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I'm reading through this arXiv paper and I ran into a problem when working through some of the RG calculations. In the supplemental info (p. 8), when evaluating the diagrams in Fig S1.

The interaction has the form $$\frac{Y}{8}\sum_{q_1,q_2,q_3,q_4} [q_{1i}P^T_{ij}q_{2j}][q_{3i}P^T_{ij}q_{4j}]f(q_1)f(q_2)f(q_3)f(q_4)$$

I understand that the first diagram will give the expression in S6, that we need to differentiate for $Z_\kappa$ (though I still have no idea where the factor of $q^4$ goes) but in $Z_Y$, I'm confused as to where the extra factor of $p^4$ is coming from.

I understand that we have the two propagators and the factor of $(1-(\hat{p}\cdot\hat{q})^2)^2$ comes from one of the vertices but where do we get an extra factor of $p^4$. I would've thought $Z_Y$ would be something along the lines of

$$\propto Y^2 \int_{\Lambda/b}^\Lambda d^2p\, (1-(\hat{p}\cdot\hat{q})^2)^4 \langle f(p)f(-p) \rangle^2$$

I know it must come from the other vertex and I thought it came from contracting two separate vertices but I can't get the math to give me a single factor of $p^4$.

Can someone tell me where I'm going wrong?

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    $\begingroup$ Are you familiar with Wilson RG (for simpler models like $\phi^4$)? Starting with a model with momentum depend vertices might be a tough one... $\endgroup$ – Adam Oct 22 at 7:11
  • $\begingroup$ Yeah I'm familiar with Wilsonian RG for simple models but yeah I'm having trouble translating those methods to use with this particular vertex $\endgroup$ – Robert Oct 22 at 17:10
  • $\begingroup$ Maybe you could write what you have tried (say, starting $\kappa$), to see where it goes wrong? It might help shed some light on what's going on there... $\endgroup$ – Adam Oct 22 at 20:13

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