Uncovering the mechanism of chiral three-nucleon force in driving spin-orbit splitting

The three-nucleon force (3NF) is crucial in shaping the shell structure of atomic nuclei, particularly impacting the enhancement of spin-orbit (SO) splitting, especially in nuclei with significant deviations from stability. Despite its importance, the specific mechanisms driving this enhancement remain unclear. In this study, we introduce a decomposition scheme based on the rank of irreducible tensors forming the 3NF, derived from chiral effective field theory at next-to-next-to-leading order, to elucidate their influence on SO splitting. Within the shell-model framework, our analysis reveals that the rank-1 component of the 3NF is the primary factor enlarging the energy gap between the 0p3/2 and 0p1/2 single-particle levels in p-shell nuclei, while the rank-2 component makes a subdominant contribution. Since the rank-1 component originates exclusively from the 2π-exchange 3NF, our finding will not depend on the choice of the low-energy constants of contact terms. We also remark on the antisymmetry of the rank-1 3NF, which can affect the quantum entanglement of spin states. This study lays the groundwork for further exploration into this field toward a microscopic understanding of the 3NF impact on the nuclear shell structure.