Double equivalent-photon approximation including radiative corrections for photon-photon collision experiments without electron tagging

It is shown that, in first approximation, radiative corrections in photon-photon collision experiments without electron tagging can be estimated by using a double equivalent-photon approximation of the same type as (but, of course, analytically much more complicated than) that used for computation of $\ensuremath{\gamma}\ensuremath{\gamma}$ cross sections without radiative corrections. It is obvious that, in such an approximation, radiative corrections\char22{}for a given beam energy ${E}_{0}$ and a given invariant mass $M$ produced\char22{}become independent of the specific process $\ensuremath{\gamma}\ensuremath{\gamma}\ensuremath{\rightarrow}X$ considered. The invariant-mass spectrum, corrected for radiation, will be written in the general form $\frac{d{\ensuremath{\sigma}}^{\mathrm{corr}}}{\mathrm{dM}}=(1+\ensuremath{\delta})\frac{d{\ensuremath{\sigma}}^{0}}{\mathrm{dM}}$ where $\frac{d{\ensuremath{\sigma}}^{0}}{\mathrm{dM}}$ is the uncorrected spectrum. Values obtained for $\ensuremath{\delta}$ at typical beam energies ${E}_{0}=1.5, 3, 15, \mathrm{and} 70$ GeV, and for $\frac{M}{{E}_{0}}$ ranging between 0.1 and 0.6, are systematically of the order of less than \ifmmode\pm\else\textpm\fi{} 1%.