Local nuclear energy density functional at next-to-next-to-next-to-leading order

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We construct nuclear energy density functionals in terms of derivatives of densities up to sixth, next-to-next-to-next-to-leading order (${\mathrm{N}}^{3}\mathrm{LO}$). A phenomenological functional built in this way conforms to the ideas of the density matrix expansion and is rooted in the expansions characteristic to effective theories. It builds on the standard functionals related to the contact and Skyrme forces, which constitute the zero-order (LO) and second-order (NLO) expansions, respectively. At ${\mathrm{N}}^{3}\mathrm{LO}$, the full functional with density-independent coupling constants, and with the isospin degree of freedom taken into account, contains 376 terms, whereas the functionals restricted by Galilean and gauge symmetries contain 100 and 42 terms, respectively. For functionals additionally restricted by the spherical, space-inversion, and time-reversal symmetries, the corresponding numbers of terms are equal to 100, 60, and 22, respectively.