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We estimate the expected precision at a multi-TeV muon collider for measuringthe Higgs boson couplings with electroweak gauge bosons, $HVV$ and $HHVV\(V=W^\pm,Z)$, as well as the trilinear Higgs self-coupling $HHH$. At very highenergies both single and double Higgs productions rely on the vector-bosonfusion (VBF) topology. The outgoing remnant particles have a strong tendency tostay in the very forward region, leading to the configuration of the "inclusiveprocess" and making it difficult to isolate $ZZ$ fusion events from the $WW$fusion. In the single Higgs channel, we perform a maximum likelihood analysison $HWW$ and $HZZ$ couplings using two categories: the inclusive Higgsproduction and the 1-muon exclusive signal. In the double Higgs channel, weconsider the inclusive production and study the interplay of the trilinear$HHH$ and the quartic $VVHH$ couplings, by utilizing kinematic information inthe invariant mass spectrum. We find that at a centre-of-mass energy of 10 TeV(30 TeV) with an integrated luminosity of 10 ab$^{-1}$ (90 ab$^{-1}$), one mayreach a 95\% confidence level sensitivity of 0.073\% (0.023\%) for $WWH$coupling, 0.61\% (0.21\%) for $ZZH$ coupling, 0.62\% (0.20\%) for $WWHH$coupling, and 5.6\% (2.0\%) for $HHH$ coupling. For dim-6 operatorscontributing to the processes, these sensitivities could probe the new physicsscale $\Lambda$ in the order of $1-10$ ($2-20$) TeV at a 10 TeV (30 TeV) muoncollider.

Electroweak Couplings of the Higgs Boson at a Multi-TeV Muon Collider