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I'm currently studying pair production by two photons (a high-energy one travelling 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 [itex] k_1^{\mu} = (\omega_1, \stackrel{\rightarrow}{k_1}), k_2^{\mu} = (\omega_2, \stackrel{\rightarrow}{k_2}) [/itex], with [itex] \omega_1 \leq \omega_2 [/itex], creating an electron-positron pair with four-momentum vectors [itex] p_{\pm}^{\mu} = (\epsilon_{\pm}, \stackrel{\rightarrow}{p_{\pm}}) [/itex].

Let [itex] \stackrel{\rightarrow}{k} = \stackrel{\rightarrow}{k_1} + \stackrel{\rightarrow}{k_2} [/itex] be the total momentum of the two-photons system, and [itex] E = \omega_1 + \omega_2 , \Delta = \omega_2 - \omega_1 [/itex].

I attached the page of the paper where my "problem" is. I understand how he gets the inequality [itex] \sqrt{ k^2 + \epsilon^2 -2kp} \leq E - \epsilon \leq \sqrt{ k^2 + \epsilon^2 +2kp} [/itex], but then even when I try to replace [itex] \epsilon [/itex] by the new variable [itex] x = \epsilon - \frac{E}{2} [/itex] and to use [itex] p = \sqrt{ \epsilon^2 -1} [/itex] (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 .

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

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