Bayesian Optimization of The Relativistic Heavy Ion Collider Luminosity via $s^*$ Control
Maximizing luminosity at the interaction point (IP) requires the collision location $s_{IP}$ to coincide with the longitudinal position of the minimum beta function, $s^$. Accurate optics measurements and control of $s^$ are therefore essential for luminosity optimization. At the Relativistic Heavy Ion Collider (RHIC), average horizontal beta-beat measurements between operating IPs are approximately $20%$, with significant variation in measured $s^*$. Precise control of the beam waist position $s^$ is particularly challenging for modern high-energy colliders with short bunch lengths and large crossing angles. We present an online Bayesian optimization (BO) application using the GPTune framework to optimize sPHENIX luminosity through $s^$ control at RHIC. GPTune was first validated at the RHIC Electron Beam Ion Source (EBIS), where it achieved up to a $70%$ increase in beam intensity over the baseline, although experienced operators could reach similar performance with longer manual tuning. The framework was subsequently deployed during sPHENIX operations. Using an intensity-normalized Zero-Degree Calorimeter (ZDC) signal as the optimization objective, due to the unavailability of the live sPHENIX MVTX signal, GPTune identified local luminosity maxima, recovered from intentionally degraded $s^$ configurations, and revealed residual horizontal and vertical waist offsets in the interaction region. These results demonstrate the robustness and efficiency of Bayesian optimization for real-time collider tuning under noisy, time-varying conditions. The $s^$ control methodology provides a promising tool for precision luminosity optimization and is particularly relevant to next-generation short-bunch colliders such as the Electron-Ion Collider (EIC).