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

Retained gauge-pole data and exact printed consequences

Retained gauge-pole data and exact printed consequences · lines 1365–1451

XVI-O1-ST@5#rec:gauge

Put $ζ = ξ - 1$ . With unchanged vertices, a graph with $n_{h}$ graviton lines has degree at most $n_{h}$ in $ζ$ under linear pole extraction.

The predecessor attributes the following to its gauge-family scripts SuppliedBot:

D03 and D11 have no graviton line and are $ξ$ -independent;
D04 has zero UV-pole polynomial in the stated family;
D06 and D13 are affine, and D01 is quadratic;
$D 06 = - 2 D 13 [K \to T (c ˉ)]$ holds coefficientwise in $ζ$ , in the incoming-antighost convention.

These remain supplied scientific records. The D01 base-point identification across all v4 artifacts is subject to Section sec:integrity.

Here

T_{mn} (c ˉ) = \frac{1}{2} (p_{m} u ˉ_{n} + p_{n} u ˉ_{m}) - \frac{1}{2} η_{mn} (p \cdot u ˉ) .

Momentum orientation and the separately disclosed D13 slot orientation must be transported consistently CheungXVIQ.

In own-rule normalization $u = i v$ , D06 is

u_{06} (ξ) = i [(- 3/8, 1/4, - 1/2)^{T} + ζ (- 7/24, - 1/12, - 1/4)^{T}] .

For $K = a_{K} \otimes a_{K}$ , set

s = a_{K} \cdot p, r = a_{K} \cdot u, t = a_{K} \cdot u, A_{K} = a_{K} \cdot a_{K}, u_{p} = u \cdot p .

On $(s Qt, s^{2} u_{p}, r s Q, A_{K} u_{p} Q)$ , D13 has coefficients

u_{13} (ξ) = i [(- 1/2, 1/4, - 3/8, 0)^{T} + ζ (- 1/4, 1/12, - 7/24, 1/12)^{T}] .

The fourth structure first appears at order $ζ$ .

For D01 write

u_{01} (ξ) = j = 1 \sum^{14} (a_{j}^{(0)} + ζ a_{j}^{(1)} + ζ^{2} a_{j}^{(2)}) B_{j} .

All printed coefficients are retained:

$j$	$a_{j}^{(0)}$	$a_{j}^{(1)}$	$a_{j}^{(2)}$
1	$27/80$	$1/2$	$1/2$
2	$- 19/60$	$- 2/3$	$- 1/2$
3	$11/48$	$1$	$5/6$
4	$71/120$	$1/2$	$1/6$
5	$- 11/10$	$- 1$	$- 1/3$
6	$4/3$	$1/2$	$0$
7	$- 19/60$	$- 2/3$	$- 1/2$
8	$13/15$	$4/3$	$2/3$
9	$- 1/6$	$- 5/4$	$- 5/6$
10	$- 4/3$	$- 1/2$	$0$
11	$11/48$	$1$	$5/6$
12	$- 1/6$	$- 5/4$	$- 5/6$
13	$1/8$	$3/2$	$5/4$
14	$3/4$	$0$	$0$

Displayed formulas in this passage

unlabeled · line 1389
$T_{mn} (c ˉ) = \frac{1}{2} (p_{m} u ˉ_{n} + p_{n} u ˉ_{m}) - \frac{1}{2} η_{mn} (p \cdot u ˉ) .$
unlabeled · line 1399
$u_{06} (ξ) = i [(- 3/8, 1/4, - 1/2)^{T} + ζ (- 7/24, - 1/12, - 1/4)^{T}] .$
unlabeled · line 1405
$s = a_{K} \cdot p, r = a_{K} \cdot u, t = a_{K} \cdot u, A_{K} = a_{K} \cdot a_{K}, u_{p} = u \cdot p .$
unlabeled · line 1413
$u_{13} (ξ) = i [(- 1/2, 1/4, - 3/8, 0)^{T} + ζ (- 1/4, 1/12, - 7/24, 1/12)^{T}] .$
unlabeled · line 1421
$u_{01} (ξ) = j = 1 \sum^{14} (a_{j}^{(0)} + ζ a_{j}^{(1)} + ζ^{2} a_{j}^{(2)}) B_{j} .$