PyR@TE 2: A Python tool for computing RGEs at two-loop

TL;DR

PyR@TE 2 is an advanced Python tool that automates the derivation of two-loop renormalization group equations for arbitrary gauge theories, incorporating new features like kinetic mixing and improved group theory capabilities.

Contribution

The paper introduces PyR@TE 2, a significantly enhanced version of the tool, with new features such as kinetic mixing, a new group theory module, and the ability to handle complex Lagrangians and representations.

Findings

Successfully derived RGEs for a model with an uncalculated two-loop set (SM+complex triplet).

Implemented two-loop RGEs for scalar and fermion anomalous dimensions.

Enabled exporting RGEs to C++ for faster numerical solutions.

Abstract

Renormalization group equations are an essential tool for the description of theories accross different energy scales. Even though their expressions at two-loop for an arbitrary gauge field theory have been known for more than thirty years, deriving the full set of equations for a given model by hand is very challenging and prone to errors. To tackle this issue, we have introduced in [1] a Python tool called PyR@TE; Python Renormalization group equations @ Two-loop for Everyone. With PyR@TE, it is easy to implement a given Lagrangian and derive the complete set of two-loop RGEs for all the parameters of the theory. In this paper, we present the new version of this code, PyR@TE 2, which brings many new features and in particular it incorporates kinetic mixing when several $\mathrm{U}(1)$ gauge groups are involved. In addition, the group theory part has been greatly improved as we introduced a new Python module dubbed PyLie that deals with all the group theoretical aspects required for the calculation of the RGEs as well as providing very useful model building capabilities. This allows the use of any irreducible representation of the $\mathrm{SU}(n)$, $\mathrm{SO}(2n)$ and $\mathrm{SO(2n+1)}$ groups. % Furthermore, it is now possible to implement terms in the Lagrangian involving fields which can be contracted into gauge singlets in more than one way. As a byproduct, results for a popular model (SM+complex triplet) for which, to our knowledge, the complete set of two-loop RGEs has not been calculated before are presented in this paper. Finally, the two-loop RGEs for the anomalous dimension of the scalar and fermion fields have been implemented as well. It is now possible to export the coupled system of beta functions into a numerical C++ function, leading to a consequent speed up in solving them.