My research hit a reef. We don't know what to do next to obtain the right answer. We've been on a calculation of the vertex correction these days, and today found that it diverges. This seems particular to the Dirac dispersion.
The physical meaning of the vertex correction is to recover the U(1) gauge invariance, which corresponds to the conservation of the charge, therefore in a brief consideration, it can't lead to the divergence.
No one around me knows the answer.
