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

TL;DR

The paper addresses finite-volume corrections in many-body cluster systems by introducing a geometric factor that arises from the configuration-space boundary structure. It develops a Jacobi-coordinate lattice framework to rigorously separate cluster-relative motion and derives how the leading volume dependence scales with a combinatorial factor G_{A,C}. Numerical verifications in light nuclei demonstrate that incorporating the geometric factor is crucial for accurately extracting asymptotic normalization constants (ANCs) and understanding breakup channels, even in the presence of non-perturbative Coulomb effects modeled by Whittaker functions. The results highlight the importance of accounting for geometric multiplicities in ab initio finite-volume studies to reliably predict nuclear structure and reaction rates.

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.

Figures (7)

• Figure 1: Iterative coordinate transformation.
• Figure 2: Coordinate transformation at the i-th iteration. This transformation is purely geometric and involves only shifting transparent regions to opaque regions of the same colors by multiples of L, indicated by the arrows in the figure. A new rectangular region that is aligned with Jacobi coordinates are then acquired and is identical to the cubic lattice under the boundary condition.
• Figure 3: Truncation dependence in IMSRG calculations. We choose the simple [4]He as our validation. Truncations of odd and even $e_{\rm max}$ are separated to extract ANCs. We ignored contribution of coulomb force, and extracted ANCs for $\isotope[4]{He}-N$ cluster where $N$ is any nucleon using the simple exponential form of truncation dependence.
• Figure 4: Volume dependence In NLEFT calculations on [4]He. In this calculation the SU(4) invariant essential interaction is used Lu:2018bat. We ignored the contribution of the Coulomb force, and extracted ANCs for $\isotope[4]{He}-N$ cluster, where $N$ is any nucleon, using the simple exponential form of volume dependence.
• Figure 5: Volume dependence in NLEFT calculations on [20]Ne. In this calculation, the same SU(4) invariant essential interaction is used as in Fig. \ref{['fig:henleft']}. Coulomb interaction between clusters is non-perturbative ($\bar{\eta} >1$) in this case, and hence the Eq. \ref{['eq:DeltaE-3D-final']} is used. We extract ANCs for $\isotope[20]{Ne}-\alpha$ cluster.