# Derivative interaction of scalar fields

I'm studying derivative interactions and I'm trying to gain a better intuition for the formal process of arriving at the vertex factor. In particular, I'm trying to evaluate the vertex factor associated with this interaction Lagrangian: $$L_{int} = (\partial_\mu\phi^2)(\partial^{\mu}\phi^2)$$ I've given it a go, but I'm just not sure if my approach is correct. I believe the first step is to write out the Fourier decomposition of the fields, like this: $$\phi_(x_1)\phi(x_2) = \int dp_1 \tilde{\phi}(p_1)e^{i(p_1x_1)} \int dp_2 \tilde{\phi}(p_2)e^{i(p_2x_2)}$$ And then apply the derivative using the product rule to get: $$\partial_{\mu}(\phi_(x_1)\phi(x_2))=\int dp_1 (ip_1) \tilde{\phi}(p_1)e^{i(p_1x_1)} \int dp_2 \tilde{\phi}(p_2)e^{i(p_2x_2)} + \int dp_1 \tilde{\phi}(p_1)e^{i(p_1x_1)} \int dp_2 (ip_2) \tilde{\phi}(p_2)e^{i(p_2x_2)}$$ And to do a similar procedure for the second two fields, but with labels $$(x_3,p_3)$$ and $$(x_4,p_4)$$, and then multiply the results. If this is correct, then my resulting vertex factor would seemingly be: $$-(p_1\cdot p_3 + p_2\cdot p_3+ p_1\cdot p_4+p_2\cdot p_4)$$

I've gathered this procedure from a variety of sources (like this question and this question), most of which just vaguely cite the fact that derivative interactions pull down factors of $$(ip)$$ when you do the Fourier decomposition. First and foremost, is the above procedure the correct approach? Am I missing steps in between? Why is it the case that I can just read off vertex factors using this method?

And second, if my resulting vertex factor is correct, I know I need to consider permutations of it for different labeling of the momenta - can someone illustrate in more detail what that means?