content_image/353919.jpg content_image/353918.jpg content_image/353917.jpg content_image/353916.jpg content_image/353915.jpg
The high energy gamma-rays produced by the ICS process e $^{±}$γ → e ± $^{′}$γ ′ have the following energy spectrum [37, 38]: 

<img src='content_image/353919.jpg'>

Since the photon may be emitted from both electron and positron, an overall factor of 2 has been multiplied in the equation. The differential energy density u$_{γ}$ ( E$_{γ}$, r, z ) of interstellar radiation field (ISRF) contains three components: the cosmic microwave background (CMB), thermal dust radiation and star light. Here we adopt the GALPROP numerical result for u$_{γ}$ in Ref. [39]. The parameter f$_{ICS}$ is defined by [37] 

<img src='content_image/353918.jpg'>

with 

<img src='content_image/353917.jpg'>

where 0 ≤ q ≤ 1. In Eq. (35), f$_{e}$ + ( E$_{e}$, r, z ) is the positron differential number density. Because of the ICS and synchrotron, high energy electrons and positrons will loss most of their energy within a kpc. Therefore one may neglect the diffusion term of Eq. (16) and approximately calculate f$_{e}$ + ( E$_{e}$, r, z ) for every point in the CR propagation region. In this case, f$_{e}$ + ( E$_{e}$, r, z ) is given by the following formulas [37, 40] 

<img src='content_image/353916.jpg'>

where the electron energy loss rate b$_{ICS}$ ( E$_{e}$, r, z ) is 

<img src='content_image/353915.jpg'>

Here we neglect the synchrotron energy loss rate b$_{syn}$ as b$_{syn}$ ≪ b$_{ICS}$ [38]. It is worthwhile to stress that the above energy loss rate b$_{ICS}$ ( E$_{e}$, r, z ) is position dependent $^{1}$. 

In order to compare with the experimental data, one needs to know the diffuse gamma- ray background which includes a galactic Φ Galactic γ contribution and an extragalactic (EG) Φ EG γ contribution. The galactic gamma-ray background Φ Galactic γ mainly comes from pion 

