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

The three-nucleon force (3NF) plays a vital role in shaping the shell structure of atomic nuclei, particularly through its enhancement of spin-orbit (SO) splitting. While its importance is well recognized, the underlying mechanism remains insufficiently understood. In this study, we analyze the contribution of the 3NF to SO splitting by decomposing the chiral 3NF at next-to-next-to-leading order into irreducible tensor components and implementing them within shell-model calculations. Focusing on p-shell nuclei, we find that the rank-1 component of the 3NF, which arises solely from the two-pion exchange term, is primarily responsible for the enlarged energy gap between the 0p3/2 and 0p1/2 orbitals. The rank-2 component gives a smaller, subleading effect, while higher-rank contributions are negligible. Since the rank-1 part is independent of the contact low-energy constants, the result is robust against their variation. In addition, preliminary analysis in the pf-shell region indicates that the conclusions drawn for p-shell nuclei persist beyond light systems. Our findings clarify the tensorial origin of the 3NF contribution to SO splitting and provide a foundation for further investigations of three-body effects in nuclear structure.