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The Georgi-Machacek (GM) model is a distinctive TeV-scale extension of theStandard Model (SM) due to the introduction of two (one real and one complex)scalar triplets to the Higgs sector. It predicts the existence of doublycharged Higgs bosons $H_5^{\pm\pm}$ and the vacuum expectation value of Higgstriplets $v_\Delta$ can reach a few tens of GeV. In this paper, we perform aparameter scan of the GM model within the H5plane benchmark scenario andinvestigate in detail the single production of a doubly charged Higgs boson inassociation with a SM-like Higgs boson via vector-boson fusion at the $14~ {\rmTeV}$ LHC and $70~ {\rm TeV}$ Super Proton-Proton Collider. Both integratedcross section and differential distributions with respect to some kinematicvariables for $pp \rightarrow W^{\pm}W^{\pm} \rightarrow H_5^{\pm\pm} h^0 + 2\,{\rm jets}$ are provided up to the QCD next-to-leading order. In thesignal-background analysis we employ the {\sc madspin} method to take intoaccount the spin correlation and finite width effects of the intermediate Higgsand $W$ bosons and present some kinematic distributions of final leptons. Thenumerical results show that the SM background can be suppressed remarkably andthe $H_5^{\pm\pm} h^0$ vector-boson fusion signal can be directly detected atfuture high-luminosity, high-energy hadron colliders by imposing a proper cuton the transverse mass $M_{T, \ell_1 \ell_2}$.

$H_5^{\pm\pm}h^0$ production via vector-boson fusion in Georgi-Machacek model at hadron colliders