Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 312 / 3 / c / Solution

The effective field theory of large-scale structure supplements the one-loop prediction by the leading deterministic counterterm and a stochastic term,where the normalization scale may be absorbed into $c_{\mathrm{eff}}^2$ and mass and momentum conservation make $P_{\mathrm{stoch}}=O(k^4)$ at small $k$. The ultraviolet part of the P13 contribution to the one-loop matter power spectrum hasIts cutoff dependence has exactly the $k^2{P}_L(k)$ form of the effective sound-speed counterterm in large-scale structure, so the running of $c_{\mathrm{eff}}^2(\Lambda )$ cancels it. The stochastic and higher-derivative counterterms similarly absorb the allowed analytic ultraviolet dependence of $P_{22}$ and higher orders.