Two vertices and two labelled contractions

For two distinct vertices, let the cross-contraction operator use $D_F$, with one derivative at each vertex per edge. The second term of its exponential carries $1/2!$. The insertion derivative is ${D_h}^2{F}_t=2t_{\mathrm{D}}$. The interaction derivative is the ordinary second derivative (eq:Hessian), with no divided-power convention.

Therefore the numerical multiplier of the two-edge contribution is

The sums over the two labelled legs are retained in the Hessian and the two kernel slots. There is no second division for tensor-slot symmetry.

This is the graph-expansion convention of FR (Eq. (82)), adopted here only for this numerical sector. Coupling powers, the vertex multiplier in an $S$-matrix, and ${\hslash }^2$ are not included in the reduced kernel. Thus the object below is not called a complete first quantum coefficient.