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.
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