Symmetry Factors of Feynman Diagrams for Scalar Fields

TL;DR

This paper derives a general formula for the symmetry factors of Feynman diagrams in scalar field theories, highlighting their role in quantum field calculations and cosmological phase transitions.

Contribution

It provides a comprehensive analysis and explicit expressions for symmetry factors, including for vacuum diagrams, in scalar quantum field theories.

Findings

Derived a general form for symmetry factors of Feynman diagrams.

Separated symmetry factors into connected and vacuum diagram components.

Highlighted the importance of vacuum diagrams in cosmological phase transitions.

Abstract

The symmetry factor of Feynman diagrams for real and complex scalar fields is presented. Being analysis of Wick expansion for Green functions, the mentioned factor is derived in a general form. The symmetry factor can be separated into two ones corresponding to that of connected and vacuum diagrams. The determination of symmetry factors for the vacuum diagrams is necessary as they play a role in the effective action and phase transitions in cosmology. In the complex scalar theory the diagrams different in topology may give the same contribution, hence inverse of the symmetry factor (1/S) for total contribution is a summation of each similar ones (1/S_i), i.e., 1/S = \sum_i (1/S_i).