Finite volume Hamiltonian method for two-particle systems containing long-range potential on the lattice

A bstract We propose a systematic method to block-diagonalize the finite volume effective Hamiltonian for two-particle systems with arbitrary spin in both the rest and moving frame. The framework is convenient and efficient for addressing the left-hand cut issue arising from long-range potential, which are challenging in the framework of standard Lüscher formula. Furthermore, the method provides a foundation for further extension to three-particle systems. We first benchmark our method by examining several toy models, demonstrating its consistency with standard Lüscher formula in the absence of long-range potential. In the presence of long-range potential, we investigate and resolve the effects and issues of left-hand cut. As a realistic application, we calculate the finite volume spectra of isoscalar $$ D{\overline{D}}^{\ast } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> <mml:msup> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> system, where the well-known exotic state χ c 1 (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 hadron physics.