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

Abstract

We derive the volume dependence of bound states from a cluster-cluster picture with nucleon degrees of freedom. A constant factor called the ``geometric factor'' appears in the generalization from point-like particles to clusters. We show that this factor becomes explicit and correct when the underlying overlap integral is evaluated directly in the original many-body space. We achieve this by constructing Jacobi coordinates on the lattice under the periodic boundary. The derivation requires only the interaction \textit{between} the clusters to be short-ranged, with the non-perturbative Coulomb force included. The factor emerges as a geometric property of the many-body configuration space rather than a multiplicity of physical channels. We validate our derivation using many-body calculations; in particular, we find this factor to be essential in extracting asymptotic normalization constants from lattice calculations of the 16O ground state.