Photoproduction and detection of $\rho'\rightarrow\pi^+\pi^-\pi^+\pi^-$ decays in ultra-peripheral collisions and at an electron-ion collider

Vector meson photoproduction is an important probe of nuclear structure. Light vector mesons are most sensitive to low$-x$ structure, as long as they are not too light for perturbative QCD calculations. The $\rho'$ is of interest as an intermediate mass state (between the $\rho$ and $J/\psi$) that is easier to detect than the $\phi$. Using HERA data on proton targets, we make projections for lead/gold targets in UPCs at the Large Hadron Collider and RHIC, and for $ep$ and $eA$ collisions at a future Electron-Ion Collider (EIC). These projections for ion targets depend on the largely-unknown $\rho'\rightarrow\pi^+\pi^-\pi^+\pi^-$ branching ratio, and use existing data to constrain that branching ratio. Current data points to a relatively low branching ratio, less than 50\%. The HERA $ep$ and ALICE UPC $e$Pb data exhibit very similar $4\pi$ mass spectra, indicating that, if the system is composed of two resonances, the products of their photon couplings with their four-pion branching ratios are similar. The predicted rates are high for both UPCs and the EIC. The $\rho'\rightarrow\pi^+\pi^-\pi^+\pi^-$ decay can be observed at the EIC with high efficiency. In $ep$ collisions at the highest energy, the forward B0 detector is needed to observe this channel down to the lowest achievable Bjorken$-x$ values.