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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.

Solidification
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  • Related: https://physics.stackexchange.com/q/526637, https://physics.stackexchange.com/q/506225 – Nihar Karve Jul 17 '21 at 15:04
  • The effective Lagrangian for such systems is not a Dirac one, but entails an additional (nonrenormalizable, but who cares?) Pauli moment term. – Cosmas Zachos Jul 17 '21 at 15:17
  • @CosmasZachos If we want to write an effective EM interaction between the EM field and protons, can we use the QED Lagrangian with $\gamma^\mu$ replaced by some other effective $\Gamma^\mu$? So the question is how to describe the EM interaction of protons (charged, composite) with the EM field in QFT, and to what extent can I use the Dirac equation. For example, I am familiar with the elastic electromagnetic scattering of electrons off fixed proton targets. – Solidification Jul 17 '21 at 15:28
  • (continued) This calculation uses an effective $ie\Gamma^\mu$ vertex for proton. This calculation involves writing $\Gamma^\mu$ in terms of $F_{1,2}(q^2)$. It seems to me that if we could calculate $F_{1,2}$ from first principles, I can use a QED-like theory of electron-proton scattering. – Solidification Jul 17 '21 at 15:29
  • @GiorgioP I have added an endnote to clarify what I am really asking. First I would like to know how correct is my impression, and then a few details to it. – Solidification Jul 17 '21 at 15:58
  • I see. I will read it more carefully and get back to this if the confusion persists. Thanks :-) – Solidification Jul 17 '21 at 16:19

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It might be up to you to recast the extra, nonminimal-coupling, unrenormalizable, gauge-invariant, hermitian, Pauli moment term, stuck in by hand, to accommodate arbitrary magnetic moments, $$ -{e\over 2M} \left (\frac{1}{2} F^{\mu\nu} \bar \psi \sigma _{\mu\nu}\psi\right )=i{e\over 2M} \left ( \partial^\nu A^\mu \bar \psi {[\gamma_\mu,\gamma_\nu]\over 4}\psi\right ), $$ by integrating by parts, $$ =-i{eA^\mu\over 8M} \partial^\nu \left ( \bar \psi [\gamma_\mu,\gamma_\nu] \psi\right ), $$ and shape your coupling vertices into your capricious language. Recall to also include the Gordon decomposition canonical part, and adjust m and M to get the ultimate moment you wish. As I insisted, the effective form I recommended is normally adhered to in effective Lagrangeanese today... But you might like this.

The magnetic moments of baryons cannot be calculated from first principles (except perhaps on the lattice), but they can be easily calculated in the effective constituent quark model, indeed one of its early triumphs! You'll find the calculation in most decent books (e.g. D Perkins') on it.

Cosmas Zachos
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