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We present first predictions of the cross sections and differentialdistributions for the central exclusive reaction $pp \to pp \pi^{+} \pi^{-}\pi^{+} \pi^{-}$ being studied at RHIC and LHC. The amplitudes for theprocesses are formulated in terms of the tensor pomeron and tensor $f_{2 R}$reggeon exchanges with the vertices respecting the standard crossing andcharge-conjugation relations of Quantum Field Theory. The $\sigma \sigma$ and$\rho \rho$ contributions to the $\pi^{+} \pi^{-} \pi^{+} \pi^{-}$ final stateare considered focussing on their specificities. The correct inclusion of thepomeron spin structure seems crucial for the considered sequential mechanismsin particular for the $\rho\rho$ contribution which is treated here for thefirst time. The mechanism considered gives a significant contribution to the$pp \to pp \pi^{+} \pi^{-} \pi^{+} \pi^{-}$ reaction. We adjust parameters ofour model to the CERN-ISR experimental data and present several predictions forthe STAR, ALICE, ATLAS and CMS experiments. A measurable cross section of orderof a few $\mu b$ is obtained including the experimental cuts relevant for theLHC experiments. We show the influence of the experimental cuts on theintegrated cross section and on various differential distributions.

physical review. d/physical review. d.

Exclusive diffractive production of $\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ via  the intermediate $\sigma\sigma$ and $\rho\rho$ states in proton-proton  collisions within tensor pomeron approach