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Background: The next-to-next-to-next-to-leading order (N3LO) nuclear energydensity functional extends the standard Skyrme functional with new termsdepending on higher-order derivatives of densities, introduced to gain betterprecision in the nuclear many-body calculations. A thorough study of thetransformation properties of the functional with respect to differentsymmetries is required, as a step preliminary to the adjustment of the couplingconstants. Purpose: Determine to which extent the presence of higher-orderderivatives in the functional can be compatible with the continuity equation.In particular, to study the relations between the validity of the continuityequation and invariance of the functional under gauge transformations. Methods:Derive conditions for the validity of the continuity equation in the frameworkof time-dependent density functional theory. The conditions apply separately tothe four spin-isospin channels of the one-body density matrix. Results: Weobtained four sets of constraints on the coupling constants of the N3LO energydensity functional that guarantee the validity of the continuity equation inall spin-isospin channels. In particular, for the scalar-isoscalar channel, theconstraints are the same as those resulting from imposing the standard U(1)local-gauge-invariance conditions. Conclusions: Validity of the continuityequation in the four spin-isospin channels is equivalent to the local-gaugeinvariance of the energy density functional. For vector and isovector channels,such validity requires the invariance of the functional under local rotationsin the spin and isospin spaces.

Continuity equation and local gauge invariance for the N3LO nuclear Energy Density Functionals