A. Zee, Quantum Field Theory in a Nutshell · Problem I.7.3

In QFT, you're building a scattering process order by order in the coupling constant $\lambda$, where each power of $\lambda$ corresponds to one interaction vertex. Since you start with two mesons and end with four, you need to create two additional mesons, which requires exactly two vertices—one at each order $\lambda^1$ and $\lambda^2$. The key insight is that each diagram topology represents a different way the interaction can happen: particles can collide directly, or intermediate virtual particles can be exchanged or created. Draw all topologically distinct ways to connect two initial mesons to four final mesons through two vertices, then read off the amplitude by assigning propagators to internal lines and vertex factors to each interaction point.

In QFT, you're building a scattering process order by order in the coupling constant $\lambda$, where each power of $\lambda$ corresponds to one interaction vertex. Since you start with two mesons and end with four, you need to create two additional mesons, which requires exactly two vertices—one at each order $\lambda^1$ and $\lambda^2$. The key insight is that each diagram topology represents a different way the interaction can happen: particles can collide directly, or intermediate virtual particles can be exchanged or created. Draw all topologically distinct ways to connect two initial mesons to four final mesons through two vertices, then read off the amplitude by assigning propagators to internal lines and vertex factors to each interaction point.