Jacobi Coordinates on Hyper-tori and Geometric Factors in the Volume Dependencies

TL;DR

This paper develops a method to analyze volume dependencies of bound states using Jacobi coordinates on lattices, introducing a geometric factor crucial for accurate extraction of physical constants from lattice calculations.

Contribution

It presents a novel construction of Jacobi coordinates on periodic lattices and introduces a geometric factor to generalize from point particles to clusters in volume dependence analysis.

Findings

The geometric factor is essential for accurate asymptotic normalization constants.

Validation through many-body calculations confirms the method's effectiveness.

Application to extsuperscript{16}O ground state demonstrates practical utility.

Abstract

We derive the volume dependence of bound states from a cluster-cluster picture with nucleon degrees of freedom. To achieve this, we demonstrate how to construct Jacobi coordinates on the lattice under the periodic boundary. A constant factor called ``Geometric factor'' appears in the generalization from point-like particles to clusters. We validate our derivation using many-body calculations, specifically, we find this factor to be essential in extracting asymptotic normalization constants from lattice calculations of \isotope[16]{O} ground state.