Extended Skyrme effective interactions with higher-order momentum dependence for transport models and neutron stars

The recently developed extended Skyrme effective interaction based on the so-called N3LO Skyrme pseudopotential is generalized to the general $\mathrm{N}n\mathrm{LO}$ case by incorporating the derivative terms up to $2n\mathrm{th}$ order into the central term of the pseudopotential. The corresponding expressions of Hamiltonian density and single-nucleon potential are derived within the Hartree-Fock approximation under general nonequilibrium conditions. The inclusion of the higher-order derivative terms provides additional higher-order momentum dependence for the single-nucleon potential, and in particular, we find that the N5LO single-nucleon potential with momentum-dependent terms up to ${p}^{10}$ can give a nice description for the empirical nucleon optical potential up to energy of $2\phantom{\rule{0.28em}{0ex}}\mathrm{GeV}$. At the same time, the density-dependent terms in the extended Skyrme effective interaction are extended correspondingly in the spirit of the Fermi momentum expansion, which allows highly flexible variation of density behavior for both the symmetric nuclear matter equation of state and the symmetry energy. Based on the Skyrme pseudopotential up to N3LO, N4LO, and N5LO, we construct a series of interactions with the nucleon optical potential having different high-momentum behaviors and with the symmetry potentials featuring different linear isospin-splitting coefficients for nucleon effective mass, by which we study the properties of nuclear matter and neutron stars. Furthermore, within the lattice Boltzmann-Uehling-Uhlenbeck transport model, some benchmark simulations with selected interactions are performed for the $\mathrm{Au}+\mathrm{Au}$ collisions at a beam energy of $1.23\phantom{\rule{0.28em}{0ex}}\mathrm{GeV}\phantom{\rule{0.28em}{0ex}}\mathrm{per}\phantom{\rule{0.28em}{0ex}}\mathrm{nucleon}$, and the predicted collective flows for protons are found to nicely agree with the data measured by HADES collaboration.