Exact factorization of finite-range three-body matrix elements between Gaussian wave packets and its application to the $0^+$ states of $^{12}C$

Abstract

We introduce finite-range three-body interactions into the $β-γ$-constrained generator coordinate method of antisymmetrized molecular dynamics. These genuine operators give saturation without an additional prescription for the projection, but require $O(A^6)$ complex exponentials at every node of the projection grid. With the graph Laplacian of the ranges, as in spectral clustering widely used in machine learning, we factorize the exponential factor of a Gaussian three-body kernel exactly into pairwise tables of $O(A^2)$ exponentials. A cofactor expansion rewrites the remaining contraction as complex matrix-matrix products. For $^{12}C$, the factorization reduces the time of the kernels with the three-body term from $530$ to $24$ times that of the two-body kernels. We apply the method to the $0^+$ states of $^{12}C$ with the Volkov, Tohsaki F1, and Kanada-En'yo--Akaishi F3B interactions, restricting the intrinsic states to time-reversal-invariant ones. Without the three-body terms of F1 and F3B, $^{12}C$ collapses far below the three $α$ threshold, whereas with them the ground state lies close to its experimental position below the threshold. When we adjust the Majorana parameter to this position, the Hoyle state lies within 0.6 MeV of experiment for all three interactions.