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

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

Jia-Shu Niu jsniu@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    Tianjun Li 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    Fang-Zhou Xu xfz14@mails.tsinghua.edu.cn Institute of Modern Physics and Center for High Energy Physics, Tsinghua University, Beijing 100084, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China
July 16, 2019
Abstract

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 deficits. With the basic physics principles such as simplicity and naturalness, we consider the , , and 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 and , respectively. These results are different from the previous analyses by neglecting the 1.4 TeV excess. Moreover, we present a 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.

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 (Shen, 1970; Harding and Ramaty, 1987; Aharonian et al., 1995; Chi et al., 1996; Zhang and Cheng, 2001) and dark matter (DM) (Bergström, 2000; Bertone et al., 2005; Bergström et al., 2009) 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 (Aharonian et al., 2008; HESS collaboration, 2009), VERITAS (Staszak and for the VERITAS Collaboration, 2015; *Holder2017), FermiLAT (Abdollahi et al., 2017; *Meehan2017), AMS-02 (AMS collaboration, 2014a; *AMS02_lepton; *AMS02_fraction01; *AMS02_fraction02), and CALET (Adriani et al., 2017). In particular, the excesses of the electrons (Chang et al., 2008; Aharonian et al., 2008; Fermi-LAT collaboration, 2009; Aguilar et al., 2014, 2014) and positrons (Adriani et al., 2009; Fermi-LAT collaboration, 2012; Aguilar et al., 2013; Accardo et al., 2014) 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 () flux from 25 GeV to 4.6 TeV with unprecedentedly high quality (Ambrosi et al., 2017). The energy resolution of the DAMPE is better than 1.5% at TeV energies, and the hadron rejection power is about . 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 (Aharonian et al., 2008; HESS collaboration, 2009). And there exists a tentative peak-like flux excess around 1.4 TeV. Thus, the DAMPE results have stimulated the extensive studies (Gu and He, 2017; Fang et al., 2017; Fan et al., 2017; Yuan et al., 2017; Duan et al., 2017; Gu, 2017a; Chao and Yuan, 2017; Tang et al., 2017; Zu et al., 2017; Liu and Liu, 2017; Cao et al., 2017; Athron et al., 2017; Chao et al., 2017; Gao and Ma, 2017; Niu et al., 2017a; Jin et al., 2017; Huang et al., 2017; Duan et al., 2017; Cao et al., 2017a; Ghorbani and Ghorbani, 2017; Nomura and Okada, 2017; Gu, 2017b; Li et al., 2017; Chen et al., 2017; Yang et al., 2017; Ding et al., 2017; Ge and He, 2017; Liu and Liu, 2017; Okada and Seto, 2017; Sui and Zhang, 2017; Cao et al., 2017b; Han et al., 2017; Niu et al., 2017a; Cholis et al., 2017; Fowlie, 2017). The spectral break can be explained by the broad distributed pulsars, pulsar wind nebulae (PWNe), supernova remnants (SNRs) (Fang et al., 2017; Yuan et al., 2017), and by the dark matter annihilation and decay in the galaxy halo (Yuan et al., 2017; Niu et al., 2017a; Athron et al., 2017; Jin et al., 2017). Also, the tentative peak is always interpreted by local pulsars, PWNe, and SNRs (Fang et al., 2017; Yuan et al., 2017; Cholis et al., 2017)), and by the DM sub-halos, clumps, and mini-spikes (Fan et al., 2017; Duan et al., 2017; Gu and He, 2017; Athron et al., 2017; Liu and Liu, 2017; Cao et al., 2017; Jin et al., 2017; Huang et al., 2017; Yang et al., 2017; Ge and He, 2017).

However, one can easily show that it is impossible to explain both the spectral break and the tentative peak simultaneously (Fang et al., 2017; Yuan et al., 2017; Cholis et al., 2017; Jin et al., 2017; Huang et al., 2017)). 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 (Ambrosi et al., 2017). The number of events and fluxes for these bins are given in Table 1. From Figure 2 of the DAMPE’s paper (Ambrosi et al., 2017), it is obvious that the 1411.4 GeV bin has a little bit more than excess, while the 1229.3 GeV bin and 1620.5 GeV bin have about 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.

Energy Bins (GeV) (original) (original) (revised) (revised)
74 92 +18 18
93 73 -20 20
33 39 +6 12

Table 1: The original and revised numbers of events and fluxes, , and for the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5 GeV bin. Here, and are the adjusted numbers of events and the numbers of events for deviations from statistical fluctuations. Thus, we should require .

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 , , and 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 and , respectively. Our results are different from the previous analyses by neglecting the 1.4 TeV excess (Niu et al., 2017a). In addition, we present a 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 (Ambrosi et al., 2017), the numbers of events and the CRE fluxes with 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 (Ambrosi et al., 2017; AMS collaboration, 2014a)

(1)

where is the number of () events, is the effective detector acceptance, is the operating time, 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. (Ambrosi et al., 2017).

Taking  days and , we can reproduce the corresponding results in the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5 GeV bin within the uncertainty . Consequently, we use the formula

(2)

in this letter to calculate the fluxes in these bins.

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

(3)

Thus, for the 1229.3 GeV bin, 1411.4 GeV bin, and 1620.5 GeV bin, we obtain , respectively. Assume , , and deviations for these bins from statistical fluctuations, we have , 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

(4)

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 1. By the way, as a cross check, with Eq. (4), we have reproduced similar statistical uncertainties of the original fluxes in the DAMPE’s paper (Ambrosi et al., 2017).

Fitting Procedure–As in Ref. (Niu et al., 2017a), we perform a global fitting on the data set including the proton fluxes from AMS-02 and CREAM (AMS collaboration, 2015a; Ahn et al., 2010) helium flux from AMS-02 and CREAM (AMS collaboration, 2015b; Ahn et al., 2010), ratio from AMS-02 (AMS collaboration, 2016), positrons flux from AMS-02 (AMS collaboration, 2014b), and CRE flux from DAMPE (Ambrosi et al., 2017), 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 . The framework of the fitting procedure is the same as our previous work (Niu et al., 2017a), 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 (Lin et al., 2015; Niu et al., 2017a). The injection spectrum of such sources is assumed to be a power law with an exponential cutoff

(5)

where is the normalization factor, is the spectral index, and is the cutoff rigidity. For the DM scenario, we employ the Einasto profile (Navarro et al., 2004; Merritt et al., 2006; Einasto, 2009; Navarro et al., 2010)

(6)

with , , and is the local DM relic density (Catena and Ullio, 2010; Weber and de Boer, 2010; Salucci et al., 2010; Pato et al., 2010; Iocco et al., 2011). And the source term, which we use to add the CRE particles from the annihilations of the Majorana DM particles, is

(7)

where is the velocity-averaged DM annihilation cross section multiplied by DM relative velocity (referred as cross section), is the DM density distribution, and is the injection energy spectrum of CREs from DM annihilating into the Standard Model (SM) final states via leptonic channels (, , and ) with (, , and ) the corresponding branching fractions. Here, we normalized as .

The parameters related to the extra source of the leptons for pulsar scenario is , and for DM scenario is .

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 fitting deviation, which do not need to employ extra local sources. For the best fit result on the DAMPE CRE spectrum, we have and for pulsar and DM scenarios, respectively.

Figure 1: The global fitting results and the corresponding residuals to the DAMPE lepton flux for pulsar scenario. The (deep red) and (light red) bounds are also shown in the figure. The three relevant bins with revised fluxes are plotted as blue dots. And we have .
Figure 2: The global fitting results and the corresponding residuals to the DAMPE lepton flux for DM scenario. The color code is the same as Fig. 1, and we have .

For the pulsar scenario, the fitting results give , which is obviously different from the fitting results in previous works (see for e.g., (Profumo, 2012)). In standard pulsar models, the injection spectrum indices of CREs from pulsars are always in the range (Reynolds, 1988; Thompson et al., 1994; Fierro et al., 1995). 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  GV. In the previous work (Niu et al., 2017a) where the 1.4 TeV peak excess was neglected, we obtained that the spectral index of the injection is and the cut-off is  GV. Thus, there exist about and deviations for and , respectively.

For the DM scenario, we obtain and . The value of is about 3 orders larger than that of thermal DM (Jungman et al., 1996). To explain this discrepancy, we will present a concrete model in the next section. Moreover, we have , , and . So the DM annihilation into is highly suppressed, which provides some hints to construct an appropriate DM model. In our previous work (Niu et al., 2017a) where the 1.4 TeV peak excess was neglected, we have , , , 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 , , and , the constraints from the Fermi-LAT observations of dwarf spheroidal galaxies Fermi-LAT collaboration (2011); Geringer-Sameth and Koushiappas (2011); Sming Tsai et al. (2012); Fermi-LAT collaboration (2015); Li et al. (2015); Profumo et al. (2017) can be avoided Yuan et al. (2017). To escape the constraints from the Planck observations of CMB anisotropies Planck collaboration (2015), we employ the Breit-Wigner mechanism Feldman et al. (2008); Ibe et al. (2008); Guo and Wu (2009); Bi et al. (2009, 2011); Hisano et al. (2011); Bai et al. (2017); Xiang et al. (2017). We consider the dark model where the SM fermions and Higgs fields are neutral under it. We introduce one SM singlet Higgs field , one chiral fermionic dark matter particle , and three pairs of the vector-like particles (, whose quantum numbers under the are

(8)

The relevant Lagrangian is

(9)

where are the right-handed charged leptons. For simplicity, we choose and . After acquires a Vacuum Expectation Value (VEV), the gauge symmetry is broken down to a symmetry under which is odd. Thus, is a DM matter candidate. For simplicity, we assume that the mass of gauge boson is about twice of mass, i.e., , while the Higgs field and vector-like particles are heavier than . Moreover, and will be mixed due to the and terms, and we obtain the mass eigenstates and by neglecting the tiny charged lepton masses

(10)

where .

Neglecting the charged lepton masses again, we obtain

(11)

where , and and are the gauge coupling and gauge boson mass for gauge symmetry.

For , decays dominantly into leptons, and the decay width is

(12)

To explain the DM best fit results, we choose

(13)

And then we obtain , and . Of course, there exists fine-tuning between and , which deserves further study. For some solutions, see Ref. Bai et al. (2017).

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  GeV and  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 , 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 (Niu et al., 2017b) proposed a novel scenario to probe the interaction between DM particles and electrons for the DM mass range .

In summary, with the simplicity and naturalness physics principle, we proposed that there exists the , , and 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 and , respectively. These results are different from the previous analyses by neglecting the 1.4 TeV excess. Moreover, we presented a 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 NLX-Preparation ().

Acknowledgement–We would like to thank Xiao-Jun Bi and Yi-Zhong Fan for helpful discussions, and thank Maurin et al. (2014) for collecting the database and associated online tools for charged cosmic-ray measurements. This research was supported in part by the Projects 11475238 and 11647601 supported by National Science Foundation of China, and by Key Research Program of Frontier Sciences, CAS. The calculation in this paper are supported by HPC Cluster of SKLTP/ITP-CAS.

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