Propagation δ K$_{0}$ (kpc $^{2}$/Myr) L (kpc) MED 0.70 0.0112 4 MAX 0.46 0.0765 15 MIN 0.55 0.00595 1 Halo Profile α β γ r$_{s}$ (kpc) NFW 1 3 1 20 Isothermal 2 2 0 5 Moore 1.5 3 1.3 30 TABLE I: Three typical combinations for the propagation parameters δ , K$_{0}$ and L from the measured B/C ratio [21, 23]. TABLE II: The values of parameters α, β, γ and r$_{s}$ for the NFW [25], Isothermal [26] and Moore [27] DM halo profiles. 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] The explicit form of the Green’s function G$_{e}$ + ( E, E $^{′}$, r, z ) is given by with 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 11 Φ DM e + ( E ) = c 4 π f$_{e}$ + ( E, r$_{⊙}$, 0) . (22) G$_{nm}$ = 2 J 2 $_{1}$( ζ$_{n}$ ) R $^{2}$L ∫ R 0 dr $^{′}$r$^{′}$J$_{0}$$^{(}$ ζ$_{n}$ r ′ R$^{)}$$^{∫}$ L − L dz $^{′}$ρ ( r $^{′}$, z $^{′}$) sin [ mπ 2 L ( z ′ − L ) ] , (21) G$_{e}$ + ( E, E $^{′}$, r, z ) = ∞ ∑ n,m =1 G$_{nm}$ τ$_{E}$ E $_{2}$J$_{0}$ $^{(}$ζ$_{n}$ r R$^{)}$ sin [ mπ 2 L ( z − L ) ] (20) × exp$^{[}$ K$_{0}$τ$_{E}$ $^{(}$ζ 2 n R 2 + m $^{2}$π2 4 L 2 $^{) (}$E δ − 1 − E ′ δ − 1 δ 1 )] , f$_{e}$ + ( E, r, z ) = 1 m$_{D}$ ∫ m$_{D}$ E dE $^{′}$G$_{e}$ + ( E, E $^{′}$, r, z ) ∑ k Γ$_{k}$ dn k e + dE ′ , (19)