjusted back-ground, for the e+e− and double τ+τ− channels.

V. THE SUBHALO

We now consider the subhalo which can produce thepeak shown above. We let the electrons and positron of asubhalo plus the background fit the points near the peak.

A. Decaying dark matter subhalo

For decaying dark matter with mass mχ = 2E0/c2,

and the lifetime tχ, the injection rate is

Qdecay =2Msub

mχtχ=

Msub

E0/c2tχ= 2.6×1039

Msub

E0tχs−1, (11)

where Msub is subhalo mass in units of solar mass, E0

in units GeV and tχ in units Universe age (4.4× 1017 s).If we adopt the life time value 7.3 × 1025 s−1, in whichthe Galactic halo contribution could produce the broadprocess, the result is shown on Fig. 11.

We see that for a reasonable distance of 0.1 kpc, therequired halo mass is about 104.5M�. Using longer darkmatter particle lifetime, for example ∼ 1028 s, we wouldhave subhalo mass as high as 106.5 M� at 0.1 kpc.

B. Annihilating dark matter subhalo

For dark matter annihilation, the density profile of thesubhalo is crucial. For the subhalo, we assume a power-law density profile (e.g. [67–69])

ρ(r) = ρ0r−α, Msub =

4πρ03− αr

3−αt , (12)

−1.5 −1.0 −0.5 0.0log(r/kpc)

3.5

4.0

4.5

5.0

5.5

6.0

6.5

log(M

sub/M�

)

1σ

2σ3σ FIG. 11: The constraints on the log(r) - log(Msub) by fitting

only the peak for decaying dark matter.

where rt is the tidal radius, and

ρ0 =(3− α)Msub

4πr3−αt

(13)

The inner cutoff radius for cuspy halo is determined byequating the annihilation time scale to dynamical timescale, τd = τanni, where

τd =rt√

GMsub/rt, τanni =

nχ(rc)

nχ(rc). (14)

Finally the injection rate from the annihilating darkmatter in the subhalo is given by

Qanni =

∫ rt

rc

2

(ρ0c

2

E0r−α

)2

〈σv〉 4πr2dr (15)

=

8π(ρ0c

2

E0

)2〈σv〉 (r3−2α

t −r3−2αc )

3−2α when α 6= 1.5

8π(ρ0c

2

E0

)2〈σv〉 [ln(rt)− ln(rc)] when α = 1.5.

For annihilation we assume 〈σv〉 ∼ 10−23 cm3 s−1,for which the annihilation of the galactic halo gives thebroad excess. Then, if the peak is due to the subhalo,the mass and distance of the subhalo is shown in Fig.12. To fit the peak, assuming the subhalo is located at∼ 0.1 kpc from the Solar system, the required subhalomass is ∼ 105M� for α = 1.2. If the halo has a steeperprofile, e.g. α = 1.7, then the halo mass could be assmall as ∼ 102.5 M�. A smaller subhalo with very steepdensity profile could be formed from the large densityperturbation peaks (δρ/ρ ∼ 0.3) in the early Universein some scenarios (e.g. [70–72].) If we instead adopta thermal cross-section, 〈σv〉 ∼ 3 × 10−26 cm3 s−1, wethen have that at 0.1 kpc, the required subhalo mass is∼ 107.5 M� for α = 1.2 and ∼ 104.5 M� for α = 1.7respectively.

11

−1.5 −1.0 −0.5 0.0log(r/kpc)

1

2

3

4

5

6

7

log(M

sub/M�

)

FIG. 12: The constraints on the log(r) - log(Msub) for darkmatter annihilation case, 〈σv〉 = 1 × 10−23 is assumed. Theblue contours for α = 1.2, red for α = 1.7.

C. Inverse Compton Scattering

If such a subhalo is located near the Solar system, itmight also produce strong γ-ray photons. Even if we con-servatively assume that photons are not produced in thedecay or annihilation, the electrons and positrons pro-duced by the subhalo would have inverse Compton scat-tering (ICS) with background photons. If a subhalo is astrong electrons/positrons source, it is also a strong ICSsource. This would helps to identify the subhalo by look-ing for nearby bright and somewhat extended gamma-raysources.

The emissivity of the up scattered photons is [73]

ε(E , E0, Q, r) =

∫dE

ψ

4π

∫dE0

3σT c

4γ2E0G(q,Γe)

dUrad

E0dE0(16)

where ψ is from Eq. 7, E0 and E are the energy of photonsbefore and after scattering, dn/dE0 is the number densityof photons per unit energy for the interstellar radiationfield.

Γe =4E0γmec2

, q =E

(γmec2 − E)Γe, (17)

and the function

G(q,Γe) = 2qlnq+ (1 + 2q)(1− q) +(Γeq)

2(1− q)2(1 + Γeq)

. (18)

From [74, 75], near our Sun, the interstellar radiationfiled has three peaks at the NIR (∼ 1µm), FIR (∼100µm) and CMB(∼ 1000µm), with energy density ∼ 0.4eV cm−3, ∼ 0.3 eV cm−3 and∼ 0.2 eV cm−3 respectively.

Adopting these values we have

ε(E , E0, Q, r) ≈1

4π

∫dEψ

∑i

3σT cUrad,i

4γ2E20,iG(qi,Γe,i)

(19)

where i is one of NIR, FIR or CMB respectively.If the line-of-sight is θ offset from the center of the

subhalo, we have the specific intensity of photons upscattered by electrons generated from a source with Qinjection rate and distance r from the observer,

Iθ(E , E0, Q, r, θ) =

∫ ∞0

ε(E , E0, Q, x′)dr′, (20)

where x′ =√r2 + r′2 − 2rr′cosθ.

We show the ICS specific intensity at 30 GeV for line-of-sight toward the center of the subhalo in Fig. 13. It ismuch smaller than the isotropic gamma radiation back-ground (IGRB) given in [76]. The expected flux increaseswith decreasing distance, this is because to achieve theDAMPE peak, the more nearby source needs to producefewer number of electrons, hence also generating less ICSradiation. It is still challenge to observe such an extendedgamma-ray source by existing or even upcoming instru-ments.

10−1 100

r [kpc]

10−10

10−9

10−8

10−7

10−6

10−5

E2I θ

(E)

[GeV

cm−

2s−

1sr−

1]

r = 0.1 kpc

nearby sourceIGRB at ∼ 30 GeV

FIG. 13: The predicted gamma-ray intensity at 30 GeV fora subhalo with electrons/positrons that fits the peak (with1σ C.L.), for line-of-sight toward the subhalo center. In thepanel we also plot the IGRB given by paper [76].

D. Anisotropy

Finally, we note that if the nearby subhalo is invoked,it may generate an large anisotropy in the CR flux. Theamplitude of the dipole anisotropy is [77](

δJ

J

)max

∼ 3Dxx

cψ

∣∣∣∣dψdr∣∣∣∣ , (21)

If we require δJ/J <∼ 1, i.e. without generating obvi-ous anisotropy in the CR flux, we find that the subhalor <∼ 0.03 kpc can be excluded.

12

VI. SUMMARY

We studied the CR electron and positron spectrumfrom dark matter annihilation or decay in the energyrange of the recent DAMPE measurement. We extractedthe broad excess in the spectrum from a few tens GeVto about 2 TeV, by using the AMS-02 data on positronspectrum. We then derive the electron and positron spec-trum using the CR propagation model. We find that forboth the W+W− channel and the combination of directe+e− and double τ+τ− channels, dark matter annihila-tion or decay in the Galactic halo can provide good fits tothe broad excess in the e+e− spectrum. This naturallyexplains the TeV break found in the CR electron andpositron spectrum. However, this require a cross sectionof 〈σv〉 ∼ 10−23cm3s−1, which is larger than the WIMPcross-section required for obtaining the correct dark mat-ter abundance through thermal decoupling in the earlyuniverse.

In the DAMPE spectrum, there is also a prominentsingle point peak at ∼1.4 TeV. Tentatively treating itas real, we investigate how such a peak could be pro-duced, in spite of the energy loss while the electrons andpositrons diffusing through the interstellar space. Weconsider the possibility that the peak is produced by anearby dark matter subhalo. We find that in the globalfit models, there is no peak in the spectrum, and inclu-sion of the nearby subhalo does not significantly improvethe fit. This is because to match the shape of broad spec-trum, a suppressed contribution from the e+e− channel isfavored by the data. As a result, a nearby subhalo is hardto generate a narrow peak, as spectrum of electrons andpositrons generated by other channels are always broad.

We then relaxed the requirement of fitting the wholespectrum data, and studied how a spectrum with a 1.4TeV peak could be generated. We find that in the caseof W+W− channel, a peak feature with gentle slope onthe low energy side and steep slope on the high energyside can be produced. In the case of direct e+e− chan-nel and double τ+τ− channel, a sharp peak feature asfound in the DAMPE data could be produced. How-

ever, in these cases, the fitting to the broad excess froma few tens to a few hundred GeVs are not very good.In these fits, however, we have assumed a backgroundderived from the AMS-02 experiment. If we rescale thisbackground, a better fit can be obtained for the doubleτ+τ− channel, even with a peak. These show that theprecisely measured spectral shape from DAMPE couldprovide stringent constraint on the model, though theo-retical uncertainty is a very important factor in makinginterpretation and obtaining the constraint.

Giving the peak strength, we estimate the mass anddistance of the subhalo. We find that if assuming thisis also the source for broad excess, the required mass fora subhalo 0.1 kpc from our Solar system is ∼ 104.5 M�for the decay case, and ∼ 105 M� (102.5 M�) for theannihilation case with density profile slope α = 1.2 (1.7).The mass could be even smaller for steeper slope. Wealso considered the inverse Compton scattering by theelectrons, and the anisotropy in CR if the subhalo is tooclose by. The inverse Compton scattering radiation isstill not observable for current γ-ray telescopes.

Postscript: While we were preparing the draft of thispaper, we noted that a paper on the theoretical inter-pretation of the DAMPE result by Q. Yuan et al. [78]appeared on the arxiv preprint server, which investigatedboth dark matter and pulsar origin of the DAMPE excessand 1.4 TeV peak, though the adopted dark matter mod-els and the methods are somewhat different from ours.

Acknowledgments

We thank Prof. Jin Chang and Dr. Xiaoyuan Huangfor helpful discussions. XLC acknowledges the support ofthe the NSFC through grant No. 1633004 and 11373030,the MoST through grant 2016YFE0100300, and the CASFrontier Science Key Project No. QYZDJ-SSW-SLH017.BY acknowledges the support of NSFC grant 11653003and the Bairen program from the Chinese Academy ofSciences (CAS).

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