Decaying Fermionic Dark Matter Search with CALET

TL;DR

This paper explores CALET's potential to detect fermionic dark matter decay signals in cosmic ray electrons, distinguishing them from astrophysical sources through spectral analysis and gamma-ray constraints.

Contribution

It demonstrates CALET's capability to differentiate dark matter decay signals from pulsar sources in cosmic ray electron spectra using spectral shape analysis.

Findings

CALET can distinguish DM decay signals from pulsars within 5 years.

DM decay models are consistent with AMS-02 positron excess data.

Gamma-ray constraints limit the parameter space for DM explanations.

Abstract

The ISS-based CALET (CALorimetric Electron Telescope) detector can play an important role in indirect search for Dark Matter (DM), measuring the electron+positron flux in the TeV region for the first time directly. With its fine energy resolution of approximately $2\%$ and good proton rejection ratio ($1:10^5$) it has the potential to search for fine structures in the Cosmic Ray (CR) electron spectrum. In this context we discuss the ability of CALET to discern between signals originating from astrophysical sources and DM decay or annihilation. We fit a parametrization of the local interstellar electron and positron spectra to current measurements, with either a pulsar or 3-body decay of fermionic DM as the extra source causing the positron excess. The expected CALET data for scenarios in which DM decay explains the excess are calculated and analyzed. The signal from this particular 3-body DM decay which can explain the recent measurements of from the AMS$-02$ experiment is shown to be distinguishable from a single pulsar source causing the positron excess by 5 years of observation with CALET, based on the shape of the spectrum. We also study the constraints from extra-galactic diffuse $\gamma$-ray data on this DM-only explanation of the positron excess and show that especially for the possibly remaining parameter space a clearly identifiable signature in the CR electron spectrum exists.

Tables

Table 1. Table 1 : Branching ratios for fit to AMS-02 ( e + + e − ) superscript 𝑒 superscript 𝑒 (e^{+}+e^{-}) and positron flux for all studied values of M DM DM {}_{\text{DM}} and E d subscript 𝐸 𝑑 E_{d} . Upper one: initial fit, lower one: fit with τ ​ τ ​ ν 𝜏 𝜏 𝜈 \tau\tau\nu component reduced. Colored boxes correspond to the examples shown in figure  1 , figure  4(b) and figure  4(a) respectively.

\rowcolorwhite                                                    Branching Ratio of DM Decay Products from Fit to AMS-02
\rowcolorwhite                                                    $\frac{\left(ee\nu /\mu \mu \nu /\tau \tau \nu \right)\to e,\mu ,\tau \text{all free}}{\left(ee\nu /\mu \mu \nu /\tau \tau \nu \right)\to \text{min.}\tau ,\text{low}\gamma }$
	1	2	5	10
2 TeV	$0.27/0/0.73$ $0.23/0.53/0.24$	$0.23/0/0.77$\cellcolorblue!25 $0/0.98/0.02$	$0.20/0/0.80$ $0.09/0.83/0.08$	$0.20/0/0.80$ $0.07/0.86/0.07$
1.5 TeV	$0.30/0/0.70$ $0.34/0.59/0.07$	$0.30/0/0.70$ \cellcolorblue!25 $0.27/0.73/0$	$0.26/0/0.74$ $0.22/0.78/0$	$0.23/0/0.77$ $0.21/0.79/0$
1 TeV	Excluded	Excluded	$0.27/0/0.73$ $0.32/0.15/0.53$	$0.26/0/0.74$ \cellcolorblue!25 $0.40/0.60/0$

Table 2. Table 2 : Average χ 2 superscript 𝜒 2 \chi^{2} obtained from the fits of the single pulsar source to the 10000 samples of simulated CALET data + positron flux from AMS-02 and the re-fit of the DM model to the same data points as a function of DM mass and background cut-off energy ( E d ) subscript 𝐸 𝑑 (E_{d}) . Colored boxes are the examples shown in figure  7 , figure  7(a) , figure  7(b) respectively. In the lower panel the number of excluded samples ( χ 2 > 95 % ​ CL ) superscript 𝜒 2 percent 95 CL (\chi^{2}>95\%\text{CL}) for each case are shown. Average Number Degree of Freedom (NDF) is 80 80 80 in these fits.

\rowcolorwhite                                                    $\frac{\text{Average}{\chi }^2\left(\text{pulsar fit/DM re-fit}\right)}{\text{No. of Samples with}{\chi }^2>95\%\left(\text{pulsar fit/DM re-fit}\right)}$
	1	2	5	10
2 TeV	$211.15/61.16$ $10000/13$	\cellcolorblue!25 $150.61/58.90$ $9958/31$	$123.87/58.31$ $8932/6$	$116.19/58.29$ $7771/1$
1.5 TeV	-	\cellcolorblue!25 $142.37/75.54$ $9943/138$	$132.98/76.06$ $9632/97$	$131.46/76.59$ $10000/49$
1 TeV	-	-	-	\cellcolorblue!25 $269.85/72.87$ $10000/49$

Table 3. Table 3 : GALDEF file parameters used for CR propagation in GALPROP.

Parameter	Value	Unit
Z${}_{\text{max}}$/$\Delta \text{Z}$	$6/0.25$	kpc
X${}_{\text{max}}$/$\Delta \text{X}$	$16/0.25$	kpc
Y${}_{\text{max}}$/$\Delta \text{Y}$	$16/0.25$	kpc
$E_{\text{min}}$	$10$	MeV
$E_{\text{max}}$	$100$	TeV
$D_0$ (Diff. coeff.)	$2.90\times {10}^{28}$	${\text{cm}}^2{\text{s}}^{-1}$
$R_0$ (ref. rigidity for diff. coeff.)	$4$	GV
${\gamma }_1/{\gamma }_2$ (injection index)	$1.70/2.45$
$R_{\gamma }$ (Break in injection Index)	$7$	GV
$\delta$ (Diff. coeff. index)	$0.40$
$v_A$ (Alfven Velocity)	$12.0$	$\text{km}{\text{s}}^{-1}$
start-timestep	$6.4\times {10}^7$	years
end-timestep	$10$	years
timestep-factor	$0.90$
timestep-repeat	$20$

Equations

$$Q=\Gamma \frac{\rho }{{\text{M}}_{\text{DM}}}\frac{dN}{dE}$$

$$\frac{1}{\Gamma }\frac{d\Gamma }{dx}=2x^2(3-2x)$$

$$\rho =\frac{{\delta }_c{\rho }_c(r/r_s)}{(1+r/r_s{)}^2}$$

$${\delta }_c=\frac{200}{3}\frac{c_v^3}{\text{ln}(1+c_v)+(c_v/1+c_v)}$$

$${\rho }_c=\frac{\frac{4}{3}\pi r_v^3}{M_v}$$

$${\varphi }_T(E)=2{\varphi }_{\text{extra}}+C_p{E}^{{\gamma }_p}\left(2\frac{C_s}{C_p}{E}^{{\gamma }_s-{\gamma }_p}+e^{(\frac{-E}{E_d})}\right)$$

$${\varphi }_{pn}=C_{pn}{E}^{{\gamma }_{pn}}{e}^{-\left(\frac{E}{E_{pn}}\right)}$$

$${\varphi }_{\text{DM}}=\frac{1}{{\tau }_e}{\varphi }_e+\frac{1}{{\tau }_{\mu }}{\varphi }_{\mu }+\frac{1}{{\tau }_{\tau }}{\varphi }_{\tau }$$

$${\varphi }_{e^{+}}={\varphi }_{\text{extra}}+C_s{E}^{{\gamma }_s}$$

$$\text{AIC}=-2L_m+m$$

$$\frac{\partial \psi }{\partial t}=Q(x,p)+\nabla \cdot (D_{xx}\nabla \psi )+\frac{\partial }{\partial p}{p}^2{D}_{pp}\frac{\partial }{\partial p}\frac{1}{p^2}\psi -\frac{\partial }{\partial p}\dot{p}\psi$$

$$D_{xx}=\beta D_0{\left(\frac{R}{R_0}\right)}^{\delta }$$

$$D_{pp}{D}_{xx}=\frac{4p^2{v}_A^2}{3\delta (4-{\delta }^2)(4-\delta )}$$

$$q=q_0{\left(\frac{d}{d_0}\right)}^{\eta }\text{exp}\left(-\zeta \frac{d-d_0}{d_0}-\frac{\left|Z\right|}{0.2}\right)$$

$$q_N=q\times \left(\sum _{i=1}^4{e}^{-\frac{(d-r_i)}{2{\sigma }^2}}\right)$$

Full text

1 Introduction

While the existence and cosmological properties of Dark Matter (DM) are well established, nature and particle properties of DM are largely unknown. Many theoretical models predict that a TeV scale Cold DM (CDM) can decay or annihilate into Standard Model (SM) particles. As a result, CDM could be detected indirectly by observing an excess in cosmic ray (CR) spectra relative to the astrophysical background [1]. Recent results from space based CR detectors such as AMS-$02$ [2] and PAMELA [3] show an increase of the positron fraction above $10$ GeV up to $300$ GeV which is not expected from the secondary production of positrons in the Interstellar Medium (ISM). This excess may be explained by an extra source emitting electron-positron pairs, such as emission from pulsars or decay and annihilation of DM [4]. To explain the positron excess with DM annihilation would require a large boost factor because the cross section of DM annihilation from relic density measurements [5] yields a positron flux which is too low to produce the excess observed in the measurements [6, 7]. The DM decay scenario can naturally explain the positron excess if the lifetime of the DM is less than $\sim 10^{25}$ s [1, 8]. Among different DM decay scenarios, a 3-body leptonic decay is favorable to explain the recent positron excess, because the 3-body decay produces a softer spectrum compared to 2-body decay. Moreover, since the decay products are only leptonic, the absence of a hadronic component allows for compatibility with the recent anti-proton measurements [9].

In this paper, we will present the prospects of discerning such a signal from decaying DM with 1–2 TeV mass from a single pulsar source in the $(e^{+}+e^{-})$ spectrum by the measurement taken with the CALorimetric Electron Telescope (CALET). CALET, in operation on the ISS since October 2015, is designed to search for signatures from nearby CR sources and DM in this spectrum with fine energy resolution of approximately $2\%$ and high proton rejection power $(1:10^{5})$ [10, 11].

We study a DM candidate undergoing 3-body decay into two charged leptons and a neutrino, as a possible extra source which can explain the excess of the positron fraction observed by AMS-02 [12]. The AMS-02 collaboration proposed an extra source emitting electron-positron pairs with an exponentially cut-off power-law spectrum [13] as an empirical model to the positron excess. This spectrum corresponds well to that of a single young pulsar [14], making it a generic scenario against which we test the DM model explaining the positron excess. This parametrization for the positron fraction is extended to the $(e^{+}+e^{-})$ flux and into the TeV region, including effects of propagation in the galaxy. The free parameters of this local CR parametrization with DM or Pulsar as extra source are determined from the best fit to AMS-02 positron flux and $(e^{+}+e^{-})$ measurements. Using this parametrization, we calculated the expected $(e^{+}+e^{-})$ spectrum for 5 years of observation with CALET for DM with a mass in the range of 1–2 TeV, and investigate the possibility of discerning this particular DM decay from a generic single pulsar source.

The recent diffuse $\gamma$-ray data measured by the Fermi-LAT experiment [15] gives a strong constraint on DM annihilation or decay in the galactic halo. We compare the $\gamma$-ray emission predicted by this DM model with the $\gamma$-ray measurement and show that $\gamma$-ray production can be reduced significantly, when the charged primary decay products from the DM are only electron and muon, excluding tau leptons. The ability of CALET to discern the DM signal from a single pulsar depends on the shape of the $(e^{+}+e^{-})$ decay spectrum, and we show this scenario with low $\gamma$-ray yield would have an especially well distinguishable signature.

2 3-Body Decay of Dark Matter and the Cosmic Ray Positron Excess

To explain the positron excess, various particle physics models with a 3-body decay of DM are proposed [16, 17, 18]. In this context, we investigate a scenario where a TeV scale DM decays to leptons $(\text{DM}\rightarrow l^{-}l^{+}\nu)$, namely a charged standard model lepton+anti-lepton pair and a neutrino. The branching ratios of the outgoing leptons are proportional to the inverse of the decay time $(\frac{1}{\tau_{e}},\,\frac{1}{\tau_{\mu}},\,\frac{1}{\tau_{\tau}})$ of the DM for the individual decay channels $(ee\nu,\,\mu\mu\nu,\,\tau\tau\nu)$. We treat these as free parameters in our study and adjust them to explain the positron excess. In a recently proposed theoretical model, this type of DM decay is predicted by extending the SM with 3 fermionic singlets $N_{L},\,\psi_{R},\,S_{R}$ and two Higgs doublets $\eta,\,\chi$ [12].

In this model, the visible matter and DM are all created from the decay of the scalar fields $(\eta,\,\chi)$, which are charged under the $U(1)_{(B-L)}$ group and created from the decay of a generic hidden sector scalar field $\phi$. These processes occur above electro-weak scale and the predicted lifetime of the DM $(5\times 10^{25}\,$–$\,6\times 10^{25}\,\text{s})$ is larger than the age of the universe if the $B-L$ symmetry is assumed to be broken above TeV scale, yielding the correct relic abundance. The smallness of neutrino masses and the matter-antimatter asymmetry also appear as consequences of this theoretical concept. The DM candidate is the lightest fermion $N_{L}$, which decays under violation of the lepton number by two units, contributing to the CR lepton spectra.

In the decaying DM scenario, the injected particles per volume and time are given by

[TABLE]

where $\Gamma$, $\text{M}_{\text{DM}}$ are the decay rate and mass of the DM respectively. Since the decay of the DM is mediated by a heavy scalar, the lifetime of the mediator is negligible, making 4-point scalar interaction a good approximation. With these assumptions the probability distribution for the momentum of the charged leptons $(e^{+}e^{-},\,\mu^{+}\mu^{-},\,\tau^{+}\tau^{-})$ is given by

[TABLE]

where $x=E/E_{\text{max}}$ and $E_{\text{max}}=0.5\,\text{M}_{\text{DM}}$. From this initial energy distributions, the $e^{+}$ and $e^{-}$ spectrum $\frac{dN}{dE}$ produced per decay is calculated using the event generator PYTHIA (Version $8.2$) [19]. The spectra for $e^{+}$ and $e^{-}$ are identical and the $e^{+}$ spectrum is propagated in GALPROP [20, 21]. The propagation parameters in GALPROP, which is modified to include the spiral arm nature of the galaxy, are determined from comparing the background CR propagation calculation (Proton spectrum and $B/C$ ratio) with AMS-02 measurements, which is discussed in Appendix A. We assume a Navarro-Frenk-White (NFW) profile [22] for the DM distribution in our galaxy.

[TABLE]

$\delta_{c}$ is defined as

[TABLE]

where $c_{v}$ is defined as the ratio of virial radius $(r_{v})$ and scale radius $(r_{s})$, and we assume $c_{v}=10$ [23]. $\rho_{c}$ is determined from the mass of the halo as

[TABLE]

where $r_{v},\,M_{v}$ are taken as $200$ kpc and $1.5\times 10^{12}\text{M}_{\odot}$ [24].

3 Parametrization of Local $e^{+}$ and $e^{-}$ Flux and Fit to Current Data

The locally observed $e^{+}$ and $e^{-}$ spectra are parametrized to reflect the variability from the free parameters of injection and propagation. Using this parametrization we determine multiple scenarios for DM as the extra source explaining the positron excess from the minimum $\chi^{2}$ in comparison with the $(e^{+}+e^{-})$ and positron flux measurements from AMS-02 [2]. The parametrization is based on the assumption that distant supernova remnants (SNR) give a power law primary electron spectrum, to which a secondary component from nuclei interactions with the ISM is added. We also assume that the injection spectrum index of electrons and nuclei is the same as they originate from the same sources. This is described by two power law indices $\gamma_{p},\,\gamma_{s}$ and two coefficients $C_{p},\,C_{s}$ which describe the relative weights of the spectra for primary electron and secondary flux. The radiative energy loss processes (such as synchrotron radiation, Inverse Compton radiation, Coulomb scattering etc.) experienced by the primary electrons are modeled as an exponential cut-off at energy $E_{d}$, which is absent for the secondary particles. With these parameters the total flux (primary+secondary) can be written as

[TABLE]

where $\phi_{\text{extra}}$ is the flux from the extra sources emitting electron-positron pairs. For the pulsar scenario we parametrize the extra source by

[TABLE]

here the weight of the diffuse spectra is given by $C_{pn}$, power law index $\gamma_{pn}$ (common for electron and positron) and a cut-off energy $E_{pn}$.

The extra source flux from DM decay is given by

[TABLE]

with $\phi_{e},\,\phi_{\mu},\,\phi_{\tau}$ being the $e^{+}$ (identical to $e^{-}$) decay spectra for $ee\nu,\,\mu\mu\nu,\,\tau\tau\nu$ channel respectively, propagated with GALPROP and $\frac{1}{\tau_{e}},\,\frac{1}{\tau_{\mu}},\,\frac{1}{\tau_{\tau}}$ are the inverse of the decay times for three leptonic decay channels.

The positron flux from eq. (3.1) can be written as

[TABLE]

This parametrization is fitted to the current measurements of the electron and positron flux to determine values for the free parameters.

In this fitting $\frac{1}{\tau_{e}},\,\frac{1}{\tau_{\mu}},\,\frac{1}{\tau_{\tau}}$ are treated as free parameters for the DM extra source in addition to the three free parameters $C_{p},\,\frac{C_{s}}{C_{p}},\,\gamma_{p}$ for the background (eq. (3.1)). Assuming a common origin for nuclei and electrons, the difference between the primary and secondary electron indices $(\gamma_{s}-\gamma_{p})$ is nearly equal to $-\delta$ and thus fixed to $-0.4$ in the fit, according to the propagation model given in Appendix A. The range of data points used for comparison with experimental results is from 15 GeV to 1 TeV. Since the CR spectra below 15 GeV are influenced by solar modulation, diffusive reacceleration and possibly a change in the injection index [25], the variability of the spectra cannot be represented by a simple parametrization. However, we apply the effect of charge independent solar modulation above 15 GeV [26] in the parametrization by assuming force field approximation with a fixed value of 500 MeV for the common $(e^{+}$ and $e^{-})$ modulation potential. The upper bound of the fit range is effectively 1 TeV as there are no high resolution data points from the AMS-02 measurements above 1 TeV. The cut-off energy $E_{d}$, which has only influence in the TeV region, cannot be determined from current experimental data and various values of $E_{d}$ (1 TeV, 2 TeV, 5 TeV, 10 TeV) are studied. To estimate the unknown $(e^{+}+e^{-})$ spectrum in the TeV region an electron-only flux from the Vela SNR, which is the most influential nearby source with distance around $1$ kpc and age less than $10^{5}$ years [27], is calculated with GALPROP for the propagation parameters as described in Appendix A. The contribution of Vela to the high energy electron spectrum may be reduced if the release of CR electrons is gradual or delayed. The parametrization reflects the variability of the contribution of Vela and also the influence of spiral arm thickness on the CR $(e^{+}+e^{-})$ spectrum (A) by choosing different values for $E_{d}$ in the range from 1-10 TeV. It should also be noted that a harder injection spectrum [28] and/or a specific energy-dependent release [29] of the electrons from Vela could create a distinct signature in the TeV region. If such a signature is found by CALET, the background model for DM search would have to be adapted.

As an example we show in figure 1 that, the fit converges at branching ratios of $0.77$ for $\tau\tau\nu$ channel and $0.23$ for $ee\nu$ channel, with no contribution from $\mu\mu\nu$ channel for a 2 TeV fermionic DM and the background cut-off energy $(E_{d})$ set to 2 TeV.

Similarly, the scenario with single pulsar as the only extra source gives a good fit to the positron flux and $(e^{+}+e^{-})$ flux in the same fit range as for DM (15 GeV - 1 TeV), shown in figure 2. Apart from the three free background parameters, the free parameters for a pulsar as extra source are $\frac{C_{pn}}{C_{p}},\,\gamma_{pn}-\gamma_{p}$. The values of the extra source (pulsar) free parameters are determined from the best fit assuming $E_{pn}=1$ TeV. Since the expected CALET data for 5 years of measurement is calculated for the DM case, the initially assumed energy cut-off for the pulsar source $(E_{pn})$ has no influence in this study, as finally when CALET’s capability to discern pulsar and DM is calculated, it is taken as a free parameter.

It is shown in a recent work [30] that there are several candidates among pulsars within a distance of $<0.5$ kpc from the solar system and with an age of $4\times 10^{4}-4.5\times 10^{5}$ years which could provide a single source explanation of the positron excess. So the single young pulsar is taken as a generic case against which we compare the DM decay model.

4 Diffuse $\gamma$-ray Constraints and Low $\gamma$-ray Flux Scenario

The decay or the annihilation of DM directly produces $\gamma$-rays in the from of Final State Radiation (FSR) and also secondary $\gamma$-rays from Inverse Compton and Bremsstrahlung processes during propagation of charged decay or annihilation products. Through these processes it is expected that the decay of the investigated DM into charged leptons $(e^{\pm},\,\mu^{\pm},\,\tau^{\pm})$ in the galactic DM-halo would produce a diffuse $\gamma$-ray flux. For DM decay which can explain the positron excess, this predicted $\gamma$-ray flux has to be compared with the Fermi-LAT [15] diffuse $\gamma$-ray measurement taken at high latitudes. Looking away from the galactic plane $(\left|b\right|>20\degree)$ strongly reduces the background from galactic astrophysical sources and thus comparison of $\gamma$-ray flux from DM with the measurement in this region gives the strongest constraint. The remaining contribution from astrophysical sources depends on the different modelings of $\gamma$-ray emission [31, 32], but the total measured flux can be considered a conservative upper bound. While the diffuse $\gamma$-ray spectrum in the relevant sky region and energy range is currently only available from Fermi-Lat, it is going to be reaffirmed by the currently operating detectors with calorimeters capable of absorbing the full shower energy up to the TeV region, such as CALET [33] and also DAMPE [34].

The $\gamma$-ray flux from DM decay depends on both the mass of the decaying DM and the decay products. As the $\tau\tau\nu$ channel produces more $\gamma$-rays compared to $ee\nu$ and $\mu\mu\nu$ channel, to study the possibility of a DM-only explanation of the positron excess compatible with the current $\gamma$-ray measurements, we reduce the tau component from the decay products of the DM. Adapting all other free parameters in each step and starting with the parameters obtained from the initial fit, we reduce the tau component in steps until the $\chi^{2}$ either positron flux or $(e^{+}+e^{-})$ flux exceeds $95\%$ CL, or the scale factor for $\tau\tau\nu$ channel reaches zero. The branching ratios for the initial fit and the fit with the reduced tau contribution are given in table 4 for different values of DM mass and cut-off energy $E_{d}$. It is shown that a good fit with completely removed $\tau\tau\nu$ channel is possible for DM with mass 1.5 TeV and 1.0 TeV, and a cut-off energy $E_{d}$ equal to or larger than 2 TeV or 10 TeV respectively. However, no good fit even including $\tau\tau\nu$ channel is possible for 1 TeV DM and $E_{d}$ equal to or smaller than 2 TeV. The chosen DM theory supports full variability of the branching fractions, which are proportional to the effective 4-point couplings for each decay mode. The effective couplings are governed by the products of the coupling constants at both vertices of the decay process which are different for each channel, making them completely free parameters also independent of the leptonic mass hierarchy [12].

The $\gamma$-ray fluxes from the FSR and decay of the primary decay products have been calculated with PYTHIA assuming NFW profile, and three different cases are plotted in figure 3 including contribution from secondary $\gamma$-rays. The charged particles from the decay of DM and their interaction with the interstellar radiation field (ISRF) produce secondary $\gamma$-rays. This isotropic diffuse $\gamma$-ray flux is calculated in GALPROP at latitudes $\left|b\right|>20\degree$, for different DM models using the default ISRF [35] provided by GALPROP. As shown in the left panel of figure 3, $\gamma$-rays from secondary production have lower energy than the primary component. For a DM of mass $2$ TeV decaying to $\tau\tau\nu$ $(73\%)$ and $ee\nu$ $(27\%)$ channel, the predicted $\gamma$-ray flux exceeds the the Fermi-LAT data significantly. However with $1.5$ TeV and 1 TeV DM decaying only to $\mu\mu\nu$ and $ee\nu$, the $\gamma$-ray fluxes from the decay are closer to the experimental data as shown in the right panel of figure 3.

The $\gamma$-ray flux from the 1 TeV DM decay scenario, as shown in figure 3, is least in conflict with the experimental data. Models with these characteristics (low DM mass, and no decay to $\tau\tau\nu$ channel) may be a unique possibility to explain the positron excess by DM, without violating the constraints from $\gamma$-ray measurements, making this model of special interest to study. For 1 TeV DM decaying only to $\mu\mu\nu$ and $ee\nu$ channel, the fit converges at branching ratios of $0.60$ for $\mu\mu\nu$ channel and $0.40$ for $ee\nu$ channel with $E_{d}$ set to 10 TeV as shown in figure 4(a). Similarly, for a $1.5$ TeV DM decaying only to $\mu\mu\nu$ and $ee\nu$ channel the best fit converges at branching ratios of $0.73$ for $\mu\mu\nu$ channel and $0.27$ for $ee\nu$ channel with 2 TeV background cut-off, shown in figure 4(b). This $1.5$ TeV fermionic DM matches best the new AMS-02 positron flux recently presented at CERN [36], making this another case to be studied. Although the predicted $\gamma$-ray flux from the 1 TeV DM is somewhat higher than the Fermi-LAT measurement, there should be an uncertainty in the lifetime of the DM, and thus the $\gamma$-ray flux, from the choice of propagation conditions used for the positrons of the DM decay. Also the shape of the DM halo may influence the charged CR $(e^{+},\,e^{-})$ and $\gamma$-ray flux. The $\gamma$-ray flux measured at higher latitudes may be reduced and the charged CR flux enhanced if the DM accumulates close to galactic plane, as in the "Dark-Disc" model [37] for partly self-interacting DM.