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For nonzero momentum, the Fourier transform of the local bias expansion isThe subtracted variance contributes only at $\mathbf{k}=0$. Because $\delta$ is a Gaussian random field, its three-point function vanishes and its four-point function factorizes by Wick theorem. At leading order one of the three galaxy fields supplies the quadratic term and the other two supply linear terms. The two cross-contractions cancel the factor $1/2$, givingThus even Gaussian matter fluctuations acquire a nonzero galaxy bispectrum through local quadratic galaxy bias.

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.