Significance of tensor force in pseudo-spin symmetry

We point out that the pseudo-spin symmetry (PSS) of nuclei significantly depends on the proton ($Z$) and neutron numbers ($N$), sometimes giving rise to the characteristic structures. By using the non-relativistic spherical Hartree-Fock calculation with a realistic tensor force, we show that the tensor force may be deeply relevant to the $Z$- and $N$-dependence of the PSS. While the PSS has often been discussed in the context of the relativistic symmetry, the tensor-force effects on the PSS sometimes look analogous to the $Z$-and $N$-dependence of the PSS in the relativistic mean-field (RMF) calculations without explicit tensor force. The observed variation of the $p0d_{3/2}$-$p1s_{1/2}$ levels from $^{40}$Ca to $^{34}$Si is consistent with the tensor-force-driven $Z$-dependence of the PSS, but not necessarily with the RMF result. Even though it is too early to be conclusive, this result elucidates the significance of the tensor force when discussing the PSS.