A unified algorithm for multi-particle correlations between azimuthal angle and transverse momentum in ultra-relativistic nuclear collisions

Multi-particle correlations between azimuthal angle and mean transverse momentum are a powerful tool for probing size and shape correlations in the initial conditions of heavy-ion collisions. These correlations have also been employed to investigate nuclear structure, including potential nuclear shape phase transitions at the energy frontier. However, their implementation is highly nontrivial, and prior studies have been mostly limited to lower-order correlations, such as the modified Pearson correlation coefficient, $ρ(v_{\rm n}^{2}, [p_{\rm T}])$. This paper presents a unified framework that employs a recursive algorithm, enabling the efficient evaluation of arbitrary-order correlations while maintaining computational efficiency. This framework is demonstrated using widely adopted transport models, including AMPT and HIJING. The proposed unified algorithm for multi-particle correlations between azimuthal angle and transverse momentum provides a systematic and efficient approach for multi-particle correlation analyses. Its application in experiments at the Relativistic Heavy Ion Collider and the Large Hadron Collider facilitates the exploration of nuclear structure at ultra-relativistic energies.