Radiative Stability of the Locking Hierarchy
Date: 17 October 2026 Branches: B01 and B05 Claim status: exact one-loop scalar counterterm mixing for the minimal quartic field theory; gauge, fermion and ultraviolet contributions remain open
1. Question
The tree-level locking potential
has one radial mode with mass proportional to \(\alpha\) and two shape modes with masses proportional to \(\beta\). The desired low-energy limit uses
The immediate naturalness question is whether setting \(\alpha=0\) is preserved by loops involving \(\beta\).
2. One-loop scalar divergence
For six canonically normalized real scalar components, the field-dependent scalar mass matrix is the Hessian
The quartic part of the one-loop ultraviolet divergence is proportional, up to the conventional common dimensional-regularization factor, to
Because \(S^2\) and \(D\) span the invariant quartics, the result must decompose back into those two structures. Exact polynomial differentiation gives
The integer coefficients refer to the normalization of \(V_4\) written above; rescaling the conventional quartic couplings rescales the beta-function coefficients but cannot remove the mixing.
3. Consequence
At \(\alpha=0\),
Therefore the shape interaction generates the radial invariant. The surface \(\alpha=0\) is not closed under scalar one-loop renormalization.
This is the precise obstruction:
The conclusion does not require assuming a particular subtraction scheme. A scheme changes finite terms and coupling coordinates; it does not erase the need for the symmetry-allowed \(S^2\) counterterm generated by \(\beta^2\).
4. Scope
This calculation does not prove that a hierarchy is impossible. It proves that the minimal field content and compact symmetry do not protect it. Viable escapes include:
- supersymmetry or another cancellation mechanism;
- compositeness that changes the ultraviolet degrees of freedom;
- a constrained nonlinear realization in which the radial coordinate is emergent;
- a fixed point or RG trajectory that predicts a useful ratio;
- accepting finite tuning over a restricted effective-field-theory range.
Gauge, Yukawa and portal interactions add further counterterms. They cannot be credited with cancelling the scalar contribution without an explicit symmetry or calculation.
5. Branch decision
The tree-level one-light-scalar branch remains open as an effective description. The stronger claim that the same minimal locking theory naturally explains the Higgs hierarchy is pinned.
Stop-everything positive: a source-derived symmetry or ultraviolet completion cancels the \(\beta^2S^2\) generation while preserving the isotropic vacuum and correct matter couplings.
Stop-everything negative: every completion preserving the required locking representation regenerates a radial mass at the heavy shape scale or violates perturbative unitarity before decoupling.
Unsafe shortcut: cite \(\beta\gg\alpha\) as an explanation merely because it is a consistent tree-level parameter choice.
6. Certificate
The one-loop mixing certificate constructs the two invariant quartics as exact six-variable polynomials, differentiates their full Hessians, decomposes all \(\alpha^2\), \(\alpha\beta\) and \(\beta^2\) contractions, and verifies zero residual on the entire polynomial.