Manuscript XVI-O1-ST@5 · XVI-O1-ST-v5.tex · section · dd45e0d1

Two-loop subgraphs and analytical insertion

Successor calculations and documentary boundaries › Two-loop subgraphs and analytical insertion · lines 2398–2437

A validated coefficient code requires an insertion map

Ins_{γ} : {validated coefficients} \to {local vertices in contracted graphs},

with derivative, tensor, antifield, Grassmann, symmetry, evanescent, and subtraction conventions.

For the declared pole extraction,

R Γ = Γ + S \sum γ \in S \prod C (γ) Γ/ S, C (γ) = - K_{pole} R γ,

R Γ = (1 - K_{pole}) R Γ.

The sum uses allowed disjoint proper UV subgraphs; nested subtractions are recursive Zimmermann,SmirnovChetyrkin. Every required proper subgraph needs its counterterm and insertion.

For ordinary gravitational vertex weights,

v \sum (n_{v} - 2) = 2 I + E - 2 V = E + 2 L - 2.

A one-loop cubic counterterm can therefore contribute to the $κ^{4}$ two-loop two-point inventory. Required Laurent depth and evanescent contributions are retained before multiplication by subdivergent poles DuganGrinstein. Separate success on gauge, mass, and external axes does not prove their simultaneous combination.

Displayed formulas in this passage

unlabeled · line 2401
$Ins_{γ} : {validated coefficients} \to {local vertices in contracted graphs},$
unlabeled · line 2410
$R Γ = Γ + S \sum γ \in S \prod C (γ) Γ/ S, C (γ) = - K_{pole} R γ,$
unlabeled · line 2417
$R Γ = (1 - K_{pole}) R Γ.$
unlabeled · line 2426
$v \sum (n_{v} - 2) = 2 I + E - 2 V = E + 2 L - 2.$