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PM 01/xx\\
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\section*{New Evaluation of the QED Running Coupling \\ and of the Muonium
Hyperfine Splitting}
\vspace*{1.0cm}
{\bf Stephan Narison}

%\vspace{0.3cm}
\nin
Laboratoire de Physique Math\'ematique,
Universit\'e de Montpellier 2,
Place Eug\`ene Bataillon 
34095 - Montpellier Cedex 05, France
and
Center for Academic Excellence on Cosmology and Particle Astrophysics (CosPA),
Department of Physics, National Taiwan University, 
Taipei, Taiwan, 10617 Republic of China.\\
Email: narison@lpm.univ-montp2.fr
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%{\bf Abstract}
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%\begin{abstract}

\nin
We present a new independent evaluation of the hadronic and QCD contributions to
the QED running coupling $\alpha(M_Z)$ and to the muonium hyperfine splitting $\nu$.
We obtain:
$\Delta\alpha_{\rm had}=2770(17)\times 10^{-5}$ and $\Delta\nu_{{\rm had}}=232.5(2.5)$
Hz. Combined with the
QED and Electroweak Standard Model contributions, they lead to:
$\alpha^{-1}(M_Z)=128.926(25)$ and to the Fermi energy splitting $\nu_F=4~459~031~783(229)~{\rm
Hz}~$, where for the latter, we have used, in addition, the precise measurement of the
muonium hyperfine splitting $\nu_{\rm exp}$. We use $\nu_F$ in order to predict
the ratios of masses $m_\mu/m_e=206.768~276(11)$ and of the magnetic moments
$\mu_\mu/\mu^e_B=4.841~970~47(25)\times 10^{-3}$, which are in excellent agreement with the ones quoted
by the Particle Data Group. These remarkable agreements can provide strong
constraints on some contributions beyond the Standard Model.
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%April 1995
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\section{Introduction}
\nin
A natural and important extension of our recent work \cite{SN} (hereafter
referred as SN) on the hadronic and QCD contributions to the anomalous magnetic
moment of the muon and tau leptons $a_l$, 
is the evaluation of the hadronic and
QCD contributions to the QED running coupling
$\alpha(M_Z)$ and to the muonium hyperfine splitting $\nu$. The analysis
is important as it will give a complete set of estimates of three independent
observables within the same inputs used in SN. These self-contained results
should pass tests from a comparison with high precision experiments and different existing
predictions in the literature.
Using a dispersion relation, it is remarkable to notice that the different 
lowest order hadronic contributions for these three processes can be
expressed in a closed form as a convolution of the
$e^+e^-\rar$ hadrons cross-section $\sigma_H(t)$ with a QED kernel function
$K(t)$ which depends on each observable:

\bea
{\cal O}_{\rm had}=\frac{1}{4\pi^3}\int_{4m^2_\pi}^\infty dt~K_{\cal O}(t)~\sigma_H(t)~,
\eea
where:
\beq
{\cal O}_{\rm had}~\equiv~
a_{l,\rm had}~,~~~\Delta\alpha_{\rm had}\times 10^5~~~{\rm or}~~~
\Delta\nu_{\rm had}~.
\eeq
$-$~For the anomalous magnetic moment $a_{l,\rm had}$, $K_{a_l}(t\geq 0)$ is
the well-known kernel function \cite{GOURDIN}:
\bea\label{kernel}
K_{a_l}(t)&=&\int_0^1 dx\frac{x^2(1-x)}{x^2+\ga{t}/{m_l^2}\dr(1-x)}~,
\eea
where $m_l$ is the lepton mass. It behaves for large  $t$ as:
\beq
K_{a_l}(t\gg m_l^2)\simeq \frac{m^2_l}{3t}~.
\eeq
$-$~For the QED running coupling $\Delta\alpha_{\rm had}\times 10^5$, the
kernel is (see e.g. \cite{JEGER}):
\beq
K_\alpha(t)=\ga\frac{\pi}{\alpha}\dr\ga\frac{M^2_Z}{ M^2_Z-t}\dr~,
\eeq
where $\alpha^{-1}(0)=137.036$ and $M_Z=91.3$ GeV. It behaves for large  $t$
like a constant.\\
$-$~For the muonium hyperfine splitting $\Delta\nu_{\rm had}$, the kernel
function is (see e.g \cite{HFS}):
\beq
K_\nu=-\rho_\nu\Bigg{[}\ga
{x_\mu}+2\dr\ln\frac{1+v_\mu}{1-v_\mu}-\ga
{x_\mu}+\frac{3}{2}\dr\ln x_\mu\Bigg{]}
\eeq
where:
\beq
\rho_\nu= 2\nu_F\frac{m_e}{m_u}~,~~~~~x_\mu=\frac{t}{4m_\mu^2}~~~~~v_\mu=\sqrt{1-\frac{1}{x_\mu}}~,
\eeq
and we take (for the moment) for a closed comparison with \cite{HFS} \footnote{In the
next section, we shall extract this value from the analysis.}, the value of the Fermi energy splitting:
\beq\label{eqhfs}
\nu_F=445~903~192~0.(511)(34)~{\rm Hz}~.
\eeq
It behaves for large  $t$ as:
\beq
K_{\nu}(t\gg m_\mu^2)\simeq \rho_\nu\ga\frac{m^2_\mu}{t}\dr\ga\frac{9}{2}\ln
\frac{t}{m^2_\mu}+\frac{15}{4}\dr~.
\eeq
The different asymptotic behaviours of these kernel functions will influence on
the relative weights of different regions contributions in the evaluation of the above integrals.
\section{Input and Numerical Strategy}
\nin
 The
different data input and QCD parametrizations  of the cross-section
$\sigma_H(t)$ have been discussed in details in SN \cite{SN} and quoted in the last column of Table 1,
corresponding to the estimate in different regions. Table 1 is analogous to
Table 2 of SN. 
We shall only sketched briefly the numerical strategy here:\\
$-$~Our result from the $I=1$
isovector channel below 3 GeV$^2$ is the mean value of the one using
$\tau$-decay and
$e^+e^-$ data. In both cases, we have used standard trapezo\"\i dal rules and/or least
square fits of the data in order to avoid theoretical model dependence parametrization of the pion form
factor. In the region
$0.6-0.8$ GeV$^2$ around the
$\omega$-$\rho$ mixing, we use in both cases $e^+e^-$ data in order to take
properly the $SU(2)_F$ mixing. The $SU(2)$ breaking in the remaining regions
are taken into account by making the average of the two results from $\tau$-decay
and $e^+e^-$ and by
adding into the errors the distance between this mean central value with the one from
each data.\\
$-$~For the $I=0$ isoscalar channel below 3 GeV$^2$, we use the contributions
of the resonances $\omega(782)$ and $\phi(1020)$ using narrow width
approximation (NWA). We add to these contributions, the
sum of the exclusive channels from 0.66 to 1.93 GeV$^2$. Above 1.93 GeV$^2$, we include
the contributions of the $\omega(1.42),~\omega(1.65)$ and
$\phi(1.68)$ using a Breit-Wigner form of the resonances. \\
$-$~For the heavy quarkonia, we include the contributions of known $J/\psi$
(1S to 4.415) and $\Upsilon$ (1S to 11.02) families and use a NWA. We have added the effect of the
$\bar tt$ bound state using the leptonic width of $(12.5\pm 1.5)$ keV given in \cite{YND}.\\
$-$~Away from thresholds, we use perturbative QCD plus negligible quark and gluon
condensate contributions, which is expected to give a good parametrization
of the cross-section. These different expressions are given in SN. However, as the relative r\^ole of
the QCD continuum is important in the estimate of $\Delta\alpha_{\rm had}$, we have added, to the usual
Schwinger interpolating factor at order $\alpha_s$ for describing the heavy quark spectral function, the
known $\alpha_s^2m_Q^2/t$ corrections given in SN. However, in the region we are working, these
corrections are tiny.
\\
$-$~On the $Z$-mass, the integral for $\Delta\alpha_{\rm had}$ has a pole, such that this contribution
has been separated in this case from the QCD continuum. Its value comes from the Cauchy principal
value of the integral.
\section{The QED Running Coupling $\alpha(M_Z)$}
The result given in Table 1 corresponds to the lowest order vacuum polarization. Radiative corrections
to this result can be taken by adding the effects of the radiative modes
$\pi^0\gamma,~\eta\gamma,\pi^+\pi^-\gamma,...$. We estimate such effects to be:
\beq
\Delta\alpha_{\rm had}=(6.4\pm 2.7)\times 10^{-5}
\eeq
by taking the largest range spanned by the two estimates in \cite{YND} and \cite{DAVIE}. Adding this
(relatively small) number to the result in Table 1, gives the total hadronic contributions:
\beq\label{alfa1}
\Delta\alpha_{\rm had}=2769.8(16.7)\times 10^{-5}~.
\eeq
Using the QED contribution to three-loops \cite{JEGER}:
\beq
\Delta\alpha_{\rm{QED}}=3149.7687\times 10^{-5}~,
\eeq
and the Renormalization Group Evolution of the QED coupling:
\beq
\alpha^{-1}(M_Z)=\alpha^{-1}(0)\Big{[}
1-\Delta\alpha_{\rm{QED}}-\Delta\alpha_{\rm had}\Big{]}~,
\eeq
\begin{table*}[H]
\setlength{\tabcolsep}{0.5pc}
\newlength{\digitwidth} \settowidth{\digitwidth}{\rm 0}
\catcode`?=\active \def?{\kern\digitwidth}
%\begin{center}
\caption{Lowest order determinations of $\Delta\alpha_{\rm had}\times 10^{5}$ and
$\Delta\nu_{\rm had}~\rm{[Hz]}$ using combined $e^+e^-$
and inclusive $\tau$ decay data (2nd and 4th columns) and averaged $e^+e^-$ data (3rd and 5th columns).}
\begin{tabular*}{\textwidth}{@{}l@{\extracolsep{\fill}}lllll}
%\begin{tabular}[h]{ccc}
\hline
&\\
\multicolumn{1}{l}{\bf Region in GeV$\bf ^2$}
 & \multicolumn{2}{c}{$\bf \Delta\alpha_{\rm had}\times 10^{5}$}
  & \multicolumn{2}{c}{$\bf \Delta\nu_{\rm had}~\rm{[Hz]}$}
& \multicolumn{1}{c}{\bf Data input} \\
&\\
\hline\\
%\cline{1-5} 
%\cline{4-4}
                 & \multicolumn{1}{l}{\bf ${\bf\tau}$+{$\bf e^+e^-$}}
& \multicolumn{1}{l}{$\bf e^+e^-$} 
                 & \multicolumn{1}{l}{\bf ${\bf\tau}$+$\bf e^+e^-$}
& \multicolumn{1}{l}{$\bf e^+e^-$}\\ 
{\bf Light Isovector}&&&\\
%&\\
$4m_\pi^2\rar 0.8$&$314.5\pm 2.3$&$302.7\pm 7.1$&$152.9\pm 1.8$&$148.4\pm
3.1$&\cite{DAVIE,ALEPH,OPAL}\\
$0.8\rar 2.1$&$77.2\pm 3.4$&$82.0\pm 5.4$&$12.1\pm 0.5$&$16.9\pm 1.9$&\cite{ALEPH,OPAL}\\
$2.1\rar 3.$&$62.3\pm 9.2$&$53.6\pm 4.9$&$7.8\pm 1.2$&$6.7\pm 0.6$&\cite{ALEPH,OPAL}\\
%&&&\\
\it Total Light I=1&$\it 454.0\pm 10.6$&$\it 438.2\pm 10.2$&$\it 172.8\pm 2.2$&$\it 172.1\pm 3.7$\\
%\hline\\
%&&\\
\multicolumn{1}{l}{\it Average }
& \multicolumn{2}{c}{$\it 446.1\pm 10.4\pm 7.9$}
& \multicolumn{2}{c}{$\it 172.5\pm 3.0\pm 0.3$}
& \multicolumn{1}{c}{} \\
%&&&\\
%&\\
\bf Light Isoscalar&&\\
{\it Below 1.93 }&\\
%&\\
\multicolumn{1}{l}{$\omega$}
& \multicolumn{2}{c}{$31.5\pm 1.1$}
& \multicolumn{2}{c}{$12.7\pm 0.4$}
& \multicolumn{1}{l}{NWA \cite{PDG}}\\
\multicolumn{1}{l}{$\phi$}
& \multicolumn{2}{c}{$52.3\pm 1.2$}
& \multicolumn{2}{c}{$13.7\pm 0.3$}
& \multicolumn{1}{l}{NWA \cite{PDG}}\\
\multicolumn{1}{l}{$0.66\rar 1.93$}
& \multicolumn{2}{c}{$11.6\pm 3.0$}
& \multicolumn{2}{c}{$2.7\pm 0.7$}
& \multicolumn{1}{l}{$\sum{\rm exclusive}$ \cite{DOL}}\\
{\it From 1.93 to 3~} &\\
\multicolumn{1}{l}{$\omega(1.42),~\omega(1.65)$}
& \multicolumn{2}{c}{$9.4\pm 1.4$}
& \multicolumn{2}{c}{$1.2\pm 0.2$}
& \multicolumn{1}{l}{BW \cite{DM2,PDG}}\\
\multicolumn{1}{l}{$\phi(1.68)$}
& \multicolumn{2}{c}{$14.6\pm 4.6$}
& \multicolumn{2}{c}{$1.7\pm .5$}
& \multicolumn{1}{l}{BW \cite{DM2,DM1,PDG}}\\
\multicolumn{1}{l}{\it Total Light I=0}
& \multicolumn{2}{c}{$119.0\pm 5.9$}
& \multicolumn{2}{c}{$32.1\pm 1.0$}
& \multicolumn{1}{c}{}\\
\bf Heavy Isoscalar&&\\
%&\\
\multicolumn{1}{l}{$J/\psi(1S\rar 4.415)$}
& \multicolumn{2}{c}{$116.3\pm 6.2$}
& \multicolumn{2}{c}{$4.0\pm 0.2$}
& \multicolumn{1}{l}{NWA \cite{PDG}}\\
\multicolumn{1}{l}{$\Upsilon(1S\rar 11.020)$}
& \multicolumn{2}{c}{$12.7\pm 0.5$}
& \multicolumn{2}{c}{$0.1\pm 0.0$}
& \multicolumn{1}{l}{NWA \cite{PDG}}\\
\multicolumn{1}{l}{$T(349)$}
& \multicolumn{2}{c}{$-(0.1\pm 0.0)$}
& \multicolumn{2}{c}{$\approx 0$}
& \multicolumn{1}{l}{NWA \cite{PDG,YND}}\\
\multicolumn{1}{l}{\it Total Heavy I=0}
& \multicolumn{2}{c}{$128.9\pm 6.2$}
& \multicolumn{2}{c}{$4.1\pm .2$}
& \multicolumn{1}{c}{}\\
%&&\\
\bf QCD continuum&&\\
\multicolumn{1}{l}{$3.\rar 4.57^2$}
& \multicolumn{2}{c}{$330.1\pm 1.0$}
& \multicolumn{2}{c}{$17.5\pm.1$}
& \multicolumn{1}{l}{$(u,d,s)$}\\
\multicolumn{1}{l}{$4.57^2\rar 11.27^2$}
& \multicolumn{2}{c}{$503.0\pm 1.0$}
& \multicolumn{2}{c}{$5.0\pm .1$}
& \multicolumn{1}{l}{$(u,d,s,c)$}\\
\multicolumn{1}{l}{$11.27^2\rar (M_Z-3~{\rm GeV})^2$}
& \multicolumn{2}{c}{$2025.7\pm 2.0$}
& \multicolumn{2}{c}{$1.3\pm 0.0$}
& \multicolumn{1}{l}{$(u,d,s,c,b)$}\\
\multicolumn{1}{l}{$(M_Z+3~{\rm GeV})^2\rar 4M_t^2$}
& \multicolumn{2}{c}{$-(794.6\pm 0.6)$}
& \multicolumn{2}{c}{$\approx 0$}
& \multicolumn{1}{c}{$-$}\\
\multicolumn{1}{l}{$Z$-pole}
& \multicolumn{2}{c}{$29.2\pm .5$}
& \multicolumn{2}{c}{$\approx 0$}
& \multicolumn{1}{l}{principal value \cite{YND}}\\
\multicolumn{1}{l}{$4M_t^2\rar\infty$}
& \multicolumn{2}{c}{$-(24.0\pm 0.1)$}
& \multicolumn{2}{c}{$\approx 0$}
& \multicolumn{1}{l}{$(u,d,s,c,b,t)$}\\
\multicolumn{1}{l}{\it Total QCD Cont.}
& \multicolumn{2}{c}{$2069.4\pm 5.2$}
& \multicolumn{2}{c}{$23.8\pm 1.4$}
& \multicolumn{1}{l}{}\\
\hline
&\\
\multicolumn{1}{l}{\bf Final value}
& \multicolumn{2}{c}{$\bf 2763.4\pm 16.5$}
& \multicolumn{2}{c}{$\bf 232.5\pm 3.2$}
& \multicolumn{1}{l}{}\\
&\\
\hline\\
\end{tabular*}
%\end{center}
\end{table*}
\nin
one obtains the final estimate:
\beq\label{alfa2}
\alpha^{-1}(M_Z)=128.926(25)~,
\eeq 
which we show in Fig 1 for a comparison with recent existing determinations. One can notice
an improved accuracy of the different recent determinations \cite{JEGER,DAVIE,YND,ALFA}
\footnote{Previous works are quoted in \cite{JEGER96}.}, which are in fair
agreement with each others. 
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=11cm]{stephan.eps}
\caption{Recent determinations of $\Delta\alpha_{\rm had}$ and $\alpha^{-1}(M_Z)$. The dashed vertical
line is the mean central value. References to the authors are in \cite{JEGER,YND,DAVIE,ALFA}.}
\end{center}
\end{figure}
\nin 
Also a detailed comparison of each region of energy with the most recent
work of \cite{YND} shows the same features (agreement and slight difference) like in the case of
$a_\mu$ in SN, due to the slight difference in the parametrization of the data and spectral function.
However, the final results are comparable. Finally, one can remark that due to the high-energy
constant behaviour of the QED kernel function in this case, the low-energy region is no longer
dominating. For
$a_\mu$, the contribution of the
$\rho$-meson below 1 GeV is 68\% of the total contribution, while the sum of the QCD continuum is only
7.4\% (see e.g. SN). Here the situation is almost reversed: the contribution of the
$\rho$-meson below 1 GeV is only 2\%, while the sum of the QCD-continuum is 73.6\%. For this reason,
improvement due to the new Novosibirsk $e^+e^-$ data \cite{NOVO} in the low-energy region will not be very
significant. At present, new BES data \cite{BES} in the $J/\psi$ region are also available, which can be
alternatively used.  Below the $J/\psi$ resonances, the BES data are in
excellent agreement with the QCD parametrization to order $\alpha_s^3$ used here for 3 flavours,
justifying the accuracy of your input. Above the $J/\psi$ resonances, the parametrization used here (sum of
narrow resonances +QCD continuum away from thresholds) can also be compared with these data. On can
notice that, in the resonance regions, the BES data are more accurate than previous ones, which may
indicate that our quoted errors in Table 1 for the $J/\psi$ family contributions are overestimated. In
addition, the threshold of the QCD continuum which we have taken above the $J/\psi$ resonances, matches
quite well with the one indicated by the BES data. We expect that with this new improved estimate of
$\alpha(M_Z)$, strong constraint on the Higgs mass can be derived.
\section{The Muonium Hyperfine Splitting}
Our final result from Table 1:
\beq\label{nu}
\Delta\nu_{\rm had}=(232.5\pm 3.2)~{\rm Hz}
\eeq
is shown in Table 2 in comparison with other determinations, where there is an excellent
agreement with the most recent determination \cite{HFS}. 
\begin{table}[hbt]
\begin{center}
% space before first and after last column: 1.5pc
% space between columns: 3.0pc (twice the above)
\setlength{\tabcolsep}{1.5pc}
% -----------------------------------------------------
% adapted from TeX book, p. 241
%\newlength{\digitwidth} \settowidth{\digitwidth}{\rm 0}
%\catcode`?=\active \def?{\kern\digitwidth}
% -----------------------------------------------------
\caption{Recent determinations of $\Delta\nu_{\rm had}$}
\label{tab:effluents}
\begin{tabular}[H]{ll}
\hline
&\\
Authors&$\Delta\nu_{\rm had}$ [Hz]\\
&\\
\hline
&\\
FKM 99 \cite{FAUST}&$240\pm 7$\\
CEK 01 \cite{HFS}&$233\pm 3$\\
SN 01 (This work)&$232.5\pm 3.2$\\
&\\
\hline
\end{tabular}
\end{center}
\end{table}
\nin
Here, due to the $(\ln t)/t$ behaviour of
the kernel function, the contribution of the low-energy region is dominant. However, the $\rho$-meson
region contribution below 1 GeV is 47\% compared with the 68\% in the case of $a_\mu$, while the QCD
continuum is about 10\% compared to 7.4\% for $a_\mu$. The accuracy of our result is mainly due to the
use of the
$\tau$-decay data, explaining the similar accuracy of our final result with the one in \cite{HFS} using
new Novosibirsk data. The agreement with \cite{HFS} canbe understood from the agreement of
the averaged correlated $e^+e^-$ and $\tau$-decay data compiled in \cite{DAVIE} with the new Novosibirsk
data used in
\cite{HFS}. We differ with DH98 \cite{ALFA} in the treatment of the QCD contribution
\footnote{For more details, see \cite{SN}.}. For light quarks,
QCD is applied in the region where non-perturbative contributions are inessential. For heavy quarks, 
perturbative QCD is applied far from heavy quark thresholds, where it can be unambiguously
used. Adding to this result, the QED contribution up to fourth order,the lowest order electroweak
contribution
\cite{HFS}, and an estimate of the higher order
weak and hadronic contributions
\cite{POPOV}:
\bea
\Delta\nu_{\rm QED}&=&4~270~819(220)~{\rm Hz}\nnb\\
\Delta\nu_{\rm weak}(l.o)&=&-\frac{G_F}{\sqrt{2}}m_em_\mu\ga\frac{3}{4\pi\alpha}\dr\nu_F
\simeq
-65~{\rm Hz}~,\nnb\\
|\Delta\nu_{\rm weak}(h.o)|&\approx& 0.7~{\rm Hz}~,\nnb\\
\Delta\nu_{\rm had}(h.o)&\simeq&7(2)~{\rm Hz}~,
\eea
one obtains the Standard Model (SM)  prediction:
\bea
\nu_{\rm SM}\equiv \nu_F+\Delta\nu_{\rm QED}+\Delta\nu_{\rm weak}+\Delta\nu_{\rm had}+
\Delta\nu_{\rm had}(h.o)~.
\eea
If one uses the relation:
\beq
\nu_F=\rho_F\ga\frac{\mu_\mu}{\mu^e_{B}}\dr \frac{1}{(1+m_e/m_\mu)^3}~,
\eeq
with:
\beq
\rho_F=\frac{16}{3}(Z\alpha)^2 Z^2cR_\infty~,
\eeq
one would obtain:
\beq
\nu_{\rm SM}=4~463~302~913(511)(34)(220)~{\rm
Hz}~,
\eeq
 where the two first errors are due to the one of the Fermi splitting energy. The first largest one
being induced by the one of the ratio of the magnetic moments. The third error is due to the 4th order QED
contribution where, one should notice that, unlike the case of $a_\mu$, the dominant errors come from the
QED calculation which should then be improved.
Unfortunately, these previous errors are still too large and obscure the effects of the electroweak and
hadronic contributions. One the opposite, the data are very precise \cite{EXP}:
\beq
\nu_{\rm exp}=4~463~302~776(51)~{\rm Hz}~.
\eeq
Therefore, at present, we find, it is more informative to extract the Fermi splitting energy $\nu_F$
from a comparison of the Standard Model (SM) prediction with the experimental value of $\nu$. Noting
that
$\nu_F$ enters as an overall factor in the theoretical contributions, one can rescale the previous
values and predict the ratio:
\bea
{\nu_{\rm SM}\over\nu_F}=1.000~957~83(5)~.
\eea
Combining this result with the previous experimental value of $\nu$, one can deduce the
SM prediction:
\beq\label{nf}                   
\nu_F=4~459~031~783(226)~{\rm Hz}~,
\eeq
where the error is dominated here by the QED contribution at fourth order. However, this result is a
factor two more precise than the determination in \cite{HFS} given in Eq. (\ref{eqhfs}), where the
main error  in Eq. (\ref{eqhfs}) comes from the input values of
the magnetic moment ratios. Using this result in Eq. (\ref{nf}) into the expression:
\beq
\nu_F=\rho_F\ga\frac{m_e}{m_\mu}\dr\frac{1}{(1+m_e/m_\mu)^3}\ga
1+a_\mu\dr~,
\eeq
where: 
\beq
\rho_F=\frac{16}{3}(Z\alpha)^2 Z^2cR_\infty~,
\eeq
and
$Z=1$ for muonium, $\alpha^{-1}(0)$=137.035 999 58(52) \cite{HUGHES}, $cR_\infty$ =3 289 841 960 368(25)
kHz
\cite{MOHR} and $a_\mu=1.165~920~3(15)\times 10^{-3}$ \cite{GM2},  
one can extract a value of the ratio of the muon over the electron mass:
\beq\label{emuon}
\frac{m_\mu}{m_e}=206.768~276(11)~,
\eeq
to be compared with the PDG value $206.768~266(13)$ using the masses in MeV units, and with the one
from
\cite{HFS}:
$206.768~276(24)$. After inserting the previous value of $m_e/m_\mu$ into the alternative (equivalent)
relation:
\beq
\nu_F=\rho_F\ga\frac{\mu_\mu}{\mu^e_{B}}\dr \frac{1}{(1+m_e/m_\mu)^3}~,
\eeq
one can deduce the ratio of magnetic moments:
\beq\label{muratio}
\frac{\mu_\mu}{\mu^e_{B}}=4.841~970~47(25)\times 10^{-3}~,
\eeq
compared to the one obtained from the PDG values of $\mu_\mu/\mu_p$ and $\mu_p/\mu^e_B$ \cite{PDG}:
$
{\mu_\mu}/{\mu^e_{B}}=4.841~970~87(14)\times 10^{-3}~.
$
In both applications, the results in Eqs. (\ref{emuon}) and (\ref{muratio}) are in excellent agreement with
the PDG values. These remarkable agreements can give strong constraints to some
contributions beyond the Standard Model and should be explored. 
%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
We have evaluated the hadronic and QCD contributions $\Delta\alpha_{\rm had}$ and $\Delta\nu_{\rm had}$
respectively to the QED
running coupling and to the Muonium hyperfine splitting. Our results shown in Eqs. (\ref{alfa1}) and
(\ref{nu}), are in excellent agreement with existing determinations shown in Fig. 1 and Table 2 and are quite
accurate. These results have been obtained  within the same strategy and data input as the one of the
anomalous magnetic moment obtained previously in SN
\cite{SN}. For this reason, they are self-contained outputs. One of the immediate consequences of
these results is the prediction of
$\alpha(M_Z)$ given in Eq. (\ref{alfa2}), while we have used the result for the muonium hyperfine
splitting for a high precision measurement of the ratios of the muon over the electron mass given in Eq.
(\ref{emuon}) and of magnetic moments given in Eq. (\ref{muratio}). These Standard Model predictions are in
excellent agreement with the ones quoted by PDG \cite{PDG}. These agreements can
be used for providing strong constraints on some model buildings beyond the Standard Model.
%%%%%%%%%%%%%%%%%%%%% 
\section*{Acknowledgements} It is a pleasure to thank W-Y. Pauchy Hwang for the hospitality at
CosPA-NTU (Taipei), where this work has been done. 
%\vfill \eject
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\end{document}

