Consider following Lagrangian for $N$ scalar fields $\phi^a, a=1, \ldots, N$ : $$ L=\frac{1}{2} \partial_\mu \phi^a \partial^\mu \phi^a-\frac{1}{2} \mu_0^2 \phi^a \phi^a-\frac{1}{8} \lambda_0\left(\phi^a \phi^a\right)^2. $$ Here the repeated index implies the summation over the index. Write down the propagator and interaction vertex for this model, and write down four-point Feynman diagrams up to one-loop level.
I was reading the solution of this problem. It went like this:
For the below picture, I've never seen any Feynman diagrams like this: For the last two graph, why is $a,b,c$ on the line? I assumed that for letters on the line it must been like momentum $\overrightarrow{p}$, but there's no arrow on the line. So what does all these graphs come from?
