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

Proposition: Scalar contact correction

Supported output-plateau inverse-quadratic inventory · lines 2518–2575 · proof included or linked

For the supported derivative profiles,

v_{0}^{ν_{1, μ}^{K}} [A] = \frac{ℓ _{μ} M _{2}}{64 π ^{2} β} A (0), v_{0}^{ν_{2}^{M}} [A] = - \frac{M _{2}}{32 π ^{2} β} A (0),

v_{1}^{ν_{2, μ}^{K}} [A] = \frac{1}{32 π ^{2} β} L_{μ} [A] .

Proof. The first two formulas follow from (eq:v0). For the third, at $δ = 0$ ,

\partial_{δ} f_{0, δ, H} = B_{2} \partial_{v} A + Z_{0}^{C} \partial_{ξ^{C}} A,

f_{1, 0, H} = \frac{1}{4} Δ_{ξ} A - \frac{1}{4 d} \partial_{v} A - \frac{m ^{2} β}{4 α} A (0),

where the last term is the massive-tail jet. Also

e^{'}_{0} = E, L^{'}_{0} = κ (- \frac{A _{1}}{α} + \frac{A _{2}}{d} - \frac{E}{β}) S - \frac{α d}{β ^{2}} ∣ Z_{1} - Z_{2} ∣^{2} .

The finite-part terms of $V_{0}^{'} (0) + V_{1} (0)$ have smoothing degree one and vanish against the order-three profile. After deleting the polynomial terms annihilated by that profile, the remaining pairing is

\frac{1}{32 π ^{2} β} \int ν_{2, μ}^{K} [(B_{2} \partial_{v} A + Z_{0}^{C} \partial_{ξ^{C}} A - \frac{E}{β} A (0) + κ S f_{1, 0, H}) lo g (- S + i 0) - \frac{∣ Z _{1} - Z _{2} ∣ ^{2}}{4 β} (S - i 0)^{- 1} A (0)] d H .

Use

\partial_{H_{2}^{μ}} = \frac{k _{μ}}{2} \partial_{A_{2}} + \frac{ℓ _{μ}}{2} \partial_{B_{2}} - (E_{C})_{μ} \partial_{Z_{2}^{C}}, ν_{2, μ}^{K} = \partial_{B_{2}}^{2} \partial_{H_{2}^{μ}} ρ .

Three integrations by parts evaluate the pairing as (eq:contact-operator).

Displayed formulas in this passage

unlabeled · line 2520
$v_{0}^{ν_{1, μ}^{K}} [A] = \frac{ℓ _{μ} M _{2}}{64 π ^{2} β} A (0), v_{0}^{ν_{2}^{M}} [A] = - \frac{M _{2}}{32 π ^{2} β} A (0),$
unlabeled · line 2526
$v_{1}^{ν_{2, μ}^{K}} [A] = \frac{1}{32 π ^{2} β} L_{μ} [A] .$
unlabeled · line 2535
$\partial_{δ} f_{0, δ, H} = B_{2} \partial_{v} A + Z_{0}^{C} \partial_{ξ^{C}} A,$
unlabeled · line 2539
$f_{1, 0, H} = \frac{1}{4} Δ_{ξ} A - \frac{1}{4 d} \partial_{v} A - \frac{m ^{2} β}{4 α} A (0),$
unlabeled · line 2546
$e^{'}_{0} = E, L^{'}_{0} = κ (- \frac{A _{1}}{α} + \frac{A _{2}}{d} - \frac{E}{β}) S - \frac{α d}{β ^{2}} ∣ Z_{1} - Z_{2} ∣^{2} .$
unlabeled · line 2556
$\frac{1}{32 π ^{2} β} \int ν_{2, μ}^{K} [(B_{2} \partial_{v} A + Z_{0}^{C} \partial_{ξ^{C}} A - \frac{E}{β} A (0) + κ S f_{1, 0, H}) lo g (- S + i 0) - \frac{∣ Z _{1} - Z _{2} ∣ ^{2}}{4 β} (S - i 0)^{- 1} A (0)] d H .$
unlabeled · line 2564
$\partial_{H_{2}^{μ}} = \frac{k _{μ}}{2} \partial_{A_{2}} + \frac{ℓ _{μ}}{2} \partial_{B_{2}} - (E_{C})_{μ} \partial_{Z_{2}^{C}}, ν_{2, μ}^{K} = \partial_{B_{2}}^{2} \partial_{H_{2}^{μ}} ρ .$

Proposition: Scalar contact correction

Displayed formulas in this passage

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