Symmetry factors of Feynman diagrams for scalar fields

Abstract

We calculate the symmetry factors of diagrams for real and complex scalar fields in general form using an analysis of the Wick expansion for Green's functions. We separate two classes of symmetry factors: factors corresponding to connected diagrams and factors corresponding to vacuum diagrams. The symmetry factors of vacuum diagrams play an important role in constructing the effective action and phase transitions in cosmology. In the complex scalar field theory, diagrams with different topologies can contribute the same, and the inverse symmetry factor for the total contribution is therefore the sum of the inverse symmetry factors.

General Formula for Symmetry Factors of Feynman Diagrams

“…This conclusion is correct for any theories. Then the method used in [14] can again be used as we will discuss next.…”

“…It will be more clear when we discuss directly on Feynman diagrams. Combining two factors of (12), (14) and the expanding factor (1/p!) in (10) we get a total factor of a particular Feynman diagram appearing in (10):…”

“…Both of them have same contributions but different S-factor values. The S-factor now is related with not only figure 7.a or 7.c but also with both of 7.b and 7.c [14]. For simplicity, we can define a diagram without directions in lines, and call it equivalent diagram.…”

“…This paper is the development of [14] in which we derived the S-factors for Feynman diagrams of theories with scalar fields. The definition of S-factor can be found in [10].…”

“…In Ref. [14] we have concentrated on two types of fields, namely real and complex scalar fields, and have noted that the distinction between these fields is very important because they contribute different factors to the formula of S-factor [14]:…”