  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.

High Energy Physics - Phenomenology