According to (4.8.15) in Section 4.8, the scattering rate for fermions is proportional to 1/2 (U_D − U_E)², where UD is the direct interaction vertex and U_E is the exchange vertex. In the diagram shown in Figure 8.32, only the direct term proportional to U²_D occurs. Show that if you account for exchange (how many additional diagrams are needed?), you get agreement of the Matusbara imaginary self energy with the out-scattering rate of (4.8.15).

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(Nk1 Nk₂ (1±Nk)(1 ± Nk) - NkNk (1 ± Nk₁)(1+Nk₂)) ~ (Nk₁) (Nk₂)(1+ (Nk)(1 ± (Nk)) - (Nk) (Nk) (1 ± (Nk₁))(1+ (Nk₂)). (4.8.15)