A search for non-resonant Higgs boson pair ($HH$) production is presented, in which one of the Higgs bosons decays to a b-quark pair ($b\bar b$) and the other decays to $WW^*$, $ZZ^*$, or $\tau^+\tau^-$, with in each case a final state with $\ell^+\ell^- +$ neutrinos ($\ell = e, \mu$). The analysis targets separately the gluon-gluon fusion and vector boson fusion production modes. Data recorded by the ATLAS detector in proton-proton collisions at a centre-of-mass energy of 13 TeV at the Large Hadron Collider, corresponding to an integrated luminosity of $140\mathrm{fb}^{-1}$, are used in this analysis. Events are selected to have exactly two $b$-tagged jets and two leptons with opposite electric charge and missing transverse momentum in the final state. These events are classified using multivariate analysis algorithms to separate the $HH$ events from other Standard Model processes. No evidence of the signal is found. The observed (expected) upper limit on the cross-section for non-resonant Higgs boson pair production is determined to be 9.7 (16.2) times the Standard Model prediction at 95% confidence level. The Higgs boson self-interaction coupling parameter $\kappa_\lambda$ and the quadrilinear coupling parameter $\kappa_{2V}$ are each separately constrained by this analysis to be within the ranges ${[-6.2, 13.3]}$ and ${[-0.17, 2.4]}$, respectively, at 95% confidence level, when all other parameters are fixed.
Pre-fit yields of the $t\bar{t}$, $Z$+HF and $Wt$ CRs, both for the ggF and VBF event selection, as well as the highest-score bins, numbered from high (VBF-SR 1 and ggF-SR 1) to low score (VBF-SR 5 and ggF-SR 7), of the BDT and DNN output distribution in the VBF and ggF event categories, respectively, as used in the final result. The shaded bands include both statistical and systematic uncertainties.
Post-fit yields from the signal+background fit of the $t\bar{t}$, $Z$+HF and $Wt$ CRs, both for the ggF and VBF event selections, as well as the highest-score bins, numbered from high (VBF-SR 1 and ggF-SR 1) to low score (VBF-SR 5 and ggF-SR 7), of the BDT and DNN output distribution in the VBF and ggF event categories respectively as used in the final result. The fit is a conditional fit with the signal strength fixed to the observed upper limit of $\mu_{HH} = 9.7$. The shaded bands include both statistical and systematic uncertainties.
Observed and expected upper limits on the ratios of the Higgs boson pair production cross-section to the corresponding Standard Model prediction $\sigma_{HH}/\sigma^\mathrm{SM}_{HH}$ for the ggF $HH$ signal only (top row), the VBF $HH$ signal only while considering ggF $HH$ as background (second row) and the combined ggF+VBF $HH$ signal considering only the ggF SR (third row) and considering all SRs (bottom row) at a 95% confidence level. The relative ratio between the ggF and VBF production modes is fixed to the SM value.
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