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 $T_{pc}$ for chiral-symmetry restoration in the presence of a magnetic field $B$. We show that the lattice predictions that $T_{pc}$ 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 $1/{\alpha }_s\sim ln(eB/{\Lambda }_{\text{QCD}}^2)$ for large values of $B$.

The Born–Oppenheimer approximation is the standard tool for the study of molecular systems. It is founded on the observation that the energy scale of the electron dynamics in a molecule is larger than that of the nuclei. A very similar physical picture can be used to describe QCD states containing heavy quarks as well as light-quarks or gluonic excitations. In this work, we derive the Born–Oppenheimer approximation for QED molecular systems in an effective field theory framework by sequentially integrating out degrees of freedom living at energies above the typical energy scale where the dynamics of the heavy degrees of freedom occurs. In particular, we compute the matching coefficients of the effective field theory for the case of the $H_2^{+}$ diatomic molecule that are relevant to compute its spectrum up to $\mathcal{O}(m{\alpha }^5)$. Ultrasoft photon loops contribute at this order, being ultimately responsible for the molecular Lamb shift. In the effective field theory the scaling of all the operators is homogeneous, which facilitates the determination of all the relevant contributions, an observation that may become useful for high-precision calculations. Using the above case as a guidance, we construct under some conditions an effective field theory for QCD states formed by a color-octet heavy quark-antiquark pair bound with a color-octet light-quark pair or excited gluonic state, highlighting the similarities and differences between the QED and QCD systems. Assuming that the multipole expansion is applicable, we construct the heavy-quark potential up to next-to-leading order in the multipole expansion in terms of nonperturbative matching coefficients to be obtained from lattice QCD.