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?



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