Two interacting scalar fields: practical renormalization

TL;DR

This paper provides a comprehensive analysis of the renormalization process for a quantum field theory with two interacting scalar fields, including one-loop and two-loop calculations and the derivation of renormalization group equations.

Contribution

It offers a detailed derivation of Feynman rules, renormalization procedures, and RG equations for a symmetric two-scalar-field model at one and two loops, utilizing computational tools.

Findings

One-loop renormalization of the two-field model is explicitly derived.

Two-loop diagrams, including the setting sun, are calculated using FeynArts.

Renormalization group equations for masses and couplings are obtained at one loop.

Abstract

The main theme of the paper is the detailed discussion of the renormalization of the quantum field theory comprising two interacting scalar fields. The potential of the model is the fourth-order homogeneous polynomial of the fields, symmetric with respect to the transformation $\phi_{i}\rightarrow{-\phi_{i}}$. We determine the Feynman rules for the model and then we present a detailed discussion of the renormalization of the theory at one loop. Next, we derive the one loop renormalization group equations for the running masses and coupling constants. At the level of two loops, we use the FeynArts package of Mathematica to generate the two loops Feynman diagrams and calculate in detail the setting sun diagram.