Impact of tensor-rank components of chiral three-nucleon forces on the single-particle structure of calcium isotopes
Background: Chiral three-nucleon forces (3NFs) play a key role in the microscopic description of nuclear shell evolution. A recent work introduced an irreducible tensor decomposition of the chiral 3NF at next-to-next-to-leading order and showed that, in $p$-shell nuclei, the enhancement of the $0p_{3/2}$--$0p_{1/2}$ spin--orbit (SO) splitting is mainly driven by its rank-1 component. Purpose: We extend the aforementioned analysis to the $0f1p$ shell to investigate whether the same mechanism persists in a heavier valence space, and how the different tensor-rank components of the 3NF affect structure properties of calcium isotopes. Methods: Effective shell-model Hamiltonians for neutrons outside the doubly magic $^{40}$Ca core are derived from chiral two-nucleon force plus 3NF. The latter is progressively included through its rank-$\lambda$ components ($\lambda=0,1,2,3$), allowing us to isolate their impact on the evolution of the neutron single-particle structure. Results: The significant enhancement of the SO splittings for both $1p$ and $0f$ orbitals produced by the chiral 3NF is mainly induced by its rank-1 component. The rank-2 term gives a smaller contribution, while the rank-3 term is negligible. The rank-0 component, and to a lesser extent the rank-1 component, are found to play an important role in determining the spacings between orbitals with different orbital angular momenta. All modifications induced by the 3NF in the single-particle structure have a relevant impact on the shell-closure properties of $^{48}$Ca. Conclusions: The dominance of the rank-1 two-pion-exchange component of the 3NF in explaining the enhancement of SO splitting -- previously identified in the $p$ shell -- persists in the $0f1p$ shell. Observed effects of the 3NF related to the different angular-momentum dependence of the orbitals are shown to arise essentially from their rank-0 and rank-1 components.