Toward extracting scattering phase shift from integrated correlation functions IV: Coulomb corrections

The formalism developed in Refs.~\cite{Guo:2023ecc,Guo:2024zal,Guo:2024pvt} that relates the integrated correlation functions for a trapped system to the infinite volume scattering phase shifts through a weighted integral is further extended to include Coulomb interaction between charged particles. The original formalism cannot be applied due to different divergent asymptotic behavior resulting from the long-range nature of the Coulomb force. We show that a modified formula in which the difference of integrated correlation functions between particles interacting with Coulomb plus short-range interaction and with Coulomb interaction alone is free of divergence, and has rapid approach to its infinite volume limit. Using an exactly solvable model, we demonstrate that the short-range potential scattering phase shifts can be reliably extracted from the formula in the presence of Coulomb interaction.