  Ionization phenomena are widely studied for decades. With the advent of
cavity technology, the question arises how quantum light affects molecular
ionization. As the ionization spectrum is recorded from the neutral ground
state, it is usually possible to choose cavities which exert negligible effect
on the neutral ground state, but have significant impact on the ion and the
ionization spectrum. Particularly interesting are cases where the ion exhibits
conical intersections between close-lying electronic states, which gives rise
to substantial nonadiabatic effects. Assuming single-molecule strong coupling,
we demonstrate that vibrational modes irrelevant in the absence of cavity play
a decisive role when the molecule is in the cavity. Here, dynamical symmetry
breaking is responsible for the ion-cavity coupling and high symmetry enables
control of the coupling via molecular orientation relative to the cavity field
polarization. Significant impact on the spectrum by the cavity is found and
shown to even substantially increase for less symmetric molecules.
