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

TL;DR

This work introduces the Finite Volume Hamiltonian (FVH) method to systematically block-diagonalize the finite-volume two-particle Hamiltonian for arbitrary spins in both rest and moving frames, directly addressing left-hand cuts from long-range potentials. By combining a projection-operator approach with an explicit irrep decomposition and detailed matrix-element formulas, the method provides a practical, model-independent bridge between finite-volume spectra and infinite-volume scattering amplitudes, even in the presence of significant long-range forces. Toy-model benchmarks demonstrate FVH's consistency with standard Lüscher results when long-range effects are absent and expose the breakdown of traditional Lüscher analyses near left-hand cuts, while a realistic application to the isoscalar $D\bar{D}^*$ system related to $\chi_{c1}(3872)$ shows qualitative agreement with lattice QCD. Overall, FVH offers a transparent, versatile framework for lattice studies of two-body and near-term three-body hadron systems with long-range interactions, with clear pathways to three-particle extensions and exotic-state investigations.

Abstract

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 $D\bar{D}^*$ system, where the well-known exotic state $χ_{c1}(3872)$ 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 hadron physics.

Figures (7)

• Figure 1: Model I and II: $k_{\text{on}}\cot\delta^1_0$ and $k^5_{\text{on}}\cot\delta^1_2$ determined by standard Lüscher formula(solid circle) and partial wave LSE(solid line). The dashed line denotes the threshold $m_1+m_2$. The employed parameters are $|c_S|=4\times10^{-5}/\text{MeV}^2$ and $|c_D|=3\times10^{-3}/\text{MeV}^2$.
• Figure 2: Model III: $k_{\text{on}}\cot\delta^1_0$ determined by standard Lüscher formula(solid circle and diamond) and partial wave LSE(solid line). The gray and green dashed lines denote the threshold and endpoint of left-hand cut, respectively. Subfigures (c) and (d) provide detailed views of regions (a) and (b) near the threshold, offering a clearer perspective of the deviations is these regions. Below the left-hand cut, $k_{\text{on}}\cot\delta^1_0$ determined by partial wave LSE acquires a non-zero imaginary component, depicted by the dot-dashed line. In subfigures (c) and (d) we zoom in the imaginary component near the cut in the upper right corner. The employed parameters are $(c_S,g_1,g_2)=(1,1,5)\times10^{-5}/\text{MeV}^2$.
• Figure 3: Model III: Same as Figs. \ref{['fig:Model3']}(c,d) with $|g|=g_2$. The solid makers denote the $k_{\text{on}}\cot\delta^1_0$ determined by standard Lüscher formula at several $L$, specifically $m_{\text{eff}}L=1.5,\,2,\,3,\,4$.
• Figure 4: Model IV: $k_{\text{on}}\cot\delta^1_0$ determined by standard Lüscher formula(solid circles) and partial wave LSE(solid line). The orange and green solid circles, which corresponds to the energy levels mainly contributed by the $S$-wave components of potential, are determined by Eq.(\ref{['eq:swave:luscher']}) and Eq.(\ref{['eq:complete:luscher']}), respectively. The employed parameters are $|c_S|=2\times10^{-5}/\text{MeV}^2$ and $|g|=0.5\times10^{-5}/\text{MeV}^2$ for the top row or $|g|=4\times10^{-5}/\text{MeV}^2$ for the bottom row.
• Figure 5: Model IV: partial wave expansion of potential $|V_{00}(k\hat{e}_z,k\hat{e_z})|$ with truncation $J\leq J_{max}$ and $c_S=0$. The $y$-axis is set as log scale. The vertical gray dashed lines from left to right denote the lowest three lattice momentum $k=0,\frac{2\pi}{L}$ and $\frac{2\sqrt{2}\pi}{L}$.