Manuscript Em01-D@2 · Em01-D-v2.tex · section · d3b49151

Grouped crossed array

The complete retained contraction › Grouped crossed array · lines 859–939

Em01-D@2#eq:C, Em01-D@2#eq:vertex, Em01-D@2#eq:vertextest

Declare

C_{I}^{μν} (x_{1}, x_{2}) = H_{T, γ}^{μν}_{I} (x_{1}) \otimes H_{T, ϕ} (x_{2}) - H_{γ}^{μν}_{I} (x_{1}) \otimes H_{ϕ} (x_{2}) .

Both tensor products exist. Their difference is kept grouped. No undeclared mixed products are introduced.

The relative map is

f (a, b, z, y) = (y - a, y - b) .

For smooth symmetric $H$ , define

Z_{I}^{ν} (H) = f^{*} (\partial_{2, μ} H_{I}^{μν} - \frac{1}{2} η_{μ λ} \partial_{2}^{ν} H_{I}^{μ λ}), M_{I} (H) = f^{*} (η_{μν} H_{I}^{μν}) .

The complete vertex is

V (K, H)^{J} = \partial_{y^{ν}} [g (y) I \sum K^{I, J} Z_{I}^{ν} (H)] - \frac{m ^{2}}{2} g (y) I \sum K^{I, J} M_{I} (H) .

Its arbitrary-test action is

⟨ i V (K, H), Φ ⟩ = - i I, J \sum ⟨ K^{I, J}, g (y) [Z_{I}^{ν} (H) \partial_{y^{ν}} Φ_{J} + \frac{m ^{2}}{2} M_{I} (H) Φ_{J}] ⟩ .

Vertex type	Crossed operator	Coefficient and output
Kinetic	$\partial_{2, μ} H_{I}^{μν}$	$\partial_{y^{ν}} (i g K_{b}^{I, J} \cdot)$ .
Trace kinetic	$\partial_{2}^{ν} H_{I}^{μ λ}$	$\partial_{y^{ν}} (- i g η_{μ λ} K_{b}^{I, J} \cdot /2)$ .
Mass	$H_{I}^{μν}$	Multiplication by $- i m^{2} g η_{μν} K_{b}^{I, J} /2$ .

Taking the product of these three types with the seven rows in (eq:Ksum) is the complete branch–vertex inventory. The internal $K_{9, mass}$ and crossed-vertex mass term are distinct and both remain present.

Fix $U ⋐ W$ and $χ \in C_{c}^{\infty} (R^{8})$ equal to one near $f (U)$ . Then (eq:target) is defined at every positive scale.

Displayed formulas in this passage

Em01-D@2#eq:C · line 862
$C_{I}^{μν} (x_{1}, x_{2}) = H_{T, γ}^{μν}_{I} (x_{1}) \otimes H_{T, ϕ} (x_{2}) - H_{γ}^{μν}_{I} (x_{1}) \otimes H_{ϕ} (x_{2}) .$
Em01-D@2#eq:vertex · line 889
$V (K, H)^{J} = \partial_{y^{ν}} [g (y) I \sum K^{I, J} Z_{I}^{ν} (H)] - \frac{m ^{2}}{2} g (y) I \sum K^{I, J} M_{I} (H) .$
Em01-D@2#eq:vertextest · line 898
$⟨ i V (K, H), Φ ⟩ = - i I, J \sum ⟨ K^{I, J}, g (y) [Z_{I}^{ν} (H) \partial_{y^{ν}} Φ_{J} + \frac{m ^{2}}{2} M_{I} (H) Φ_{J}] ⟩ .$
unlabeled · line 874
$f (a, b, z, y) = (y - a, y - b) .$
unlabeled · line 878
$Z_{I}^{ν} (H) = f^{*} (\partial_{2, μ} H_{I}^{μν} - \frac{1}{2} η_{μ λ} \partial_{2}^{ν} H_{I}^{μ λ}), M_{I} (H) = f^{*} (η_{μν} H_{I}^{μν}) .$