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In this paper we estimate the production of dimuons ($\mu^+ \mu^-$) inexclusive photon -- photon ($\gamma \gamma$) and diffractive Pomeron - Pomeron$(I\!P I\!P)$, Pomeron - Reggeon $(I\!P I\!R)$ and Reggeon - Reggeon $(I\!RI\!R)$ interactions in $pp$ collisions at the LHC energy. The invariant mass,rapidity and tranverse momentum distributions are calculated using the ForwardPhysics Monte Carlo (FPMC), which allows to obtain realistic predictions forthe dimuon production with two leading intact hadrons. In particular,predictions taking into account the CMS and LHCb acceptances are presented.Moreover, the contribution of the single diffraction for the dimuon productionalso is estimated. Our results demonstrate that the experimental separation ofthese different mechanisms is feasible. In particular, the events characterizedby pairs with large squared transverse momentum are dominated by diffractiveinteractions, which allows to investigate the underlying assumptions present inthe description of these processes.

Exclusive and diffractive $\mu^+ \mu^-$ production in $pp$ collisions at the LHC