Manuscript II-B@1 · II-B-v1.tex · section · bfd2eb3c

Ward reduction and the target function

The frozen response problem › Ward reduction and the target function · lines 610–683

II-B@1#eq:ward3, II-B@1#eq:ward4

For a gauge parameter $ζ$ , let

G_{ζ} = \partial ζ + (\partial ζ)^{T},

(R_{ζ} h)_{μν} = ζ^{ρ} \partial_{ρ} h_{μν} + h_{ρ ν} \partial_{μ} ζ^{ρ} + h_{μ ρ} \partial_{ν} ζ^{ρ} .

The variation $δ h = G_{ζ} + κ R_{ζ} h$ is the scaled Lie derivative of the metric. Invariance of the integrated Einstein density modulo a divergence gives, after polarization,

v_{3} (G_{ζ}, u, v) v_{4} (G_{ζ}, u, v, w) = - ⟨ K_{0} u, R_{ζ} v ⟩ - ⟨ K_{0} v, R_{ζ} u ⟩, = - v_{3} (R_{ζ} u, v, w) - v_{3} (u, R_{ζ} v, w) - v_{3} (u, v, R_{ζ} w) .

The pairings mean integrated coefficients or stripped conserved-momentum coefficients. The identities follow by differentiating $δ_{0} E_{3} + δ_{1} S_{2} = 0$ and $δ_{0} E_{4} + δ_{1} E_{3} = 0$ , respectively.

Define the cubic current by

⟨ J (v, w), z ⟩ = v_{3} (z, v, w) .

For free $v, w$ , (eq:ward3) with arbitrary $ζ$ implies conservation of $J (v, w)$ . Lemma lem:inverse therefore applies at the physical end of each exchange.

The stripped tree prescription is

T_{4} = V_{4} - {ij ∣ k l} \in {12∣34, 13∣24, 14∣23} \sum V_{3} (i, j, \cdot) D (k_{i} + k_{j}) V_{3} (\cdot, k, l) .

The common momentum delta, $λ^{2}$ , and overall Feynman factor $i$ are removed.

Take leg one replaced. Put

(R_{1 i})_{μν} = (ξ_{1} \cdot k_{i}) ϵ_{i μν} + k_{1 μ} ξ_{1}^{ρ} ϵ_{i ρ ν} + k_{1 ν} ξ_{1}^{ρ} ϵ_{i μ ρ},

and define

W_{2} = v_{3} (R_{12}, ϵ_{3}, ϵ_{4}), W_{3} = v_{3} (R_{13}, ϵ_{2}, ϵ_{4}), W_{4} = v_{3} (R_{14}, ϵ_{2}, ϵ_{3}) .

The momentum of $R_{1 i}$ is $k_{1} + k_{i}$ ; that insertion is generally off shell.

Write

D_{s} = M \sum d_{J_{1} J_{2} M} d_{J_{3} J_{4} M}, D_{t} = M \sum d_{J_{1} J_{3} M} d_{J_{2} J_{4} M}, D_{u} = M \sum d_{J_{1} J_{4} M} d_{J_{2} J_{3} M},

and $D = (D_{s} + D_{t} + D_{u}) /3$ .

Displayed formulas in this passage

II-B@1#eq:ward3 · line 625
$v_{3} (G_{ζ}, u, v) v_{4} (G_{ζ}, u, v, w) = - ⟨ K_{0} u, R_{ζ} v ⟩ - ⟨ K_{0} v, R_{ζ} u ⟩, = - v_{3} (R_{ζ} u, v, w) - v_{3} (u, R_{ζ} v, w) - v_{3} (u, v, R_{ζ} w) .$
II-B@1#eq:ward4 · line 625
$v_{3} (G_{ζ}, u, v) v_{4} (G_{ζ}, u, v, w) = - ⟨ K_{0} u, R_{ζ} v ⟩ - ⟨ K_{0} v, R_{ζ} u ⟩, = - v_{3} (R_{ζ} u, v, w) - v_{3} (u, R_{ζ} v, w) - v_{3} (u, v, R_{ζ} w) .$
unlabeled · line 613
$G_{ζ} = \partial ζ + (\partial ζ)^{T},$
unlabeled · line 616
$(R_{ζ} h)_{μν} = ζ^{ρ} \partial_{ρ} h_{μν} + h_{ρ ν} \partial_{μ} ζ^{ρ} + h_{μ ρ} \partial_{ν} ζ^{ρ} .$
unlabeled · line 642
$⟨ J (v, w), z ⟩ = v_{3} (z, v, w) .$
unlabeled · line 650
$T_{4} = V_{4} - {ij ∣ k l} \in {12∣34, 13∣24, 14∣23} \sum V_{3} (i, j, \cdot) D (k_{i} + k_{j}) V_{3} (\cdot, k, l) .$
unlabeled · line 659
$(R_{1 i})_{μν} = (ξ_{1} \cdot k_{i}) ϵ_{i μν} + k_{1 μ} ξ_{1}^{ρ} ϵ_{i ρ ν} + k_{1 ν} ξ_{1}^{ρ} ϵ_{i μ ρ},$
unlabeled · line 666
$W_{2} = v_{3} (R_{12}, ϵ_{3}, ϵ_{4}), W_{3} = v_{3} (R_{13}, ϵ_{2}, ϵ_{4}), W_{4} = v_{3} (R_{14}, ϵ_{2}, ϵ_{3}) .$
unlabeled · line 675
$D_{s} = M \sum d_{J_{1} J_{2} M} d_{J_{3} J_{4} M}, D_{t} = M \sum d_{J_{1} J_{3} M} d_{J_{2} J_{4} M}, D_{u} = M \sum d_{J_{1} J_{4} M} d_{J_{2} J_{3} M},$

Ward reduction and the target function

Displayed formulas in this passage

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