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The Born--Oppenheimer approximation is the standard tool for the study ofmolecular systems. It is founded on the observation that the energy scale ofthe electron dynamics in a molecule is larger than that of the nuclei. A verysimilar physical picture can be used to describe QCD states containing heavyquarks as well as light-quarks or gluonic excitations. In this work, we derivethe Born--Oppenheimer approximation for QED molecular systems in an effectivefield theory framework by sequentially integrating out degrees of freedomliving at energies above the typical energy scale where the dynamics of theheavy degrees of freedom occurs. In particular, we compute the matchingcoefficients of the effective field theory for the case of the $H^+_2$ diatomicmolecule that are relevant to compute its spectrum up to ${\cal O}(m\alpha^5)$.Ultrasoft photon loops contribute at this order, being ultimately responsiblefor the molecular Lamb shift. In the effective field theory the scaling of allthe operators is homogeneous, which facilitates the determination of all therelevant contributions, an observation that may become useful forhigh-precision calculations. Using the above case as a guidance, we constructunder some conditions an effective field theory for QCD states formed by acolor-octet heavy quark-antiquark pair bound with a color-octet light-quarkpair or excited gluonic state, highlighting the similarities and differencesbetween the QED and QCD systems. Assuming that the multipole expansion isapplicable, we construct the heavy-quark potential up to next-to-leading orderin the multipole expansion in terms of nonperturbative matching coefficients tobe obtained from lattice QCD.

The Born-Oppenheimer approximation in an effective field theory language