We make predictions for cross sections of $rho$ and $phi$ vector meson photoproduction in ultraperipheral Xe-Xe collisions at $sqrt{s_{NN}}=5.44$ TeV. Analyzing the momentum transfer distribution of $rho$ mesons in this process, we explore the feasibility of extracting the nuclear density of $^{129}$Xe, which is needed in searches for dark matter with Xenon-based detectors.
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