Executable amplitude rules

Builder/output: feynman.py, shadow.py, amplitude_rules.py, P14_amplitude_rules.json, exports/P14_vertex_tensors.json, exports/P14_integrands/*.json.

Rules.

• Extraction: ordered, $V[L_1..L_n]={{\partial }^L}_{L_n}\cdots {{\partial }^L}_{L_1}S$, with leg 1 leftmost; the Feynman vertex is $i{\kappa }^{n}V$.

• Graviton legs: independent components; the multiplicity (1 diagonal, 2 off-diagonal) enters only the tensor form.

• Ghost: $\bar{c}$ is on the left in $\bar{c}ch$ and in $M_0$; a closed ghost loop carries $-1$.

• Source: $K^{\prime }$ sits left of $c$ in $K^{\prime }ch$ and in the ${\Gamma }_1$ coefficient ${K^{\prime }}_{a^{\prime }}{G}_{a^{\prime }\nu }{c}^{\nu }$.

• Symmetry factors: $S_f=\frac{1}{2}$ for D01, D02, D11, D12, D14a, D14c and $S_f=1$ for D03, D13, D14b.

• Ghost-loop sign: $-1$ for D03 and D14b.

The vertex tensors include every derivative term of the ghost-graviton vertex ($\bar{c}ch=2ia{D}^{\rho }[{\mathcal{L}}_{c}h]$, including the transport term $c^r{\partial }_{r}h$) and the $K^{\prime }$ vertex.

Controls.

W1.

Berezin/Wick shadow theory: the symmetry factors and Grassmann signs are fixed independently.

W2.

A tensor-route recomputation of the multiplicities.

W3.

Closed forms of the vertices.

W4.

The primary evidence is the generic symbolic off-shell Ward identities at ${\kappa }^1$ and ${\kappa }^2$. A supplementary check is the “Ward contraction with two free on-shell graviton legs and an off-shell gauge leg”. W4 also checks Bose symmetry.

W5.

Independent sympy regeneration of D14a, D14b, D14c, D11 and D13 only, plus a Bose-symmetry check of D01.

W6 (p14r).

Independent regeneration of D01, D02, D03 and D12 from the one-loop expansion of the functional (super)trace log of the field-dependent Hessian of $S_{\Psi }$ (hessian_route.py), not from the Feynman-rule assembler:

$$i{\Gamma }^{AB}=-\frac{1}{2}\mathrm{Tr}(G{U}_2^{AB})+\frac{1}{2}\mathrm{Tr}(G{U}_1^{A}G{U}_1^B)\text{(bosons)},i{\Gamma }^{AB}=-\mathrm{Tr}(G{U}_1^{A}G{U}_1^B)\text{(ghosts)}.$$

The ingredients are an own monomial differentiation of ${\ell }_3,{\ell }_4$, an own inverse of the full $14\times 14$$(h,b)$ Hessian, and the ghost and scalar Hessians from the $S_{\Psi }$ densities. Structural checks: $X^{(1)}$ has no field-only term and $X^{(2)}=0$, so there is no $\bar{c}chh$ vertex. The route equals the exported integrands exactly, component by component (D01 100, D02 94, D03 100, D12 94; 0 mismatches), and it also rejects the tadpole-factor, ghost-sign, multiplicity and transport-term mutations.

Mutations that must be rejected (all rejected): wrong tadpole factor, wrong ghost-loop sign, off-diagonal multiplicity, reversed source/ghost order, and omitted ghost transport term. D13 has priority; its integrand is exported with prefactor $i{\kappa }^2$ and order $(K^{\prime },c)$. Integrands are symbolic (not integrated); the per-graph table is in the summary above.