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This Letter reports on a search for off-shell production of the Higgs bosonusing 139 $\textrm{fb}^{-1}$ of $pp$ collision data at $\sqrt{s}=$ 13 TeVcollected by the ATLAS detector at the Large Hadron Collider. The signature isa pair of $Z$ bosons, with contributions from both the production andsubsequent decay of a virtual Higgs boson and the interference of that processwith other processes. The two observable final states are $ZZ\rightarrow 4\ell$and $ZZ\rightarrow 2\ell2\nu$ with $\ell = e$ or $\mu$. In the $ZZ\rightarrow4\ell$ final state, a dense Neural Network is used to enhance analysissensitivity with respect to matrix element-based discrimination. Thebackground-only hypothesis is rejected with an observed (expected) significanceof 3.3 (2.2) standard deviations, representing experimental evidence foroff-shell Higgs boson production. Assuming that no new particles enter theproduction of the virtual Higgs boson, its total width can be deduced from themeasurement of its off-shell production cross-section. The measured total widthof the Higgs boson is $4.5^{+3.3}_{-2.5}$ MeV, and the observed (expected)upper limit on the total width is found to be 10.5 (10.9) MeV at 95% confidencelevel.

Evidence of off-shell Higgs boson production from $ZZ$ leptonic decay channels and constraints on its total width with the ATLAS detector