Determine the physical higher-dimensional matching for the restricted loop mechanism

Derive the complete mixed visible–dark on-shell operator basis and matching coefficients for the restricted renormalizable light–heavy loop mechanism, beginning at dimension nine after the vanishing of the χF³ structure and the redundancy of the apparent dimension-seven derivative–Higgs operator.

Background

The selected Class-B light kernel has an exact one-χ zero, but the previously considered estimate based on (H†H)χF³ is invalid because the χF³ operator vanishes identically. Multiplying the EOM-redundant dimension-seven representative by visible scalars also does not provide a valid replacement, because integration by parts generates additional terms involving derivatives of the visible spurions.

The authors therefore leave the physical mixed-sector matching unresolved. This calculation is distinct from the generic dimension-five parent connectors, which remain allowed and can dominate unless an additional ultraviolet selection rule removes them.

References

We therefore keep the restricted renormalizable-loop contribution explicitly open at dimension nine and above rather than attach a lifetime to an unreduced representative.

— Dark matter glueball candidate from a $G(2)$--$E_6$--$E_7$ exceptional grand unified theory: $G$-parity, spin, mass and stability  (2609.04296 - Masi, 3 Sep 2026) in Section 2, subsection “Restricted renormalizable loops: what the light-kernel zero actually proves”; subsection “Accidental dark G_D: what survives at the selected EFT level”