%Title: Modeling the Pion and Kaon Form Factors in the Timelike Region
%Author: Christine Bruch, Alexander Khodjamirian, Johann H. Kühn
%Published: * Eur.Phys.J. * ** C39 ** (2005) 41-54.
%hep-ph/0409080
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\begin{document}
\begin{titlepage}
\begin{flushright}
TTP04-20\\
SI-HEP-2004-09\\
\end{flushright}
\vfill
\begin{center}
{\Large\bf Modeling the Pion and Kaon Form Factors in the Timelike Region}
\\[2cm]
{\large\bf Christine Bruch~$^{a)}$, Alexander Khodjamirian~$^{b)}$,
Johann~H.~K\"uhn~$^{a)}$ }\\[0.5cm]
{\it $^a)$Institut f\"ur Theoretische Teilchenphysik, Universit\"at
Karlsruhe,\\ D-76128 Karlsruhe, Germany}\\[0.2cm]
{\it $^b)$ Theoretische Physik 1, Fachbereich Physik, Universit\"at Siegen,\\
D-57068 Siegen, Germany}\\
\end{center}
\vfill
\begin{abstract}
New, accurate measurements of the pion and kaon electromagnetic form factors
are expected in the near future from experiments at electron-positron
colliders,
using the radiative return method. We construct a model for the timelike pion
electromagnetic form factor, that is valid also at momentum transfers
far above the $\rho$ resonance. The ansatz
is based on vector dominance and includes a pattern of radial excitations
expected from dual resonance models.
The form factor is fitted to the existing data in the timelike region, continued
to the spacelike region and compared with
the measurements there and with the QCD predictions.
Furthermore, the model is extended to the kaon
electromagnetic form factor. Using isospin
and SU(3)-flavour symmetry relations
we extract the isospin-one contribution and predict the kaon weak form factor
accessible in semileptonic $\tau$ decays.
\end{abstract}
\vfill
\end{titlepage}
\section{Introduction}
The pion electromagnetic (e.m.) form factor $F_\pi(s)$,
one of the traditional study objects in hadron physics,
nowadays plays an essential role for the
precise determination of electroweak observables.
An accurate knowledge of $F_\pi(s)$ at timelike momentum
transfers $s>4m_\pi^2$ is needed
to calculate the hadronic loop contribution
to the muon anomalous magnetic moment and to the running of the e.m. coupling
(see \cite{g2,g21} for the current status).
New accurate data on the pion form factor in the timelike region
have recently been obtained by the
CMD-2 collaboration \cite{CMD2002} measuring the
$e^+e^-\to \pi^+\pi^-$ cross section at $\sqrt{s}= 0.61\div 0.96$ GeV
(for an update of these data see \cite{CM2update}).
In this region, it is successfully described (fitted)
using models based
on $\rho$-meson dominance, with a small but clearly visible
$\omega$-meson admixture. Above 1~GeV
data on $e^+e^-\to \pi^+\pi^-$ \cite{ADONE,Barkov,DM2}
exist but are not that accurate.
Employing isospin symmetry one also gains independent
information from the measurements of
$\tau \to \pi^-\pi^0 \nu_\tau$ at $sm_\rho^2$ is a
complicated object determined by
a large or even infinite amount of hadronic parameters
not accessible at present in a rigorous theoretical framework.
On the other hand, at sufficiently large $s$
one expects $F_\pi(s)\sim \alpha_s(s)/s$, as predicted from
perturbative QCD \cite{exclusive} for
the pion form factor in the spacelike region at $s\to -\infty$,
analytically continued to $s \to +\infty$.
In other words, the overlap of many
intermediate hadronic states has to build up a smooth,
power-behaved function.
One might consider using this QCD prediction in the region of
present interest, that is, at intermediate timelike $s$.
However, data on the form factor in the spacelike
region, $s<0$, indicate that the onset of this asymptotic behaviour is
far from the ``few GeV$^2$'' region. Preasymptotic contributions
$\sim 1/s^n$ with $n>1$, stemming from the end-point, soft mechanism
\cite{endpoint} are essential for $s\sim 1-10 $ GeV$^2$.
Approximate methods valid at intermediate spacelike momenta, for example
QCD sum rules \cite{sr,lcsr,BK}, allow to calculate
$F_\pi(s)$ including soft effects. However, a straightforward
analytic continuation of $F_\pi(s<0)$ to large $s>0$ is difficult.
The timelike form factor will suffer
from uncertainties in
the analytic continuation of soft parts,
Sudakov logs and of $\alpha_s(s)$ (for a
discussion see \cite{GP,BRS}). Hence, QCD
calculations cannot be directly used
in the large $s>0$ region, e.g., for
estimating the ``tail'' of higher resonances
in the form factor.
In this paper we therefore prefer to adopt a model for the pion form-factor
formulated entirely in terms of hadronic degrees of freedom.
Importantly, the form factor,
as obtained from the model, fitted in the time-like region and
extrapolated to spacelike momenta,
has a proper power-law behaviour which can be compared with the
data at $s<0$ and used to test various QCD-based predictions.
Models of hadronic amplitudes, where an infinite
series of resonances at $s>0$ is summed to yield
a power-law behaviour at $s<0$, are rooted in
the Veneziano amplitude and dual resonance models formulated long
before the advent of QCD.
Importantly, the pattern of infinite zero-width resonances
is predicted in the $N_c=\infty$ limit of QCD.
Recently, the model for the pion form factor
using the masses and coefficients
chosen according to Veneziano amplitude was considered in \cite{Dom}.
Earlier, similar analyses of the pion form factor can be found in
\cite{olddual}. Models of dual-resonance type with infinite
number of resonances are widely used also for other hadronic problems,
some recent works can be found in \cite{newdual}.
We will use the dual-QCD$_{N_c=\infty}$ model \cite{Dom}
as a starting point, modifying it
for the first few $\rho$ resonances, by keeping their parameters
(masses, widths and coefficients) free and fitting them
to experiment. In this way,
the complicated effects of $\rho$ resonances coupled
to multihadron ($2\pi,4\pi$ etc.) states
are implicitly taken into account.
It is remarkable that the gross features of the model are well
reproduced with the (fitted) resonance parameters.
Since in the dual-resonance amplitude
the coefficients of higher resonance contributions decrease
with the resonance number,
the corresponding modifications for individual higher states
are not important, and the ``tail' of resonances
is treated as in \cite{Dom}.
The model for the pion form factor is also analytically continued to
the spacelike region and compared with the data there and with the
QCD predictions on $F_\pi(s)$ at large spacelike $s$.
Furthermore, we extend the model to the kaon
form factor, employing an SU(3)-generalization of the pion
amplitude. Fitting the charged and neutral
kaon form factors to the data, and using flavour symmetries,
we predict the weak kaon
form factor relevant for $\tau$ semileptonic decays.
Let us also mention at this point that the decomposition of the form factors
into their isospin zero and one components respectively, is also of relevance for
a model independent evaluation of $\gamma$-Z-mixing \cite{Jegerlehner}
where the two amplitudes contribute with a different relative
weight.
The plan of the paper is as follows. In Section 2 we summarize the phenomenology
of $F_\pi(s)$ recalling the derivation of the
$\rho$-meson contribution, whereas in Section 3 the contributions
of excited $\rho$ resonances are discussed. The model for the pion
form factor is introduced in Section 4 and its parameters are fitted to the data.
In Section 5 we proceed to the kaon e.m. form factors and
Section 6 is devoted to the weak kaon form factor in $\tau$ decays.
Section 7 contains our summary and conclusions.
\section{The $\rho$-meson contribution to the pion form factor}
The pion e.m. form factor is defined in the standard way,
\be
\langle \pi^+(p_1)\pi^-(p_2)\mid j^{em}_\mu \mid 0 \rangle
=(p_1-p_2)_\mu F_\pi(s)
\label{formf}
\ee
The quark e.m. current
$
j^{em}_\mu=\sum\limits_{q=u,d,s} e_q \bar{q}\gamma_\mu q\,
\label{jem}
$
can be decomposed into isospin one and zero
components respectively. At the quark level this corresponds to
\be
j_\mu^{em} =\frac{1}{\sqrt{2}} j_\mu^{3} + \frac{1}{3\sqrt{2}} j_\mu^{I=0} -
\frac{1}{3} j_\mu^{s}
\ee
with
\be
j_\mu^3 = (\bar u\gamma_\mu u - \bar d \gamma_\mu d )/\sqrt{2}\,,~~
j_\mu^{I=0} = (\bar u\gamma_\mu u + \bar d\gamma_\mu d )/\sqrt{2}\,,~~
j_\mu^{s} = \bar s \gamma_\mu s\,.
\label{components}
\ee
The isotriplet partners of the current $j_\mu^3$ form the
charged weak current
\be
j_\mu^-= (j^1_\mu+ij^2_\mu)/\sqrt{2}=\bar{u}\gamma_\mu d .
\label{charged}
\ee
In the isospin symmetry limit, the $I=0$ and $s$-quark components
of the current do not contribute to $F_\pi$. In Eq.~(\ref{formf}),
$s=(p_1+p_2)^2$ is the timelike momentum
transfer squared, $s \geq 4m_\pi^2$. The form factor $F_\pi(s)$,
being analytically continued to the spacelike region $s< 0$,
corresponds to the
hadronic matrix element $ \langle \pi^+(p_1)\mid j_\mu^{em} \mid \pi^+(-p_2)\rangle$
related to Eq.~(\ref{formf}) by crossing-symmetry.
There are very
few model-independent relations determining or constraining
the pion form factor. One of them is the normalization condition for the pion electric charge,
\be
F_{\pi}(0)=1\,.
\label{norm}
\ee
An important role is played by the dispersion relation,
\be
F_{\pi}(s)=\frac1{\pi}\!\!\int\limits_{4m_\pi^2}^\infty
\!\!\!ds \frac{\mbox{Im}F_{\pi}(s')}{s'-s-i\epsilon}\,.
\label{disp}
\ee
The asymptotic behaviour
expected from perturbative QCD \cite{exclusive},
\be
\lim\limits_{s \to -\infty}
F_{\pi}(s) \sim \frac{\alpha_s}{s}\,,
\ee
allows for an unsubtracted dispersion relation.
The application of Eq.~(\ref{disp})
is based on an independent equation for the imaginary part of the
form factor derived from the unitarity condition,
\be
2\mbox{Im}F_{\pi}(s)(p_1-p_2)_\mu =
\sum\limits_h \int d\tau_h \langle \pi^+(p_1)\pi^-(p_2)\mid h\rangle \langle h
\mid j^{em}_\mu|0\rangle^*\,,
\label{unitar}
\ee
where all possible hadronic states $h$ with
$J^{PC}(I^G)=1^{--}(1^{+})$ are inserted. Each term in the sum in
Eq.~(\ref{unitar}) includes the integration over the phase space and
the summation over
polarizations of the intermediate state $h$. Since isospin symmetry is not
exact, there could also be a small admixture
of the isospin-zero $J^{PC}=1^{--}$ states, e.g. the $\omega$ and
its radial excitations.
Experimental data reveal that at low $s\leq$ 1 GeV$^2$
the most important contribution to Eq.~(\ref{unitar}) stems from
$\rho$ meson. The $\rho$-meson decay constant:
\be
\langle \rho^0 \mid j_\mu^{em}\mid 0 \rangle =
\frac{m_\rho f_\rho}{\sqrt{2}}\epsilon^{(\rho)*}_\mu \,,
\label{vect}
\ee
and the strong $\rho\pi\pi$ coupling:
\be
\langle \pi^+(p_1)\pi^-(p_2) |\rho^0\rangle=
(p_2-p_1)^\alpha\epsilon^{(\rho)}_\alpha g_{\rho \pi\pi}\,,
\label{coupl}
\ee
(where $\epsilon^{(\rho)}$ is
the $\rho$-meson polarization four-vector) determine
the $\rho$-contribution to the
imaginary part of the pion form factor in the narrow width approximation:
\be
\mbox{Im}F^{(\rho)}_{\pi}(s)= \frac{m_\rho f_\rho}{\sqrt{2}}g_{\rho\pi\pi}\pi\delta(s-m_\rho^2)\,.
\ee
Substituting this into the dispersion relation (\ref{disp}) gives:
\be
F^{(\rho)}_\pi(s)=
\frac{m_\rho f_\rho g_{\rho\pi\pi}}{\sqrt{2}(m_\rho^2-s-i\epsilon)}\,.
\label{Fpirho}
\ee
The excited $\rho', ...$ resonances have contributions of the same form,
with the decay constants $f_{\rho',..}$ and strong couplings
$g_{\rho'\pi\pi},...$.
The pion form factor constructed by adding up the zero-width
$\rho,\rho',...$ -resonances
is an oversimplified ansatz which cannot be used at $s>0$ where
the experimentally observable large widths of these resonances
are important. The widths are generated
by the contributions of multi-hadron states to the
imaginary part of $F_\pi$, starting from the lowest two-pion state.
The contribution of the latter to the unitarity relation
\be
2\mbox{Im}F^{(2\pi)}_{\pi}(s)(p_1-p_2)_\mu=
\int d\tau_{2\pi} (p'_1-p'_2)_\mu A_{\pi\pi}(s)F_\pi^*(s)
\,,
\label{unitar2pi}
\ee
involves the two-pion phase space :
$$
d\tau_{2\pi}=\frac{d^3p'_1}{(2\pi)^3 2E'_1}\frac{d^3p'_2}{(2\pi)^3 2E'_2}
(2\pi)^4\delta^4(p'_1+p'_2-p_1-p_2)
$$
where $(p'_1+p'_2)^2=s$ and $A_{\pi\pi}(s)$ is the amplitude of
the strong pion-pion P-wave elastic scattering:
\be
A_{\pi\pi}(s)=\langle \pi^+(p_1)\pi^-(p_2) | \pi^+(p'_1)\pi^-(p'_2)\rangle_{I=1,J=1}\,.
\label{2piampl}
\ee
In the low-energy region $4m_\pi^2~~4}$ states
is taken as in the dual-QCD$_{N_c=\infty}$ model.
However, having in mind the insufficient precision of the current data at $\sqrt{s}>$
1 GeV, we restrict the number of free fit parameters
to the coefficients $c_{0,1,2}$,
the masses $m_0$ and $m_1$ and the total widths $\Gamma_0$ and $\Gamma_1$.
The values of $\Gamma_2$ and $m_2$ are taken from \cite{PDG}.
The coefficient $c_3$ of $\rho_3$ is fixed from
the normalization condition for the form factor:
\be
c_3=1-(c_0+c_1+c_2)_{fit} -\left(\sum_{n=4}^{\infty} c_n
\right)_{dual-QCD_{N_c=\infty}}\,.
\label{c3}
\ee
All remaining parameters in Eq.~(\ref{themodel}),
that is $c_{n\geq 4}$, $m_{n\geq 3}$,
and $\Gamma_{n\geq 3} $ are
calculated from Eqs.~(\ref{cn}), (\ref{mn}) and (\ref{gamman}), respectively.
Let us emphasize the main qualitative features of this model.
It nicely matches the existing $\rho$-dominance
models at $\sqrt{s}<$ 1 GeV, such as the ones considered
in \cite{KS}, simply because the $\rho_{n>2}$ states play a minor role
in the $\sqrt{s}<1$ GeV region. The model (\ref{themodel})
is flexible, that is, it allows to vary the proportion of fitted
and modelled resonances above $\rho$.
E.g., with sufficiently precise data at higher energies one can include also the $\rho_4$-state
into the 'fit' part. Alternatively, $\rho_{3}$ can be removed from the 'fit'
part and added to the dual-QCD$_{N_c=\infty}$ part. Furthermore, as
mentioned already, $F_\pi(s)$ as given by Eq.~(\ref{themodel}), can be easily continued
to $s<0$ and compared with the experimental data
and QCD predictions in the spacelike region.
Accordingly, one gets a smooth power-like behaviour at
asymptotically large timelike $s$. Altogether, the model contains a
reasonably small amount of free parameters: three
per each fitted resonance (the mass, coefficient and total width)
and three ``global'' parameters $\alpha',\beta,\gamma$,
with $\alpha'=1/(2m_\rho^2$) taken from the Regge trajectory,
$\beta$ taken from Eq.~(\ref{cn})
with $c_0$ derived from the fit, and with $\gamma =0.2$.
In principle, one can try
to relax and independently fit also the three ``global'' parameters,
but we prefer to keep them fixed.
Needless to say, the suggested ansatz has considerable room
for improvement, especially concerning the treatment
of the effective $s$-dependent resonance widths.
We have fitted the model (\ref{themodel}) to the existing data
\cite{CMD2002,ADONE,Barkov,DM2}
for the timelike pion form factor.
The results of the fit for the two different versions of
BW-formulae: KS (Eq.~(\ref{KS1}) for all resonances) and GS (Eq.~(\ref{GS1})
for the first three resonances $\rho_{0,1,2}$), together with
the relevant input parameters are presented in Table~\ref{tab:pion} and compared
with the predictions of the dual-QCD$_{N_c=\infty}$ model.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[h]
%\begin{center}
%\vskip0.2cm
\begin{tabular}{|c|c|r|r|r|r|}
\hline
Parameter & Input & Fit(KS) & Fit(GS) & dual- &
PDG value \\
&&&&QCD$_{N_c=\infty}$& \cite{PDG}\\
\hline
$m_\rho$ & - & 773.9\,$\pm$\,0.6& 776.3\,$\pm$\,0.6&input & 775.5\,$\pm$\,0.5 \\
$\Gamma_{\rho}$ & - & 144.9\,$\pm$\,1.0& 150.5\,$\pm$\,1.0 & input & 150.3\,$\pm$\,1.6 \\
\hline
$m_{\omega}$ & 783.0 & - & - & - & 782.59\,$\pm$\,0.11 \\
$\Gamma_{\omega}$ & \,\,\,\,\,\,8.4 &- &- & - & 8.49\,$\pm$\,0.08 \\
\hline
$m_{\rho'}$ &- & 1357\,$\pm$\,18 & 1380\,$\pm$\,18& 1335 & 1465\,$\pm$\,25 \\
$\Gamma_{\rho'}$ &- & 437\,$\pm$\,60& 340\,$\pm$\,53& 266 & 400\,$\pm$\,60 \\
\hline
$m_{\rho''}$ & 1700 &- &- &1724 & 1720$\,\pm$\,20 \\
$\Gamma_{\rho''}$ & \,\,\,240 &- &- & 344& 250\,$\pm$\,100 \\
\hline
$m_{\rho'''}$& - &- &- & 2040 &-\\
$\Gamma_{\rho'''}$ &- &- &- & 400& - \\
\hline
\hline
$c_0$ &-& 1.171$\pm$0.007&1.098$\pm$0.005&1.171 &-\\
$\beta$& $c_0$ and Eq.~(\ref{cn})
&2.30$\pm$0.01&2.16$\pm0.015$ & 2.3(input)&-\\
$c_\omega$&0.00184(KS)& -&-& -&-\\
&0.00195(GS)& & & &-\\
\hline
$c_1$&-&-0.119\,$\pm$\,0.011&-0.069\,$\pm$\,0.009&-0.1171&-\\
$c_2$&-&0.0115\,$\pm$\,0.0064&0.0216\,$\pm$\,0.0064&-0.0246&\\
$c_3$&Eq.~(\ref{c3})&-0.0438\,$\mp$\,0.02&-0.0309\,$\mp\,0.02$ &-0.00995&-\\
$\sum\limits_{n=4}^{\infty} c_n$& -0.01936
&-&-&-0.01936&-\\
\hline
$\chi^2/d.o.f.$&- &155/101& 153/101 &-&-\\
\hline
\end{tabular}
\caption{{\it Parameters of the pion form factor (\ref{themodel})
and results of the fit to the data. Masses and widths
are given in MeV. The row 'Fit KS(GS)' contains the fitted values
for the case where all $BW_n$ are taken as in Eq.~(\ref{KS1})
($BW_{0,1,2}$ taken as in Eq.~(\ref{GS1})). The sum over $c_n$
in the last line is calculated from Eq.~(\ref{cn}). The PDG parameters
for $\rho(770)$ are those listed in \cite{PDG}
for ``Charged only, $\tau$ decays and $e^+e^-$''. The parameter $c_\omega$ is taken from \cite{KS}.}}
\label{tab:pion}
%\end{center}
\end{table}
%%%%%%%%%%%%%%
The results for $|F_\pi(s)|^2$ are plotted in Fig.~1, separately
for the $\sqrt{s}<1$ GeV and $\sqrt{s}>1$ GeV regions.
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering
\vspace{-1cm}
%\subfigure[]{
%\hspace{-1.0cm}
\includegraphics[width=0.7\textwidth]{fig1a.ps}\label{fig-Fpitmla}
%}%
(a)
\\
%\subfigure[]{
%\hspace{-1.2cm}
\includegraphics[width=0.7\textwidth]{fig1b.ps}
\label{fig-Fpitmlb}
%}%
(b)
\caption{
\it The pion form factor squared $|F_\pi(s)|^2$ as a function of $\sqrt{s}$
fitted to the data in the region near (a) and above (b) $\rho$ resonance.
The solid (dashed) line
corresponds to the KS(GS) parameterization of BW formula. Data
are taken from \cite{CMD2002,Barkov}(crosses),\cite{ADONE}(stars),
and \cite{DM2}(squares). The triangle point is the value of the form factor
extracted from $J/\psi\to 2\pi$.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
A few comments are in order.
The most spectacular result of the fit is the change of the
$\rho''$ coefficient $c_2$
with respect to the model \cite{Dom} and to the earlier
fits \cite{KS}, from negative values $\sim -(0.02-0.04)$ to smaller but
positive values $\sim 0.01$-$0.02$. The positive sign is a direct consequence
of the dip in the cross section around 1.6 GeV
(also the earlier fit of the data in \cite{DM2} revealed a similar pattern).
Furthermore, the fitted mass of $\rho'$ gets shifted
with respect to the PDG value, the latter
obtained by adding together data from all decay
channels of $\rho'$. Note that a lower $m_{\rho'}$
consistent with our fit is also obtained by the fits in \cite{KS}
and predicted by the dual-QCD$_{N_c=\infty}$ model. It is quite probable
that a more elaborated model of the total width of $\rho'$
including multiparticle thresholds will shift the mass.
We note that the values of the masses and widths of $\rho$, $\rho'$,
$\rho''$ as well as $c_0$
and $c_1$ are in the ballpark of the dual-QCD$_{N_c=\infty}$ model.
The same is true for the magnitude of $c_2$, its positive sign is
enforced by the dip around 1.6 GeV and might be a consequence of
the strong mixing between $\rho''$ and nearby resonances.
In general, the $\rho'$, $\rho''$ and $\rho''$ terms
and their interplay with the contribution of $\rho$
in both imaginary and real parts of the form factor
are the main effects which determine the behavior
of $|F_\pi(s)|^2$
at $\sqrt{s}>1~\mbox{GeV}$.
In particular,
the dip in the form factor observed near $\sqrt{s}=1.6 $ GeV
can only be described by altering the sign of $c_2$.
The role of the summed ``tail''
of $\rho_{n\geq 4}$ states is less important.
One has to admit that the quality of the fit is not very high,
(we get typically $\chi^2/d.o.f.\simeq 1.5$), in fact
this could simply indicate inconsistencies in the normalization of
the various pieces of existing data in the timelike region.
Moreover, the data points with large errors at $\sqrt{s}=2.5\div 3$ GeV
are systematically higher than the fitted curve, and the value
of $F_\pi(\sqrt{s}=m_{J/\psi})$ extracted from the $J/\psi\to 2 \pi$
partial width is larger than the model prediction
by a factor of about three (this point was not included into the fit).
This disagreement deserves a comment.
Note that the extraction of $F_\pi(m_{J/\psi}^2)$ is
``theoretically biased'', because one tacitly assumes that
the intermediate photon exchange is the only mechanism in this
decay. Other mechanisms such as one-photon plus two
intermediate gluons could also be important in the $J/\psi \to
2\pi$. Hence, the hadronic matrix element
in this isospin-violating transition could be actually different
from the pion form factor. Although estimates \cite{Milana} of gluonic effects
based on the perturbative charmonium annihilation
are in favour of their smallness, we still think there is
room for nonperturbative effects which are however not easily
assessed. For the time being it seems difficult to accommodate a pion
form factor as large as the one derived from $J/\psi$ decay.
(It will be interesting to check for a similar effect
in the $4\pi$ mode.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG.2
\begin{figure}
\vspace{-1cm}
\centering
%\subfigure[]{\hspace{-1.0cm}
\includegraphics[width=0.7\textwidth]{fig2a.ps}\label{fig-Fpispcla}
%}%
(a)
\\
%\subfigure[]{\hspace{-1.2cm}
\includegraphics[width=0.7\textwidth]{fig2b.ps}\label{fig-Fpispclb}
(b)
%}%
\caption{
\it The pion form factor squared
$|F_\pi(s)|^2$ as a function of $s$
in the spacelike region at low $|s|$ (a) and large $|s|$ (b).
The solid (dashed) line corresponds
to the analytic continuation of the timelike form factor
(\ref{themodel}) with the KS(GS) parameterizations of the BW formula. Data
are taken from \cite{Amendolia}(crosses), \cite{Jlab}(open circles)
and \cite{Bebek} (full squares).
Dotted lines at $|s|> 1 \mbox{GeV}\,^2$ represent the interval derived from
QCD light-cone sum rule predictions \cite{BK}.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
To check our model further we continue $F_\pi(s)$ to $s<0$ and compare
the result with the data there (see Fig. 2). The form factor
obtained from direct electron-pion scattering
at small $s<0$ (up to $|s|= 0.25~\mbox{GeV}^2$)~\cite{Amendolia}
is not sensitive to the contributions
of higher than $\rho$ resonances, provided the
normalization to one at $s=0$ is imposed.
Note that the fitted form factor (\ref{themodel}) predicts
$\langle r_\pi \rangle^2 $= 0.440(0.426) fm$^2$ for the KS(GS) versions,
close to the dual-QCD$_{N_c=\infty}$ prediction (\ref{rpion})
and to the experimental value \cite{Amendolia} quoted above.
At larger $|s|$ the old data \cite{Bebek} obtained
from pion electroproduction have large errors and suffer from
some intrinsic uncertainties \cite{Carlson}. More accurate data
obtained recently at JLab \cite{Jlab}
at $|s|<1.6 ~\mbox{GeV}^2$ are in agreement with the model, but not sensitive
to the details of the fit.
Altogether there is reasonable agreement
between the model and the QCD light-cone sum rule (LCSR) predictions
\cite{lcsr}. The latter are taken from \cite{BK} with their
estimated theoretical uncertainty. The pion form factor
calculated from 3-point QCD sum rules
\cite{sr} in the region of their validity $|s|\sim 1-4~ \mbox{GeV}^2$
is within the LCSR interval and therefore not shown separately.
If we try to artificially enhance the form factor
at large timelike region, e.g. by enhancing
the contribution of the ``tail'' trying to fit also
the point at $\sqrt{s}=m_{J/\psi}$ the form factor at spacelike $s<0$
increases correspondingly, and the general agreement between data
and QCD sum rule predictions is lost.
We conclude therefore, just as before, that it is implausible for
the form factor obtained on the basis of dual
resonance approach
to reach values $|F_\pi(s)|^2 \geq 0.01$ at
$\sqrt{s}= 2.5\div 3 $ GeV without conflicting with
the spacelike data and especially with QCD predictions.
This statement is independent of many details
involved in the timelike form factor model and in the QCD calculations.
It is therefore extremely interesting
to obtain new accurate data at least up to $\sqrt{s}=2.5$ GeV
to check this conjecture.
\section{Charged and neutral kaon e.m. form factors}
We now adopt the analogous strategy to describe the kaon form
factors. Combining information on $K^+K^-$ and $K^0 \bar K^0$ production
with constraints from isospin symmetry, and using assumptions deduced
from the quark model and the OZI rule, it is possible to separate the
$I=1$ and $I=0$ amplitudes in the form factor, and even the $\omega$- and
$\phi$-components of the $I=0$ part. The $I=1$ part can then be used to
predict the rate for $\tau\to \nu K^- K^0$.
The electromagnetic form factors for charged and neutral kaons
defined similar to Eq.~(\ref{formf}):
\be
\langle K^+(p_1) K^-(p_2) | j^{em}_\mu | 0 \rangle = (p_1-p_2)_\mu \,F_{K^+}(s)
\label{formfKpl}
\ee
\be
\langle K^0(p_1)\bar K^0(p_2) | j^{em}_\mu | 0 \rangle = (p_1-p_2)_\mu\,F_{K^0}(s)\,,
\label{formfK0}
\ee
obey the constraints
\be
F_{K^+}(0)=1,~~F_{K^0}(0)=0\,.
\label{Knorm}
\ee
They can be separated into their isospin one and zero parts,
\be
F_{K^+(K^0)}=F_{K^+(K^0)}^{(I=1)} + F_{K^+(K^0)}^{(I=0)}.
\ee
From isospin invariance one derives
\be
F_{K^+}^{(I=0)}=F_{K^0}^{(I=0)},~~F_{K^+}^{(I=1)}=-F_{K^0}^{(I=1)},
\ee
and the $I=1$ part can furthermore be used to predict the charged
current matrix element:
\be
\langle K^+(p_1) \bar{K}^0(p_2) | j^-_\mu | 0 \rangle =
(p_1-p_2)_\mu 2F_{K^+}^{(I=1)}(s)\,.
\label{KplK0}
\ee
A simultaneous fit to the two electromagnetic form factors leads,
therefore, to a direct prediction for the rate for $\tau\to \nu K^-
K^0$, without any further assumption.
In the context of vector dominance, combined with the quark model, the
kaon form factors
are saturated by $\rho$, $\omega$, $\phi$
and their radial excitations,
\be
F_K(s)=\sum_{V=\rho,\omega,\phi,\rho',\omega',\phi',...}\frac{\kappa_Vf_V g_{V K\bar{K}}m_V}{m_V^2-s-im_V\Gamma_V}\,,
\label{FPV}
\ee
and it is the $\rho$-mediated $I=1$ part which
enters both the e.m. and the charged current matrix elements.
We define the decay constants of the vector mesons via
\be
\langle V|j_\mu^{em}|0\rangle =\kappa_V
m_V f_V \epsilon_{(V)}^{\mu*}\,,
\label{vect1}
\ee
where $\epsilon_{V}$ is the polarization vector of $V$,
and the coefficients
$\kappa_\rho=1/\sqrt{2}$ (see Eq.~(\ref{vect})),
$\kappa_\omega=1/(3\sqrt{2})$ and
$\kappa_\phi=-1/3$ reflect the valence quark content
of these mesons corresponding to ``ideal'' mixing:
\be
\rho^0=\frac{\bar{u}u-\bar{d}d }{\sqrt{2}},~~
\omega=\frac{\bar{u}u+\bar{d}d }{\sqrt{2}},~~
\phi= \bar{s}s\,.
\label{content}
\ee
The strong coupling is defined as in Eq.~(\ref{coupl}):
\be
\langle K(p_1)\bar{K}(p_2)|V\rangle= (p_2-p_1)^\nu\epsilon_{\nu(V)}g_{VK\bar{K}}\,.
\label{coupl1}
\ee
In the flavour SU(3)-symmetry limit one evidently
has
\be
F_{K^+}(s)=F_{\pi}(s),~~ F_{K^0}(s)=0\,.
\label{su3}
\ee
Since this symmetry
is broken by the strange-nonstrange quark-mass difference,
one has to expect quite noticeable deviations from Eq.~(\ref{su3}),
in particular the $K^0$ form factor does not vanish
at nonzero $s$. Moreover, since $BR(\phi\to K^+K^-)\sim
BR(\phi\to K^0\bar{K}^0)$, the neutral and charged kaon
form factors have almost equal magnitudes near the $\phi$ resonance.
In fact, the SU(3)-breaking also manifests itself
in the valence quark content of vector mesons given in Eq.~(\ref{content})
and in the mass splitting between $m_\rho\simeq m_\omega$ and
$m_\phi$.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.6\textwidth]{fig3.eps}
\end{center}
%\vspace{-1cm}
\caption{{\em The quark diagrams corresponding to the
contributions to the pion and kaon form factors with various flavour combinations. }}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
In order to derive the kaon form factors in
terms of separate vector meson contributions, it is
convenient to consider a generic quark diagram of the
strong $VP\bar{P}$ coupling and distinguish these diagrams
by the presence and position of $s$ quarks.
In the isospin symmetry limit, which we adopt here, there are
three different diagrams depicted in Fig.~3:
1) without strange quarks (upper), 2) with $s$ and $\bar{s}$
in the $P\bar{P}$ state only (middle) and 3) with $s$ and $\bar{s}$
in the $V$ and the $P\bar{P}$ state (bottom).
We denote the corresponding
hadronic invariant amplitudes $A_{qq}$, $A_{qs}$ and $A_{sq}$.
The diagrams with charge conjugated quark lines
lead to the same amplitudes with an additional minus sign,
taking into account the negative C-parity of neutral vector mesons.
The strong couplings of $\rho,\omega,\phi$
are then expressed in terms of diagrams:
\ba
&&g_{\rho^0\pi^+\pi^-}\equiv g_{\rho\pi\pi}= \sqrt{2}A_{qq}\,,\nonumber\\
&&g_{\rho^0 K^+ K^-}= g_{\omega K^+K^-}=\frac{1}{\sqrt{2}}A_{qs}\,,~~
g_{\rho^0 K^0\bar{K}^0}=-g_{\omega K^0\bar{K}^0}=
-\frac{1}{\sqrt{2}}A_{qs}\,,\nonumber\\
&&g_{\phi K^+K^-}=g_{\phi K^0\bar{K^0}}=- A_{sq}\,.
\label{strongV}
\ea
Thus, from the
quark model one expects approximately equal $K\bar K \rho$ and $K\bar K
\omega$ couplings. It is easy to check that this simple formalism
correctly reproduces $g_{\omega\pi^+\pi^-}=g_{\rho\pi^0\pi^0}=0$,
as well as SU(3)-symmetry relations between the couplings.
In what follows we also use the relation between the
decay constants of $\omega$ and $\rho$ following from
Eq.~(\ref{vect1}): $f_\omega=f_\rho$.
Substituting the decay constants and hadronic couplings
(\ref{strongV}) to Eq.~(\ref{FPV}) we obtain
the desired decompositions of the kaon form factors
in terms of vector meson contributions:
\be
F_{K^+}(s)= \frac{f_\rho A_{qs}}{2 m_\rho}BW_\rho(s)+
\frac{f_\rho A_{qs}}{6 m_\omega}BW_\omega(s)
+\frac{f_\phi A_{sq}}{3 m_\phi}BW_\phi(s)\,,
\label{Kformfpm}
\ee
\be
F_{K^0}(s)= -\frac{f_\rho A_{qs}}{2 m_\rho}BW_\rho(s)+
\frac{f_\rho A_{qs}}{6 m_\omega}BW_\omega(s)
+\frac{f_\phi A_{sq}}{3 m_\phi}BW_\phi(s)\,.
\label{Kformf0}
\ee
Written in the same terms the pion form factor is
\be
F_{\pi}(s)= \frac{f_{\rho} A_{qq}}{m_\rho}BW_\rho(s)\,,
\label{pionformf2}
\ee
and $f_\rho A_{qq}/m_\rho=1$ in the simplest version of VDM.
In the SU(3) limit
$f_\rho=f_\phi$, $m_\rho=m_\omega= m_\phi$ , $A_{qq}=A_{qs}=A_{sq}$
and the relations (\ref{su3}) are reproduced. In Eqs.~(\ref{Kformfpm}) and
(\ref{Kformf0}) we will adopt the KS version of BW formulae for $\rho$,
and the analogous $s$-dependent width for $\phi$,
\be
\Gamma_\phi(s)=
\frac{m_\phi^2}{s}\left(\frac{p^K(s)}{p^K(m_\phi^2)}\right)^3\Gamma_\phi\,.
\label{gammaphi}
\ee
with $\Gamma_\phi\equiv \Gamma^{tot}(\phi)$ and $p^K(s)=(s-4m_K^2)^{1/2}/2$.
For simplicity, we assume that the effective threshold of all
$\phi$ decay modes including $\phi\to 3\pi$ is approximated
by Eq.~(\ref{gammaphi}), having in mind that
the non-$K\bar{K}$ channels give only about 20\% of $\Gamma_\phi$.
Naturally, the possibility to use the GS-form in
Eqs.~(\ref{Kformfpm}) and (\ref{Kformf0}) exists,
yielding inessential differences. For simplicity, to avoid complicated
$3\pi$-threshold factors we will use constant
widths for $\omega$, having in mind that the
thresholds are much lower than the boundary of the physical region
of the form factor: $9m_\pi^2\ll4m_K^2$. This is a good
approximation at least for the narrow $\omega$ resonance.
Adding radial excitations to all ground-state
vector mesons is the next natural step.
From the pion form factor analysis
we already learned that the ``tails'' of higher resonances
are numerically inessential. For the kaon form factor we therefore
restrict the analysis to the
excited states $\rho'$, $\rho''$,
$\omega' \equiv \omega(1420)$ , $\omega''\equiv \omega(1650)$
and $\phi'\equiv \phi(1680)$ \cite{PDG}.
Higher excitations, as well as more elaborated $s$-dependent widths,
can be installed in the future when more accurate data will be available.
Since the products of decay constants and strong couplings
in Eqs.~(\ref{Kformfpm}) and (\ref{Kformf0})
will not be separated and have to be fitted as a whole,
it is convenient to introduce again the normalization factors
$c_V^K$ instead of these products. The ansatz
for the kaon form factors
thus reads:
\ba
F_{K^+}(s)= \frac12 (c^K_\rho BW_{\rho}(s)+
c^K_{\rho'} BW_{\rho'}(s)+c^K_{\rho''} BW_{\rho''}(s))
\nonumber
\\+
\frac{1}{6}(c_\omega^K BW_\omega(s)+c_{\omega'}^K BW_{\omega'}(s)+
c_{\omega''}^K BW_{\omega''}(s))\nonumber \\
+\frac{1}3(c_{\phi} BW_{\phi}(s)+c_{\phi'} BW_{\phi'}(s))\,,
\label{Kformfpm1}
\ea
\ba
F_{K^0}(s)= -\frac12 (c^K_\rho BW_{\rho}(s)+
c^K_{\rho'} BW_{\rho'}(s)+c^K_{\rho''} BW_{\rho''}(s))\nonumber\\
+\frac{1}{6}(c_\omega^K BW_\omega(s)+c_{\omega'}^K BW_{\omega'}(s)+
c_{\omega''}^K BW_{\omega''}(s))
\nonumber \\
+\frac{1}3(\eta_\phi c_{\phi} BW_{\phi}(s)+c_{\phi'} BW_{\phi'}(s))\,,
\label{Kformf01}
\ea
The widths are with $p$-wave factors for $\rho$ and
$\phi$ states as explained above, and constant for $\omega$-states,
which is however a rather crude approximation for $\omega',\omega''$.
The ansatz in Eqs.~(\ref{Kformfpm1}) and (\ref{Kformf01}) reflects
isospin invariance and the hierarchy
of vector meson contributions according to their valence-quark
content, however, it allows for the possibility of SU(3) violations
which could and will manifest in differences between the fitted
normalization coefficients.
The additional factor $\eta_\phi$ in Eq.~(\ref{Kformf01})
takes into account the
isospin-breaking difference between the charged and neutral kaon couplings
to $\phi$:
\be
\eta_\phi\equiv \frac{g_{\phi K^0\bar{K}^0}}{g_{\phi K^+K^-}}=
\left(\frac{
BR(\phi\to K^0 \bar{K}^0)(m_\phi^2-4m_{K^+}^2)^{3/2}}{
BR(\phi\to K^+ K^-)(m_\phi^2-4m_{K^0}^2)^{3/2}}
\right)^{1/2}\,.
\label{ratiophiKK}
\ee
According to \cite{PDG} the central value of this factor
slightly deviates from the unit:
\be
\eta_{\phi}= 1.027\pm 0.01\,.
\label{etaphiexp}
\ee
In the vicinity of the $\phi$ resonance this small effect
is noticeable in the fit, and as far as the branching ratio is concerned,
is dominated by the phase space factor.
The factor $\eta_\phi$ also takes care of Coulomb-rescattering
and other isospin-violating differences
between charged and neutral modes (see also \cite{Voloshin,Bramonetal}).
To ensure the proper normalizations
$F_{K^0}(0)=0$ and $F_{K^+}(0)=1$, we introduce an additional
energy dependence with a simple step-function
\be
\eta_\phi(s)= 1+(\eta_\phi-1)\theta(\sqrt{s}-(m_\phi-\Gamma_\phi))
\theta(m_\phi+\Gamma_\phi-\sqrt{s})\,,
\label{etaphi}
\ee
which in future, after this effect is better understood both experimentally
and theoretically, can be replaced by an appropriate analytical
energy-dependence.
We have fitted the model (\ref{Kformfpm1})
and (\ref{Kformf01}) to the available data on charged
\cite{AchasovK,AkhmetshinK,Ivanov,Dolinsky,BiselloK} and neutral
\cite{CM2update,AchasovK,AkhmetshinK0,Mane} kaon form factors.
The masses and widths of $\rho$, $\omega$ and their excitations
are taken from \cite{PDG} and are listed in the Table~\ref{tab:kaon}.
Two different variants of the fit are carried out:
(1) the {\em constrained} fit
(motivated by the quark model) where the normalization factors
for $\omega$ resonances are fixed:
$c^K_{\omega,\omega',\omega''} = c^K_{\rho,\rho',\rho''}$
and only the normalization factors for the $\rho$ resonances are fitted;
(2) the {\em unconstrained} fit, where $\omega$- and $\rho$-
factors are fitted as independent parameters.
First, the mass and width of $\phi$ as well as the coefficient
$\eta_\phi$ are fitted in the region around $\phi$ resonance.
We obtain: $\eta_\phi= 1.011 \pm 0.009$ ($1.019 \pm 0.009$ ) for
the constrained (unconstrained) fit, in a good agreement with the
experimental value (\ref{etaphiexp}).
Fixing $\eta_\phi$ and using $m_\phi$ and $\Gamma_\phi$ as starting values,
the data in the whole region of $\sqrt{s}$ are then fitted.
The results of the fit are collected in Table~\ref{tab:kaon}.
%%%%%%%%%%%TABLE 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
%[t]
\begin{center}
%\vskip0.2cm
\begin{tabular}{|c|r|r|r|r|}
\hline
Parameter & Input & Fit(1)& Fit(2) & PDG value\cite{PDG}\\
&&&&\\
\hline
$m_\phi$ & - & 1019.372\,$\pm$\,0.02&1019.355\,$\pm$\,0.02& 1019.456\,$\pm$\,0.02\,\\
$\Gamma_{\phi}$ & - & 4.36\,$\pm$\,0.05&4.29\,$\pm$\,0.05 & 4.26\,$\pm$\,0.05\\
\hline
$m_{\phi'}$ & 1680 &-&- &1680\,$\pm$\,20\\
$\Gamma_{\phi'}$ & 150 &- &-& 150\,$\pm$\,50\\
\hline
\hline
%%%%%%%%%%%%%
$m_\rho$ & 775 &- &-&775.8\,$\pm$\,0.5 \\
$\Gamma_{\rho}$ & 150 &- &- &150.3\,$\pm$\,1.6 \\
\hline
$m_{\rho'}$ & 1465 &-&- & 1465\,$\pm$\,25 \\
$\Gamma_{\rho'}$ & 400 &- &-& 400\,$\pm$\,60 \\
\hline
$m_{\rho''}$ & 1720 &- &-& 1720$\,\pm$\,20 \\
$\Gamma_{\rho''}$ & \,\,\,250 &-&- & 250\,$\pm$\,100 \\
%%%%%%%%%%%%%%%
\hline
$m_{\omega}$ & 783.0 & - &-& 782.59\,$\pm$\,0.11 \\
$\Gamma_{\omega}$ & \,\,\,\,\,\,8.4 & - &-& 8.49\,$\pm$\,0.08 \\
\hline
$m_{\omega'}$ & 1425 & - &-& 1400-1450 \\
$\Gamma_{\omega'}$ &215 & - &-& 180-250\\
\hline
$m_{\omega''}$ & 1670 & - &-& 1670\,$\pm$ 30\, \\
$\Gamma_{\omega''}$ & \,\,\,315 &-& - & 315\,$\pm$\,35 \\
\hline
\hline
$c_{\phi}$& - & 1.018\,$\pm$\,0.006 &0.999\,$\pm$\,0.007
& -\\
$c_{\phi'}$& $1-c_\phi^K$ & -0.018\,$\mp$\,\,0.006 &0.001\,$\mp$\,0.007&-\\
\hline
$c_\rho^K$ &- &1.195\,$\pm$\,0.009 &1.139\,$\pm$\,0.010 &-\\
$c_{\rho'}^K$ & -& -0.112\,$\pm$\,0.010 & -0.124\,$\pm$\,0.012 &-\\
$ c_{\rho''}^K$& $1-c_\rho^K-c_{\rho'}^K$&-0.083\,$\mp$\,0.019 &-0.015\,$\mp$\,0.022 &-\\
\hline
\hspace{0.5cm}$c_\omega^K$(1)& $c^K_\rho$ &1.195\,$\pm$\,0.009 &-&-\\
\hspace{0.5cm}$c_{\omega}^K$(2)&-&-& 1.467\,$\pm$\,0.035&-\\
\hspace{0.5cm}$c_{\omega'}^K$(1)& $c^K_{\rho'}$ &-0.112\,$\pm$\,0.010&-&-\\
\hspace{0.5cm}$c_{\omega'}^K$(2)&-&-&-0.018\,$\pm$\,0.024 &-\\
$ c_{\omega''}^K$ & $1-c_{\omega }^K-c_{\omega'}^K$&-0.083\,$\mp$\,0.019 &-0.449\,$\mp$\,0.059 &-\\
\hline
$\chi^2/d.o.f.$&- & 328/242 &281/240&-\\
\hline
\end{tabular}
\caption{{\it Parameters of the kaon form factors
and results of the fit to the data. Masses and widths
are given in MeV. The row 'Fit(1)' (Fit(2))contains the values
of the constrained (unconstrained) fits.}}.
\label{tab:kaon}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%
The best (i.e., stable and physically plausible) results for both variants of the fit are obtained if
data on $F_{K^+}$ and $F_{K^0}$ are fitted simultaneously.
Thus, predicting $F_{K^0}$ from $F_{K^+}$ with the
currently available data is not yet possible.
The resulting curves for the form factors are plotted in Figs.~4,5.
Most importantly, fitting the kaon form factor
above $\phi$ resonance, it is indeed possible
to extract separate $\rho,\omega,\phi$ components,
which was not possible in the $\phi$ region due to
the dominance of this resonance.
We also find the pattern of the normalization factors
$c_{\rho,\rho'}^K$ for the first two $\rho$-resonance
to be very similar
to the corresponding values $c_{0,1}$ obtained in the pion form factor fits.
These factors can be immediately translated into the strong
couplings dividing out the decay constants of vector mesons.
The latter are independently measured in the leptonic decays revealing
a very mild SU(3) breaking, at the level of 10 \%;
according to the data in \cite{PDG}: $f_\rho\simeq$ 220 MeV,
$f_\omega\simeq$ 195 MeV and $f_\phi\simeq$ 228 MeV.
The SU(3)-violating difference between the couplings
of $\rho$ and $\phi$ to kaons
estimated from comparing $c_\rho^K$ and $c_\phi$
is also moderate in both versions of the fit.
As already noticed above, the constraint $c_\omega^K=c_\rho^K$
naturally follows from the valence quark content of both mesons
and we consider this constraint as a part of our model.
The fact that the unconstrained fit gives about 25\% difference
between these two coefficients,
a noticeable deviation from the quark-diagram relation,
should be taken with caution, having in mind poor quality of data.
Also $\chi^2$'s of both fits are in the same ballpark,
so that from the fitting point of view we cannot yet give
any preference to the version with the
``floating'' couplings of $\omega$-resonances.
On the other hand, this difference indicates that the fit is able
to resolve also the ``fine structure'' of the couplings.
The differences between the normalization factors
given by the fit for excited resonances: $c_{\rho'}^K$
vs. $c_{\omega'}^K$ (in the unconstrained
fit), $c_{\rho'}^K$ vs. $c_{\phi'}$, etc.
are generally large, which is not surprising,
in view of the complicated mixing between all these states.
Including in the future more precise data and switching on the ``tails'' of
the dual QCD$_{N_c=\infty}$ amplitudes in the kaon form factors
for all three vector mesons will allow to reveal
these differences more accurately.
Furthermore, an indication for an excess of the measured charged kaon
form factor vs. the model is present in Fig.~4b in the region around 2 GeV,
although the experimental errors are large. Remember, that we have not
included in our fit the contribution of the second excited
$\phi''$ state with a mass around 2 GeV, which might be responsible for this
potential difference. However, we refrain from further investigation
before more accurate data are available.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG.4
\begin{figure}
\vspace{-1cm}
\centering
%\subfigure[]{
\hspace{-0.6cm}
\includegraphics[width=0.7\textwidth]{fig4a.ps}\label{fig-FKpla}
%}%
(a)
\\
%\subfigure[]{\hspace{-1.2cm}
\includegraphics[width=0.7\textwidth]{fig4b.ps}\label{fig-FKplb}
(b)
%}%
\caption{
\it
The charged kaon form factor squared
$|F_{K^+}(s)|^2$ as a function of $\sqrt{s}$
fitted to the data taken from \cite{AchasovK} (crosses),
\cite{AkhmetshinK} (open squares), \cite{Ivanov} (open circles),
\cite{Dolinsky} (full squares) and \cite{BiselloK} (full circles). }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%FIG.5
\begin{figure}
\vspace{-1cm}
\centering
%\subfigure[]{\hspace{-1.0cm}
\includegraphics[width=0.7\textwidth]{fig5a.ps}\label{fig-FK0a}
%}%
(a)
\\
%\subfigure[]{\hspace{-1.2cm}
\includegraphics[width=0.7\textwidth]{fig5b.ps}\label{fig-FK0b}
(b)
%}%
\caption{
\it
The neutral kaon form factor squared
$|F_{K^0}(s)|^2$ as a function of $\sqrt{s}$
fitted to the data taken from \cite{CM2update}(triangles), \cite{AchasovK}(crosses),
\cite{AkhmetshinK0}(open squares) and
\cite{Mane}(full circles).}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The mean-squared charge radius of the $K^+$ obtained
in our model: $\sqrt{\langle r_K^2\rangle}= 0.56$ fm
(for both fits and with a small error), is in a good
agreement with the experimental value \cite{Dally}
$\sqrt{\langle r_K^2\rangle}_{exp}= 0.53\pm 0.05$ fm.
We have also checked that, being analytically continued to
large $s<-1 \mbox{GeV}^2$, the charged kaon form factor agrees
with the LCSR prediction obtained in \cite{BK}.
Finally, as mentioned in the Introduction, the separate
reconstruction of $I=0$ and $I=1$ spectral functions
might be a useful ingredient for various phenomenological
analyses. In Fig.~6
%\ref{fig:rho}
we display
the spectral functions defined as
\be
\rho_{K\bar{K}}^{(I=0,1)}(s)=
\frac{1}{12\pi}
\left| \frac{F_{K^+}(s)\pm F_{K^0}(s)}{2}\right|^2
\left(\frac{2p_K(s)}{\sqrt{s}}\right)^3\,,
\label{spectral}
\ee
noticing that this observable is quite sensitive to the
pattern of resonances in the form factor.
%%%%%%%%%%%%%%%%%%%%FIG.6
\begin{figure}
\vspace{-5cm}
\centering
%\subfigure[]{\hspace{-1.0cm}
\hspace{-0.8cm}
\includegraphics[width=0.75\textwidth]{fig6a.ps}\label{fig-spectr_a}
%}%
(a)
\\
%\subfigure[]{\hspace{-1.2cm}
\includegraphics[width=0.7\textwidth]{fig6b.ps}\label{fig-spectr_b}
(b)
%}%
\caption{
\it
The spectral functions (\ref{spectral}) with $I=0$ (a)
and $I=1$ (b) obtained from the fitted kaon form factors.
The solid(dashed) lines correspond to the constrained (unconstrained)
fit.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Predicting $\tau\to K^- K^0\nu_\tau$ decay distribution and rate}
As emphasized above, the isospin one part of the e.m. kaon form factor, together
with the isospin-symmetry relation
(\ref{KplK0}), can be used to predict the $\tau\to K^- K^0\nu_\tau$ decay width.
The differential decay distribution in $\sqrt{Q^2}$
(the invariant mass of the kaon pair),
normalized to the leptonic width of $\tau$ reads:
\ba
&&\left(\frac{1}{BR(\tau\to \mu^-\bar{\nu}_\mu \nu_\tau)}\right)
\frac{dBR(\tau\to K^-K^0\nu_\tau)}{d\sqrt{Q^2}}
\nonumber\\
&&=
\frac{|V_{ud}|^2}{2 m_\tau^2}
\left(1+\frac{2Q^2}{m_\tau^2}\right)\left(1-\frac{Q^2}{m_\tau^2}\right)^2
\left(1-\frac{4m_K^2}{Q^2}\right)^{3/2}\sqrt{Q^2}\,|F_{K^-K^0}(Q^2)|^2\,.
\label{distr}
\ea
In accordance with the isospin limit, we neglect the mass
difference between charged and neutral kaons and the
effect of the scalar form factor.
Using Eq.~(\ref{KplK0}) we have $ F_{K^-K^0}=-2F_{K^+}^{(I=1)}$,
hence
\be
|F_{K^-K^0}(Q^2)|^2= |c^K_\rho BW_{\rho}(Q^2)+
c^K_{\rho'} BW_{\rho'}(Q^2)+c^K_{\rho''} BW_{\rho''}(Q^2)|^2\,.
\label{FKminK0}
\ee
The fitted values for $c_{\rho,\rho',\rho''}^K$
from Table~\ref{tab:kaon} thus allow us to
calculate the decay distribution (\ref{distr}).
The normalized distribution is plotted in Fig.~7 and
(qualitatively) compared with the
event distribution in the kaon pair mass,
measured by CLEO Collaboration \cite{CLEOtau}. Another measurement
of this distribution by ALEPH Collaboration can be found in \cite{ALEPHtau}.
Integrating over $\sqrt{Q^2}$ from $2m_K$ to $m_\tau$
we obtain the branching ratio
\be
BR(\tau\to K^-K^0 \nu_\tau )= 0.19\pm 0.01\% ~~(0.13 \pm 0.01 \%)
\ee
for the constrained (unconstrained) fit,
to be compared with the experimentally measured value \cite{PDG}
\be
BR(\tau\to K^-K^0 \nu_\tau)= 0.154\pm 0.016\%.
\ee
We see that both the decay distribution and the decay width
are very sensitive to the pattern of $\rho$ resonances
in the isospin-1 form factor. Generally, the width grows
with the increase of the excited $\rho$ contributions,
an effect observed earlier in \cite{FKM} (see also \cite{EidI}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FIG.7
\begin{figure}
\vspace{-1cm}
\centering
%\subfigure[]{
\hspace{-0.6cm}
\includegraphics[width=0.7\textwidth]{fig7.ps}\label{fig-tau-a}
%}%
%(a)
%\\
%\subfigure[]{\hspace{-1.2cm}
%\includegraphics[width=1.0\textwidth]{fig7bb.ps}\label{fig-tau-b}
%(b)
%}%
\caption{\it The normaized distribution
$\frac{d\Gamma(\tau\to K^-K^0\nu_\tau)/d\sqrt{Q^2}}{
\Gamma(\tau\to K^-K^0\nu_\tau)}$
in the kaon pair invariant mass $\sqrt{Q^2}$ in units of $\mbox{GeV}^{-1}$
obtained from the fitted kaon form factor;
the solid (dashed) line corresponds to the constrained (unconstrained)
fit. The event distribution measured by CLEO Collaboration \cite{CLEOtau}
and normalized, dividing by the total number of events, is shown with points. }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
In this paper we considered the models of timelike form factors of pions
and kaons in \,anticipation of new and more accurate data
in the region above 1 GeV,
from $e^+e^-$ machines using the radiative return method.
We introduced an ansatz for the pion form factor which is based
on dual-resonance models and Veneziano amplitude.
We argued that the parameters of the ground-state
and first excited states can deviate from the model
prediction due to effects of mixing with multiparticle (e.g., two -
or 4 $\pi$ states), therefore have to be fitted independently
as free parameters. From the fit to the available pion form factor
data we have found that
the main contribution to the form factor originates from
the ground state plus the first two-three radially excited states.
The tail from the infinite series of resonances produces inessential,
but visible effects. The sign and value of the coefficients at certain
excited resonances are shifted with respect to the dual
QCD$_{N_c=\infty}$ ansatz signaling large mixing effects.
Possible checks of the model are provided by the spacelike form factors,
the pion charge radius and the behaviour for large $s$ in the timelike region.
In particular around $\sqrt{s}=3$ GeV we predict a value smaller
than the one anticipated from $J/\psi$ decay.
On the other hand, data fitted to this model can be used for
important tests of quark-hadron duality,
and of the QCD calculations of the pion form factor
in the spacelike region.
Furthermore, we formulated an analogous model for the kaon form factor
and demonstrated that the contributions of
$\phi$ , $\omega$ and $\rho$ resonances
(or, alternatively, the isospin 0 and 1 components) can be
separated by the fit. Interestingly, the $\tau\to K ^-K^0\nu_\tau$-decay
distribution and partial width predicted from the model
manifest a substantial sensitivity
to the pattern of $\rho$ resonances in the isospin-1 part of the form factor.
The model still has considerable room for improvement. In particular,
a more detailed kinematical and dynamical analysis
of total widths in the Breit-Wigner factors would allow
to implement a more accurate energy-dependence in these widths.
\bigskip
{\bf Acknowledgements}
We are grateful to S.~Eidelman for a useful discussion.
This work is supported by the
German Ministry for Education and Research (BMBF).
\begin{thebibliography}{99}
\bibitem{g2}
%\cite{Davier:2003pw}
%\bibitem{Davier:2003pw}
M.~Davier, S.~Eidelman, A.~Hocker and Z.~Zhang,
%``Updated estimate of the muon magnetic moment using revised results from e+
%e- annihilation,''
Eur.\ Phys.\ J.\ C {\bf 31} (2003) 503; hep-ph/0308213(v2).
%%CITATION = HEP-PH 0308213;%%
%\cite{Hagiwara:2003da}
\bibitem{g21}
K.~Hagiwara, A.~D.~Martin, D.~Nomura and T.~Teubner,
%``Predictions for g-2 of the muon and alpha(QED)(M(Z**2)),''
Phys.\ Rev.\ D {\bf 69} (2004) 093003.
%[arXiv:hep-ph/0312250].
%%CITATION = HEP-PH 0312250;%%
%\cite{Akhmetshin:2001ig}
\bibitem{CMD2002}
R.~R.~Akhmetshin {\it et al.} [CMD-2 Collaboration],
%``Measurement of e+ e- $\to$ pi+ pi- cross section with CMD-2 around rho meson,''
Phys.\ Lett.\ B {\bf 527} (2002) 161.
%[arXiv:hep-ex/0112031].
%%CITATION = HEP-EX 0112031;%%
%\cite{Akhmetshin:2003zn}
\bibitem{CM2update}
R.~R.~Akhmetshin {\it et al.} [CMD-2 Collaboration],
%``Reanalysis of hadronic cross section measurements at CMD-2,''
Phys.\ Lett.\ B {\bf 578} (2004) 285.
%[arXiv:hep-ex/0308008].
%%CITATION = HEP-EX 0308008;%%
%\cite{Bollini:1975pn}
\bibitem{ADONE}
D.~Bollini, P.~Giusti, T.~Massam, L.~Monari, F.~Palmonari, G.~Valenti and A.~Zichichi,
%``The Pion Electromagnetic Form-Factor In The Timelike Range 1.44-Gev**2 - 9.0-Gev**2,''
Lett.\ Nuovo Cim.\ {\bf 14} (1975) 418.
%%CITATION = NCLTA,14,418;%%
\bibitem{Barkov}
%\cite{Barkov:ac}
%\bibitem{Barkov:ac}
L.~M.~Barkov {\it et al.},
%``Electromagnetic Pion Form-Factor In The Timelike Region,''
Nucl.\ Phys.\ B {\bf 256}, 365 (1985).
%%CITATION = NUPHA,B256,365;%%
%\cite{Bisello:1988hq}
\bibitem{DM2}
D.~Bisello {\it et al.} [DM2 Collaboration],
%``The Pion Electromagnetic Form-Factor In The Timelike Energy Range 1.35-Gev <= S**(1/2) <= 2.4-Gev,''
Phys.\ Lett.\ B {\bf 220} (1989) 321.
%%CITATION = PHLTA,B220,321;%%
%\cite{Binner:1999bt}
\bibitem{BinnKuhnMeln}
S.~Binner, J.~H.~Kuhn and K.~Melnikov,
%``Measuring sigma(e+ e- $\to$ hadrons) using tagged photon,''
Phys.\ Lett.\ B {\bf 459} (1999) 279.
%[arXiv:hep-ph/9902399].
%%CITATION = HEP-PH 9902399;%%
\bibitem{KLOE}
%\cite{Aloisio:2003sh}
%\bibitem{Aloisio:2003sh}
A.~Aloisio {\it et al.} [KLOE Collaboration],
%``Measurement of sigma(e+ e- $\to$ pi+ pi-) at DAPHNE with the radiative
%return,''
arXiv:hep-ex/0312056.
%%CITATION = HEP-EX 0312056;%%
\bibitem{BABAR}
%\cite{Solodov:2002xu}
%\bibitem{Solodov:2002xu}
E.~P.~Solodov [BABAR collaboration],
%``Study of e+ e- collisions in the 1.5-GeV - 3-GeV cm energy region using ISR
%at BaBar,''
in {\it Proc. of the $e^+ e^-$ Physics at Intermediate Energies Conference } ed. Diego Bettoni, eConf {\bf C010430} (2001) T03
[arXiv:hep-ex/0107027].
%%CITATION = HEP-EX 0107027;%%
\bibitem{GT}
%\cite{Geshkenbein:qn}
%\bibitem{Geshkenbein:qn}
B.~V.~Geshkenbein and M.~V.~Terentev,
%``The Phenomenological Formula For The Pion Electromagnetic Form-Factor,''
Sov.\ J.\ Nucl.\ Phys.\ {\bf 40}, 487 (1984).
%%CITATION = YAFIA,40,758;%%
%\cite{Kuhn:1990ad}
\bibitem{KS}
J.~H.~K\"uhn and A.~Santamaria,
%``Tau Decays To Pions,''
Z.\ Phys.\ C {\bf 48} (1990) 445.
%%CITATION = ZEPYA,C48,445;%%
\bibitem{exclusive}
G.~P.~Lepage and S.~J.~Brodsky,
%``Exclusive Processes In Quantum Chromodynamics: Evolution Equations For
%Hadronic Wave Functions And The Form-Factors Of Mesons,''
Phys.\ Lett.\ {\bf 87B} (1979) 359;
%%CITATION = PHLTA,B87,359;%%
%G.~P.~Lepage and S.~J.~Brodsky,
%``Exclusive Processes In Perturbative Quantum Chromodynamics,''
Phys.\ Rev.\ {\bf D22} (1980) 2157;\\
%%CITATION = PHRVA,D22,2157;%%
A.~V.~Efremov and A.~V.~Radyushkin,
%``Factorization And Asymptotical Behavior Of Pion Form-Factor In QCD,''
Phys.\ Lett.\ {\bf B94} (1980) 245;
%%CITATION = PHLTA,B94,245;%%
%A.~V.~Efremov and A.~V.~Radyushkin,
%``Asymptotical Behavior Of Pion Electromagnetic Form-Factor In QCD,''
Theor.\ Math.\ Phys.\ {\bf 42} (1980) 97;\\
%% English version of Teor.Mat.Fiz.42:147-166,1980
%%CITATION = TMPHA,42,97;%%
V.~L.~Chernyak and A.~R.~Zhitnitsky,
%``Asymptotic Behavior Of Hadron
%Form-Factors In Quark Model. (In Russian),''
JETP Lett.\ {\bf 25} (1977) 510;
%V.~L.~Chernyak and A.~R.~Zhitnitsky,
%``Asymptotics Of Hadronic Form-Factors In The Quantum Chromodynamics.
%(In Russian),''
Sov.\ J.\ Nucl.\ Phys.\ {\bf 31} (1980) 544;\\
%%CITATION = JTPLA,25,510;%%
%%CITATION = SJNCA,31,544;%%
%\cite{Farrar:1979aw}
%\bibitem{Farrar:1979aw}
G.~R.~Farrar and D.~R.~Jackson,
%``The Pion Form-Factor,''
Phys.\ Rev.\ Lett.\ {\bf 43} (1979) 246.
%%CITATION = PRLTA,43,246;%%
\bibitem{endpoint}
%\cite{Isgur:cy}
%\bibitem{Isgur:cy}
N.~Isgur and C.~H.~Llewellyn Smith,
%``Perturbative QCD In Exclusive Processes,''
Phys.\ Lett.\ B {\bf 217}, 535 (1989);\\
%%CITATION = PHLTA,B217,535;%%
%\cite{Radyushkin:vi}
%\bibitem{Radyushkin:vi}
A.~V.~Radyushkin,
%``Exclusive Processes In Perturbative QCD,''
Nucl.\ Phys.\ A {\bf 527}, 153C (1991);
%%CITATION = NUPHA,A527,153C;%%
%\cite{Radyushkin:1990te}
%\bibitem{Radyushkin:1990te}
%A.~V.~Radyushkin,
%``Hadronic Form-Factors: Perturbative QCD Versus QCD Sum Rules,''
%Nucl.\ Phys.\
A {\bf 532}, 141 (1991);\\
%%CITATION = NUPHA,A532,141;%%
%\cite{Jakob:1993iw}
%\bibitem{Jakob:1993iw}
R.~Jakob and P.~Kroll,
%``The Pion form-factor: Sudakov suppressions and intrinsic transverse
%momentum,''
Phys.\ Lett.\ B {\bf 315}, 463 (1993)
[Erratum-ibid.\ B {\bf 319}, 545 (1993)].
%[arXiv:hep-ph/9306259].
%%CITATION = HEP-PH 9306259;%%
\bibitem{sr}
%\cite{Ioffe:qb}
%\bibitem{Ioffe:qb}
B.~L.~Ioffe and A.~V.~Smilga,
%``Meson Widths And Form-Factors At Intermediate Momentum Transfer In
%Nonperturbative QCD,''
Nucl.\ Phys.\ B {\bf 216}, 373 (1983);\\
%%CITATION = NUPHA,B216,373;%%
%\cite{Nesterenko:1982gc}
%\bibitem{Nesterenko:1982gc}
V.~A.~Nesterenko and A.~V.~Radyushkin,
%``Sum Rules And Pion Form-Factor In QCD,''
Phys.\ Lett.\ B {\bf 115}, 410 (1982);\\
%%CITATION = PHLTA,B115,410;%%
%\cite{Braguta:2004ck}
%\bibitem{Braguta:2004ck}
V.~V.~Braguta and A.~I.~Onishchenko,
%``Pion form factor and QCD sum rules: Case of axial current,''
Phys.\ Lett.\ B {\bf 591} (2004) 267.
%[arXiv:hep-ph/0403240].
%%CITATION = HEP-PH 0403240;%%
\bibitem{lcsr}
%\cite{Braun:ij}
%\bibitem{Braun:ij}
V.~M.~Braun and I.~E.~Halperin,
%``Soft Contribution To The Pion Form-Factor From Light Cone QCD Sum Rules,''
Phys.\ Lett.\ B {\bf 328}, 457 (1994);\\
%[arXiv:hep-ph/9402270]
%%CITATION = HEP-PH 9402270;%%
%\cite{Braun:1999uj}
%\bibitem{Braun:1999uj}
V.~M.~Braun, A.~Khodjamirian and M.~Maul,
%``Pion form factor in QCD at intermediate momentum transfers,''
Phys.\ Rev.\ D {\bf 61}, 073004 (2000).
%[arXiv:hep-ph/9907495].
%%CITATION = HEP-PH 9907495;%%
\bibitem{BK}
%\cite{Bijnens:2002mg}
J.~Bijnens and A.~Khodjamirian,
%``Exploring light-cone sum rules for pion and kaon form factors,''
Eur.\ Phys.\ J.\ C {\bf 26} (2002) 67.
%[arXiv:hep-ph/0206252].
%%CITATION = HEP-PH 0206252;%%
%\cite{Gousset:1994yh}
\bibitem{GP}
T.~Gousset and B.~Pire,
%``Timelike form-factors at high-energy,''
Phys.\ Rev.\ D {\bf 51} (1995) 15.
%[arXiv:hep-ph/9403293].
%%CITATION = HEP-PH 9403293;%%
\bibitem{BRS}
A.~P.~Bakulev, A.~V.~Radyushkin and N.~G.~Stefanis,
%``Form factors and QCD in spacelike and timelike regions,''
Phys.\ Rev.\ D {\bf 62} (2000) 113001.
%[arXiv:hep-ph/0005085].
%%CITATION = HEP-PH 0005085;%%
%\cite{Dominguez:jw}
\bibitem{Dom}
C.~A.~Dominguez,
%``Pion Form-Factor In Large N(C) QCD,''
Phys.\ Lett.\ B {\bf 512} (2001) 331.
%%CITATION = PHLTA,B512,331;%%
\bibitem{olddual}
%\cite{Frampton:1969ry}
%\bibitem{Frampton:1969ry}
P.~H.~Frampton,
%``Generalized Vector Dominance Theory And Vertex Functions,''
Phys.\ Rev.\ D {\bf 1}, 3141 (1970);\\
%%CITATION = PHRVA,D1,3141;%%
%\cite{Tokuda:nv}
%\bibitem{Tokuda:nv}
N.~Tokuda and Y.~Oyanagi,
%``Meson Form-Factors And The Veneziano Model,''
Phys.\ Rev.\ D {\bf 1}, 2626 (1970);\\
%%CITATION = PHRVA,D1,2626;%%
%\cite{Urrutia:1973sc}
%\bibitem{Urrutia:1973sc}
L.~F.~Urrutia,
%``A Fit To The Pion Electromagnetic Form-Factor,''
Phys.\ Rev.\ D {\bf 9}, 3213 (1974);\\
%%CITATION = PHRVA,D9,3213;%%
R.~Jengo and E.~Remiddi,
Nuovo Cimento Lett. 18, 922 (1969).
\bibitem{newdual}
%\cite{Shifman:2000jv}
%\bibitem{Shifman:2000jv}
M.~A.~Shifman, in 'At the Frontier of Particle Physics/Handbook of QCD',
ed. M. Shifman (World Scientific, Singapore, 2001).
%``Quark-hadron duality,''
arXiv:hep-ph/0009131;\\
%%CITATION = HEP-PH 0009131;%%
%\cite{Geshkenbein:1994wg}
%\bibitem{Geshkenbein:1994wg}
B.~V.~Geshkenbein and V.~L.~Morgunov,
%``Hadronic contributions to the anomalous magnetic moment of the muon by QCD
%model with infinite number of vector mesons,''
arXiv:hep-ph/9410252;\\
%%CITATION = HEP-PH 9410252;%%
%\cite{Polyakov:1995vh}
%\bibitem{Polyakov:1995vh}
M.~V.~Polyakov and V.~V.~Vereshagin,
%``Effective Chiral Lagrangian from Dual Resonance Models,''
Phys.\ Rev.\ D {\bf 54}, 1112 (1996);\\
%%CITATION = HEP-PH 9509259;%%
%\cite{Peris:2000tw}
%\bibitem{Peris:2000tw}
S.~Peris, B.~Phily and E.~de Rafael,
%``Tests of large-N(c) QCD from hadronic tau decay,''
Phys.\ Rev.\ Lett.\ {\bf 86}, 14 (2001);\\
%[arXiv:hep-ph/0007338].
%%CITATION = HEP-PH 0007338;%%
%\cite{Afonin:2004yb}
%\bibitem{Afonin:2004yb}
S.~S.~Afonin, A.~A.~Andrianov, V.~A.~Andrianov and D.~Espriu,
%``Matching Regge theory to the OPE,''
arXiv:hep-ph/0403268.
%%CITATION = HEP-PH 0403268;%%
%\cite{Jegerlehner:1985gq}
\bibitem{Jegerlehner}
F.~Jegerlehner,
%``Hadronic Contributions To Electroweak Parameter Shifts: A Detailed
%Analysis,''
Z.\ Phys.\ C {\bf 32}, 195 (1986).
%%CITATION = ZEPYA,C32,195;%%
%\cite{Leutwyler:2002hm}
\bibitem{Leutw}
H.~Leutwyler,
%``Electromagnetic form factor of the pion,'',
in Continuous advances in QCD 2002, pp.23-40, World Scientific (2002),
arXiv:hep-ph/0212324.
%%CITATION = HEP-PH 0212324;%
%\cite{Gounaris:1968mw}
\bibitem{GS}
G.~J.~Gounaris and J.~J.~Sakurai,
%``Finite Width Corrections To The Vector Meson Dominance Prediction For Rho $\to$ E+ E-,''
Phys.\ Rev.\ Lett.\ {\bf 21} (1968) 244.
%%CITATION = PRLTA,21,244;%%
%\cite{Eidelman:2004wy}
\bibitem{PDG}
S.~Eidelman {\it et al.} [Particle Data Group Collaboration],
%``Review of particle physics,''
Phys.\ Lett.\ B {\bf 592} (2004) 1.
%%CITATION = PHLTA,B592,1;%%
%K.~Hagiwara {\it et al.} [Particle Data Group Collaboration],
%%``Review Of Particle Physics,''
%Phys.\ Rev.\ D {\bf 66} (2002) 010001;
(see also http://pdg.lbl.gov/2004/listings/)
%%CITATION = PHRVA,D66,010001;%%
\bibitem{CzK}
%\cite{Czyz:2000wh}
%\bibitem{Czyz:2000wh}
H.~Czyz and J.~H.~Kuhn,
%``Four pion final states with tagged photons at electron positron colliders,''
Eur.\ Phys.\ J.\ C {\bf 18}, 497 (2001.)
%[arXiv:hep-ph/0008262].
%%CITATION = HEP-PH 0008262;%%
\bibitem{AchasovKozh}
%\cite{Achasov:1996gw}
%\bibitem{Achasov:1996gw}
N.~N.~Achasov and A.~A.~Kozhevnikov,
%``Rho primes in analyzing e+ e- annihilation, MARK III, LASS and ARGUS data.
%(Updated version),''
Phys.\ Rev.\ D {\bf 55} (1997) 2663.
%[arXiv:hep-ph/9609216].
%%CITATION = HEP-PH 9609216;%%
\bibitem{Milana}
%\cite{Milana:1993wk}
%\bibitem{Milana:1993wk}
J.~Milana, S.~Nussinov and M.~G.~Olsson,
%``Does J / psi $\to$ pi+ pi- fix the electromagnetic form-factor F-pi(t) at t =
%M2 (J / psi)?,''
Phys.\ Rev.\ Lett.\ {\bf 71}, 2533 (1993).
%[arXiv:hep-ph/9307233].
%%CITATION = HEP-PH 9307233;%%
%\cite{Amendolia:1986wj}
\bibitem{Amendolia}
S.~R.~Amendolia {\it et al.} [NA7 Collaboration],
%``A Measurement Of The Space - Like Pion Electromagnetic Form-Factor,''
Nucl.\ Phys.\ B {\bf 277} (1986) 168.
%%CITATION = NUPHA,B277,168;%%
%\cite{Bijnens:2002hp}
\bibitem{BijnensT}
J.~Bijnens and P.~Talavera,
%``Pion and kaon electromagnetic form factors,''
JHEP {\bf 0203} (2002) 046.
%[arXiv:hep-ph/0203049].
%%CITATION = HEP-PH 0203049;%%
%\cite{Bebek:pe}
\bibitem{Bebek}
C.~J.~Bebek {\it et al.},
%``Electroproduction Of Single Pions At Low Epsilon And A Measurement Of The Pion Form-Factor Up To Q**2 = 10-Gev**2,''
Phys.\ Rev.\ D {\bf 17} (1978) 1693.
%%CITATION = PHRVA,D17,1693;%%
%\cite{Carlson:zn}
\bibitem{Carlson}
C.~E.~Carlson and J.~Milana,
%``Difficulty In Determining The Pion Form-Factor At High Q**2,''
Phys.\ Rev.\ Lett.\ {\bf 65}, 1717 (1990).
%%CITATION = PRLTA,65,1717;%%
%\cite{Volmer:2000ek}
\bibitem{Jlab}
J.~Volmer {\it et al.} [The Jefferson Lab F(pi) Collaboration],
%``New results for the charged pion electromagnetic form-factor,''
Phys.\ Rev.\ Lett.\ {\bf 86}, 1713 (2001).
%[arXiv:nucl-ex/0010009].
%%CITATION = NUCL-EX 0010009;%%
%papers on the kaon form factor experimental data
%\cite{Voloshin:2004nu}
\bibitem{Voloshin}
M.~B.~Voloshin,
%``Relative yield of charged and neutral heavy meson pairs in e+ e- annihilation
%near threshold,''
arXiv:hep-ph/0402171.
%%CITATION = HEP-PH 0402171;%%
%\cite{Bramon:2000qe}
\bibitem{Bramonetal}
A.~Bramon, R.~Escribano, J.~L.~Lucio M. and G.~Pancheri,
%``The ratio Phi $\to$ K+ K- / K0 anti-K0,''
Phys.\ Lett.\ B {\bf 486} (2000) 406.
%[arXiv:hep-ph/0003273].
%%CITATION = HEP-PH 0003273;%%
%\cite{Achasov:ni}
\bibitem{AchasovK}
M.~N.~Achasov {\it et al.},
%``Measurements Of The Parameters Of The Phi(1020) Resonance Through Studies Of
%The Processes E+ E- $\to$ K+ K-, Kskl, And Pi+ Pi- Pi0,''
Phys.\ Rev.\ D {\bf 63}, 072002 (2001).
%%CITATION = PHRVA,D63,072002;%%
%\cite{Akhmetshin:vz}
\bibitem{AkhmetshinK}
R.~R.~Akhmetshin {\it et al.},
%``Measurement Of Phi Meson Parameters With Cmd-2 Detector At Vepp-2m
%Collider,''
Phys.\ Lett.\ B {\bf 364}, 199 (1995).
%%CITATION = PHLTA,B364,199;%%
%\cite{Ivanov}
\bibitem{Ivanov}
P.~M.~Ivanov {\it et al.},
%``Measurement Of The Charged Kaon Form-Factor In The Energy Range 1.0-Gev To
%1.4-Gev,''
Phys.\ Lett.\ B {\bf 107}, 297 (1981).
%%CITATION = PHLTA,B107,297;%%
%\cite{Dolinsky}
\bibitem{Dolinsky}
S.~I.~Dolinsky {\it et al.},
%``Summary Of Experiments With The Neutral Detector At The E+ E- Storage Ring
%Vepp-2m,''
Phys.\ Rept.\ {\bf 202}, 99 (1991).
%%CITATION = PRPLC,202,99;%%
%\cite{Bisello:1988ez}
\bibitem{BiselloK}
D.~Bisello {\it et al.} [DM2 Collaboration],
%``Study Of The Reaction E+ E- $\to$ K+ K- In The Energy Range 1350 <= S**(1/2)
%<= 2400-Mev,''
Z.\ Phys.\ C {\bf 39}, 13 (1988).
%%CITATION = ZEPYA,C39,13;%%
%\cite{Akhmetshin:2002vj}
\bibitem{AkhmetshinK0}
R.~R.~Akhmetshin {\it et al.},
%``Study of the process e+ e- $\to$ K0(L) K0(S) in the CM energy range 1.05-GeV
%to 1.38-GeV with CMD-2,''
Phys.\ Lett.\ B {\bf 551} (2003) 27.
%[arXiv:hep-ex/0211004].
%%CITATION = HEP-EX 0211004;%%
%\cite{Mane}
\bibitem{Mane}
F.~Mane, D.~Bisello, J.~C.~Bizot, J.~Buon, A.~Cordier and B.~Delcourt,
%``Study Of The Reaction E+ E- $\to$ K0(S) K0(L) In The Total Energy Range
%1.4-Gev To 2.18-Gev And Interpretation Of The K+ And K0 Form-Factors,''
Phys.\ Lett.\ B {\bf 99}, 261 (1981).
%%CITATION = PHLTA,B99,261;%%
%\cite{Dally:1980dj}
\bibitem{Dally}
E.~B.~Dally {\it et al.},
%``Direct Measurement Of The Negative Kaon Form-Factor,''
Phys.\ Rev.\ Lett.\ {\bf 45} (1980) 232.
%%CITATION = PRLTA,45,232;%%
%\cite{Coan:1996iu}
\bibitem{CLEOtau}
T.~E.~Coan {\it et al.} [CLEO Collaboration],
%``Decays of tau leptons to final states containing K(s)0 mesons,''
Phys.\ Rev.\ D {\bf 53} (1996) 6037.
%%CITATION = PHRVA,D53,6037;%%
\bibitem{ALEPHtau}
%\cite{Barate:1997tt}
%\bibitem{Barate:1997tt}
R.~Barate {\it et al.} [ALEPH Collaboration],
%``K0(S) production in tau decays,''
Eur.\ Phys.\ J.\ C {\bf 4} (1998) 29.
%%CITATION = EPHJA,C4,29;%%
\bibitem{FKM}
%\cite{Finkemeier:1996hh}
%\bibitem{Finkemeier:1996hh}
M.~Finkemeier, J.~H.~K\"uhn and E.~Mirkes,
%``Theoretical aspects of tau $\to$ K h(h)nu/tau decays and experimental
%comparisons,''
Nucl.\ Phys.\ Proc.\ Suppl.\ {\bf 55C}, 169 (1997).
%[arXiv:hep-ph/9612255].
%%CITATION = HEP-PH 9612255;%%
%\cite{Eidelman:1990pb}
\bibitem{EidI}
S.~I.~Eidelman and V.~N.~Ivanchenko,
%``E+ E- Annihilation Into Hadrons And Exclusive Tau Decays,''
Phys.\ Lett.\ B {\bf 257} (1991) 437.
%%CITATION = PHLTA,B257,437;%%
\end{thebibliography}
\end{document}
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