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Will charge be confined, as colour is in QCD?

A simple argument, based on non-relativistic quantum mechanics, is that the binding energy of an electron in a hidrogen atom is given by E= - 1/2 (alpha)² m_e c². If alpha=2, instead of alpha=1/137, one could produce a pair electron-positron, bind the electron to the proton, and put the positron faw away. This could be a mechanism to surround any free particle with electrons or positrons (or muons, taus ..).

Of course, this argument does not take into account relativity, or quantum field effects. So, my question is whether it is there some theoretical treatment of non-perturbative QED, hinting to the presence of confinement .

I guess that I would like to understand if confinement is something related to the strength of the coupling constant, or something associated to the non-abelian nature of theories such as QCD.

Thanks.