Hamiltonian Effective Field Theory for two-particle system containing long-range potential on the lattice

We propose a systematic method to block-diagonalize the finite volume effective Hamiltonian for two-particle system with arbitrary spin in both the rest and moving frames. The framework is convenient and efficient for addressing the left-hand cut issue arising from the long-range potential, which are challenging in the framework of standard L\"ushcer formula. Furthermore, the method provides a foundation for further extension to the three-particle system. We first benchmark our approach by examining several toy models, demonstrating its consistency with the standard L\"ushcer formula in the absence of long-range interactions. In the presence of long-range potential, we investigate and resolve the effect of left-hand cut. As a realistic application, we calculate the finite volume spectrum of isoscalar $D\bar{D}^*$ system, where the well-known exotic state $\chi_{c1}(3872)$ is observed. The results are qualitatively consistent with the lattice QCD calculation, highlighting the reliability and potential application of our framework to the study of other exotic states in the hadron physics.
Comment: 46 pages, 7 figures