The BaBar radiative return program aims at the study of all significant hadronic processes in
electron-positron annihilation, $e^+e^-\to hadrons$, for energies
from threshold up to about 4.5 GeV. Moreover, hadron spectroscopy of the initial
J$^{PC} = 1^{--}$ states, which are produced in $e^+e^-$ collisions, and of their
decay products is performed. In this chapter BaBar results for
processes with 3, 4, 5 and 6 hadrons in the final state, as well as measurements of
baryon form factors in the time-like region are reported.
A precision analysis of the pion form factor,
i.e. of the cross section $e^+e^-\to \pi^+\pi^-$, which is essential for an improved
determination of the hadronic contribution to the anomalous magnetic moment of the muon,
appeared most recently~\cite{Aubert:2009fg}.
The results presented in this chapter are based on a total integrated luminosity
of 230 fb$^{-1}$, except for the $3\pi$ and 4 hadron channels of Ref.~\cite{Aubert:2005eg}, which
were analysed using a data sample of 90 fb$^{-1}$. The total BaBar data sample
collected between the years 1999 to 2008 amounts to 530 fb$^{-1}$.
A typical feature common to all radiative return analyses at BaBar is
a wide coverage of the entire mass range of interest
in one single experiment, with reduced point-by-point uncertainties compared
to previous experiments.
\\
\\
{\bf \it $e^+e^-\to 3$ pions}
\\
The $\pi^+\pi^-\pi^0$ mass spectrum has been measured from $1.05$ GeV up to the $J/\psi$ mass region
with a systematic error of $\sim 5\%$ below $2.5$ GeV, and up to $\sim 20\%$ at
higher masses~\cite{Aubert:2004kj}.
The spectrum is dominated by the $\omega$, $\phi$ and $J/\psi$ resonances.
The BaBar measurement was able to significantly improve the world knowledge on the
excited $\omega$ states. The spectrum has been fitted
up to $1.8$ GeV and the following results for the masses and widths of the
$\omega^\prime$ and $\omega^{\prime\prime}$ states have been found:
%\footnote{in this fit
%$M_{\omega,\phi}$ and the phases between $\omega$, $\phi$ with respect to
%$\omega^\prime$, $\omega^{\prime\prime}$ are fixed.}
$M(\omega^\prime)=(1350\pm20\pm20)$ MeV, $\Gamma(\omega^\prime)=(450\pm70\pm70)$ MeV,
$M(\omega^{\prime\prime})=(1660\pm10\pm2)$ MeV,
$\Gamma(\omega^{\prime\prime})=(230\pm30\pm20)$ MeV.
Note that below $1.4$ GeV the results from BaBar are in good agreement
with those from SND~\cite{Achasov:2002ud}, while above this
energy the cross sections measured by BaBar are much higher than those
from DM2~\cite{Antonelli:1992jx}.
%Fig.~\ref{fig.3pi} shows the BaBar data points
%together with previous measurements in the energy range relevant
%for the exited states $\omega^\prime$ and $\omega^{\prime\prime}$.
%A clear disagreement with DM2~\cite{dm23pi} is seen above $1.4$ GeV.
%The BaBar spectrum has been fitted
%up to $1.8$ GeV and the following results for the masses and widthes of the
%$\omega^\prime$ and $\omega^{\prime\prime}$ have been found\footnote{in this fit
%$M_{\omega,\phi}$ and the phases between $\omega$, $\phi$ with respect to
%$\omega^\prime$, $\omega^{\prime\prime}$ are fixed.}:
%$M(\omega^\prime)=(1350\pm20\pm20)$ MeV, $\Gamma(\omega^\prime)=(450\pm70\pm70)$ MeV,
%$M(\omega^{\prime\prime})=(1660\pm10\pm2)$ MeV,
%$\Gamma(\omega^{\prime\prime})=(230\pm30\pm20)$ MeV.
\begin{figure}[t]
\begin{center}
\epsfig{file=xs4pi_isr_2.2.eps, width=6.cm}
%\epsfig{file=xs_2k2pi_all.eps, width=6.cm}
%\vspace{-1.0cm}
\caption{\small{BaBar measurement of the energy dependence of the
$e^+e^-\to \pi^+\pi^-\pi^+\pi^-$
%and the $e^+e^-\to K^+K^-\pi^+\pi^-$ (lower plot)
cross section obtained
by radiative return in comparison with the world data set.}}
\label{fig.4pi}
\end{center}
\end{figure}
\\
\\
{\bf \it $e^+e^-\to 4$ hadrons}
\\
The $\pi^+\pi^-\pi^+\pi^-$, $K^+K^-\pi^+\pi^-$ and $K^+K^-K^+K^-$ exclusive final states
have been measured from threshold up to $4.5$ GeV with systematic errors of $5\%$, $15\%$ and
$25\%$, respectively~\cite{Aubert:2005eg}.
The $K^+K^-K^+K^-$ measurement is the first measurement of this process at all.
Figure~\ref{fig.4pi} shows the mass distribution of the $\pi^+\pi^-\pi^+\pi^-$ channel.
We identify an impressive improvement with respect to previous experiments.
Background is relatively low for all channels under study
(e.g. a few percent at $1.5$ GeV for $\pi^+\pi^-\pi^+\pi^-$)
and is dominated by ISR-events of higher multiplicities and by continuum non-ISR
events at higher masses.
The $\pi^+\pi^-\pi^+\pi^-$ final state is dominated by the two-body intermediate state $a_1(1260)\pi$;
the $K^+K^-\pi^+\pi^-$ final state shows no significant two-body states, but a rich three-body
structure, including $K^*(890)K\pi$, $\phi\pi\pi$, $\rho K K$ and $K_2^*(1430)K\pi$.
\\
Figure~\ref{fig2n} shows BaBar preliminary results for the process
$e^+e^-\to\pi^+\pi^-\pi^0\pi^0$. The current systematic error of the measurement varies from 8\% around
the peak of the cross section to 14\% at 4.5 GeV.
%\begin{figure}
%\includegraphics[width=0.4\textwidth]{figs/Fig1.eps}
%\caption{\label{fig1}The $e^+e^-\to\pi^+\pi^-\pi^0\pi^0$ cross section, measured by BaBar.}
%\end{figure}
BaBar results are in agreement with
SND~\cite{AchasovSND:2001zz} in the energy range below 1.4 GeV and show a significant improvement for higher
energies ($>$ 1.4 GeV). In the energy range above 2.5 GeV this is the first measurement at all.
\begin{figure}
\begin{center}
\epsfig{file=Fig2_babar.eps, width=6.cm}
\caption{\label{fig2n} Preliminary BaBar data for the $e^+e^-\to\pi^+\pi^-\pi^0\pi^0$ cross section
in comparison with previous experiments.}
\end{center}
\end{figure}
The $e^+e^-\to\pi^+\pi^-\pi^0\pi^0$ final state is dominated by the
$\omega\pi^0$, $a_1(1260)\pi$ and $\rho^+\rho^-$ intermediate channels, where the
latter channel has been observed for the first time.
\\
A specific analysis was devoted to the intermediate structures in the
$e^+e^-\to K^+K^-\pi^+\pi^-$ and $e^+e^-\to K^+K^-\pi^0\pi^0$ channels~\cite{Aubert:2007ur}.
Of special interest is
the intermediate state $\phi f_0(980)$, where the decays $f_0(980)\to\pi^+\pi^-$ and
$f_0(980)\to\pi^0\pi^0$ have been looked at.
A peak is observed in the $\phi f_0(980)$ channel at a mass $M = 2175\pm 18$ MeV and a width $\Gamma = 58\pm 2$ MeV.
The new state is usually denoted as Y(2175) and is also clearly visible in the $K^+K^-f_0$ spectrum.
\\
\\
{\it \bf $e^+e^-\to 2(\pi^+\pi^-)\pi^0$, $2(\pi^+\pi^-)\eta$}
\\
The $e^+e^- \to 2(\pi^+\pi^-)\pi^0$ cross section has been measured by BaBar
from threshold up to 4.5 GeV~\cite{Aubert:2007ef}. A large coupling of the $J/\psi$ and $\psi(2S)$ to this channel is observed.
The systematic error of the measurement is about 7\% around the peak of the mass spectrum.
In the $\pi^+\pi^-\pi^0$ mass distribution the $\omega$ and
$\eta$ peaks are observed; the rest of the events have a $3\pi\rho$ structure.
%The $\eta\pi^+\pi^-$ cross section is shown
%in Fig.~\ref{fig6} and compared to previous data sets.
%\begin{figure}
%\includegraphics[width=0.4\textwidth]{figs/Fig4.eps}
%\caption{The $e^+e^-\to 2(\pi^+\pi^-)\pi^0$ cross section measured by BaBar.}
%\label{fig4}
%\end{figure}
%\begin{figure}
%\includegraphics[width=0.4\textwidth]{figs/Fig5.eps}
%\caption{The $e^+e^-\to \eta\pi^+\pi^-$ cross section measured by BaBar, the comparison
%with previous measurements is also shown.}
%\label{fig6}
%\end{figure}
%\begin{figure}
%\includegraphics[width=0.4\textwidth]{figs/Fig7.eps}
%\caption{The $e^+e^-\to 2(\pi^+\pi^-)\eta$ cross section measured by BaBar.}
%\label{fig7}
%\end{figure}
\\
BaBar performed also the first measurement of the $e^+e^-\to 2(\pi^+\pi^-)\eta$ cross section.
A peak value of about 1.2 nb at about 2.2 GeV is observed, followed by a monotonic decrease towards
higher energies. Three intermediate states are seen: $\eta\rho(1450)$, $\eta^{\prime}\rho(770)$ and
$f_1(1285)\rho(770)$.
\begin{figure}[t]
\begin{center}
\epsfig{file=xs_6pi_babar.eps, width=6.cm}
\epsfig{file=xs_4pi2pi0_babar.eps, width=6.cm}
%\vspace{-1.0cm}
\caption{\small{The energy dependence of the cross sections for
$e^+e^-\to 3(\pi^+\pi^-)$ (upper plot) and $e^+e^-\to
2(\pi^+\pi^-)2\pi^0$ (lower plot), obtained by BaBar (filled circles)
by radiative return, in comparison with previous data.}}
\label{fig.6pi}
\end{center}
\end{figure}
\\
\\
{\bf \it $e^+e^-\to 6$ hadrons}
\\
The $6$ hadron final state has been measured in the exclusive channels
$3(\pi^+\pi^-)$, $2(\pi^+\pi^-)2\pi^0$ and $K^+K^-2(\pi^+\pi^-)$ \cite{Aubert:2006jq}.
The cross section in the last case has never been measured before; the precision
in the first two cases is $\sim 20\%$, which is a large improvement with respect to existing data.
Again, the entire energy range from threshold up to $4.5$ GeV is measured
in a single experiment.
The distributions for the final states $3(\pi^+\pi^-)$ and $2(\pi^+\pi^-)2\pi^0$
are shown in Fig.~\ref{fig.6pi}.
A clear dip is visible at about $1.9$ GeV in both pion
modes. A similar feature was already seen
by FOCUS~\cite{Frabetti:2001ah} in the diffractive photo-production of six charged pions.
The spectra are fitted by BaBar using the sum of a Breit-Wigner resonance function
and a Jacob-Slansky continuum shape. For the
$3(\pi^+\pi^-)$ ($2(\pi^+\pi^-)2\pi^0$) mode, BaBar obtains values of $1880 \pm 30$ MeV
($1860 \pm 20$ MeV) for the resonance peak, $130 \pm 30$ MeV
($160 \pm 20$ MeV) for the resonance width and $21^{\rm o} \pm 14^{\rm o}$
($-3^{\rm o} \pm 15^{\rm o}$) for the phase shift between the resonance and continuum.
%The width values differ significantly from the FOCUS result of $29 \pm 14$ MeV.
\\
\\
{\it \bf $e^+e^-\to K^+K^-\pi^0, K^+K^-\eta$, $K_S K^{\pm}\pi^{\mp}$}
\\
A recent BaBar ISR-analysis is dedicated to three hadrons in the final state, including a pair of
kaons ($K^+K^-\pi^0$, $K K_S \pi$); a peak near 1.7 GeV, which is mainly due to
the $\phi^{\prime}$(1680) state, is observed.
A Dalitz plot analysis shows that the $KK^{*}$(892) and $KK^{*}_2$(1430)
intermediate states are dominating the $K\bar{K}\pi$ channel.
A fit to the $e^+e^-\to K\bar{K}\pi$ cross section assuming
the expected contributions from the
$\phi,\phi^{\prime},\phi^{\prime\prime},\rho^{0},\rho^{\prime},\rho^{\prime\prime}$
states was performed. The parameters of the $\phi^{\prime}$ and other excited
vector meson states are compatible with PDG values.
%\begin{figure}
%\includegraphics[width=0.4\textwidth]{figs/Fig8.eps}
%\caption{The $\phi f_0(980)$ cross section measured in the $K^+K^-\pi^+\pi^-$ (circles) and
%the $K^+K^-\pi^0\pi^0$ (squares) final states. The hatched histogram shows the
%simulated cross section, assuming no resonant structure. The solid (dashed) line shows
%the result of a one-resonance (no-resonance) fit to data.}
%\label{fig8}
%\end{figure}
%\begin{figure}
%\includegraphics[width=0.4\textwidth]{figs/Fig9.eps}
%\caption{BaBar measurement of the $e^+e^-\to K_S K^{\pm}\pi^{\mp}$ cross section.}
%\label{fig9}
%\end{figure}
%\begin{figure}
%\includegraphics[width=0.4\textwidth]{figs/Fig10.eps}
%\caption{The Dalitz plot for the $K_S K^{\pm}\pi^{\mp}$ state show that the
%K$\bar{K}\pi$ channel is dominated by the intermediate states K$^{*}$(892) and K$^{*}_2$(1430).}
%\label{fig10}
%\end{figure}
\\
\\
{\bf \it Time-like proton form factor $e^+e^-\to p \bar{p}$,
hyperon form factors $e^+e^-\to\Lambda^0\bar{\Lambda^0},\Lambda^0\bar{\Sigma^0},\Sigma^0\bar{\Sigma^0}$}
\\
\begin{figure}[t]
\begin{center}
\epsfig{file=envect2.eps, width=6.8cm}
%\vspace{-1.0cm}
\caption{\small{The $e^+e^-\to p\bar{p}$ cross section measured by BaBar (filled circles)
in comparison with data from other $e^+e^-$ colliders (blue points) and from $\bar{p}p$
experiments (red points).}}
\label{fig.ppbar}
\end{center}
\end{figure}
BaBar has also performed a measurement of the $e^+e^-\to p\bar{p}$ cross section~\cite{Aubert:2005cb}.
This time-like process is parametri\-sed by the electric
and magnetic form factors, $G_E$ and $G_M$:
\begin{eqnarray*}
\sigma_{e^+e^- \to p \bar{p}} & = & \frac {4 \pi \alpha^2 \beta C}{3s} \\
& \times & (|G_M|^2 + \frac{2m_p^2}{s} |G_E|^2),
\end{eqnarray*}
where $\beta=\sqrt{1-4m_p^2/s}$ and the
factor $C=y/(1-e^{-y})$ (with $y=\pi \alpha m_p/(\beta \sqrt{s})$) accounts for the Coulomb interaction of the final state particles.
The proton helicity angle $\theta_p$ in the $p\bar{p}$ rest frame can be used to
separate the $|G_E|^2$ and $|G_M|^2$ terms. Their respective variations are approximately
$\sim \sin^2\theta_p$ and $\sim (1+\cos^2\theta_p)$.
By fitting the $\cos\theta_p$ distribution to a sum of the two terms,
the ratio $|G_E|/|G_M|$ can be extracted. This is done separately in six bins
of $M_{p\bar{p}}$. The results disagree significantly
with previous measurements from LEAR~\cite{Bardin:1994am} above threshold. BaBar observes a ratio
$|G_E|/|G_M|>1$ above threshold, while at larger values of
$M_{p\bar{p}}$ the BaBar measurement finds $|G_E|/|G_M| \approx 1$. LEAR data, on the
contrary, show a behaviour $|G_E|/|G_M|<1$ above threshold.
\\
In order to compare the cross section measurement with previous data ($e^+e^-$ and $\bar{p}p$
experiments), the {\it effective} form factor $G$ is introduced:
$G = \sqrt{|G_E|^2 + 2m_p^2/s |G_M|^2}$. The BaBar measurement of
$G$ is in good agreement with existing results, as can be seen in Fig.~\ref{fig.ppbar}. The structure of
the form factor is rather complicated; the following observations can be made:
(i) BaBar confirms an increase of $G$ towards threshold as seen before by other experiments;
(ii) two sharp drops of the spectrum at $M_{p\bar{p}}=2.25$ and $3.0$ GeV are observed;
(iii) data at large values
$M_{p\bar{p}}>3$ GeV are in good agreement with the prediction from perturbative QCD.
\\
A continuation of the ISR program with baryon final states is the
measurement of the $e^+e^-\to\Lambda\bar{\Lambda}$ cross section~\cite{Aubert:2007uf}.
So far only one data point from DM2~\cite{Bisello:1990rf} was existing for this channel, which is in good
agreement with BaBar data. About 360 $\Lambda\bar{\Lambda}$ events could be selected
using the $\Lambda\to p\pi$ decay.
In two invariant mass bins an attempt has been made to extract the
ratio of the electric to magnetic form factor $|G_E|/|G_M|$.
In the mass range below $2.4$ GeV this ratio is above unity --
as in the proton case -- with
a significance of one standard deviation ($|G_E|/|G_M|=1.73^{+0.99}_{-0.57}$).
Above $2.4$ GeV the ratio is consistent with unity ($|G_E|/|G_M|=0.71^{+0.66}_{-0.71}$).
Also the $\Lambda$ polarisation and the phase between $G_E$ and $G_M$ was studied
using the slope of the angle between the polarisation axis
and the proton momentum in the $\Lambda$ rest frame.
The following limit on $\Lambda$ polarisation is obtained: $-0.22 < \zeta < 0.28$; the relative phase
between the two form factors is measured as $-0.76 < \sin(\phi) < 0.98$, which is
not yet significant due to limited statistics.
%\begin{figure}
%\includegraphics[width=0.4\textwidth]{figs/Fig12.eps}
%\caption{BaBar measurement of the $\Lambda \bar{\Lambda}$ effective form factors, i.e.
%assuming that the electric and magnetic form factors $G_E$ and $G_M$ are equal.}
%\label{fig12}
%\end{figure}
%\begin{figure}
%\includegraphics[width=0.4\textwidth]{figs/Fig13.eps}
%\caption{The time-like form factors measured by BaBar for the proton, $\Lambda$, $\Sigma^0$ and
%$\Lambda\Sigma^0$; all form factors show a similar behaviour.}
%\label{fig13}
%\end{figure}
\\
Finally, the first measurements of the
$e^+e^-\to\Sigma^0\bar{\Sigma^0}$ and $e^+e^-\to\Sigma^0\bar{\Lambda}(\Lambda\bar{\Sigma^0})$ cross
sections were performed. For the detection of the $\Sigma^0$ baryon, the decay
$\Sigma^0\to\Lambda\gamma\to p\pi\gamma$ was used.
About 40 candidate events were selected for the reaction
$\Sigma^0\bar{\Sigma^0}$ and about
20 events for $\Lambda\bar{\Sigma^0}$.
All baryon form factors measured by BaBar have a similar size and mass shape, namely
a rise towards threshold. The reason for this peculiar behaviour is not
understood.