Introduction

One of the main goals of nuclear theory is to build the unified microscopic framework for heavy nuclei in which the bulk nuclear properties, nuclear excitations, and nuclear reactions can be described on the same footing. Microscopic theory also provides the solid foundation for phenomenological models and coupling schemes which have been applied so successfully to explain specific nuclear properties. Exotic short-lived nuclei are very important in this quest. The abnormal neutron-to-proton ratios of these nuclei isolate and amplify important features, which are not clearly visible in stable systems.

For medium-mass and heavy nuclei, a critical challenge is the quest for the universal energy density functional, which will be able to describe properties of finite nuclei as well as extended asymmetric nucleonic matter (e.g., as found in neutron stars). Self-consistent methods based on the density functional theory have already achieved a level of sophistication and precision which allows analyses of experimental data for a wide range of properties and for arbitrarily heavy nuclei. For instance, self-consistent HF and HFB models are now able to reproduce measured nuclear binding energies with an impressive rms error of $\sim$700 keV [1,2]. However, much work remains to be done. Developing a universal nuclear density functional will require a better understanding of the density dependence, isospin effects, pairing, as well as an improved treatment of many-body correlations. All those aspects are essential for the structure of proton-rich nuclei with $N$$\approx$$Z$, which are expected to exhibit proton-neutron (pn) pairing [3]; it is precisely in those nuclei that the state-of-the-art microscopic mass formula needs to be supplemented by a phenomenological Wigner term [1,2].

In spite of an impressive experimental progress in the heavy $N$$\approx$$Z$ region, it is still unclear (i) what the specific fingerprints of the pn pairing are and (ii) what is the interplay between the like-particle and pn ($T$=0,1) p-h and p-p channels. Before attempting to answer these questions, established theoretical models of nuclear pairing need to be generalized to properly account for pn correlations. The present work is a step in this direction. We propose the general HFB formalism which fully incorporates the pn mixing on the mean-field level. The resulting density matrices have a very rich spin-isospin structure, which, in the presence of static pn pairing, can produce novel mean-fields and deformations.

The paper is organized as follows. Section 2 contains a brief review of the pn pairing. Section 3 discusses the density matrices (scalar, vector, and tensor), both in the p-h and p-p channel. The discussion is based on the coordinate-space HFB formalism [4,5,6], which was introduced earlier to describe pairing correlations between like nucleons. This method is the tool of choice when dealing with weakly-bound heavy nuclei [7]. The energy functional is constructed in Sec. 4, the associated mean fields are derived in Sec. 5, and Sec. 6 deals with the resulting coordinate-space HFB equations. In the discussion of pn pairing, the notion of self-consistent symmetries, especially those associated with charge invariance and time reversal, is crucial, and we devote Sec. 7 to this topic. Finally, conclusions are contained in Sec. 8.