Nonlinear fate of Branch II

Determine whether Branch II of ghost-free quasidilaton massive gravity is genuinely strongly coupled or instead constitutes a true constraint branch in the full nonlinear theory, beyond the linear perturbative mode count.

Background

On Branch II, defined by J=0, the gravitational vector kinetic coefficient K_V vanishes, while the scalar perturbation analysis produces a locking relation that removes the longitudinal Stückelberg field from the reduced linear equations. These features could indicate either infinite strong coupling of modes whose kinetic terms vanish on the exact FLRW background or the genuine elimination of those modes through nonlinear constraints.

The paper establishes only the linear behavior on an exact FLRW background. The authors explain that fluctuations away from the exact Branch II solution could regenerate kinetic terms through higher-order interactions, but they do not determine whether the apparent missing modes persist nonlinearly as strongly coupled degrees of freedom or are actually absent from the theory.

References

Together with the two missing vector kinetic terms, this is suggestive, but a linear mode count cannot tell us whether Branch~II is strongly coupled or is a true constraint branch. That question remains open.

— Vector Perturbations in Ghost-Free Quasidilaton Massive Gravity  (2608.13529 - Kulchoakrungsun et al., 13 Aug 2026) in Section 5, “Interpretation of the J=0 branch” (label: section:branchII)

A key open question is the extension of (shift symmetric) area metric actions to non-linear order and to non-flat backgrounds.

— Gravitational wave signatures from area metric gravity  (2608.16046 - Dittrich, 17 Aug 2026) in Section 6, Discussion and outlook

Whether this freezing property persists at nonlinear orders requires a higher-order perturbative analysis or a full Hamiltonian analysis, which we leave for future work.

— Effective Field Theory for Freezing Gravity with Minimally Coupled Matter  (2608.20085 - Yao, 20 Aug 2026) in Section 3.3, subsection “Ghost stability conditions”; Section 5, Conclusions