Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Supported assembly and its leading coefficient

Supported assembly and its leading coefficient · lines 1256–1332

Em01-G@5#eq:combined-test, Em01-G@5#eq:supported-amplitudes, Em01-G@5#eq:exact-supported, Em01-G@5#eq:supported-coefficients

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At small scales the exact combined test is

U_{δ}^{I, J} [Φ] = g (y) [(e^{I})^{μν} f^{*} (\partial_{2, μ} d_{δ}^{ρ}) \partial_{y^{ν}} Φ_{J} - \frac{m ^{2}}{2} ϑ_{I} f^{*} d_{δ}^{ρ} Φ_{J}] .

The crossed factor is independent of $r$ . At fixed characteristic tangential coordinates, $\partial_{u} = - k^{μ} \partial_{2, μ}$ on that factor. The normal derivatives commute with Cartesian output derivatives.

Let

w (ϑ; v, ξ) = (a, a + β ℓ, a, a + α ℓ + v k + ξ E) .

For $0 \leq b \leq j \leq 2$ , use the amplitudes

A_{ζ, j, b, I, J, μ, ν}^{K} A_{ζ, j, b, I, J}^{M} = (- 1)^{∣ ζ ∣ + j} (b j) c_{ζ, j}^{I, J} (e^{I})^{μν} g (y) (\partial_{r}^{ζ} \partial_{u}^{j - b} \partial_{y^{ν}} Φ_{J}) (w), = (- 1)^{∣ ζ ∣ + j} (b j) c_{ζ, j}^{I, J} (- \frac{m ^{2}}{2} ϑ_{I}) g (y) (\partial_{r}^{ζ} \partial_{u}^{j - b} Φ_{J}) (w),

and profiles

ν_{b, μ}^{K} = (- k \cdot \nabla_{H_{2}})^{b} \partial_{H_{2}^{μ}} ρ, q_{K} = b + 1, ν_{b}^{M} = (- k \cdot \nabla_{H_{2}})^{b} ρ, q_{M} = b .

For the resulting finite list $L$ , the exact expression is

t_{δ, supp}^{app} (Φ) = λ^{3} ℏ^{2} σ \in L \sum \int δ^{- q_{σ}} J_{δ}^{ν_{σ}} [A_{σ}] d ϑ, q_{σ} \leq 3.

Both root rows and both vertex terms are included.

The coefficients are

A_{- p}^{supp} (Φ) A_{l o g}^{supp} (Φ) T_{supp} (Φ) = σ \in L q_{σ} \geq p \sum \int v_{q_{σ} - p}^{ν_{σ}} [A_{σ}] d ϑ, p = 1, 2, 3, = σ \in L \sum \int u_{q_{σ}}^{ν_{σ}} [A_{σ}] d ϑ, = σ \in L \sum \int v_{q_{σ}}^{ν_{σ}} [A_{σ}] d ϑ .

Displayed formulas in this passage

Em01-G@5#eq:combined-test · line 1260
$U_{δ}^{I, J} [Φ] = g (y) [(e^{I})^{μν} f^{*} (\partial_{2, μ} d_{δ}^{ρ}) \partial_{y^{ν}} Φ_{J} - \frac{m ^{2}}{2} ϑ_{I} f^{*} d_{δ}^{ρ} Φ_{J}] .$
Em01-G@5#eq:supported-amplitudes · line 1280
$A_{ζ, j, b, I, J, μ, ν}^{K} A_{ζ, j, b, I, J}^{M} = (- 1)^{∣ ζ ∣ + j} (b j) c_{ζ, j}^{I, J} (e^{I})^{μν} g (y) (\partial_{r}^{ζ} \partial_{u}^{j - b} \partial_{y^{ν}} Φ_{J}) (w), = (- 1)^{∣ ζ ∣ + j} (b j) c_{ζ, j}^{I, J} (- \frac{m ^{2}}{2} ϑ_{I}) g (y) (\partial_{r}^{ζ} \partial_{u}^{j - b} Φ_{J}) (w),$
Em01-G@5#eq:exact-supported · line 1305
$t_{δ, supp}^{app} (Φ) = λ^{3} ℏ^{2} σ \in L \sum \int δ^{- q_{σ}} J_{δ}^{ν_{σ}} [A_{σ}] d ϑ, q_{σ} \leq 3.$
Em01-G@5#eq:supported-coefficients · line 1317
$A_{- p}^{supp} (Φ) A_{l o g}^{supp} (Φ) T_{supp} (Φ) = σ \in L q_{σ} \geq p \sum \int v_{q_{σ} - p}^{ν_{σ}} [A_{σ}] d ϑ, p = 1, 2, 3, = σ \in L \sum \int u_{q_{σ}}^{ν_{σ}} [A_{σ}] d ϑ, = σ \in L \sum \int v_{q_{σ}}^{ν_{σ}} [A_{σ}] d ϑ .$
unlabeled · line 1275
$w (ϑ; v, ξ) = (a, a + β ℓ, a, a + α ℓ + v k + ξ E) .$
unlabeled · line 1296
$ν_{b, μ}^{K} = (- k \cdot \nabla_{H_{2}})^{b} \partial_{H_{2}^{μ}} ρ, q_{K} = b + 1, ν_{b}^{M} = (- k \cdot \nabla_{H_{2}})^{b} ρ, q_{M} = b .$

Supported assembly and its leading coefficient

Displayed formulas in this passage

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