The Importance of Asymptotic Freedom for the Pseudocritical Temperature   in Magnetized Quark Matter
  Although asymptotic freedom is an essential feature of QCD, it is absent in effective chiral quark models like the Nambu--Jona-Lasinio and linear sigma models. In this work we advocate that asymptotic freedom plays a key role in the recently observed discrepancies between results of lattice QCD simulations and quark models regarding the behavior of the pseudocritical temperature $T_{\rm pc}$ for chiral symmetry restoration in the presence of a magnetic field $B$. We show that the lattice predictions that $T_{\rm pc}$ decreases with $B$ can be reproduced within the Nambu--Jona-Lasinio model if the coupling constant $G$ of the model decreases with $B$ and the temperature. Without aiming at numerical precision, we support our claim by considering a simple ansatz for $G$ that mimics the asymptotic freedom behavior of the QCD coupling constant $1/\alpha_s \sim \ln (eB/\Lambda^2_{QCD})$ for large values of $B$. 
R.L.S. Farias, K.P. Gomes, G.I. Krein and M.B. Pinto
hep-ph