Introducing RGBeta: a Mathematica package for the evaluation of renormalization group $$ \beta $$-functions

Abstract

In completely generic four-dimensional gauge-Yukawa theories, the renormalization group $$ \beta $$ β -functions are known to the 3–2–2 loop order in gauge, Yukawa, and quartic couplings, respectively. It does, however, remain difficult to apply these results to realistic models without the use of dedicated computer tools. We describe a procedure for extracting $$ \beta $$ β -functions using the general results and introduce , a dedicated Mathematica package for extracting the $$ \overline{\text {MS}} $$ MS ¯ $$ \beta $$ β -functions in broad classes of models. The package and example notebooks are available from the GitHub repository.

General gauge-Yukawa-quartic β-functions at 4-3-2-loop order

“…We have made the results of our computation including all TSs and coefficients directly available in the appendix. Appreciating, however, that it requires a lot of effort to utilize this information directly in computations, we have also implemented them directly in the latest update of RGBeta (v1.1.0) [25].…”

“…Recent years have also seen new developments in hitherto neglected aspects of β-functions, namely, the β-functions for the relevant operators [13][14][15], the running of the kinetic-mixing parameters between Abelian gauge fields [16][17][18], and the extension of the 3-loop gauge β-function to semi-simple gauge groups [18,19]. All these developments have been collated in a number of accessible computer tools -SARAH 4 [20,21], PyR@TE 3 [22,23], ARGES [24], and RGBeta [25] -for JHEP01(2022)051 the application of the general formulas to specific theories, which have opened the doors for the wider application of higher-loop general results in the community. The bespoke tool RGE++ [26] has also been made available for fast evaluation of the resulting RG equations.…”

“…However, both computations rely on the β-function parametrization of ref. [18] and the tensor implementation of RGBeta [25], making for a single point of failure. That said, such a failure appears highly unlikely due to the required self-consistency of both computations.…”

Four-Loop Gauge and Three-Loop Yukawa Beta-Functions in a General Renormalizable Theory

“…The U (1) charges satisfy Q h + Q ψ = Q Ψ , and the sums run over i = 1, 2. This choice of models is also motivated by the fact that we can easily implement them both in RGBeta [33] and DIANA [35]. We use the former to obtain the beta functions in terms of unknown coefficients, while the latter allows us to utilize our standard setup [13,17] and compute required two-and three-point functions with FORCER [36].…”

“…[25] as a guide, but it is fair to say that we found the required models by trial and error. We use TSs implemented in the prominent code RGBeta [33] to speed up our investigation.…”