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Hadrons are strongly bound, yet their spectra can be classified as for atoms (and molecules). The apparent dominance of the valence ($q\bar q$, $qqq$) components of QCD bound states is paradoxical: The strong gluon field would be expected to generate an abundance of quark and gluon constituents.
The atomic features of hadrons motivates a closer look at the bound state methods of QED. They are based on a perturbative expansion quite unlike that of the perturbative S-matrix. Bound state perturbation theory starts from an approximate bound state (given, e.g., by the Schrödinger equation), whose wave function is non-polynomial in the coupling $\alpha$. The series may be reordered by shifting powers of $\alpha$ between the initial wave function and its perturbative corrections. The expansion in $\alpha$ (and $\log\alpha$) of physical quantities such as binding energies is nevertheless unique.
QCD expansions starting with freely propagating quarks and gluons (Feynman diagrams) neglect confinement in the $in$- and $out$- states. Bound state constituents interact at all times. Valence quark dominance requires instantaneous interactions, which arise when the gauge is fixed over all space at an instant of time. In temporal gauge ($A^0(t,\vec x)=0$) Gauss' law is implemented as a constraint on physical states. The instantaneous positions of the charges determine the longitudinal electric field $E_L$, which gives a classical instantaneous potential.
The QCD lagrangian does not determine the hadronic scale $\Lambda_{QCD} \simeq 1$ fm$^{-1}$. This scale can be introduced via the temporal gauge fixing mentioned above. The requirement of translation and rotation symmetry determines $E_L$ uniquely, up to a universal constant $\Lambda$ of O($\alpha_s^0$). Only (globally) color singlet states are allowed when $\Lambda \neq 0$. For $q\bar q$ states $E_L$ gives a linear O($\alpha_s^0$) potential $V(r) = \Lambda^2 r$. The confining potential for other states ($qqq,\ q\bar qg,\ gg$) is likewise uniquely determined.
The O($\alpha_s^0$) bound states define the lowest order of a formally exact bound state expansion. Higher Fock states with transversally polarized (propagating) gluons and $q\bar q$ pairs are generated as power corrections in $\alpha_s$ by the interaction terms of $H_{QCD}$. As in any perturbative expansion each order of $\alpha_s$ must incorporate the exact symmetries of the theory, such as Poincaré invariance and unitarity.
Hadron dynamics is simplified at O($\alpha_s^0$) but is still non-trivial, due to relativistic effects and hadron loops. Mesons lie on linear Regge trajectories (at small quark masses) and there are features of duality as observed in experiments. 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].
