A simple and natural interpretations of the DAMPE cosmic-ray electron/positron spectrum within two sigma deviations
Jia-Shu Niu Tianjun Li Fang-Zhou Xu
hep-ph astro-ph.HE








jsniu@itp.ac.cn
Institute of Theoretical Physics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan, 030006, China
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China

 

tli@itp.ac.cn
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China


Institute of Modern Physics and Center for High Energy Physics, Tsinghua University, Beijing 100084, China


             


  The DArk Matter Particle Explorer (DAMPE) experiment has recently announced the first results
for the measurement of total electron plus positron fluxes between 25 GeV and 4.6 TeV. A spectral break
at about 0.9 TeV and a tentative peak excess around 1.4 TeV have been found. However, it is very difficult 
to reproduce both the peak signal and the smooth background including spectral break simultaneously. 
We point out that the numbers of events in the two energy ranges (bins) close to the 1.4 TeV excess have 
1σ deficits. With the basic physics principles such as simplicity and naturalness, we consider
the  -2σ, +2σ, and -1σ deviations due to statistical fluctuations for the 
1229.3 GeV bin, 1411.4 GeV bin,  and 1620.5 GeV bin. Interestingly, we show that 
all the DAMPE data can be explained consistently via both the continuous distributed pulsar and 
dark matter interpretations, which have χ^2≃ 17.2 and χ^2≃ 13.9 (for all the 38 points in DAMPE electron/positron spectrum with 3 of them revised), respectively. 
These results are different from the previous analyses by neglecting the 1.4 TeV excess.
 At the same time, we do a similar global fitting on the newly released CALET lepton data, which could also be interpreted by such configurations.
Moreover, we present a U(1)_D dark matter model with Breit-Wigner mechanism, which can provide
the proper dark matter annihilation cross section and escape the CMB constraint.
Furthermore, we suggest a few ways to test our proposal.




                             


                              
A Simple and Natural Interpretations of the DAMPE Cosmic Ray Electron/Positron Spectrum within Two Sigma Deviations
    Fang-Zhou Xu
    December 30, 2023
===================================================================================================================







§ INTRODUCTION


Because of the strong radiative cooling via synchrotron
and inverse Compton scattering (ICS) processes, 
the TeV electrons can only travel a short distance of about 
 a few kpc in the Milky Way. Therefore,
the nearby Cosmic Ray (CR) sources such as pulsars <cit.> and dark matter (DM) <cit.> 
can be probed via the high energy electrons and positrons. 
The spectra of the cosmic ray electrons and positrons (CREs)  
have been measured up to TeV energy scales by the ground-based and space-borne experiments,
for example, HESS <cit.>, VERITAS <cit.>, FermiLAT <cit.>, AMS-02 <cit.>, and CALET <cit.>. 
In particular, the excesses of the electrons <cit.> and positrons <cit.> have
been discovered as well.

Recently, the DArk Matter Particle Explorer (DAMPE), which is a new generation space-borne 
experiment to measure CRs and was launched in December 2015,
has announced the first results of high energy CR electron plus positron
(e^-+e^+) flux from 25 GeV to 4.6 TeV with unprecedentedly high quality <cit.>. 
The energy resolution of the DAMPE is better than 1.5% at TeV energies,
and the hadron rejection power is about 10^5. Thus, DAMPE is able to 
reveal (fine) structures of the electron and positron fluxes.
The main DAMPE spectrum can be fitted by a smoothly broken power-law model with
a spectral break around 0.9 TeV, which confirms the previous results by
HESS experiment <cit.>. And there exists a tentative peak-like flux excess around 1.4 TeV.
Thus, the DAMPE results have stimulated the extensive studies <cit.>.
The spectral break can be explained by the broad distributed pulsars, pulsar wind nebulae (PWNe), 
supernova remnants (SNRs) <cit.>, and by the  
dark matter annihilation and decay in the galaxy halo <cit.>. 
Also, the tentative peak is always interpreted by local pulsars, PWNe, 
and SNRs <cit.>), and by the DM sub-halos, clumps, and
 mini-spikes <cit.>. 
 Another important interpretation ascribes observed CR spectrum puzzling features to a nearly 2-3 Myr Super Nova <cit.>, which could naturally explain not only proton to helium ratio, positron and anti-proton fluxes, but also plateau in the cosmic ray dipole anisotropy.

However, one can easily show that it is impossible to explain both the spectral break and the
tentative peak simultaneously <cit.>). 
In addition, we have 74, 93, and 33 events for three continuous bins or energy ranges
[1148.2, 1318.3] GeV, [1318.3, 1513.6] GeV, and [1513.6, 1737.8] GeV, respectively,
which for simplicity we shall call 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5 GeV bin <cit.>.
The  number of events and fluxes for these bins are given in Table <ref>.
From Figure 2 of the DAMPE's paper <cit.>, it is obvious that the 1411.4 GeV bin 
has a little bit more than 3σ
excess, while the 1229.3 GeV bin and 1620.5 GeV bin have about 1σ deficits.
Therefore, it is very difficult to explain the events in these three bins, especially
the first two, no matter by the pulsar or dark matter interpretations.



From the theoretical physics point of view, we would like to explain nature with basic
principles such as simplicity and naturalness, or say truth and beauty! 
In the words of Sir Isaac Newton,  
“Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things.”
Therefore, to explain all the DAMPE data via a simple and natural way, we propose that
the excess in the 1411.4 GeV bin and the deficits in the 1229.3 GeV bin and 1620.5 GeV bin
arise from the +2σ, -2σ, and -1σ deviations due to
statistical fluctuations, which happened frequently in collider experiments. Remarkably,
we can indeed explain all the DAMPE data consistently via the pulsar and dark matter interpretations,
which have χ^2≃ 17.2 and χ^2≃ 13.9 (for all the 38 points in DAMPE electron/positron spectrum with 3 of them revised), respectively. 
Our results are different from the previous analyses by neglecting the 1.4 TeV excess <cit.>.
 As a comparison, the newly released CALET lepton data is used to do a similar global fitting, which could give us some more supports on the origin of the lepton excess.
In addition, we present a U(1)_D dark matter model with Breit-Wigner mechanism, which can provide
the proper dark matter annihilation cross section and escape the CMB constraint.
Furthermore, we suggest a few ways to test our proposal as well as the 1.4 TeV excess.







§ STATISTICAL FLUCTUATIONS


In the DAMPE's paper <cit.>, the numbers of events and 
the CRE fluxes with 1σ statistical and systematic errors have been given in its Table 1. 
To evaluate the
uncertainties for numbers of the events, we need to understand their relations.
The relation between the number of events and fluxes in each energy bin is <cit.>

    Φ() = N(E) · (1 - (E))/(E) · T ·Δ E·(E),

where N is the number of () events,  is the effective detector acceptance, 
T is the operating time, Δ E is the energy range of the bin, 
 is the background fraction of the events, and  represents the effects caused 
by other mechanisms which were not given in the Table 1 of Ref. <cit.>.

Taking T = 530 days and = 1.3, we can reproduce the corresponding results 
in the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5 GeV bin
 within the uncertainty < 0.1 %. Consequently, we use the formula

    Φ() = N(E) · (1 - (E))/(E) · T ·Δ E· 1.3

in this letter to calculate the fluxes in these bins.

We calculate the 2σ deviations for the number of events (Δ N_2σ) 
from the flux statistical fluctuations as follows

    Δ N_2σ = ΔΦ()_2 σ_stat/Φ()· N .

Thus, for the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5 GeV bin,
we obtain Δ N_2σ = ± 18, ± 20, ± 12, respectively. 
Assume  -2σ, +2σ, and -1σ deviations for
these bins from statistical fluctuations, 
we have Δ N = + 18, - 20, + 6, respectively. 
Therefore, the revised numbers of events for the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5 GeV bin,
 are 92, 73, and 39, respectively.


Furthermore, we reestimate the statistical uncertainties in these bins based on the revised 
numbers of events via the formula

    Δ N_1σ≃1/√(N) ,

and then calculate the corresponding fluxes and their statistical uncertainties.
The systematical uncertainties are assumed to be invariant. All the detailed information for
 these three bins are given in Table <ref>. By the way, as a cross check,
with Eq. (<ref>), we have reproduced similar 1σ statistical uncertainties 
of the original fluxes
in the DAMPE's paper <cit.>.






§ FITTING PROCEDURE




In CR theory, the CR electrons are considered to be accelerated during the acceleration of CR nuclei at the sources, e.g. SNRs. On the other hand, the CR positrons are produced as secondary particles from CR nuclei interaction with the interstellar medium (ISM) <cit.>. From the observed spectra of positrons and electrons <cit.>, we can conclude that there should exist some extra sources producing electron-positron pairs. As we stated in the first section, these extra sources could be astrophysical sources or DM annihilation or decay. As a result, the CREs data contains (i) the primary electrons; (ii) the secondary electrons; (iii) the secondary positrons; (iv) the extra source of electron-positron pairs. If we want to study the properties of the extra source, we should deduct the primary electrons and secondary electrons/positrons first.

The primary electrons are always assumed to have a power-law injection and the secondary electrons/positrons are determined mainly by the CR proton and helium nucleus interact with ISM. Consequently, we should do global fitting to all these related data simultaneously which can avoid the bias of choosing the lepton background parameters.

The public code  dragon [https://github.com/cosmicrays/DRAGON] <cit.> was used to do numerical calculations. Some custom modifications are performed in the original code, such as the possibility to use specie-dependent injection spectra, which is not allowed by default in dragon.

In view of some discrepancies when fitting  the new data <cit.>, we use a factor c_ to re-scale the helium-4 abundance (which has a default value of 7.199 × 10^4), which helps us to get a better global fitting.






 §.§ Background



In this work, we use the widely used  diffusion-reacceleration model which can give a consistent fitting results to the AMS-02 nuclei data (see for e.g., <cit.>). In the whole propagation region, a uniform diffusion coefficient (D_xx = D_yy= D_zz = D_0β( R/R_0 )^δ) is employed to describe the propagation .

The hardening of the nuclei spectra at ∼ 300 (which has been observed by ATIC-2 <cit.>, CREAM <cit.>, PAMELA <cit.>, and AMS-02 <cit.>) is considered by adding breaks in the primary source injections. At the same time, considering the observed significant difference in the slopes of proton and helium (of about ∼ 0.1 <cit.>), we use separate primary source spectra settings for proton and helium. In summary, for nuclei primary source injections, each of them has 2 breaks at rigidity R_1 and R_2. The corresponding slopes are ν_1 (R ≤ R_1), ν_2 (R_1 < R ≤ R_2) and ν_3 (R > R_3).


For CR electrons primary source, we use one break R_e for electron primary source, and the corresponding slopes are ν_e1 (R ≤ R_e) and ν_e2 ((R > R_e)).
 

In order to  take into account the uncertainties when calculating the secondary CR particles' fluxes, we employ parameters c_ and  to re-scale the calculated secondary flux to fit the data <cit.>. Note that the above mentioned uncertainties may not be simply represented by a constant factor, but most probably are  energy dependent <cit.>. Here we expect that a constant factor could be a simple assumption. 

The force-field approximation <cit.> is used to describe the effects of solar modulation effects. ,  and  are used to modulate the local interstellar spectra of nuclei (proton and helium), anti-proton and positrons respectively, which based on the charge-sign dependence of solar modulation. On the other hand, Because the DAMPE lepton data ≳ 20, we did not consider the modulation effects on electrons (or leptons).






 §.§ Extra Sources



We consider  both pulsar and DM scenarios to generate the CRE excesses in the observed spectrum 
by the DAMPE experiment. 
For the pulsar scenario, a continuous distributed pulsar background was used <cit.>. 
The injection spectrum of such sources is  assumed to be a power law with an exponential cutoff

    q_e^(p) = N_(R/10)^-ν_exp(-R/R_c),

where N_ is the normalization factor, ν_ is the spectral index, and 
R_c is the cutoff rigidity. 
For the DM scenario, we employ the Einasto profile <cit.>

    ρ(r)=ρ_⊙exp[
    -( 2/α)
    (r^α-r_⊙^α/r_s^α)
    ] ,

with α≈ 0.17, r_s≈ 20, and ρ_⊙≈ 0.39  ^-3 is 
the local DM relic density <cit.>. And the source term, which
 we use to add the CRE particles from the annihilations of the Majorana DM particles, is

    Q(r,p)=ρ(r)^2/2 m_χ^2⟨σ v ⟩∑_fη_fdN^(f)/dp ,

where ⟨σ v ⟩ is the velocity-averaged DM annihilation cross section multiplied by DM relative velocity (referred as cross section), ρ(r) is the DM density distribution, and dN^(f)/dp is the injection energy spectrum  of CREs from DM annihilating into the Standard Model (SM) final states via leptonic channels 
ff̅ (, , and ) with η_f (, , and ) 
the corresponding branching fractions. Here, we normalized η_f as +  +  = 1.


The parameters related to the extra source of the leptons for pulsar scenario is (N_, ν_, R_c), and for DM scenario is (, , , , ).






 §.§ Data Sets and Parameters



As in Ref. <cit.>, we perform a global fitting on the data set including the proton fluxes from AMS-02 and CREAM <cit.> helium flux from AMS-02 and CREAM[The CREAM data was used as the supplement of the AMS-02 data because it is more compatible with the AMS-02 data when R ≳ 1.] <cit.>,  ratio from AMS-02 <cit.>, positrons flux from AMS-02 <cit.>, and CRE flux from DAMPE <cit.>, which could account for the primary electrons, the secondary leptons, and the extra leptons in a self-consistent way [The errors used in our global fitting are the quadratic summation over statistical and systematic errors.]. Moreover, the employed AMS-02 positron flux is used to calibrate the positron contribution in the DAMPE CRE flux in energy region ≲ 300[The systematics between AMS-02 and DAMPE are dealt with by employing a re-scale factor  on positron flux.]. The framework of the fitting procedure is the same as our previous work <cit.>, where the details can be found.

Considering the systematics between different CREs spectra observed by different experiments (see in Fig. <ref> and <ref>), we take the newly released CREs spectrum from CALET <cit.> as a comparison to do global fitting as that on DAMPE CREs spectrum.
Because both of these experiments are implemented in space and have a similar ability to obtain CREs data, a reasonable explanation on lepton excess should explain both of them simultaneously.
In this case, DAMPE and CALET could be considered as the maximum and minimum CREs flux examples.



Altogether, the data set in our global fitting is 
 
    D =    {D^AMS-02_, D^AMS-02_,  D^AMS-02_, D^CREAM_, 
       D^CREAM_, D^AMS-02_, D^DAMPE_ / D^CALET_} .


The parameter sets for pulsar scenario is 

    θ_ =     { D_0, δ, z_h, v_A, | N_, R_1, R_2, ν_1, ν_2, ν_3, 
         R_1,  R_2, ν_1, ν_2, ν_3, |  , , c_, c_,  | 
        N_, R_1, ν_1, ν_2, |   
        N_, ν_, R_c, | 
       , c_} ,

for DM scenario is 

    θ_ =     { D_0, δ, z_h, v_A, | N_, R_1, R_2, ν_1, ν_2, ν_3, 
         R_1,  R_2, ν_1, ν_2, ν_3, |  , , c_, c_, | 
        N_, R_1, ν_1, ν_2, |   
       , , η_e, η_μ, η_τ,  | 
       , c_} .


Note that, most of these 2 scenarios' parameters in the set θ_ and θ_ is the same with each other except those which account the extra sources of lepton.










§ RESULTS




When the Markov Chains have reached their equilibrium state, we take the samples of the parameters as their posterior probability distribution functions. 
The best-fit values, statistical mean values, standard deviations and allowed intervals at 95 % CL for parameters in set θ_ and θ_ for DAMPE and CALET   are shown in Appendix, Table <ref> (DAMPE, pulsar scenario), Table <ref> (CALET, pulsar scenario),  Table <ref> (DAMPE, DM scenario), and Table <ref> (CALET, DM scenario), respectively. For best fit results of the global fitting, we got χ^2/d.o.f = 243.13/299 (DAMPE, pulsar scenario), χ^2/d.o.f = 229.98/301 (CALET, pulsar scenario), χ^2/d.o.f = 262.94/297 (DAMPE, DM scenario), and χ^2/d.o.f = 265.03/299 (CALET, DM scenario). 






 §.§ Background



The best-fitting results and the corresponding residuals of the proton flux, helium flux and  ratio for pulsar scenario is showed in Appendix, Fig. <ref>, for DM scenario is showed in Appendix, Fig. <ref>. In these figures, we can see that the nuclei data is perfectly reproduced, which would provide a good precondition for the  fitting on the lepton data.


Although the main purpose of this work does not focus on the CR propagation models, we would like to emphasize some points here: (i) As shown in Appendix, Table <ref>, Table <ref>, Table <ref>, and Table <ref>, we got larger best-fit values of D_0 and z_h than previous works <cit.>. This mainly because the newly released AMS-02 nuclei spectra favor large values of D_0 and z_h.[A similar global fitting result can be found in Ref. <cit.>, which use a different numerical tool galprop to do calculation.] Moreover, the employed 2 breaks in nuclei primary source injection strengthen the classical degeneracy between D_0 and z_h based on the data set we used in this work (without B/C), which both got larger best-fit values in this work. (ii) The employed 2 breaks in the nuclei primary source injection accounted for the observed hardening in the observed spectra, other than use only one break and let δ compromise the different slopes in high energy regions, which lead to a smaller value of δ and fitting uncertainties on δ (≲ 0.01) than previous works.







 §.§ Extra Sources



The fitting results of the pulsar and DM scenario on DAMPE and CALET CREs spectrum are given in Figs. <ref> and <ref> respectively, which also shows some  CREs spectrum from other experiments. Cleaner fitting results are shown in Figs. <ref> and <ref>.[Here, one should note that both proton and positron flux from AMS-02 are well fitted in such configurations, but the coincidence of the proton and positron flux in this work is caused by chance.]  From these figures, we can conclude that both scenarios could provide a excellent fittings to the DAMPE CREs spectrum within 3σ fitting deviation, which do not need to employ extra local sources. At the same time, the CALET CREs spectrum could also be fitted by both scenarios, although the fitting result is not that good because of a suspected bump at about 0.9 - 1 .


For the best fit result on the DAMPE and CALET CREs spectrum, we got χ^2≃ 17.2 (DAMPE, pulsar scenario), χ^2≃ 16.1 (CALET, pulsar scenario), χ^2≃ 13.9 (DAMPE, DM scenario) and χ^2≃ 25.7 (CALET, DM scenario). This indicate that CALET CREs data disfavors the DM scenario because of the defective fit on the suspected bump at about 0.9 - 1 . This need more events accumulated in the future. 


The detailed results of the constraints on the parameters could be found in Appendix, Table <ref>, Table <ref>, Table <ref>, and Table <ref>. Most of the parameters are slightly different between DAMPE and CALET fitting because of the systematics between them. One point we want to mention is that in the fitting results of CALET CREs data, DM scenario, ≃ 0.930, ≃ 0.042, and ≃ 0.028, which is largely different from the DAMPE results.


In the following of this work, we will focus on the analysis of the DAMPE results, whose method could be extended to deal with the CALET results without difficulties. 





















For the pulsar scenario, the fitting results give ν_≃ 0.62, which is obviously different 
from the fitting results in previous works (see for e.g., <cit.>). In standard pulsar models, 
the injection spectrum indices of CREs from pulsars are always 
in the range ν_∈ [1.0,2.4] <cit.>. As a result, 
 more attention should be paid in future researches. This may indicate: 
(i) there is something wrong or inaccuracy with the classical pulsar CRE injection model; 
(ii) the CRE excess is not contributed primarily by pulsars.
Moreover, the cut-off is R_c ≃ 692 GV.
In the previous work <cit.> where  the 1.4 TeV peak excess was neglected,
we obtained that the spectral index of the injection is
ν_≃ 0.65 and the cut-off is R_c ≃ 650 GV.  Thus, there exist about +5% and -5% deviations
for ν_ and R_c, respectively.


For the DM scenario, we obtain ≃ 4.07 × 10^-23^2^-1 and ≃ 1884. 
The value of  is about 3 orders larger than that of thermal DM <cit.>. To explain
this discrepancy, we will present a concrete model in the next section. Moreover, 
we have ≃ 0.465, ≃ 0.510, and ≃ 0.025. So the DM annihilation into
 is highly suppressed, which provides some hints to construct an appropriate DM model.
In our previous work <cit.> where  the 1.4 TeV peak excess was neglected,
we have ≃ 1.48 × 10^-23^2^-1, ≃ 1208, ≃≃ 0.5,
while  is highly suppressed. Thus, we have similar results on branching fractions,
but different DM masses and annihilation cross sections.






§ MODEL BUILDING


Because we have ∼ 0.465, ∼ 0.510, and ∼ 0.025,
the constraints from the Fermi-LAT observations of 
dwarf spheroidal galaxies <cit.> 
can be avoided <cit.>. 
To escape the constraints from the Planck observations of CMB anisotropies <cit.>, we employ
the Breit-Wigner mechanism <cit.>.  
We consider the dark U(1)_D model where the SM fermions and Higgs fields
are neutral under it. We introduce one SM singlet Higgs field S, one chiral fermionic dark matter particle
χ, and three pairs of the vector-like particles (XE_i, XE_i^c), whose quantum numbers under
the SU(3)_C× SU(2)_L × U(1)_Y × U(1)_D are

    S: (1, 1, 0, 2) ,   χ: (1, 1, 0, -1) 
       XE_i: (1, 1, -1, -2) ,   XE^c_i: (1, 1, 1, 2)  .

The relevant Lagrangian is 

    - L   =    -m_S^2 |S|^2 + λ/2 |S|^4 + (M^V_ijXE^c_i XE_j
    . 
          . + y_ij S E_i^c XE_j +y S  χχ +  H.C.)  ,

where E_i^c are the right-handed charged leptons.
For simplicity, we choose M^V_ij = M^V_iδ_ij and y_ij = y_i δ_ij.
After S acquires a Vacuum Expectation Value (VEV), the U(1)_D gauge symmetry is broken
down to a Z_2 symmetry under which χ is odd. Thus, χ is a DM matter candidate.
For simplicity, we assume that the mass of U(1)_D gauge boson is about twice of χ mass, 
i.e., M_Z'≃ 2 m_χ, while the Higgs field S and vector-like particles
are heavier than M_Z'.
Moreover,  E_i^c and XE_i^c will be mixed due to
the M^V_iXE^c_i XE_i and y_i S E_i^c XE_i terms, and we obtain 
the mass eigenstates E_i^c and XE_i^c by neglecting the tiny charged lepton masses

    (
    [  E_i^c; XE_i^c ])=
    (
    [  cosθ_i  sinθ_i; -sinθ_i  cosθ_i ])
    (
    [    E_i^c; XE_i^c ' ])
     ,

where tanθ_i = -y⟨ S ⟩/M^V_i.

Neglecting the charged lepton masses again, we obtain

    σ v = ∑_i=1^3g'^4sin^2θ_i/6πs-m_χ^2/(s-m_Z'^2)^2+(m_Z'Γ_Z')^2 ,

where m_χ = y ⟨ S ⟩, and g' and M_Z' are the gauge coupling and gauge boson mass
for U(1)_D gauge symmetry.

For m_Z'≃ 2m_χ, Z' decays dominantly into leptons, and the decay width is 

    Γ_Z'=∑_i=1^3 g'^2sin^2θ_i/6π m_Z' .



To explain the DM best fit results, we choose

    g' ≃ 0.028,  m_χ≃ 1884 , m_Z'-2m_χ/m_Z'≃ 3.0× 10^-6, 
          sinθ_e ≃ 0.21 ,   sinθ_μ≃ 0.22 ,   sinθ_τ≃ 0.05 .

And then we obtain ⟨σ v⟩≃ 4.07× 10^-23^3^-1,  
and η_e :η_μ : η_τ≃ 0.465 : 0.510 : 0.025. Of course,
there exists fine-tuning between m_Z' and m_χ , which deserves further study.
For some solutions, see Ref. <cit.>.




§ DISCUSSIONS AND CONCLUSION


First, we would like to point out that
if the numbers of events in the 1229.3 GeV bin and 1411.4 GeV bin are exchanged, 
we can also explain the DAMPE's data similarly. Of course,
the most important question is how to test our proposal that
there exists statistical fluctuations in the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5 GeV bin. 
For the data analyses, we suggest that one chooses different energy ranges to study the data again.
For example, we can shift the energy ranges by ± 50 GeV and ± 100 GeV 
for the high energy bins, and then study the corrsponding events and fluxes.
In the future, DAMPE will provide us more accurate spectrum data reaching up 
to ∼ 10, which can give us a unprecedented opportunity to study the origin and 
propagation of CREs. We predict that the CRE spectrum would be more continuous. In particular,
the peak excess in the 1411.4 GeV bin as well as the deficits in the 1229.3 GeV bin and 1620.5 GeV bin 
will all decrease! Moreover,
if the 1.4 TeV peak signal was proved to be correct, we do need a local source of high energy CREs. 
Other experiment is needed as a cross check if such signal arises from DM annihilation, 
for example, our recent work <cit.> proposed a novel scenario to probe 
the interaction between DM particles 
and electrons for the DM mass range 5 ≲≲ 10.


In summary, with the simplicity and naturalness physics principle, we proposed that there exists
the  -2σ, +2σ, and -1σ deviations due to statistical fluctuations for the 
1229.3 GeV bin, 1411.4 GeV bin,  and 1620.5 GeV bin of the DAMPE data. Interestingly, we showed that 
all the DAMPE data can be explained consistently via both the pulsar and 
dark matter interpretations, which have χ^2≃ 17.2 and χ^2≃ 13.9 (for all the 38 points in DAMPE electron/positron spectrum with 3 of them revised), respectively. 
These results are different from the previous analyses by neglecting the 1.4 TeV excess. 
 At the same time, we employed the newly released CALET CREs spectrum to do a similar global fitting, which cold also be fitted by continuous distributed pulsar and DM scenarios.
Moreover, we presented a U(1)_D dark matter model with Breit-Wigner mechanism, which can provide
the proper dark matter annihilation cross section and escape the CMB constraint.
Furthermore, we suggested a few ways to test our proposal. 







We would like to thank Xiao-Jun Bi and Yi-Zhong Fan for helpful discussions, and 
thank D. Maurin, et al. <cit.> for collecting the database and associated online tools  
for charged cosmic-ray measurements.
This research was supported in part by the Projects 11475238 and 11875062 (from National  
Science Foundation of China), and the Projects 11747601 (from Key Research Program of Frontier Sciences, Chinese Academy of Sciences). 
The calculation in this paper are supported by HPC Cluster of SKLTP/ITP-CAS.



























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  author author F. A. Aharonian, author A. M. Atoyan,  and author H. J. Voelk, @noop  journal journal
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[Chi et al.(1996)Chi,
  Cheng, and Young]Chi1996
  author author X. Chi, author K. S. Cheng,
   and author E. C. M. Young,  10.1086/309943 journal journal
   volume 459, pages L83
  (year 1996)NoStop
[Zhang and Cheng(2001)]Zhang2001
  author author L. Zhang and author K. S. Cheng, 10.1051/0004-6361:20010021 journal journal  volume 368,  pages 1063 (year 2001)NoStop
[Bergström(2000)]Bergstrom2000
  author author L. Bergström, 10.1088/0034-4885/63/5/2r3 journal journal Reports on Progress in Physics volume 63, pages 793 (year 2000),  http://arxiv.org/abs/hep-ph/0002126 hep-ph/0002126 NoStop
[Bertone et al.(2005)Bertone, Hooper, and Silk]Bertone2005
  author author G. Bertone, author D. Hooper,
   and author J. Silk, 10.1016/j.physrep.2004.08.031 journal journal  volume 405, pages
  279 (year 2005), http://arxiv.org/abs/hep-ph/0404175 hep-ph/0404175 NoStop
[Bergström et al.(2009)Bergström, Edsjö, and Zaharijas]Bergstrom2009
  author author L. Bergström, author J. Edsjö,  and author G. Zaharijas, 10.1103/PhysRevLett.103.031103 journal journal Physical Review Letters volume 103, eid 031103 (year 2009), http://arxiv.org/abs/0905.0333 arXiv:0905.0333 [astro-ph.HE] NoStop
[Aharonian et al.(2008)Aharonian, Akhperjanian, Barres de
  Almeida, Bazer-Bachi, Becherini,
  Behera, Benbow, Bernlöhr, Boisson, Bochow, and  et al.]Aharonian2008
  author author F. Aharonian, author A. G. Akhperjanian, author U. Barres de Almeida, author A. R.  Bazer-Bachi, author Y. Becherini, author B. Behera, author W. Benbow, author K. Bernlöhr, author C. Boisson, author A. Bochow,  and author et al., 10.1103/PhysRevLett.101.261104 journal journal Physical Review Letters volume 101, eid 261104 (year 2008), http://arxiv.org/abs/0811.3894 arXiv:0811.3894 NoStop
[HESS collaboration(2009)]Aharonian2009
  author author HESS
  collaboration, 10.1051/0004-6361/200913323 journal journal  volume 508,  pages 561 (year 2009), http://arxiv.org/abs/0905.0105 arXiv:0905.0105 [astro-ph.HE] NoStop
[Staszak and for the VERITAS
  Collaboration(2015)]Staszak2015
  author author D. Staszak and author for the VERITAS
  Collaboration, @noop  journal journal
  ArXiv e-prints  (year 2015), http://arxiv.org/abs/1508.06597 arXiv:1508.06597 [astro-ph.HE]
  NoStop
[Holder(2017)]Holder2017
  author author J. Holder, in 10.1063/1.4968898 booktitle 6th International Symposium on High Energy Gamma-Ray
  Astronomy, series American Institute of Physics Conference
  Series, Vol. volume 1792 (year 2017) p.  pages 020013, http://arxiv.org/abs/1609.02881
  arXiv:1609.02881 [astro-ph.HE] NoStop
[Abdollahi et al.(2017)Abdollahi et al.]Abdollahi2017
  author author S. Abdollahi et al. (collaboration Fermi-LAT), 10.1103/PhysRevD.95.082007 journal journal Phys. Rev. volume D95, pages
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  NoStop

[Meehan et al.(2017)Meehan, Vandenbroucke, and for the
  Fermi-LAT Collaboration]Meehan2017
  author author M. Meehan, author J. Vandenbroucke,  and author for the
  Fermi-LAT Collaboration, @noop  journal journal ArXiv e-prints  (year 2017), http://arxiv.org/abs/1708.07796 arXiv:1708.07796 [astro-ph.HE]
  NoStop
[AMS
  collaboration(2014a)]AMS02_lepton_sum
  author author AMS collaboration,  10.1103/PhysRevLett.113.221102 journal
  journal Physical Review Letters volume
  113, eid 221102 (year
  2014a)NoStop
[AMS
  collaboration(2014b)]AMS02_lepton
  author author AMS collaboration,  10.1103/PhysRevLett.113.121102 journal
  journal Physical Review Letters volume
  113, eid 121102 (year
  2014b)NoStop
[AMS
  collaboration(2013a)]AMS02_fraction01
  author author AMS collaboration,  10.1103/PhysRevLett.110.141102 journal
  journal Physical Review Letters volume
  110, eid 141102 (year
  2013a)NoStop
[AMS
  collaboration(2014c)]AMS02_fraction02
  author author AMS collaboration,  10.1103/PhysRevLett.113.121101 journal
  journal Physical Review Letters volume
  113, eid 121101 (year
  2014c)NoStop
[collaboration(2017a)]CALET2017
  author author C. collaboration (collaboration CALET Collaboration), 10.1103/PhysRevLett.119.181101 journal journal Phys. Rev. Lett. volume 119, pages 181101 (year 2017a)NoStop
[Adriani et al.(2018)Adriani, Akaike, Asano, Asaoka, Bagliesi, Berti, Bigongiari, Binns, Bonechi, Bongi, Brogi, Buckley, Cannady, Castellini, Checchia,
  Cherry, Collazuol, Di Felice,
  Ebisawa, Fuke, Guzik,
  Hams, Hareyama, Hasebe,
  Hibino, Ichimura, Ioka,
  Ishizaki, Israel, Kasahara,
  Kataoka, Kataoka, Katayose,
  Kato, Kawanaka, Kawakubo,
  Kohri, Krawczynski, Krizmanic, Lomtadze, Maestro, Marrocchesi, Messineo, Mitchell,
  Miyake, Moiseev, Mori,
  Mori, Mori, Motz,
  Munakata, Murakami, Nakahira,
  Nishimura, de Nolfo, Okuno,
  Ormes, Ozawa, Pacini,
  Palma, Papini, Penacchioni,
  Rauch, Ricciarini, Sakai,
  Sakamoto, Sasaki, Shimizu,
  Shiomi, Sparvoli, Spillantini, Stolzi, Suh, Sulaj, Takahashi, Takayanagi, Takita, Tamura, Tateyama, Terasawa, Tomida, Torii, Tsunesada, Uchihori, Ueno, Vannuccini, Wefel, Yamaoka, Yanagita, Yoshida, and Yoshida]CALET2018
  author author O. Adriani, author Y. Akaike,
  author K. Asano, author Y. Asaoka, author
  M. G. Bagliesi, author
  E. Berti, author G. Bigongiari, author W. R.  Binns, author S. Bonechi, author M. Bongi,
  author P. Brogi, author J. H. Buckley, author
  N. Cannady, author G. Castellini, author C. Checchia, author M. L.  Cherry, author G. Collazuol, author V. Di Felice, author K. Ebisawa,
  author H. Fuke, author
  T. G. Guzik, author
  T. Hams, author M. Hareyama, author N. Hasebe, author K. Hibino, author M. Ichimura, author K. Ioka, author W. Ishizaki, author M. H. Israel, author K. Kasahara,
  author J. Kataoka, author R. Kataoka, author
  Y. Katayose, author
  C. Kato, author N. Kawanaka, author Y. Kawakubo, author K. Kohri, author H. S.  Krawczynski, author J. F.  Krizmanic, author T. Lomtadze, author P. Maestro,
  author P. S. Marrocchesi,
  author A. M. Messineo, author J. W. Mitchell, author
  S. Miyake, author A. A. Moiseev, author K. Mori, author M. Mori, author N. Mori, author H. M. Motz, author
  K. Munakata, author
  H. Murakami, author
  S. Nakahira, author
  J. Nishimura, author
  G. A. de Nolfo, author
  S. Okuno, author J. F. Ormes, author S. Ozawa, author L. Pacini, author F. Palma, author P. Papini, author A. V.  Penacchioni, author B. F.  Rauch, author S. B. Ricciarini, author K. Sakai,
  author T. Sakamoto, author M. Sasaki, author
  Y. Shimizu, author A. Shiomi, author R. Sparvoli, author P. Spillantini, author F. Stolzi, author J. E.  Suh, author A. Sulaj, author I. Takahashi,
  author M. Takayanagi, author M. Takita, author
  T. Tamura, author N. Tateyama, author T. Terasawa, author H. Tomida, author S. Torii, author Y. Tsunesada, author Y. Uchihori, author S. Ueno, author E. Vannuccini, author J. P. Wefel, author K. Yamaoka,
  author S. Yanagita, author A. Yoshida,  and author K. Yoshida (collaboration CALET Collaboration), 10.1103/PhysRevLett.120.261102 journal journal
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[Chang et al.(2008)Chang, Adams, Ahn, Bashindzhagyan, Christl, Ganel,
  Guzik, Isbert, Kim,
  Kuznetsov, Panasyuk, Panov, Schmidt, Seo, Sokolskaya, Watts, Wefel,
  Wu, and Zatsepin]Chang2008
  author author J. Chang, author J. H. Adams, author H. S. Ahn,
  author G. L. Bashindzhagyan,
  author M. Christl, author O. Ganel, author
  T. G. Guzik, author
  J. Isbert, author
  K. C. Kim, author
  E. N. Kuznetsov, author
  M. I. Panasyuk, author
  A. D. Panov, author
  W. K. H. Schmidt, author
  E. S. Seo, author
  N. V. Sokolskaya, author
  J. W. Watts, author
  J. P. Wefel, author
  J. Wu,  and author
  V. I. Zatsepin, 10.1038/nature07477 journal journal   volume 456, pages 362 (year
  2008)NoStop
[Fermi-LAT collaboration(2009)]Abdo2009
  author author Fermi-LAT
  collaboration, 10.1103/PhysRevLett.102.181101
  journal journal Physical Review Letters  volume 102, eid 181101 (year
  2009), http://arxiv.org/abs/0905.0025 arXiv:0905.0025
  [astro-ph.HE] NoStop
[Aguilar et al.(2014)Aguilar
  et al.]Aguilar2014
  author author M. Aguilar et al. (collaboration AMS), 10.1103/PhysRevLett.113.121102 journal journal
  Phys. Rev. Lett. volume 113, pages
  121102 (year 2014)NoStop

[Adriani et al.(2009)Adriani, Barbarino, Bazilevskaya,
  Bellotti, Boezio, Bogomolov, Bonechi, Bongi,
  Bonvicini, Bottai, and et al.]Adriani2009
  author author O. Adriani, author G. C. Barbarino, author G. A. Bazilevskaya, author R. Bellotti, author M. Boezio, author E. A. Bogomolov, author L. Bonechi, author M. Bongi,
  author V. Bonvicini, author S. Bottai,  and author et al., 10.1038/nature07942
  journal journal  volume 458, pages 607 (year 2009), http://arxiv.org/abs/0810.4995 arXiv:0810.4995 NoStop
[Fermi-LAT collaboration(2012)]Ackermann2012
  author author Fermi-LAT
  collaboration, 10.1103/PhysRevLett.108.011103
  journal journal Physical Review Letters  volume 108, eid 011103 (year
  2012), http://arxiv.org/abs/1109.0521 arXiv:1109.0521
  [astro-ph.HE] NoStop
[Aguilar et al.(2013)Aguilar
  et al.]Aguilar2013
  author author M. Aguilar et al. (collaboration AMS), 10.1103/PhysRevLett.110.141102 journal journal
  Phys. Rev. Lett. volume 110, pages
  141102 (year 2013)NoStop

[Accardo et al.(2014)Accardo
  et al.]Accardo2014
  author author L. Accardo et al. (collaboration AMS), 10.1103/PhysRevLett.113.121101 journal journal
  Phys. Rev. Lett. volume 113, pages
  121101 (year 2014)NoStop

[Ambrosi et al.(2017)Ambrosi
  et al.]DAMPE2017
  author author G. Ambrosi et al. (collaboration DAMPE collaboration),  10.1038/nature24475 journal journal Nature  (year 2017), 10.1038/nature24475,  http://arxiv.org/abs/1711.10981 arXiv:1711.10981 [astro-ph.HE]
  NoStop

[Gu and He(2017)]Gu2017
  author author P.-H. Gu and author X.-G. He, @noop  journal journal ArXiv
  e-prints  (year 2017), http://arxiv.org/abs/1711.11000 arXiv:1711.11000 [hep-ph] NoStop
[Fang et al.(2017)Fang,
  Bi, and Yin]Fang2017
  author author K. Fang, author X.-J. Bi,   and author P.-F. Yin,  @noop  journal journal ArXiv e-prints 
  (year 2017), http://arxiv.org/abs/1711.10996
  arXiv:1711.10996 [astro-ph.HE] NoStop
[Fan et al.(2017)Fan,
  Huang, Spinrath, Sming
  Tsai, and Yuan]Fan2017
  author author Y.-Z. Fan, author W.-C. Huang,
  author M. Spinrath, author Y.-L. Sming Tsai,  and  author Q. Yuan, @noop 
  journal journal ArXiv e-prints  (year 2017), http://arxiv.org/abs/1711.10995 arXiv:1711.10995
  [hep-ph] NoStop
[Yuan et al.(2017a)Yuan, Feng,
  Yin, Fan, Bi,
  Cui, Dong, Guo,
  Fang, Hu, Huang,
  Lei, Li, Lin,
  Liu, Ma, Peng,
  Qiao, Shen, Su,
  Wei, Xu, Yue,
  Zang, Zhang, Zhang,
  Zhang, Zhang, and Zhang]Yuan2017_dampe
  author author Q. Yuan, author L. Feng,
  author P.-F. Yin, author Y.-Z. Fan, author
  X.-J. Bi, author
  M.-Y. Cui, author
  T.-K. Dong, author
  Y.-Q. Guo, author
  K. Fang, author H.-B. Hu, author X. Huang, author S.-J.  Lei, author X. Li, author S.-J. Lin,
  author H. Liu, author P.-X. Ma, author
  W.-X. Peng, author
  R. Qiao, author Z.-Q. Shen, author M. Su, author Y.-F.  Wei, author Z.-L. Xu, author C. Yue,
  author J.-J. Zang, author C. Zhang, author
  X. Zhang, author Y.-P. Zhang, author Y.-J.  Zhang,  and author Y.-L.  Zhang, @noop  journal journal ArXiv e-prints  (year 2017a),  http://arxiv.org/abs/1711.10989 arXiv:1711.10989 [astro-ph.HE]
  NoStop
[Duan et al.(2017)Duan,
  Feng, Wang, Wu,
  Yang, and Zheng]Duan2017
  author author G. H. Duan, author L. Feng,
  author F. Wang, author L. Wu, author
  J. M. Yang,  and author
  R. Zheng, @noop  journal journal ArXiv e-prints  (year
  2017), http://arxiv.org/abs/1711.11012 arXiv:1711.11012
  [hep-ph] NoStop
[Gu(2017a)]Gu2017a
  author author P.-H. Gu, @noop  journal journal ArXiv
  e-prints  (year 2017a), http://arxiv.org/abs/1711.11333 arXiv:1711.11333 [hep-ph] NoStop

[Chao and Yuan(2017)]Chao2017
  author author W. Chao and author Q. Yuan,  @noop  journal journal ArXiv e-prints 
  (year 2017), http://arxiv.org/abs/1711.11182
  arXiv:1711.11182 [hep-ph] NoStop

[Tang et al.(2017)Tang,
  Wu, Zhang, and Zheng]Tang2017
  author author Y.-L. Tang, author L. Wu, author M. Zhang,  and author
  R. Zheng, @noop  journal journal ArXiv e-prints  (year
  2017), http://arxiv.org/abs/1711.11058 arXiv:1711.11058
  [hep-ph] NoStop

[Zu et al.(2017)Zu,
  Zhang, Feng, Yuan, and  Fan]Zu2017
  author author L. Zu, author C. Zhang, author L. Feng, author
  Q. Yuan,  and author
  Y.-Z. Fan, @noop  journal journal ArXiv e-prints  (year
  2017), http://arxiv.org/abs/1711.11052 arXiv:1711.11052
  [hep-ph] NoStop

[Liu and Liu(2017)]Liu2017
  author author X. Liu and author Z. Liu,  @noop  journal journal ArXiv e-prints 
  (year 2017), http://arxiv.org/abs/1711.11579
  arXiv:1711.11579 [hep-ph] NoStop
[Cao et al.(2017)Cao,
  Feng, Guo, Shang,
  Wang, and Wu]Cao2017
  author author J. Cao, author L. Feng,
  author X. Guo, author L. Shang, author
  F. Wang,  and author
  P. Wu, @noop  journal journal ArXiv e-prints  (year
  2017), http://arxiv.org/abs/1711.11452 arXiv:1711.11452
  [hep-ph] NoStop
[Athron et al.(2017)Athron, Balazs, Fowlie, and  Zhang]Athron2017
  author author P. Athron, author C. Balazs,
  author A. Fowlie,  and  author Y. Zhang, @noop
   journal journal ArXiv e-prints  (year 2017), http://arxiv.org/abs/1711.11376 arXiv:1711.11376
  [hep-ph] NoStop
[Chao et al.(2017)Chao,
  Guo, Li, and Shu]Chao2017a
  author author W. Chao, author H.-K. Guo,
  author H.-L. Li,  and author J. Shu, @noop  journal journal ArXiv e-prints  (year
  2017), http://arxiv.org/abs/1712.00037 arXiv:1712.00037
  [hep-ph] NoStop

[Gao and Ma(2017)]Gao2017
  author author Y. Gao and author Y.-Z. Ma,  @noop  journal journal ArXiv e-prints 
  (year 2017), http://arxiv.org/abs/1712.00370
  arXiv:1712.00370 [astro-ph.HE] NoStop

[Niu et al.(2018a)Niu, Li,
  Ding, Zhu, Xue, and  Wang]Niu2017_dampe
  author author J.-S. Niu, author T. Li,
  author R. Ding, author B. Zhu, author
  H.-F. Xue,  and author
  Y. Wang, 10.1103/PhysRevD.97.083012 journal journal
   volume 97, eid 083012 (year 2018a), http://arxiv.org/abs/1712.00372
  arXiv:1712.00372 [astro-ph.HE] NoStop
[Jin et al.(2017)Jin,
  Yue, Zhang, and Chen]Jin2017_dampe
  author author H.-B. Jin, author B. Yue,
  author X. Zhang,  and  author X. Chen, @noop 
  journal journal ArXiv e-prints  (year 2017), http://arxiv.org/abs/1712.00362 arXiv:1712.00362
  [astro-ph.HE] NoStop
[Huang et al.(2017)Huang, Wu, Zhang, and Zhou]Huang2017
  author author X.-J. Huang, author Y.-L. Wu,
  author W.-H. Zhang,  and  author Y.-F. Zhou,  @noop  journal journal ArXiv e-prints 
  (year 2017), http://arxiv.org/abs/1712.00005
  arXiv:1712.00005 [astro-ph.HE] NoStop
[Duan et al.(2017)Duan,
  Feng, Wang, Wu, Yang, and Zheng]Duan2017a
  author author G. H. Duan, author L. Feng, author F. Wang, author
  L. Wu, author J. M. Yang,  and author R. Zheng, @noop  journal journal ArXiv e-prints  (year 2017), http://arxiv.org/abs/1711.11012 arXiv:1711.11012 [hep-ph] NoStop

[Cao et al.(2017a)Cao, Feng, Guo, Shang,
  Wang, Wu, and Zu]Cao2017a
  author author J. Cao, author L. Feng, author X. Guo, author
  L. Shang, author F. Wang, author P. Wu,  and author L. Zu,  @noop  journal journal ArXiv e-prints 
  (year 2017a), http://arxiv.org/abs/1712.01244 arXiv:1712.01244 [hep-ph] NoStop

[Ghorbani and Ghorbani(2017)]Ghorbani2017
  author author K. Ghorbani and author P. H. Ghorbani, @noop  journal journal
  ArXiv e-prints  (year 2017), http://arxiv.org/abs/1712.01239 arXiv:1712.01239 [hep-ph] NoStop

[Nomura and Okada(2017)]Nomura2017
  author author T. Nomura and author H. Okada,  @noop  journal journal ArXiv e-prints 
  (year 2017), http://arxiv.org/abs/1712.00941
  arXiv:1712.00941 [hep-ph] NoStop

[Gu(2017b)]Gu2017b
  author author P.-H. Gu, @noop  journal journal ArXiv
  e-prints  (year 2017b), http://arxiv.org/abs/1712.00922 arXiv:1712.00922 [hep-ph] NoStop

[Li et al.(2017)Li,
  Okada, and Shafi]Li2017
  author author T. Li, author N. Okada,  and  author Q. Shafi, @noop 
  journal journal ArXiv e-prints  (year 2017), http://arxiv.org/abs/1712.00869 arXiv:1712.00869
  [hep-ph] NoStop

[Chen et al.(2017)Chen,
  Chiang, and Nomura]Chen2017
  author author C.-H. Chen, author C.-W. Chiang,   and author T. Nomura,  @noop  journal journal ArXiv e-prints 
  (year 2017), http://arxiv.org/abs/1712.00793
  arXiv:1712.00793 [hep-ph] NoStop

[Yang et al.(2017)Yang,
  Su, and Zhao]Yang2017
  author author F. Yang, author M. Su,  and  author Y. Zhao, @noop 
  journal journal ArXiv e-prints  (year 2017), http://arxiv.org/abs/1712.01724 arXiv:1712.01724
  [astro-ph.HE] NoStop
[Ding et al.(2017)Ding,
  Han, Feng, and Zhu]Ding2017
  author author R. Ding, author Z.-L. Han,
  author L. Feng,  and author B. Zhu, @noop  journal journal ArXiv e-prints  (year
  2017), http://arxiv.org/abs/1712.02021 arXiv:1712.02021
  [hep-ph] NoStop

[Ge and He(2017)]Ge2017
  author author S.-F. Ge and author H.-J. He, @noop  journal journal ArXiv
  e-prints  (year 2017), http://arxiv.org/abs/1712.02744 arXiv:1712.02744 [astro-ph.HE]
  NoStop
[Okada and Seto(2017)]Okada2017
  author author N. Okada and author O. Seto,  @noop  journal journal ArXiv e-prints 
  (year 2017), http://arxiv.org/abs/1712.03652
  arXiv:1712.03652 [hep-ph] NoStop

[Sui and Zhang(2017)]Sui2017
  author author Y. Sui and author Y. Zhang,  @noop  journal journal ArXiv e-prints 
  (year 2017), http://arxiv.org/abs/1712.03642
  arXiv:1712.03642 [hep-ph] NoStop

[Cao et al.(2017b)Cao, Guo, Shang, Wang,  and Wu]Cao2017b
  author author J. Cao, author X. Guo, author L. Shang, author
  F. Wang,  and author
  P. Wu, @noop  journal
  journal ArXiv e-prints  (year
  2017b), http://arxiv.org/abs/1712.05351
  arXiv:1712.05351 [hep-ph] NoStop

[Han et al.(2017)Han,
  Wang, and Ding]Han2017
  author author Z.-L. Han, author W. Wang,  and  author R. Ding, @noop 
  journal journal ArXiv e-prints  (year 2017), http://arxiv.org/abs/1712.05722 arXiv:1712.05722
  [hep-ph] NoStop

[Cholis et al.(2017)Cholis, Karwal, and Kamionkowski]Cholis2017
  author author I. Cholis, author T. Karwal,
   and author M. Kamionkowski,  @noop  journal journal ArXiv e-prints 
  (year 2017), http://arxiv.org/abs/1712.00011
  arXiv:1712.00011 [astro-ph.HE] NoStop
[Fowlie(2017)]Fowlie:2017fya
  author author A. Fowlie, @noop  journal journal
  ArXiv e-prints  (year 2017), http://arxiv.org/abs/1712.05089 arXiv:1712.05089 [hep-ph] NoStop
[Kachelrieß et al.(2015)Kachelrieß, Neronov, and Semikoz]KNS2015
  author author M. Kachelrieß, author A. Neronov,  and author D. V. Semikoz, 10.1103/PhysRevLett.115.181103 journal journal Physical Review Letters volume 115, eid 181103 (year 2015), http://arxiv.org/abs/1504.06472 arXiv:1504.06472 [astro-ph.HE]
  NoStop
[Kachelrieß et al.(2018)Kachelrieß, Neronov, and Semikoz]KNS2018
  author author M. Kachelrieß, author A. Neronov,  and author D. V. Semikoz, 10.1103/PhysRevD.97.063011 journal journal  volume 97,  eid 063011 (year 2018), http://arxiv.org/abs/1710.02321 arXiv:1710.02321 [astro-ph.HE]
  NoStop
[AMS collaboration(2013b)]AMS2013
  author author AMS collaboration,  10.1103/PhysRevLett.110.141102 journal
  journal Physical Review Letters volume
  110, eid 141102 (year
  2013b)NoStop
[HEAT collaboration(1997)]Barwick1997
  author author HEAT
  collaboration, 10.1086/310706 journal
  journal  volume 482, pages L191 (year 1997), http://arxiv.org/abs/astro-ph/9703192 astro-ph/9703192 NoStop
[AMS-01 collaboration(2007)]AMS01
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[AMS
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[Planck collaboration(2015)]Ade:2015xua
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[Guo and Wu(2009)]Guo:2009aj
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[Bi et al.(2011)Bi,
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[Hisano et al.(2011)Hisano, Kawasaki, Kohri,
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[Bai et al.(2017)Bai,
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[Xiang et al.(2017)Xiang, Bi, Lin, and Yin]Xiang:2017jou
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§ APPENDIX


































































