Conformal Triangles and Zig-Zag Diagrams

We consider the convenient operator representation for so called zig-zag diagrams in the 4-dimensional $\varphi^4$ quantum field theory. The explicit diagonalization of the graph-building operator allows to prove in a very simple way the Broadhurst-Kreimer conjecture about the values of zig-zag multi-loop diagrams which make a significant contribution to the renormalization group $\beta$-function.
The talk is based on our work with A. P. Isaev and L. Shumilov Phys.Lett.B 830 (2022) 137150