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For example, let's consider the electromagnetic interaction between a massless fermion and a electromagnetic externel sourse $A^\mu$, then the lagrangian is $$\mathcal{L}=\bar{\psi}\gamma^\mu\partial_\mu\psi+ieA_\mu\bar{\psi}\gamma^\mu\psi$$ and how can we calculate the Feynman diagram and $\mathcal{M}$ of it?

But according to relativity, the equation of motion of a particle with charge is $$e F^a\ _bU^b=U^b\partial_bP^a$$ where $F_{ab}$ is the electromagnetic tensor, $U^a$ is the 4-velocity of the particle and $P^a$ is the 4-momentum of the particle. If the particle is massless, then the speed will be $1$ and this equation fails.

So can a massless particle have charge and interaction with externel sourse? If could, for example, how can I calculate the force and equation of motion? If couldn't, why?

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I think, you are mixing quantum electrodynamics with classical (relativistic) electrodynamics. The equation of motion you stated above is from the classical electrodynamics where particles are considered point charges. In quantum electrodynamics particles are just field excitation, so you cannot ask for their four-velocity but rather for the time evolution of the field configuration of the corresponding field. In this case you are interested in the fermionic field $\Psi$ which is described by the Dirac equation. You can derive it from the lagrangian above by applying the Euler Lagrange equations. When solving the Dirac equation you will typically be forced to go into Fourier space and therefore deal with Fourier modes rather than with the field configuration in space-time, which usually leads to computing cross sections of scattering between plain waves (i.e. Fourier modes) of the $A$ and the $\Psi$ field.

To answer your actual question, indeed I cannot think of an electromagnetically charged massless particle, but for example gluons have colour charge and are massless. They can interact among themselves and with quarks, so interaction of charged massless particles is possible in principle.

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