Exclusive \rho \rho production in two-photon collisions is studied at LEP for quasi-real photons (\gamma \gamma , centre-of-mass energies 161 GeV \le \sqrt {s} \le 209 GeV, total integrated luminosity L=698 {pb}-1) and one virtual photon (\gamma \gamma *, 89 GeV \le \sqrt {s} \le 209 GeV L=855 {pb}-1). The cross sections of the \rho \rho production processes are determined as a function of the photon virtuality, Q2, and the two-photon centre-of-mass energy, W\gamma \gamma , in the kinematic region: Q2 \le 30 GeV2 and 1 GeV \le W\gamma \gamma \le 3 GeV.
PACS numbers: 13.60.Le
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