Diphoton production in lead-lead and proton-proton UPC

We discuss diphoton semi(exclusive) production in ultraperipheral $PbPb$ collisions at energy of $\sqrt{s_{NN}}=$ 5.5 TeV (LHC). The nuclear calculations are based on equivalent photon approximation in the impact parameter space. The cross sections for elementary $\gamma \gamma \to \gamma \gamma$ subprocess are calculated including three different mechanisms: box diagrams with leptons and quarks in the loops, a VDM-Regge contribution with virtual intermediate hadronic excitations of the photons and the two-gluon exchange contribution (formally three-loops) to elastic photon-photon scattering in the high-energy approximation. We got relatively high cross sections in $PbPb$ collisions ($306$ nb). This opens a possibility to study the $\gamma \gamma \to \gamma \gamma$ (quasi)elastic scattering at the LHC. We find that the cross section for elastic $\gamma\gamma$ scattering could be measured in the lead-lead collisions for the diphoton invariant mass up to $W_{\gamma\gamma} \approx 15-20$ GeV. We identify region(s) of phase space where the two-gluon exchange contribution becomes important ingredient compared to box and nonperturbative VDM-Regge mechanisms. We perform a similar analysis for the $pp \to pp\gamma\gamma$ reaction at energy of $\sqrt{s_{NN}}=7$ and $100$ TeV.