No-lose theorem for discovering the new physics of $(g-2)_μ$ at muon colliders

We perform a model-exhaustive analysis of all possible beyond Standard Model (BSM) solutions to the $(g-2)_μ$ anomaly to study production of the associated new states at future muon colliders, and we formulate a no-lose theorem for the discovery of new physics if the anomaly is confirmed and weakly coupled solutions below the GeV scale are excluded. Our goal is to find the highest possible mass scale of new physics subject only to perturbative unitarity, and optionally the requirements of minimum flavor violation and/or naturalness. We prove that a 3 TeV muon collider is guaranteed to discover all BSM scenarios in which Δaμ is generated by SM singlets with masses above $∼GeV$; lighter singlets will be discovered by upcoming low-energy experiments. If new states with electroweak quantum numbers contribute to $(g-2)_μ$, then the minimal requirements of perturbative unitarity guarantee new charged states below $O(100 TeV)$, but this is strongly disfavoured by stringent constraints on charged lepton flavor violating (CLFV) decays. Reasonable BSM theories that satisfy CLFV bounds by obeying minimal flavor violation and avoid generating two new hierarchy problems require the existence of at least one new charged state below $∼10 TeV$. This strongly motivates the construction of high-energy muon colliders, which are guaranteed to discover new physics: either by producing these new charged states directly, or by setting a strong lower bound on their mass, which would empirically prove that the Universe is fine-tuned and violates the assumptions of minimal flavor violation while somehow not generating large CLFVs. The former case is obviously the desired outcome, but the latter scenario would perhaps teach us even more about the Universe by profoundly revising our understanding of naturalness, cosmological vacuum selection, and the SM flavor puzzle.