Search in diphoton and dielectron final states for displaced production of Higgs or Z bosons with the ATLAS detector in $\sqrt{s} = 13$ TeV $pp$ collisions

A search is presented for displaced production of Higgs bosons or $Z^0$ bosons, originating from the decay of a neutral long-lived particle (LLP) and reconstructed in the decay modes $H\rightarrow \gamma\gamma$ and $Z\rightarrow ee$. The analysis uses the full Run 2 data set of proton--proton collisions delivered by the LHC at an energy of $\sqrt{s}=13$ TeV between 2015 and 2018 and recorded by the ATLAS detector, corresponding to an integrated luminosity of 139 fb$^{-1}$. Exploiting the capabilities of the ATLAS liquid argon (LAr) calorimeter to precisely measure the arrival times and trajectories of electromagnetic objects, the analysis searches for the signature of pairs of photons or electrons which arise from a common displaced vertex and which arrive after some delay at the calorimeter. The results are interpreted in a gauge-mediated supersymmetry breaking model where the LLPs are pair-produced and each LLP subsequently decays into either a Higgs boson or a $Z^0$ boson. The final state includes a particle that escapes direct detection, giving rise to missing transverse momentum. No significant excess is observed above the background expectation. The results are used to set upper limits on the cross section for LLP pair-production, for signals with an LLP mass between 100 and 705 GeV and lifetime between 0.25 and 1000 ns. A model-independent limit is also set on the production of pairs of photons or electrons with a significant delay of arrival at the calorimeter.