Background: The next-to-next-to-next-to-leading order (N3LO) nuclear energy
density functional extends the standard Skyrme functional with new terms
depending on higher-order derivatives of densities, introduced to gain better
precision in the nuclear many-body calculations. A thorough study of the
transformation properties of the functional with respect to different
symmetries is required, as a step preliminary to the adjustment of the coupling
constants. Purpose: Determine to which extent the presence of higher-order
derivatives in the functional can be compatible with the continuity equation.
In particular, to study the relations between the validity of the continuity
equation and invariance of the functional under gauge transformations. Methods:
Derive conditions for the validity of the continuity equation in the framework
of time-dependent density functional theory. The conditions apply separately to
the four spin-isospin channels of the one-body density matrix. Results: We
obtained four sets of constraints on the coupling constants of the N3LO energy
density functional that guarantee the validity of the continuity equation in
all spin-isospin channels. In particular, for the scalar-isoscalar channel, the
constraints are the same as those resulting from imposing the standard U(1)
local-gauge-invariance conditions. Conclusions: Validity of the continuity
equation in the four spin-isospin channels is equivalent to the local-gauge
invariance of the energy density functional. For vector and isovector channels,
such validity requires the invariance of the functional under local rotations
in the spin and isospin spaces.Comment: 12 Latex pages, submitted to Physical Review
