A late-time connected four-point function of a scale-invariant field in three spatial dimensions, after removing its momentum-conserving delta function, must obey
$$
B_4(\Lambda\mathbf k_1,\ldots,\Lambda\mathbf k_4)
=\Lambda^{-9}B_4(\mathbf k_1,\ldots,\mathbf k_4).
$$
In both contributions, $s/(k_1k_2k_3k_4)$ scales as $\Lambda^{-3}$ and every energy-denominator term scales as $\Lambda^{-6}$. Hence the complete $s$-channel <primordial trispectrum> scales as $\Lambda^{-9}$, as required by scale invariance.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 1 / v / Solution

{title2=$T(\mathbf k_1,\mathbf k_2,\mathbf k_3,\mathbf k_4)$}
{wiki=Trispectrum}

The primordial trispectrum is the connected four-point function with its momentum-conserving delta function removed. Exchange diagrams split naturally into $s$, $t$, and $u$ channels according to which pair of external momenta meets at one cubic vertex.


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 Incoming links: Primordial trispectrum