Second-order density Laplacian counterterm ($C(\Lambda ){\partial }_x^2{\delta }^{(2)}$)

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 $C(\Lambda )=-I_{\Lambda }/2+C_{\mathrm{finite}}$. This sign follows from ${\partial }_x^2\leftrightarrow -k^2$; the finite coefficient describes unresolved short-scale dynamics.