The clockwork mechanism is a means of naturally generating exponential hierarchies in theories without significant hierarchies among fundamental parameters. We emphasize the role of interactions in the clockwork mechanism, demonstrating that clockwork is an intrinsically abelian phenomenon precluded in non-abelian theories such as Yang-Mills, non-linear sigma models, and gravity. We also show that clockwork is not realized in extra-dimensional theories through purely geometric effects, but may be generated by appropriate localization of zero modes.

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.

Dark matter (DM) without a matter asymmetry is studied in the context of Twin Higgs (TH) theories in which the LHC naturalness problem is addressed. These possess a twin sector related to the Standard Model (SM) by a (broken) Z2 symmetry, and interacting with the SM via a specific Higgs portal. We focus on the minimal realisation of the TH mechanism, the Fraternal Twin Higgs, with only a single generation of twin quarks and leptons, and SU (3) ×SU (2) gauge group. We show that a variety of natural twin-WIMP DM candidates are present (directly linked to the weak scale by naturalness), the simplest and most attractive being the τ lepton with a mass m τ > mHiggs/2, although spin-1 W ± DM and multicomponent DM are also possible (twin baryons are strongly disfavoured by tuning). We consider in detail the dynamics of the possibly (meta)stable glueballs in the twin sector, the nature of the twin QCD phase transition, and possible new contributions to the number of relativistic degrees of freedom ∆N eff . Direct detection signals are below current bounds but accessible in near future experiments. Indirect detection phenomenology is rich and requires detailed studies of twin hadronization and fragmentation to twin glueballs and quarkonia and their subsequent decay to SM, and possible light twin sector states.

We study Asymmetric Dark Matter (ADM) in the context of the minimal (Fraternal) Twin Higgs solution to the little hierarchy problem, with a twin sector with gauged SU (3) × SU (2) , a twin Higgs, and only third generation twin fermions. Naturalness requires the QCD scale Λ QCD 0.5 − 20 GeV, and t to be heavy. We focus on the light b quark regime, m b Λ QCD , where QCD is characterised by a single scale Λ QCD with no light pions. A twin baryon number asymmetry leads to a successful DM candidate: the spin-3/2 twin baryon, ∆ ∼ b b b , with a dynamically determined mass (∼ 5Λ QCD ) in the preferred range for the DM-to-baryon ratio ΩDM/Ω baryon 5. Gauging the U (1) group leads to twin atoms (∆ -τ bound states) that are successful ADM candidates in significant regions of parameter space, sometimes with observable changes to DM halo properties. Direct detection signatures satisfy current bounds, at times modified by dark form factors.