= Solution
The leading infrared terms cancel in the observable sum:
$$
P_{22,{\rm IR}}+2P_{13,{\rm IR}}
=\left(\frac13-2\frac16\right)k^2P(k)
\int\frac{dq}{2\pi^2}P(q)=0.
$$
This <infrared cancellation in large-scale structure> follows from the <Equivalence principle>: a sufficiently long-wavelength displacement translates short-scale structure without changing an equal-time power spectrum.
The ultraviolet part of $2P_{13}$ is proportional to $k^2P(k)$ and depends on the cutoff $\Lambda$. The leading deterministic <effective field theory of large-scale structure> counterterm has exactly this shape,
$$
\boxed{P_{\rm ct}(k)=-2a^4c_{\rm eff}^2(\Lambda)k^2P(k)}.
$$
Its cutoff-dependent part can be chosen as
$$
c_{\rm eff}^2(\Lambda)\big|_{\rm cutoff}
=-\frac{61}{630}\int^{\Lambda}\frac{dq}{2\pi^2}P(q),
$$
which cancels the stated $2P_{13,{\rm UV}}$. The $P_{22}$ ultraviolet contribution begins at $k^4$ and, when cutoff sensitive, is absorbed by higher-derivative or stochastic counterterms.
Physically, coarse graining the matter equations does not justify setting the short-scale stress $\sigma_{ij}$ to zero. Unresolved multistreaming and nonlinear motion generate an effective pressure and viscosity. Their leading derivative contribution to the Euler equation is proportional to $-c_{\rm eff}^2\nabla_i\delta$, producing the required $k^2\delta$ correction and making long-distance predictions independent of the arbitrary cutoff.
Back to article page
