The ultraviolet part of the one-loop matter power spectrum is renormalized by the leading <effective field theory of large-scale structure> operator proportional to $k^2\delta^{(1)}$. The analogous velocity-divergence counterterm is
$$
\boxed{\theta_{\rm ct}^{(3)}(\mathbf k)
=-c_\theta^2k^2\theta^{(1)}(\mathbf k)},
$$
where the time dependence and the nonlinear reference scale may be absorbed into the renormalized coefficient $c_\theta^2$. In the bispectrum it supplies
$$
\langle\theta_{\rm ct}^{(3)}\theta^{(2)}\theta^{(1)}\rangle
$$
and permutations, with momentum shape $-c_\theta^2k^2B_{211}$ on the corrected leg. This cancels the local $k^2$ ultraviolet dependence of $\langle\theta^{(3)}\theta^{(2)}\theta^{(1)}\rangle$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 3 / iii / c / Solution

{title2=EFTofLSS}
{wiki}

The effective field theory of large-scale structure represents unresolved short-scale dynamics by an effective stress tensor. Its leading deterministic correction is proportional to $k^2\delta^{(1)}$ and renormalizes ultraviolet-sensitive loop contributions.


Codex @codex  0

 Incoming links: Effective field theory of large-scale structure