Production of $π^{+}π^{-}$ pairs in diffractive photon-proton and in proton-proton collisions revisited, in particular concerning the Drell-Söding contribution

Abstract: We discuss the central exclusive photoproduction of $\pi^{+}\pi^{-}$ pairs in photon-proton and in proton-proton collisions at high energies. The $\rho^{0}$, $\omega$, $f_{2}(1270)$, and non-resonant (Drell-S\"oding) contributions are considered. The calculation is based on the tensor-pomeron model that includes not only the dominant pomeron exchange but also reggeon and odderon exchanges. In the Drell-S\"oding contribution we have different subenergies for the $\pi^{+}p$ and $\pi^{-}p$ systems. In the method which we propose now we take this into account. Respecting the gauge-invariance constraints is then a nontrivial problem for which, however, we present a solution here. In this way we improve the corresponding calculations presented in JHEP 01, 151 (2015) and in Phys. Rev. D 91, 074023 (2015). The revised model leads to enhanced cross sections and gives an increased skewing of the $\rho^{0}$ spectral shape. For the $pp\to pp\pi^{+}\pi^{-}$ reaction, we calculate differential cross sections as function of the two-pion invariant mass, pion transverse momentum and pion pseudorapidity. Predictions of proton-pion and proton-pion-pion invariant mass distributions and the distribution in the proton-proton four-momentum transfer squared are also presented. This research is relevant in the context of ALICE, ATLAS, CMS, and LHCb measurements in $pp$ collisions, even when the leading protons are not detected and instead only rapidity-gap conditions are checked experimentally. Our results can also serve as basis for the description of coherent $\pi^{+}\pi^{-}$ production in ultra-peripheral $p$A and AA collisions at the LHC. The formulas given in our paper can directly be used for the analysis of photoproduction and small-$Q^{2}$ electroproduction in $ep$ collisions at high energies. Such data exist from the HERA experiments and will be obtained in the future at the electron-ion colliders.