Exact Combinatorics of Bern-Kosower-type Amplitudes for Two-Loop $\Phi^3$ Theory

TL;DR

This paper develops a precise combinatorial framework for Bern-Kosower-type amplitudes in two-loop $\

Contribution

It introduces a method to determine exact combinatorics of world-line formulas for two-loop $\

Findings

Provides a systematic way to count contributions in world-line formulas.

Derives compact world-line representations for the effective action.

Connects string theory amplitudes with Feynman diagram combinatorics.

Abstract

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.