γ γ cross-sections and γ γ colliders

We summarize the predictions of different models for total $\gamma \gamma$ cross-sections. The experimentaly observed rise of $\sigma_{\gamma \gamma}$ with $\sqrt{s_{\gamma \gamma}}$, faster than that for $\sigma_{\bar p p}$, $\sigma_{\gamma p}$ is in agreement with the predictions of the Eikonalized Minijet Models as opposed to those of the Regge-Pomeron models. We then show that a measurement of $\sigma_{\gamma \gamma}$ with an accuracy of $\lessapprox 8-9% (6-7%)$ is necessary to distinguish among different Regge-Pomeron type models (among the different parametrisations of the EMM models) and a precision of $\lessapprox$ 20% is required to distinguish among the predictions of the EMMs and of those models which treat like 'photon like a proton', for the energy range $300 < \sqrt{s_{\gamma \gamma}} < 500$ GeV. We further show that the difference in model predictions for $\sigma_{\gamma \gamma}$ of about a factor 2 at $\sqrt{s_{\gamma \gamma}} = 700 $ GeV reduces to $\sim$ 30% when folded with bremsstrahlung $\gamma$ spectra to calculate $\sigma ({e^+e^- \to e^+e^-\gamma \gamma \to e^+e^-X})$. We point out then the special role that $\gamma \gamma$ colliders can play in shedding light on this