Exact Combinatorics of Bern-Kosower-type Amplitudes for Two-Loop   $\Phi^3$ Theory
  Counting the contribution rate of a world-line formula to Feynman diagrams in $\phi^3$ theory, we explain the idea how to determine precise combinatorics of Bern-Kosower-like amplitudes derived from a bosonic string theory for $N$-point two-loop Feynman amplitudes. In this connection we also present a method to derive simple and compact world-line forms for the effective action. 
Haru-Tada Sato and Michael G. Schmidt