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).

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