Complex scalar field coupled to real scalar field - how are amplitudes non-zero?
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Given a Lagrangian coupling a complex scalar fieldψψ to a real scalar field ϕϕ:
L=12∂μϕ∂μϕ+∂μψ∂μψ∗+12m2ϕ2+12M2ψ2+gϕψψ∗L=12∂μϕ∂μϕ+∂μψ∂μψ∗+12m2ϕ2+12M2ψ2+gϕψψ∗
I'm struggling to see how there can be any non-zero Feynman diagrams for the ψψ→ψψψψ→ψψscattering. (As in figure 9 of Tong's notes)
If we have two internal vertices I understand that we need to calculate the quantity:
<0|T{ψ(x1)ψ(x2)ψ(y1)ψ(y2) ∫ϕψψ∗∫ϕψψ∗}|0><0|T{ψ(x1)ψ(x2)ψ(y1)ψ(y2) ∫ϕψψ∗∫ϕψψ∗}|0>
Now By Wick's theorem this should be given by the sum of all possible contractions. However the contraction of ψψ with ψψ is zero so I can't see anyway to completely contract this?
L=12∂μϕ∂μϕ+∂μψ∂μψ∗+12m2ϕ2+12M2ψ2+gϕψψ∗L=12∂μϕ∂μϕ+∂μψ∂μψ∗+12m2ϕ2+12M2ψ2+gϕψψ∗
I'm struggling to see how there can be any non-zero Feynman diagrams for the ψψ→ψψψψ→ψψscattering. (As in figure 9 of Tong's notes)
If we have two internal vertices I understand that we need to calculate the quantity:
<0|T{ψ(x1)ψ(x2)ψ(y1)ψ(y2) ∫ϕψψ∗∫ϕψψ∗}|0><0|T{ψ(x1)ψ(x2)ψ(y1)ψ(y2) ∫ϕψψ∗∫ϕψψ∗}|0>
Now By Wick's theorem this should be given by the sum of all possible contractions. However the contraction of ψψ with ψψ is zero so I can't see anyway to completely contract this?
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