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\def\lamb#1#2{$^{#1}_{\Lambda}${#2}} %  A < 10
\def\lam#1#2{$^{#1}_{~\Lambda}${#2}} %  A .ge. 10

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% \title{
% {\bf E94-107: High-resolution hypernuclear 1p shell spectroscopy} \\
% }
\subsection{E94-107}

% \begin{document}


\begin{center}
{\bf High Resolution Hypernuclear 1p-shell Spectroscopy}\\
F. Garibaldi, S. Frullani, J. J. LeRose, P. Markowitz, spokespersons\\
and\\
the Hall A collaboration
\end{center}

\subsubsection{The experiment}
Hypernuclei provide us with a variety of nuclear
phenomena. Hypernuclei are unique laboratories for studying the
$\Lambda N$ interaction.  Experiment E94-107 started a systematic
study of high-resolution hypernuclear spectroscopy on $p$-shell
targets, $^{12}$C, $^{9}$Be, $^{16}$O, providing crucial information
complementary to the studies performed with hadron probes and
$\gamma$-ray spectroscopy.  The first part of the experiment on
$^{12}$C and $^{9}$Be targets was performed in January and April-May
2004 in Hall A. The second part of the experiment on $^{16}$O was
carried out in June 2005.  The physics motivation, the experimental
challenges and the improvements of the Hall A apparatus (septum
magnets, optics, aerogel detectors, RICH detectors, waterfall target)
were described in previous Hall A status reports. The update of the
status of analysis is presented here.

\subsubsection{Present status and outlook}

\subsubsection*{High Resolution Spectroscopy of \lam{16}{N} by $K^+$ 
Electroproduction on $^{16}$O}

The results of \lam{12}{B} by $K^+$ electroproduction on $^{12}$C are
published on \cite{Carbon}.

Experiment E94-107 used a waterfall target for hypernuclear production
on Oxygen nuclei and simultaneously for the measurement of the
elementary cross section in p$(e,\,e'{\rm K}^+)\Lambda$.

Kinematics were set to electron detection at $6^\circ$ for scattered
electrons with momentum of 1.44~GeV/c, incident beam energy of
3.66~GeV, virtual photon energy of 2.2 GeV with Q$^{2} = 0.06$
GeV$^{2}$.  Scattered kaons were detected with momenta of 1.96~GeV/c
at $6^\circ$ .


Figure~\ref{oxy1} shows the cross-section for the \lamb{16}{N}
hypernuclear spectrum produced on Oxygen nuclei at
$\theta_{K^+,e}=6^\circ$ . The dotted line is a result of the
theoretical calculation using the SLA model for the elementary cross
section and J.~Millener calculations for the hypernuclear
structure. The red line is the fit of the curve with same procedure
used for Carbon spectrum.  The overall picture shows good agreement
between the data and the calculations in terms of positions and
relative strength of the levels, as reported in the Table
\ref{oxytable}.  The agreement also provides a upper limit on the
charge-dependent effects in hyperon-nucleon interactions.  The
position of the fourth peak has a significant discrepancy between
theory and experiment, providing an indication of very small
spin-orbit splitting of the $p_{\Lambda}$ orbits.


The waterfall thickness used during the experiment was
75~mg/cm$^2$. This value was determined using the known
cross-section of the elastic reaction on hydrogen.

\begin{table}
\centering
\caption[E94-107: $^{16}$O$(e,e'K^+)$\lam{16}{N} levels and cross sections compared with theoretical predictions]{%
Levels and cross sections obtained by fitting the
$^{16}$O$(e,e'K^+)$\lam{16}{N} spectrum compared with theoretical
predictions.}

\begin{tabular}{ccccccc}
$E_x/E_\Lambda$ &  Width  &  Cross section & $E_x$  & Wave
function & $J^\pi$ & Cross section \\
 \units{[MeV]}  &  \units{[FWHM, MeV]} & \units{[nb/sr^2/GeV]} & \units{[MeV]} &   &   &
 \units{[nb/sr^2/GeV]}\\
 \hline
 0.0/13.76 $\pm$ 0.16   & 1.71 & 1.45$\pm$ 0.26 & 0.00 & $p^{-1}_{1/2}
\otimes s_{1/2\Lambda}$& $0^-$ & 0.002 \\
   &  &  & 0.03 & $p^{-1}_{1/2}\otimes s_{1/2\Lambda}$ & $1^-$ & 1.45 \\
 & & & & & & \\
6.83 $\pm$ 0.06 & 0.88  & 3.16 $\pm$ 0.35 & 6.71 &
$p^{-1}_{3/2}\otimes s_{1/2\Lambda}$ & $1^-$ & 0.80 \\
   &  &  & 6.93 &$p^{-1}_{3/2}\otimes s_{1/2\Lambda}$ & $2^-$   &2.11 \\
 & & & & & & \\
10.92 $\pm$ 0.07 & 0.99 & 2.11 $\pm$ 0.37 & 11.00 &
$p^{-1}_{1/2}\otimes p_{3/2\Lambda}$ & $2^+$ & 1.82 \\
  & & & 11.07 & $p^{-1}_{1/2}\otimes p_{1/2\Lambda}$ & $1^+$ & 0.62 \\
 & & & & & & \\
17.10 $\pm$ 0.07 & 1.00 & 3.44 $\pm$ 0.52 & 17.56 &
$p^{-1}_{3/2}\otimes p_{3/2\Lambda}$ & $3^+$ & 2.10 \\
  & &  & 17.57 & $p^{-1}_{3/2}\otimes p_{1/2\Lambda}$ & $2^+$ & 2.26 \\
\hline
\end{tabular}
\label{oxytable}
\end{table}


\begin{figure}[ht!]
%  \centerline{\psfig{figure=fig_oxy_new.eps,width=12.0cm}}
 \centerline{\showfig{width=12.0cm}{E94107_fig_oxy_new}}
\caption[E94-107: Experimental data compared to new theoretical model for the \lam{16}{N} binding-energy spectrum]{%
Experimental data for the \lam{16}{N} binding-energy spectrum compared
with a new theoretical model (dashed curve). The solid curve
represents the fit of the data with Voigt functions.}
\label{oxy1}
\end{figure}


These results are significantly refined with respect to those
published on the previous Hall A Report and presented at HYP 2006
Conference \cite{JJL}.  The differences in the measured cross-sections
are mainly due to the precise calculation of the target thickness,
significantly different than the previous, expected value which was
based on a old calibration performed in a different laboratory.  In
order to understand and explain the new values of the cross-sections,
the calculation of the $K^+$ optical potential was improved. In
details, in the Distorted Wave Impulse Approximation (DWIA), kaon
distortion in the final state is usually described by simple first
order optical potential constructed from elementary t-matrices of
kaon-baryon scattering.  The optical potential used in previous
calculation was based an a so-called Martin parametrization that was
found to have limitations at the Kaon energy of the E94-107
experiment.  Therefore a more realistic model, based on recent CERN
and HERA data, with a stronger kaon absorption, is now used and
compared with the experimental data, showing a good general agreement.
This new optical potential changes the theoretical calculations
published on \cite{Carbon}, but there is still a good agreement with
experimental data, as shown in Figure~\ref{newC}.

\begin{figure}[ht!]
%  \centerline{\psfig{figure=prova.eps,width=11.0cm}}
 \centerline{\showfig{width=11.0cm}{E94107_prova}}
\caption[E94-107: Experimental data compared to new theoretical model for the \lam{12}{B} binding-energy spectrum]{%
Experimental data for the \lam{12}{B} binding-energy spectrum compared
with a new theoretical model (dashed curve). The solid curve
represents the fit of the data with Voigt functions.}
\label{newC}
\end{figure}


The measured $\Lambda$ binding energy in \lam{16}{N}, $B_\Lambda =
13.76\pm 0.16$ MeV obtained from the first peak is a previously not
well known quantity. There are few emulsion events for the heavier
p-shell hypernuclei and these events tend to have ambiguous
interpretations~\cite{davis91}. Also, the reactions involving the
production of a $\Lambda$ from a neutron are more difficult to
normalize.  For these reasons, data on the mirror hypernucleus have a
difficult interpretation.  Therefore the present measurement may help
in shedding light on this issue, providing the most precise data point
and considering the charge-dependent effect for the other peaks.

The preliminary results on the simultaneous measurement of the
elementary reaction on hydrogen also shows a similar agreement between
the data ($\sigma$=30~nb/sr$^2$/GeV $\pm$ 20\%) and the predicted
cross section of 36~nb/sr$^2$/GeV according to the SLA model.


\subsubsection*{High Resolution Spectroscopy of \lam{9}{Li} by $K^+$ 
Electroproduction on $^{9}$Be}

Preliminary results of the \lam{9}{Li} are shown in Figure~\ref{berguid}.

\begin{figure}[ht!]
%  \centerline{\psfig{figure=BerylliumFig3ExEn.eps,width=12.0cm}}
 \centerline{\showfig{width=12.0cm}{E94107_BerylliumFig3ExEn}}
\caption[E94-107: Preliminary data for \lam{9}{Li}]{%
Preliminary data for \lam{9}{Li}. The dashed line represents a
theoretical model, the solid curve is a fit performed with Voigt
functions as for the previous data. The number of peaks to be fitted is
determined by a peak-search algorithm.}
\label{berguid}
\end{figure}

The first-step analysis shown in Fig.~\ref{berguid} is driven by a
peak-search and $\chi^2$ minimization procedures. A second step is the
attempt of fitting the spectrum according to the expected structure of
the energy levels, as shown in Fig.~\ref{ber}.

\begin{figure}[ht!]
%  \centerline{\psfig{figure=berRound11EnEx.eps,width=11.0cm}}
 \centerline{\showfig{width=11.0cm}{E94107_berRound11EnEx}}
\caption[E94-107: Preliminary data for \lam{9}{Li}]{%
Preliminary data for \lam{9}{Li}. The number of peaks to be fitted
is determined according to the theoretical models and to the $\chi^2$
minimization.}
\label{ber}
\end{figure}

\begin{table}
\centering
\caption[E94-107: $^{9}$Be$(e,e'K^+)$\lam{9}{Li} levels and relative strength]{%
Levels and relative strength obtained by fitting the
$^{9}$Be$(e,e'K^+)$\lam{9}{Li} spectrum shown in Fig.~\ref{ber}.}
%
\begin{tabular}{ccccc}
 Peak &  Strength   &  Position & FWHM & Integral \\
 (a. u.)  &  (a. u.) & \units{[MeV]} & \units{[MeV]} & (a. u.)\\
\hline
 0 & 19.15 $\pm$ 4.89   & 6.83 $\pm$ 0.08 & 0.52 $\pm$ 0.07 & 19.0 \\
 1 & 14.60 $\pm$ 4.39   & 7.30 $\pm$ 0.12 & 0.52 $\pm$ 0.07 & 14.5 \\
 2 & 14.63 $\pm$ 2.37   & 8.11 $\pm$ 0.05 & 0.50 $\pm$ 0.07 & 14.5  \\
 3 & 12.99 $\pm$ 2.40   & 8.64 $\pm$ 0.06 & 0.53 $\pm$ 0.07 & 12.9 \\
 4 & 11.81 $\pm$ 2.02   & 9.24 $\pm$ 0.05 & 0.45 $\pm$ 0.07 & 11.7 \\
\hline
\end{tabular}
%
\end{table}

This spectrum would be able to provide information on the spin-spin
part of the $\Lambda N$ potential, the most interesting term. The
choice of five peaks is guided by the $\chi^2$ test and takes into
account the theoretical predictions. Continuous improvements in the
analysis is presently going and Fig.~\ref{ber} has to be considered as
a intermediate step, even if close to final results.


\begin{thebibliography}{99}

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               {\bf 117}, 151 (1994).

\bibitem{SLA}  J.C. David, C. Fayard, G.-H. Lamot, B. Saghai,
               Phys. Rev. C {\bf 53}, 2613 (1996);
               T. Mizutani, C. Fayard, G.-H. Lamot, B. Saghai,
               Phys. Rev. C {\bf 58}, 75 (1998).

\bibitem{modJM} M. Ukai \etal, Phys. Rev. C \textbf{73},
012501 (2006).

\bibitem{hashtam} O. Hashimoto and H. Tamura, Prog. Part. Nucl. Phys.
\textbf{57}, 405 (2006). 

\bibitem{davis91} D. H. Davis, in \textit{LAMPF Workshop on $(pi,K)$ Physics}, edited by B. F.
Gibson, W. R. Gibbs, and M. B. Johnson, AIP Conf. Proc. \textbf {224}, 38 (1991).

\bibitem{Carbon} M. Iodice \etal, {\it Phys Rev Lett} 99, 052501 (2007).

\bibitem{JJL} J. J. LeRose \etal, {\it Nucl Phys A} 804, 116-124 (2008).

\end{thebibliography}

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