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 α_C = 13/120 for the Weyl-squared coefficient and α_R(ξ) = 2(ξ − 1/6)² for the R² coefficient, where ξ is the…