# What would be the effect of a complex vector field in the dirac equation?

$$i\overline{\psi}\gamma^\mu A_\mu\psi + i\psi \gamma^\mu A_\mu \overline{\psi}$$
$$\overline{\psi}\gamma^\mu B_\mu\psi - \psi \gamma^\mu B_\mu \overline{\psi}$$
But this time without the $$i$$ factor.