# How to find Lagrangian density from a vertex factor

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1. Mar 31, 2016

### Maurice7510

1. The problem statement, all variables and given/known data
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).

2. Relevant 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.

3. 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.

2. Apr 5, 2016