Addendum: 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 update our analyses concerning the non-resonant (Drell-S\"oding) contribution to the reaction $\gamma^{(*)} p \to \pi^{+} \pi^{-} p$, 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-S\"oding 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.