Conclusions

Experimental studies of the heavy $N$$\sim$$Z$ nuclei have sparked renewed interest in physics of pn correlations, especially pn pairing. While the appearance of the $T$=1 pn pairing is a simple consequence of the charge invariance, in spite of vigorous research, no hard evidence for the elusive $T$=0 pairing phase has yet been found. There are conflicting messages coming from calculations based on the quasiparticle approach. In some models, the $T$=0 and $T$=1 pairing modes are mutually exclusive, while in others they are not. What is clear, however, that predictions of calculations that impose some symmetry constraints (which can rule out the presence of some pairing fields), should be taken with the grain of salt.

In this work, we propose the most general nuclear energy density functional which is quadratic in isoscalar and isovector densities. To this end, we discuss the isospin structure of the density matrices and self-consistent mean fields that appear in the coordinate-space HFB theory allowing for a microscopic description of pairing correlations in all isospin channels. The resulting expressions incorporate an arbitrary mixing between protons and neutrons. No particular self-consistent symmetries of the energy density functional have been imposed, however, the consequences of the time reversal and proton-neutron symmetry are discussed. The obtained nuclear energy density functional (77-80) does not have to be related to any given local potential. However, if the underlying potential is local and velocity-independent, the potential energy density is invariant with respect to a local gauge transformation. The resulting densities appear in certain gauge-invariant combinations (100,104) which lead to a significant simplification of the functional.

The self-consistent wave functions obtained by solving the generalized HFB equations are not eigenstates of isospin. This is a serious drawback of the quasiparticle approach. To cure this problem, isospin should be restored by means of, e.g., projection techniques. While this can be carried out in a straightforward manner for energy functionals that are related to a two-body potential, the restoration of spontaneously broken symmetries of a general density functional poses a conceptional dilemma [185,186,187,188] and a serious challenge that is left for the future work.