General formula for symmetry factors of Feynman diagrams

TL;DR

This paper derives a universal formula for symmetry factors of Feynman diagrams involving high-spin fields, showing they are independent of spin and simplifying calculations across various quantum field theories.

Contribution

It provides a general formula for symmetry factors applicable to high-spin fields, extending scalar theory results to more complex theories.

Findings

Symmetry factors do not depend on field spins.

S-factors of diagrams with three different fields are unity.

Inverse S-factors of combined diagrams sum over individual inverse S-factors.

Abstract

General formula for symmetry factors (S-factor) of Feynman diagrams containing fields with high spins is derived. We prove that symmetry factors of Feynman diagrams of well-known theories do not depend on spins of fields. In contributions to S-factors, self-conjugate fields and non self-conjugate fields play the same roles as real scalar fields and complex scalar fields, respectively. Thus, the formula of S-factors for scalar theories --- theories include only real and complex scalar fields --- works on all well-known theories of fields with high spins.Two interesting consequences deduced from our result are : (i) S-factors of all external connected diagrams consisting of only vertices with three different fields, e.g., spinor QED, are equal to unity; (ii) some diagrams with different topologies can contribute the same factor, leading to the result that the inverse S-factor for the total contribution is the sum of inverse S-factors, i.e., 1/S = \sum_i (1/S_i).