Soft gluon resummation for gluon fusion $ZH$ production

We examine the effects of soft gluons on Higgs boson production in association with a $Z$ boson at the Large Hadron Collider (LHC). Utilizing the universal cusp anomalous dimensions and splitting kernels, we analyze effects of soft gluons on the gluon fusion $ZH$ process, focusing on the total production cross-section as well as the invariant mass distribution at the next-to-leading logarithmic level. Additionally, we estimate the next-to-soft effects on this subprocess to the same level of accuracy. A detailed phenomenological analysis is performed for the $13.6$ TeV LHC. Finally, combining these results with those from other subprocesses, we provide comprehensive predictions for the $ZH$ production cross-section and the invariant mass distribution that will be valuable for comparison with experimental data from the upcoming LHC run as well as the future hadron colliders.