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\begin{document}
\title{ \vspace{1cm} New Nuclei around the N = Z in the A = 80-90 region}
\author{N.\ Marginean,$^{1,2}$ C.\ Rossi Alvarez,$^3$ D.\ Bucurescu,$^2$ C. A.\
Ur,$^{3,2}$\\
A.\ Gadea,$^4$ S.\ Lunardi,$^3$ D.\ Bazzacco,$^3$ G.\ de Angelis,$^1$
M.\ Axiotis,$^1$\\
M.\ De Poli,$^1$ E.\ Farnea,$^{1,3}$ M.\ Ionescu-Bujor,$^2$
A.\ Iordachescu,$^2$ S. M.\ Lenzi,$^3$\\
Th.\ Kr\"oll,$^{1,3}$ T.\
Martinez,$^1$ R.\ Menegazzo,$^3$ D. R.\ Napoli,$^1$\\
G.\ Nardelli,$^5$ P.\
Pavan,$^3$ B.\ Quintana,$^{3,6}$ P.\ Spolaore$^1$\\
\\
$^1$INFN, Laboratori Nazionali di Legnaro, Italy\\
$^2$H. Hulubei National Inst. for Phys. and Nucl. Eng., Bucharest,
Romania\\
$^3$Dipartimento di Fisica dell'Universit\`a and INFN, Sez. di Padova,
Italy\\
$^4$Instituto de Fisica Corpuscular, Valencia, Spain\\
$^5$Dip. di Chi. Fis. dell'Universit\`a di Venezia and INFN, Sez. di Padova,
Italy\\
$^6$ Grupo de Fisica Nuclear, Universidad de Salamanca, Spain}
\maketitle
\begin{abstract} Correlations in the nuclear wave-function beyond the mean-field
or Hartree-Fock approximation are very important to describe basic properties of
nuclear structure. Various approaches to account for such correlations are
described and compared to each other. This includes the hole-line expansion, the
coupled cluster or ``exponential S'' approach, the self-consistent evaluation of
Greens functions, variational approaches using correlated basis functions and
recent developments employing quantum Monte-Carlo techniques. Details of these
correlations are explored and their sensitivity to the underlying
nucleon-nucleon interaction. Special
attention is paid to the attempts to investigate these correlations in
exclusive nucleon knock-out experiments induced by electron scattering.
Another important issue of nuclear structure physics is the role of relativistic
effects as contained in phenomenological mean field models. The sensitivity of
various nuclear structure observables on these
relativistic features are investigated. The report includes the discussion of
nuclear matter as well as finite nuclei.
\end{abstract}
%\eject
%\tableofcontents
\section{Introduction}
One of the central challenges of theoretical nuclear physics is the attempt to
describe the basic properties of nuclear systems in terms of a realistic
nucleon-nucleon (NN) interaction. Such an attempt typically contains two major
steps. In the first step one has to consider a specific model for the NN
interaction. This could be a model which is inspired by the
quantum-chromo-dynamics\cite{faes0}, a meson-exchange or One-Boson-Exchange
model\cite{rupr0,nijm0} or a purely phenomenological ansatz in terms of
two-body spin-isospin operators multiplied
by local potential
functions\cite{argo0,urbv14}. Such models are considered as a realistic
description of the NN interaction, if the adjustment of parameters within the
model yields a good fit to the NN scattering data at energies below the
threshold for pion production as well as energy and other observables of the
deuteron.
After the definition of the nuclear hamiltonian,
the second
step implies the solution of the many-body problem of $A$ nucleons interacting
in terms of such a realistic two-body NN interaction. The simplest approach to
this many-body problem of interacting fermions one could think of would be the
mean field or Hartree-Fock approximation. This procedure yields very good
results for the bulk properties of nuclei, binding energies and radii,
if one employs simple phenomenological NN forces like e.g.~the Skyrme forces,
which are adjusted to describe such nuclear structure data\cite{skyrme}.
However, employing realistic NN interactions the Hartree-Fock approximation
fails very badly: it leads to unbound nuclei\cite{art99}.
\begin{figure}[tb]
%\epsfysize=9.0cm
\begin{center}
\begin{minipage}[t]{8 cm}
\epsfig{file=emblem.ps,scale=0.5}
\end{minipage}
\begin{minipage}[t]{16.5 cm}
\caption{Cartoon of a nucleus, displaying the size of the nucleons as compared
to the typical distance to nearest neighbors. Also indicated are the internal
structure of nucleons and mesons.\label{fig1}}
\end{minipage}
\end{center}
\end{figure}
The calculation scheme discussed so far, determine the interaction of two
nucleons in the vacuum in a first step and then solve the many-body problem of
nucleons interacting by such realistic potentials in a second step, is of course
based on the picture that nucleons are elementary particles with properties,
which are not affected by the presence of other nucleons in the nuclear medium.
One knows, of course, that this is a rather simplified picture: nucleons are
built out of quarks and their properties might very well be influenced by the
surrounding medium. A cartoon of this feature is displayed in Fig.~\ref{fig1}.
\section{Many-Body Approaches}
\subsection{\it Hole - Line Expansion \label{sec:holeline}}
As it has been discussed already above one problem of nuclear structure
calculations based on realistic NN interactions is to deal with the strong
short-range components contained in all such interactions. This problem is
evident in particular when
so-called hard-core potentials are employed, which are infinite for relative
distances smaller than the radius of the hard core $r_c$. The matrix elements
of such a potential $V$ evaluated for an uncorrelated two-body wave function
$\Phi (r)$ diverges since $\Phi (r)$ is different from zero also for relative
distances $r$ smaller than the hard-core radius $r_c$ (see the schematic picture
in Fig.~\ref{fig3}. A way out of this problem is to account for the two-body
correlations induced by the NN interaction in the correlated wave function
$\Psi (r)$ or by defining an effective operator, which acting on the
uncorrelated wave function $\Phi (r)$ yields the same result as the bare
interaction $V$ acting on $\Psi (r)$. This concept is well known for example in
dealing with the scattering matrix $T$, which is defined by
\be
<\Phi \vert T \vert \Phi > = <\Phi \vert V \vert \Psi > \; . \label{eq:tmat}
\ee
As it is indicated in the schematic Fig.~\ref{fig3}, the correlations tend to
enhance the amplitude
of the correlated wave function $\Psi$ relative to the uncorrelated one
at distances $r$ for which the interaction is attractive. A reduction of
the amplitude is to be expected for small distances for which $V(r)$ is
repulsive. From this discussion we see that the
correlation effects tend to make the matrix elements of $T$ more attractive
than those of the bare potential $V$. For two nucleons in the vacuum the $T$
matrix can be determined by solving a Lippmann-Schwinger equation
\bea
T \vert \Phi > &= &V \left\{ \vert \Phi > + \frac{1}{\omega - H_0 +
i\epsilon } V \vert \Psi >\right\}\nonumber \\
& = & \left\{ V + V \frac{1 }{\omega - H_0 +i\epsilon } T\right\} \vert
\Phi >\, . \label{eq:lipschw}
\eea
\begin{figure}[tb]
%\epsfysize=9.0cm
\begin{center}
\begin{minipage}[t]{8 cm}
\epsfig{file=fig3.eps,scale=0.7}
\end{minipage}
\begin{minipage}[t]{16.5 cm}
\caption{Schematic picture of a NN interaction with hard core and its effect on
the correlated NN wave function $\Psi(r)$. \label{fig3}}
\end{minipage}
\end{center}
\end{figure}
Therefore it seems quite natural to define the single-particle potential $U$ in
analogy to the Hartree-Fock definition with the bare interaction $V$ replaced
by the corresponding $G$-matrix. To be more precise, the Brueckner-Hartree-Fock
(BHF) definition of $U$ is given by
\be
<\alpha \vert U \vert \beta> = \cases{ \sum_{\nu \le F} <\alpha \nu \vert
\frac{1} {2} \left( G(\omega_{\alpha \nu}) + G(\omega_{\beta \nu}) \right)
\vert \beta \nu >, & if $\alpha$ and $\beta$ $\le F$ \cr \sum_{\nu \le F}
<\alpha \nu \vert G(\omega_{\alpha \nu}) \vert \beta \nu >, & if $\alpha\le F$
and $\beta > F$ \cr 0 & if $\alpha$ and $\beta$ $>F$, \cr}\, . \label{eq:ubhf}
\ee
\subsection{\it Many-Body Theory in Terms of Green's Functions
\label{subsec:green}}
The two-body approaches discussed so far, the hole-line expansion as well as the
CCM, are essentially restricted to the evaluation of ground-state properties.
The Green's function approach, which will
shortly be introduced in this section
also yields results for dynamic properties like e.g.~the single-particle
spectral function which is closely related to the cross section of particle
knock-out and pick-up reactions. It is based on the time-dependent
perturbation expansion and also assumes a separation of the total hamiltonian
into an single-particle part $H_0$ and a perturbation $H_1$.
A more detailed description can be found e.g.~in the textbook
of Fetter and Walecka\cite{fetwal}.
\section{Effects of Correlations derived from Realistic Interactions}
\subsection{\it Models for the NN Interaction\label{sec:nninter}}
In our days there is a general agreement between physicists working on this
field, that quantum chromo dynamics (QCD) provides the basic theory of
the strong
interaction. Therefore also the roots of the strong interaction between two
nucleons must be hidden in QCD. For nuclear structure calculations, however, one
needs to determine the NN interaction at low energies and momenta, a region in
which one cannot treat QCD by means of perturbation theory. On the other hand,
the
system of two interacting nucleons is by far too complicate to be treated by
means of lattice QCD calculations. Therefore one has to consider
phenomenological models for the NN interaction.
With the OBE ansatz one can now solve the Blankenbecler--Sugar or a
corresponding scattering equation and adjust the parameter of the OBE model to
reproduce the empirical NN scattering phase shifts as well as binding energy and
other observables for the deuteron. Typical sets of parameters resulting from
such fits are listed in table~\ref{tab:obe}.
\begin{table}
\begin{center}
\begin{minipage}[t]{16.5 cm}
\caption{Parameters of the realistic OBE potentials Bonn $A$, $B$ and $C$ (see
table A.1 of \protect{\cite{rupr0}}).
The second column displays the type of
meson: pseudoscalar (ps), vector (v) and scalar (s) and the third its
isospin $T_{\rm iso}$.}
\label{tab:obe}
\end{minipage}
\begin{tabular}{rrrr|rr|rr|rr}
\hline
&&&&&&&&&\\[-2mm]
&&&&\multicolumn{2}{c}{Bonn A}&\multicolumn{2}{c}{Bonn
B}&\multicolumn{2}{c}{Bonn C}\\
Meson &&$T_{\rm iso}$&$m_{\alpha}$&$g^2_{\alpha}/4\pi$&$\Lambda_{\alpha}$
&$g^2_{\alpha}/4\pi$&$\Lambda_{\alpha}$&$g^2_{\alpha}/4\pi$&$\Lambda_{\alpha}$\\
&&&[MeV]&&[MeV]&[MeV]&[MeV]\\
&&&&&&&&&\\[-2mm]
\hline
&&&&&&&&&\\[-2mm]
$\pi$ & ps & 1 & 138.03 & 14.7 & 1300 & 14.4 & 1700 & 14.2 & 3000\\[2mm]
$\eta$ & ps & 0 & 548.8 & 4 & 1500 & 3 & 1500 & 0 & -\\[2mm]
$\rho$ & v & 1 & 769 & 0.86$^{\rm a}$ & 1950 & 0.9$^{\rm a}$ & 1850 &
1.0$^{\rm a}$ & 1700 \\[2mm]
$\omega$ & v & 0 & 782.6 & 25$^{\rm a}$ & 1350 & 24.5$^{\rm a}$ & 1850 &
24$^{\rm a}$ & 1400\\[2mm]
$\delta$ & s & 1 & 983 & 1.3 & 2000 & 2.488 & 2000 & 4.722 & 2000\\[2mm]
$\sigma^{\rm b}$ & s & 0 & 550$^{\rm b}$ & 8.8 & 2200 & 8.9437 & 1900 & 8.6289 &
1700\\
&&&(710-720)$^{\rm b}$ & 17.194 & 2000 & 18.3773 & 2000 & 17.5667 & 2000\\
&&&&&&&&&\\[-2mm]\hline
\end{tabular}
%noalign{\smallskip\hrule}\cr}
\begin{minipage}[t]{16.5 cm}
\vskip 0.5cm
\noindent
$^{\rm a}$ The tensor coupling constants are $f_{\rho}$=6.1 $g_{\rho}$
and $f_{\omega}$ = 0. \\
$^{\rm b}$ The $\sigma$ parameters in the first line apply for NN channels
with isospin 1, while those in the second line refer to isospin 0 channels. In
this case the masses for the $\sigma$ meson of 710 (Bonn A) and 720 MeV (Bonn B
and C) were considered.
\end{minipage}
\end{center}
\end{table}
\subsection{\it Ground state Properties of Nuclear Matter and Finite Nuclei}
In the first part of this section we would like to discuss the convergence of
the many-body approaches and compare results for nuclear matter as obtained from
various calculation schemes presented in section 2.
The convergence of the hole-line expansion for nuclear matter has been
investigated during the last few years in particular by the group in
Catania\cite{song1,song2}. Continuing the earlier work of Day\cite{day81} they
investigated the effects of the three-hole-line contributions for various
choices of the auxiliary potential $U$ (see Eq.~\ref{eq:ubhf}). In particular
they considered the standard or conventional choice, which assumes a
single-particle potential $U=0$ for single-particle states above the Fermi level,
and the so-called ``continuous choice''. This continuous choice supplements the definition of the
auxiliary potential of the hole states in Eq.~(\ref{eq:ubhf}) with a
corresponding definition (real part of the BHF self-energy) also for the
particle states with momenta above the Fermi momentum, $k >k_F$. In this way
one does not have any gap in the single-particle spectrum at $k=k_F$.
\section{Conclusion}
The main aim of this review has been to demonstrate that nuclear systems
are very intriguing many-body systems. They are non-trivial
systems in the sense that they require the treatment of correlations beyond the
mean field or Hartree-Fock approximation. Therefore, from the point of view of
many-body
theory, they can be compared to other quantum many-body systems like liquid He,
electron gas, clusters of atoms etc. A huge amount of experimental data is
available for real nuclei with finite number of particles as well as for the
infinite limit of nuclear matter or the matter of a neutron star.
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\itemsep -2pt
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\end{thebibliography}
\end{document}