Precision Higgs Width and Couplings with a High Energy Muon Collider

The interpretation of Higgs data is typically based on different assumptions about whether there can be additional decay modes of the Higgs or if any couplings can be bounded by theoretical arguments. Going beyond these assumptions requires either a precision measurement of the Higgs width or an absolute measurement of a coupling to eliminate a flat direction in precision fits that occurs when $|g_{hVV}/g_{hVV}^{SM}|>1$, where $V=W^\pm, Z$. In this paper we explore how well a high energy muon collider can test Higgs physics without having to make assumptions on the total width of the Higgs. In particular, we investigate off-shell methods for Higgs production used at the LHC and searches for invisible decays of the Higgs to see how powerful they are at a muon collider. We then investigate the theoretical requirements on a model which can exist in such a flat direction. Combining expected Higgs precision with other constraints, the most dangerous flat direction is described by generalized Georgi-Machacek models. We find that by combining direct searches with Higgs precision, a high energy muon collider can robustly test single Higgs precision down to the $\mathcal{O}(.1\%)$ level without having to assume SM Higgs decays. Furthermore, it allows one to bound new contributions to the width at the sub-percent level as well. Finally, we comment on how even in this difficult flat direction for Higgs precision, a muon collider can robustly test or discover new physics in multiple ways. Expanding beyond simple coupling modifiers/EFTs, there is a large region of parameter space that muon colliders can explore for EWSB that is not probed with only standard Higgs precision observables.