  We demonstrate theoretically that the interaction of electrons in gapped
Dirac materials (gapped graphene and transition-metal dichalchogenide
monolayers) with a strong off-resonant electromagnetic field (dressing field)
substantially renormalizes the band gaps and the spin-orbit splitting.
Moreover, the renormalized electronic parameters drastically depend on the
field polarization. Namely, a linearly polarized dressing field always
decreases the band gap (and, particularly, can turn the gap into zero), whereas
a circularly polarized field breaks the equivalence of valleys in different
points of the Brillouin zone and can both increase and decrease corresponding
band gaps. As a consequence, the dressing field can serve as an effective tool
to control spin and valley properties of the materials and be potentially
exploited in optoelectronic applications.
