Contradicting effective mass scalings in the single-particle spectra calculated using the Skyrme energy density functional method

The problem of the effective mass scaling in the single-particle (s.p.) spectra calculated within the Skyrme energy density functional (EDF) method is studied. It is demonstrated that for specific pairs of orbitals like $1d_{3/2}\text{−}1f_{7/2}$ the commonly anticipated isoscalar effective mass ($m^{*}$) scaling of the s.p. level splittings is almost canceled by an implicit $m^{*}$ scaling present in the two-body spin-orbit (SO) strength. On the other hand, the $\nu f_{7/2}\text{−}\nu f_{5/2}$ SO splitting depends solely on the SO strength. Hence, two conflicting scaling properties appear to be at work in standard Skyrme EDF, making the theory internally inconsistent with respect to s.p. energies. It is argued that this unphysical property is, to a large extent, a consequence of the strategies and data sets used to fit these functionals. The inclusion of certain s.p. spin-orbit splittings to fit the two-body spin-orbit and the tensor interaction strengths reinstates the conventional $m^{*}$ scaling and improves the performance of the Skyrme EDF.