Corrections to Equivalent-Photon Approximation for Two-Photon Processes in Colliding Beams

The cross section for the two-photon process $\mathrm{ee}\to \mathrm{ee}\Gamma$ in colliding electron beams, in the limit $E\to \infty$ with $\frac{E^2}{s}$ fixed ($E$ being the energy per beam of the colliding electrons of mass $m$, and $\sqrt{s}$ the invariant mass of the state $\Gamma$), is well known to have a leading term of order ${\left[\ln \right(\frac{4E^2}{m^2}\left)\right]}^2$. This leading term is given correctly by the well-known Weizsäcker-Williams approximation or the equivalent-photon approximation. There are sizable corrections coming from nonleading terms, of which the first, namely the term of order $\ln \left(\frac{4E^2}{m^2}\right)$, has been obtained for an arbitrary hadronic final state $\Gamma$. The approximations made in getting the Weizsäcker-Williams result and the correction terms due to each of them have been exhibited.