A Simple and Natural Interpretations of the DAMPE Cosmic Ray Electron/Positron Spectrum within Two Sigma Deviations

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 $1\sigma$ deficits. With the basic physics principles such as simplicity and naturalness, we consider the $-2\sigma$, $+2\sigma$, and $-1\sigma$ 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 $\chi^{2} \simeq 17.2 $ and $\chi^{2} \simeq 13.9$ (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.

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 [22]. 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 [9,10]. And there exists a tentative peak-like flux excess around 1.4 TeV. Thus, the DAMPE results have stimulated the extensive studies . The spectral break can be explained by the broad distributed pulsars, pulsar wind nebulae (PWNe), supernova remnants (SNRs) [24,26], and by the dark matter annihilation and decay in the galaxy halo [26,34,37,38]. Also, the tentative peak is always interpreted by local pulsars, PWNe, and SNRs [24,26,54]), and by the DM sub-halos, clumps, and mini-spikes [23, 25, 27, 32-34, 38, 39, 47, 49].
However, one can easily show that it is impossible to explain both the spectral break and the tentative peak simultaneously [24,26,38,39,54] [22]. The number of events and fluxes for these bins are given in Table I. From Figure  2 of the DAMPE's paper [22], 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, respectively. Our results are different from the previous analyses by neglecting the Energy Bins (GeV) N (original) Φ(e − + e + ) ± σstat ± σsys (original) N (revised) Φ(e − + e + ) ± σstat ± σsys (revised)  1.4 TeV excess [37]. 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 [22], 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 [13,22] Φ where N is the number of (e − + e + ) events, A eff is the effective detector acceptance, T is the operating time, ∆E is the energy range of the bin, ε bg is the background fraction of the events, and ε other represents the effects caused by other mechanisms which were not given in the Table 1 of Ref. [22]. Taking T = 530 days and ε other = 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 in this letter to calculate the fluxes in these bins. We calculate the 2σ deviations for the number of events (∆N 2σ ) from the flux statistical fluctuations as follows Thus, for the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5 GeV bin, we obtain ∆N 2σ = ±18, ± 20, ± 12, respectively. Assume −2σ, +2σ, and −1σ deviations for these bins from statistical fluctuations, we have ∆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 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 I. By the way, as a cross check, with Eq. (4), we have reproduced similar 1σ statistical uncertainties of the original fluxes in the DAMPE's paper [22]. Fitting Procedure-As in Ref. [37], we perform a global fitting on the data set including the proton fluxes from AMS-02 and CREAM [56,57] helium flux from AMS-02 and CREAM [57,58],p/p ratio from AMS-02 [59], positrons flux from AMS-02 [13], and CRE flux from DAMPE [22], which could account for the primary electrons, the secondary leptons, and the extra leptons in a self-consistent way. Moreover, the employed AMS-02 positron flux is used to calibrate the positron contribution in the DAMPE CRE flux in energy region < ∼ 300 GeV. The framework of the fitting procedure is the same as our previous work [37], where the details can be found.
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 [37,60]. The injection spectrum of such sources is assumed to be a power law with an exponential cutoff where N psr is the normalization factor, ν psr is the spectral index, and R c is the cutoff rigidity. For the DM scenario, we employ the Einasto profile [61][62][63][64] with α ≈ 0.17, r s ≈ 20 kpc, and ρ ⊙ ≈ 0.39 GeV cm −3 is the local DM relic density [65][66][67][68][69]. And the source term, which we use to add the CRE particles from the annihilations of the Majorana DM particles, is where σv 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 ff (e − e + , µμ, and ττ ) with η f (η e , η µ , and η τ ) the corresponding branching fractions. Here, we normalized η f as η e + η µ + η τ = 1.
The parameters related to the extra source of the leptons for pulsar scenario is (N psr , ν psr , R c ), and for DM scenario is (m χ , σv , η e , η µ , η τ ).
Results-The fitting results of the pulsar and DM scenario on the DAMPE CRE spectrum are given in Figs. 1  and 2, respectively. From these figures, we can conclude that both scenarios could provide the excellent fittings to the DAMPE CRE spectrum within 3σ fitting deviation, which do not need to employ extra local sources. For the best fit result on the DAMPE CRE spectrum, we have χ 2 ≃ 17.2 and χ 2 ≃ 13.9 for pulsar and DM scenarios, respectively. For the pulsar scenario, the fitting results give ν psr ≃ 0.62, which is obviously different from the fitting results in previous works (see for e.g., [70]). In standard pulsar models, the injection spectrum indices of CREs from pulsars are always in the range ν psr ∈ [1.0, 2.4] [71][72][73]. 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 [37] where the 1.4 TeV peak excess was neglected, we obtained that the spectral index of the injection is ν psr ≃ 0.65 and the cut-off is R c ≃ 650 GV. Thus, there exist about +5% and −5% deviations for ν psr and R c , respectively.
For the DM scenario, we obtain σv ≃ 4.07 × 10 −23 cm 2 s −1 and m χ ≃ 1884 GeV. The value of σv is about 3 orders larger than that of thermal DM [74]. To explain this discrepancy, we will present a concrete model in the next section. Moreover, we have η e ≃ 0.465, η µ ≃ 0.510, and η τ ≃ 0.025. So the DM annihilation into ττ is highly suppressed, which provides some hints to construct an appropriate DM model. In our previous work [37] where the 1.4 TeV peak excess was neglected, we have σv ≃ 1.48 × 10 −23 cm 2 s −1 , m χ ≃ 1208 GeV, η e ≃ η µ ≃ 0.5, while η τ is highly suppressed. Thus, we have similar results on branching fractions, but different DM masses and annihilation cross sections.
The relevant Lagrangian is where E c i are the right-handed charged leptons. For simplicity, we choose M V ij = M V i δ 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 ′ ≃ 2m χ , while the Higgs field S and vector-like particles are heavier than M Z ′ . Moreover, E c i and XE c i will be mixed due to the M V i XE c i XE i and y i S E c i XE i terms, and we obtain the mass eigenstates E c i and XE c i by neglecting the tiny charged lepton masses where tan θ i = −y S /M V i . Neglecting the charged lepton masses again, we obtain 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 To explain the DM best fit results, we choose sin θ e ≃ 0.21 , sin θ µ ≃ 0.22 , sin θ τ ≃ 0.05 .
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 TeV, 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 [90] proposed a novel scenario to probe the interaction between DM particles and electrons for the DM mass range 5 GeV < ∼ m χ < ∼ 10 TeV. 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, respectively. These results are different from the previous analyses by neglecting the 1.4 TeV excess. 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. The details for global fittings will be given elsewhere [91].