Figure 1: Examples of one-loop (first-order), two-loop (second-order), and three-loop (third-order) Feynman diagrams used to calculate the fermionic self-energy $\mathrm{\Sigma }(k,\omega )$ (and thus the fermion lifetime) for a model of fermions interacting with a massless bosonic field. Solid lines represent fermions, dashed lines are bosons ($p$ and $l$ are fermionic and bosonic momenta, respectively). The loop expansion is expressed in powers of $1/N$, where $N$ is the number of fermionic flavors, artificially extended from $N=1$ to $N>>1$. By a naive power counting, the three-loop self-energy at zero temperature should scale as $1/N^3$. In reality, some of three-loop diagrams (including the one shown) do not contain $1/N$ in the prefactor. Moreover, for this diagram $d\mathrm{\Sigma }(k,\omega )/dk$ diverges logarithmically when $\omega$ and $k-k_F$ vanish. These diagrams contain only backscattering and forward scattering and represent hidden 1D processes which, as it turned out, play a crucial role in the behavior of 2D systems. (Figure adapted from [2].)