Low-momentum Hyperon-Nucleon Interactions
B.-J. Schaefer
We present a first exploratory study for hyperon-nucleon interactions using renormalization group techniques. The effective two-body low-momentum potential $\vlowk$ is obtained by integrating out the high-momentum components from realistic Nijmegen $YN$ potentials. A $T$-matrix equivalence approach is employed, so that the low-energy phase shifts are reproduced by $\vlowk$ up to a momentum scale $\Lambda \sim 500$ MeV. Although the various bare Nijmegen models differ somewhat from each other, the corresponding $\vlowk$ interactions show convergence in some channels, suggesting a possible unique $YN$ interaction at low momenta.
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\begin{document}
\title{Low-momentum Hyperon-Nucleon Interactions}

\author{B.-J. Schaefer}
\email[E-Mail:]{bernd-jochen.schaefer@physik.tu-darmstadt.de}
\author{M. Wagner}
\affiliation{Institut f\"{u}r Kernphysik, TU Darmstadt, D-64289 Darmstadt, Germany}
\author{J. Wambach}
\affiliation{Institut f\"{u}r Kernphysik, TU Darmstadt, D-64289 Darmstadt, Germany}
\affiliation{Gesellschaft f\"{u}r Schwerionenforschung GSI, D-64291 Darmstadt, Germany}
\author{T.T.S. Kuo}
\author{G.E. Brown}
\affiliation{Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794-3800, USA}

\date{\today}

\pacs{13.75.Ev}
% 13.75.Ev      Hyperon-nucleon interactions

\begin{abstract}
  We present a first exploratory study for hyperon-nucleon
  interactions using renormalization group techniques. The effective
  two-body low-momentum potential $\vlowk$ is obtained by integrating
  out the high-momentum components from realistic Nijmegen $YN$
  potentials. A $T$-matrix equivalence approach is employed, so that
  the low-energy phase shifts are reproduced by $\vlowk$ up to a
  momentum scale $\Lambda \sim 500$ MeV. Although the various bare
  Nijmegen models differ somewhat from each other, the corresponding
  $\vlowk$ interactions show convergence in some channels, suggesting
  a possible unique $YN$ interaction at low momenta.
\end{abstract}

\maketitle
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Introduction
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Starting from modern nucleon-nucleon interactions and performing a
renormalization group (RG) decimation it has become possible to derive
a unique low-momentum effective interaction $\vlowk$~\cite{SKB03}. The
basic idea is to integrate out the short-distance physics encoded in
hard-core interactions which are not well constrained by the available
phase shift data. The resulting effective interactions form the
starting point for ab-initio nuclear structure calculations in
few-body systems~\cite{ANSKB04}, shell-model
studies~\cite{Andreozzi:1996tf} and mean-field treatments via
density-functional methods~\cite{Bogner:2005sn}.  They also serve as
input for the derivation of Landau Fermi-liquid interactions and
provide predictions of pairing gaps in nuclei and homogeneous neutron
matter~\cite{Schwenk2002}.

In this paper we generalize the $\vlowk$ approach to the
hyperon-nucleon ($YN$) sector. The ultimate goal is to provide
effective potentials of similar quality as in the $NN$ case that could
serve as the starting point for realistic calculations of the
structure of hypernuclei and homogeneous hyperonic matter. At present
such a program is hampered by the lack of a comparable data base and
the collapse to a unique low-momentum potential is far from
obvious. In a first exploratory study we wish to address this point by
considering the low-momentum decimation of various potentials by the
Nijmegen group. We focus on hyperons with strangeness $S=-1$ for which
$I=1/2$ and $I=3/2$ isospin states are available. For $I=1/2$, several
hyperon-nucleon channels occur which require new technical
developments for the coupled RG flow equations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Two-body low momentum interaction
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{1ex}

The effective, low-momentum potential $\vlowk$ for elastic two-body
scattering is obtained by integrating out high-momentum components of
a realistic bare potential $V$ interaction. This is achieved by
imposing a cutoff $\Lambda$ on all loop integrals in the half-on-shell
(HOS) $T$-matrix equation and replacing the bare potential $V$ with
the effective $\vlowk$ potential. Since the physical low-energy
quantities must not depend on the cutoff, the HOS $T$-matrix should be
preserved for relative three-momenta $k'$,$k \leq \Lambda$. This
results in a modified Lippmann-Schwinger equation with a
cutoff-dependent effective potential $\vlowk$
\[
  T(k'\!\!,k;k^2)\!=\!\vlowk(k'\!\!,k)+\frac 2 \pi{\cal
P}\!\!\!\int\limits_0^{\Lambda}\!\! q^2 dq \frac{\vlowk (k'\!\!,q) T
(q,\!k;k^2)} {k^2-q^2}.
\]

By demanding $dT(k',k;k^2)/d\Lambda = 0$, an exact renormalization
group (RG) flow equation for $\vlowk$ can be obtained
\cite{SKB+01}
\begin{equation}
\label{rgeq}
\frac {d}{d \Lambda} \vlowk (k',k) = \frac 2 \pi \frac{\vlowk (k',
  \Lambda) T(\Lambda,k;\Lambda^2)}{1-k^2/\Lambda^2}\ .
\end{equation}

Integrating this flow equation with a given initial bare potential at
a large cutoff (small distance) one obtains the physically equivalent
effective theory ($\vlowk$) at a smaller cutoff $\Lambda$ (larger
distance).

Instead of solving the RG equation (\ref{rgeq}) directly as a
differential equation with e.g. standard Runge-Kutta methods, we use
the Andreozzi-Lee-Suzuki (ALS) iteration method, which is based on a
similarity transformation \cite{andreozzi,suzuki80}. With folded
diagram techniques it has been shown \cite{bogner02} that this
iteration method indeed yields a solution of Eq.~(\ref{rgeq}).

By construction, the resulting $\vlowk$ for the $NN$ interaction
reproduces the empirical deuteron binding energy and scattering phase
shifts up to $E_{lab}= 2\hbar^2\Lambda^2/M$. Most importantly, it is
found that for $\Lambda < 2$ fm$^{-1}$ ($E_{lab}< 330$ MeV) $\vlowk$
is independent of the particular $V_{NN}$ model, i.e. all diagonal
matrix elements of the different high-precision potentials collapse to
a single unique low-momentum effective potential~\cite{SKB+01}. This
can be largely attributed to the long-range one-pion exchange (OPE)
which is common to all realistic potentials and dominates the
low-momentum scattering. The main effect of the RG evolution is a
constant shift of the bare matrix elements which removes the
ambiguities in the short-range part of the potential.

The energy-independent $\vlowk$ is non-Hermitian. This can readily be
seen from the RG equation~(\ref{rgeq}) because the momenta are treated
asymmetrically. With a second similarity transformation the
non-Hermiticity of $\vlowk$ can be eliminated.  Phase shifts are
preserved by this second transformation and there are a number of such
phase shift equivalent transformations such as the well-known Okubo
one~\cite{holt04}. In the present work we have used the Okubo
transformation to obtain the Hermitian $\vlowk$. For the $NN$
interaction, the diagonal matrix elements are almost unchanged by the
second transformation.
%\vspace{1ex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Towards the low momentum hyperon nucleon interaction
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As realistic $YN$ interactions we use the soft-core potentials by the
Nijmegen group~\cite{rijken}. They are based on one-boson-exchange
models (OBE) of the $NN$ potential and use $SU(3)_F$-symmetry to infer
the coupling vertices in the presence of a hyperon. Since the flavor
symmetry is broken by the finite quark masses the pertinent coupling
strengths have to be adjusted to data. Six different fits are
available referred to NSC97a - NSC97f in the following.  Each
potential comes in two basis representations (isospin- and physical
particle-basis). In this paper we work on the isospin basis which was
also originally used for the potential construction by the Nijmegen
group. Therefore all isomultiplets are degenerate. The corresponding
isospin-averaged masses are given by $m_N = 938.9$ MeV, $m_\Lambda =
1115.7$ MeV and $m_\Sigma = 1193.1$ MeV. In the $I=3/2$ channel no
$\Lambda$-hyperon is involved. 


The different Nijmegen fits describe the known $YN$ cross section data
equally well ($\chi^2/N \sim 0.55$) but exhibit differences on a more
detailed level. Due to the few available data points (only 35
altogether), the phase shifts and some scattering lengths exhibit
large variations for different fits~\cite{nnon}. This is in marked
contrast to the $NN$ case where the wealth of the data base allows for
high-precision potentials. These differ only in their short-range
properties which are not constrained by the available data.  The
collapse of the low-momentum potential $\vlowk$ for the $NN$
interaction to a 'unique' effective potential, observed after the RG
decimation~\cite{SKB03} is basically driven by the precision of the
measured phase shifts. Due to the scarceness and uncertainties in the
$YN$ data we cannot expect a unique low-momentum $YN$ $\vlowk$
potential.

In addition, for the $YN$ interaction the choice of the cutoff
$\Lambda $ is not so obvious. In this work a cutoff around $\Lambda
\sim 500$ MeV is chosen which corresponds to $2.5$ fm$^{-1}$ in the
configuration space.
\begin{figure}[!htb]
\centerline{\hbox{
  \includegraphics[width=0.5\columnwidth]{psl500_1s0}
  \includegraphics[width=0.5\columnwidth]{sl500_vkk1s0}
  }}
\caption{\label{fig:sig1s0} Left panel: The corresponding $^1S_0$
phase shifts for the six different potentials as a function of the
momentum in the lab frame $p_{\rm LAB}$. Right panel: Three diagonal
bare potentials $V_{\rm bare}$ (NSC97a,c,f; dashed lines) and three
(Hermitian) $\vlowk$ matrix elements (dotted) for $\Sigma N \to \Sigma
N$ ($I = 3/2$, $^1S_0$) versus the relative momentum $k$. }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% I=3/2 channel
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{1ex}



We first consider the $I=3/2$ case which corresponds to the $\Sigma N
\to \Sigma N$ channel. The ALS iteration is exactly the same as for
the $NN$ interaction albeit with the appropriate substitution of the
hyperon mass. For the ALS iteration we have used a cutoff $\Lambda
\equiv\Lambda_P = 500$ MeV in the model space ($P$-space) with $64$
grid points and a cutoff $\Lambda_Q = 10$ GeV for the complementary
$Q$-space also with $64$ grid points. In order to verify the
insensitivity of the results on these quantities we have varied the
grid points in an interval $[32,70]$ and the cutoff $\Lambda_Q$ in
between the range $\Lambda_Q\pm 2$ GeV. The standard ALS method
converges rapidly. Details will be presented in~\cite{2letter}.
%(e.g.~less than 300 iterations needed in order to
%fulfill the decoupling equation with an accuracy of
%approx.~$10^{-5}$. Details will be presented in~\cite{2letter}).

The results for the $^1S_0$ partial wave are shown in
Fig.~\ref{fig:sig1s0}.  In the right panel the diagonal matrix
elements for three different bare potentials (dashed lines) and the
corresponding RG evolved $\vlowk$ potentials are displayed versus the
relative momentum $k$. The RG decimation yields a soft-core $\vlowk$
potential which is more attractive (dotted curves). As already
mentioned, the $\vlowk$'s are basically non-Hermitian after the RG
decimation. By means of an Okubo-transformation we obtain Hermitian
$\vlowk$ potentials. As in the $NN$ case~\cite{SKB03} the differences
are negligible.

By construction, the on-shell $T$-matrix is phase-shift equivalent
which must result in identical phase shifts for a given bare potential
fit and momenta below the cutoff $\Lambda$.  This is demonstrated in
the left panel of Fig.~\ref{fig:sig1s0}. This comparison also serves
as a test for our numerics. For the $^1S_0$ partial wave, the bare
potentials do not differ strongly and thus yield almost the same
$\vlowk$ for all fits considered.
\begin{figure}[!htb]
  \centerline{\hbox{
  \includegraphics[width=0.5\columnwidth]{psl500_3s1}
  \includegraphics[width=0.5\columnwidth]{sl500_vkk3s1}
  }}
\caption{\label{fig:sigtensor} The same as in Fig.~\ref{fig:sig1s0}
for the partial wave $^3S_1$. Left panel: nuclear bar phase shift.}
\end{figure}
As discussed below, this changes for higher partial waves where the
different bare potential fits deviate significantly and therefore no
'unique' $\vlowk$ is found.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% tensor coupling
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As in the $NN$ case, the $YN$ interaction contains tensor components
and hence partial waves can mix. The generalization of the ALS
iteration in the presence of tensor forces is straightforward. For
$S=1$ one has to enlarge the $T$-matrix to a $(2\times 2)$ block
structure in the standard way, corresponding to the orbital angular
momentum combinations $L=J\pm 1$. Also for this case we have verified
that $\vlowk$ is phase-shift equivalent to the bare potentials, as is
shown in left panel Fig.~\ref{fig:sigtensor} for the $^3S_1$ partial
wave.  The right panel displays the corresponding $\vlowk$ potentials
together with the bare ones.  They again become more attractive for
all Nijmegen fits. Due to strong differences in the phase shifts we do
not find a collapse of $\vlowk$ to one potential, especially at larger
momenta where deviations are most pronounced.

The RG flow equation (\ref{rgeq}) implies a pole at the cutoff
boundary $k=\Lambda$. In the vicinity of this pole the slope for the
$\vlowk$ diverges which can be clearly seen in the right panel of
Fig.~\ref{fig:sigtensor}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% spin-spin  coupling
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!htb]
  \centerline{\hbox{
  \includegraphics[width=0.5\columnwidth]{psl500_1p1}
  \includegraphics[width=0.5\columnwidth]{sl500_vkk1p1}
        }}\caption{\label{fig:sig1p1} Same as
        Fig.~\ref{fig:sig1s0} for the partial wave $^1P_1$ and
        $I=3/2$.}
\end{figure}

To complete the analysis for the $I=3/2$ channel we show in
Fig.~\ref{fig:sig1p1} the $^1P_1$ partial wave which serves as an
example for the spin singlet-spin triplet transition. Such transitions
which are induced by the tensor force are negligible in the $NN$
interaction due to the small mass differences. However, for the $YN$
interaction this is not the case anymore and these transitions can be
significant. Due to the smaller deviations in the phase shift, the
corresponding $\vlowk$ interactions collapse to single potential
(especially for small momenta).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% I = 1/2 channel
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{1ex}


The treatment of of the $I=1/2$ channel is much more complicated.  For
this isospin four coupled channels available corresponding to the
transitions: $(\Lambda N,\Sigma N )\to(\Lambda N,\Sigma N )$. This is
a completely new situation for the $\vlowk$ approach. Since now
channels with different masses couple, new phenomena are to be
expected concerning e.g.~the convergence behavior of the ALS
iteration.

In flavor space the Lippmann-Schwinger equation becomes a coupled
($2\times 2$) matrix equation where the diagonal matrix elements
describe respectively the $\Lambda N \to\Lambda N$ and $\Sigma N \to
\Sigma N$ channels while the off-diagonal elements describe the
$\Lambda N \to \Sigma N$ and $\Sigma N \to\Lambda N$ transitions.
Using a notation where we only list the hyperons ($Y,Y'=\Lambda,
\Sigma$)
\begin{eqnarray}
\label{LS}
T^{Y'Y}(k',k;E_k^Y) = V^{Y'Y}(k',k) \hspace*{3cm}&& \nonumber \\
+\sum_{Z=\Lambda,\Sigma}
\frac{2}{\pi} \mathcal{P}\!\! \int\limits_0^\infty\!\! dq q^2
\frac{V^{Y'Z}(k',q) T^{ZY}(q,k;E_k^Y)}
{E^Y_k -H^Z_0 }&&
\end{eqnarray}
with the free Hamiltonian $H^Y_0(q)=\frac {q^2}{2\mu_Y} + m_Y + m_N$,
the energy $E_k^Y = \frac {k^2}{2\mu_Y} + m_Y + m_N$ and the reduced
mass $\mu_Y = \frac {m_Y m_N}{m_Y+m_N}$ we have (with the inclusion of
tensor forces) in general four coupled equations to solve. For some
channels, the mass difference $m_\Sigma - m_\Lambda$ enters in the
denominator of Eq.~(\ref{LS}) which induces e.g.~a threshold behavior
for the $\Sigma$ hyperon. The mass differences which are not present
in the $NN$ interaction enter also in the ALS iteration. It is found
that the standard ALS iteration procedure does not converge to the
proper consecutive set of eigenvalues. As a consequence, a wrong
sorting of the eigenvalues in the different $P$- and $Q$-spaces
emerges. Using a modified ALS iteration or by introducing an energy
cutoff solves this problem and convergence to the correct eigenvalues
can be found~\cite{2letter}.

\begin{figure}[!htb]
  \centerline{\hbox{
  \includegraphics[width=0.5\columnwidth]{pll500_1s0}
  \includegraphics[width=0.5\columnwidth]{ll500_vkk1s0} }}
  \caption{\label{fig:lam1s0} $^1S_0$ partial wave for the $I=1/2$
  $\Lambda N \to \Lambda N$ channel. Left panel: Corresponding phase
  shifts.  Right panel: bare potentials and $\vlowk$ potentials. The
  labeling in both panels is the same as in Fig.~\ref{fig:sig1s0}.}
\end{figure}

As an example we show in Fig.~\ref{fig:lam1s0} the $^1S_0$ partial
wave for the $\Lambda N \to \Lambda N$ channel. All $\vlowk$
potentials are again more attractive and a more narrow grouping as
compared to the bare potentials can be observed.  One also observes
that the shift of the $\vlowk$ potential is largest for the bare
NSC97f potential and smallest for the NSC97a potential in contrast to
all other partial waves. The $\vlowk$'s are shifted in such a way that
the potentials collapse for relative momenta near the cutoff
reflecting the corresponding trends in the phase shifts.
\begin{figure}[!htb]
\centerline{\hbox{
\includegraphics[width=0.5\columnwidth]{pll500_3s1}
\includegraphics[width=0.5\columnwidth]{ll500_vkk3s1}
}} \caption{\label{fig:lamtensor} Same as Fig.~\ref{fig:lam1s0} but
for the partial wave $^3S_1$. Left panel: nuclear bar phase shifts.}
\end{figure}

In Fig.~\ref{fig:lamtensor} the $\Lambda N \to \Lambda N$ channel with
tensor coupling is shown for the $^3S_1$ partial wave . Here again the
RG decimation pushes the bare potentials down basically by a (large)
constant to attractive $\vlowk$'s. Since this channel includes the
$\Sigma N$ transition, the $\Sigma$ threshold is visible in the phase
shifts for lab-momenta above $600$ MeV. The jump in the phase shift at
the threshold depends strongly on the model.
\begin{figure}[!htb]
\centerline{\hbox{
\includegraphics[width=0.5\columnwidth]{pll500_1p1}
\includegraphics[width=0.5\columnwidth]{ll500_vkk1p1}
}}\caption{\label{fig:lam1p1} Same as Fig.~\ref{fig:lam1s0} but for
the partial wave $^1P_1$. Left panel: nuclear bar phase shift.}
\end{figure}

To complete this analysis results for the spin singlet - spin triplet
transition in the $\Lambda N \to \Lambda N$ channel for the $^1P_1$
partial wave are presented in Fig.~\ref{fig:lam1p1}. For small momenta
no RG decimation takes place. For this partial wave the bare
interaction is repulsive and so is $\vlowk$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% summary and conclusion
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{1ex}


Recently, it has been shown that the $\vlowk$ approach which is based
on RG techniques provides a novel and powerful tool to obtain
phase-shift equivalent low-momentum $NN$ interactions. After the RG
decimation a unique low-momentum potential $\vlowk$ for different
high-precision $NN$ interactions was found. This model-independence of
the diagonal matrix elements of the $\vlowk$ is an important property
which is basically driven by the phase shift equivalence of the input
models.

In the present work the model-dependence of the $\vlowk$ for the $YN$
interaction is investigated. Because of coupled-channel effects in
flavor space that are not present in the $NN$ case, the RG evolution
is technically more complicated but can be treated. Due to the few
experimental data currently available, the model fits by the Nijmegen
group do not allow for a unique low-momentum $YN$ interaction. It is
therefore of importance to calculate $\vlowk$ for other $YN$ models
such as the revised J\"ulich potential~\cite{JHWM01}. Such
calculations are in preparation~\cite{2letter}.

Although no unique $YN$ low-momentum interaction is obtained at
present, a convergence of the different $\vlowk$'s is seen generally
for all the Nijmegen potentials. Especially, for partial waves which
do not deviate strongly for different bare potentials the uniqueness
of $\vlowk$ is pronounced. All $\vlowk$ potentials are much softer
than the bare ones. Softer interactions lead to stronger binding which
should be of relevance in microscopic hypernuclei calculations.


By construction, all low-energy two-body observables are
cutoff-independent. Bogner et al. argue that any (new) induced cutoff
dependence is due to higher-body forces~\cite{ANSKB04}. For the $NN$
interaction they conclude that such contributions are rather
small. For the $YN$ interaction this is still an open issue and should
be tested in light hypernuclei.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% acknowledgments
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We thank A. Schwenk for helpful discussions. One of the authors (BJS)
would also like to thank S.K.~Bogner for numerous enlightening
discussions. He also expresses his gratitude to G.E.~Brown and
T.T.S.~Kuo for the invitation to Stony Brook where this work was
initiated. MW is supported by BMBF Grand No.~06DA116.  Partial support
to TTSK and GEB from the US Department of Energy under contract
DE-FG02-88ER/40388 is gratefully acknowledged.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% references via bibtex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% use revtex style
%\bibliographystyle{vlowk}
%\bibliography{vlowk_refs}
\begin{thebibliography}{14}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem[{\citenamefont{Bogner et~al.}(2003)\citenamefont{Bogner, Kuo, and
  Schwenk}}]{SKB03}
\bibinfo{author}{\bibfnamefont{S.~K.} \bibnamefont{Bogner}},
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\bibitem[{\citenamefont{Nogga et~al.}(2004)\citenamefont{Nogga, Bogner, and
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\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Nogga}},
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\bibitem[{\citenamefont{Bogner et~al.}(2005)\citenamefont{Bogner, Schwenk,
  Furnstahl, and Nogga}}]{Bogner:2005sn}
\bibinfo{author}{\bibfnamefont{S.~K.} \bibnamefont{Bogner}},
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  \bibinfo{author}{\bibfnamefont{R.~J.} \bibnamefont{Furnstahl}},
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\bibitem[{\citenamefont{Schwenk et~al.}(2002)\citenamefont{Schwenk, Brown, and
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\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Schwenk}},
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  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Friman}},
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\bibitem[{\citenamefont{Bogner et~al.}(2001)\citenamefont{Bogner, Schwenk, Kuo,
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\bibinfo{author}{\bibfnamefont{S.~K.} \bibnamefont{Bogner}},
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\bibitem[{\citenamefont{Andreozzi}(1996)}]{andreozzi}
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\bibitem[{\citenamefont{Suzuki and Lee}(1980)}]{suzuki80}
\bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Suzuki}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{S.~Y.} \bibnamefont{Lee}},
  \bibinfo{journal}{Progress of Theoretical Physics}
  \textbf{\bibinfo{volume}{64}}, \bibinfo{pages}{2091} (\bibinfo{year}{1980}).

\bibitem[{\citenamefont{Bogner et~al.}(2002)\citenamefont{Bogner, Kuo,
  Coraggio, Covello, and Itaco}}]{bogner02}
\bibinfo{author}{\bibfnamefont{S.~K.} \bibnamefont{Bogner}},
  \bibinfo{author}{\bibfnamefont{T.~T.~S.} \bibnamefont{Kuo}},
  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Coraggio}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Covello}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Itaco}},
  \bibinfo{journal}{Phys. Rev. C} \textbf{\bibinfo{volume}{65}},
  \bibinfo{pages}{051301(R)} (\bibinfo{year}{2002}).

\bibitem[{\citenamefont{Holt et~al.}(2004)\citenamefont{Holt, Kuo, and
  Brown}}]{holt04}
\bibinfo{author}{\bibfnamefont{J.~D.} \bibnamefont{Holt}},
  \bibinfo{author}{\bibfnamefont{T.~T.~S.} \bibnamefont{Kuo}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{G.~E.} \bibnamefont{Brown}},
  \bibinfo{journal}{Phys. Rev. C} \textbf{\bibinfo{volume}{69}},
  \bibinfo{pages}{034329} (\bibinfo{year}{2004}).

\bibitem[{\citenamefont{Rijken et~al.}(1999)\citenamefont{Rijken, Stoks, and
  Yamamoto}}]{rijken}
\bibinfo{author}{\bibfnamefont{T.~A.} \bibnamefont{Rijken}},
  \bibinfo{author}{\bibfnamefont{V.~G.~J.} \bibnamefont{Stoks}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Yamamoto}},
  \bibinfo{journal}{Phys. Rev. C} \textbf{\bibinfo{volume}{59}},
  \bibinfo{pages}{21} (\bibinfo{year}{1999}).

\bibitem[{nno()}]{nnon}
\emph{\bibinfo{title}{N{N}-{O}nline}}, \urlprefix\url{http://nn-online.org}.

\bibitem[{\citenamefont{Schaefer et~al.}()\citenamefont{Schaefer, Wagner,
  Wambach, Kuo, and Brown}}]{2letter}
\bibinfo{author}{\bibfnamefont{B.-J.} \bibnamefont{Schaefer}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Wagner}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Wambach}},
  \bibinfo{author}{\bibfnamefont{T.~T.~S.} \bibnamefont{Kuo}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{G.~E.} \bibnamefont{Brown}},
  \emph{\bibinfo{title}{to be published}}.

\bibitem[{\citenamefont{Haidenbauer et~al.}(2001)\citenamefont{Haidenbauer,
  Melnitchouk, and Speth}}]{JHWM01}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Haidenbauer}},
  \bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Melnitchouk}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Speth}},
  \bibinfo{journal}{AIP Conf. Proc.} \textbf{\bibinfo{volume}{603}},
  \bibinfo{pages}{421} (\bibinfo{year}{2001}), \eprint{nucl-th/0108062}.

\end{thebibliography}

% any troubles w/ bibtex? then include only letter.bbl and remove command
% \bibliography{vlowk}

\end{document}








nucl-th0506065.extracted_bib
\begin{thebibliography}{14}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
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  \def\bibnamefont#1{#1}\fi
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  \def\bibfnamefont#1{#1}\fi
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  \def\citenamefont#1{#1}\fi
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  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem[{\citenamefont{Bogner et~al.}(2003)\citenamefont{Bogner, Kuo, and
  Schwenk}}]{SKB03}
\bibinfo{author}{\bibfnamefont{S.~K.} \bibnamefont{Bogner}},
  \bibinfo{author}{\bibfnamefont{T.~T.~S.} \bibnamefont{Kuo}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Schwenk}},
  \bibinfo{journal}{Phys. Rept.} \textbf{\bibinfo{volume}{386}},
  \bibinfo{pages}{1} (\bibinfo{year}{2003}), \eprint{nucl-th/0305035}.

\bibitem[{\citenamefont{Nogga et~al.}(2004)\citenamefont{Nogga, Bogner, and
  Schwenk}}]{ANSKB04}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Nogga}},
  \bibinfo{author}{\bibfnamefont{S.~K.} \bibnamefont{Bogner}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Schwenk}},
  \bibinfo{journal}{Phys. Rev.} \textbf{\bibinfo{volume}{C70}},
  \bibinfo{pages}{061002} (\bibinfo{year}{2004}), \eprint{nucl-th/0405016}.

\bibitem[{\citenamefont{Andreozzi et~al.}(1996)}]{Andreozzi:1996tf}
\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Andreozzi}}
  \bibnamefont{et~al.}, \bibinfo{journal}{Phys. Rev.}
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\bibitem[{\citenamefont{Bogner et~al.}(2005)\citenamefont{Bogner, Schwenk,
  Furnstahl, and Nogga}}]{Bogner:2005sn}
\bibinfo{author}{\bibfnamefont{S.~K.} \bibnamefont{Bogner}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Schwenk}},
  \bibinfo{author}{\bibfnamefont{R.~J.} \bibnamefont{Furnstahl}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Nogga}}
  (\bibinfo{year}{2005}), \eprint{nucl-th/0504043}.

\bibitem[{\citenamefont{Schwenk et~al.}(2002)\citenamefont{Schwenk, Brown, and
  Friman}}]{Schwenk2002}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Schwenk}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Brown}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Friman}},
  \textbf{\bibinfo{volume}{703}}, \bibinfo{pages}{745} (\bibinfo{year}{2002}).

\bibitem[{\citenamefont{Bogner et~al.}(2001)\citenamefont{Bogner, Schwenk, Kuo,
  and Brown}}]{SKB+01}
\bibinfo{author}{\bibfnamefont{S.~K.} \bibnamefont{Bogner}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Schwenk}},
  \bibinfo{author}{\bibfnamefont{T.~T.~S.} \bibnamefont{Kuo}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{G.~E.} \bibnamefont{Brown}}
  (\bibinfo{year}{2001}), \eprint{nucl-th/0111042}.

\bibitem[{\citenamefont{Andreozzi}(1996)}]{andreozzi}
\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Andreozzi}},
  \bibinfo{journal}{Phys. Rev. C} \textbf{\bibinfo{volume}{54}},
  \bibinfo{pages}{684} (\bibinfo{year}{1996}).

\bibitem[{\citenamefont{Suzuki and Lee}(1980)}]{suzuki80}
\bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Suzuki}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{S.~Y.} \bibnamefont{Lee}},
  \bibinfo{journal}{Progress of Theoretical Physics}
  \textbf{\bibinfo{volume}{64}}, \bibinfo{pages}{2091} (\bibinfo{year}{1980}).

\bibitem[{\citenamefont{Bogner et~al.}(2002)\citenamefont{Bogner, Kuo,
  Coraggio, Covello, and Itaco}}]{bogner02}
\bibinfo{author}{\bibfnamefont{S.~K.} \bibnamefont{Bogner}},
  \bibinfo{author}{\bibfnamefont{T.~T.~S.} \bibnamefont{Kuo}},
  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Coraggio}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Covello}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Itaco}},
  \bibinfo{journal}{Phys. Rev. C} \textbf{\bibinfo{volume}{65}},
  \bibinfo{pages}{051301(R)} (\bibinfo{year}{2002}).

\bibitem[{\citenamefont{Holt et~al.}(2004)\citenamefont{Holt, Kuo, and
  Brown}}]{holt04}
\bibinfo{author}{\bibfnamefont{J.~D.} \bibnamefont{Holt}},
  \bibinfo{author}{\bibfnamefont{T.~T.~S.} \bibnamefont{Kuo}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{G.~E.} \bibnamefont{Brown}},
  \bibinfo{journal}{Phys. Rev. C} \textbf{\bibinfo{volume}{69}},
  \bibinfo{pages}{034329} (\bibinfo{year}{2004}).

\bibitem[{\citenamefont{Rijken et~al.}(1999)\citenamefont{Rijken, Stoks, and
  Yamamoto}}]{rijken}
\bibinfo{author}{\bibfnamefont{T.~A.} \bibnamefont{Rijken}},
  \bibinfo{author}{\bibfnamefont{V.~G.~J.} \bibnamefont{Stoks}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Yamamoto}},
  \bibinfo{journal}{Phys. Rev. C} \textbf{\bibinfo{volume}{59}},
  \bibinfo{pages}{21} (\bibinfo{year}{1999}).

\bibitem[{nno()}]{nnon}
\emph{\bibinfo{title}{N{N}-{O}nline}}, \urlprefix\url{http://nn-online.org}.

\bibitem[{\citenamefont{Schaefer et~al.}()\citenamefont{Schaefer, Wagner,
  Wambach, Kuo, and Brown}}]{2letter}
\bibinfo{author}{\bibfnamefont{B.-J.} \bibnamefont{Schaefer}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Wagner}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Wambach}},
  \bibinfo{author}{\bibfnamefont{T.~T.~S.} \bibnamefont{Kuo}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{G.~E.} \bibnamefont{Brown}},
  \emph{\bibinfo{title}{to be published}}.

\bibitem[{\citenamefont{Haidenbauer et~al.}(2001)\citenamefont{Haidenbauer,
  Melnitchouk, and Speth}}]{JHWM01}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Haidenbauer}},
  \bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Melnitchouk}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Speth}},
  \bibinfo{journal}{AIP Conf. Proc.} \textbf{\bibinfo{volume}{603}},
  \bibinfo{pages}{421} (\bibinfo{year}{2001}), \eprint{nucl-th/0108062}.

\end{thebibliography}