content_image/339725.jpg content_image/339724.jpg content_image/339723.jpg
## A. Electrons and positrons 

In the Milky Way, the propagation of cosmic-ray (CR) particles can be approximated by a two-zone propagation model, which includes two cylinders centered at the galactic center. A cylinder of radius R = 20 kpc with height 2 L ( L = 1 − 20 kpc) in the z direction delimits the CR propagation region. A smaller cylinder with the same radius but with thickness 2 h$_{z}$ = 0 . 2 kpc models the galactic plane. The solar system is located at r$_{⊙}$ = 8 . 5 kpc and z$_{⊙}$ = 0. In general, the CR propagation equations contain terms for convection and reacceleration effects [20]. However, for electrons and positrons with energy E ≳ 10 GeV, these effects can safely be neglected [21]. Thus one only needs to consider the diffusion effect. In this case, the steady-state diffusion equation for positron is given by 

<img src='content_image/339725.jpg'>

where f$_{e}$ + ( E, r, z ) is the positron differential number density in cylinder coordinate and K ( E ) = K$_{0}$ ( E/ GeV) δ is the spatial diffusion coefficient. Through fitting the measured ratio of boron to carbon (B/C), the propagation parameters δ , K$_{0}$ and L can be determined [22]. In Table. I, we list three typical combinations of the propagation parameters [21, 23]. The parameter set MIN and MAX corresponds to the minimal and maximal positron flux respectively, while the MED scenario best fits the B/C ratio and produce the moderate flux. The energy loss rate b ( E ) are mainly due to the ICS and synchrotron processes. The two processes combined give b ( E ) = E $^{2}$/ (GeV · τ$_{E}$ ) with τ$_{E}$ = 10 16 s [24]. For decaying DM model, the source term Q is given by 

<img src='content_image/339724.jpg'>

where Γ$_{k}$ is the decay width and k runs over all the channels with positrons in the final states. dn k e + /dE is the positron energy spectrum per DM decay. The DM halo profile ρ ( r ) is usually parameterized as a spherically symmetric form 

<img src='content_image/339723.jpg'>

with local DM density ρ$_{⊙}$ = 0 . 3 GeVcm − $^{3}$. Possible DM halo profile parameters α, β, γ and r$_{s}$ are listed in Table. II. In this paper, we adopt the NFW profile and MED propagation model in numerical calculations. 

