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We discuss diffractive processes in proton-proton collisions at smallmomentum transfers without and with photon radiation. We consider the softexclusive reactions $pp \to pp$, $p\bar{p} \to p\bar{p}$, and $pp \to pp\gamma$ within the tensor-pomeron and vector-odderon approach. We compare ourresults with the data for $pp$ and $p\bar{p}$ total cross sections, for theratio of real to imaginary part of the forward scattering amplitude, and forthe elastic $pp$ cross sections, especially those from TOTEM. To describe thelow-energy data more accurately the secondary reggeons must be included. Wewrite down the amplitudes for the photon bremsstrahlung in high-energyproton-proton collisions using the tensor-pomeron model. These results arerelevant for the c.m. energies presently available at the Relativistic HeavyIon Collider and at the LHC. We present predictions for the proposedmeasurements of soft photons with the planned future upgrade of the ALICEexperiment at the LHC. We investigate the limits of applicability of thesoft-photon approximation (SPA) based on Low's theorem. The corresponding SPAresults are compared to those obtained from our complete model. The regions ofphase space are given quantitatively where SPA and our complete tensor-pomeronresults are close to each other. As an example, let $k_{\perp}$, $\rm y$, and$\omega$, be the absolute value of the transverse momentum, the rapidity, andthe energy of the photon, respectively, in the overall c.m. system. For theregion $1 \; {\rm MeV} < k_{\perp} < 100 \; {\rm MeV}$ and $3.5 < |\rm y| <5.0$, we find that the SPA Ansatz with only the pole terms $\propto\omega^{-1}$ agrees at the percent level with our complete model result up to$\omega \cong 2$ GeV.

Soft-photon radiation in high-energy proton-proton collisions within the tensor-Pomeron approach: Bremsstrahlung