Domain-First Rigidity of the Higgs Potential: A Fluctuation Principle and the Tree-Level Prediction κ_λ^vac = −1/3

抄録

Given the electroweak vacuum location and Higgs mass, what physical postulates can relate nonlinear Higgs self-interactions left undetermined in a general effective description? We propose a statistical-response construction based on a fixed positive dimensionless state intensity X. Its mean is identified with the Higgs background, while constrained forward Kullback–Leibler (KL) minimization selects the corresponding state distribution and its minimum cost is mapped to the leading-order potential energy. Requiring constant relative variance along the minimizing linear-source family characterizes, under stated regularity conditions, a gamma reference law and selects V_LO(ψ) − V_LO(1) = ¼m_h²v²(ψ − 1 − ln ψ). With standard kinetic and gauge–Yukawa interactions retained, this gives the vacuum coupling ratios (κ_λ^vac, κ₄^vac) = (−1/3, 1) relative to Standard Model tree values and fixes the higher vacuum derivatives of the potential with respect to the canonical radial Higgs field. A correlated Gaussian state model realizes the response with a unique KL minimum over all admissible distributions and shows that the retained observable, not Gaussianity alone, is essential. The cubic ratio remains invariant under a specified class of smooth nonlinear energy maps, while comparison models expose finite-order identification limits. Scalar one-loop nonclosure requires an enlarged effective field theory and finite matching beyond vacuum–curvature alignment. The microscopic realization of the proposed correspondence in the physical quantum vacuum remains open. The resulting action and sensitivity relations provide a concrete starting point for gg → hh rate and invariant-mass calculations.