In a problem in my QFT course i am asked to write down the most general posible Lorentz invarient Lagrangian density for a vector field (at most quadratic in the field and in the derivatives) and to find what it reduces to if we ask for gauge invariance.
This is my work so far:
The most general Lorentz invariant Lagrangian density is:
$ \mathcal{L} = \alpha A^{\mu}A_{\mu} + \beta \partial_{\mu}A^{\mu}\partial_{\nu}A^{\nu} + \gamma \partial_{\nu}A_{\mu}\partial^{\nu}A^{\mu}$
we want this to be invariant under a gauge transformation ($ A_{\mu} \xrightarrow{} A_{\mu} + \partial_{\mu}\xi$)
$ \mathcal{L^{'}} = \alpha (A^{\mu}+\partial^{\mu}\xi)(A_{\mu}+\partial_{\mu}\xi) + \beta \partial_{\mu}(A^{\mu}+\partial^{\mu}\xi)\partial_{\nu}(A^{\nu}+\partial^{\nu}\xi)) +\\ \gamma \partial_{\nu}(A_{\mu}+\partial_{\mu}\xi)\partial^{\nu}(A^{\mu}+\partial^{\mu}\xi)$
I think what I have to obtain is that the mass term must be zero ($\alpha=0$) in order for the equations of motion to be invariant under such a tranformation. However, if I expand each term I get:
$\mathcal{L^{'}} = \alpha\left[ A^{\mu}A_{\mu}+A^{\mu}\partial_{\mu}\xi + \partial^{\mu}\xi A_{\mu}+\partial^{\mu}\xi\partial_{\mu}\xi\right] \\+ \beta (\partial_{\mu}A^{\mu}\partial_{\nu}A^{\nu}+\partial_{\mu}A^{\mu}\partial_{\nu}\partial^{\nu}\xi+\partial_{\mu}\partial^{\mu}\xi\partial_{\nu}A^{\nu}+\partial_{\mu}\partial^{\mu}\xi\partial_{\nu}\partial^{\nu}\xi) \\+ \gamma (\partial_{\nu}A_{\mu}\partial^{\nu}A^{\mu}+\partial_{\nu}A_{\mu}\partial^{\nu}\partial^{\mu}\xi+\partial_{\nu}\partial_{\mu}\xi\partial^{\nu}A^{\mu}+\partial_{\nu}\partial_{\mu}\xi\partial^{\nu}\partial^{\mu}\xi) $
How can I simplify this expresion? I suppose that I may have to ask for a condition for $\gamma$ and $\beta$ as well, but I am honestly stuck and kind of lost with all the indexes. Any help is appreciated. Thanks!
edit:
I can write $\mathcal{L}^{'}$ as:
$\mathcal{L^{'}} = \mathcal{L} + \alpha\left[A^{\mu}\partial_{\mu}\xi + \partial^{\mu}\xi A_{\mu}+\partial^{\mu}\xi\partial_{\mu}\xi\right] \\+ \beta (\partial_{\mu}A^{\mu}\partial_{\nu}\partial^{\nu}\xi+\partial_{\mu}\partial^{\mu}\xi\partial_{\nu}A^{\nu}+\partial_{\mu}\partial^{\mu}\xi\partial_{\nu}\partial^{\nu}\xi) \\+ \gamma (\partial_{\nu}A_{\mu}\partial^{\nu}\partial^{\mu}\xi+\partial_{\nu}\partial_{\mu}\xi\partial^{\nu}A^{\mu}+\partial_{\nu}\partial_{\mu}\xi\partial^{\nu}\partial^{\mu}\xi) $
But I still dont see how I can get this to be zero by selecting the appropiate $\alpha$, $\beta$ and $\gamma$. I have a question. First, from what I undsrstand, terms like $\partial_{\nu}A_{\mu}\partial^{\nu}\partial^{\mu}\xi$ and $\partial_{\nu}\partial_{\mu}\xi\partial^{\nu}A^{\mu}$ are equal, is this correct? I find this by lowering al indexes using the metric tensor, $g_{\mu\nu}$ and $g_{\mu\nu}g_{\mu^{'}\nu^{'}}=1$. Next, I feel like the terms multiplied by $\gamma$ and $\beta$ should somehow cancel out (choosing $\gamma=\pm\beta$), but I cant see how to do this.