Appendix

Following the same steps as in the Appendix B, we can construct diagrammatic series for five order matrix element $T^{k k_1 k_2}_{k_3 k_4}$. We have to combine all third and fourth order vertexes in such a way that we have 2 "incoming" arguments and 3 "out-coming" arguments. The result of course will be the same if we would consider 3 "incoming" arguments and 2 "out-coming" arguments. Considering all possible topologies consistent with definitions of vertexes and Green functions and without internal loops, we conclude, that there exist 60 diagrams constructed from three third order verticesand 2 Green functions and 20 diagrams constructed from one three order vertex, one four order vertex and one Green function. We call these two groups "3+3+3" and "4+3" correspondingly.

Below are the diagrams and analytical expressions for "3+3+3" and "4+3" terms. Together with the bare fifth order vertex $Q^{k_1 k_2 k_3}_{p q}$ this sum gives the full fifth-order interaction matrix element $T^{k_1 k_2 k_3}_{p q}$ or $T_{k_1 k_2 k_3}^{p q}$. We used these expressions as an input to Matematica, therefore notation here is slightly different.