Self-consistent symmetries in the proton-neutron Hartree-Fock-Bogoliubov approach

Symmetry properties of densities and mean fields appearing in the nuclear density functional theory with pairing are studied. We consider energy functionals that depend only on local densities and their derivatives. The most important self-consistent symmetries are discussed: spherical, axial, space-inversion, and mirror symmetries. In each case, the consequences of breaking or conserving the time-reversal and/or proton-neutron symmetries are discussed and summarized in a tabulated form, useful in practical applications. Particular attention is paid to the case of broken proton-neutron symmetry, especially in the context of isoscalar pairing. We demonstrate that isoscalar pairing fields have geometrical properties markedly different from those of the usual isovector pairing fields and their theoretical treatment requires great care.