Quantum Field Theory

Notice that the momentum dependence in the secondterm is different from that of nucleon-nucleonscattering (3.56), reflecting the different Feynman diagramthat contributes to the process. In the centerof mass frame, ⃗p 1 = −⃗p 2 , the denominator of the secondterm is 4(M 2 + ⃗p 1) 2 − m 2 . If m < 2M, then thisterm never vanishes and we may drop the iǫ. In contrast,if m > 2M, then the amplitude correspondingto the second diagram diverges at some value of ⃗p.In this case it turns out that we may also neglect theiǫ term, although for a different reason: the mesonis unstable when m > 2M, a result we derived in(3.30). When correctly treated, this instability adds Figure 14:a finite imaginary piece to the denominator whichoverwhelms the iǫ. Nonetheless, the increase in the scattering amplitude which we seein the second diagram when 4(M 2 + ⃗p 2 ) = m 2 is what allows us to discover new particles:they appear as a resonance in the cross section. For example, the Figure 14 showsthe cross-section (roughly the amplitude squared) plotted vertically for e + e − → µ + µ −scattering from the ALEPH experiment in CERN. The horizontal axis shows the centerof mass energy. The curve rises sharply around 91 GeV, the mass of the Z-boson.Meson ScatteringFor φφ → φφ, the simplest diagram we can write/k + p 1pdown has a single loop, and momentum conservation at 1each vertex is no longer sufficient to determine every/k + p 1 − p k1momentum passing through the diagram. We chooseto assign the single undetermined momentum k to thek− p / 2right-hand propagator. All other momenta are then determined.The amplitude corresponding to the diagramshown in the figure isFigure 15:∫ d(−ig) 4 4 k1(2π) 4 (k 2 − M 2 + iǫ)((k + p ′ 1) 2 − M 2 + iǫ)1×((k + p 1 ′ − p 1 ) 2 − M 2 + iǫ)((k − p 2) ′ 2 − M 2 + iǫ)p 2p 1/These integrals can be tricky. For large k, this integral goes as ∫ d 4 k/k 8 , which is atleast convergent as k → ∞. But this won’t always be the case!p / 2– 65 –