  We calculate differential distributions for the $p p \to p p \gamma$ reaction
at the LHC energy $\sqrt{s} = 14$ TeV. We consider diffractive classical
bremsstrahlung mechanisms including effects of non point-like nature of
protons. In addition, we take into account (vector meson)-pomeron, photon-pion
as well as photon-pomeron exchange processes for the first time in the
literature. Predictions for the total cross section and several observables
related to these processes e.g. differential distributions in pseudorapidities
and transverse momenta of photons or protons are shown and discussed. The
integrated diffractive bremsstrahlung cross section ($E_{\gamma}>100$ GeV) is
only of the order of $\mu$b. We try to identify regions of the phase space
where one of the mechanisms dominates. The classical bremsstrahlung dominates
at large forward/backward photon pseudorapidities, close to the
pseudorapidities of scattered protons. In contrast, the photon-pomeron
(pomeron-photon) mechanism dominates at midrapidities but the related cross
section is rather small. In comparison the virtual-omega rescattering mechanism
contributes at smaller angles of photons (larger photon rapidities). Photons in
the forward/backward region can be measured by the Zero Degree Calorimeters
(ZDCs) installed in experiments at the LHC while the midrapidity photons are
difficult to measure (small cross section, small photon transverse momenta).
Protons could be measured by ALFA detector (ATLAS) or TOTEM detector at CMS.
The exclusivity could be checked with the help of main central detectors.
