Manuscript II@2 · II.tex · section · 50d64def

Channel reduction

Ward reduction and coupling factorization › Channel reduction · lines 812–844

II@2#eq:R

Take leg one replaced. For $ζ = - i ξ_{1} e^{i k_{1} \cdot x}$ , define

(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} .

Its momentum is $k_{1} + k_{i}$ . Put

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 composite 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 $D = (D_{s} + D_{t} + D_{u}) /3$ . If $X_{i}$ denotes the unweighted exchange contraction before its tree minus sign, the preceding identity gives

X_{i} = - W_{i} .

Displayed formulas in this passage

II@2#eq:R · line 816
$(R_{1 i})_{μν} = (ξ_{1} \cdot k_{i}) ϵ_{i μν} + k_{1 μ} ξ_{1}^{ρ} ϵ_{i ρ ν} + k_{1 ν} ξ_{1}^{ρ} ϵ_{i μ ρ} .$
unlabeled · line 825
$W_{2} = v_{3} (R_{12}, ϵ_{3}, ϵ_{4}), W_{3} = v_{3} (R_{13}, ϵ_{2}, ϵ_{4}), W_{4} = v_{3} (R_{14}, ϵ_{2}, ϵ_{3}) .$
unlabeled · line 833
$D_{s} = M \sum d_{J_{1} J_{2} M} d_{J_{3} J_{4} M}, D_{t} = M \sum d_{J_{1} J_{3} M} d_{J_{2} J_{4} M}, D_{u} = M \sum d_{J_{1} J_{4} M} d_{J_{2} J_{3} M},$
unlabeled · line 841
$X_{i} = - W_{i} .$

Channel reduction

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