= B411 contribution to the one-loop matter bispectrum
{c}
{title2=$B_{411}$}
The connected correlation of one fourth-order and two first-order <density contrasts> contracts a pair of inputs at the fourth-order vertex into a loop. For each choice of the fourth-order external field there are twelve <Wick contractions>. In one dimension,
$$
B_{411}=12P_L(k_1)P_L(k_2)\int\frac{dq}{2\pi}F_4(-k_1,-k_2,q,-q)P_L(q)+\text{two cyclic terms}.
$$
For the <one-dimensional cosmological density kernel>, $F_4=-k_3^4/(24k_1k_2q^2)$ exactly. Its leading cutoff dependence is therefore proportional to $k_3^2F_2(k_1,k_2)P_L(k_1)P_L(k_2)\int dq\,P_L(q)/q^2$.
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