Local density approximation for proton-neutron pairing correlations: Formalism

In the present study we generalize the self-consistent Hartree-Fock-Bogoliubov (HFB) theory formulated in the coordinate space to the case which incorporates an arbitrary mixing between protons and neutrons in the particle-hole $(p\text{\ensuremath{-}}h)$ and particle-particle ($p\text{\ensuremath{-}}p$ or pairing) channels. We define the HFB density matrices, discuss their spin-isospin structure, and construct the most general energy-density functional that is quadratic in local densities. The consequences of the local gauge invariance are discussed and the particular case of the Skyrme energy-density functional is studied. By varying the total energy with respect to the density matrices the self-consistent one-body HFB Hamiltonian is obtained and the structure of the resulting mean fields is shown. The consequences of the time-reversal symmetry, charge invariance, and proton-neutron symmetry are summarized. The complete list of expressions required to calculate total energy is presented.