Calculating the scalar self-force experienced by extreme-mass-ratio binaries during $r\theta$-resonances*

A vast majority of extreme-mass-ratio black hole binaries (EMRIs) will encounter at least one strong $r\theta$-resonance as they evolve through LISA's passband. These resonances occur when the frequencies of the librating radial and polar motion of the EMRI's smaller body form a low-integer ratio, and they drive significant `kicks' in the amount of energy and angular momentum that EMRIs radiate through gravitational waves. These kicks, if not properly accounted for, can amplify errors in modeled EMRI waveforms by factors of $\sim 100$. Despite the importance of modeling these resonant dynamics, researchers have not yet calculated the gravitational self-force experienced by EMRIs during $r\theta$-resonances. As a first step in quantifying these effects, we calculate the scalar self-force (the scalar analog to the gravitational self-force) experienced by a scalar-charged particle following an $r\theta$-resonant geodesic around a Kerr black hole. We present how local and global radiation-reaction effects vary with respect to initial conditions. We also demonstrate, numerically, that conservative self-force effects do not contribute to the leading-order evolution of the system, as hypothesized by previous researchers.