Hierarchy Problem

Higgs Protection via K3 Geometry

EEq Regularization Framework
Analyzing how spectral regularization on Ricci-flat K3 surfaces eliminates quadratic divergences in the Higgs mass while preserving threshold sensitivity. A mathematically rigorous approach that organizes the naturalness problem.
Honest scope. This framework does not solve naturalness by itself. Heavy thresholds with mass M and coupling αº still induce finite corrections α´α¼Â² ~ (αº/16π²)M². EEq removes regulator artifacts but preserves genuine threshold sensitivity.
[EEq] Spectral heat-kernel/zeta regularization on compact Ricci-flat K3 (χ=24) eliminates α›Â² terms while making topological contributions explicit.

The Hierarchy Problem & EEq Organization

Core Problem & Mathematical Framework

The Standard Model Higgs mass receives dangerous quadratic corrections α´m_h² ∝ α›Â² in cutoff regularization. Our approach: compute one-loop corrections using spectral regularization on a compact Ricci-flat K3 background, eliminating power divergences while preserving physical threshold effects.

Standard Hierarchy Problem

In conventional cutoff regularization, the Higgs mass parameter receives corrections:

$$\delta m_h^2 = \frac{\Lambda^2}{16\pi^2}\Big[6\lambda + \frac{9}{4}g^2 + \frac{3}{4}g'^2 - 6y_t^2\Big] + \text{logs} + \cdots$$

The Veltman combination in brackets is non-zero with measured couplings, creating severe UV sensitivity for α› â‰« v.

EEq Framework Advantages

What EEq@K3 provides: Geometric, covariant organization of loop physics with explicit topological bookkeeping via χ = 24

No α›Â² divergences
Manifest covariance
Threshold transparency

EEq Background Selection

Resistance Functional Extremization
$$\mathcal{R}[g,\phi] = \int_M \big(\|\text{Ric}(g)\|^2 + \|\nabla\phi\|^2 + V(\phi)\big) \text{dvol}_g + \mathcal{T}(M)$$

Over compact, KΩ¤hler, c₁(M)=0, simply-connected 4-manifolds

Selected Background
$$\text{Ric}(g_\star) = 0, \quad \nabla\phi = 0, \quad V'(\phi_0) = 0$$

In 4D this selects K3 surfaces, yielding Ricci-flat backgrounds

Consequence for Higgs Sector

Since R = 0 on the chosen K3 background, any non-minimal coupling α¾R|H|² vanishes, leaving the standard minimal Higgs sector structure for loop calculations.

Heat Kernel Regularization on K3

One-Loop Effective Action
$$\Gamma^{(1)} = \pm\frac{1}{2}\text{Tr}\ln(-\nabla^2 + \mathcal{M}^2) = \mp\frac{1}{2}\int_0^{\infty} \frac{ds}{s} \text{Tr} e^{-s(-\nabla^2 + \mathcal{M}^2)}$$

Zeta function regularization avoids explicit cutoff parameters

Heat Kernel Expansion
$$\text{Tr} e^{-s\Delta} \sim \frac{1}{(4\pi s)^2}(a_0 + a_1 s + a_2 s^2 + \cdots)$$

Seeley-DeWitt coefficients determine divergence structure

Ricci-Flat Simplification
$$R = 0, \quad R_{\mu\nu} = 0 \quad \Rightarrow \quad a_1 = 0$$

Linear divergences automatically vanish

Topological Contribution

$$\int_{\text{K3}} (R_{\mu\nu\rho\sigma}^2 - 4R_{\mu\nu}^2 + R^2) \text{dvol} = 32\pi^2 \chi(\text{K3}) = 32\pi^2 \times 24$$

The Euler characteristic χ = 24 appears only in gravitational log sectors via the Gauss-Bonnet density, not directly in Higgs mass corrections.

One-Loop Effective Potential

Standard Coleman-Weinberg Structure

On the Ricci-flat K3 background, the effective potential takes the standard form:

$$V_1(h) = \sum_i \frac{n_i}{64\pi^2} m_i^4(h)\left(\ln\frac{m_i^2(h)}{\mu^2} - c_i\right)$$

with multiplicities n_W=6, n_Z=3, n_t=-12, n_h=1, n_G=3 and standard constants c_gauge=5/6, c_scalar,fermion=3/2.

Logarithmic Mass Correction
$$\delta m_h^2\Big|_{\log} = \frac{1}{16\pi^2}\left[6\lambda m_h^2 + \frac{9}{4}g^2 m_W^2 + \frac{3}{4}g'^2 m_Z^2 - 6y_t^2 m_t^2\right]\ln\frac{\mu^2}{\mu_0^2}$$

This is the usual flat-space structure, valid because R = 0 on the K3 background.

RG Equation for Higgs Mass Parameter
$$\mu \frac{d\mu^2}{d\mu} = \frac{1}{16\pi^2}\left[6\lambda \mu^2 + \frac{9}{4}g^2 m_W^2 + \frac{3}{4}g'^2 m_Z^2 - 6y_t^2 m_t^2\right] + O(\text{2-loop})$$

Standard one-loop RG evolution; EEq/K3 removes regulator α›Â² artifacts but preserves physical running.

Gauge Dependence Notes

Heavy Threshold Reality Check

Threshold Sensitivity Persists

For any heavy state of mass M coupling to H with strength αº:

$$\boxed{\delta\mu^2 \sim \frac{\kappa}{16\pi^2}M^2}$$

Quantitative Examples

GUT-scale state (M = 10¹⁶ GeV): To keep α´m_h ≲ 100 GeV requires αº â‰² 2.5Ω—10⁻²⁶ - extremely small.

TeV-scale state (M = 3 TeV, αº = 0.1): Gives α´m_h ~ 75 GeV - already significant naturalness pressure.

Fundamental Limitation

Naturalness pressure is dominated by real thresholds, not regulator choice. EEq removes α›Â² artifacts but cannot suppress (αº/16π²)M² without additional physics beyond geometric organization.

Euclidean→’Lorentzian Mapping

Operational Plan

1
Local Operator Content Compute counterterms and finite parts of α“_eff^E on K3
2
Analytical Continuation Continue couplings/fields to Lorentz signature using Schwinger-Keldysh formalism
3
FRW Matching Expand around homogeneous/isotropic metrics for corrected Friedmann equations
4
Modest Claims Gauss-Bonnet is topological in 4D; expect tiny loop-level anomaly effects

Background Independence

Current calculations are one-loop around a fixed Ricci-flat saddle. Full background independence would require integration over metrics/moduli, with graviton fluctuations entering at higher orders suppressed by M_Pl.

Observable Handles & Testable Predictions

✓“ Curvature-Squared Coefficients

Topological weights in gravitational sector from χ = 24

✓“ Absence of α›Â² Artifacts

No scheme-independent quadratic divergences

✓“ Threshold Transparency

Explicit exposure of heavy state couplings

Potential Probes

Note: All extremely challenging and model-dependent

EEq vs Dimensional Regularization

Aspect Dimensional Regularization EEq on K3 Advantage
α›Â² Terms Absent Absent Equal
Covariance Manifest Geometric/manifest EEq more natural
Topological Input Hidden in counterterms Explicit χ = 24 EEq transparent
Framework Consistency Independent choice Aligned with meta-law EEq coherent
Naturalness Threshold problem remains Threshold problem remains No advantage

Assessment & Deliverables

Minimal, Reproducible Claims

Mathematical Framework
Divergence Organization
Naturalness Solution

Scientific Status

Speculative theoretical framework; pre-peer review. Mathematically consistent within EEq and standard QFT. Does not solve naturalness but provides geometric organization separating regulator artifacts from genuine threshold sensitivity.