The fate of $s_{\pm}$ superconductivity in the bilayer nickelate La$_3$Ni$_2$O$_7$

Motivated by the recent measurement of high-temperature superconductivity in the bilayer nickelate system La$_3$Ni$_2$O$_7$ in the bulk as well as thin film samples, we seek to understand the role of the interlayer coupling in this system in dictating the pairing symmetry. Many proposals posit the existence of an $s_\pm$ superconducting phase in close competition with the mroe conventional cuprate-like $d$-wave state in both weak- and strong-coupling theories. However, at least in the latter, the effects of Coulomb repulsion and the double occupancy constraint have been largely neglected. In this manuscript, we propose a mean-field fractionalization treatment that captures the full $SU(2)$ particle-hole symmetry of the t-J model at half-filling, and show how a systematic treatment of this new symmetry leads to an added Coulomb pseudopotential constraint - one that is absent in the usual $U(1)$ treatment. We then apply this new fractionalization scheme to a simplified bilayer t-J model of La$_3$Ni$_2$O$_7$, showing how the Coulomb pseudopotential suppresses the existence of $s_\pm$ symmetry in favor of $d$-wave pairing.