= Second-order density Laplacian counterterm
{title2=$C(\Lambda)\partial_x^2\delta^{(2)}$}
For the one-dimensional <B411 contribution to the one-loop matter bispectrum>, contracting two hard inputs in the fourth-order field gives $-(I_\Lambda/2)k^2\delta^{(2)}$, where $I_\Lambda=\int_{q_h<|q|<\Lambda}dq\,P_L(q)/(2\pi q^2)$. A <counterterm> $C(\Lambda)\partial_x^2\delta^{(2)}$ cancels this cutoff-dependent part when $C(\Lambda)=-I_\Lambda/2+C_{\rm finite}$. This sign follows from $\partial_x^2\leftrightarrow-k^2$; the finite coefficient describes unresolved short-scale dynamics.
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