A recent study investigated transverse vector perturbations in an extended, ghost-free quasidilaton massive gravity model, which lacks a kinetic term for the quasidilaton. The research focused on the presence of minimal matter and aimed to understand the behavior of gravitational vector modes within this theoretical framework. The findings are crucial for assessing the viability of certain solution branches in massive gravity theory.

In vacuum, the researchers confirmed that the kinetic coefficient ($K_V$) of the gravitational vector modes vanishes on the self-accelerating branch ($J=0$). This implies that these modes are infinitely strongly coupled at linear order, posing a perturbative health issue. Upon introducing a canonical scalar field and an Abelian vector field (Maxwell or Proca), it was observed that the value of $K_V$ remained unchanged, as in vacuum. The scalar matter did not generate transverse perturbations, and although its terms appeared in the auxiliary shift constraint, they canceled out when the Friedmann equation was applied.

A Maxwell or Proca field with a vanishing isotropic background also did not mix with the gravitational vectors at quadratic order. This leads to the conclusion that ordinary minimal matter is insufficient to render the vector sector perturbatively healthy on this specific branch (Branch II). Consequently, to obtain healthy gravitational vector modes at the linear level, Branch I of the theory should be considered. The study underscores the importance of selecting the correct solution branch in massive gravity to avoid infinite couplings and ensure theoretical consistency.