The Interplay Between Electromagnetic Fields and Baryon Stopping in a Hydrodynamic Model for Charged Flow

Abstract: Charge-dependent directed flow provides a sensitive probe of early electromagnetic fields and baryon stopping in relativistic heavy-ion collisions. Recent STAR measurements show a centrality-dependent sign change in the directed flow splitting (the difference between the directed flow of protons and antiprotons), indicating that electromagnetic effects alone are not sufficient to describe this observable and that the baryon stopping $-$ in particular the component of the stopped proton distribution that is odd in rapidity and odd under reflection in the impact parameter direction $-$ must also be included. We develop a semi-analytic hydrodynamic framework that combines spectator-induced electromagnetic fields with a Glauber-based description of baryon stopping, built upon an analytic solution for the background hydrodynamic flow due to Gubser together with the simplifying assumption of a constant electrical conductivity. For Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV, we find that baryon stopping gives a positive contribution to the directed flow splitting that decreases for more peripheral collisions, while electromagnetic fields give a negative contribution that is larger for more peripheral collisions. The competition between these two effects naturally reproduces the observed sign change in Au+Au collisions as a function of centrality, describes the observed rapidity dependence in the $50$--$80\%$ centrality interval, and reproduces trends seen in U+U collisions. Although the simplifying assumptions that we have made regarding the analytic background and constant conductivity limit our ability to make quantitative comparisons, our model provides a transparent explanation of how transported baryon number and spectator-induced electromagnetic fields jointly shape charge-dependent directed flow.