Dirac equation describes elementary fermions (e.g., electrons, muons, etc). When coupled to electromagnetic fields, the $g$-value comes out to be equal to $2$ which is exactly the value for the electron, muon, etc at the leading order. Radiative corrections at the sub-leading order can also be calculated using QED. This is a standard calculation in QED.
However, in order to describe the interaction of charged composite spin-1/2 fermions (e.g, protons) with EM field, the Dirac equation cannot be used in its unblemished form because their $g-$values significantly differ from $2$. Therefore, I think, we have to modify the Dirac equation first or replace it with an "effective Dirac equation" to make room for $g\neq 2$ at the leading order itself.
Can we do so by writing the Dirac equation in terms of a $\Gamma^\mu$ matrix, in place of $\gamma^\mu$? Can somebody sketch the method of calculating this $\Gamma^\mu$ for the proton from first principles and where it differs from the method of calculating $\Gamma^\mu$ for electrons? Thanks!
Let me write a few more lines to clarify the question. It seems to me that if we want to write an effective description for the EM interaction of protons, we can use the QED Lagrangian with $\gamma^\mu$ replaced by some other effective $\Gamma^\mu$. Let me take a concrete example. The elastic electromagnetic scattering of electrons off fixed proton targets. This calculation (See Halzen & Martin) uses an effective $ie\Gamma^\mu$ QED vertex for proton and involves writing the most general form of $\Gamma^\mu$ in terms of $F_{1}(q^2)$ and $F_2(q^2)$ without calculating them from first principles. However, I think, if we could calculate $F_{1}(q^2)$ and $F_2(q^2)$ from first principles, I can use a QED-like theory of electron-proton scattering. How wrong or right is this impression? Can we calculate $F_{1}(q^2)$ and $F_2(q^2)$ from first principles? The answer here does not really address this.
