Quantum Field Theory

• Each internal line gets a factor of the relevant propagator.pifor scalarsp 2 − µ 2 + iǫp i( /p + m)for fermions (5.44)p 2 − m 2 + iǫThe arrows on the fermion lines must flow consistently through the diagram (thisensures fermion number conservation). Note that the fermionic propagator is a4×4 matrix. The matrix indices are contracted at each vertex, either with furtherpropagators, or with external spinors u, ū, v or ¯v.• Impose momentum conservation at each vertex, and integrate over undeterminedloop momenta.• Add extra minus signs for statistics. Some examples will be given below.5.7.1 ExamplesLet’s run through the same examples we did for the scalar Yukawa theory. Firstly, wehaveNucleon ScatteringFor the example we worked out previously, the two lowest order Feynman diagrams areshown in Figure 25. We’ve drawn the second Feynman diagram with the legs crossedq,rp,s/p,s/q,r/ /+q,r/ // /q,rp,sp,sFigure 25: The two Feynman diagrams for nucleon scatteringto emphasize the fact that it picks up a minus sign due to statistics. (Note that theway the legs point in the Feynman diagram doesn’t tell us the direction in which theparticles leave the scattering event: the momentum label does that. The two diagramsabove are different because the incoming legs are attached to different outgoing legs).Using the Feynman rules we can read off the amplitude.( )[ūA = (−iλ) 2 s ′ (⃗p ′ ) · u s (⃗p)] [ū r′ (⃗q ′ ) · u r (⃗q)] (⃗p ′ ) · u r (⃗q)] [ū− [ūs′ r′ (⃗q ′ ) · u s (⃗p)](5.45)(p − p ′ ) 2 − µ 2 (p − q ′ ) 2 − µ 2– 118 –