The <effective field theory of large-scale structure> supplements the one-loop prediction by the leading deterministic counterterm and a stochastic term,
$$
P_{\rm EFT}(k)=P_{11}+P_{22}+2P_{13}
-2c_{\rm eff}^2k^2P_{11}(k)+P_{\rm stoch}(k)+\cdots,
$$
where the normalization scale may be absorbed into $c_{\rm eff}^2$ and mass and momentum conservation make $P_{\rm stoch}=O(k^4)$ at small $k$. The ultraviolet part of the <P13 contribution to the one-loop matter power spectrum> has
$$
P_{13}^{\rm UV}(k)\propto
P_L(k)\int^{\Lambda}\frac{d^3q}{(2\pi)^3}
\frac{k^2}{q^2}P_L(q)
=k^2P_L(k)\frac1{2\pi^2}
\int^{\Lambda}dq\,P_L(q).
$$
Its cutoff dependence has exactly the $k^2P_L(k)$ form of the <effective sound-speed counterterm in large-scale structure>, so the running of $c_{\rm eff}^2(\Lambda)$ cancels it. The stochastic and higher-derivative counterterms similarly absorb the allowed analytic ultraviolet dependence of $P_{22}$ and higher orders.

Solution

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

{title2=$-2c_{\rm eff}^2k^2P(k)$}

The leading deterministic power-spectrum counterterm in the effective field theory of large-scale structure is proportional to $-2c_{\rm eff}^2k^2P(k)$. Its coefficient depends on the coarse-graining convention and runs so that it cancels the cutoff dependence of the one-loop $P_{13}$ contribution.


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 Incoming links: Effective sound-speed counterterm in large-scale structure