This paper presents a measurement of the electroweak production of two jets in association with a $Z\gamma$ pair, with the $Z$ boson decaying into two neutrinos. It also presents a search for invisible or partially invisible decays of a Higgs boson with a mass of 125 GeV produced through vector-boson fusion with a photon in the final state. These results use data from LHC proton-proton collisions at $\sqrt{s}$ = 13 TeV collected with the ATLAS detector and corresponding to an integrated luminosity of 139 fb$^{-1}$. The event signature, shared by all benchmark processes considered for the measurements and searches, is characterized by a significant amount of unbalanced transverse momentum and a photon in the final state, in addition to a pair of forward jets. Electroweak $Z\gamma$ production in association with two jets is observed in this final state with a significance of 5.2 (5.1 expected) standard deviations. The measured fiducial cross-section for this process is 1.31$\pm$0.29 fb. An observed (expected) upper limit of 0.37 ($0.34^{+0.15}_{-0.10}$) at 95% confidence level is set on the branching ratio of a 125 GeV Higgs boson to invisible particles, assuming the Standard Model production cross-section. The signature is also interpreted in the context of decays of a Higgs boson into a photon and a dark photon. An observed (expected) 95% CL upper limit on the branching ratio for this decay is set at 0.018 ($0.017^{+0.007}_{-0.005}$), assuming the Standard Model production cross-section for a 125 GeV Higgs boson.
Post-fit results for all $m_\text{jj}$ SR and CR bins in the EW $Z \gamma + \text{jets}$ cross-section measurement with the $\mu_{Z \gamma_\text{EW}}$ signal normalization floating. The post-fit uncertainties include statistical, experimental, and theory contributions.
Post-fit results for all DNN SR and CR bins in the search for $H \to \text{inv.}$ with the $\mathcal{B}_\text{inv}$ signal normalization set to zero. For the $Z_\text{Rev.Cen.}^\gamma$ CR, the third bin contains all events with DNN output score values of 0.6-1.0. The $H \to \text{inv.}$ signal is scaled to a $\mathcal{B}_\text{inv}$ of 37%. The post-fit uncertainties include statistical, experimental, and theoretical contributions.
Post-fit results for the ten [$m_\text{jj}$, $m_\text{T}$] bins constituting the SR and CRs defined for the dark photon search with the $\mathcal{B}(H \to \gamma \gamma_\text{d})$ signal normalization set to zero. A $H \to \gamma \gamma_\text{d}$ signal is shown for two different mass hypotheses (125 GeV, 500 GeV) and scaled to a branching ratio of 2% and 1%, respectively. The post-fit uncertainties include statistical, experimental, and theoretical contributions.
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