  Diagrammatic perturbation theory is a powerful tool for the investigation of
interacting many-body systems, the self-energy operator $\Sigma$ encoding all
the variety of scattering processes. In the simplest scenario of correlated
electrons described by the $GW$ approximation for the electron self-energy, a
particle transfers a part of its energy to neutral excitations. Higher-order
(in screened Coulomb interaction $W$) self-energy diagrams lead to improved
electron spectral functions (SF) by taking more complicated scattering channels
into account and by adding corrections to lower order self-energy terms.
However, they also may lead to unphysical negative spectral function. The
resolution of this difficulty has been demonstrated in our previous works. The
main idea is to represent the self-energy operator in a Fermi Golden rule form
which leads to the manifestly positive definite SF and allows for a very
efficient numerical algorithm. So far, the method has only been applied to 3D
electron gas, which is a paradigmatic system, but a rather simple one. Here, we
systematically extend the method to 2D including realistic systems such as mono
and bilayer graphene. We focus on one of the most important vertex function
effects involving the exchange of two particles in the final state. We
demonstrate that it should be evaluated with the proper screening and discuss
its influence on the quasiparticle properties.
