Researchers have developed an improved finite-sum expression for the partial-wave projection of the one-particle exchange (OPE) in relativistic three-body scattering amplitudes. This advancement is crucial for analyzing systems where three particles interact at near-light speeds, a fundamental scenario in particle and nuclear physics. The new formulation is valid for three massive, spinless particles in arbitrary angular momentum configurations, extending its applicability and precision compared to previous methods.

The one-particle exchange is a critical component in describing how particles interact in scattering processes. In the context of three-body amplitudes, OPE describes the situation where a single particle is exchanged between incoming and outgoing two-body subsystems. The challenge lies in the rapid proliferation of partial waves contributing to any given total angular momentum (J), making calculations computationally intensive.

The new expression has been validated through a series of analytical and numerical checks, demonstrating that it reproduces the known properties of the partial-wave OPE, including its threshold scaling behavior and singularity structure. Compared to previous work, this formulation more efficiently handles the rapid proliferation of partial waves, enabling the construction of robust wavesets for three-body amplitude analyses. This is essential for unraveling fundamental interactions in complex systems, such as those studied in particle accelerators.