Three-nucleon contact forces in the Jacobi partial-wave basis

TL;DR

The paper develops a permutation-invariant, zero-range framework to construct three-nucleon contact interactions in the Jacobi partial-wave basis, formulating separable W ∝ |H_c⟩⟨H_d| + h.c. and solving a finite-dimensional eigenproblem for the cyclic permutation P_123 to obtain permutation-invariant states. By organizing these zero-range 3N states into finite subspaces Γ_c and classifying them by ν = l + λ + 2(m + n), the authors identify all relevant channels up to O(Q^2) and assemble the 3N contact potentials as outer products of H_c and H_d, conserving J, T, and parity. In the T = 1/2 and T = 3/2 sectors, explicit matrix elements are provided for a set of O(Q^0) and O(Q^2) operators, with detailed plane-wave to Jacobi-basis transformations demonstrating consistency with known operator structures. They report 13 independent O(Q^2) 3N contact operators, aligning with Girlanda, and discuss the framework’s potential applications to Faddeev–Yakubovsky equations and no-core shell model calculations, as well as its limitations and avenues for further refinement. The approach thus offers a robust, exchange-symmetric foundation for 3N contact forces in both Pionless EFT and chiral EFT contexts, enabling systematic, symmetry-respecting implementations in few- and many-body nuclear calculations.

Abstract

We construct the three-nucleon contact potentials in the Jacobi partial-wave basis. The potentials are built in the separable form as the products of the antisymmetrized three-nucleon states in which the nucleons are arbitrarily close to each other. We compile the three-nucleon contact potentials up to $\mathcal{O}(Q^2)$. These contact potentials are by construction independent operators under exchanges of any nucleon pairs.