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\title{Measurement of longitudinal spin transfer to the
$\Lambda$-hyperon at HERMES}
\author{Yu. Naryshkin on behalf of the HERMES
collaboration \address[MCSD]{
Petersburg Nuclear Physics Institute,
Gatchina, St. Petersburg, \\ }}%
\runtitle{Measurement of longitudinal spin transfer to the
$\Lambda$-hyperon at HERMES}
\runauthor{Yu. Naryshkin}
\begin{document}
\begin{abstract}
The spin transfer to the $\Lambda$-hyperon from a longitudinally
polarized positron beam has been measured at the \mbox{HERMES} experiment
in the semi-inclusive reaction $\vec e+p(d) \to e'+ \vec \Lambda +X$
in deep inelastic scattering regime with negative four-momentum
transfer squared of the exchanged virtual photon \mbox{$Q^2>1$~GeV$^2$}.
The data were accumulated in the years 1996-2000 using
the 27.5~GeV polarized HERA positron beam
and an unpolarized internal gas target in a storage cell.
The longitudinal spin transfer coefficient $D^{\Lambda}_{LL'}$
is found to be $0.11 \pm 0.10(stat) \pm 0.03(syst)$
at the average fractional $\Lambda$ energy $\langle z \rangle = 0.45$.
\vspace{1pc}
\end{abstract}
% typeset front matter (including abstract)
\maketitle
\section{Introduction}
Longitudinal spin transfer from a
polarized positron to a $\Lambda$ hyperon produced in the
deep-inelastic scattering process is sensitive
to two unknowns: the spin structure of the lightest hyperon,
and the spin-dependent dynamics of the fragmentation process in
deep-inelastic scattering.
Given the non-trivial spin structure of the proton \cite{SpinReview},
it is of interest to consider the spin
structure of other baryons.
In the naive Constituent Quark Model the spin of the
$\Lambda$ hyperon is
entirely carried by the $s$ quark:
${\Delta q^\Lambda_s = 1}$,
while the $ud$ pair is in a spinless
(singlet) state, i.e.,
${\Delta q^\Lambda_u=\Delta q^\Lambda_d=0}$. Here ${\Delta
q^\Lambda_f \equiv q^{\Lambda+}_f - q^{\Lambda-}_f}$,
where $q^{\Lambda+}_f$ and $q^{\Lambda-}_f$
are the probabilities to find a quark with the spin
parallel and anti-parallel to the
the hyperon spin direction, respectively,
and $q^\Lambda_f$ ($f=u,d,s$)
is the number density for quarks plus antiquarks of flavor $f$ in
the $\Lambda$ hyperon,
while the unpolarized number density is
${ q^\Lambda_f \equiv q^{\Lambda+}_f + q^{\Lambda-}_f}$.
Alternatively, one can use SU(3)-flavor
symmetry in conjunction with the experimental results on the proton
to estimate the first moments of the
helicity-dependent quark distributions in the $\Lambda$ hyperon. Using
such assumptions
Burkardt and Jaffe found
$\Delta q^\Lambda_u = \Delta q^\Lambda_d =-0.23 \pm 0.06$
and
$\Delta q^\Lambda_s=0.58 \pm 0.07$~\cite{BJ}.
According to this estimate, the spins of
the $u$ and $d$ quarks and antiquarks
are directed predominantly opposite to the spin of the $\Lambda$
hyperon resulting in a weak but non-zero net polarization.
If such an SU(3)-flavor rotation (see Eq.~3 of Ref.~\cite{QCDSF}, for
example) is applied to the recent semi-inclusive
data on the nucleon~\cite{HERMES_semi}, the values
$\Delta q^\Lambda_u = \Delta q^\Lambda_d = -0.09 \pm 0.06$ and
$\Delta q^\Lambda_s = 0.47 \pm 0.07$
are obtained instead, favoring a much smaller polarization of the
$u$ and $d$ quarks and antiquarks.
A lattice-QCD calculation~\cite{QCDSF} also
finds small light-quark polarizations,
$\Delta q^\Lambda_u = \Delta q^\Lambda_d = -0.02 \pm 0.04$ and
$\Delta q^\Lambda_s = 0.68 \pm 0.04$.
Finally, other
authors~\cite{Ma00,Boros,Liang} have employed phenomenological models
to explore the dependence of
$\Delta q_f^\Lambda (x)$ on the Bjorken scaling variable
$x$. These models predict a large positive polarization of the
$u$ and $d$ quarks in the kinematic region $x > 0.3$.
Longitudinal spin transfer to
$\Lambda$ hyperons has previously been explored
by the LEP
experiments OPAL and ALEPH at an energy corresponding to the $Z^0$
pole~\cite{OPAL,ALEPH}.
In these experiments the $\Lambda$ hyperons are predominantly
produced via the decay $Z^0 \rightarrow s \overline{s}$, in which the
primary strange
quarks from the decay are strongly (and negatively) polarized at the level
of $-91$\%. The OPAL and ALEPH data show a $\Lambda$ polarization of
about $-30$\% at $z>0.3$.
The polarization of final-state $\Lambda$ hyperons can be measured via
the
weak decay channel $\Lambda^0 \rightarrow p \pi^-$
through the angular distribution of the final-state particles:
\begin{equation}
\frac{dN}{d\Omega_p}
\propto 1 + \alpha \vec{P}_{\Lambda} \cdot \hat{k}_p.
\label{eq:decay}
\end{equation}
Here $\frac{dN}{d\Omega_p}$ is the angular distribution of
the protons,
$\alpha = 0.642 \pm 0.013$ is the asymmetry parameter of the
parity-violating weak decay, $\vec{P}_{\Lambda}$ is the
polarization of the $\Lambda^0$, and $\hat{k}_p$ is the unit
vector along the proton momentum in the rest frame of the
$\Lambda^0$. Because of the parity-violating nature of this decay,
the proton is preferentially emitted along the spin direction of
its parent, thus offering access to spin degrees of freedom in
the deep-inelastic scattering final state.
The component of the polarization transferred along the direction
$L'$ from the virtual photon to the produced $\Lambda$ is given by
\begin{equation}
P^\Lambda_{L'} = P_b D(y) D_{LL'}^\Lambda,
\label{eq:depol}
\end{equation}
\noindent
where $P_b$ is the longitudinal polarization of the beam,
$y = \nu/E$ is the fractional energy carried
by the photon in the target rest frame, $\nu=E-E'$ is the energy transfer
of the virtual photon to the nucleon,
$E(E')$ is the energy of the positron before (after) the scattering process
and $D(y) \simeq [ 1-(1-y)^2 ] / [1 + (1-y)^2 ]$ is the
depolarization factor taking into account the loss of polarization of the
virtual photon as compared to that of the incident positron,
$L$ is the primary quantization axis, directed along
the virtual photon momentum.
The spin transfer coefficient $D_{LL'}^\Lambda$ in Eq.~\ref{eq:depol}
describes the probability that the polarization of the struck quark
is transferred to the $\Lambda$ hyperon along a secondary
quantization axis $L'$.
In this analysis, the quantization axis $L'$ is chosen along the
virtual photon direction $L$.
\section{The HERMES experiment and data analysis}
The $\Lambda$ electroproduction data were
accumulated by the HERMES experiment at DESY. In this experiment,
the 27.6 GeV longitudinally-polarized positron beam \cite{HERA}
of the HERA $e$-$p$ collider passes through
an open-ended tubular storage cell into which polarized or
unpolarized target atoms in undiluted gaseous form are continuously
injected. The HERMES detector is described in detail in
Ref. \cite{HERMES_spectro}.
The data presented here were recorded using positron beams
during two two-year periods: 1996-1997 and 1999-2000.
A variety of unpolarized target gases were used in the analysis.
Most of the data were collected from hydrogen and deuterium, but
$^{3}$He, $^{4}$He, $^{14}$N, $^{20}$Ne and $^{84}$Kr
targets were also included, and the data from all targets were combined.
An average beam polarization of about 55\% was typical during data taking.
The scattered positrons and the $\Lambda$ decay products were
detected by the HERMES spectrometer
in the polar-angle range from 40 to 220 mrad.
A positron trigger was formed from a coincidence between three
scintillator
hodoscope planes and a lead-glass calorimeter. The trigger required
a minimum energy deposit in the calorimeter of 3.5 GeV for the
data employed in this analysis.
The $\Lambda$ hyperons were identified in the analysis through their
$p \pi^-$ decay channel. Events were selected by requiring
the presence of at least three reconstructed tracks: a positron track and
two hadron candidates of opposite charge. If more than one positive or
negative hadron was found in one event, all possible combinations of
positive and negative hadrons were used. The requirements
$Q^2 > 0.8$ GeV$^2$ and $W > 2$ GeV, where $-Q^2$ is the four-momentum
transfer squared of the exchanged virtual photon and $W$ is the
invariant mass of the photon-nucleon system, were imposed
on the positron kinematics to ensure that the events
originated from the deep-inelastic scattering domain.
In addition, the requirement $y=1-E'/E < 0.85$ was imposed
to exclude the large contribution of radiative corrections.
Two spatial vertices were reconstructed for each event by determining
the intersection (i.e., point of closest approach) of pairs of
reconstructed tracks. The primary (production) vertex was determined
from the intersection of the beamline and the scattered beam lepton, while
the secondary (decay)
vertex was determined from the intersection of the proton and pion tracks.
In both cases, the distance of closest approach was required to be less
than 1.5 cm. For tracks fulfilling these requirements the invariant mass
of the hadron pair was evaluated, under the assumption that the
high-momentum leading hadron is the proton while the low-momentum hadron
is the pion. In order to suppress the background from hadrons emitted from the
the primary vertex, a vertex separation requirement of
$z_2 - z_1 > 10$~cm has been applied, where $z_1$ and $z_2$ represent
the coordinates of the primary and secondary vertex positions along the
beam direction. In addition to that a RICH
(or \v{C}erenkov) based particle identification has been used for the leading particle.
The spectrum with and without these background suppression constraints
is displayed in Fig.~\ref{fig:invmass}.
%
\begin{figure}[h]
\vspace{9pt}
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\includegraphics[width=\columnwidth]{mass_peaks.eps}
\vspace{-2cm}
\caption{The yield of semi-inclusively produced $\Lambda$ hyperons in
deep-inelastic scattering. The left (right) panel shows the
invariant-mass spectrum before (after) the application of
background suppression cuts. The vertical lines show
the boundaries at $\pm 3.3\,\sigma$.}
\label{fig:invmass}
\end{figure}
Without background suppression, 8,200 $\Lambda$
events were extracted from all unpolarized data,
while the final data sample,
with all requirements imposed, contained 7,300 $\Lambda$ events.
The HERMES spectrometer has a limited
acceptance for the reconstruction of $\Lambda$ hyperons.
In order to cancel the effect of the acceptance on the result, the spin
transfer to
the $\Lambda$ has been determined by combining the two data sets
measured with opposite beam polarizations into one helicity-balanced
data sample, in which the luminosity-weighted average
beam polarization for the selected data is
$ \overline{P_b} \equiv \frac{1}{L} \int P_b\,dL= 0.$
Here $L = \int dL$ is the integrated luminosity.
A detailed derivation based on the method of maximum likelihood
leads to the relation~\cite{SBform98}:
\begin{equation}
D_{L L'}^\Lambda =
\frac{1}{\alpha \overline{P_b^2}} \cdot
\frac{\sum_{i=1}^{N_\Lambda} P_{b,i}\,D(y_i) \cos\Theta^i_{pL'}}
{\sum_{i=1}^{N_\Lambda} D^2(y_i) \cos^2\Theta^i_{pL'}}.
\label{eq:extract}
\end{equation}
Here $\Theta_{pL'}$ is the angle between the proton
momentum in the $\Lambda$
rest frame and the $\Lambda$ spin quantization axis $L'$, where
$L'$ is chosen along the virtual photon direction $L$.
\section{Results}
In the forward-production region (\mbox{$x_F > 0$},
where $x_F$ is the fractional longitudinal momentum of
the $\Lambda$ hyperon) at ${\langle x_F \rangle =0.31}$,
and at the average fractional $\Lambda$ energy
$\langle z \rangle = 0.45$, the spin-transfer coefficient
${D_{LL'}^\Lambda = 0.11 \pm 0.10\ \mathrm{(stat)} \pm 0.03\
\mathrm{(syst)}}$ was obtained.
\begin{figure}[h]
\vspace{9pt}
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\includegraphics[width=\columnwidth]{plotz1.eps}
\vspace{-1.4cm}
\caption{
Dependence of the longitudinal spin-transfer coefficient
$D_{LL'}^\Lambda$
on $z$, for $x_F > 0$.
The curves represent the phenomenological model calculations
of Refs.~\protect\cite{Ma00,Ma02}.
Error bars are statistical only.}
\label{fig:dll_z}
\end{figure}
This value is in a good agreement with the NOMAD result \cite{NOMAD}
${-P^\nu_{\Lambda} =0.09 \pm 0.06\
\mathrm{(stat)} \pm 0.03\ \mathrm{(syst)}}$ measured for ${x_F > 0}$
(${\langle x_F \rangle =0.21}$), where
$-P^\nu_{\Lambda}=D_{LL'}^\Lambda$ \cite{HERMES_DLL_new}.
% The spin-transfer coefficient
%consistent with zero and is in contrast with the large $\Lambda$
%polarization observed in $e^+ e^-$ annihilation at OPAL and ALEPH.
%This difference is not unexpected, as $\Lambda$
% production in the reaction $e^+ e^- \rightarrow Z^0 \rightarrow
%\Lambda X$ is dominated by the fragmentation of strange quarks, while in
%deep-inelastic scattering the fragmentation of $u$ and $d$ quarks,
%weakly polarized in the $\Lambda$ hyperon,
%is the principal source.
\begin{figure}[h]
\vspace{9pt}
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\includegraphics[width=\columnwidth]{plotxF1.eps}
\vspace{-1.5cm}
\caption{
Dependence of the longitudinal spin-transfer coefficient
$D_{LL'}^\Lambda$
on $x_F$. The HERMES measurements~\cite{HERMES_DLL_new} are represented by the
solid circles, while the open symbols represent data from
NOMAD~\cite{NOMAD} (squares) and
E665~\cite{E665} (circles). Error bars are statistical only.}
\label{fig:dll_xF}
\end{figure}
The dependence of $D_{LL'}^\Lambda$ on $z$
with the requirement $x_F > 0$ imposed
is presented in Fig.~\ref{fig:dll_z}.
Superimposed on the data are the
phenomenological model calculations of Ref.~\cite{Ma00} (pQCD and
quark-diquark models) which predict a pronounced rise of the spin
transfer at high values of $z$.
The pQCD prediction is disfavoured by the data.
Also shown is the model of Ref.~\cite{Ma02} (SU(3)-flavor
rotation of proton values) which predicts a more gradual increase.
The HERMES data \cite{HERMES_DLL_new} as a function of $x_F$ are presented in
Fig.~\ref{fig:dll_xF} together with data obtained
by the NOMAD experiment~\cite{NOMAD} at CERN
with a 43 GeV $\nu_\mu$-beam and
the Fermilab E665 experiment~\cite{E665} obtained
with a 470 GeV polarized muon beam. As it can be see from
Fig.~\ref{fig:dll_xF} the \mbox{NOMAD} and HERMES results
are compatible in the kinematic region of overlap $ -0.1 < x_F < 0.3$.
\begin{thebibliography}{9}
%1
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Adv. Nucl. Phys. 26 (2001) 1.
%2
\bibitem{BJ} M.~Burkardt and R.L.~Jaffe, Phys. Rev. Lett. 70 (1993) 2537.
%3
\bibitem{QCDSF} QCDSF Collaboration, M.~G\"ockeler et al.,
Phys. Lett. B 545 (2002) 112.
%4
\bibitem{HERMES_semi} HERMES Collaboration, A.~Airapetian et al.,
Phys. Rev. Lett. 92 (2004) 012005;
Phys. Rev. D 71 (2005) 012003.
%5
\bibitem{Ma00} B.-Q.~Ma, J.-J.~Yang, I.~Schmidt,
Phys. Lett. B 477 (2000) 107.
%6
\bibitem{Boros} C.~Boros, J.T.~Londergan, and A.W.~Thomas,
Phys. Rev. D 61 (2000) 014007.
%7
\bibitem{Liang} C.~Liu and Z.~Liang,
Phys. Rev. D 62 (2000) 094001.
%8
\bibitem{OPAL} OPAL Collaboration, K.~Ackerstaff et al.,
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%9
\bibitem{ALEPH} ALEPH Collaboration, D.~Buskulic et al.,
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%10
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%11
\bibitem{HERMES_spectro} HERMES Collaboration, K.~Ackerstaff et al.,
Nucl. Instrum. Methods A 417 (1998) 230.
%12
\bibitem{SBform98} S.~Belostotski in {\it Proc. of the IX-Workshop on
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%13
\bibitem{Ma02} B.-Q.~Ma, I.~Schmidt, J.~Soffer, and J.-J.~Yang,
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%14
\bibitem{NOMAD} NOMAD Collaboration, P.~Astier et al.,
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%15
\bibitem{HERMES_DLL_new} HERMES Collaboration, A.~Airapetian et al.,
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%16
\bibitem{E665} E665 Collaboration, M.R.~Adams et al.,
Eur. Phys. J. C 17 (2000) 263.
\end{thebibliography}
\end{document}