Higgs Physics at a $\sqrt{s} = 10$ TeV Muon Collider

This contribution discusses the physics potential of a future muon collider operating at a center-of-mass energy of $\sqrt{s} = 10$ TeV for precision studies in the Higgs sector. Using a detailed detector simulation that incorporates the dominant sources of machine-induced background, the expected sensitivity to key Higgs processes is evaluated. These include the measurement of production cross sections for $H\to b\bar{b}$, $H\to WW^*$, and double-Higgs production $H\!H\to b\bar{b}b\bar{b}$. A central focus of the study is the determination of the Higgs boson trilinear self-coupling, a critical parameter for understanding the structure of the Higgs potential and electroweak symmetry breaking. The analysis is based on the MUSIC multi-purpose detector concept, specifically optimized for the muon collider environment, and assumes an integrated luminosity of $10$ ab$^{-1}$ collected over five years. The results presented highlight the exceptional prospects of a multi-TeV muon collider for exploring the Higgs potential with a level of precision unattainable by any other proposed future collider within a comparable timeframe.