A search for Higgs boson decays into a $Z$ boson and a light resonance in two-lepton plus jet events is performed, using a $pp$ collision dataset with an integrated luminosity of 139 fb$^{-1}$ collected at $\sqrt{s}=13$ TeV by the ATLAS experiment at the CERN LHC. The resonance considered is a light boson with a mass below 4 GeV from a possible extended scalar sector, or a charmonium state. Multivariate discriminants are used for the event selection and for evaluating the mass of the light resonance. No excess of events above the expected background is found. Observed (expected) 95$\% $ confidence-level upper limits are set on the Higgs boson production cross section times branching fraction to a $Z$ boson and the signal resonance, with values in the range 17 pb to 340 pb ($16^{+6}_{-5}$ pb to $320^{+130}_{-90}$ pb) for the different light spin-0 boson mass and branching fraction hypotheses, and with values of 110 pb and 100 pb ($100^{+40}_{-30}$ pb and $100^{+40}_{-30}$ pb) for the $\eta_c$ and $J/\psi$ hypotheses, respectively.
Observed number of data events and expected number of background events in the signal region.
Efficiencies of the MLP selection, complete selection and total expected signal yields for each signal sample, assuming B$(H\to Z(Q/a))=100\%$ and $\sigma(pp\to H) = \sigma_\text{SM}(pp\to H)$. Pythia 8 branching fractions of $a$ are assumed using a $\tan\beta$ value of 1. The MLP efficiencies, total efficiencies, and expected yields are determined using MC samples, with uncertainties due to MC sample statistics, except for the expected background yield. The expected background yield and its uncertainty is calculated as described in the main text of the paper.
Expected and observed 95% CL upper limits on $\sigma(pp\to H)B(H\to Za)/$pb. These results are quoted for $B(a\to gg)=100\%$ and $B(a\to s\bar{s})=100\%$ for each signal sample. The smaller (larger) quoted ranges around the expected limits represent $\pm 1\sigma$ ($\pm 2\sigma$) fluctuations.
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