Subleading D-like Three-Nucleon Interactions

TL;DR

This work advances the high-order description of three-nucleon forces in chiral EFT by deriving the subleading D-like one-pion-exchange–contact 3NF at N$^4$LO, showing it requires 16 low-energy constants $F_i$. Through three complementary methods—direct momentum-operator parametrization, a non-relativistic πNN Lagrangian with reparametrization invariance, and a covariant Lagrangian NR reduction—the authors demonstrate a consistent operator basis and express the 3NF as $V rac{g_A^2}{4F_\pi^2}rac{oldsymbol{\sigma}_3 oldsymbol{ m dot} oldsymbol{q}_3}{q_3^2+M_\pi^2} imes ig[ ext{linear combination of } F_i ig]$ with five permutations. The isospin analysis shows all 16 LECs can, in principle, be constrained by nucleon-deuteron scattering, while unitary ambiguities from short-range NN forces can shift several LECs and push some contributions to higher orders; resonance saturation with the Δ(1232) suggests a reduced, Delta-dominated description in terms of four α-parameters that parameterize NN→NΔ short-range transitions. Overall, the results bolster a path toward high-precision 3NFs and underscore the role of Δ dynamics and UT considerations in shaping subleading 3NF structures.

Abstract

We consider subleading contributions to the three-nucleon force from tree-level diagrams involving a single-pion exchange and a contact interaction between two nucleons, which appear at fifth order in the chiral expansion. We show that the corresponding D-like three-nucleon potential depends on 16 low-energy constants, which need to be determined from few-body data. Assuming that their numerical values are governed by the intermediate Δ(1232) excitation mechanism, the considered three-nucleon force can be approximated using 4 low-energy constants that parametrize the short-range nucleon-nucleon to nucleon-Δ transition amplitude.