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

Abstract

We update our analyses concerning the non-resonant (Drell-Söding) contribution to the reaction ${\gamma }^{(\ast )}p\to {\pi }^{+}{\pi }^{-}p$, now including general vertex functions for the coupling of pomeron, $f_{2\mathbb{R}}$, and ${\rho }_{\mathbb{R}}$, to pions. This takes into account the off-shellness of the pions. We present a comparison of our tensor-pomeron model, including our new Drell-Söding and $\rho$, $\omega$ vector-meson contributions, with data for the $\gamma p\to {\pi }^{+}{\pi }^{-}p$ reaction measured by the ZEUS and H1 Collaborations. We find very good agreement of our theory with the experimental data for the ${\pi }^{+}{\pi }^{-}$ invariant mass ($M_{\pi \pi }$) and for angular distributions. Here we restrict our analysis to the region $M_{\pi \pi }\lesssim 1.2$ GeV. For higher values of $M_{\pi \pi }$ we will have to consider production of higher-mass mesons. The purpose of the present article is to show that our tensor-pomeron model gives a very good fit to the so-called skewing of the $\rho$-resonance contribution in photoproduction. In fact, it is an old problem/puzzle to understood the difference of the shapes of the $M_{\pi \pi }$ distributions in the region of the $\rho (770)$ resonance in $e^{+}{e}^{-}$ annihilation and photoproduction. We think that with our results we have obtained for this problem a solution which is very satisfactory from the Quantum-Field-Theory point of view. Our results are also relevant for central-exclusive ${\pi }^{+}{\pi }^{-}$ production in ultraperipheral $pp$, A$p$, and AA collisions at the LHC and for photo- and electroproduction of ${\pi }^{+}{\pi }^{-}$ at future electron-proton/ion colliders, EIC and LeHC.

Comments: 14 pages, 7 figures, Addendum to JHEP 06 (2026) 015 (arXiv:2508.06334v2 [hep-ph])