Hexadecapole response to tensor terms in Skyrme functionals: quadrupole rearrangement and functional dependence

Abstract: Tensor terms can change a nuclear hexadecapole moment by rearranging the density at fixed quadrupole shape or by shifting that shape along a coupled valley. Axial Skyrme--Hartree--Fock calculations with BCS pairing reveal contrasting roles for these routes. In a 21-nucleus SLy5/SLy5T survey, excluding three Zr nuclei with changed reference branches leaves 18 nuclei: 12 have $|\Delta\beta_4|<0.005$, five exceed $0.01$, and one lies between. Constrained paths show that quadrupole readjustment dominates the reduction of $\beta_4$ by about $0.040$ in $^{28}$Si and supplies the larger part of the cooperative enhancement in $^{116}$Xe. In $^{154}$Sm, the fixed-shape contribution instead dominates; quadrupole readjustment offsets about one third of it, leaving a net increase of $0.0123$. In $^{156}$Gd, SLy5T increases the quadrupole-relaxed hexadecapole energy gain from $1.78$ to $3.18$ MeV while increasing the local curvature from $165$--$169$ to $240$--$257$ MeV. The greater gain thus accompanies a stiffer deformed minimum. Matched TIJ comparisons show that refitting the EDF background substantially reduces the $\beta_4$ suppression produced by the same tensor terms on the SLy5 background. Rotational $E4$ estimates increase with SLy5T in seven rare-earth nuclei without uniformly improving agreement with experiment. The equilibrium moment change alone therefore reveals neither the balance of the two response routes nor the associated change in energetic preference.