Pair Production by two photons : energy range of the electron created

[Tabatta]

Hi all,

I'm currently studying pair production by two photons (a high-energy one traveling in a isotropic field of low-energy ones), and I'm trying to understand the energy range of the electron created by this phenomenon.
For this, I'm studying an old paper from Aharonian 1983, "Photoproduction of electron-positron pairs in compact x-ray sources".
The situation is the following : we consider a cloud of isotropically distributed photons with four-momentum vectors $k_1^{\mu} = (\omega_1, \stackrel{\rightarrow}{k_1}), k_2^{\mu} = (\omega_2, \stackrel{\rightarrow}{k_2})$, with $\omega_1 \leq \omega_2$, creating an electron-positron pair with four-momentum vectors $p_{\pm}^{\mu} = (\epsilon_{\pm}, \stackrel{\rightarrow}{p_{\pm}})$.
Let $\stackrel{\rightarrow}{k} = \stackrel{\rightarrow}{k_1} + \stackrel{\rightarrow}{k_2}$ be the total momentum of the two-photons system, and $E = \omega_1 + \omega_2 , \Delta = \omega_2 - \omega_1$.
I attached the page of the paper where my "problem" is. I understand how he gets the inequality $\sqrt{ k^2 + \epsilon^2 -2kp} \leq E - \epsilon \leq \sqrt{ k^2 + \epsilon^2 +2kp}$, but then even when I try to replace $\epsilon$ by the new variable $x = \epsilon - \frac{E}{2}$ and to use $p = \sqrt{ \epsilon^2 -1}$ (in natural units), I don't get the equations (21).

If some of you had some ideas of how to get them, it would help me a lot, 'cause it's kind of obsessing me right now ! Thank you for reading this message anyway .

 

[mfb]

What do you get, instead of the equations 21?
As they don't have $\epsilon$ and p any more, an obvious solving method is to replace them in the inequality, and then solve it for x and k respectively.

 

[Tabatta]

Thanks, I finally managed to get the first equation of 21 and then I realized that the two other ones just come from this one ... Thanks again !