We lay out a comprehensive physics case for a future high-energy muon collider, exploring a range of collision energies (from 1 to 100 TeV) and luminosities. We highlight the advantages of such a collider over proposed alternatives. We show how one can leverage both the point-like nature of the muons themselves as well as the cloud of electroweak radiation that surrounds the beam to blur the dichotomy between energy and precision in the search for new physics. The physics case is buttressed by a range of studies with applications to electroweak symmetry breaking, dark matter, and the naturalness of the weak scale. Furthermore, we make sharp connections with complementary experiments that are probing new physics effects using electric dipole moments, flavor violation, and gravitational waves. An extensive appendix provides cross section predictions as a function of the center-of-mass energy for many canonical simplified models.

The Higgs self-interactions play a crucial role in exploring the underlying mechanisms of electroweak symmetry breaking and the nature of the phase transition involved. In this article, we propose to probe the quartic Higgs self-interaction at lepton and hadron colliders, via the di-Higgs productions. We analyze the contributions of the quartic Higgs coupling, including the renormalization of the cubic Higgs coupling and the modification of the VVhh form factor, to the vector boson-fusion and the vector boson-associated di-Higgs productions at one-loop level. Such an effect is independent of the choice of gauge fixing, if the quartic Higgs coupling is decoupled from other couplings in the contexts considered. Notably, a combination of these two di-Higgs productions is important for optimizing the collider sensitivities to probe the quartic Higgs coupling. With this guideline, we explore the International Linear Collider and Compact Linear Collider sensitivities, and find that the International Linear Collider has the potential to measure the quartic Higgs coupling, normalized by its standard model value, with a precision of ∼ AE25 (500 GeV, 4 ab −1 þ 1 TeV, 2.5 ab −1) and ∼ AE20 (500 GeV, 4 ab −1 þ 1 TeV, 8 ab −1), at 1σ C:L:, after marginalizing the cubic Higgs coupling in the χ 2 analysis. The dependence on the renormalization scheme of the cubic Higgs coupling is discussed.

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