# Constraints on box-shaped cosmic ray electron feature from dark matter annihilation with the AMS-02 and DAMPE data

###### Abstract

Precise measurements of spectra of cosmic ray electrons/positrons can effectively probe the nature of dark matter (DM) particles. In a class of models that DM particles first annihilate into a pair of intermediate particles which then decay into standard model particles, box-shaped spectra can be produced. Such energy spectra of cosmic rays are distinct from astrophysical backgrounds, and can probably be regarded as characteristic features of the DM annihilation. In this work, we search for such spectral feature in the total electron plus positron fluxes by AMS-02 and DAMPE. We find no significant signal of the DM annihilation into box-shaped electrons/positrons. The 95% confidence level upper limits of the velocity-weighted cross section are derived, which range from for DM mass of 50 GeV to for DM mass of 10 TeV.

###### pacs:

11.15.Kc## I Introduction

Dark Matter (DM) is a very important concept in physics and astrophysics. Several methods were proposed to detect a kind of weakly interacting massive particle which is one of the most popular DM candidates, including the collider detection DM_co (); DM_co1 (), direct detection DM_de (); DM_de1 (), and indirect detection DM_in (); DM_in1 (). The indirect detection of DM particles is to search for unexpected “excess” in the energy spectra of cosmic rays or -rays box1 (); box2 (); box3 (); box4 (). Recent progresses include the significant excesses of positron fraction and spectrum observed by PAMELA pamela2 () and AMS-02 ams02 (); ams02-2 (), and the total electron plus positron spectra observed by ATIC atic (), Fermi-LAT fermi (), and also AMS-02 ams02-3 (), when compared with the predictions of cosmic ray propagation model. The model prediction of antiproton fluxes is largely consistent with the data pamela-antiproton1 (); pamela-antiproton2 (); ams-antiproton () (see, however, cui2017 (); cuoco2017 ()).

Either the DM annihilation/decay models Bringmann2008 (); Barger2009 (); Cirelli2009 (); Yin2009 (); Zhang2009 (); feng2014 (); yuan2015 (); Kopp2013 (); Cholis2013 (); jin2013 () or astrophysical sources yuksel (); Hooper2009 (); Profumo2009 (); Malyshev2009 (); hu2009 (); Blasi (); Linden2013 (); Yin2013 () have been proposed to interpret the electron and positron excesses. For DM annihilation models, the annihilation cross section needs to be about which is too large to give the correct (observed) relic density and a large boost factor () is needed Cirelli2009 (); Nima2008 (). For DM decay models, the lifetime of DM is about s. Such models have been stringently constrained by observations of -rays gammaconstrains1 (); gammaconstrains2 (); gammaconstrains3 (), radio emission Bertone2009 (); Bergstrom2009 (), and the cosmic macrowave background (CMB; cmbconstrains1 (); cmbconstrains2 ()).

In this work, we focus on the scenario that DM particles annihilate/decay into a pair of intermediate particles which then decay into Standard Model particles box6 (); box7 (). It can also produce box-shaped spectrum to get distinguished from other astrophysical process efficiently. This model has been used to explain the antiproton data of AMS-02 box8 (). The box-shaped -ray feature was searched with the Fermi-LAT data and constraints were reported box9 (); box10 (). In this work, we search for such spectral feature in the total electron plus positron fluxes by AMS-02 and DArk Matter Particle Explorer (DAMPE) dampe-data (). The mass ratio of the intermediate particle and DM has been taken as a free parameter (if only the mass ratio can lead to a box-shaped spectrum). DAMPE is a new cosmic ray telescope with very high energy resolution (better than @1TeV for electrons and -rays) and excellent hadron rejection power (higher than ) dampe-detector0 (); dampe-detector (). Such improvements are very helpful in identifying the unique structure of cosmic-ray spectrum and the measurements are expected to be able to tightly constrain the theoretical models. Recently, the DAMPE collaboration has published their first result, i.e., the electron plus positron spectrum dampe-data (). The DAMPE data cover the energy range from 25 GeV to 4.6 TeV, which provides us a good probe of the properties of DM particles.

This work is organized as follows: In Section II, we describe the energy spectrum of electrons and positrons generated by the DM two-step cascade annihilation. In Section III A, we explore such feature in the AMS-02 and DAMPE data and derive the constrains on the parameter space. Then we focus on the local subhalo scenario with box-shaped spectrum in Section III B. Our conclusions are summarized in Section IV.

## Ii Box Shaped spectra

We focus on the two-step annihilation where DM annihilates into a pair of scalar particles which then decay into electrons and positrons. In the rest frame of , the energy of the final electron and positron is . Since we focus on high energy cosmic rays, it is reasonable to assume that where is the mass of DM particles. If the mass of is comparable with or , a narrow peak spectrum can be generated box8 (). In the lab frame, where the DM particles are non-relativistic and , the energy of the final electron/positron is

(1) |

where , is the angle between the outgoing electron/positron and the parent scalar in the lab frame. To get the real solution(box-shaped SED), we let and then . Here we just consider the situation that and it is reasonable because for and for . From the above equation we see that the highest (lowest) energy corresponds to (). Since the intermediator particle is a scalar particle, the distribution of outgoing angle of electrons/positrons in its rest frame is isotropic. Therefore the spectrum of the final electrons/positrons in the lab frame is a box-shaped spectrum in an allowed energy range

(2) |

where is Heaviside function and . We get the same result as that in Refs. box6 (); box7 () if we set in rest intermediator frame. And if , it returns to the photon emission spectrum box9 (). Note that here we neglect the contribution of electroweak corrections. In Ref. PPPC4 (), the authors calculated such processes in detail using the Pythia package pythia (), assuming a light intermediate particle. We will compare our results with that (labelled as “PPPC4”) in the following.

From Eq. (2), we can see that there are two free parameters ( and ) to determine the spectrum. In the following discussion, we use and the mass ratio to present the results. If , it degenerates to the line spectrum. On the other hand, if (but still ), the produced box-shaped spectrum is very broad. Fig. 1 shows illustrations of the energy spectra of electrons for TeV and mass ratios of 0.1, 0.5, and 0.9. The result from PPPC4 is also presented in Fig. 1. We see clearly the effect on the broadening of the energy spectrum as the mass ratio decreases.

## Iii Results

### iii.1 The smooth dark matter distribution scenario

To compare with the observational data, we also need to calculate the propagation of electrons/positrons in the Galaxy. There are some numerical codes developed to compute the propagation of cosmic rays, such as GALPROP galprop () and DRAGON dragon (). Here we adopt the LikeDM package likedm (), which employs a Green’s function method (i.e., with quasi-monochromatic injection spectrum at a series of energy grids) to deal with the propagation of cosmic rays based on numerical tables obtained with GALPROP for given source spatial distribution, to calculate the propagation of electrons/positrons and the corresponding likelihood (or ). There are several sets of density profiles of DM and propagation parameters in LikeDM. In this study we adopt the isothermal profile iso (), and the propagation parameters with halo height of kpc. See Ref. likedm () for more details of the parameters.

The background includes the conventional primary electrons from e.g., supernova remnants, secondary electrons and positrons from the inelastic collisions between cosmic ray nuclei and the interstellar medium. Both components are phenomenologically described with broken power-law functions. For the primary electron spectrum, an additional cutoff is applied according to the DAMPE data dampe-data (). A primary source of electron/positron pair from e.g., pulsars, is further assumed to account for the electron/positron excesses. This is because the DM interpretations of such excesses are strongly disfavored by the -ray and CMB observations gammaconstrains1 (); gammaconstrains2 (); gammaconstrains3 (); cmbconstrains1 (); cmbconstrains2 (). The fluxes of the background electrons and positrons are described as likedm ()

(3) | |||||

(4) |

The spectrum of the extra source is assumed to be an exponential cutoff power law

(5) |

The total background energy spectrum of is then

(6) |

In the second term of the right hand side, represents the secondary electrons which are less than secondary positrons due to the charge conservation of the neutron production process Kamae2006 ().

The data used include the AMS-02 positron fraction ams02 (), the AMS-02 fluxes in the range of ams02-3 (), and the DAMPE fluxes dampe-data (). The best-fit parameters of the backgrounds are given in Table 1. Note that they are somewhat different from that given in Ref. likedm () because of different data sets used in the fitting.

(GeVmssr) | (GeV) | (GeV) | |||
---|---|---|---|---|---|

… | |||||

… | … |

When the DM contribution is added in the model, we enable the backgrounds to vary to some degree through multiplying adjustment factors , with , on , , and , respectively. The parameters and are optimized using the profile likelihood method. We scan the and parameter plane of DM. If no signal is found, the 95% upper limits of the annihilation cross section for each DM mass is given by the condition that . The 1-dimensional upper limits on the cross section and 2-dimensional likelihood maps on the plane for several values of the mass ratio and for the energy spectrum given by PPPC4 code are shown in Figs. 2 and 3.

The upper limits of the DM annihilation cross section range from at to at . The upper limits for different mass ratios do not differ much from each other. For the results are slightly lower than the others. We find that the upper limits of become weaker for TeV. From Fig. 3 we can see that adding the DM component for TeV can improve the fit to the data slightly. The test statistic (TS) value of such a DM signal is about 2.8. We show in Fig. 4 the comparison between the best-fit model and the data, for TeV, , and . The DM contribution can slightly improve the match with the small bump of the DAMPE data around GeV. However, the significance is too small for the current data.

### iii.2 Local DM subhalo scenario

Yuan et al. Yuan2017 () proposed a local DM sub-halo scenario to explain the tentative peak structure at 1.4 TeV in the electron and positron spectrum of DAMPE. In this subsection we examine whether a box-shaped electron/positron spectrum can reasonably reproduce the data. Since the propagation of the electrons and positrons is usually expected to be independent of the initial spectrum, we take the parameters of the distance and the mass of the hypothesized subhalo as and , the same as those of Yuan2017 (). We show our likelihood map in Fig.5 with the mass ratio and . And in Fig.6, we plot the best fit result for , GeV and . In order to explain the tentative peak structure of DAMPE data at 1.4 TeV, we need a high mass ratio of , which almost leads to a line spectrum that is un-distinguishable from normal scenario without the intermediates.

## Iv Conclusion

In this work, we use the AMS-02 and DAMPE data to search for possible DM annihilation signal from a class of models with two-step annihilation process including intermediate scalar particles. The resulting electron/positron spectrum from such DM models looks like a box shape, and may be imprinted in high energy resolution spectra of cosmic ray electrons/positrons (by e.g., DAMPE).

We do not find any significant annihilation signal with such box-shaped spectrum. The upper limits of the DM annihilation cross section range from to for the DM particle mass between 50 GeV and 10 TeV. A 2 TeV DM component is found to improve slightly the fit to the DAMPE data around 700 GeV, but the significance is very low. With a local DM subhalo, as assumed in (Yuan2017, ), we find out that a high mass ratio (for which the resulting spectrum is line like) can improve the fit. In view of the narrow energy range of the tentative peak at 1.4 TeV, such a result is actually anticipated. Future measurements of the fine structures of the electrons/positrons with higher precision may shed new light on the indirect detection of DM with characteristic spectral features.

###### Acknowledgements.

This work is supported by the National Key Research and Development Program of China (No. 2016YFA0400200), National Natural Science of China (Nos. 11722328, 11773075, U1738206, U1738210), the Youth Innovation Promotion Association of Chinese Academy of Sciences (No. 2016288), the Natural Science Foundation of Jiangsu Province (No. BK20151608), and the 100 Talents program of Chinese Academy of Sciences.## References

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