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\title{$\Lambda^0$ Polarization in exclusive and inclusive p-Nucleus reactions}
\author{J. F\'elix\address{Instituto de F\'{\i}sica, Universidad de Guanajuato. Lomas del
Bosque 103, Frac. Lomas del Campestre, Le\'{o}n GTO. 37150, M\'{e}xico. felix@fisica.ugto.mx}%
\thanks{Thanks to the BNL e766 and FNAL e690 collaborations (http://www-e690.fnal.gov).
This research was supported in part by CONACYT, M\'{e}xico, under grant No.
2002-C01-39941, CONCYTEG 06-16-K117-99, Apoyo a la investigaci\'{o}n cient\'{\i}fica UGTO 2006,
and PROMEP-2005.}}
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\runtitle{$\Lambda^0$ Polarization in exclusive and inclusive
p-Nucleus reactions} \runauthor{J. F\'elix}
\begin{document}
\begin{abstract}
Among all properties of baryons, the polarization they acquire when
created from unpolarized p-Nucleus collisions is the most recent
discovered one; so far, the origin of baryon polarization remains
unexplained in spite of the experimental evidences accumulated in
the past thirty years. Up to these days, $\Lambda^0$ is the most
studied baryon for polarization, because it is very easy to produce
at the energies of the principal high energy physics accelerators of
the world. This paper is a review of the experimental information
accumulated on the polarization of $\Lambda^0$ in unpolarized
exclusive and inclusive p-Nucleus collisions as function of $x_F$,
$P_T$, and diffractive mass of the $\Lambda^0K^+$ system in the past
thirty years. \vspace{1pc}
\end{abstract}
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\section{INTRODUCTION}
It is an experimental evidence that baryons in general are created
polarized, when they are produced from unpolarized {\it
particle-Nucleus} collisions. For instance $\Xi$, $\Sigma$,
$\Lambda^0$, etc., are produced polarized in exclusive as well as in
inclusive reactions at different energies\cite{1,2,3}. In the same
experimental circumstances, it appears that $\Omega^-$\cite{1}(K.B.
Luk et al) and ${\overline\Lambda^0}$\cite{1} \cite{500}(K. Heller
et al) do not appear polarized. However, it appears that
${\overline\Lambda^0}$ is created polarized in $K^-p$
reactions\cite{600}. The origin of baryon polarization is not known
up to these days.\
From the above mentioned baryons, $\Lambda^0$ is the most studied
one for polarization, because it is very easy to produce and analyze
for polarization. In many experiments $\Lambda^0$'s from unpolarized
$pp$ inclusive and exclusive collisions, at different energies, are
produced polarized\cite{2}. This polarization depends on $x_F$,
$P_T$, and $\Lambda^0K^+$ invariant mass\cite{3}.\
Based on the above experimental evidences, some authors have
proposed many theoretical ideas -in the context of the Lund model,
parton recombination, multiple scattering of the strange quark,
gluon fusion, Regge theory, coherent scattering, low and high order
QCD calculations, valence quark effects, and quark condensates-
trying to understand $\Lambda^0$ polarization\cite{4}. See reference
\cite{5} for a review of the theoretical ideas proposed to explain
$\Lambda^0$ polarization. To date, there is no a satisfactory
explanation for baryon polarization in general.\
This paper is a summary of the experimental results on $\Lambda^0$
polarization, as function of $x_F$, $P_T$, and $M_{\Lambda^0K^+}$,
in the specific final states
\begin{equation}\label{eq:1}
pp\to p\Lambda^{0}K^{+}(\pi ^{+}\pi ^{-})^N; N=1,2,3,4,5,
\end{equation}
at $27.5~GeV$, from the BNL e766 collaboration. And in the specific
final state
\begin{equation}\label{eq:2}
pp\to p\Lambda^{0}K^{+},
\end{equation}
at $800~GeV$, from the FNAL e690 collaboration.\
FNAL e690 and BNL e766 collaborations are described in detail
elsewhere\cite{6}.\
\section{TECHNIQUE TO MEASURE $\Lambda^0$ POLARIZATION}
%For $\Lambda^0$ weakly decays into $p\pi^-$ and the angular
%distribution of the protons from $\Lambda^0$'s, in the reference
%system where $\Lambda^0$ is at rest, is very well known\cite{800},
%the measurement of $\Lambda^0$ polarization consists in determining
%the angular distribution of the proton from the $\Lambda^0$ and
%fitting it to a straight line. For an ample description of this
%technique, see reference\cite{90}.\
The polarization of $\Lambda^0$ is measured with respect to the
normal vector to the $\Lambda^0$ production plane defined as
\begin{equation}\label{eq:3}
\hat{n} \equiv \frac{\vec{P}_{beam} \times \vec{P}_{\Lambda^0}}{
\mid \vec{P}_{beam} \times \vec{P}_{\Lambda^0} \mid},
\end{equation}
where $\vec{P}_{beam}$ and $\vec{P}_{\Lambda^0}$ are the momentum
vectors of the incident beam proton and of the $\Lambda^0$,
respectively.\
The angular distribution of the proton from $\Lambda^0$, in the
$\Lambda^0$ rest frame, is described by this expression\cite{800}:
\begin{equation}\label{eq:4}
\frac{dN}{d\Omega} = N_0(1+ \alpha \mathcal{P} cos\theta),
\end{equation}
where $\alpha$ is the $\Lambda^0$ decay asymmetry parameter ($0.642
\pm 0.013$\cite{7}) and $\mathcal{P}$ is the $\Lambda^0$
polarization. The polarization is determined by a linear fit of a
function of the above form to the $cos \theta$ distribution of the
proton -after Monte Carlo detector acceptance corrections- for two
free parameters: Intercepts ($N_0$), and slope $N_0 \alpha
\mathcal{P}$. From these parameters the $\Lambda^0$ polarization and
its statistical error are measured. The systematic errors are
obtained simulating the detector acceptance and efficiency using the
Monte Carlo technique.\
\section{$\Lambda^0$ POLARIZATION AS FUNCTION OF $x_F$ AND $P_T$}
The discovery of $\Lambda^0$ polarization in inclusive $pp$
reactions showed that it depends linearly on $P_T$ and that it is
negative with respect to the $\Lambda^0$ production plane\cite{8}:
At $P_T=0.0$, $\mathcal{P}=0.0$, and it decreases as $P_T$ increases
up to close $\mathcal{P}= -0.25$ at $P_T=1.2~GeV.$ Also it was
determined in inclusive $pp$ reactions that $\Lambda^0$ polarization
depends on $x_F.$ See Reference \cite{9} for a review of the
experimental results on $\Lambda^0$ polarization.\
In the reaction $pp \to p \Lambda^0 K^+ (\pi^+ \pi^-)^2$ at
$27.5~GeV$, $\Lambda^0$ polarization in a function of $P_T$ and
$x_F$\cite{10}. Reference\cite{10} was the first report on
$\Lambda^0$ polarization in exclusive reactions at high energies.\
%Fig. 1 shows the results as function of $P_T$.
$\Lambda^0$ polarization as function of $P_T$ is given by
$\mathcal{P}=(-0.250\pm0.067)P_T+(0.063\pm0.041)$
-$\frac{\chi^2}{N_{DOF}}=1.04$, for $x_F>0.0$-; by
$\mathcal{P}=(0.147\pm0.056)P_T+(-0.007\pm0.033)$
-$\frac{\chi^2}{N_{DOF}}=1.04$, for $x_F<0.0$-; and by
$\mathcal{P}=(-0.189\pm0.042)P_T+(0.029\pm0.025)$
-$\frac{\chi^2}{N_{DOF}}=1.68$, for $x_F>0.0$ and $x_F<0.0$-
combined.\
%Fig. 2 shows the results as function of $x_F$.
The average $\Lambda^0$ polarization is roughly linear as function
of $x_F$. This is given by
$\mathcal{P}=(-0.20\pm0.06)x_F+(-0.032\pm0.018)$
-$\frac{\chi^2}{N_{DOF}}=0.247$, for $x_F>0.0$ and $x_F<0.0$
combined-. The average $\Lambda^0$ polarization changes sign at
$x_F\simeq0.0$.\
A complete analysis of $\Lambda^0$ polarization in the particular
reactions described by Eq. 1, with $N=1,2,3,4$, was performed at
high statistics, searching for $\Lambda^0$ polarization as function
of $x_F$ and $P_T$\cite{11} together. To analyze the data,
$\mathcal{P}$ is parameterized as a function of $x_F$ and $P_T$
simultaneously: $\mathcal{P} = \mathcal{P}(x_F,P_T)$. The parameters
of this function are determined using the maximum likelihood
method\cite{40} with Eq. 4 as the probability distribution for
having $dN$ protons in a solid angle $d\Omega$. For the maximum
likelihood analysis, the chosen function that represents the
simplest bilinear combination of $x_F$ and $P_T$ is as follows:
\begin{equation}\label{eq:5}
\mathcal{P}(x_F, P_T) = (-0.443\pm0.037)x_F P_T,
\end{equation}
for -$1.0 \leq x_F\leq1.0$ and $0.0\leq P_T \leq 1.8~GeV/c$-.
The conclusion from this result is as follows: The mechanism
responsible for $\Lambda^0$ polarization is independent of the
$\Lambda^0$ production mechanism, at least for the examined final
state reactions.\
Furthermore, in general, equation
\begin{equation}\label{eq:6}
\mathcal{P}(x_F, P_T) = (-a)x_F P_T,
\end{equation}
for $-1.0 \leq x_F \leq 1.0$ and $0.0 \leq P_T \leq 1.8~GeV/c$, is
valid for $\Lambda^0$ polarization from exclusive and inclusive
$p-Nucleus$ reactions\cite{100} with the proper $a$ value that
depends on the particular beam and target reaction.\
Both $\Lambda^0$ polarization from inclusive and exclusive
$p-nucleus$ reactions follow the same trend as function of $x_F$ and
$P_T$: $\mathcal{P}(x_F,P_T)=(a)x_FP_T$. This equation is valid for
$-1.0