Researchers have successfully generalized Weinberg's compositeness relations, a fundamental concept in nuclear and particle physics, to include p-wave bound states near the decay threshold. This advance is crucial for understanding the nature of composite particles, such as exotic hadrons, which do not fit into the simple quark-antiquark model. The original Weinberg relations primarily applied to s-wave states, and their extension to p-wave opens new avenues for analyzing systems with non-zero orbital angular momentum.

The generalization establishes connections between the p-wave effective range expansion parameters, the binding energy (B) of the state, and the field renormalization constant (Z). The constant Z is a key indicator of the state's nature: a value close to zero suggests a molecular particle composed of other particles, while a value close to one indicates an elementary particle or a strongly compact state. The authors have also provided the corresponding Feynman rules, which will allow this formalism to be applied regardless of whether the near-threshold state is a pure molecular state or an elementary multiquark state.

This development is particularly relevant in the context of exotic hadron physics, where numerous states have been observed that cannot be easily explained as mesons (quark-antiquark) or baryons (three quarks). The ability to determine the compositeness of these states, whether tetraquarks, pentaquarks, or hadronic molecules, is essential for advancing our understanding of the strong interaction and quantum chromodynamics. The proposed methodology offers a robust theoretical tool for interpreting future and current experimental data from particle accelerators.