I am stuck at the beginning of a problem where I am given an interaction term that modifies the regular QED Lagrangian. It involves the interaction between a photon field and a massive vector boson: $\mathcal{L}_{int_1} = \frac{1}{4}g_1G_{\mu \nu}F^{\mu \nu}$. Here, $F^{\mu \nu}$ is the electromagnetic field tensor. Similarly, $G_{\mu \nu} = \partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}$ where $B_{\mu}$ is the massive vector field.
I am trying to derive the Feynman rule for this vertex. I am not sure what to do. Usually, I just bring down a 4-momentum when I have a field with derivatives in the Lagrangian, but in this case, it looks like two of the 4-momenta would end up forming a dot product together, while the other two would not, and I would be left with a term without any indices and two terms with indices, which doesn't make sense. Does anyone know what the Feynman rule for this vertex is and why it is so?
EDIT: I think the Feynman rule might be $i\mathcal{M} = -i g_1 (g_{\mu \nu}-\frac{k_\mu a_\nu}{k \cdot q})$. I obtained this by an analogous computation for finding the photon propagator. Is doing this allowed?
The main problem that I am trying to solve involves the diagram $f\bar{f} \to \phi\phi^*$. From comments, I realize that I have two options: 1) I can diagonalize the mass matrix and use the new mass as the mass in the propagator between the two end vertices or 2) I can consider the infinite series of $A$ propagator > $B$ propagator > ... > $A$ propagator > $B$ propagator and obtain the same answer. The problem I am having with diagonalizing the mass matrix is that in previous problems where I have diagonalized a mass matrix, my interaction term did not contain derivative terms. So my attempt at a solution (which I am trying now) is to just bring down a 4-momentum and go from there. But then wouldn't that make my mass momentum dependent? With the second method, I think I would need to still know the feynman rule for the interaction vertex, which circles back to my original problem.