**F. Raimondi, B. G. Carlsson, J. Dobaczewski, J. Toivanen**

**November 11, 2011**

**Background:**- The next-to-next-to-next-to-leading order (NLO) 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 NLO 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.

- introduction
- Continuity equation in the EDF approach
- Time evolution of a spin- particle
- Time-dependent density functional theory
- The NLO quasilocal functional

- Continuity equations in the four spin-isospin channels
- Constraints for zero-order terms
- Constraints for the scalar-isoscalar channel
- Constraints for the scalar-isovector channel
- Constraints for the vector-isoscalar channel
- Constraints for the vector-isovector channel

- conclusions
- Constraints for the scalar-isovector channel (fourth and sixth orders)
- Constraints for the vector-isoscalar channel (fourth and sixth orders)
- Constraints for the vector-isovector channel (fourth and sixth orders)
- Bibliography
- About this document ...