How to find Lagrangian density from a vertex factor

[Maurice7510]

Homework Statement

This isn't a homework problem, per se, in that it's not part of a specific class. That being said, the question I would like help with is finding a Lagrangian density from the vertex factor, $$-ig_a\gamma^{\mu}\gamma^5.$$ This vertex would be identical to the QED vertex except with the photon replaced by a massive spin-1 particle (which obviously couples to $ \gamma^5 $). I was told that there is a one-to-one correspondence between vertex factors and Lagrangian densities, and so having the vertex factor should give me the density (I don't believe they actually meant the relationship was bijective, but it shouldn't make much difference).

Homework Equations

I have only thus far been able to find ways to go from a Lagrangian density to the vertex factor, a method for which the are many sources. The only equation (other than the vertex factor, above) that I can think might be useful would be the vector field $$ A^{\mu}(x) = \sum_{\lambda} \int \frac{d^3p}{\sqrt{(2\pi)^32E}}\left(\epsilon^{\mu}ae^{-ipx}+\epsilon^{*\mu}a^{\dagger}e^{ipx}\right)$$
where $$p_{\mu}\epsilon^{\mu} = 0$$ for massive particles.

The Attempt at a Solution

Since the particle interacts with the electron in the vertex via QED I would assume we would have something like $$\mathcal{L} = \mathcal{L}_{QED} + \mathcal{L}_{int}, $$ and the interaction density should presumably come from the vertex.

 

[BigJDubs]

I would assume this is true because the particle is obviously massive and thus it should interact with the electron explicitly in the Lagrangian density. This would lead me to believe that $$\mathcal{L}_{int} = -ig_aA^{\mu}(x)\bar{\psi}(x)\gamma^{\mu}\gamma^5\psi(x).$$ However, I'm not sure if this is actually correct (or the best way to go about it). Any help would be greatly appreciated!