  We present a strategy to study QCD non-perturbatively on the lattice at very
high temperatures. This strategy exploits a non-perturbative, finite-volume,
definition of the strong coupling constant to renormalize the theory. As a
first application we compute the flavor non-singlet mesonic screening masses in
a wide range of temperature, from $T\sim 1 $ GeV up to $\sim 160 $ GeV with
three flavors in the chiral limit of QCD. Our results show very interesting
features of the screening spectrum at very high temperatures. On one hand the
mass splitting between the vector and the pseudoscalar screening masses is
clearly visible up to the electroweak scale and cannot be explained by the
known 1-loop perturbative result. On the other hand the restoration of chiral
symmetry manifests itself through the degeneracy of the pseudoscalar and the
scalar channels and of the vector and the axial-vector ones. This degeneracy
pattern is the one expected by Ward identities associated to the presence of
chiral symmetry.
