Nonlocal one-loop form factors of the spectral action with Standard Model content

We compute the complete nonlocal one-loop form factors F₁ (□/Λ²) and F₂ (□/Λ², ξ) of the curvature-squared sector of the spectral action S = Tr f (D²/Λ²) for the full Standard Model particle content: 4 real scalars (Higgs), 45/2 Dirac-equivalent fermions (3 generations), and 12 gauge bosons (SU (3) × SU (2) × U (1) ). Using the Barvinsky–Vilkovisky covariant perturbation theory and the Codello–Zanusso diagrammatic heat kernel, we derive closed-form results for each spin sector (0, 1/2, 1) in the C², R² Weyl basis and assemble the Standard Model totals. The local limits yield the parameter-free prediction αC = 13/120 for the Weyl-squared coefficient and αR (ξ) = 2 (ξ − 1/6) ² for the R² coefficient, where ξ is the Higgs non-minimal coupling. Both form factors are shown to be entire functions of □/Λ², guaranteeing ghost-freedom of the one-loop effective action. We derive the c₁/c₂ ratio in the R², R²_μν basis, the scalar graviton decoupling condition at conformal coupling ξ = 1/6, and the UV asymptotic behavior. The form factors yield a modified Newtonian potential with calculable effective masses m₂ = Λ√ (60/13) and m₀ = Λ/√ (6 (ξ − 1/6) ²), connecting the spectral action framework to solar-system phenomenology. All results are verified by independent multi-precision numerical evaluation.