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The amplitudes for the $p p \to p p \pi^{0}$ process applicable at highenergy are discussed in detail. Both diffractive bremsstrahlung(Drell-Hiida-Deck--type model), photon-photon and photon-omega exchangemechanisms are included in the calculation. Large cross sections of the orderof mb are predicted. The corresponding differential cross sections inrapidities and transverse momenta of outgoing protons and pions as well asrelative azimuthal angle between outgoing protons are calculated for RHIC andLHC energies. The hadronic bremsstrahlung contributions dominate at large(forward, backward) pion rapidities. The diffractive nonresonant backgroundcontributes at small $\pi^0 p$ invariant mass and could be thereforemisinterpreted as the Roper resonance. We predict strong dependence of theslope in $t$ (squared four-momentum transfer between ingoing and outgoingproton) on the mass of the supplementary excited $\pi^{0}p$ system. At highenergies and midrapidities, the photon-photon contribution dominates over thediffractive components, however, the corresponding cross section is rathersmall. The photon-odderon and odderon-photon contributions are included inaddition and first estimates (upper limits) of their contributions to rapidityand transverse momentum distribution of neutral pions are presented. We suggesta search for the odderon contribution at midrapidity and at $p_{\perp,\pi^{0}}\sim$ 0.5 GeV. Our predictions are ready for verification at RHIC and LHC. Thebremsstrahlung mechanisms discussed here contribute also to the $pp \to p(n\pi^{+})$ reaction. Both channels give a sizable contribution to the low-masssingle diffractive cross section and must be included in extrapolating themeasured experimental single diffractive cross section.

Exclusive $p p \to p p \pi^{0}$ reaction at high energies