Two-Photon Processes in Colliding-Beam Experiments

The general structure of the two-photon process in colliding-beam experiments, $e+e\to e+e+\Gamma$, is studied for an arbitrary hadron final state $\Gamma$. The dependence of the scattering amplitudes on the lepton variables are explicitly factored out from the helicity amplitudes for the basic (hadronic) process $\gamma +\gamma \to \Gamma$. General formulas are given for the differential cross section as well as for important special cases. The most important inclusive channel ($\gamma +\gamma \to \mathrm{anything}$) and exclusive channel ($\gamma +\gamma \to \pi +\pi$) are studied in some detail. The first case can yield information on the fundamental process $\gamma +\gamma \to \gamma +\gamma$. The second case provides a clean method for extracting the $S$-wave $\pi -\pi$ phase shifts.