Precision Higgs Physics and the Reconstruction of the Higgs Potential

The discovery of a Higgs boson established the existence of a scalar state associated withelectroweak symmetry breaking, but did not by itself constitute a reconstruction of theHiggs potential. More than a decade of LHC measurements has transformed the observed125 GeV state into a precision object: its pole mass is known at the sub-per-mille level,while representative global signal-strength determinations have reached uncertainties oforder five percent. The scalar self-interactions remain far less constrained. Higgs-pair production provides leading-amplitude access to the cubic interaction, and the joint ATLASCMS Run-2 interpretation yields −0.71 < κλ < 6.1 at 95% CL, although Standard Modelnonresonant HH production has not yet been observed. Triple-Higgs production extendssuch access to the quartic interaction, for which present constraints remain extremelybroad. We review the experimental and theoretical status of Higgs-potential reconstruction within an inference-centered framework that distinguishes experimental precision,leading-amplitude access to a specified interaction, and the assumptions required forinterpretation. We trace the resulting inference chain from collider observables to localscalar interactions, effective field theories, ultraviolet dynamics, and finite-temperaturephysics, identifying where the inverse map becomes nonunique or framework dependent.Current data tightly anchor the physical mass scale associated with the local quadraticstructure of the observed Higgs state, provide direct-at-leading-amplitude but still weakconstraints on the cubic self-interaction, and have initiated direct-at-leading-amplitudeexperimental exploration of the quartic interaction. A reconstruction of the global scalarpotential, its ultraviolet origin, or the electroweak thermal history remains substantiallymore assumption dependent.