Compton Scattering Sum Rules for Massive Vector Bosons

2.5 Feynman Rules for L Eff
iel 2 〈W α |W ∗ ρτW σ ∂ σ F ρτ∣ ∣W β A µ 〉 (2.32)
= iel 2
((
ip
′
ρ δ α τ − ip ′ τ δ α ρ
)
δ
β
σ (−iq σ ) (−iq ρ g τµ + iq τ g ρµ ) ) 〈0|0〉
= 2el 2
(
p′ · q g αµ − p ′µ q α) q β ,
−iel 2 〈W α |Wσ ∗ W ρτ ∂ σ F ρτ∣ ∣ W β A µ 〉 (2.33)
( ( )
= −iel 2 δ
α
σ −ipρ δτ β + ip τ δρ
β (−iq σ ) (−iq ρ g τµ + iq τ g ρµ ) )
(
= 2el 2 p · q g βµ − p µ q β) q α .
So, put together, this results in the 3-vertex Feynman rule
)
⇒ Γ αβµ (p, p ′ ) = − e
(g αβ P µ − p ′β g αµ − p α g βµ
(2.34)
+ ( q β g αµ − q α g βµ) l 1
− 2 ( q α q β P µ − p · qq α g βµ − p ′ · qq β g αµ) l 2 ,
where P µ = p µ + p ′µ .
γγWW–Vertex
Γ αβµν (p ′ , p) := 〈W α (p ′ ), A ν (k)|L Eff |W β (p)A µ (q)〉 (2.35)
First, we focus on the direct interaction parts. Note that we have to consider crossing,
which means here that the photons can couple to both the initial and the final state.
−e 2 〈W α (p ′ )A µ |A 2 W ∗ τ W τ∣ ∣ W β A ν 〉 = −2e 2 ( g µν g αβ) , (2.36)
e 2 〈W α (p ′ )A µ |A ρ W ∗ τ A τ W ρ∣ ∣ W β A ν 〉 + e 2 〈W α (p ′ )A µ |A ρ W ∗ τ A τ W ρ∣ ∣ W β A ν 〉 (2.37)
= e 2 ( g αµ g βν + g αν g βµ) ,
e 2 l 2
1
2 〈W α A µ |T ∗ ρτW σ ∂ σ F ρτ∣ ∣ W β A ν 〉 + e 2 l 2
1
2 〈W α A µ |T ∗ ρτW σ ∂ σ F ρτ∣ ∣ W β A ν 〉 (2.38)
= −e 2 l 2
(
2q α q β g µν − q β q µ g αν − q β q ν g αµ) ,
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