# Physical meaning of minimal coupling assumption

In SM, both for the gauge field $$A_{\mu}$$ associated with photon and for the gauge fields $$W^{j}_{\mu}$$, to restore the invariance of the lagrangian after the changing of the global transformation to local gauge transformation you introduce under the assumption of minimal coupling between fermion fields and gauge boson fields the covariant derivative $$D_{\mu}=\partial_{\mu}+iqA_{\mu}$$ $$D_{\mu}=\partial_{\mu}-ig\frac{\sigma_j}{2}W^j_{\mu}$$

I read on Wikipedia that minimal coupling between fields assume that only the charge distribution and not higher multipole moments is involved in the interaction.

1. Maybe I have gaps in classic electromagnetism but I don't know intuitively what this means?

2. What is the physical assumption behind the minimal coupling hypothesis and how is it verifiable experimentally?

• Possibly this answer is of relevance; non minimal-couplings typically involve magnetic moment g s of the Pauli type, i.e. with values different than the Dirac value (minimal coupling). – Cosmas Zachos Feb 1 '20 at 19:59
• "minimal coupling" means absence of non-renormalizable terms such as mass dimension 5 terms delineated in the link provided by Cosmas Zachos above and mass dimension 6 terms here: physics.stackexchange.com/questions/513424/… – MadMax Feb 3 '20 at 15:10

For a point-like charge (electron) in an external field $$A_{\mu}^{\text{ext}}$$ this works to certain limitations (Classic Electrodynamics = CED), but it fails even in CED when one applies this way of interaction to the "self-action" $$A_{\mu}$$.