Domain-First Rigidity of the Higgs Potential: Scale Covariance, a Positive-Domain Log Barrier, and the Prediction κλ^vac = −1/3

抄録

Vacuum-local polynomial ansätze for the Higgs potential are locally valid, but they do not by themselves fix the status of the gauge-invariant positive boundary ρ ≡ 2H†H → 0+. We instead specify the vacuum sector on the interior ratio domain ψ ≡ ρ/v² > 0, anchored at ψ = 1, by requiring the Hessian metric induced by the excess-to-equilibrium functional to be invariant under the multiplicative action of the domain. Three inputs are declared rather than derived: that the requirement acts on the induced geometry; that the affine structure is invariant as well, confining the coordinate to the family u ∝ ψ^s, within which κλ^vac = (2s−3)/3 sweeps the real line; and that s = 1, on operator-dimension grounds—the declaration on which the measurable prediction rests. The generator is then fixed up to scale to the Kullback–Leibler-type convex excess Δ(ψ) = a(ψ − 1 − ln ψ). One consequence is independent of the third declaration: κ4 = 3κλ² + 2/3 throughout the family—a relation the Standard Model point (1, 1) does not satisfy, though it is not measurable at the High-Luminosity LHC.

With (mh, v) fixed, the selected shape correlates the entire near-vacuum derivative tower and yields the tree-level prediction κλ^vac ≡ λ3/λ3^SM = −1/3, with no coefficient left to adjust, alongside κV = κ2V = κt = κb = 1 at tree level, a pattern that holds to the extent the required completion decouples. The tree-level target is sharp within the declared specification; its physical, template-level realization is not yet a sharp number: the one-loop corrections computed here (the Goldstone contribution excluded) are about a quarter of the tree-level trilinear—two to three times the corresponding Standard Model fraction, since the trilinear is three times smaller—and in the minimal single-operator stabilization estimate the completion indicated below a few TeV shifts it by a further six to thirteen per cent. These effects are distinct in kind rather than a single statistical band, and already place the template-level realization at the tens-of-per-cent level of theoretical sensitivity; generic completions may shift it further. The boundary barrier is a tree-level feature, and cosmology is outside the scope of the proposal.

The prediction sits at the edge of current sensitivity: in the August 2026 ATLAS single-plus-double-Higgs fit, whose assumptions the hypothesis satisfies at tree level, −1/3 lies just outside the observed 68% CL interval and well inside the 95% CL interval [−1.5, 6.5], while the implied σ(gg→hh) ≈ 2.6–2.9 times the Standard Model rate is at the level of the Standard-Model-like signal-strength limit. The hypothesis is under active experimental test but not excluded, and a protocol-complete comparison further awaits the loop-order-matched map κλ^vac → κλ^tmpl → κλ^fit, the first open calculation of this programme. The primary falsification handle is the triangle–box interference pattern in gg → hh, testable already with the Run 2 and Run 3 combinations.