%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
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\begin{document}
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{\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}
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\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}
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\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.
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\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[1]:= <