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This paper presents the measurement of fiducial and differential crosssections for both the inclusive and electroweak production of a same-sign$W$-boson pair in association with two jets ($W^\pm W^\pm jj$) using 139fb$^{-1}$ of proton-proton collision data recorded at a centre-of-mass energyof $\sqrt{s}=13$ TeV by the ATLAS detector at the Large Hadron Collider. Theanalysis is performed by selecting two same-charge leptons, electron or muon,and at least two jets with large invariant mass and a large rapiditydifference. The measured fiducial cross sections for electroweak and inclusive$W^\pm W^\pm jj$ production are $2.92 \pm 0.22\, \text{(stat.)} \pm 0.19\,\text{(syst.)}$ fb and $3.38 \pm 0.22\, \text{(stat.)} \pm 0.19\,\text{(syst.)}$ fb, respectively, in agreement with Standard Model predictions.The measurements are used to constrain anomalous quartic gauge couplings byextracting 95% confidence level intervals on dimension-8 operators. A searchfor doubly charged Higgs bosons $H^{\pm\pm}$ that are produced in vector-bosonfusion processes and decay into a same-sign $W$ boson pair is performed. Thelargest deviation from the Standard Model occurs for an $H^{\pm\pm}$ mass near450 GeV, with a global significance of 2.5 standard deviations.

Measurement and interpretation of same-sign $W$ boson pair production in  association with two jets in $pp$ collisions at $\sqrt{s} = 13$ TeV with the  ATLAS detector