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Version 3 of {\tt RunDec} and {\tt CRunDec}
Florian Herren, Matthias Steinhauser
TL;DR
RunDec and CRunDec address the need for precise QCD running, decoupling, and heavy-quark mass transformations across scales, flavour numbers, and schemes. Version 3 implements higher-order corrections and new mass-conversion routines, extending running to five loops and decoupling to four loops. The package therefore provides updated procedures for these calculations, with examples of practical use.
Problem
Precise relations are needed because αs and quark-mass values depend on renormalization scale, active flavours, and the heavy-quark mass scheme.
Method
RunDec and CRunDec collect procedures for QCD running, heavy-quark decoupling, and conversions among MS, on-shell, and short-distance mass schemes.
Results
Five-loop running and four-loop decoupling are implemented, alongside four-loop heavy-quark mass relations and threshold-mass conversions.
Takeaways & Limitations
Version 3 provides updated tools for evaluating αs and quark masses at different scales and active-flavour numbers and transforming heavy-quark mass definitions.
Takeaways & Limitations
Non-zero light-quark masses require a changed C++ parameter format incompatible with older versions, and AsRunDec is not maintained because scale-based decoupling extraction can be inconsistent.
Abstract
from arXiv · showhide
We present new versions of the packages {\tt RunDec} and {\tt CRunDec} which can be used for the running and decoupling of the strong coupling constant and quark masses. Furthermore several conversion formulae for heavy quark masses are implemented. The new versions include five-loop corrections of the QCD beta function and four-loop decoupling effects. Furthermore, various relations between the heavy quark mass defined in the $\overline{\rm MS}$ and other short-distance schemes are implemented to next-to-next-to-next-to-leading order. We discuss in detail the improvements and provide several examples which show how {\tt RunDec} and {\tt CRunDec} can be used in frequently occurring situations.
Program summary
RunDec and CRunDec provide procedures for evaluating the QCD coupling and quark masses across scales and flavour numbers, with support for mass-scheme conversions. They use state-of-the-art corrections and are available as Mathematica and C++ programs.
- The programs support running quantities with dependence on the energy scale µ and active-flavour number nf.
- RunDec and CRunDec evaluate the QCD coupling and quark masses using collected formulae and convenient procedures.
- Heavy-quark masses can be converted from the MS scheme into other short-distance schemes or the on-shell definition.
- CRunDec solves the relevant differential equations with an adaptive Runge-Kutta procedure, while RunDec uses Mathematica functions.
- Individual operations generally run below a millisecond, although some cases can take approximately a second.
1. Introduction
RunDec and CRunDec address the need for precise QCD running, decoupling, and heavy-quark mass-scheme transformations when scales and active-flavour numbers change. Version 3 improves these capabilities while retaining accessible Mathematica and C++ interfaces.
- QCD coupling and quark-mass values depend on the renormalization scale µ and the number of active quark flavours nf.
- Precise relations are needed to move αs and mq between renormalization scales and active-flavour numbers used in predictions.
- Heavy-quark masses use on-shell, MS, and other short-distance schemes, making accurate inter-scheme relations important for preserving precision.
- Version 3 implements high-order QCD corrections for running and decoupling αs and mq and for converting heavy-quark masses among schemes.
- RunDec is loaded as a Mathematica package, while CRunDec provides C++ functions accessible through object-based code.
2. New and updated routines in RunDec and CRunDec
Version 3 updates running, decoupling, and heavy-quark mass-conversion routines with higher-order QCD corrections and new threshold-mass relations. It also extends light-quark effects and documents compatibility and uncertainty controls.
- Running and decoupling of αs and mq: Five-loop running and four-loop decoupling are available for αs, mq, and Λ-related routines.The updated functions include combined running-and-decoupling operations usable at five-loop order.
- Quark mass relations: Four-loop MS–on-shell heavy-quark mass relations and conversions to PS, 1S, RS, and RSp threshold masses are implemented.
- Quark mass relations: The unknown non-logarithmic four-loop term is assigned an uncertainty of about 0.2%, with the fdelm argument controlling its variation.The default fdelm value is 1, and fdelm=0.998 produces a 0.2% variation.
- Quark mass relations: Light-quark mass effects in heavy-quark relations are extended from two-loop to three-loop order.The light-quark masses are specified in the MS scheme at their own renormalization scales.
- Quark mass relations: CRunDec requires an array of four mass–scale pairs or a null pointer when non-zero light-quark masses are supplied.This parameter change is incompatible with older versions in that case.
- Quark mass relations: The AsRunDec routine is not maintained because extracting decoupling steps from input scales can produce inconsistencies.The authors recommend AlL2AlH or AlH2AlL instead.
3. Useful examples
The examples demonstrate how RunDec and CRunDec handle high-energy running, threshold decoupling, uncertainty estimation, and conversions involving αs, Λ, and quark masses.
- 3.1.1. αs at high energies: RunDec combines running and decoupling routines to evolve αs between theories with different active-flavour numbers and renormalization scales.The examples include five-loop running, threshold matching, and explicit uncertainty estimates from input, truncation, and scale variation.
- 3.1.1. αs at high energies: Five-loop running produces decoupling-scale-independent behaviour, while the variation of α(5)s(MZ) remains about 0.3%.The examples vary the decoupling scale around the heavy-quark thresholds and compare loop orders.
- 3.1.1. αs at high energies: 0.00030 is the charm-threshold scale uncertainty in α(5)s(MZ) extracted from α(3)s(mτ), dominating the quoted theoretical uncertainty.The result is α(5)s(MZ) = 0.1201 ± 0.0019 ± 0.0003, with the second uncertainty decomposed into truncation, bottom-scale, and charm-scale contributions.
- 3.1.3. Λ and αs: The package supports extracting Λ(nf) from αs and comparing explicit and implicit procedures, whose difference decreases with increasing loop order.The examples also compare two routes for obtaining α(3)s(mτ), finding consistent central values but different truncation uncertainties.
- 3.1.3. Λ and αs: 0.3119 ± 0.0073 ± 0.0040 is obtained for α(3)s(mτ) using one Λ-based procedure, while the alternative gives 0.3125 ± 0.0074 ± 0.0019.The two results are consistent, but their truncation uncertainties differ by a factor of two.
- 3.2. Heavy-quark mass conversions: Three-loop mass corrections exceed two-loop corrections, whereas decoupling heavy quarks and using α(3)s improves the situation.The examples also report heavy-quark mass values, including mMS_t = 163.508 ± 0.095δmPS_t ± 0.043δαs GeV and mb(Mt) = 2670 MeV.
4. Summary
Version 3 of RunDec and CRunDec extends precision and mass-scheme coverage for QCD running, decoupling, and heavy-quark mass conversions. The packages also provide examples illustrating their Mathematica and C++ use.
- Version 3 implements five-loop corrections for running the strong coupling constant and quark masses.
- The packages include four-loop corrections for decoupling heavy quarks from the running.
- Heavy-quark mass relations between the MS and on-shell schemes, and between these and threshold masses, are implemented.
- Light-quark mass effects are included in the MS-on-shell relation through three loops.
- Examples demonstrate both the Mathematica and C++ versions, and the routines can be modified for inclusion in other codes.