# Renormalization of Bending Young's Modulus and Diagrams

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?

• 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... – Adam Oct 22 at 7:11
• 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 – Robert Oct 22 at 17:10
• 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... – Adam Oct 22 at 20:13