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RunDec: a Mathematica package for running and decoupling of the strong coupling and quark masses

K. G. Chetyrkin, J. H. Kuehn, M. Steinhauser

TL;DR

This paper compiles and implements the full set of high-precision formulae needed to run the QCD coupling and light-quark masses across energy scales, including heavy-quark threshold decoupling. It delivers a Mathematica package, RunDec, that provides explicit procedures for running $\alpha_s^{(n_f)}(\mu)$ and $m_q^{(n_f)}(\mu)$, with MS-bar and on-shell mass definitions, as well as scheme conversions and decoupling relations up to four-loop running and three-loop decoupling. The work emphasizes consistent threshold matching, presents detailed decoupling constants $\zeta_g$ and $\zeta_m$ across MS, SI, and OS schemes, and demonstrates stability improvements when higher-order terms are included. The result is a practical, high-precision tool for QCD phenomenology across a wide range of energies, including lattice-QCD contexts via RI-mass definitions and explicit automation of parameter evolution.

Abstract

In this paper the formulae are collected which are needed for the computation of the strong coupling constant and quark masses at different energy scales and for different number of active flavours. All equations contain the state-of-the-art QCD corrections up to three- and sometimes even four-loop order. For the practical implementation {\tt Mathematica} is used and a package containing useful procedures is provided.

RunDec: a Mathematica package for running and decoupling of the strong coupling and quark masses

TL;DR

This paper compiles and implements the full set of high-precision formulae needed to run the QCD coupling and light-quark masses across energy scales, including heavy-quark threshold decoupling. It delivers a Mathematica package, RunDec, that provides explicit procedures for running and , with MS-bar and on-shell mass definitions, as well as scheme conversions and decoupling relations up to four-loop running and three-loop decoupling. The work emphasizes consistent threshold matching, presents detailed decoupling constants and across MS, SI, and OS schemes, and demonstrates stability improvements when higher-order terms are included. The result is a practical, high-precision tool for QCD phenomenology across a wide range of energies, including lattice-QCD contexts via RI-mass definitions and explicit automation of parameter evolution.

Abstract

In this paper the formulae are collected which are needed for the computation of the strong coupling constant and quark masses at different energy scales and for different number of active flavours. All equations contain the state-of-the-art QCD corrections up to three- and sometimes even four-loop order. For the practical implementation {\tt Mathematica} is used and a package containing useful procedures is provided.

Paper Structure

This paper contains 5 sections, 31 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Running strong coupling constant
  3. Quark masses in the $\overline{\rm MS}$ and on-shell scheme
  4. Decoupling at flavour thresholds
  5. Description of the main procedures

Figures (2)

  • Figure 1: $\mu^{(5)}$ dependence of $\alpha_s^{(5)}(M_Z)$ calculated from $\alpha_s^{(4)}(M_\tau)=0.36$ and $M_b=4.7$ GeV using Eq. (\ref{['eq:defbeta']}) at one (dotted), two (dashed), three (dot-dashed), and four (solid) loops in connection with Eq. (\ref{['eq:invzetagOS']}) at the respective orders.
  • Figure 2: $\mu^{(5)}$ dependence of $m_c^{(5)}(M_Z)$ calculated from $\mu_c=m_c^{(4)}(\mu_c)=1.2$ GeV, $M_b=4.7$ GeV and $\alpha_s^{(5)}(M_Z)$ using Eq. (\ref{['eq:defgamma']}) at one (dotted), two (dashed), three (dot-dashed), and four (solid) loops in connection with Eq. (\ref{['eq:invzetamOS']}) at the respective orders.