content_image/338614.jpg content_image/338615.jpg content_image/338626.jpg content_image/338625.jpg content_image/338624.jpg content_image/338623.jpg
<img src='content_image/338614.jpg'>

<img src='content_image/338615.jpg'>

Eq. (16) can be solved analytically from the above considerations. For the decaying dark matter, the solution of the positron diffusion can be written as [23, 28] 

<img src='content_image/338626.jpg'>

The explicit form of the Green’s function G$_{e}$ + ( E, E $^{′}$, r, z ) is given by 

<img src='content_image/338625.jpg'>

with 

<img src='content_image/338624.jpg'>

where J$_{0(1)}$ is the zeroth(first)-order Bessel function. ζ$_{n}$ are successive zeros of the function J$_{0}$ . At the heliosphere boundary ( r = r$_{⊙}$ and z = 0), the interstellar positron flux per unit energy from the DM decay is 

<img src='content_image/338623.jpg'>

