Curvature bispectrum from a zeta zeta-prime-squared interaction ($B_{\zeta \zeta {\prime }^2}$)

For $H_{\mathrm{int}}=-M_{\mathrm{Pl}}^2\int d^{3}x{a}^3{ϵ}^2\zeta {\dot{\zeta }}^2$ and the standard Gaussian de Sitter modes, the tree in-in formalism gives $B=H^4{\sum }_{\mathrm{cyc}}{k}_2^2{k}_3^2(K^{-1}+k_1{K}^{-2})/[16ϵM_{\mathrm{Pl}}^4(k_1{k}_2{k}_3{)}^3]$, where $K={\sum }_i{k}_i$. The result includes six connected Wick contractions and a vacuum contour.

The comoving curvature perturbation is dimensionless. With spatial coordinates and time carrying inverse mass dimensions, the cubic interaction $\int dt{d}^{3}x{a}^3{ϵ}^2\zeta {\dot{\zeta }}^2$ needs a mass-squared coefficient. Its standard gravitational normalization is $M_{\mathrm{Pl}}^2$. Six De Sitter curvature mode functions supply $M_{\mathrm{Pl}}^{-6}$; this vertex then supplies $M_{\mathrm{Pl}}^2$, leaving the physical $M_{\mathrm{Pl}}^{-4}$ curvature primordial bispectrum. The Wick-pairing multiplicity for a zeta zeta-prime-squared vertex is independently six: three choices of undifferentiated external leg times two differentiated-leg assignments.

The undifferentiated soft-leg permutation gives $B(q,k,k)\simeq (ϵ/2)P(q)P(k)$ as $q/k\to 0$. It retains a squeezed $q^{-3}$ enhancement, but corresponds to local amplitude $f_{\mathrm{NL}}^{\mathrm{effective}}=5ϵ/24$. Soft momentum placed only at derivative slots would miss this dominant term.

Three external fields can be joined to the three cubic vertex fields in six connected Wick contractions. There are three choices for the external field at the undifferentiated slot and two exchanges of the differentiated slots. Three cyclic terms therefore require an explicit factor of two. Ordered contractions with an earlier vertex use complex-conjugated mode functions.