Debye (mass?) in Rindler space

On one hand, a uniformly accelerated observer perceives the Minkowski vacuum as a thermal bath with an Unruh temperature. On the other hand, massless particles are widely known to acquire the effective mass due to interaction with a thermal bath — in other words, mass generation is an important property of thermal states. We try to figure out whether the Unruh state reproduces this property, whether it changes the spectrum of particles, and what physical meaning this effect has.