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\begin{center}
{\large\bf $\alphas$ from $\sigma(\epem\to$~hadrons)}\\[0.5cm]
J. K\"uhn\\[0.3cm]
Institution, City, Country
\end{center}

\vspace{1.cm}

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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 a region 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 betwee the predictions based on ``Fixed Order�� and 
``Contour Improved�� perturbation theory [we,old Pich, ref therein]  and will 
be discussed in more detail in  [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,
$\Gamma(Z\to hadrons)$ and the closely related quantities for the $W-$ or the
Higgs-boson.proceeds though

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

The $R$-ratio, given by $Sigma(e^+e^-\to hadrons)/\sigma_pt$ 
can be measured at vastly different energies.
In the low energy region the annihilation the annihilation proceeds through
the virtual photon only. A combined fit to CLEO 
\cite{Kuhn:2007tc}

%[J.H. K\"uhn, M. Steinhauser, T. Teubner, Phys.Rev. D 076, 074003 (2007), 
%arXiv:0707.2589]

 and BESS [Phys.Lett. B 641(2006)145, 
based on four-loop accuracy, gives

\beq
\alpha_s^{(4)}(9^2 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[REF], MD-1 [REF] and CLEO [REF], and leads to [we] 
\beq
\alpha_s^{(4)}(9^2 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.

Let us mention that this result is in nice agreement with a more recent
measurement of of the BESS collaboration
\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 $e^+e^-\to hadrons$ at the $Z$ resonance}
 
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 ${\cal O}(\alpha_s^4)$.
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 (FIGs):

i.) The non-singlet vector and axial vector correlators $r_{NS}$

\beq
r_{NS}=
\eeq


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

iii.)A contribution to the axial correlator,resulting from the top-bottom
doublet with $4 m_t^2\ll s \ll 4 m_b^2$, which starts in order $\alpha_s^2$.

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 keV,\\\\\ \delta\Gamma^V_S=2.7 keV,\\\\\ 
\delta\Gamma^A_S=42 keV
\eeq
and, correspondingly, to a variation of the strong coupling around 
$3\dot 10^{-4}$. 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
$\delta\Gamma_{had}\approx 100 keV$ expected for an analysis at the FCC-ee.

Various additional small corrections must be applied, which, however, have 
been calculated some time ago [ref]: quark mass corrections up to order
$m_b^2\alpha_s^4$ and $m_b^4\alpha_s^3$ (????) and mixed QCD and electroweak
corrections that will be discussed in the following.

B.) Mixed electrowaek and QCD corrections

For light quarks terms of order $\alpha\alpha_s$ have been evaluated some 
time ago [czarnecki+jk]. The difference between the correct two-loop 
correction and the product of the corresponding two one-loop corrections
\beq
\Delta \Gamma \equiv \Gamma(two~loop(EW+QCD))_{EW}
- \Gamma_{Born} \delta_{EW}^{NLO}\delta_{QCD}^{NLO} = -0.59(3) MeV
\eeq
is again of high importance, if a precision of 
$\delta \Gamma \approx 0.1 MeV$ 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,
$\Gamma(Z\to b \bar b) \equiv R_b = 0.21629 \pm 0.00066$, obtained
at LEP (check???) corresponds to an error of about 1.6~MeV. 
Pushing the relative error $\delta\Gamma(Z\to b \bar b) / \Gamma_Z$
down to $2\dot{}^{-5}$, would imply a precision of 0.05~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)}) MeV
\eeq
separated into $m_t^2$ enhanced terms [Fleischer+..] and the rest 
[Harlander,..hep-ph/9712228].

The size of these terms has even motivated the evaluation of the 
$m_t^2$-enhanced corrections [chet+Steinh hep-ph/990480]
of order $G_F m_t^2\alpha_s^2$ with
the result $\delta \Gamma_b \approx 0.1 MeV$
for the non-singlet contributions (singlet????), a result drasically 
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 $\overline{MS}$-mass
[karlsruhe1999, Marquard,smirnov,Smirnov,Steinhauser2015]
\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 $\overline{MS}$-mass may well be directly accessible
with high precision, e.g. through a measurement of the potential 
subtracted mass with a precision of 20 -- 30 MeV.

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

The precision of the present $M_W$-measurement amounts to 30~MeV from
LEP only, which can be improved to 15~MeV when considering the combination 
of LEP and Tevatron data. (check)
For the FCC-ee option a precision of 0.5 to 1~MeV is forseen.
In the Standard Model $M_W$ can be considered a derived quantity  term of order
that can be calculated from the following chain of equations
\beq
M_W^2 = .....
\delta M_W=....
\delta\rho_t =
\eeq

In this case the three-loop term amounts to 9.5~MeV, the four-loop term
to 2.1~MeV [chet, Boug]. In three loop approximation also
purely weak ($\propto x_t^3$ and mixed (QCD*electroweak) 
three-loop terms are available, corresponding to
$\delta M_W = 0.2 MeV$ and 2.5~MeV respectively.
These are shown in Fig. xxx, together with the terms of order   .....

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

\section{Perspectives for $e�+e^- \to hadrons$ 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 beteen 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 not prediction for the $R$-ratio at
a level rquired for a precise determination of the $Zf\bar f$ coupling is
available at present.

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

a.) $H\to b\bar b$
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 $(1+17/3 \alpha_s/\pi)$.
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 
$m_b(10GeV)=3610- \frac{\alpha_s - 0.1189}{0.002}$.
Running $m_b$ to the scale of $M_H$ one obtains    
$m_b(M_H)=2759 \pm 8\mid_{m_b} \pm 27\mid_{\alpha_s} MeV$.

While the quark mass anomallous dimension $\gamma_4$ has been
obtained in five loop approximation aready some time ago
[Baikov, Chet, JK], 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)} = eq...
\eeq
which should be compared to $....?=2.0\times 10^{-4}$ 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}}
= ....
\eeq
b.) $ 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 [ref] to order $\alpha_s^5$:
\beq
\Gamma(H\to gg) = K \Gamma_{Born}(H\to gg)\\
\Gamma_{Born}(H\to gg) = ....\\
K= ....= ....
\eeq
where $M_t=175 GeV$, $m_H=120 GeV$ and $a_s'= \alpha_s^{(5)}(M_H)/\pi$
had been assumed. Note that an esperimental precision for
$\sigma(HZ)\times Br(H\to gg)$ of $1.4\%$ has been claimed in ?????.
Mass corrections to this result can be found in [Steinhauser + Schreck].

\section{Summary and Conclusions}

Improved measurements of the total cross section for electron-positron
annihilation just below the charm and the bottom threshoicipatedld would lead to
model independent tests ofthe running of $\alpha_s$ overa 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~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.
 



\bibliographystyle{elsarticle-num}

\bibliography{chet}


\end{document}






smirnov_books_JHEP,lit,laporta,tentyukov_verm,gm5,higgs_as5,dim_reg,JJ,QGRAF_EXP,chet,baikov,asmirnov,smirnov,vladimirov,vermaseren,masters,%
surguladze,gorishnii,tarasov,bierenbaum,kataev,czakon,kazakov,david_dirk_qQED,other_masters,%
sector_decom,lee,kotikov_before1995,remiddi_1997-2000,broadhurst,gracey,velizhanin,%
DIS,DIS2,DIS4,acat,penin,grozin}







\ice{
Ideally your talk and written contribution should include the following elements regarding your $\alphas$
determination method:
\begin{itemize}
\item state of the art of the latest theoretical/experimental developments in this topic, and relevant biblio references
\item size of current experimental/theoretical uncertainties: missing higher orders, electroweak corrections,
    power corrections, hadronization corrections, experimental uncertainties, ... 
\item Expected $\alphas$ uncertainty in ~10 years from now: new theoretical developments, plus O(1 ab-1)
    p-p at 14 TeV at the LHC 
\item Expected improvements brought about by the FCC-ee: 10$^{12}$ Z's at sqrt(s)=mZ; 10$^8$ W's at
    sqrt(s)=mWW; similar orders-of-magnitude for quarkonia, tau, jets,... 
\item Expected systematic uncertainties that the FCC-ee detectors should target in order to match the expected
statistical precision, or where that is not possible, important theoretical targets that should be met or exceeded.
\end{itemize}

}
\vspace{1.cm}


You can include tables such as Table~\ref{tab:YourName:1} and figures such as Fig.~\ref{fig:YourName:1}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\renewcommand{\arraystretch}{1.2}
\begin{table*}[ht]
{\small 
\begin{center}
\begin{tabular}{|l|c|c||c|c|}\hline 
Process & Q [GeV] &  $\alpha_s(m_Z)$ & excl. mean $\alpha_s(m_Z)$ & std. dev.\\\hline 
$\tau$-decays        & 1.78     &  $0.1197 \pm 0.0016$  & $0.11818 \pm 0.00070$ & 0.9 \\
DIS [$F_2$] & 2 - 170 &  $0.1142 \pm 0.0023$ & $0.11876 \pm 0.00123$ & 1.7 \\
DIS [e-p $\rightarrow$ jets] & 6 - 100 &  $0.1198 \pm 0.0032$ & $0.11836 \pm 0.00069$ & 0.4 \\
Lattice QCD & 7.5 &  $0.1183 \pm 0.0008$  & $0.11862 \pm 0.00114$ & 0.2 \\
$\Upsilon$ decays & 9.46 & $0.119^{+0.006}_{-0.005}$ &$ 0.11841 \pm 0.00070$ & 0.1 \\
$\rm e^+\rm e^-$ [jets \& shps] & 14 - 44 &  $0.1172\pm0.0051$ & $0.11844 \pm 0.00076$ & 0.2 \\
$\rm e^+\rm e^-$ [ew prec. data] & 91.2 &  $0.1193 \pm 0.0028$ & $0.11837 \pm 0.00076$ & 0.3 \\
$\rm e^+\rm e^-$ [jets \& shps] & 91 - 208  &  $0.1224 \pm 0.0039$ & $0.11831 \pm 0.00091$ & 1.0 \\ \hline
\hline
\end{tabular}
\end{center}
\caption{\small PLACEHOLDER TABLE \label{tab:YourName:1}.}
}
\end{table*}
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\begin{figure}[htbp!]
\begin{center}
\resizebox{0.70\textwidth}{!}{%  
  \includegraphics{alphas_placeholder.pdf}}
\caption{PLACEHOLDER FIGURE. \label{fig:YourName:1}}
\end{center}
\vskip-1cm
\end{figure}
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\begin{thebibliography}{99}

%\cite{Agashe:2014kda}%``Review of Particle Physics,''
\bibitem{PDG}K.~A.~Olive {\it et al.} [PDG Collab.], Chin.\ Phys.\ C {\bf 38} (2014) 090001.
%%CITATION = CHPHD,C38,090001;%%
\\[-5mm] 

\bibitem{sb-fin} ....\\[-5mm] 


\end{thebibliography}




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