Quantum Field Theory

The denominators in each term are due to the meson propagator, with the momentumdetermined by conservation at each vertex. This agrees with the amplitude wecomputed earlier using Wick’s theorem.Nucleon to Meson ScatteringLet’s now look at ψ ¯ψ → φφ. The two lowest order Feynman diagrams are shown inFigure 26.q,rp,s/p,s/q,r/ /+q,r/ // /q,rp,sp,sFigure 26: The two Feynman diagrams for nucleon to meson scatteringApplying the Feynman rules, we have(¯v r (⃗q)[γ µ (pA = (−iλ) 2 µ − p ′ µ ) + m]us (⃗p)+ ¯vr (⃗q)[γ µ (p µ − q µ ′ ) + )m]us (⃗p)(p − p ′ ) 2 − m 2 (p − q ′ ) 2 − m 2Since the internal line is now a fermion, the propagator contains γ µ (p µ −p ′ µ )+m factors.This is a 4 × 4 matrix which sits on the top, sandwiched between the two externalspinors. Now the exchange statistics applies to the final meson states. These arebosons and, correspondingly, there is no relative minus sign between the two diagrams.Nucleon-Anti-Nucleon ScatteringFor ψ ¯ψ → ψ ¯ψ, the two lowest order Feynman diagrams are of two distinct types, justlike in the bosonic case. They are shown in Figure 27. The corresponding amplitudeis given by,()A = (−iλ) 2 (⃗p ′ ) · u s (⃗p)] [¯v r (⃗q) · v− [ūs′ r′ (⃗q ′ )]+ [¯vr (⃗q) · u s (⃗p)] [ū s′ (⃗p ′ ) · v r′ (⃗q ′ )](5.46)(p − p ′ ) 2 − µ 2 (p + q) 2 − µ 2 + iǫAs in the bosonic diagrams, there is again the difference in the momentum dependencein the denominator. But now the difference in the diagrams is also reflected in thespinor contractions in the numerator.– 119 –