# Which term in QED Lagrangian represents the muon?

I often heard that electron-muon scattering is a QED process. So I suppose that muon is a field in QED. However, when looking at the QED Lagrangian, it is basically

F^2 + psi D_A psi

where D_A is the gauge covariant derivative (with gamma matrices). Since psi already represents the electron, my question is, where is the muon? Is it just "another copy" of Fermionic field, so that the Lagrangian looks something like the following?

F^2 + psi_1 D_A psi_1 + psi_2 D_A psi_2

• The problem is that, in order to have an accurate picture of the muon, you have to include muon decay, which is a weak interaction and requires all of the machinery of the weak Lagrangian to make sense. Or are you operating in a hypothetical universe where the muon doesn't decay? Commented Sep 15, 2020 at 16:47
• Also, does D_A include the mass term? Commented Sep 15, 2020 at 16:52
• Thanks for your reply. I was assuming there's no muon decay, and just wanted to understand a scattering such as e e -> mu mu. And yes D_A includes mass (I should have written D_A+m) Commented Sep 15, 2020 at 20:45

Assuming there's no muon decay or other weak or strong interactions (i.e. the only interactions were electromagnetic), then yes, the Lagrangian would consist of the EM tensor term and two charged Dirac fermion terms as you have written above. The only difference would be the mass: $$m_e$$ in one and $$m_\mu$$ in the other.