# 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

\[
V_4(K)=\alpha S^2+\beta D,
\qquad
S=\operatorname{tr}(K^TK),
\qquad
D=S^2-4\det(K^TK),
\]

has one radial mode with mass proportional to \(\alpha\) and two shape modes with masses proportional to \(\beta\). The desired low-energy limit uses

\[
\beta\gg\alpha.
\]

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

\[
H_V=\frac{\partial^2V_4}{\partial K_A\partial K_B}.
\]

The quartic part of the one-loop ultraviolet divergence is proportional, up to the conventional common dimensional-regularization factor, to

\[
\operatorname{Tr}(H_V^2).
\]

Because \(S^2\) and \(D\) span the invariant quartics, the result must decompose back into those two structures. Exact polynomial differentiation gives

\[
\boxed{
\begin{aligned}
\operatorname{Tr}(H_V^2)
={}&
\left(224\alpha^2+128\alpha\beta+128\beta^2\right)S^2\\
&+\left(192\alpha\beta+96\beta^2\right)D.
\end{aligned}
}
\]

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 \(\alpha=0\),

\[
\operatorname{Tr}(H_V^2)
=128\beta^2S^2+96\beta^2D.
\]

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:

\[
\boxed{
\beta\gg\alpha
\text{ is a valid tree-level decoupling limit but is not technically natural in the minimal scalar theory.}
}
\]

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](../../../verification/verify/verify-locking-one-loop-mixing.mjs) 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.

## Companion documents

- [Locking-potential Hessian](81_locking_potential_hessian_and_scalar_decoupling.md)
- [Locking-field spectrum](80_locking_field_gauge_and_scalar_spectrum.md)
- [Locking gauge mixing and hypercharge gate](82_locking_gauge_mixing_and_hypercharge_gate.md)