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We present the explicit form of the next-to-next-to-leading order (N$^2$LO)Skyrme interaction in momentum space by including the fourth-order gradientpotentials to the standard Skyrme interaction. With the N$^2$LO Skyrmeinteraction, we evaluate the second-order corrections to the nuclear bulkquantities of nuclear matter: equation of state (EoS) of isospin symmetric andpure neutron matter, density-dependent in-medium effective nucleon mass,isospin-asymmetry energy, pressure and incompressibility. These second-ordercontributions are ultraviolet (UV) divergent due to the zero range character ofthe interaction and renormalized using the techniques of dimensionalregularization (DR) with the minimal subtraction scheme (MS). We adjust the 18parameters of the interaction by performing a global fit to the nuclear bulkquantities. Besides the too strong dependence $k_F^{12}$ of severalsecond-order corrections, a very good reproduction of a realistic nuclearmatter saturation curve with all the nuclear bulk quantities in the densityregion $0.12<\rho<0.20\;$ fm$^{-3}$ is obtained.

Next-to-next-to-leading order Skyrme interaction in nuclear matter: Nuclear bulk quantities at second order in perturbation theory