Symmetry Factors of Feynman Diagrams and the Homological Perturbation Lemma

TL;DR

This paper derives a general formula for symmetry factors of Feynman diagrams in scalar field theories and proves their correct inclusion in scattering amplitude calculations using the homological perturbation lemma.

Contribution

It provides a concise formula for symmetry factors and a direct proof of their correct application in homological perturbation-based amplitude computations.

Findings

Derived a general formula for symmetry factors

Proved correct inclusion of diagrams with symmetry factors in homological perturbation

Clarified the role of symmetry factors in scalar field theory diagrams

Abstract

We discuss the symmetry factors of Feynman diagrams of scalar field theories with polynomial potential. After giving a concise general formula for them, we present an elementary and direct proof that when computing scattering amplitudes using the homological perturbation lemma, each contributing Feynman diagram is indeed included with the correct symmetry factor.