Interpretation of the cosmic ray positron and electron excesses with an annihilating-decaying dark matter scenario

TL;DR

This paper proposes a dark matter model where particles both annihilate and decay, explaining cosmic ray positron and electron excesses and spectral features observed by DAMPE and AMS-02, consistent with existing constraints.

Contribution

It introduces a combined annihilating-decaying dark matter scenario to explain cosmic ray excesses and spectral features, with a specific realization in a spin-1 vector dark matter model.

Findings

The model fits DAMPE and AMS-02 data well.

The spectral peak is attributed to nearby dark matter annihilation.

The broad excesses are due to dark matter decay in the Milky Way.

Abstract

The precise measurements of energy spectra of cosmic ray positrons and/or electrons by recent experiments show clear excesses above 10 GeV. Moreover, a potential sharp spectral feature was suggested by the Dark Matter Particle Explorer (DAMPE) data. These results inspire quite a number of discussions on the connection with either the annihilation/decay of dark matter (DM) or the astrophysical origins. Here we discuss a DM scenario in which DM particles could annihilate and decay into standard model particle pairs simultaneously. In this model, the peak structure is due to the DM annihilation in a nearby subhalo and the broad positron/electron excesses are due to the decay of DM in the Milky Way. This model can reasonably explain the DAMPE and AMS-02 data of the total $e^+e^-$ spectra and the positron fraction, with model parameters being consistent with existing constraints. A simple realization of such a DM model is the spin-1 vector DM model.

Tables

Table 1. Table 1: Propagation parameters. The benchmark setting is model II which is shown in bold.

Model	$D_0^a$	$z_h$	$v_A$	$d{V}_c/dz$	$\delta$
	$\left({10}^{28}{\mathrm{cm}}^2{s}^{-1}\right)$	(kpc)	($\mathrm{km}{s}^{-1}$)	($\mathrm{km}{s}^{-1}{\mathrm{kpc}}^{-1}$)
I	2.7	2.0	35.0	0	0.33
II	5.3	4.0	33.5	0	0.33
III	9.4	10.0	28.6	0	0.33
IV	4.2	10.9	0	5.4	0.59

Table 2. Table 2: The ADDM model parameters and required subhalo masses for different propagation parameters. The DM annihilation/decay branching ratios are assumed to be e : μ = 1 : 1 : 𝑒 𝜇 1 : 1 e:\mu=1:1 .

Model	$m_{\text{DM}}$	$⟨\sigma v⟩$	$\tau$	$M_{\mathrm{sub}}$
	(TeV)	(${10}^{-24}$ cm3 s-1)	(${10}^{26}$ s)	(${10}^4$ M⊙)
I	1.5	2.8	8.1	3.5
II	1.5	2.8	9.0	5.0
III	1.5	2.8	11.3	8.0
IV	1.5	2.8	9.0	4.5

Table 3. Table 3: The injection spectral parameters of the background electrons, the renormalization factor of the background positrons, and the force-field potential of the solar modulation.

Model	${\nu }_1$	${\nu }_2$	${\nu }_3$	$R_{\mathrm{br},1}$	$R_{\mathrm{br},2}$	$R_{\mathrm{cut}}$	$c_{e^{+}}$	$\Phi$
				(GV)	(GV)	(GV)		(GV)
I	1.48	3.11	2.61	2.2	41.0	$4.0\times {10}^4$	2.72	1.83
II	1.48	3.00	2.56	2.3	55.0	$1.1\times {10}^4$	3.27	1.42
III	1.53	2.93	2.46	2.7	79.6	$4.4\times {10}^3$	3.84	1.15
IV	1.63	2.90	2.48	3.6	46.0	$2.3\times {10}^4$	2.80	0.88

Equations

$$\frac{\partial \psi }{\partial t}$$

$$q(E,r)=\frac{⟨\sigma v⟩}{2m_{\text{mbox}DM}^2}\frac{\mathrm{d}N}{\mathrm{d}E}\times {\rho }^2(r),$$

$$q(E,r)=\frac{\rho (r)}{m_{\text{mbox}DM}\tau }\frac{\mathrm{d}N}{\mathrm{d}E},$$

$${\rho }_{\mathrm{mw}}(r)=\frac{{\rho }_s}{1+(r/r_s{)}^2},$$

$${\rho }_{\mathrm{sub}}\left(r\right)=\frac{{\delta }_c{\rho }_{\mathrm{crit}}}{\left(r/r_{\mathrm{s}}\right){\left(1+r/r_{\mathrm{s}}\right)}^2},$$

$${\delta }_c=7.213{\delta }_V=\frac{200}{3}\frac{c^3}{\ln (1+c)-c/(1+c)}.$$

$${\delta }_V=1.2\times 10^6{\left(\frac{M_{\mathrm{sub}}}{10^6{M}_{\odot }}\right)}^{-0.18},$$

$$-{\mathcal{L}}_1=\left(\frac{1}{2}{m}_V{V}^2+m_F\bar{F}F\right)+\left(g_V{V}_{\mu }\bar{\ell }{\gamma }^{\mu }{P}_{L}F+h.c.\right),$$

$$⟨{\sigma }_{\bar{\ell }\ell }v⟩=\frac{g_V^4}{9\pi }\frac{1}{m_V^2}\frac{r^4}{(r^2+1{)}^2},$$

$$g_{\ell }-2=-|g_V{|}^2\frac{m_{\ell }^2}{m_V^2}\frac{5-14r^2+39r^4-38r^6+8r^8+18r^4\ln r^2}{96{\pi }^2(1-r^2{)}^4},$$

$${\mathcal{O}}_V=g_5\frac{e}{4\pi }\frac{{\alpha }_V}{m_F^2}{\partial }_{\mu }{V}^{\alpha }{\partial }_{\alpha }{V}_{\nu }{F}_{\rho \sigma }{ϵ}^{\mu \nu \rho \sigma },$$

$${\sigma }_p\sim 0.5\times 10^{-13}{\left(\frac{1.5\mathrm{TeV}}{m_F}\right)}^4{\left(\frac{{\alpha }_V}{1.0}\right)}^2{\left(\frac{v}{10^{-3}}\right)}^{2}pb.$$

$$-{\mathcal{L}}_{\mathrm{decay}}={ϵ}_V{V}_{\mu }\bar{\ell }{\gamma }^{\mu }\ell .$$

Full text

††arxiv: 1902.09235

1 Introduction

Dark matter (DM) particles may annihilate or/and decay into standard model particles such as pairs of electrons/positrons and protons/anti-protons, and can hence give rise to excesses in the cosmic ray spectra. Identification of such excesses is one of the most important goals of the indirect detection experiments of DM such as PAMELA Adriani:2008zr , ATIC Chang:2008aa , Fermi-LAT Abdo:2009zk ; Abdollahi:2017nat , and AMS-02 Accardo:2014lma ; Aguilar:2014fea . Some progresses have been made in recent years. The most widely-known phenomena are perhaps the spectral anomalies (excesses) of positrons and the electron plus positron spectra. Either astrophysical source(s)/process(es) Yuksel:2008rf ; Hooper:2008kg ; Profumo:2008ms ; Malyshev:2009tw ; Hu:2009bc ; Blasi:2009hv ; Linden:2013mqa ; Yin:2013vaa or the DM annihilation/decay Bergstrom:2008gr ; Barger:2008su ; Cirelli:2008pk ; Yin:2008bs ; Zhang:2008tb ; Sergio:2010 ; Feng:2013zca ; Blum:2013zsa ; Bergstrom:2013jra ; Kopp:2013eka ; Cholis:2013psa ; Yuan:2013eja ; Jin:2013nta ; Dev:2013 ; Belotsky:2015 ) were proposed to account for the data.

The DArk Matter Particle Explorer (DAMPE, Chang:2014 ; TheDAMPE:2017dtc ), launched on December 17, 2015, is a high-energy particle detector dedicated to DM indirect detection and cosmic ray physics. The DAMPE collaboration reported the precise measurement of the cosmic ray $e^{+}+e^{-}$ (CREs) spectrum from 25 GeV to 4.6 TeV Ambrosi:2017wek . The DAMPE data confirms the spectral hardening of CREs around 50 GeV Abdollahi:2017nat , and reveals clearly a spectral softening at $\sim 0.9$ TeV Ambrosi:2017wek . These results are consistent with previously reported CRE excesses Chang:2008aa ; Abdo:2009zk ; Abdollahi:2017nat ; Aguilar:2014fea . Moreover, there might be a sharp peak at $\sim 1.4$ TeV. Although the significance of the current data is relatively low ($\sim 2.3\sigma$ considering the look-elsewhere effect Fowlie:2017fya ), this structure, if confirmed, should imply the existence of nearby quasi-monoenergetic electron sources Yuan:2017ysv .

If the DM annihilation or decay is employed to account for the peak, the DM particles should annihilate or decay dominantly into leptons, and the annihilation or decay should occur in local regions not far away from the solar system. This is because that TeV CREs lose their energies very quickly when travelling in the Milky Way, and can not reach us if they were generated far away. The quark channels should be dramatically suppressed due to the constraints of anti-protons. Such models have been proposed in literature Jin:2017qcv ; Zu:2017dzm ; Fan:2017sor ; Gu:2017gle ; Duan:2017pkq ; Tang:2017lfb ; Chao:2017yjg ; Gu:2017bdw ; Athron:2017drj ; Cao:2017ydw ; Duan:2017qwj ; Liu:2017rgs ; Huang:2017egk ; Gao:2017pym ; Niu:2017hqe ; Chen:2017tva ; Li:2017tmd ; Zhu:2017tvk ; Gu:2017lir ; Nomura:2017ohi ; Ghorbani:2017cey ; Cao:2017sju ; Yang:2017cjm ; Ding:2017jdr ; Liu:2017obm ; Ge:2017tkd ; Zhao:2017nrt ; Sui:2017qra ; Okada:2017pgr ; Chao:2017emq ; Cao:2017rjr ; Han:2017ars ; Niu:2017lts ; Nomura:2018jkd ; Wang:2018pcc ; Pan:2018lhc ; Liu:2019iik . See Ref. Yuan:2018rys for a special review of relevant studies.

The main purpose of this work is to interpret both the sub-TeV positron and electron excesses and the DAMPE peak simultaneously in an annihilating plus decaying DM (ADDM) model. It is natural to speculate that DM particles with a limited lifetime can at the same time annihilate with each other. While the annihilation products, presumed to be leptons, can explain the peak excess around 1.4 TeV, the decay products with lower energies can account for the sub-TeV electron/positron excesses. We will investigate whether such a scenario can be realized to account for the data, without violating existing constraints from e.g., $\gamma$-ray observations Chen:2009uq ; Hutsi:2010ai ; Cirelli:2012ut ; Cheng:2016slx and the cosmic microwave background (CMB) data Ade:2015xua .

This paper is organized as follows. In Section II, we compare ADDM model prediction with the AMS-02 positron fraction and the AMS-02/DAMPE CRE data. In Section III, we discuss a possible theoretical realization of such an ADDM scenario. Our conclusions are summarized in Section IV.

2 CREs from ADDM

2.1 Cosmic ray propagation

The general form of cosmic ray propagation equation reads Strong:2007nh :

[TABLE]

where $\psi=\psi({\bf r},p,t)$ is the phase space density, $q({\bf r},p)$ denotes the source function, $D_{xx}=\beta D_{0}(E/4\,{\rm GeV})^{\delta}$ is the spatial diffusion coefficient, ${\bf V}=dV/dz\cdot{\bf z}$ represents the convection velocity, $D_{pp}$ denotes the diffusion coefficient in the momentum space which is characterized by the Alfvenic speed $v_{A}$ and is to describe the reacceleration of particles in the interstellar medium, $\dot{p}=dp/dt$ is the momentum loss rate, and $\tau_{f}$ ($\tau_{r}$) is the time scale of the fragmentation (the radioactive decay).

Typically the Boron-to-Carbon ratio, the radioactive secondary abundances, as well as diffuse $\gamma$-rays are used to constrain the cosmic ray propagation parameters Strong:2007nh ; Yuan:2017ozr ; Yuan:2018vgk ; Ackermann:2012rg . In this work, we consider a few propagation model configurations and parameter settings, which are summarized in Table 1. For the diffusion-reacceleration configuration, we also adopt several values of the halo height following Ref. Ackermann:2012pya . As a benchmark setting, we adopt the diffusion-reacceleration model with a half-height of the propagation cynlinder $z_{h}=4$ kpc, as shown in bold in Table 1 (model II). Different propagation models affect mainly the low-energy spectra of electrons and positrons, and have very minor effects on the DM model which is the focus of this work. This is because the main propagation effect of high-energy CREs is radiative cooling rather than the diffusion, convection, and reacceleration. High-energy CREs can only propagate a very limited distance in the Galaxy before they get cooled down. As an illustration, the cooling time for TeV CREs is about $3\times 10^{5}$ years and the corresponding diffusion length is about 1 kpc for the diffusion parameter setting II.

This propagation equation can be solved numerically, by e.g., GALPROP Strong:1998pw and DRAGON Evoli:2008dv . In this work we use the LikeDM package Huang:2016pxg , which tabulates the outputs from GALPROP and enables fast computation, to calculate the propagation of CREs for all the background (non-DM) components and the Milky Way DM annihilation/decay component of CREs. Note that we will also deal with the CRE propagation from a nearby DM subhalo. For this particular case, we simplify the propagation equation, keeping only the diffusion and energy loss terms, and adopt the analytical Green’s function to solve the CRE propagation in a spherically symmetric geometry with infinite boundary conditions Atoyan:1995ux .

2.2 DM distribution

For the annihilation process, the CRE source function is

[TABLE]

where $m_{\mbox{\tiny DM}}$ is the mass of the DM particle and ${\rm d}N/{\rm d}E$ is the spectrum of CREs per annihilation. For the decaying case, the source term is

[TABLE]

where $\tau$ is the lifetime of the DM particle.

The DM distribution $\rho(r)$ in the Milky Way is assumed to be an isothermal distribution Bahcall:1980fb

[TABLE]

where $\rho_{s}=1.16$ GeV cm*-3* denotes the finite central density and $r_{s}=5$ kpc represents the core radius. As for the subhalo, we assume an NFW distribution Navarro:1996gj

[TABLE]

where $r_{s}$ and $\delta_{c}$ are the scale radius and characteristic density, $\rho_{\rm crit}=3H^{2}/8\pi G$ is the critical density of the Universe. The $\delta_{c}$ parameter relates with the subhalo concentration parameter $c=r_{v}/r_{s}$, where $r_{v}$ is the virial radius, as Springel:2008cc

[TABLE]

For subhalos in the solar neighborhood, we have approximately Springel:2008cc

[TABLE]

where $M_{\rm sub}$ is the mass of the subhalo. The tidal force of the Milky Way DM halo would remove the DM beyond a so-called tidal radius from the subhalo. Adopting the method of Ref. Springel:2008cc , the tidal radius is found to be roughly 0.2 times of the original virial radius of a subhalo Yuan:2017ysv . Therefore, the DM distribution of the subhalo is an NFW distribution truncated at the tidal radius $r_{t}$.

2.3 Results

The background electrons include primary electrons accelerated from conventional cosmic ray sources, whose injection spectrum is parameterized as a three-segment broken power-law with an exponential cutoff Yuan:2017ysv , and secondary electrons from inelastic collisions of cosmic ray nuclei and the interstellar medium. The background electron spectral parameters are tuned to match the data. The background positrons are mainly from the inelastic collisions between cosmic ray nuclei and the medium. The detailed parameters of the CR electron and positron backgrounds are presented in Table 3 in the Appendix. The annihilation and decay contributions from DM in both the Milky Way halo and the nearby subhalo are also added to the model.

Here we take the propagation model setting II as an example to discuss the fitting results. Figure 1 shows the CRE spectrum (left panel) and the positron fraction (right panel) from the model prediction for the propagation model setting II, compared with the measurements Accardo:2014lma ; Aguilar:2014fea ; Ambrosi:2017wek . Here the ADDM model parameters are: the mass of the DM particle is $m_{\mbox{\tiny DM}}={\rm 1.5~{}TeV}$, the annihilation cross section is $\langle\sigma v\rangle={\rm 2.8\times 10^{-24}~{}cm^{3}~{}s^{-1}}$, the decay lifetime is $\tau=9\times 10^{26}$ s, and the branching ratios are $e:\mu=1:1$. The subhalo mass is $M_{\rm sub}=5\times 10^{4}$ M*⊙*, and the distance to the subhalo center is $d=0.1$ kpc. Figure 2 shows a slightly improved fitting with annihilation/decay branching ratios $e:\mu=1:3$. It is shown that the model prediction matches well with the data. The parameter values for the other propagation models are compiled in Table 2, which show minor variations among different models. For reference, the comparisons between the model and data for the other three propagations models are given in Figure 5 in the Appendix.

The spike structure around 1.4 TeV is due to the annihilation of DM in the subhalo, and the sub-TeV broad excesses of positrons and CREs are mainly due to the decay of DM in the Milky Way halo. To clearly see this, we plot in Figure 3 the CRE fluxes from the DM annihilation or decay in either the Milky Way or the subhalo separately. For the parameters we adopt, the subhalo contribution is dominated by the DM annihilation, which gives rises to the 1.4 TeV peak shown in the data. For the Milky Way components, the decay component is slightly larger than the annihilation one. Note that here we need a relatively high contribution from the DM annihilation in the Milky Way (and hence a relatively large cross section), otherwise the CRE data from 700 GeV to TeV cannot be well reproduced.

The annihilation cross section and decaying lifetime are marginally consistent with the constraints from $\gamma$-rays Chen:2009uq ; Hutsi:2010ai ; Cirelli:2012ut ; Cheng:2016slx and CMB Ade:2015xua . As shown in Ref. Yuan:2017ysv , the upper limits of the annihilation cross section from CMB are about $3\times 10^{-24}$ cm3 s*-1* for the $\mu^{+}\mu^{-}$ channel and $10^{-24}$ cm3 s*-1* for the $e^{+}e^{-}$ channel. The lower limits on the decay lifetime from the extragalactic diffuse $\gamma$-ray background are about $4\times 10^{26}$ s for the $\mu^{+}\mu^{-}$ channel and $10^{27}$ s for the $e^{+}e^{-}$ channel. Therefore, the model parameters derived in this work are not excluded by the current data.

The DM annihilation and decay in a nearby subhalo should also be constrained by $\gamma$-rays and CRE anisotropies Yuan:2017ysv ; Coogan:2019 . In addition, the profile of the line-like feature can also constrain the parameters of the subhalo Huang:2017egk ; Coogan:2019 . A large distance of subhalo ($\gtrsim 0.3$ kpc) is ruled out by the width of DAMPE CRE excess, and the halo size is bounded by the Fermi $\gamma$-ray observations Coogan:2019 . In a forthcoming work, we will show that the $\gamma$-ray emission from this postulated DM subhalo is consistent with the current Fermi data at least in some low-latitude sky regions Ge:2020 . Finally, we comment that the probablity of finding such a massive subhalo within a short distance from the Earth is relatively low Yuan:2017ysv ; zhao:2019cpc . According to Fig. 8 of Ref. Yuan:2017ysv , we estimate that such a probability is about $0.1\%$ for $d=0.1$ kpc and $M_{\rm sub}=5\times 10^{4}$ M*⊙*.

3 Model

According to the results in the previous section, we need a primary spectrum which includes contributions from both annihilation and decay of DM into a pair of light charged leptons (referring to $e$ and $\mu$, which are commonly denoted as $\ell$ hereafter). The simplest candidate is supposed to be a spin-0 particle $S$. However, this scenario does not work naturally. We first consider that $S$ annihilates into $\bar{\ell}_{i}\ell_{j}$ via a $t$-channel charged fermionic mediator. To avoid the $p$-wave suppression, it requires a large chiral violation which at the same time gives rise to either a large lepton flavor violation $\bar{\ell}_{i}\rightarrow\ell_{j}+\gamma$ or a large $g_{\ell}-2$; both have been ruled out. Then we consider the $s$-channel annihilation exchanging a spin-0 or spin-1 mediator $X$. The latter is again $p$-wave suppressed, whereas the former is not well motivated.

A spin-1 vector DM (VDM) $V_{\mu}$ may works. The reason is that its annihilation into $\bar{\ell}\ell$ is not $p$-wave suppressed even in the absence of chiral violation. In this work we are not aiming at constructing a complete model. We just consider the following minimal $Z_{2}$-invariant effective model

[TABLE]

where the $Z_{2}$-odd fermion $F$ is a Dirac fermion, mediating the $t$-channel annihilation $VV\to\bar{\ell}\ell$. Here the VDM is leptophilic because it couples dominantly to light leptons. To suppress the lepton flavor violation, we further require that each lepton flavor $\ell_{i}$ has its own partner $F_{i}$, and they do not give rise to new flavor violation. Different from the scalar DM case, the annihilation cross section has unsuppressed $s$-wave contribution

[TABLE]

with $r\equiv m_{V}/m_{F}<1$. Without chiral violation, the contribution to $g_{\ell}-2$ from Eq. (8) is not significant, given by Agrawal:2014ufa

[TABLE]

which is suppressed by the light fermion mass square. We display the numerical results of Eq. (9) and Eq. (10) in Fig. 4. We find that for $\langle\sigma_{\bar{\ell}\ell}v\rangle$ as large as $\sim 100$ pb, the resulting $g_{\ell}-2$ can still lie below the current uncertainties, which are $\sim 10^{-12}$ and $10^{-9}$ for electrons and muons, respectively. The conclusion is particularly true in the region with $r\sim 1$. Note that for an annihilation cross section as large as 100 pb, the relic density after freezing-out will be too small to account for the total DM budget. One way to overcome this problem is assuming that the early universe after reheating does not reach a very high temperature, but much below the TeV scale. Consequently, the VDM and as well the mediator $F$ never reach full thermal equilibrium, so one can not calculate VDM relic density via the usual freeze-out dynamics. Instead, one should track all productions of DM via the scattering like $\ell\bar{\ell}\rightarrow VV$, namely through freeze-in Chen:2017kvz

It is seen that $g_{V}\sim 4$ is needed to make $\langle\sigma_{\ell\bar{\ell}}v\rangle\sim 100$ pb, so one should worry if such a VDM is subjected to a strong constraint by direct detection. Since this VDM is leptonic and moreover self-conjugate (a real vector), its leading interaction with the quark electromagnetic current is via the anapole momentum of VDM, described by the dimension-6 operator Agrawal:2014ufa

[TABLE]

with $F_{\rho\sigma}$ the Maxwell field strength tensor. The operator coefficient is estimated from the one-loop diagram with a charged loop, giving $\alpha_{V}=g_{V}^{2}/4\pi\sim 1$ and $g_{5}\sim{\cal O}(1)$. The resulting VDM-nucleon scattering is suppressed by $p$-wave in the nonrelativistic limit, which is explicit in the following estimation on the cross section 111We did not make a solid calculation, but borrowed the result from a similar model studied in Ref. Ho:2012bg , based on the Majorana fermion anapole dark matter. ,

[TABLE]

It is four orders of magnitude below the current strongest bound from XENON1T. Actually, it already goes beneath the neutrino floor and therefore probably can not be probed in the direct detection experiments

Actually, a structure similar to Eq. (8) is presented in the little Higgs model with $T$-parity Birkedal:2006fz or the extra dimensional theory with KK-pairty Servant:2002aq ; Cheng:2002ej , where the lightest $T/{\rm KK}$-odd vector boson is the VDM candidate. In these models all left-handed fermions are accompanied with $T/{\rm KK}$-odd heavy fermions. But after identifying $g_{V}$ with the $U(1)_{Y}$ gauge coupling $g^{\prime}$, the cross section is far from the required value to explain the CRE data. From the required large $g_{V}$ value, it is of interest to build VDM in the context of the composite Higgs model.

Now we discuss the VDM decay. To make it decay into a pair of light leptons, the simplest approach is to introduce a term which breaks $Z_{2}$ explicitly as the following:

[TABLE]

The resulting lifetime of $V$ is $\tau_{V}\sim 10^{26}$ s  $(10^{-26}/\epsilon_{V})^{2}~{}(1.5~{}{\rm TeV}/m_{V})$. To understand the smallness of $\epsilon_{V}$, one needs the completion of the effective model and specifies the origin of $Z_{2}$. For instance, it might be due to a spontaneous but tiny break of $Z_{2}$ by a scalar field; such a break results in a tiny mixing between $V$ and some leptophilic gauge bosons, then giving rise to the effective operator in Eq. (13) through this tiny mixing. Alternatively, if $Z_{2}$ is identified with the $T$-parity, its violation may be due to anomaly Hill:2007zv .

To end up this section we make a comment on the spin-1/2 candidate $\chi$. It may also give the desired CRE spectrum if its interaction is largely specified by a spin-1 intermediate state $X_{\mu}$ which is merely slightly lighter than $\chi$ and dominantly couples to $\ell$222The $t$-channel annihilation again suffers from the $p$-wave suppression as the scalar case Baek:2015fma .. In such a case, the DM cascade decay $\chi\to\nu+X(\to\bar{\ell}\ell)$ produces the $e^{+}e^{-}$ spectrum similar to that from the two-body decay. Moreover, the annihilation $\bar{\chi}+\chi\to\bar{\ell}_{i}+\ell_{j}$ is not $p$-wave suppressed. We leave this scenario for future studies.

4 Conclusion

In this work, we propose an ADDM model to interpret the positron and CRE data from AMS-02 and DAMPE. In this model we assume that DM particles could annihilate and decay into light leptons (electrons and muons) at the same time. It is shown that the annihilation in a nearby subhalo can explain the potential peak structure of the CRE spectrum, and the decay in the Milky Way halo can explain the broad sub-TeV excesses of both positrons and CREs. The model parameters, although somehow tuned to explain the data, are consistent with the current $\gamma$-ray and CMB constraints.

A spin-1 VDM was proposed as a particle realization of the model. The annihilation into leptons of the VDM is not $p$-wave suppressed even in the absence of chiral violation. At the same time, its contribution to $g-2$ is not significant.

Acknowledgements.

We thank Dr. Ran Ding for helpful discussions and suggestions. This work was supported in part by the National Key Research and Development Program of China (No. 2016YFA0400200), the National Natural Science Foundation of China (Nos. 11773075, 11525313, 11722328, U1738210, U1738209, U1738206) and the Youth Innovation Promotion Association of Chinese Academy of Sciences (No. 2016288). QY is also supported by the 100 Talents Program of Chinese Academy of Sciences.

Appendix: results for the other cosmic ray propagation parameters

The injection spectrum of the background electrons is parameterized as a three-piece broken power-law with an exponential cutoff form, with spectral indices $\nu_{1}$, $\nu_{2}$, $\nu_{3}$, break rigidities $R_{\rm br,1}$, $R_{\rm br,2}$, and cutoff rigidity $R_{\rm cut}$. The background positron spectrum is calculated according to the interaction between CR nuclei and the interstellar medium Yuan:2017ozr . A renormalization factor ($c_{e^{+}}$) has been multiplied to the secondary positron spectrum in order to fit the low-energy positron fraction and CRE spectra simultaneously. We further apply a force-field solar modulation model on the propagated electron and positron spectra before comparing with the measurements carried out on top of the atmosphere Gleeson:1968zza .

The comparisons between the model and data for the other three propagations models are given in Figure 5 for DM annihilation/decay branching ratios $e:\mu=1:1$. From the figure, we can see that these three propagation models match with the data very well. Similar to CR propagation model II, the annihilation cross section and decaying lifetime as shown in in Table 2 are all consistent with the current $\gamma$-rays Chen:2009uq ; Hutsi:2010ai ; Cirelli:2012ut ; Cheng:2016slx and CMB Ade:2015xua observations.