introduction

In recent years, methods using energy density functionals (EDFs) [1] to describe nuclear properties are being developed in three complementary directions. First, the ideas of effective theories [G.P. Lepage, Lectures given at the VIII Jorge Andre Swieca Summer School (Brazil, 1997), nucl-th/9706029(),2] are employed in determining the EDFs from first principles [3,4,5,Stoitsov et al.(2010)Stoitsov, Kortelainen, Bogner, Duguet, Furnstahl, Gebremariam, and Schunck]. These developments are supplemented by a renewed interest [Dobaczewski et al.(2010)Dobaczewski, Carlsson, and Kortelainen,Carlsson and Dobaczewski(2010),Gebremariam et al.(2010)Gebremariam, Duguet, and Bogner] in the density-matrix expansion (DME) methods [Negele and Vautherin(1972),Negele and Vautherin(1975)], which allow for treating exchange correlations in terms of (quasi)local functionals. Second, the coupling constants of the well-known EDFs undergo a thorough scrutiny, including an advanced work on the readjustment of parameters [Erler et al.(2010)Erler, Klüpfel, and Reinhard,6] and study of inter-parameter correlations [7]. Finally, the standard functionals are extended by adding new terms [Carlsson et al.(2008)Carlsson, Dobaczewski, and Kortelainen,9,8,Raimondi et al.(2011)Raimondi, Carlsson, and Dobaczewski], so as to gain increased precision of description and predictability, in quest for the spectroscopic-quality [10] and universal [Bertsch et al.(2007)Bertsch, Dean, and Nazarewicz] EDFs.

In the present work we study properties of EDFs [Carlsson et al.(2008)Carlsson, Dobaczewski, and Kortelainen] and pseudopotentials [Raimondi et al.(2011)Raimondi, Carlsson, and Dobaczewski] extended by adding terms that depend on higher-order derivatives up to sixth, next-to-next-to-next-to-leading order (N$^{3}$LO). Such extensions lead to self-consistent mean-field Hamiltonians that are sixth-order differential operators [11], that is, they depend on up to sixth power of the momentum operator. This makes them unusual objects, in the sense that standard second-order one-body Hamiltonians contain only the Laplace operator in the kinetic-energy term and possibly the angular-momentum operator in the spin-orbit term. The main question we address here is whether the presence of higher powers of momenta is compatible with the continuity equation (CE).

The CE is a differential equation that describes a conservative transport of some physical quantity [K. F. Riley, M. P. Hobson and S. J. Bence,Mathematical Methods For Physics And Engineering, (Cambridge Univ. Press, Cambridge, 2008)()]. In quantum mechanics, it relates the time variation of the probability density to the probability current [12]. In our case, it appears when the N$^{3}$LO EDFs or pseudopotentials are employed within a time-dependent theory. For the standard Skyrme (NLO) functional, the validity of the CE has been checked explicitly [Engel et al.(1975)Engel, Brink, Goeke, Krieger, and Vautherin]. Our goal here is to derive constraints on the coupling constant of the N$^{3}$LO EDF or parameters of the pseudopotential that would guarantee the validity of the CE. Apart from linking the CE to the local gauge symmetry [13], we also analyze the CEs in vector and isovector channels and link them to the local non-abelian gauge symmetries.

The paper is organized as follows. In Sec. 2.1 we present the standard quantal CE for a single particle and introduce the vector CE. Then, in Sec. 2.2 we discuss the CEs within the time-dependent density functional theory and in Sec. 2.3 we specify the case to the N$^3$LO quasilocal functional. The main body of results obtained for the CEs in the four spin-isospin channels is presented in Sec. 3 and Appendices A-C. Finally in Sec. 4 we formulate the conclusions of the present study.