## 2. KinematicsWe consider an isotropic photon field where is the dimensionless photon energy in a rest frame which we call the laboratory frame. The Lorentz invariant scalar product of the four-momenta of two photons having energies colliding under an angle of cosine in the laboratory frame is then given by Here, is the photon energy in the center-of-momentum frame. In order to allow for the possibility to create an electron-positron pair, conservation of energy implies , and the condition determines the pair-production threshold. is the Lorentz factor of the electron/positron in the cm frame where the produced electrons move with speed and . The definition of the angle variables needed in this calculation is illustrated in Fig. 1.
The cm frame moves relative to the laboratory frame with velocity and Lorentz factor . The four velocity of the laboratory frame ( in the cm frame) is denoted by . The Lorentz factors of the produced pairs in the laboratory frame are related to the cm quantities by Evaluating the Lorentz invariant scalar product in the laboratory and the cm-frame, respectively, we find and Inserting Eq. (4) into Eq. (2) and using energy conservation
() fixes the angle cosine The differential cross section for - pair production (see Eq. [11]) depends on © European Southern Observatory (ESO) 1997 Online publication: April 28, 1998 |