%Title: RunDec: a Mathematica package for running and decoupling of the strong coupling and quark masses %Author: K.G. Chetyrkin, J.H. K\"uhn, M. Steinhauser %Published: Comput. Phys. Commun. 133 (2000) 43-65. %hep-ph/0004189 %\documentclass[12pt]{article} %\usepackage{a4wide,epsfig} \documentstyle[12pt,epsf,a4]{article} \voffset0cm \hoffset0cm \oddsidemargin0cm \evensidemargin0cm \topmargin0cm \textwidth16.cm \textheight22cm \newcommand{\bv}{\begin{verbatim}} \newcommand{\ev}{\end{verbatim}} \newcommand{\smM}{\mbox{\small{\it M}}} \renewcommand{\thefootnote}{\fnsymbol{footnote}} \setcounter{footnote}{2} \renewcommand{\textfraction}{0} \renewcommand{\topfraction}{1} \renewcommand{\bottomfraction}{1} \newcommand{\note}{{\tiny (note)}\marginpar {\scriptsize #1}} \newcommand{\li}{\mathop{{\mbox{Li}}_4}\nolimits} \newcommand{\gsim}{\;\rlap{\lower 3.5 pt \hbox{$\mathchar \sim$}} \raise 1pt \hbox {$>$}\;} \newcommand{\lsim}{\;\rlap{\lower 3.5 pt \hbox{$\mathchar \sim$}} \raise 1pt \hbox {$<$}\;} \newcommand{\qsla}{q\hspace{-.5em/\hspace{.5em}}} \newcommand{\lnmum}{l_{\mu M}} \newcommand{\lmM}{l_{\mu M}} \sloppy %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{\vskip-3cm{\baselineskip14pt \centerline{\normalsize\hfill DESY 00--034} \centerline{\normalsize\hfill TTP00--05} \centerline{\normalsize\hfill hep-ph/0004189} \centerline{\normalsize\hfill March 2000} } \vskip.7cm {\tt RunDec}: a {\tt Mathematica} package for running and decoupling of the strong coupling and quark masses \vskip.3cm } \author{ {K.G. Chetyrkin}$^{a,}$\thanks{Permanent address: Institute for Nuclear Research, Russian Academy of Sciences, 60th October Anniversary Prospect 7a, Moscow 117312, Russia.} \,, {J.H. K\"uhn}$^a$ \,and {M. Steinhauser}$^b$ \\[3em] { (a) Institut f\"ur Theoretische Teilchenphysik,}\\ { Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany} \\[.5em] { (b) II. Institut f\"ur Theoretische Physik,}\\ { Universit\"at Hamburg, D-22761 Hamburg, Germany} } \date{} \maketitle \begin{abstract} \noindent 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. \vspace{.2cm} \noindent PACS numbers: 12.38.Aw 12.38.-t 14.65.-q \end{abstract} \thispagestyle{empty} \newpage \setcounter{page}{1} \renewcommand{\thefootnote}{\arabic{footnote}} \setcounter{footnote}{0} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Program Summary} \begin{itemize} \item[]{\it Title of program:} {\tt RunDec} \item[]{\it Available from:}\\ {\tt http://www-ttp.physik.uni-karlsruhe.de/Progdata/ttp00/ttp00-05/ } \item[]{\it Computer for which the program is designed and others on which it is operable:} Any work-station or PC where {\tt Mathematica} is running. \item[]{\it Operating system or monitor under which the program has been tested:} UNIX, Mathematica~4.0 \item[]{\it No. of bytes in distributed program including test data etc.:} $65000$ \item[]{\it Distribution format:} ASCII \item[]{\it Keywords:} Quantum Chromodynamics, running coupling constant, running quark mass, on-shell mass, $\overline{\rm MS}$ mass, decoupling of heavy particles \item[]{\it Nature of physical problem:} The values for the coupling constant of Quantum Chromodynamics, $\alpha_s^{(n_f)}(\mu)$, actually depends on the considered energy scale, $\mu$, and the number of active quark flavours, $n_f$. The same applies to light quark masses, $m_q^{(n_f)}(\mu)$, if they are, e.g., evaluated in the $\overline{\rm MS}$ scheme. In the program {\tt RunDec} all relevant formulae are collected and various procedures are provided which allow for a convenient evaluation of $\alpha_s^{(n_f)}(\mu)$ and $m_q^{(n_f)}(\mu)$ using the state-of-the-art correction terms. \item[]{\it Method of solution:} {\tt RunDec} uses {\tt Mathematica} functions to perform the different mathematical operations. \item[]{\it Restrictions on the complexity of the problem:} It could be that for an unphysical choice of the input parameters the results are nonsensical. \item[]{\it Typical running time:} For all operations the running time does not exceed a few seconds. \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} Quantum Chromodynamics (QCD) is nowadays well established as the theory of strong interaction within the Standard Model of elementary particle physics. In recent years there has been a wealth of theoretical results (for a review see~\cite{ckk96}). At the same time perturbative QCD has been extremely successful in describing the experimental data with high precision. The fundamental quantity of QCD is the so-called beta function which connects the value of the strong coupling constant, $\alpha_s(\mu)$, at different energy scales $\mu$. It is thus particularly important to know the beta function as precise as possible. In~\cite{RitVerLar97} the four-loop corrections were evaluated allowing for a consistent running at order $\alpha_s^4$. In the majority of all computations performed in QCD the $\overline{\rm MS}$ renormalization scheme~\cite{BarBurDukMut78} is adopted. In this scheme the Appelquist-Carazzone decoupling theorem~\cite{AppCar75} is not directly applicable. When crossing flavour thresholds, it is thus important to perform the decoupling by hand''. In order to be consistent, four-loop running must go along with the three-loop decoupling relation which was evaluated in~\cite{CheKniSte97}. Similar considerations are also valid for quark masses. Also here the renormalization group function is available up to the four-loop level~\cite{Che97LarRitVer972} and the corresponding decoupling relation up to order $\alpha_s^3$~\cite{CheKniSte98} (see also~\cite{Rod9397}). In this paper all relevant formulae are collected which are necessary for the running and decoupling of $\alpha_s$ and for quark masses. Their proper use is discussed and easy-to-use {\tt Mathematica}~\cite{math} procedures collected in the package {\tt RunDec} are provided. Their handling is described and examples are given. The outline of the paper is as follows. In the next Section the formulae are presented which are needed for the running of the strong coupling constant up to the four-loop level. The corresponding equations for the quark masses are presented in Section~\ref{sec:masses}. In addition the conversion formulae between the $\overline{\rm MS}$ and on-shell scheme are discussed in some detail. Section~\ref{sec:dec} is concerned with the decoupling of the strong coupling and quark masses. Finally, in Section~\ref{sec:desc}, the most important procedures of the package {\tt RunDec} are described in an easy-to-use way. For most practical applications they should be sufficient. In the Appendix the complete collection of procedures is given. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Running strong coupling constant} The beta function governing the running of the coupling constant of QCD is defined through \begin{eqnarray} \mu^2\frac{{\rm d}}{{\rm d}\mu^2} \frac{\alpha_s^{(n_f)}(\mu)}{\pi} &=& \beta^{(n_f)}\left(\alpha_s^{(n_f)}\right) \,\,=\,\, - \sum_{i\ge0} \beta_i^{(n_f)}\left(\frac{\alpha_s^{(n_f)}(\mu)}{\pi}\right)^{i+2} \,, \label{eq:defbeta} \end{eqnarray} where $n_f$ is the number of active flavours. The coefficients are given by~\cite{gro,jon,tar,RitVerLar97} \begin{eqnarray} \beta_0^{(n_f)} &=&\frac{1}{4}\left[ 11 - \frac{2}{3} n_f\right] \,, \nonumber\\ \beta_1^{(n_f)} &=&\frac{1}{16}\left[ 102 - \frac{38}{3} n_f\right] \,, \nonumber \\ \beta_2^{(n_f)} &=&\frac{1}{64}\left[\frac{2857}{2} - \frac{5033}{18} n_f + \frac{325}{54} n_f^2\right] \,, \nonumber \\ \beta_3^{(n_f)} &=&\frac{1}{256}\left[ \frac{149753}{6} + 3564 \zeta_3 + \left(- \frac{1078361}{162} - \frac{6508}{27} \zeta_3 \right) n_f \right.\nonumber\\&&\mbox{} + \left( \frac{50065}{162} + \frac{6472}{81} \zeta_3 \right) n_f^2 +\left. \frac{1093}{729} n_f^3\right] \,. \label{eq:betafct} \end{eqnarray} $\zeta$ is Riemann's zeta function, with values $\zeta_2=\pi^2/6$ and $\zeta_3\approx1.202\,057$. It is convenient to introduce the following notation: \begin{eqnarray} b_i^{(n_f)} &=& \frac{\beta_i^{(n_f)}}{\beta_0^{(n_f)}} \,, \nonumber\\ a^{(n_f)}(\mu) &=& \frac{\alpha^{(n_f)}(\mu)}{\pi} \,. \label{eq:bidef} \end{eqnarray} In the following the labels $\mu$ and $n_f$ are omitted if confusion is impossible. Integrating Eq.~(\ref{eq:defbeta}) leads to \begin{eqnarray} \ln\frac{\mu^2}{\Lambda^2}&=&\int\frac{da}{\beta(a)} %\label{eq:lam_int} \nonumber \\ &=&\frac{1}{\beta_0}\left[\frac{1}{a}+b_1\ln a+(b_2-b_1^2)a + \left(\frac{b_3}{2}-b_1b_2+\frac{b_1^3}{2}\right)a^2\right]+C, \label{eq:lamexpl} \end{eqnarray} where an expansion in $a$ has been performed. The integration constant is conveniently split into $\Lambda$, the so-called asymptotic scale parameter, and $C$. The conventional $\overline{\rm MS}$ definition of $\Lambda$, which we shall adopt in the following, corresponds to choosing $C=(b_1/\beta_0)\ln\beta_0$~\cite{BarBurDukMut78,FurPet82}. Iteratively solving Eq.~(\ref{eq:lamexpl}) yields~\cite{CheKniSte97} \begin{eqnarray} a&=&\frac{1}{\beta_0L}-\frac{b_1\ln L}{(\beta_0L)^2} +\frac{1}{(\beta_0L)^3}\left[b_1^2(\ln^2L-\ln L-1)+b_2\right] \nonumber\\ &&\mbox{}+\frac{1}{(\beta_0L)^4}\left[ b_1^3\left(-\ln^3L+\frac{5}{2}\ln^2L+2\ln L-\frac{1}{2}\right) -3b_1b_2\ln L+\frac{b_3}{2}\right], \label{eq:alp} \end{eqnarray} where $L=\ln(\mu^2/\Lambda^2)$ and terms of ${\cal O}(1/L^5)$ have been neglected. $\Lambda$ is defined in such a way that Eq.~(\ref{eq:alp}) does not contain a term proportional to $({\rm const.}/L^2)$~\cite{BarBurDukMut78}. The canonical way to compute $a(\mu_2)$ when $a(\mu_1)$ is given for a fixed number of flavours is as follows: \begin{enumerate} \item Determine $\Lambda$. There are several possibilities to do this. One could, e.g., use the explicit solution given in Eq.~(\ref{eq:lamexpl}). Another possibility is the use of (\ref{eq:alp}) and solve the equation iteratively for $\Lambda$. Furthermore the first line of~(\ref{eq:lamexpl}) could be used and the integral could be solved numerically without performing any expansion in $\alpha_s$. We will see in the examples below that the numerical differences are small. \item $a(\mu_2)$ is computed with the help of Eq.~(\ref{eq:alp}) where the value of $\Lambda$ is inserted and $\mu$ is set to $\mu_2$. \end{enumerate} It is also possible to avoid the introduction of $\Lambda$ in intermediate steps and to solve the differential equation~(\ref{eq:defbeta}) numerically using $a(\mu)|_{\mu=\mu_1}=a(\mu_1)$ as initial condition. This convention requires the knowledge of both $\alpha_s$ and the scale $\mu$ in order to determine $\alpha_s$ at the new scale. Frequently, $\mu = M_Z$ is used as reference scale. On the other hand $\Lambda$ plays the role of an universal parameter which at the same time sets the characteristic scale of QCD. At this point it is instructive to consider an example. Let us assume that $\alpha_s$ is given at the $Z$-boson scale: $\alpha_s^{(5)}(M_Z)=0.118$. Let us further assume that it is determined from the experiment with three-loop accuracy, which means that in the $\beta$ function~(\ref{eq:betafct}) only the coefficients up to $\beta_2$ are considered and $\beta_3$ is neglected. Let us now evaluate the strong coupling at the scale $\mu=M_b$ and compare the results obtained with the different strategies outlined above. In the following a possible {\tt Mathematica} session is shown. {\tt NumDef} is a set of {\tt Mathematica} rules which assigns typical values to the physical parameters used in our procedures. The numbers used in this paper can be found in Eq.~(\ref{eq:numdef}) and the procedures are described in the Appendix. \begin{verbatim} In:= <