Following the same steps as in the Appendix B, we can construct diagrammatic series for five order matrix element . 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
this sum gives the full fifth-order
interaction matrix element
or
.
We used these expressions as an input to Matematica, therefore
notation here is slightly different.
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