Production of $\pi^{+}\pi^{-}$ pairs in diffractive photon-proton and in proton-proton collisions revisited, in particular concerning the Drell-S\"oding contribution

Abstract

We discuss the 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. 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 the present paper we give in this way a substantial improvement of the calculations for real photoproduction of $\pi^{+}\pi^{-}$ from JHEP 01, 151 (2015), and we extend the calculations to low $Q^{2}$ electroproduction, $0 \leqslant Q^{2} \leqslant 0.5$ GeV$^{2}$. The photo- and electroproduction amplitudes are then the basis for the calculation of central exclusive production (CEP) of $\pi^{+}\pi^{-}$ pairs in $pp$ collisions, where at least one proton participates in the CEP via a virtual-photon emission. 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. This research is relevant in the context of ALICE, ATLAS, CMS, and LHCb measurements in $pp$ collisions. 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 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.