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With $e^-$ constituents in a classical potential atoms are at the borderline between hard processes and soft classical physics. Data and phenomenology indicate that hadrons are similar. The importance of this cannot be overemphasized: It suggests a first-principles, analytic QCD approach to strong dynamics analogous to that long developed for QED atoms.
Bound state perturbation theory is sometimes characterized as an ''art''. Atoms may be considered ''non-perturbative'', yet their binding energies can be expanded in powers of $\alpha$ and $\log\alpha$. The essential difference compared to standard perturbation theory is that the ''lowest order'' wave function (given, e.g., by the Schrödinger or Bethe-Salpeter equation) already has all powers of $\alpha$. Hence the ordering of the expansion is not unique, and may be chosen for optimal convergence.
Gauge theories have instantaneous interactions in non-local gauges. In temporal $(A^0=0)$ gauge the longitudinal electric field $\boldsymbol{E}_L$ provides a potential also for relativistic constituents. Gauss' law determines $\boldsymbol{\nabla}\cdot\boldsymbol{E}_L$ for each state from the instantaneous positions of its (color) charged constituents. Poincaré invariance allows including a homogeneous solution of Gauss' law for color singlet states in QCD. This gives rise to an ${\cal O}(\alpha_s^0)$ confining potential that is fully determined for any quark and gluon state, up to a universal scale $\Lambda$.
In a ''Bound Fock expansion'' the constituents of each state propagate in their instantaneous potential. Requiring the valence state ($e^+e^-$ for Positronium, $q\bar q$ for mesons, $qqq$ for baryons) to be an eigenstate of the Hamiltonian $\mathcal{H}$ determines its wave function. The bound state equation reduces to the Schrödinger equation in the non-relativistic limit. For quarkonia one obtains the ''Cornell'' potential with linear confinement. Higher Fock states are determined perturbatively by the transverse gauge boson interactions of $\mathcal{H}$. The QCD coupling $\alpha_s(Q) \simeq 0.5$ is frozen at hadronic scales $Q < \Lambda$, where the classical $\boldsymbol{E}_L^a$ field dominates. Considering all Fock states one obtains a formally exact expression for a hadron in any frame.
Mesons have no transverse gluon constituents at $\ {\cal O}(\alpha_s^0)$, but $\ q\bar q$ pairs arise due to time ordered ''$Z$''-diagrams. These have the properties of sea quarks and contribute to Deep Inelastic Scattering at low $x_{Bj}$. For small quark masses the meson states lie on linear Regge trajectories with parallel daughter trajectories. Features of the parton model and duality appear in the relevant limits. The approach seems promising, although many aspects remain to be investigated [1,2].
[1] Paul Hoyer, ``Journey to the Bound States'', SpringerBriefs in Physics (Springer, 2021) arXiv:2101.06721 [hep-ph].
[2] Paul Hoyer, “Hadrons as QCD Bound States,” in 14th Conference on Quark Confinement and the Hadron Spectrum (2021) arXiv:2109.06257 [hep-ph].