%Title: The radiative return and form factors at large $Q^2$
%Author: J.H. Kuhn
%Published: *Proceedings of the Eighth International Workshop on Tau Lepton Physics (TAU 04)* 14-17 September 2004, Nara, Japan.
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\title{The radiative return and form factors at large $Q^2$}
\author{J.H.~K\"uhn
\address{Institut f\"ur Theoretische Teilchenphysik, Universit\"at
Karlsruhe,\\ D-76128 Karlsruhe, Germany.}
\thanks{Work supported in part by BMBF under grant number 05HT9VKB0,
and SFB-TR/9.} }
\begin{document}
\begin{abstract}
The measurement of the pion form factor and, more
generally, of the cross section for electron--positron annihilation
into hadrons through the radiative return has become an important
task for high luminosity colliders such as the $\Phi$- or
$B$-meson factories. For a detailed understanding and analysis
of this reaction, the construction of a Monte Carlo program,
PHOKHARA, has been undertaken. A brief overview of
the most recent developments is presented, which include the combination
of initial- and final-state radiation in the case of two-pion
production. Baryon pair production is now included and
the pion form factor is extended into the region of large $Q^2$.
A model for the kaon
form factor is discussed which is
fitted to $K^+K^-$ and $K^0\bar{K}^0$ data and which
leads to predictions for the $\tau$-decay into $K^-K^0\nu$.
\vspace{1pc}
\end{abstract}
% typeset front matter (including abstract)
\maketitle
\section{Introduction}
Electron--positron annihilation into hadrons is one of the basic
reactions of particle physics, crucial for the understanding
of hadronic interactions \cite{CKK}.
The recent advent of
$\Phi$- and $B$-meson factories with their enormous luminosities
allows us to exploit the radiative return to explore the whole energy
region from threshold up to the nominal energy of the collider.
Photon radiation from the initial state reduces the cross section by a factor
${\cal O}(\alpha/\pi)$. However, this is easily compensated by the
enormous luminosity of `factories' and the advantage of performing
the measurement over a wide range of energies in one homogeneous data
sample~\cite{Binner:1999bt} (for an early proposal along these lines,
see~\cite{Zerwas}). In principle the reaction
$e^+e^- \to \gamma + {\mathrm {hadrons}}$ receives contributions from
both initial- and final-state radiation. Only the former is of interest
for the radiative return; the latter has to be eliminated by
suitably chosen cuts. The proper analysis thus requires the construction
of Monte Carlo event generators.
The event generator EVA was based on a leading order treatment of ISR and FSR,
supplemented by an approximate inclusion of additional collinear radiation
based on structure functions and included two-pion~\cite{Binner:1999bt} and
four-pion final states~\cite{Czyz:2000wh}.
Subsequently the event generator PHOKHARA was developed; it is based
on a complete next-to-leading order (NLO) treatment of
ISR~\cite{Rodrigo:2001jr,Rodrigo:2001kf,Kuhn:2002xg,Czyz:2002np}.
In its version 2.0 it included ISR at NLO and FSR at leading order (LO)
for $\pi^+ \pi^-$ and $\mu^+ \mu^-$ final states and four-pion
final states without FSR. A version which also includes $K\bar{K}$
and three-pion final states is currently inder construction \cite{prac}.
Recent preliminary experimental results indeed demonstrate the power
of the method and seem to indicate that a precision of one per cent
or better is within
reach~\cite{CDKMV2000}.
In view of this progress a further improvement of theoretical
understanding and MC-programs is necessary.
PHOKHARA version 3.0~\cite{Czyz:PH03}
allows for the `simultaneous' emission of one photon from
the initial and one photon from the final state, requiring only one
of them to be hard. This includes in particular the radiative return
to $\pi^+ \pi^- (\gamma)$ and thus allows the measurement of the (one-photon)
inclusive $\pi^+ \pi^-$ cross section.
\section{Final state radiation in next-to-leading order}
The issue of photon radiation from the final states is closely
connected to the question of $\pi^+ \pi^- (\gamma)$
contributions to $a_{\mu}$.
Soft photon emission is clearly described by
the point-like pion model. Hard photon emission, with
$E_{\gamma} \geq {\cal O}$(100 MeV), however, might be sensitive
to unknown hadronic physics. Therefore the size of
virtual, soft and hard corrections has to be studied
separately. In \cite{Czyz:PH03} it has been argued that contributions from
the hard region, above 100 MeV, are small with respect
to the present experimental and theory-induced uncertainty.
Let us start with the process in leading order:
\begin{equation}
e^+ \ e^- \to \pi^+(p^+) \ \pi^-(p^-) \ \gamma~.
\end{equation}
The amplitude of interest describes radiation from
the initial state (ISR). It is proportional to
the pion form factor, evaluated at
$Q^2=(p^+ + p^-)^2$. However, radiation from the charged pions (FSR)
evidently leads to the same final state and must be suppressed
and controlled with sufficient precision. A variety of methods
are described in more detail in
~\cite{Binner:1999bt,Rodrigo:2001kf,Czyz:2002np,Czyz:PH03}.
The fully differential cross section describing photon emission can
be split into three pieces
\begin{equation}
d\sigma = d\sigma_\mathrm{ISR} + d\sigma_\mathrm{FSR} + d\sigma_\mathrm{INT}~,
\end{equation}
which originate from the squared ISR and FSR amplitudes respectively,
plus the interference term.
Let us summarise the main points of the `leading order'
discussion~\cite{Czyz:PH03}:
\begin{itemize}
\item[1.]{The cross sections for photon emission from
ISR and FSR can be disentangled as a consequence of the marked
difference in angular distributions between the two processes.
This observation is completely general
and does not rely on any model like sQED for FSR. This
allows us to measure the cross section for
$\sigma(e^+e^-\to\gamma^*\to\pi^+\pi^-\gamma)$ directly for fixed $s$ as
a function of $E_\gamma$, an important ingredient in the analysis of
hadronic contributions to $a_\mu$.
Alternatively we can employ suitable angular
cuts to separate the two components.}
\item[2.]{Various charge-asymmetric distributions can be used for independent
tests of the FSR model amplitude, with the forward--backward
asymmetry as simplest example. Typically the charge asymmetry is
large in the region where ISR and FSR are comparable in size and
small when the separation is clean and simple from general
considerations. A typical case for this second possibility is the
radiative return at $\sqrt{s} = 10.52$~GeV, at the $B$-meson
factories, where the $\pi^+\pi^-\gamma$ final state is completely
dominated by ISR.}
\end{itemize}
Intuitively it should also be possible to exploit the radiative return
for the extraction of FSR for arbitrary invariant mass $\sqrt{s'}$
of the $\pi^+\pi^-\gamma$ system through the reaction
\begin{equation}
e^+e^-\to \gamma\gamma^*(\to \pi^+\pi^-\gamma) \ .
\label{eq:twostep}
\end{equation}
The implementation of this two-step process has been achieved in
PHO\-KHA\-RA 3.0 and the impact of the new corrections on a
few selected distributions has been discussed in~\cite{Czyz:PH03}.
Let us recall the meaning of various
abbreviations:
ISRNLO corresponds to the ISR cross section calculated at NLO without
any FSR. IFSLO includes in addition FSR at LO. Finally, IFSNLO stands for ISR
and FSR at NLO as implemented in the new version of PHOKHARA (version 3.0).
For a cms energy of 1.02 GeV, relevant for the
KLOE experiment, the IFSNLO correction to the cross section
is relatively big
at low \(Q^2\), if no cuts are applied.
Below the \(\rho\) resonance, they grow from zero at the resonance to
10\% of the IFSLO cross section near the production threshold,
while they remain small above the \(\rho\) resonance. This is due to the fact
that the ISR leading to the \(\rho\) meson is strongly enhanced.
Subsequently the \(\rho\) decays into \(\pi^+\pi^-\gamma\), with a large
contribution from the region where \(Q^2\), the
invariant mass of the \(\pi^+\pi^-\), is low. Of course
this is smeared by the width of the \(\rho\), but
the above discussion remains valid and the contribution from the
newly implemented diagrams, through the reaction
\(e^+e^-\to\gamma\rho(\to \gamma\pi^+\pi^-)\), is sizeable.
The standard KLOE cut on the
track mass reduces FSR further, to less than 1\% for most
of the \(Q^2\) range, apart of the high values of \(Q^2\), where the
corrections remain at the level of 2\%-3\% (Fig.~\ref{fig:ifs+lo_cut},
lower curve). The behaviour of the additional contribution
at $\sqrt{s}$ = 10.52 GeV is
qualitatively similar.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=7.5cm,height=6cm]{v_trkmass_50.ps}
\vskip-1cm
\caption{Comparison of the \(Q^2\) differential cross sections
for \(\sqrt{s} = 1.02\)~ GeV. Cuts on
the missing momentum direction and the track mass are imposed
as discussed in the text.}
\vskip-0.9cm
\label{fig:ifs+lo_cut}
\end{center}
\end{figure}
\section{Nucleon form factors at ${\boldmath{B}}$-meson factories }
The importance of measuring the nucleon form factor has been
repeatedly emphasized in the literature (e.g.
\cite{iachello,brodsky,Baldini} and refs therein).
Recent experiments at Jefferson Lab have explored the ratio of electric and
magnetic form factors in the space-like region
up to ~5.6~GeV$^2$, using the recoil polarization method; their results show
disagreement with data obtained with the Rosenbluth method.
Data in the time-like region based on electron--positron annihilation
into proton--antiproton, and the
inverse reactions only extend to 6~GeV${}^2$.
These latter measurements exhibit fairly large errors and do not
provide a separation of the contributions from the two form factors.
In \cite{nucleon} the potential of the radiative return at $B$-meson
factories has been be explored. These measurements will
cover the region from threshold up to 8 or even 9 GeV${}^2$ with
sufficient counting rates.
Not surprisingly, many aspects of the radiative return are quite similar
to those of the reaction $e^+e^-\to p\bar p$. In particular, it is
again the square of the electric and magnetic form factors, which can
be determined separately through the analysis of angular distributions.
Radiative corrections are indispensable for a precise interpretation
of the data. Furthermore, given the limited acceptance and the
asymmetric kinematic configuration at present $B$-meson factories,
a Monte Carlo event generator is required to demonstrate the feasibility
of the measurement. This has motivated the extension of the
generator
to a new version, PHOKHARA 4.0.
In this case photon emission from the initial
state only (ISR) was considered. Final-state radiation
is small if the reaction is considered at a
centre of mass (cms) collider energy of 10.5~GeV.
Cross sections and distributions for $e^+e^-\to N\bar N$
are conveniently expressed in terms of the
electric and magnetic form factors $G_M^N$ and $G_E^N$,
where N stands for proton or neutron.
The choice of form factors adopted in the program is described
in detail in \cite{nucleon}. It is, however, set up
in a modular form such that the two form factors can be easily
modified and replaced by the user.
% ----------------------------
A qualitative discussion of form factor measurements
through the radiative return, can be based on the leading order process
\begin{equation}
e^+(p_1)+e^-(p_2) \to \bar{N}(q_1)+ N(q_2)+\gamma(k)~.
\label{proc}
\end{equation}
From the simple analytical results it is straightforward to
evaluate production rates, to understand the qualitative aspects of
angular distributions, and to develop strategies for the separation of
the electric and the magnetic form factor.
The differential cross section for
reaction (\ref{proc}) (with ISR only) can be written as
%
\bea
d\sigma &=& \frac{1}{2s}L_{\mu\nu}H^{\mu\nu} \nonumber \\
&& d\Phi_2(p_1+p_2;Q,k)
d\Phi_2(Q;q_1,q_2)\frac{dQ^2}{2\pi},
\eea
%
where $L_{\mu\nu}$ and $H^{\mu\nu}$ are the leptonic and hadronic tensors
respectively. For notation, definitions and an explicit form of the
leptonic tensor, see for instance \cite{Czyz:2000wh,Kuhn:2002xg}.
The hadronic tensor is given by
\bea
H_{\mu\nu}&=&2|G_M^N|^2(Q_\mu Q_\nu - g_{\mu\nu} Q^2) \nonumber \\
&&- \frac{8\tau}{\tau -1}\left(|G_M^N|^2-\frac{1}{\tau}
|G_E^N|^2\right)q_\mu q_\nu~.
\label{had_tens}
\eea
%
%
The nucleon and antinucleon
momenta are denoted by $q_1$ and $q_2$ respecti\-vely.
$Q=q_1+q_2$, $\tau = Q^2/4m_{N}^2$ and $q=(q_2-q_1)/2$.
Integrating over the whole range of nucleon angles and the
photon azimuthal angle, and restricting the photon polar angle
within $\frac{m_e}{\sqrt{s}} \ll
\theta_\gamma^{\rm min} < \theta_\gamma < \pi -\theta_\gamma^{\rm min} $,
the differential cross section factorizes into the cross section for
electron--positron annihilation into hadrons and a flux factor that
depends on $s$ and $Q^2$ only \cite{Binner:1999bt}:
%
\bea
\frac {{\rm d}\sigma}{{\rm d} Q^2} = \frac
{4 \alpha^3}{3 s Q^2 } R(Q^2)
\biggl\{ \frac {(1+\frac{Q^4}{s^2})}{(1-\frac{Q^2}{s})}
\log \frac {1+c_m}
{1-c_m} \nonumber \\
- \left(1-\frac{Q^2}{s}\right) c_m \biggl\}~,
\label{rad_ret}
\eea
%
and
%
\bea
R(Q^2)&=&\frac{\sigma(e^+e^-\rightarrow N \bar{N})}{\sigma_{\rm point}}
\nonumber \\
&=& \frac{\beta_N}{2}\left(2|G_M^N|^2+\frac{1}{\tau}|G_E^N|^2\right)~,
\label{rr}
\eea
where $c_m=\cos\theta_\gamma^{\rm min}$ and $\beta_N^2 = 1 - 4m_N^2/Q^2$.
After integration over the whole range of nucleon angles,
the separation of electric and magnetic form factors is no longer
feasible. However, the modulus of their ratio can be determined
from the study of properly chosen angular distributions.
The close connection between this analysis and the one based on the
reaction $e^+e^-\to N\bar N$ becomes apparent once the result
is expressed in terms of the polar and azimuthal angles $\hat\theta$ and
$\hat\varphi$, which characterize the nucleon direction in the rest frame
of the hadronic system, with the $z$-axis opposite to the direction of
the real photon momentum and the $y$-axis in the plane spanned by the
beam and the photon direction.
%
In the limit $Q^2\ll s$
%
\begin{eqnarray}
L_{\mu\nu}H^{\mu\nu} &\approx& \frac{(4\pi\alpha)^3}{Q^2}
\frac{(1+\cos^2\theta_\gamma)}{(1-\cos^2\theta_\gamma)}
\label{eq:LH3}
\\ & \kern-60pt
\times & \kern-40pt
4\left(|G_M^N|^2(1+\cos^2\hat\theta) + \frac{1}{\tau}|G_E^N|^2\sin^2\hat\theta\right).
\nonumber
\end{eqnarray}
(A more detailed discussion of angular distribution can be found in
\cite{nucleon}.)
It is instructive to compare the angular distribution in Eq.~(\ref{eq:LH3})
with the angular distribution from $e^+e^-\to N\bar N$:
\begin{eqnarray}
\frac{d\sigma}{d\Omega}=\frac{\alpha^2 \beta_N}{4 Q^2}
\biggl( |G_M^N|^2(1+\cos^2\theta) \nonumber \\
+ \frac{1}{\tau}|G_E^N|^2\sin^2\theta\biggr).
\label{sigdir}
\end{eqnarray}
%
The close relation between Eqs.~(\ref{eq:LH3}) and (\ref{sigdir})
is clearly visible.
\hskip-0.2cm
As is already evident from the hadronic tensor in Eq.~(\ref{had_tens}),
the phases of the form factors are, however, not accessible by this method.
Their determination would require the measurement of the polarization of the
nucleons in the final state \cite{DDR,Rock}. PHOKHARA can easily be
extended to describe this additional degree of freedom.
This option is of particular interest for $\Lambda \bar \Lambda$
production, with its self-analysing decay mode.
As will be discussed below, the event rate drops rapidly with increasing
$Q^2$ and only few events will be observed for invariant hadron masses
around 3~GeV. However, the rate for baryon pair production from $J/\psi$
decays will be significantly enhanced, all considerations given above
for continuum production do apply, and a precise measurement of the
branching ratio of $J/\psi$ into baryon pairs and the relative strength
of magnetic and electric coupling is within reach.
% -------------
Data giving information about the form factors in the time-like region
originate from the reactions
$e^+e^- \to p\bar p$ (and $p\bar p \to e^+e^-$) as
shown in Fig.~\ref{fig:1}.
The parametrization of the form factors used in PHOKHARA
has been adopted from Ref. \cite{ia-ja-la},
and is described in detail in
\cite{nucleon}.
%
A free parameter, denoted $\theta$ is set to
$\theta =\pi/4$ or $ \theta = 19\pi/64$ as recommended in \cite{iachello}.
%
\begin{figure}[htb]
\includegraphics[width=7.5cm,height=6.5cm]{praca_dsigeepp.ps}
\vskip-0.9cm
\caption{Comparison of the measured
\cite{antonelli98,antonelli94,antonelli93,bisello90,bisello83,delcourt79,castellano73}
$e^+e^- \to p\bar p$ cross section with the model from
Ref.\cite{ia-ja-la}. Predictions are given
for two different values ($\pi/4$ - curve A and $19\pi/64$ - curve B)
of the parameter $\theta$.}
\vskip-0.7cm
\label{fig:1}
\end{figure}
\begin{figure}
\includegraphics[width=7.5cm,height=6.5cm]{praca_protNLO_zcieciami_i_bez.ps}
\vskip-0.9cm
\caption{Differential cross section of the process
$e^+e^- \to p \bar p \gamma (\gamma)$ for two different sets of cuts.}
\vskip-0.7cm
\label{fig:5}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=7.5cm,height=6cm]{mQframe_GR_GMGE_qq4.5-5_fot25-155_hist-p.ps}
\vskip-0.9cm
\caption{Angular distributions in the polar angle of the proton
a typical different range of $Q^2$, with and without the constraint
$G_M^p = \mu_p G_E^p $ (see text for a more detailed explanation).}
\vskip-0.9cm
\label{fig:15a}
\end{center}
\end{figure}
Only measurements of the total cross section are available.
Form factors, if available at all, are extracted under the assumption
$G_M^p = \mu_p G_E^p$, and no true separation of the
electric and magnetic form factors has been performed.
The poor experimental knowledge can be substantially improved by using
the radiative return at $B$-meson
factories and the separation of electric and magnetic form factors
may be envisaged.
The differential cross section as a function of $Q^2$, as observed through
the radiative return, is shown in Fig.~\ref{fig:5}
for $p\bar{p}\gamma$. Similar results are obtained for $n\bar{n}\gamma$.
The upper curve represents the cross section after integration over all
proton angles and over photon angles down to $5^\circ$ w.r.t. the beam
direction. The lower curve is valid for cuts corresponding
to the BaBar acceptance region transformed to the $e^+e^-$ cms.
The event rates accumulated at $B$-factories
already now, may be large enough to extract and separate
$|G_M^N|$ and $|G_E^N|$.
To explore the sensitivity to the form
factors individually, we compare the differential cross sections obtained
for the model described above with the distribution obtained under
the assumption $G_M^p = \mu_p G_E^p $, subject, however, to
the constraint that $\sigma(e^+e^-\rightarrow p \bar{p})$
(see Eq.~(\ref{rr})) remains unaffected.
The angular distribution of the baryons, if defined in the laboratory
or cms frame, is strongly affected by the boost. The difference between
different choices for the form-factor ratios is expected to be more
pronounced for the proton angular distribution in the hadronic rest
frame, with the $z$-axis aligned with the direction of the photon and
the $y$-axis in the plane spanned by the beam and the photon directions
(see Eq.(\ref{eq:LH3}) above). For
$Q^2\ll s$, the case relevant to the present discussion,
the pronounced dependence of the
$\cos\hat\theta$ distribution on the choice of the form factors is
clearly visible (Fig. \ref{fig:15a}).
The importance of the radiative corrections
has been studied in \cite{nucleon}.
The use of the NLO generator is
highly recommended.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Pion and Kaon Form Factors at large $Q^2$}
In the near future the experimental knowledge on
the pion and kaon form factors $F_{\pi,K}(s)$
will be substantially improved, due to new data
to be obtained using the radiative return method.
The first measurements of $F_\pi$
with this technique in the $\rho$ region
have already been performed by the KLOE Collaboration \cite{KLOE}.
At larger energies, up to 2.0-2.5~GeV, perhaps even 3~GeV,
accurate measurements of $F_{\pi,K}$ are anticipated from the
BABAR experiment.
The high rates expected
at $s\gg m_\rho^2$ demand phenomenological models
more elaborated than the simple
$\rho$- ($\rho,\omega,\phi$-) dominance models for $F_\pi$ ($F_K$).
In \cite{PionKaon} an ansatz was suggested
that is valid in the region below at, and far
above $\rho$-resonance and extended to the kaon form factor.
It obeys the constraints from analyticity and
isospin-symmetry and incorporates the proper behaviour at high
energies, consistent with perturbative QCD, and the correct
normalization at $s=0$. Furthermore it is based on plausible
assumptions derived from the quark model, moderate SU(3)-breaking, vector
dominance and a pattern of radial excitations expected from dual
resonance models.
Above the $\rho$-resonance the excited $\rho'(1450)$ and $\rho''(1700)$
are expected to play an important role.
Already now these two states are indispensable, if one wants to
accommodate the measured parameters of the $\rho$-resonance with the
correct normalization of $F_\pi(s)$ at $s=0$.
In general,
to include all possible intermediate hadronic states in
the $\gamma^*\to \pi^+\pi^-$ transition amplitude,
one has to take into account an infinite series of
radially excited $\rho$'s. In addition, there are
multi-hadron intermediate states with
$J^P=1^-$ and $I=1$ ($2\pi, 4\pi, K\bar{K} $ etc.).
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 space-like region at $s\to -\infty$,
analytically continued to $s \to +\infty$.
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.
Recently, the model for the pion form factor
using the masses and coefficients
chosen according to Veneziano amplitude was considered in \cite{Dom}.
In \cite{PionKaon} the dual-QCD$_{N_c=\infty}$ model was used
as a starting point, modified
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 multi-hadron ($2\pi,4\pi$ etc.) states
were implicitly taken into account.
\subsection{The pion form factor}
%{\em The pion form factor}
The following model for the pion form factor has been adopted:
\begin{eqnarray}
F_\pi(s)&=& \left[\sum\limits_{n=0}^3 c_n BW_n(s)\right]_{fit} \nonumber \\
&+& \left[\sum\limits_{n=4}^{\infty}c_n BW_n(s)
\right]_{dual-QCD_{N_c=\infty}},
\label{themodel}
\end{eqnarray}
where in the $\rho$ contribution ($n=0$) the $\rho-\omega$ mixing is
included.
The parameters of the four lowest $\rho_{0,1,2,3}$
states (i.e. $\rho,\rho'(1450)$, $\rho''(1700)$ and $\rho'''$) are
fitted to data and $BW_n$ denotes a Breit-Wigner resonance amplitude.
Let us emphasize the main qualitative features of this model.
It nicely matches the existing $\rho$-dominance
models at $\sqrt{s}<$ 1 GeV, simply because the $\rho_{n>2}$
states play a minor role
in the $\sqrt{s}<1$ GeV region. It
is flexible, that is, it allows to vary the proportion of fitted
and modeled 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 above, can be easily continued
to $s<0$ and compared with the experimental data
and QCD predictions in the space-like region.
Accordingly, one gets a smooth power-like behaviour at
asymptotically large time-like $s$.
We have fitted the model (\ref{themodel}) to the existing data
for the time-like pion form factor.
The results of the fit for two different versions of
BW-formulae (KS and GS
for the first three resonances $\rho_{0,1,2}$),
are plotted in Fig. \ref{fig-Fpitmlb}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=7.5cm,height=5.8cm]{fig1a.ps} \\
\vskip-1cm (a) \\
\vskip-0.3cm
\includegraphics[width=7.5cm,height=5.8cm]{fig1b.ps} \\
\vskip-1cm (b)
\vskip-0.5cm
\caption{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.}
\vskip-0.8cm
\label{fig-Fpitmlb}
\end{center}
\end{figure}
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.
Furthermore, the fitted mass of $\rho'$ (1360-1380 MeV) gets shifted
with respect to the PDG value (1465$\pm$25 MeV).
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.
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).
For the time being it seems difficult to accommodate a pion
form factor as large as the one derived from $J/\psi$ decay.
To check our model further $F_\pi(s)$ was continued to $s<0$ and
successfully compared with the data there.
Furthermore, reasonable agreement
between the model and the QCD light-cone sum rule (LCSR) predictions
\cite{lcsr,BK} was observed.
\subsection{Charged and neutral kaon e.m. form factors}
The analogous strategy was adopted 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 ansatz for the kaon form factors adopted in \cite{PionKaon} reads:
\begin{eqnarray}
&&\kern-20pt F_{K^+/K^0}(s)= \nonumber \\
&&\kern-20pt\pm\frac12 (c^K_\rho BW_{\rho}(s)+
c^K_{\rho'} BW_{\rho'}(s)+c^K_{\rho''} BW_{\rho''}(s))
\nonumber
\\&&\kern-20pt+
\frac{1}{6}(c_\omega^K BW_\omega(s)+c_{\omega'}^K BW_{\omega'}(s)+
c_{\omega''}^K BW_{\omega''}(s))\nonumber \\
&&\kern-20pt+\frac{1}3(c_{\phi} BW_{\phi}(s)+c_{\phi'} BW_{\phi'}(s)).
\label{Kformfpm1}
\end{eqnarray}
%
\begin{figure}[ht]
\begin{center}
\vskip-0.2cm
\includegraphics[width=7.5cm,height=5.8cm]{fig4b.ps} \\
\vskip-0.7cm
\includegraphics[width=7.5cm,height=5.8cm]{fig5b.ps} \\
\vskip-1.35cm
\caption{The charged and neutral kaon form factor squared as a function
of $\sqrt{s}$ in the region above the $\phi$ resonance.}
\vskip-0.9cm
\label{fig-F}
\end{center}
\end{figure}
%
\begin{figure}[ht]
\begin{center}
\includegraphics[width=7.5cm,height=6cm]{fig7.ps}
\vskip-1.4cm
\caption{ Normalized distribution
in the kaon pair invariant mass $\sqrt{Q^2}$
obtained from the fitted kaon form factor;
the solid (dashed) line corresponds to the constrained (unconstrained)
fit.}
\vskip-1.cm
\label{fig-distr}
\end{center}
\end{figure}
The resulting curves for the form factors are plotted
in Fig. \ref{fig-F}.
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.
Two different variants of the fit were 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
(solid line);
(2) the {\em unconstrained} fit, where $\omega$- and $\rho$-
factors are fitted as independent parameters (dashed line).
We 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.
\subsection{Predicting $\tau\to K^- K^0\nu_\tau$ decay distribution and rate}
The isospin one part of the e.m. kaon form factor,
can be used to predict the $\tau\to K^- K^0\nu_\tau$ decay width.
The fitted values for $c_{\rho,\rho',\rho''}^K$
from $e^+e^-$ annihilation thus allow to
calculate the decay rate and distribution.
The normalized distribution is plotted in Fig.\ref{fig-distr} and
(qualitatively) compared with the
event distribution in the kaon pair mass,
measured by CLEO Collaboration \cite{CLEOtau,ALEPHtau}.
The branching ratio prediction for the
$ BR(\tau\to K^-K^0 \nu_\tau )=0.19\pm 0.01\% ~~(0.13 \pm 0.01 \%) $
for the constrained (unconstrained) fit can
be compared with the experimentally measured value \cite{PDG}
$BR(\tau\to K^-K^0 \nu_\tau)= 0.154\pm 0.016\%$.
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 increasing contributions from the excited $\rho$-mesons,
an effect observed earlier in \cite{FKM}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}
This contribution is based on work performed in collaboration with
C.~Bruch, H.~Czy{\.z}, A.~Grzeli{\'n}ska, A.~Khodjamirian and G.~Rodrigo.
I would like to thank A.~Denig, W.~Kluge, and S.~M\"uller for discussion of
experimental aspects and A.~Grzeli{\'n}ska for help with the preparation
of the manuscript.
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\end{document}