Algorithms for partial wave amplitudes under covariant $L$-$S$ scheme

With the continuous accumulation of data from hadron collision experiments, efficient Partial Wave Analysis tools are indispensable for constructing a clear hadron spectrum. Currently, automated computations of scattering amplitudes primarily use the helicity scheme and covariant effective Lagrangian method. The automated calculations under the covariant $L$-$S$ scheme, which is one of the commonly used partial wave analysis schemes, have not been fully realized. In this work, we provide a general algorithm for computing partial wave amplitudes under the covariant $L$-$S$ scheme. This will lay the foundation for automated computation of partial wave amplitudes under the covariant $L$-$S$ scheme.