Double diffractive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho$\end{document}-production in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma^*\gamma^*$\end{document} collisions

We present a first estimate of the cross-section for the exclusive process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma^{*}_{\mathrm {L}} (Q_1^2)\gamma^{*}_{\mathrm {L}}(Q_2^2) \to \rho_{\mathrm {L}}^0 \rho_{\mathrm {L}}^0$\end{document}, which will be studied in the future high energy e + e--linear collider. As a first step, we calculate the Born order approximation of the amplitude for longitudinally polarized virtual photons and mesons, in the kinematical region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s \gg -t, Q_1^2 , Q_2^2$\end{document}. This process is completely calculable in the hard region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Q_1^2, Q_2^2 \gg \Lambda^2_{\mathrm {{QCD}}}$\end{document}. We perform most of the steps in an analytical way. The resulting cross-section turns out to be large enough for this process to be measurable with foreseen luminosity and energy, for Q12 and Q22 in the range of a few GeV2.