  We study the mixed system of correlation functions involving a scalar field
charged under a global $U(1)$ symmetry and the associated conserved spin-1
current $J_\mu$. Using numerical bootstrap techniques we obtain bounds on new
observables not accessible in the usual scalar bootstrap. We then specialize to
the $O(2)$ model and extract rigorous bounds on the three-point function
coefficient of two currents and the unique relevant scalar singlet, as well as
those of two currents and the stress tensor. Using these results, and comparing
with a quantum Monte Carlo simulation of the $O(2)$ model conductivity, we give
estimates of the thermal one-point function of the relevant singlet and the
stress tensor. We also obtain new bounds on operators in various sectors.
