Neutron–proton pairing correlations: Theoretical foundations, open questions and confrontation of BCS and exact solutions

Neutron–proton pairing correlations occupy a central position in nuclear structure physics. While likeparticle pairing between neutrons or protons in the isovector ([Formula: see text]) channel is firmly established and accounts for many systematic features of nuclear spectra, the role of neutron–proton correlations, particularly in the isoscalar ([Formula: see text]) channel, remains a subject of ongoing debate in both experiment and theory. In this review, we examine the theoretical foundations and current understanding of neutron–proton pairing, beginning with the conceptual distinctions among several phenomena that are often conflated in the literature: the residual neutron–proton interaction, isovector pairing correlations and possible isoscalar pairing condensates. We discuss how these different forms of correlation emerge in the shell–model framework and how they are represented in BCS mean-field approaches. Special attention is given to the pedagogical aspects of single-j pairing models, algebraic approaches, as well as the shell-model-like exact pairing diagonalization algorithms, which provide valuable benchmarks for understanding the limitations of mean-field descriptions and the interplay between isovector and isoscalar pairing channels. We also review experimental signatures that have been proposed as evidence for neutron–proton pairing, including spectroscopic patterns in [Formula: see text] nuclei, mass systematics and transfer reactions and discuss the extent to which these observations support or challenge current theoretical interpretations. The review highlights both the progress achieved in clarifying the mechanisms of neutron–proton correlations and the major open questions that remain, particularly regarding the existence and manifestation of collective [Formula: see text] pairing modes in finite nuclei. Finally, a practical computational scheme for the exact diagonalization of neutron–proton pairing is presented.