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The nonlinear wavenumber is the inverse scale at which linear density contrast fluctuations become of order unity. For the one-dimensional convention $⟨\delta (k)\delta (k^{\prime })⟩=2\pi {\delta }^{(D)}(k+k^{\prime })P_L(k)$, a usual estimate is $k_{\mathrm{NL}}{P}_L(k_{\mathrm{NL}})/\pi \sim 1$. Loop momenta above this scale require the effective field theory of large-scale structure even when external modes are perturbative.

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.

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.

The leading infrared terms cancel in the observable sum: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 $k^{2}P(k)$ and depends on the cutoff $\Lambda$. The leading deterministic effective field theory of large-scale structure counterterm has exactly this shape,Its cutoff-dependent part can be chosen aswhich cancels the stated $2P_{13,\mathrm{UV}}$. 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 $-c_{\mathrm{eff}}^2{\nabla }_i\delta$, producing the required $k^2\delta$ correction and making long-distance predictions independent of the arbitrary cutoff.

In standard perturbation theory in cosmology, ${\delta }^{(n)}$ is an $n$-leg vertex carrying the symmetrized standard perturbation theory density kernel $F_n$. For Gaussian random fields, a connected four-point diagram with $L$ loops contains $3+L$ linear power spectra, hence $6+2L$ linear fields.

At tree level the required perturbative-order partitions areThe first is a star with one $F_3$ vertex and three linear external legs. The second has two $F_2$ vertices joined by one internal power-spectrum line, with one linear external leg attached to each vertex. At one loop all five partitions of eight into four positive parts are required:These are conventionally denoted $T_{5111}$, $T_{4211}$, $T_{3311}$, $T_{3221}$, and $T_{2222}$; all inequivalent placements of external legs and all Wick contractions are understood. This list is the complete set of one-loop matter trispectrum topologies.

Let $P_a=P_L(k_a)$ and ${\mathbf{k}}_{ab}={\mathbf{k}}_a+{\mathbf{k}}_b$. With ${\sum }_a{\mathbf{k}}_a=0$, the star contribution iswhere $\{a,b,c\}=\{1,2,3,4\}\setminus \{d\}$. For the exchange contribution, sum over the six unordered choices $\{a,b\}$ of linear external legs and let $\{c,d\}$ be the complementary pair:Thus the connected tree trispectrum is $T_{\mathrm{tree}}=T_{3111}+T_{2211}$.

The pressureless single-stream equations cease to be a complete one-loop prediction because loop momenta probe short nonlinear scales and produce ultraviolet-sensitive terms. A consistent result must include the effective field theory of large-scale structure: effective-stress counterterms, including their insertions into the tree topologies, and stochastic terms at the order allowed by mass conservation and momentum conservation. Their coefficients absorb short-scale dependence and must be fitted or matched. If the observable is a biased tracer rather than matter itself, the corresponding renormalized bias expansion is also required.

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 iswhere the time dependence and the nonlinear reference scale may be absorbed into the renormalized coefficient $c_{\theta }^2$. In the bispectrum it suppliesand permutations, with momentum shape $-c_{\theta }^2{k}^2{B}_{211}$ on the corrected leg. This cancels the local $k^2$ ultraviolet dependence of $⟨{\theta }^{(3)}{\theta }^{(2)}{\theta }^{(1)}⟩$.