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\title{$\Lambda^{0}$ polarization and its relation with nucleonic resonances}
\author{Victor M. Castillo-Vallejo\address[MCSD]{Departamento de F\'{\i}sica Aplicada, CINVESTAV-IPN, Unidad
M\'{e}rida\\
Ant. Carr. Progreso Km 6, A.P. 73 Cordemex 97310, M\'{e}rida, Yucat\'{a}n, M\'{e}xico }%
\thanks{Present address: Centro de Investigaciones en Optica (CIO), Loma del Bosque 115, Lomas del Campestre,
Le\'on
Guanajuato, 37150, M\'{e}xico, Email: victorcv@cio.mx}
Juli\'an F\'elix \address{ Instituto de F\'{\i}sica, Universidad de Guanajuato,\\ Loma del Bosque 103,
Lomas del Campestre
Le\'{o}n, Guanajuato, 37150, M\'{e}xico}
and
Virendra Gupta\addressmark[MCSD]}
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\runauthor{V\'{\i}ctor M. Castillo-Vallejo et al}
\begin{document}
\begin{abstract}
In the sequential reactions $pp \to pN^{*}$ and $N^{*} \to
\Lambda^{0} K^{+}$ $\Lambda^{0}$ polarization vector is proportional
to $N^{*}$ po\-la\-ri\-za\-tion vector. This result is valid
whatever the value of the spin of $N^{*}$ and for both,
conserving-parity decays and non-conserving-parity decays. Besides,
for the particular case of $N^{*}$ having spin 1/2, $N^{*}$ and
$\Lambda^{0}$ production planes coincide each other. Based on these
results, a technique to evidence nucleonic resonances in $pp \to
p\Lambda^{0} K^{+}$ is established, which is independent of the
mechanism responsible of this reaction. \vspace{1pc}
\end{abstract}
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\maketitle
\section{Introduction}
Hyperons polarization studies contribute to understand the spin
dependence of hadronic interactions. From all the hyperons,
$\Lambda^{0}$ is the most stu\-died in
both theoretical and experimental a\-na\-ly\-sis %\cite{**}
and, in spite of $\Lambda^{0}$ po\-la\-ri\-za\-tion problem is not
completely solved yet, some clues have been established about the
origin of its polarization; for instance, in exclusive $pp$
reactions, $\Lambda^{0}$ polarization seems to be related to its
production me\-cha\-nism\cite{0.2}. In order to completely solve the
pro\-blem of $\Lambda^{0}$ po\-la\-ri\-za\-tion in $pp$ reactions,
it is ne\-ce\-ssa\-ry to discriminate the contributions to
$\Lambda^{0}$ polarization from $\Lambda^{0}$ produced directly
from those produced indirectly. In this work it is a\-na\-ly\-zed
the case when $\Lambda^{0}$ is created through the decay of one
nucleonic resonance from un\-po\-la\-ri\-zed $pp \to
p\Lambda^{0}K^{+}$ reactions. The problem is analyzed using only
kinematical information of the reaction and, hence, the conclusions
are independent of any dynamical model which could explain the
interaction of the initial protons. The reaction is split out in two
sequential reactions: $pp \to pN^{*}$ and $N^{*} \to \Lambda^{0}
K^{+}$. A relation between polarization vectors of the resonance
$N^{*}$ and baryon $\Lambda^{0}$ is derived. From this relation, a
graphic method to detect the presence of nu\-cleo\-nic resonances in
$pp \to p \Lambda^{0}K^{+}$ is established.
\section{POLARIZATION VECTOR IN STRONG INTERACTIONS}
It is known that polarization vector is a pseudovector and it
changes its sign under time reversal operation. If the reaction of
interest follows parity and time reversal symmetries, any component
of polarization vector with properties di\-ffe\-rent to the above
exposed should be suppressed. A first assertion coming as a
consequence of the above arguments is the next:
\begin{statement}\label{sta1}
In $pp \to pN^{*}$, $N^{*}$ spin should point along its
production-plane normal. In the decay $N^{*}\to \Lambda^{0}K^{+}$,
$\Lambda^{0}$ is created with its spin pointing along the normal to
its own production plane.
\end{statement}
We use this basic result (which is valid due to both the parity and
the time reversal symmetries) to demonstrate in Section \ref{sec4}
a technique to detect resonances.
\section{RELATION BETWEEN $\Lambda^{0}$ AND $N^{*}$ POLARIZATION VECTORS}
Statement \ref{sta1}, in spite of its simplicity, allow to establish
restrictions over angular distributions and over the relation
between polarization vectors of $N^{*}$ and $\Lambda^{0}$, such as
it is shown in the following sections.
\subsection{Case 1. $N^{*}$ spin equals $\frac{1}{2}$}
In $N^{*}\to \Lambda^{0}K^{+}$, $\Lambda^{0}$ polarization is
related to $N^{*}$ polarization through the following equations:
\begin{equation}\label{eq3}
\vec{\mathcal{P}}_{\Lambda^{0}}=(\vec{\mathcal{P}}_{N^{*}}\cdot
\hat{p})\hat{p}+\hat{p}\times (\vec{\mathcal{P}}_{N^{*}}\times
\hat{p}).
\end{equation}
\begin{equation}\label{eq4}
\vec{\mathcal{P}}_{\Lambda^{0}}=(\vec{\mathcal{P}}_{N^{*}}\cdot
\hat{p})\hat{p}-\hat{p}\times (\vec{\mathcal{P}}_{N^{*}}\times
\hat{p}).
\end{equation}
\noindent where $\hat{p}$ is a unit vector pointing along the motion
of $\Lambda^{0}$ in the frame where $N^{*}$ is at rest. Eqs.
(\ref{eq3}) and (\ref{eq4}) are obtained from the most general
formula relating polarization vector of daughter and parent baryons
in the decay of a 1/2-spin baryon into a 1/2-spin baryon and a
0-spin meson due to Lee and Yang\cite{5,6.5}, by requiring the
parity to be conserved
In the decay $N^{*} \to \Lambda^{0} K^{+}$, if the component of
$\Lambda^{0}$ polarization along its motion direction is
res\-tric\-ted to be zero, according to Statement \ref{sta1}, then
in Eqs. (\ref{eq3}) and (\ref{eq4}) the term
$\vec{\mathcal{P}}_{N^{*}}\cdot \hat{p}$ \hspace{2mm} should be
zero and hence $\Lambda^{0}$ momentum and $N^{*}$ polarization are
perpendicular each other. Therefore, for the particular case of a
1/2-spin particle strongly decaying into one 1/2-spin baryon and a
0-spin meson, the production planes of the parent and daughter
particles are the same. This shows the validity of the next
statement:
\begin{statement}\label{sta3}
In $pp \to pN^{*}$, $N^{*}\to \Lambda^{0}K^{+}$, when $N^{*}$ has
$S=1/2$, the production plane of $\Lambda^{0}$ and the production
plane of $N^{*}$ are the same.
\end{statement}
\subsection{Case 2. $N^{*}$ spin higher that 1/2}
Since strong decays conserve parity, when the spin and parity of
$N^{*}$ are fixed, its decay occurs through one channel of parity.
In order to separately study the cases for both values of $N^{*}$
pa\-ri\-ty, two channels are defined as follows:
\noindent \textbf{Channel 1.} $l=S-1/2$ and then, parity of $N^{*}$
can be expressed as $\wp(N^{*})=(-1)^{S+1/2}$.
\noindent \textbf{Channel 2.} $l=S+1/2$ and then, parity of
$N^{*}$ can be expressed as $\wp(N^{*})=(-1)^{S+3/2}$. \\
\noindent where $l$ is the orbital angular momentum of $\Lambda^{0}$
and $K^{+}$ measured in the $N^{*}$ rest frame and which takes the
values $l=S\pm1/2$. The following relations between $N^{*}$ and
$\Lambda^{0}$ polarization vectors are obtained when $\Lambda^{0}$
polarization vector is integrated over $\Lambda^{0}$ angular
distribution measured in $N^{*}$ rest frame.
\noindent \textbf{Channel 1}
\begin{equation}\label{eq21}
\vec{\mathcal{P}}(\Lambda^{0})=\vec{\mathcal{P}}(N^{*}).
\end{equation}
\textbf{Channel 2}
\begin{equation}\label{eq20}
\vec{\mathcal{P}}(\Lambda^{0})=\left(-\frac{S}{S+1}\right)\vec{\mathcal{P}}(N^{*}).
\end{equation}
These results allow to establish the next statement:
\begin{statement}\label{sta4}
Polarization vector of $\Lambda^{0}$, when it is integrated over the
angular distribution of $\Lambda^{0}$ measured in the $N^{*}$ rest
frame, is proportional to polarization vector of $N^{*}$.
\end{statement}
Eqs. (\ref{eq20}) and (\ref{eq21}) are valid when the decay
conserves parity. In a more general situation, the decay of one
resonance could violate parity, i.e., it could have interference
between both of the channels above defined. In this case, the more
general relation between the polarization vector of $N^{*}$ and that
of $\Lambda^{0}$ is as follows:
\begin{equation}\label{eq24}
\vec{\mathcal{P}}(\Lambda^{0})=\left(\frac{1+(2S+1)\gamma}{2(S+1)}\right)\vec{\mathcal{P}}(N^{*}).
\end{equation}
The parameter $\gamma$ is related to the parity violation of the
decay and it is defined in Ref. \cite{5}. This equation reduces to
Eq. \ref{eq20} (\ref{eq21}) when $\gamma=-1(+1)$. When the decaying
particle has $S=3/2$, the above expression coincides with that
reported in literature for the decay $\Omega^{-} \to
\Lambda^{0}K^{-}$\cite{7,8,9}.
\section{NUCLEONIC RESONANCES IN $pp \to
p\Lambda^{0}K^{+}$}\label{sec4}
Known nucleonic resonances decaying in $\Lambda^{0} K^{+}$ channel
are broad (typical widths are $100 -200$ $MeV$) and they are located
inside of a short range of masses (from $1650$ until $2200$ $MeV$).
This causes they overlap, making difficult the characterization of
them, i. e., it is hard a clear identification of each resonance. A
different graphical method to detect the presence of nucleonic
resonances in the above reaction, based in the validity of Statement
\ref{sta4}, can be established as follows.
Assuming that parity is conserved in $pp \to pN^{*}$, $N^{*} \to
\Lambda^{0}K^{+}$, the angle between $\Lambda^{0}$ and $N^{*}$
production planes can be calculated, on one hand, with the next
equation
\begin{equation}
\hat{n}_{N^{*}}\cdot\hat{n}_{\Lambda^{0}}=\frac{\vec{\mathcal{P}}_{N^{*}}\cdot
\vec{\mathcal{P}}_{\Lambda^{0}}}{|\vec{\mathcal{P}}_{N^{*}}\cdot
\vec{\mathcal{P}}_{\Lambda^{0}}|},
\end{equation}
\noindent where $\hat{n}_{N^{*}}$ and $\hat{n}_{\Lambda^{0}}$ are
the normal to $N^{*}$ and $\Lambda^{0}$ production planes
respectively, $\vec{\mathcal{P}}_{N^{*}}$ and
$\vec{\mathcal{P}}_{\Lambda^{0}}$ are the polarization vectors. Due
to Statement \ref{sta4}, that expression is restricted to be
\begin{equation}
\hat{n}_{N^{*}}\cdot\hat{n}_{\Lambda^{0}}=\pm 1,
\end{equation}
\noindent independently of $N^{*}$ spin and when $\Lambda^{0}$
polarization vector is integrated over its angular distribution in
$N^{*}$ rest frame . On the other hand, for fixed direction of
$N^{*}$ and independently of its e\-ner\-gy,
$\hat{n}_{N^{*}}\cdot\hat{n}_{\Lambda^{0}}$ can be measured by
taking into account that
\begin{eqnarray}
\nonumber \hat{n}_{N^{*}}&=& \frac{(\vec{p}_{beam} \times
\vec{p}_{N^{*}})}{|\vec{p}_{beam} \times \vec{p}_{N^{*}})|} ,\\
\hat{n}_{\Lambda^{0}}&=& \int \mathrm{d}
\Omega_{\Lambda^{0}}\frac{(\vec{p}_{N^{*}} \times
\vec{p}_{\Lambda^{0}})}{|\vec{p}_{N^{*}} \times
\vec{p}_{\Lambda^{0}})|},
\end{eqnarray}
\noindent where $\vec{p}_{beam}$, $\vec{p}_{N^{*}}$, and
$\vec{p}_{\Lambda^{0}}$ are the momenta of beam, $N^{*}$ and
$\Lambda^{0}$ measured in a fixed reference frame (CMF for
instance), $\hat{n}^{'}$ is the normal to $\Lambda^{0}$ production
plane for each event and the integration is done over the angular
distribution of $\Lambda^{0}$ in any, the fixed reference frame
(CMF) or the $N^{*}$ rest frame. Experimentally, $\vec{p}_{N^{*}}$
is defined by $\vec{p}_{N^{*}}=\vec{p}_{\Lambda^{0}} +
\vec{p}_{K^{+}}$. The analysis is performed in bins of momenta of
$N^{*}$.
Using the above considerations, a plot of \mbox{
$\hat{n}_{N^{*}}\cdot \hat{n}_{\Lambda^{0}}$} $vs$ $M(\Lambda^{0}
K^{+})$, where $M(\Lambda^{0} K^{+})$ is the invariant mass of
$\Lambda^{0}K^{+}$ system, would reveal the presence of resonant
states of the proton de\-ca\-ying through that channel. Those
resonances would show a particular shape in that plot: It should
have a concentration of events near of $\hat{n}_{N^{*}}\cdot
\hat{n}_{\Lambda^{0}}= \pm 1$, in contrast to non-resonant states,
which would be uniformly distributed along all the va\-lues of
$\hat{n}_{N^{*}}\cdot \hat{n}_{\Lambda^{0}}$ because non-resonant
states don't have definite spin nor parity. This method can be
implemented in an analysis of the above reaction in a more practical
way by imposing a cut in the variable $\hat{n}_{N^{*}}\cdot
\hat{n}_{\Lambda^{0}}$ as follows:
\begin{eqnarray}\label{eq6.28}
\nonumber \textrm{Resonant events (principally).} \\
\nonumber a) \ \ \hat{n}_{N^{*}}\cdot \hat{n}_{\Lambda^{0}} \ge
0.999, \ \textrm{or} \ \hat{n}_{N^{*}}\cdot \hat{n}_{\Lambda^{0}}
\le -0.999.
\\ \nonumber \textrm{Non-resonant events.} \\ b) -0.999 <
\hat{n}_{N^{*}}\cdot \hat{n}_{\Lambda^{0}} < 0.999.
\end{eqnarray}
The above cut would isolate events with $\Lambda^{0}$ coming from
the decay of one resonance and hence, it would allow to measure the
properties of one resonance directly from the spectrum of
$M(\Lambda^{0} K^{+})$ after the events of non-resonant contribution
(background) are separated from the rest. This technique can be
considered as one complementary to a partial wave analysis. This
method can not be used if $\vec{\mathcal{P}}_{N^{*}}=0$, i.e., if
$N^{*}$ is created unpolarized, because in this particular case
$\hat{n}_{N^{*}}\cdot\hat{n}_{\Lambda^{0}}=0$ and hence, there is
not criterium to isolate the resonant events.
\section{CONCLUSIONS}
In the sequential reactions $pp \to p N^{*}$, $N^{*} \to \Lambda^{0}
K^{+}$, restrictions from symmetries on the resonances and
$\Lambda^{0}$ spins become to establish a relation between the
production planes and the polarization vectors $\vec{\mathcal{P}}$
of $N^{*}$ and $\Lambda^{0}$ allowing isolate resonant events from
those non-resonant such as Statements \ref{sta1}-\ref{sta4}
establish.
\begin{thebibliography}{9}
\bibitem{0.2} J. F\'elix \textit{et al}, Phys. Rev. Lett. 88, (2002) 061801.
\bibitem{5} D. E. Groom \textit{et al}, \textit{Eur. Phys. Jour.} \textbf{C15}, 1 (2000).
\bibitem{6.5} T. Lee and C. Yang, \textit{Phys. Rev.} \textbf{108}, 1645 (1957).
\bibitem{7} J. Kim \textit{et al}, \textit{Phys. Rev.} \textbf{D46}, 1060 (1992).
\bibitem{8} K. B. Luk \textit{et al}, \textit{Phys. Rev.} \textbf{D38}, 19 (1988).
\bibitem{9} Kam Biu Luk, \textit{Ph.D. Thesis}, Rutgers University, (1983), New Jersey.
\end{thebibliography}
\end{document}