From the $\gamma \gamma \to p \bar{p}$ reaction to the production of $p \bar{p}$ pairs in ultraperipheral ultrarelativistic heavy-ion collisions at the LHC

In this paper we consider the production of proton-antiproton pairs in two-photon interactions in electron-positron and heavy-ion collisions. We try to understand the dependence of the total cross section on the photon-photon c.m. energy as well as corresponding angular distributions measured by the Belle Collaboration for the $\gamma \gamma \to p \bar{p}$ process. To understand the Belle data we include the proton-exchange, the $f_2(1270)$ and $f_2(1950)$ $s$-channel exchanges, as well as the hand-bag mechanism. The helicity amplitudes for the $\gamma \gamma \to f_{2} \to p \bar{p}$ process are written explicitly based on a Lagrangian approach. The parameters of vertex form factors are adjusted to the Belle data. Having described the angular distributions for the $\gamma \gamma \to p \bar{p}$ process we present first predictions for the ultraperipheral, ultrarelativistic, heavy-ion reaction $^{208}\!Pb \, ^{208}\!Pb \to$ $^{208}\!Pb \, ^{208}\!Pb \,p \bar{p}$. Both, the total cross section and several differential distributions for experimental cuts corresponding to the ALICE, ATLAS, CMS, and LHCb experiments are presented. We find the total cross section 100 $\mu$b for the ALICE cuts, 160 $\mu$b for the ATLAS cuts, 500 $\mu$b for the CMS cuts, and 104 $\mu$b taking into account the LHCb cuts. This opens a possibility to study the $\gamma \gamma \to p \bar{p}$ process at the LHC.