Hidden charm molecules in finite volume

TL;DR

This paper investigates methods to extract properties of hidden charm meson molecules from finite volume energy levels, proposing efficient strategies for analyzing synthetic lattice QCD data to determine bound states and scattering parameters.

Contribution

It introduces and compares novel strategies for extracting bound state energies and phase shifts from finite volume data, improving accuracy and efficiency in the analysis of hidden charm molecules.

Findings

Effective range fit with two levels is more efficient.

Potential-based fit yields accurate infinite volume results.

Gaussian wave function regularization is computationally efficient.

Abstract

In the present paper we address the interaction of pairs of charmed mesons with hidden charm in a finite box. We use the interaction from a recent model based on heavy quark spin symmetry that predicts molecules of hidden charm in the infinite volume. The energy levels in the box are generated within this model, and from them some synthetic data are generated. These data are then employed to study the inverse problem of getting the energies of the bound states and phase shifts for $D \bar D$ or $D^* {\bar D}^*$. Different strategies are investigated using the lowest two levels for different values of the box size, carrying a study of the errors produced. Starting from the upper level, fits to the synthetic data are carried out to determine the scattering length and effective range plus the binding energy of the ground state. A similar strategy using the effective range formula is considered with a simultaneous fit to the two levels, one above and the other one below threshold. This method turns out to be more efficient than the other one. Finally, a method based on the fit to the data by means of a potential and a loop function conveniently regularized, turns out to be very efficient and allows to produce accurate results in the infinite volume starting from levels of the box with errors far larger than the uncertainties obtained in the final results. A regularization method based on Gaussian wave functions turns out to be rather efficient in the analysis and as a byproduct a practical and fast method to calculate the L\"uscher function with high precision is presented.