Title: Even-parity stability of hairy black holes in $U(1)$ gauge-invariant scalar-vector-tensor theories

Abstract: The $U(1)$ gauge-invariant scalar-vector-tensor theories, which catches five degrees of freedom, are valuable for its implications to inflation problems, generation of primordial magnetic fields, new black hole (BH) and neutron star solutions, etc. In this paper, we derive conditions for the absence of ghosts and Laplacian instabilities of nontrivial BH solutions dressed with scalar hair against both odd- and even-parity perturbations on top of the static and spherically symmetric background in the most general $U(1)$ gauge-invariant scalar-vector-tensor theories with second-order equations of motion. In addition to some general discussions, several typical concrete models are investigated. Specially, we show that the stability against even-parity perturbations is ensured outside the event horizon under certain constraints to these models. This is a crucial step to check the self-consistency of the theories and to shed light on the physically accessible models of such theories for future studies.