Quantum Field Theory

p 1p 1/p 1p 1/p 2p 2/p 2p 2/Figure 10: A contribution at O(g 4 ). Figure 11: A contribution at O(g 6 )AmplitudesOur final result for the nucleon scattering amplitude 〈f|S − 1 |i〉 at order g 2 was[]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)′The δ-function follows from the conservation of 4-momentum which, in turn, followsfrom spacetime translational invariance. It is common to all S-matrix elements. We willdefine the amplitude A fi by stripping off this momentum-conserving delta-function,〈f|S − 1 |i〉 = i A fi (2π) 4 δ (4) (p F − p I ) (3.57)where p I (p F ) is the sum of the initial (final) 4-momenta, and the factor of i out frontis a convention which is there to match non-relativistic quantum mechanics. We cannow refine our Feynman rules to compute the amplitude iA fi itself:• Draw all possible diagrams with appropriate external legs and impose 4-momentumconservation at each vertex.• Write down a factor of (−ig) at each vertex.• For each internal line, write down the propagator• Integrate over momentum k flowing through each loop ∫ d 4 k/(2π) 4 .This last step deserves a short explanation. The diagrams we’ve computed so far haveno loops. They are tree level diagrams. It’s not hard to convince yourself that intree diagrams, momentum conservation at each vertex is sufficient to determine themomentum flowing through each internal line. For diagrams with loops, such as thoseshown in Figures 10 and 11, this is no longer the case.– 63 –