Quantum Field Theory

3.5 Examples of Scattering AmplitudesLet’s apply the Feynman rules to compute the amplitudes for various processes. Westart with something familiar:Nucleon Scattering RevisitedLet’s look at how this works for the ψψ → ψψ scattering at order g 2 . We can writedown the two simplest diagrams contributing to this process. They are shown in Figure9.p 1/p 1k/p 2p 2+p 1/p2kp / 1p 2Figure 9: The two lowest order Feynman diagrams for nucleon scattering.Applying the Feynman rules to these diagrams, we get[]i(−ig) 2 1(p 1 − p 1 ′)2 − m + 1(2π) 4 δ (4) (p 2 (p 1 − p 2 ′)2 − m 2 1 + p 2 − p ′ 1 − p′ 2 ) (3.56)which agrees with the calculation (3.51) that we performed earlier. There is a nicephysical interpretation of these diagrams. We talk, rather loosely, of the nucleonsexchanging a meson which, in the first diagram, has momentum k = (p 1 −p ′ 1) = (p 2 −p ′ 2).This meson doesn’t satisfy the usual energy dispersion relation, because k 2 ≠ m 2 : themeson is called a virtual particle and is said to be off-shell (or, sometimes, off massshell).Heuristically, it can’t live long enough for its energy to be measured to greataccuracy. In contrast, the momentum on the external, nucleon legs all satisfy p 2 = M 2 ,the mass of the nucleon. They are on-shell. One final note: the addition of the twodiagrams above ensures that the particles satisfy Bose statistics.There are also more complicated diagrams which will contribute to the scatteringprocess at higher orders. For example, we have the two diagrams shown in Figures10 and 11, and similar diagrams with p ′ 1 and p′ 2 exchanged. Using the Feynman rules,each of these diagrams translates into an integral that we will not attempt to calculatehere. And so we go on, with increasingly complicated diagrams, all appearing at higherorder in the coupling constant g.– 62 –