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 Σ(k,ω) (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 dΣ(k,ω)/dk diverges logarithmically when ω 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].)