A muon collider provides an interesting opportunity to test various aspects of Higgs physics and potential BSM models. For a muon collider, Vector-boson fusion provides the dominant channel for the production of Higgs bosons. We calculate the lowest and higher order Higgs jet distribution as a function of jet invariant mass for the super-renormalizable splitting $h\rightarrow hh$ and compare it to the background QCD jet distribution calculated from the NLL resummed cross section for $e^+$ $e^-$ annihilation. The qualitative difference between the two distributions shows up distinctly at collider center of mass energies greater than $10$ TeV as the peak of the QCD jets is pushed off to higher invariant jet masses, making it easier to observe the super-renormalizable and ultra-collinear $h\rightarrow hh$ jet distribution. This can also prove to be an important channel to test potential BSM models.
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