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\newcommand{\pp}{p-p}
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\begin{center}
{\large\bf $\mathbf{\alpha}_s$ from $\mathbf{\sigma}(\epem\to $~hadrons)}\\[0.5cm]
K. G.  Chetyrkin and J. H. K\"uhn \footnote{Presented by Johann  H. K\"uhn}\\[0.3cm]
Institut f\"ur Theoretische Teilchenphysik, Karlsruher
  Institut f\"ur  Technologie (KIT), D-76128 Karlsruhe, Germany





\end{center}

\vspace{1.cm}

\newcommand{\prd}{\partial}
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\section{Introduction}


The cross section for electron-positron annihilation into hadrons is one 
of the cleanest ways for measuring the strong coupling constant.
In principle a wide range of energies exists which can be exploited for 
this purpose, ranging from the region below the charm threshold, say around 3~GeV,
up to the highest energies available at electron-positron colliders.

Apart from the regions  very close to the charm-, bottom- and top-threshold 
the cross section can be evaluated by taking the limit of massless quarks.
The leading term for QCD corrections is thus given by the extremely simple 
form $(1+a_s)$ with $a_s\equiv \alpha_s/\pi$. Higher order terms, however,
exhibit a non-trivial dependence on the number of active flavours. Also 
the difference between vector- and axial vector current induced reactions 
starts (in the massless limit) in order $\alpha_s^2$. Closely related to
this quantity are QCD-corrections to $\tau$-lepton decays. Although in 
principle extemely clean and simple, the predictions are, in this case,	
at the limit of applicability of perturbative QCD. This fact is well visible
from the difference  between  the predictions based on ``Fixed Order'' and 
``Contour Improved'' 
(see \cite{Baikov:2008jh} and references  therein)
and will be discussed in more detail in  \cite{pich}.
%(A.Pich, these Proceedings).

The following discussion will, therefore, be limited to decays of the virtual
photon into hadrons, i.e. the famous $R$-ratio,  or the corresponding object 
at higher energies, in particular the decay rate of the $Z$-boson into hadrons,
$\Gamma(Z\to \mbox{hadrons})$, and the closely related quantities for the $W$ or the
Higgs-boson decay.

\section{The R-ratio below the bottom threshold}

The $R$-ratio, given by $\sigma(e^+e^-\to {\mathrm{ hadrons}})/\sigma_{\mathrm pt}$, 
can be measured at vastly different energies.
In the low energy region the annihilation  proceeds through
the virtual photon only. A combined fit \cite{Kuhn:2007tc}  to CLEO 
%[J.H. K\"uhn, M. Steinhauser, T. Teubner, Phys.Rev. D 076, 074003 (2007), 
%arXiv:0707.2589]
%cite{Ablikim2006145}
results \cite{Besson:2007aa},
%BESS [Phys.Lett. B 641(2006)145, 
based on four-loop accuracy, gives
\beq
\alpha_s^{(4)}(9^2\,  \mbox{GeV}^2) = 0.160\pm 0.024\pm 0.024
\eeq
which, at the scale of $M_Z$, corresponds to
\beq
\alpha_s^{(5)}(M^2_Z) = 
0.110 \left(\begin{array}{cc}+0.014\\-0.017\end{array}\right)
{}.
\eeq

This CLEO result, in turn, can be combined with earlier measurements by 
BESS 
\cite{Bai:2001ct}, 
MD-1 \cite{Blinov:1993fw}
and 
CLEO \cite{Ammar:1997sk}, 
and leads to \cite{Kuhn:2007tc}
\beq
\alpha_s^{(4)}(9^2\,  \mbox{GeV}^2) 
= 0.182\left(\begin{array}{cc} +0.022 \\ -0.025\end{array}\right)
\eeq
corresponding to
\beq
\alpha_s^{(5)}(M_Z^2)
=0.119 \left(\begin{array}{cc} +0.009 \\ -0.011\end{array}\right)
{}.
\eeq

This result is in nice agreement with a more recent
measurement  of the BESS collaboration \cite{Ablikim:2006aj}
\beq
R_{uds}=2.224 \pm 0.019 \pm 0.089
\eeq 
taken at energy values below the charm threshold.

Let us emphasize again that more precise measurements in this region, 
based on the enormous statistic of the BESS and BELLE experiment,
would be highly welcome.

\section{Status and perspectives for $e^+e^-\to \mbox{hadrons}$ at the $Z$ resonance}
 
{\bf  a.) QCD corrections}

At present the most precise determination of $R$ is based on the 
measurement of $Z$ decays, more precisely on the measurement of
$\Gamma_{had}$ and $\Gamma_{had}/\Gamma_{lept}$, where corrections 
have been calculated to ${\cal O}(\alpha_s^4)$
\cite{Baikov:2008jh,Baikov:2010je,Baikov:2012er,Baikov:2012zn}.
Effects from non-vanishing charm and bottom quark masses will be 
mentioned below.

Even in the massless limit there are three different types of 
contributions 
\begin{figure}[here]
\begin{center}
\begin{minipage}{0.54\linewidth}
\raisebox{-2.4mm}{\includegraphics[width=.3\linewidth]{figures/PropNS.eps}}
\includegraphics[width=.3\linewidth]{figures/PropSV.eps}
\includegraphics[width=.3\linewidth]{figures/PropSA.eps}
\end{minipage}
\end{center}
\caption{
Non-singlet \& singlet, vector \& axial correlators.
}
\end{figure}

i.) The non-singlet vector and axial vector correlators $r_{NS}$
\cite{Baikov:2008jh,Baikov:2010je}.

ii.) The singlet correlator $r^V_S$, which starts in ${\cal O}(\alpha_s^3)$ 
and is present for the vector current only \cite{Baikov:2012er,Baikov:2012zn}.

iii.)A contribution to the axial correlator, $r_S^A$, resulting from the top-bottom
doublet with $4 m_t^2  \gg s \gg 4 m_b^2$, which starts in order $\alpha_s^2$
\cite{Kniehl:1989bb,Chetyrkin:1993ug,Baikov:2012zn}.

The corresponding predictions are available to order $\alpha_s^4$. For an 
extremely conservative error estimate based on the 
variation of the scale parameter $\mu$ between $M_Z/3$ and $3 M_Z$ this
corresponds to the variation
\beq
\delta\Gamma_{NS}=101 \, \mbox{KeV},\\\\\ \delta\Gamma^V_S=2.7 \, \mbox{KeV},\\\\\ 
\delta\Gamma^A_S=42\,  \mbox{KeV}
\eeq
and, correspondingly, to a variation of the strong coupling around 
$3 \cdot 10^{-4}$.   This theory error is completely neglegeable 
compared to the present experimental uncertainty around 0.026.
In fact, it is even comparable to the error of
$\delta\Gamma_{had}\approx 100 \, \mbox{KeV}$ expected for an analysis at the FCC-ee.
\begin{figure}
\includegraphics[width=.8\linewidth]{figures/rns.eps}
\caption{Predictions for $r_{NS}$ in  increasing orders of $\alpha_s$ \cite{Baikov:2012er}.}
\end{figure}
Various additional small corrections must be applied, which, however, have 
been calculated some time ago: quark mass corrections up to order
$m_b^2\alpha_s^4$ and $m_b^4\alpha_s^3$
\cite{Baikov:2004ku,Chetyrkin:2000zk} 
and mixed QCD and electroweak
corrections that will be discussed in the following.



{\bf b.) Mixed electroweak and QCD corrections}

For light quarks terms of order $\alpha\alpha_s$ have been evaluated some 
time ago \cite{Czarnecki:1996ei}.
The difference between the correct two-loop 
terms  and the product of the corresponding two one-loop terms 
\beq
\Delta \Gamma\equiv\Gamma(\text{two loop (EW $\star$
  QCD)})-\Gamma_{\text{Born}}\delta_{\text{EW}}^{\text{NLO}}\delta_{\text{QCD}}^{\text{NLO}}
=-0.59(3)
\eeq
\ice{\beq
\Delta \Gamma \equiv \Gamma(two~loop(EW+QCD))_{EW}
- \Gamma_{Born} \delta_{EW}^{NLO}\delta_{QCD}^{NLO} = -0.59(3) \, \mbox{MeV}
\eeq
}
is again of high importance, if a precision of 
$\delta \Gamma \approx 0.1 \mbox{MeV}$ should be reached at some point 
in the future.
The next, presently unknow three-loop term is smaller by another 
factor of order $\alpha_s/\pi$, times an unknown coefficient
which should not exceed 5 in order to keep this term under control.


Similar comments apply to $\Gamma(Z\to b \bar b)$. The present result,
$\Gamma(Z\to b \bar b)/\Gamma_Z \equiv R_b = 0.1512  \pm 0.0005$, obtained
at LEP  corresponds to an error of about 1.3~\mbox{MeV}. 
Pushing the relative error $\delta\Gamma(Z\to b \bar b) / \Gamma_Z$
down to $2 \cdot 10^{-5}$, would imply a precision of 0.05~\mbox{MeV} for the 
partial width $\Gamma(Z\to b \bar b)$. This should be compared to the
present result of order $\alpha\alpha_s$, which is given by
\beq
\Gamma(Z  \to b\bar b) - \Gamma(Z\to d \bar d) =
(-5.69 - 0.79)_{{\cal O}(\alpha)} + 
(+0.50 + 0.06)_{{\cal O}(\alpha\alpha_s)}) \, \mbox{MeV}
\eeq
separated into $m_t^2$ enhanced terms \cite{Fleischer:1992fq} and the rest 
\cite{Harlander:1997zb}.
%\newcommand{\MSbar}{\ensuremath{\overline{\text{MS}}}}


The size of these terms has even motivated the evaluation of the 
$m_t^2$-enhanced corrections \cite{Chetyrkin:1999za} 
of order $G_F m_t^2\alpha_s^2$ with
the result $\delta \Gamma_b \approx 0.1 \, \mbox{MeV}$
for the non-singlet contributions, a result drastically 
below the current precision and comparable in size to the planned 
precision of the FCC-ee.

Let us mention that many corrections seem to be significantly 
smaller if expressed in terms of the $\MSbar$-mass
\cite{Chetyrkin:1999qi,Melnikov:2000qh,Marquard:2015qpa}
%[karlsruhe1999, Marquard,smirnov,Smirnov,Steinhauser2015a]
\beq
\bar m_t(\bar m_t) = m_{pole} (1-1.33 \, a_s - 6.46 \, a_s^2 - 
60.27 \, a_s^3 - 704.28 \, a_s^4)
{}.
\eeq
Note that the $\MSbar$-mass may well be directly accessible
at an $e^+ e^-$ collider with high precision, e.g. through a measurement of the potential 
subtracted mass with a precision of 20 -- 30 \mbox{MeV}.

\section{ $M_W$ from $G_F$, $M_Z$, $\alpha$ and the rest}

The precision of the present $M_W$-measurement amounts to 23~\mbox{MeV} 
\ice{
from
LEP only, which can be improved to 15~\mbox{MeV} when considering the combination 
of LEP and Tevatron data. (check)
}
For the FCC-ee option a precision of 0.5 to 1~\mbox{MeV} is forseen.
In the Standard Model $M_W$ can be considered a derived quantity  
that can be calculated from 
$G_F, \ M_Z, \ M_t,\ \Delta \alpha$ and $m_H$. Let us, for the moment,  focus
on  the top mass dependence,  which is obtained from
the following chain of equations
\ice{
\bea
&{}& M_W^2 = 
\delta M_W=....
\delta\rho_t =
\eeq
}
%
\bea
{} M_W^2 &=& f(G_F,M_Z,m_t,\Delta\alpha,\ldots) \\
&=& \frac{M_Z^2}{2(1-\delta\rho)}
\left(1+\sqrt{1-\frac{4\pi\alpha(1-\delta\rho)}{\sqrt{2}G_FM_Z^2}\left(\frac{1}{1-\Delta\alpha}+\ldots\right)}\right)
{},
\\
\delta M_W &\approx& M_W \frac{1}{2} \frac{\cos^2 \theta_w}{\cos^2\theta_w-\sin^2\theta_w}\delta\rho\approx5.7\times 10^4\,\delta\rho [MeV]
{},
\\
\delta\rho_t &=& 3 X_t\left(1-2.8599\left(\frac{\alpha_s}{\pi}\right)-14.594\left(\frac{\alpha_s}{\pi}\right)^2-93.1\left(\frac{\alpha_s}{\pi}\right)^3\right)
{}.
\eea
{
\begin{center}
\begin{minipage}{0.5\textwidth}
\rule{0mm}{6cm}

\begin{picture}(60,-50)
\put(0,-20){\includegraphics[width=1.2\textwidth]{figures/shifts.eps}}
%\put(210,215){$\alpha_s X_t^2$}
\put(140,66){${\alpha_s^2 X_t}$}
\put(175,120){$X_t^3$}
\put(120,130){$\alpha_s X_t^2$}
%\put(210,45){$X_t^2$}
\put(140,15){$X_t^2$}
\end{picture}
\\
\begin{center}
Fig. 3:
{Shifts in $M_W$ and in $\sin^2\Theta_W$ from various  corrections to
$\Delta \rho$ \cite{Faisst:2003px}.}
\end{center}
\end{minipage}
\end{center}


In this case the three-loop term amounts to 9.5~\mbox{MeV}, the four-loop term
to 2.1~\mbox{MeV} \cite{Chetyrkin:2006bj,Boughezal:2006xk}. In three loop approximation also
purely weak ($\sim  X_t^3$) and mixed (QCD*electroweak) of order $\alpha_s X_t^2$  
three-loop terms are available, corresponding to
$\delta M_W = 0.2\,\mbox{MeV}$ and 2.5~\mbox{MeV} respectively.
These are shown in Fig.~3  together with the terms of order  
$\alpha_s X_t^2$  and  $X_t^2$.
%\vspace*{18cm}

Let us emphasize that there are still various uncalculated terms of order
0.1~\mbox{MeV} -- 0.5~\mbox{MeV}, for example the four-loop term of order
$\alpha_s^2 X_t^2$ or higher order terms relating pole and 
$\MSbar$-mass.




\section{Perspectives for $e�+e^- \to \mbox{hadrons}$ above the $Z$-resonance}

No detailled analysis of the cross section exists in the moment
for the energy region above the $Z$-resonance. The situation is 
significantly more complicated than the one around the $Z$-boson or at low
energies:

As a consequence of Furry's theorem the interference between Born and 
one-loop correction vanishes, as long as only vector currents are involved. 
This is no longer true for a mixture of vector and axial currents, such that
the interference (relative to the Born term) is proportional to order
$\alpha$. In combination with the large radiative tail this enhances the 
radiative corrections drastically and no prediction for the $R$-ratio at
a level required for a precise determination of the $Zf\bar f$ coupling is
available at present.

\section{Perspectives for $e^+e^- \to Z + H(\to \mbox{hadrons})$}

{ \bf   a.) $ \mathbf{H\to b\bar b} $}

Up to this point only QCD corrections for vector and axial currents 
have been considered, where the first terms are equal for both cases 
and given by $(1+\alpha_s/\pi)$. In contrast, for the case of Higgs 
boson decay into a quark-antiquark pair, the leading QCD correction
is of the form $(1+17/3 \cdot  \alpha_s/\pi)$.
The full correction for the decay rate into $b$ quarks is given by
\beq
\Gamma(H\to b \bar b) = 
\frac{G_F M_H}{4\sqrt{2}\pi} m_b^2(M_H) R^S(s=M_H^2, \mu^2=M_H^2)
\eeq
\beq
R^S(M_H)= 1 + 1 + 5.667 \, a_s + 29.147 \, a_s^2 + 41.758 \, a_s^3 - 825.7 \, a_s^4
=1.2298
\eeq
for $\alpha_s(M_H)=0.108$, corresponding to $\alpha_s(M_Z)=0.118$. 

The theory uncertainty of the correction factor has been reduced 
from 5 (four loop) to 1.5 (five loop) permille, where a variation of $\mu$
between $M_H/3$ and $3M_H$ has been assumed and $\alpha_s(M_H)$ 
has been fixed to 0.108.

The parametric uncertainties can be traced to the value of $\alpha_s$, 
of $m_b$ at low energies, say 10~GeV, and to the running of $m_b$ from
10~GeV to $M_H$. 
For the values of $\alpha_s$ and $m_b$ we adopt
$\alpha_s(M_Z)=0.1189\pm0.002$     and 
$m_b(10 \, \mbox{GeV})=3610- \frac{\alpha_s - 0.1189}{0.002}$.
Running $m_b$ to the scale of $M_H$ one obtains    
$m_b(M_H)=2759 \pm 8\mid_{m_b} \pm 27\mid_{\alpha_s} \, \mbox{MeV}$.


While the quark mass anomallous dimension $\gamma_4$ has been
obtained in five loop approximation aready some time ago
\cite{Baikov:2014qja} the corresponding result for the  
QCD beta-function is still missing at present.
This corresponds to an uncertainty
\beq
\frac{\delta m_b^2(M_H)}{m_b^2(M_H)}=-1.4\times 10^{-4}\; (\frac{\beta_4}{\beta_0}=0)\quad |\quad
-4.3\times 10^{-4}\; (\frac{\beta_4}{\beta_0}=100) \quad |\quad -7.3\times 10^{-4}\;
 (\frac{\beta_4}{\beta_0}=200)
\label{MH_uncert}
\eeq
which should be compared to 
$\delta\Gamma(H \to b\bar{b})/\Gamma(H \to b\bar{b})=2.0 \times 10^{-4}$ 
expected for the FCC-ee. Assuming an improvement of our knowledge of the strong coupling
by a factor 10, corresponding to $\delta \alpha_s = 2\times 10^{-4}$
and for $m_b$ by a factor 4 ($\delta m_b(M_H) / m(M_H) = 10^{-3}$)
on finds
\beq
\frac{\delta\Gamma_{H \to b\bar{b}}}{\Gamma_{H \to b\bar{b}}}=
\pm 2\times 10^{-3}|_{m_b} 
\pm 1.3\times 10^{-3}|_{\alpha_s,\text{running}}
\pm 1\times 10^{-3}|_{\text{theory}}
{}.
\eeq



{\bf b.)} $\mathbf{ H\to gg}$

Another hadronic decay mode of the Higgs boson with a large 
branching ratio is the one into two gluons, which starts in order $\alpha_s^2$
and is presently known \cite{Baikov:2006ch}  to order $\alpha_s^5$:
\bea
\Gamma(H\to gg) &=& K\cdot\Gamma_{\text{Born}}(H\to gg)
\\
K =
1 &+& 17.9167 \, a'_s +   (156.81 - 5.7083 \ln\frac{M_t^2}{M_H^2}) (a'_s)^2 \nonumber \\ &+& 
     (467.68 - 122.44\ln\frac{M_t^2}{M_H^2}  + 10.94 \ln^2\frac{M_t^2}{M_H^2} )\ (a'_s)^3 \nonumber
{}.
%\eea
% \end{widetext}
%For $M_t = 175 $ GeV, $M_H =120$ GeV and $a'_s=\alpha_s^{(5)}(M_H)/\pi = 0.0363$ we have
%\bea
\\
K &=& 1 + 17.9167 \, a'_s +   152.5 (a'_s)^2 + 
    381.5 (a'_s)^3 
\nonumber
\\
&=& 1 + 0.65038  + 0.20095   + 0.01825 \nonumber
{}.
\eea
where $M_t=175 GeV$, $m_H=120 GeV$ and $a_s'= \alpha_s^{(5)}(M_H)/\pi$
had been assumed. Note that an experimental precision for
$\sigma(HZ)\times Br(H\to gg)$ of $1.4\%$ has been claimed in \cite{Gomez-Ceballos:2013zzn}.
Mass corrections to the Higgs-boson decay rate result can be found in \cite{Schreck:2007um}.


\section{Summary and Conclusions}

Improved measurements of the total cross section for electron-positron
annihilation just below the charm and the bottom threshold would lead to
model independent tests of the running of $\alpha_s$ over a wide energy range.
At high energies the precise control of QCD corrections is crucial for 
the detailled and precise comparison of theory predictions with experimental
results at a future circular or linear electron-positron collider.

A significant improvement in the determination of the $Z$ decay rate 
would immediately lead to a correspondingly improved value of the 
strong coupling constant.
In combination with an improved determination of the fine structure constant
at the scale of the $Z$-boson this would lead to a significantly improved prediction 
for $M_W$ at the level of 1~\mbox{MeV}, corresponding to the anticipated
precision of a future measurement.

A collider with an energy around 250~GeV would be ideally suited for 
the measurement of $e^+e^-\to ZH$, leading to the precise determination of the branching 
ratios of the Higgs boson into its various decay channels. Examples
are presented which involve corrections up to order $\alpha_s^4$ 
for the decay into $b\bar b$ and order $\alpha_s^5$
for decays into $gg$. 
These demonstrate that predictions with a precision around one percent are
well within reach for the dominant decay modes.
 




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%}


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