N$^3$LO soft-gluon corrections for $Zγ$ production

We calculate soft-gluon corrections for $Z\gamma$ production through next-to-next-to-next-to-leading-order (N$^3$LO) in one-particle-inclusive (1PI) kinematics. We use these results to produce approximate N$^3$LO (aN$^3$LO) QCD predictions for $Z\gamma$ production cross sections in proton-proton collisions at LHC energies. Differential cross sections in photon transverse momentum $p_T^\gamma$ and rapidity $y^\gamma$ are computed at 13 TeV and 13.6 TeV collider energies. Higher-order $K$-factors are examined systematically, and scale and PDF uncertainties are quantified at each perturbative order. Soft-gluon corrections are found to be significant in the high-$p_T$ region, with good perturbative convergence observed at aN$^3$LO.