%Title: Matching Coefficients for the Strong Coupling in the Standard Model and MSSM 
%Author: Matthias Steinhauser
%Published: <EM> Contribution to Loops and Legs 2006, Eisenach, Germany, 23-28 April 2006 </EM> <B>  </B> () . 
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\title{Matching Coefficients for the Strong Coupling in the Standard
       Model and MSSM
\mbox{}\hfill {\small [TTP06-20]}
}

\author{Matthias Steinhauser\address{Institut f{\"u}r
    Theoretische Teilchenphysik,
    Universit{\"a}t Karlsruhe, 76128 Karlsruhe, Germany}}
       
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\runtitle{Matching Coefficients in the SM and MSSM}
\runauthor{M. Steinhauser}

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\begin{document}

%- {{{ abstract:

\begin{abstract}
In this contribution we review the status of the decoupling
coefficients for $\alpha_s$ both in the Standard Model (SM) and its
minimal supersymmetric extension, the so-called MSSM. We stress the
importance of a consistent treatment, in particular within the
framework of the MSSM. 
\vspace{1pc}
\end{abstract}

\maketitle

%- }}}
%- {{{ Motivation:

\section{Motivation}

The strong coupling constant, $\alpha_s$, constitutes a fundamental 
parameter in
the Standard Model (SM) and thus its precise numerical value is very important 
for many physical predictions. An interesting property of $\alpha_s$
is its scale dependence, in particular its strong rise for low
and its small value for high energies which make perturbative
calculations within the framework of QCD possible.
The scale dependence is governed by the $\beta$ function. However, in
order to relate $\alpha_s$ at two different scales it is also necessary 
to incorporate threshold effects of heavy quarks which is achieved with
the help of the so-called matching or decoupling relations.
Thus, when specifying $\alpha_s$ it is necessary to indicate next to the 
scale also the number of active flavours.

A consistent treatment of the running and decoupling procedure
requires the use of $(N-1)$-loop decoupling formulae together with 
$N$-loop expressions for the $\beta$ function.
As far as QCD is concerned, recently the four-loop matching relations
have been computed by two independent
groups~\cite{Schroder:2005hy,Chetyrkin:2005ia}. Formally this would
require the knowledge of the five-loop $\beta$ function, however,
currently only the four-loop terms
are known~\cite{vanRitbergen:1997va,Czakon:2004bu}.

The situation is different in the context of supersymmetry. In the
limit where the number of active fermions and sfermions is equal 
the $\beta$ function up to three loops is available since already
quite some time~\cite{Jack:1996vg}, however, until recently only the
one-loop matching conditions have been available in the literature.
This gap has been closed in Ref.~\cite{Harlander:2005wm} where
two-loop decoupling relations in the framework of the MSSM have been
studied.

In this contribution we provide a brief overview on the current
status, both for QCD, which is treated in Section~\ref{sec::qcd},
and SUSY-QCD, which can be found in Section~\ref{sec::mssm}.
%Section~\ref{sec::concl} contains the conclusions.

%- }}}
%- {{{ QCD:

\section{\label{sec::qcd}Decoupling in QCD}

A typical problem where the question of decoupling plays a crucial
role occurs when two values of $\alpha_s$, defined for different
number of active flavours, shall be related to each other. E.g., consider
the case where $\alpha_s^{(5)}(M_Z)$ shall be computed from the 
knowledge of $\alpha_s^{(4)}(M_\tau)$. In this example 
the decoupling constant has to provide a relation between
$\alpha_s^{(4)}(\mu_b)$ and $\alpha_s^{(5)}(\mu_b)$ where $\mu_b$ is
the matching scale where the bottom quark is decoupled.

In a more general way we define the decoupling constant via
\begin{eqnarray}
  g_s^{0\prime}  &=& \zeta_g^0 g_s^0\,,
\end{eqnarray}
where $g_s^0=\sqrt{4\pi\alpha_s^0}$ is the bare strong coupling.
In close analogy to the renormalization procedure it is possible to
relate $\zeta_g^0$ to the decoupling constants of the ghost-gluon
vertex, the gluon propagator and the ghost propagator:
\begin{eqnarray}
  \zeta_g^0 &=& \frac{\tilde{\zeta}_1^0}{\tilde{\zeta}_3^0\sqrt{\zeta}_3^0}
  \,.
  \label{eq::zetag0}
\end{eqnarray}
It is convenient to define the $\zeta$'s in a first step for bare
quantities since then one-to-one relations between the $N$-loop
contributions to the decoupling constants and the $N$-loop vacuum
integrals can be established~\cite{Chetyrkin:1997un}.
Sample diagrams up to four-loop order can be found in
Fig.~\ref{fig::diags}(a), (b) and (c).
The technology up to three-loop order is well documented in the
literature and publicly available programs exist (see, e.g.,
Refs.~\cite{Steinhauser:2002rq,Steinhauser:2000ry}).  
The development of the four-loop technology is quite new. The results
of Ref.~\cite{Schroder:2005hy} are based on a
program~\cite{Schroder:2002re} where the so-called 
Laporta algorithm has been implemented in FORM~\cite{Vermaseren:2000nd}.

Once the bare four-loop result for the decoupling relation is
available it is straightforward to obtain the renormalized expression
with the help of the usual renormalization constants 
(see, e.g., Ref.~\cite{Chetyrkin:2004mf})
\begin{eqnarray}
  \zeta_g &=& \frac{Z_g}{Z_g^\prime} \zeta_g^0
  \,.
\end{eqnarray}

The analytical result for $\zeta_g$ can be found in
Ref.~\cite{Schroder:2005hy} where only one constant, which was denoted
by $X_0$, has not been known in analytical form but with high
numerical precision. Meanwhile also this constant
is available analytically~\cite{Schroder_X0} with the result
\begin{eqnarray}
  X_0 &=&
  - 384 a_5 
  + \frac{873}{2}\zeta_5
  - \frac{53}{15}\pi^4\ln 2
  \nonumber\\&&\mbox{}
  - \frac{16}{3}\pi^2\ln^2 2
  + \frac{16}{5}\ln^5 2
  \nonumber\\&\approx&
  1.8088795462083347414\ldots
  \,,
\end{eqnarray}
where $a_5=\mbox{Li}_5(1/2)$ and $\zeta_5\approx 1.036927755\ldots$\,.


\begin{figure}[b]
  \begin{tabular}{c}
  \leavevmode
  \epsfxsize=.43\textwidth
  \epsffile[80 380 600 430]{diags.ps}
  \\
  \mbox{}\hfill(a)\hfill(b)\hfill(c)\hfill(d)\hfill(e)\hfill\mbox{}
  \\[-2em]
  \end{tabular}
  \caption{
    Sample diagrams for the gluon (a) and ghost (b) propagator
    and the ghost-gluon vertex (c).
    In (d) the lowest-order diagram is shown mediating the Higgs-gluon
    coupling in the SM and (e) shows an example for a
    five-loop diagram contributing to the coefficient function $C_1$.
  }
  \label{fig::diags}
\end{figure}


In order to study the numerical impact of our result
we consider the evaluation of $\alpha_s^{(5)}(M_Z)$
from $\alpha_s^{(4)}(M_\tau)$, i.e. we apply our formalism to the
crossing of the bottom quark threshold.
In general one assumes that the value of the scale 
$\mu_b$, where the matching has to be performed, is of order $m_b$.
However, it is not determined by theory. Thus this uncertainty
contributes significantly to the error of physical predictions.
On general grounds one expects that 
while including higher order perturbative corrections
the relation
between $\alpha_s^{(4)}(M_\tau)$ and $\alpha_s^{(5)}(M_Z)$
becomes insensitive to the choice of the
matching scale which we want to demonstrate in the following 
up to the five-loop order.

Starting from $\alpha_s^{(4)}(M_\tau)=0.36$ we compute in a first step
$\alpha_s^{(4)}(\mu_b)$ using $N$-loop accuracy.
Afterwards $\alpha_s^{(5)}(\mu_b)$ is obtained where for consistency
the $(N-1)$-loop matching equations have to be used.
Finally, we compute $\alpha_s^{(5)}(M_Z)$ using again the $N$-loop in
the $\beta$ function.

In Fig.~\ref{fig::asMZ} the result for $\alpha_s^{(5)}(M_Z)$ as a
functions $\mu_b$ is displayed for the one- to five-loop analysis.
For illustration, $\mu_b$ is varied rather extremely, by almost two orders
of magnitude.
While the leading-order result exhibits a strong logarithmic behaviour, the
analysis is gradually getting more stable as we go to higher orders.
The five-loop curve is almost flat for $\mu_b \ge 1$~GeV and
demonstrates an even more stable behaviour than the four-loop analysis
of Ref.~\cite{Chetyrkin:1997un}. It should be noted that around $\mu_b \approx
1$~GeV both the three-, four- and five-loop curves show a strong
variation which can be interpreted as a sign for the breakdown of
perturbation theory.
Note that for this analysis the unknown five-loop coefficient of the 
$\beta$ function is set to zero.
%Besides the $\mu_b$ dependence of $\alpha_s^{(5)}(M_Z)$, also its absolute 
%normalization is significantly affected by the higher orders.
%At the central matching scale $\mu_b=M_b$, we encounter a rapid
%convergence behaviour.


\begin{figure}[t]
\begin{center}
  \epsfxsize=\textwidth
  \epsfig{figure=asmz.eps,width=.46\textwidth}

  \vspace*{-2em}

  \caption{$\mu_b$ dependence of $\alpha_s^{(5)}(M_Z)$ calculated from
    $\alpha_s^{(4)}(M_\tau)=0.36$ and $m_b=4.7$~GeV. The procedure is
    described in the text. The dotted, short-dashed, long-dashed and
    dash-dotted line corresponds to 
    one- to four-loop running. The solid curve 
    includes the effect of the new four-loop matching term.
  }
  \label{fig::asMZ}
\end{center}
\end{figure}


An interesting connection between the decoupling constant discussed so
far, which constitutes a fundamental quantity of QCD, and a building
block for the production and decay of a SM Higgs boson
is established by the all-order low-energy
theorem~\cite{Chetyrkin:1997un} given by
\begin{eqnarray}
  C_1 &=& -\frac{1}{2}\,\frac{m_t^2\partial}{\partial m_t^2}\,
  \ln\zeta_g^2
  \,.
  \label{eq::c1let1}
\end{eqnarray}
Eq.~(\ref{eq::c1let1}) relates $\zeta_g$ to the coefficient function
appearing in the effective Lagrangian of an intermediate-mass
Higgs boson to two, three and four gluons. It is conveniently written
in the form
\begin{eqnarray}
  {\cal L}_{\rm eff} &=& -\frac{H^0}{v^0} C_1 {\cal O}_1
  \,,
  \label{eq::Leff}
\end{eqnarray}
where the effective operator ${\cal O}_1$ is given by
\begin{eqnarray}
  {\cal O}_1 &=& \left(G^{a}_{\mu\nu}\right)^2\,,
\end{eqnarray}
The appealing feature of Eq.~(\ref{eq::c1let1}) is the derivative with
respect to $m_t^2$ since the $(N+1)$-loop dependence of $\zeta_g$ can
be reconstructed from the knowledge of $\zeta_g$ to $N$-loop order. 
This is possible since $m_t$ only enters in the combination $\ln\mu^2/m_t^2$.
The five-loop result for $C_1$, which contains diagrams like the one
in Fig.~\ref{fig::diags}(e), can be found in
Ref.~\cite{Schroder:2005hy}. It only contains the fermionic part of
the five-loop $\beta$ function as unknown coefficient.

%- }}}
%- {{{ MSSM:

\section{\label{sec::mssm}Decoupling in the MSSM}

In extensions of the SM similar considerations hold as for
QCD. E.g., a typical problem in the minimal extension of
the SM is the relation of the strong coupling at the GUT
scale and the electroweak scale. In contrast to QCD many mass scales
enter the decoupling procedure and thus beyond one-loop order 
both the calculation and the
formulae become significantly more complicated although the very
formalism drafted in the beginning of the previous Section remains the same.
A further difference to QCD occurs in connection to the regularization
scheme: since dimensional regularization breaks supersymmetry in this
Section dimensional reduction is used (see
Ref.~\cite{Harlander:2005wm} for more details). 

At one-loop order only one mass scale enters at a time in the Feynman
diagram and thus the exact formula is available. It is given by
\begin{eqnarray} 
  (\zeta_g^{\rm SUSY})^2 &=& 1 + \frac{\alpha_s^{(\rm SUSY)}}{\pi}
  \Bigg[-
  \frac{1}{6} L_t -  \frac{1}{12} \sum_{s=1}^{n_s} L_{\tilde{q}_s}
  \nonumber\\&&\mbox{}-
  \frac{n_{\tilde{g}}}{2}
  L_{\tilde{g}}
  \Bigg]
  + {\cal O}\left(\alpha_s^2\right)
  \,,
  \label{eq::zetag1l}
\end{eqnarray}
where $L_x=\ln(\mu^2/m_x^2)$. Already in this formula one can see that
a single choice for $\mu$ does not avoid possible large logarithms.

Sample Feynman diagrams for the two-loop calculation can be found in
Fig.~\ref{fig::diagrams}. As can be seen, in general several mass
scales appear which make the analysis more involved. 
In Ref.~\cite{Harlander:2005wm} several scenarios have
been considered in order to demonstrate the numerical effects. In particular
for the following cases the decoupling constant has been evaluated
\begin{itemize}
\item[(A)]
  $m_{\tilde{u}}$, \ldots, $m_{\tilde{b}}$ 
  $\gg$ 
  $m_{\tilde{t}}$, $m_{\tilde{g}}$, $m_t$
  $\gg m_b$
\item[(B)]
  $m_{\tilde{u}}$, \ldots, $m_{\tilde{b}}$, $m_{\tilde{t}}$ 
  $\gg$ 
  $m_{\tilde{g}}$, $m_t$
  $\gg m_b$
\item[(C)]
  $m_{\tilde{u}}$, \ldots, $m_{\tilde{b}}$, $m_{\tilde{t}}$, $m_{\tilde{g}}$
  $\gg$ 
  $m_t$
  $\gg m_b$
\item[(D)]
  $m_{\tilde{u}}$, \ldots, $m_{\tilde{b}}$, $m_{\tilde{t}}$, $m_{\tilde{g}}$,
  $m_t$
  $\gg m_b\,,$
\end{itemize}  
where $m_{\tilde{u}}$, \ldots, $m_{\tilde{t}}$ are the squark masses,
$m_{\tilde{g}}$ is the gluino mass and $m_t$, $m_b$ are the top and the
bottom quark mass, respectively. Masses separated by a comma are
assumed to have the same order of magnitude and for each ``$\gg$''
sign a decoupling constant and a corresponding matching scale have to
be introduced.
Note that case~(B) corresponds to
so-called split SUSY and in case~(C) all supersymmetric
masses are considered as heavy with respect to the SM ones.  Squark
masses are always taken to be larger than the top quark mass.
Analytic results for the individual decoupling constants can be found
in Ref.~\cite{Harlander:2005wm}.

\begin{figure}[t]
  \begin{center}
    \begin{tabular}{c}
\unitlength=.9bp%
\begin{feynartspicture}(350,70)(5,1)
\input  diagrams
\end{feynartspicture}  
    \end{tabular}

    \vspace*{-3em}
    
      \caption[]{\label{fig::diagrams}\sloppy Sample diagrams
        contributing to the decoupling constants 
	$\zeta_3^0$, $\tilde{\zeta}_3^0$ and $\tilde{\zeta}_1^0$.}
  \end{center}
\end{figure}


\begin{figure}[b]
  \begin{center}

    \vspace*{-5em}

    \begin{tabular}{c}
      \epsfig{figure=asC.ps,width=18em}
    \vspace*{-10em}
    \end{tabular}

    \vspace*{-10em}

    {\Large $\alpha_s(\mu_{\rm GUT})$}\\[9em]
    \mbox{}\vspace*{2em}
         {\Large $\mu_{\rm th,2}$~(GeV) \hfill $\mu_{\rm th,1}$~(GeV)}

    \vspace*{-3em}

      \caption[]{\label{fig::as3D}\sloppy
$\alpha_s(\mu_{\rm GUT})$ for $\mu_{\rm GUT}=10^{16}$~GeV as a function 
of the two matching scales $\mu_{\rm th,1}$ and $\mu_{\rm th,2}$.
The lower (green), middle (blue) and upper (red) planes correspond to
one-, two- and three-loop running accompanied with the corresponding 
order in the decoupling relation for the scenario (C).
        }
  \end{center}
\end{figure}

Let us in the following discuss the numerical effects for various
examples.

In Fig.~\ref{fig::as3D} which shows $\alpha_s(\mu_{\rm GUT})$ for
scenario~(C) as a function of the two matching scales 
$\mu_{\rm th,1}$ and $\mu_{\rm th,2}$,
the dependence on the matching scales and the loop order is exemplified.
Each matching scale is varied by a factor of ten around the default
value 1000~GeV and 400~GeV, respectively.

One can clearly see how the result stabilizes as soon as two-loop running
and one-loop matching (middle plane) is included.  The three-loop
corrections stabilize the dependence of $\alpha_s$ on $\mu_{\rm th,1}$
and $\mu_{\rm th,2}$ even more (upper plane), such that the result is
practically independent of the matching scales within a wide range
around their default values. Within the considered range of $\mu_{\rm
th,1}$ and $\mu_{\rm th,2}$, the variation of $\alpha_s(\mu_{\rm GUT})$
is 13.8/1.6/0.3\% at 1-/2-/3-loop order.

\begin{figure}[t]
  \begin{center}
    \begin{tabular}{c}
      \epsfig{figure=asilcth2c.eps,width=18em}
    \end{tabular}

  \vspace*{-2em}

      \caption[]{\label{fig::asilc}\sloppy
        $\alpha_s(800~\mbox{GeV})$ as a function 
        of $\mu_{\rm th,1}$ for scenario (C).
        The dotted, dashed and full lines correspond to
        one-, two- and three-loop running.
        }

  \vspace*{-2em}

  \end{center}
\end{figure}

Another practical application of our results is shown in
Fig.~\ref{fig::asilc}\footnote{Note that the plot in
  Ref.~\cite{Harlander:2005wm} corresponds to $\alpha_s(M_Z)=0.118$
  and not to $\alpha_s(M_Z)=0.120$ which was used for the 
  strong coupling in the $\overline{\rm DR}$ scheme.}, 
where the strong coupling is computed for
$\sqrt{s}=800$~GeV which is the proposed
center-of-mass energy of a future International
Linear Collider (ILC) is computed.  While for $\mu_{\rm th,2}\approx 200$~GeV
the two- and three-loop result practically give the same value for
$\alpha_s$, for $\mu_{\rm th,2}=2000$~GeV the shift induced by the new
three-loop terms is comparable with the uncertainty induced by the
experimental value for $\alpha_s(M_Z)$. The latter is given by
$\pm 0.0012$ assuming an error of $\pm0.002$ for $\sqrt{s}=M_Z$.
This underlines once more the importance
of the two-loop decoupling and three-loop running terms.

\begin{figure}[b]
\begin{center}

  \vspace*{-6em}

  \begin{minipage}{20em}
    \epsfxsize=.5\textwidth
    \epsfig{figure=couplings.B.ps,width=.8\textwidth}
    \\[-33em]
    \epsfxsize=\textwidth
    \epsfig{figure=couplings.C.ps,width=.8\textwidth}
    \\[-32em]
  \end{minipage}
  \caption{\label{fig::couplings}
        Running of the inverse couplings $1/\alpha_i$, $i=1$ 
        (see main text for details).
}
\end{center}
\end{figure}

In Fig.~\ref{fig::couplings} we combine our 
findings for the evolution of the inverse strong coupling  
at the GUT scale (yellow band) with the  two-loop order results
for the strong ($\alpha_s$, red band), the electro-magnetic ($\alpha_1$,
blue band) and the weak coupling ($\alpha_2$, green band) 
(for a precise definition, see Ref.~\cite{Allanach:2004ud}, and
references therein). The wide error bands
correspond to the present experimental accuracy for the gauge couplings
and the supersymmetric mass spectrum of the SPS1a reference point.
The inner bands account for the  expected improvements in 
the absolute errors due to the future experimental analysis at LHC+LC. 

Depending on the scenario, i.e. the particle masses in the MSSM
respectively their hierarchy, a significant numerical effect can be
observed. Whereas for scenario~B (upper plot) the shift due to the
three-loop is relatively small the yellow band for scenario~C (lower plot)
is outside the LHC error band. This demonstrates that it is important
to include two-loop corrections in the analyses considering the
unification of couplings but also the evaluation of low-energy
parameters from unified parameters at high energies.

%- }}}
%- {{{ Conclusions:

%%%\section{\label{sec::concl}Conclusions}



%- }}}
%- {{{ Acknowledgements:

\section*{Acknowledgements}

I would like to thank R. Harlander, L. Mihaila and Y. Schr\"oder
for the pleasant collaboration leading to the results presented in this
contribution. I would like to thank the organizers of Loops and Legs
2006 for the pleasant atmosphere during the conference.
This work was supported by the DFG through SFB/TR~9.

%- }}}
%- {{{ references:

\begin{thebibliography}{99}

%
% ll06_ref.tex -- generated by sortref-2.3.5  
% ((C) R. Harlander, http://www.robert-harlander.de/software/)
% on Tue Jun 27 09:46:06 CEST 2006
%

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%2
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%3
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%13
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%14
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%%CITATION = HEP-PH 0403133;%%



\end{thebibliography}

%- }}}

\end{document}
