Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

The exact 42-entry audit expansion

The exact 42-entry audit expansion · lines 2491–2555

Em01-F@3#eq:vertex, Em01-F@3#eq:Wk, Em01-F@3#eq:Wt, Em01-F@3#eq:Wmass, Em01-F@3#eq:ledgerentry

Write $C = C_{+} - C_{-}$ , where

C_{+}^{μν}_{I} = ℏ^{2} N^{μν}_{I} (D_{T, 0} \otimes D_{T, m}), C_{-}^{μν}_{I} = ℏ^{2} N^{μν}_{I} (W_{0} \otimes W_{m}) .

Set

H_{δ}^{\pm} = ρ_{δ} * (χ C_{\pm}), C_{δ} = H_{δ}^{+} - H_{δ}^{-} .

For a smooth symmetric array $H$ , define

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

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) .

This is the coefficient from the mixed matter derivative after transferring its remaining scalar derivative.

The three compact joint tests are

W_{k}^{I, J} (H, Φ) W_{t}^{I, J} (H, Φ) W_{m}^{I, J} (H, Φ) = - i g (y) f^{*} (\partial_{2, μ} H_{I}^{μν}) \partial_{y^{ν}} Φ_{J}, = \frac{i}{2} g (y) η_{μα} f^{*} (\partial_{2}^{ν} H_{I}^{μα}) \partial_{y^{ν}} Φ_{J}, = - \frac{i m ^{2}}{2} g (y) f^{*} (η_{μν} H_{I}^{μν}) Φ_{J} .

For $ϵ_{+} = 1, ϵ_{-} = - 1$ , put

L_{j, v}^{ε} (δ; Φ) = ϵ_{ε} I, J \sum ⟨ K_{j}^{I, J}, W_{v}^{I, J} (H_{δ}^{ε}, Φ)⟩ .

Displayed formulas in this passage

Em01-F@3#eq:vertex · line 2521
$V (K, H)^{J} = \partial_{y^{ν}} [g (y) I \sum K^{I, J} Z_{I}^{ν} (H)] - \frac{m ^{2}}{2} g (y) I \sum K^{I, J} M_{I} (H) .$
Em01-F@3#eq:Wk · line 2532
$W_{k}^{I, J} (H, Φ) W_{t}^{I, J} (H, Φ) W_{m}^{I, J} (H, Φ) = - i g (y) f^{*} (\partial_{2, μ} H_{I}^{μν}) \partial_{y^{ν}} Φ_{J}, = \frac{i}{2} g (y) η_{μα} f^{*} (\partial_{2}^{ν} H_{I}^{μα}) \partial_{y^{ν}} Φ_{J}, = - \frac{i m ^{2}}{2} g (y) f^{*} (η_{μν} H_{I}^{μν}) Φ_{J} .$
Em01-F@3#eq:Wt · line 2532
$W_{k}^{I, J} (H, Φ) W_{t}^{I, J} (H, Φ) W_{m}^{I, J} (H, Φ) = - i g (y) f^{*} (\partial_{2, μ} H_{I}^{μν}) \partial_{y^{ν}} Φ_{J}, = \frac{i}{2} g (y) η_{μα} f^{*} (\partial_{2}^{ν} H_{I}^{μα}) \partial_{y^{ν}} Φ_{J}, = - \frac{i m ^{2}}{2} g (y) f^{*} (η_{μν} H_{I}^{μν}) Φ_{J} .$
Em01-F@3#eq:Wmass · line 2532
$W_{k}^{I, J} (H, Φ) W_{t}^{I, J} (H, Φ) W_{m}^{I, J} (H, Φ) = - i g (y) f^{*} (\partial_{2, μ} H_{I}^{μν}) \partial_{y^{ν}} Φ_{J}, = \frac{i}{2} g (y) η_{μα} f^{*} (\partial_{2}^{ν} H_{I}^{μα}) \partial_{y^{ν}} Φ_{J}, = - \frac{i m ^{2}}{2} g (y) f^{*} (η_{μν} H_{I}^{μν}) Φ_{J} .$
Em01-F@3#eq:ledgerentry · line 2548
$L_{j, v}^{ε} (δ; Φ) = ϵ_{ε} I, J \sum ⟨ K_{j}^{I, J}, W_{v}^{I, J} (H_{δ}^{ε}, Φ)⟩ .$
unlabeled · line 2495
$C_{+}^{μν}_{I} = ℏ^{2} N^{μν}_{I} (D_{T, 0} \otimes D_{T, m}), C_{-}^{μν}_{I} = ℏ^{2} N^{μν}_{I} (W_{0} \otimes W_{m}) .$
unlabeled · line 2505
$H_{δ}^{\pm} = ρ_{δ} * (χ C_{\pm}), C_{δ} = H_{δ}^{+} - H_{δ}^{-} .$
unlabeled · line 2511
$Z_{I}^{ν} (H) = f^{*} (\partial_{2, μ} H_{I}^{μν} - \frac{1}{2} η_{μα} \partial_{2}^{ν} H_{I}^{μα}), M_{I} (H) = f^{*} (η_{μν} H_{I}^{μν}),$

The exact 42-entry audit expansion

Displayed formulas in this passage

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