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QED bound states (atoms) are “non-perturbative” in the sense that no finite order Feynman diagram has a bound state pole. Wave functions are typically gauge dependent and exponential in $\alpha$. Physical binding energies on the other hand do have a perturbative expansion in $\alpha$ (and log $\alpha$). The hyperfine splitting of Positronium has been evaluated to O($\alpha^7$log$\alpha$) and agrees with data, whose accuracy is of O($10^{-10}$). Thus “non-perturbative” need not imply that a perturbative expansion is irrelevant. This may hold also for QCD hadrons: Their initial wave functions can have non-perturbative features (confinement, chiral symmetry breaking), with corrections expressed in powers of $\alpha_s/\pi \sim 0.16$.
The standard method for atoms is to evaluate Feynman diagram corrections to an assumed wave function (typically the Schrödinger one). In a Hamiltonian approach the expansion is instead formulated in terms of Fock states. In temporal ($A^0 = 0$) gauge higher Fock states are suppressed by powers of $\alpha$. For Positronium the $|e^+e^-\rangle$ Fock state is bound by the instantantaneous longitudinal electric field $E_L$, giving a Schrödinger wave function. The $|e^+e^-\gamma\rangle$ state with a transverse photon is suppressed by $\alpha$ and gives rise to hyperfine splitting, and so on.
In temporal gauge Gauss’ law appears as a constraint on physical states, which serves to completely fix the gauge, i.e., $E_L$. In QCD the Gauss constraint allows solutions with a spatially constant, O($\alpha_s^0$) longitudinal color electric field $E_L^a$ for each Fock component, e.g., for a $|q^C\bar{q}^C\rangle$ meson Fock state. The field vanishes when summed over the color $C$ for (globally) color singlet states, and is thus invisible to an external observer. However, it generates a linear potential for each $|q^C\bar{q}^C\rangle$ component. The outcome is reminiscent of the MIT bag model, which postulates a non-vanishing field energy density for the vacuum. Now there is no fixed bag boundary so Poincar\'e invariance is intact: the quarks move in the confining field rather than in a field-free bubble. The single energy scale $\Lambda$ is fixed by the vacuum field energy density.
The strong O($\alpha_s^0$) potential confines charges even though $\alpha_s$ is small. The potential of any Fock state, e.g., $|q^Aq^Bq^C\rangle$ (baryons),$|q^Aq^Bg^a\rangle$ (mesons with a transverse gluon) and $|g^ag^a\rangle$ (glueballs) are given in terms of $\Lambda$ by the Gauss constraint. The Hamiltonian defines hadrons of any momentum $P$. The potential is determined by the instantaneous positions of the charges and is thus independent of $P$. Details may be found in arXiv:1807.05598v2.