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Use signed one-dimensional external momenta $k_1+k_2+k_3=0$, with nonzero $k_i$. Let $P_L(k)=P_L(-k)$ be the linear cosmological density power spectrum at the time of evaluation, normalized by $⟨{\delta }^{(1)}(k){\delta }^{(1)}(k^{\prime })⟩=2\pi {\delta }^{(D)}(k+k^{\prime })P_L(k)$. Using $P_L$ absorbs the factor $a^6$ multiplying a product of three initial power spectra. All kernels below are symmetrized one-dimensional cosmological density kernels.

The connected B222 contribution to the one-loop matter bispectrum has a triangular Feynman diagram: each second-order external field has two Gaussian inputs, joined pairwise to the other two vertices. The connected B411 contribution to the one-loop matter bispectrum has a fourth-order vertex joined to both linear external fields, with its remaining two inputs contracted into a loop at that vertex. External stubs indicate the measured momenta; blue internal lines represent linear power spectra.

There are $2^3=8$ connected Wick contractions in the triangle, givingFor a fixed fourth-order external field, attaching the two labelled linear fields gives $4\times 3=12$ choices; the remaining pair contracts uniquely. ThereforeThese are contributions to the reduced connected one-loop matter bispectrum, with its external momentum-conserving Dirac delta function removed. Disconnected contractions have a zero external momentum and are excluded.

For the ultraviolet behaviour, the three $F_2$ denominators multiply to a negative square. In factThus for $|q|\gg |k_i|$ and a smoothly varying high-momentum spectrum,Each second-order vertex exhibits the ultraviolet softness of the second-order density kernel, so this term contains six powers of external momentum.

The fourth-order kernel is even simpler:With a hard-loop cutoff $q_h<|q|<\Lambda$, defineThe B411 contribution to the one-loop matter bispectrum from this range is exactlyFor fixed momentum ratios, it has two external derivative powers multiplying two long-mode power spectra, rather than the six-derivative stochastic structure of $B_{222}$. It is the leading deterministic UV-sensitive contribution for external modes below the nonlinear wavenumber. This is a comparison of derivative orders and generic cutoff sensitivity; an arbitrary specially chosen spectrum or vanishing shape can change their numerical ranking. A convergent integral can still depend on nonperturbative short scales. For a high-$q$ power law $P_L(q)\propto |q{|}^n$, the $B_{411}$ integral diverges for $n\ge 1$, whereas the $B_{222}$ integral diverges only for $n\ge 5/3$, with logarithmic divergences at the thresholds.

At the field level, there are six ways to contract a hard pair within ${\delta }^{(4)}$. Comparing $6F_4$ with $F_2$ givesThe appropriate second-order density Laplacian counterterm is thereforeIndeed its contribution with two linear fields is $-2C(\Lambda ){\sum }_{\mathrm{cyclic}}{k}_3^2{F}_2{P}_L(k_1)P_L(k_2)$, which cancels the displayed hard-loop term. The counterterm sign follows from ${\partial }_x^2\leftrightarrow -k^2$. Its finite coefficient must be matched to short-scale dynamics, as in the effective field theory of large-scale structure. A term proportional to ${\partial }_x^2{\delta }^{(1)}$ similarly renormalizes a linear response, but a purely linear insertion cannot by itself cancel this $⟨{\delta }^{(4)}{\delta }^{(1)}{\delta }^{(1)}⟩$ shape, since an odd Gaussian three-point function vanishes.

Use the one-loop matter power spectrum convention in which $P_{13}$ is one ordering of $⟨{\delta }^{(1)}{\delta }^{(3)}⟩$. Thenwith the P22 contribution to the one-loop matter power spectrumand the P13 contribution to the one-loop matter power spectrumThe factors two and three count Wick contractions. The mixed $P_{12}$ contains an odd number of zero-mean Gaussian fields, so $P_{12}=0$.

For $q\gg k$, the ultraviolet softness of the second-order density kernel gives $F_{2,s}=O(k^2/q^2)$, soThe third-order kernel has $F_{3,s}(\mathbf{k},\mathbf{q},-\mathbf{q})=O(k^2/q^2)$, givingThese integrals expose the sensitivity to unresolved short-scale dynamics. The effective sound-speed counterterm in large-scale structure has exactly the required deterministic shape:Here $c_{\mathrm{eff}}^2$ absorbs the time and normalization conventions and has dimensions of length squared. A stochastic correction begins at $k^4$ and can absorb the corresponding short-scale sensitivity of $P_{22}$.

Put $\mathbf{p}=\mathbf{k}-\mathbf{q}$, $p^2=k^2-2kq\mu +q^2$, and $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{q}}$. Direct substitution into $F_2$ givesUsing $d^{3}q=2\pi q^{2}dqd\mu$ in part ii therefore yieldsFor $q\ll k$, $F_2(\mathbf{q},\mathbf{k}-\mathbf{q})\sim k\mu /(2q)$. Including the equal soft region $|\mathbf{k}-\mathbf{q}|\to 0$ and using ${\int }_{-1}^1{\mu }^{2}d\mu =2/3$ givesFor $q\gg k$, the constant and linear hard-momentum terms cancel, displaying the ultraviolet softness of the second-order density kernel. Since $p\sim q$,The angular integral ${\int }_{-1}^1(3-10{\mu }^2{)}^{2}d\mu =18$ then gives

The ultraviolet softness of the second-order density kernel requiresfor fixed $\mathbf{k}$ as $q\to \infty$. In this limit the two hard vectors are opposite,The constant part of the stated kernel is thereforeIt must vanish, givingThis recovers the standard standard perturbation theory density kernel $F_2$.