Photon-photon scattering at high energy and fixed angle

In this paper, we study the photon-photon elastic scattering amplitude in the limit of the incident energy $\omega \to \infty$ with the scattering angle $\theta$ fixed. In the lowest (4th) order, this amplitude is a function of $\theta$ only. We calculate explicitly these scaling functions of $\theta$ and find that they are surprisingly simple. The higher-order photon-photon scattering amplitudes have $\ln s$ factors. Thus scaling is violated. These $\ln s$ factors come from self-energy diagrams and vertex diagrams only and can therefore be absorbed into the running coupling constant. The photon-photon scattering amplitude of the $n\mathrm{th}$ perturbative order is hence of the form of $e^4{\mathrm{}\left[e\right(t\left)\right]}^{n-4\mathrm{}}$ times a function of $\theta$.