Validity of the equivalent-photon approximation for virtual photon-photon collisions

For virtual photon-photon collisions in electron storage rings we derive the equivalent-photon approximation from a helicity treatment and present it in two forms, involving, respectively (i) polarized transverse photons ("transverse-photon approximation") and (ii) unpolarized ones ("Williams-Weizsäcker approximation"). We first postulate the conditions of validity of the approximation on the basis of analytic considerations, and then check them numerically in the case of the process $\mathrm{ee}\to \mathrm{ee}{\mu }^{+}{\mu }^{-}$. For this check we consider the completely differentiated cross section as far as approximation (i) is concerned, and in the case of approximation (ii), the cross section differentiated with respect to all variables except the azimuthal angles. Our results are given in the form of tables showing the lower and higher limit of the error involved in the approximation for a large variety of kinematic configurations (i.e., energy losses and scattering angles of both electrons). These tables are discussed in detail, and conclusions are drawn as to the applicability of the equivalent-photon approximation to future experiments to be performed with high-energy electron storage rings.