  We study the stability of persistent currents in a coherently coupled
quasi-2D Bose-Einstein condensate confined in a ring trap at T=0. By
numerically solving Gross-Pitaevskii equations and by analyzing the excitation
spectrum obtained from diagonalization of the Bogoliubov-de Gennes matrix, we
describe the mechanisms responsible for the decay of the persistent currents
depending on the values of the interaction coupling constants and the Rabi
frequency. When the unpolarized system decays due to an energetic instability
in the density channel, the spectrum may develop a roton-like minimum, which
gives rise to the finite wavelength excitation necessary for vortex nucleation
at the inner surface. When decay in the unpolarized system is driven by
spin-density excitations, the finite wavelength naturally arises from the
existence of a gap in the excitation spectrum. In the polarized phase of the
coherently coupled condensate, there is an hybridization of the excitation
modes that leads to complex decay dynamics. In particular, close to the phase
transition, a state of broken rotational symmetry is found to be stationary and
stable.
