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\noindent
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%{\it {\large version of \today}} \\[.3em] 
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\multicolumn{4}{l}{{\bf
                32st International Conference on High Energy Physics, ICHEP04},
                Aug.~16-23,~2004,~Beijing} \\
                 & Abstract:        & {\bf 756,758}    &\\
                 & Parallel Session & {\bf 4,5}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
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\vspace*{2cm}

\begin{center}
  \Large
  {\bf 
   First Measurement of the Polarisation Dependence of the Total Charged 
Current Cross Sections}

  \vspace*{1cm}
    {\Large H1 Collaboration} 
\end{center}

\begin{abstract}

\noindent
Data taken with the H1 detector, for longitudinally polarised positrons 
in left and right handed states in collision with
unpolarised protons at HERA, are used to measure the total charged
current cross section for $Q^2>400$\,GeV$^2$ and inelasticity $y<0.9$.
The polarisation dependence of the total charged current cross section
is compared with Standard Model expectations. The data are used to
obtain an extrapolated total charged current cross section for a fully
left handed positron beam. This extrapolation is found to be
consistent with the expectations of the Standard Model.


\noindent
\end{abstract}


\end{titlepage}

\pagestyle{plain}

\section{Introduction}
In Autumn 2003 the HERA accelerator started operation of the second
phase of its Deep Inelastic Scattering (DIS) programme. The $ep$
collision data collected since then has been taken with longitudinally
polarised positrons for the first time.  The polaristion of the lepton
beam allows HERA to further constrain the parton densities of the
proton through measurements of polarisation asymmetries~\cite{klein}, as well as
continuing to test the Electroweak part of the Standard Model at the
energy frontier. 

The two contributions to DIS, neutral current (NC),
\mbox{$ep \rightarrow eX$}, and charged current (CC) interactions,
\mbox{$ep \rightarrow \nu X$}, can both be measured at HERA and provide
complementary information on the Standard Model, which makes clear
predictions of the polaristion dependence of these cross sections.
Specifically, it predicts that the CC cross section should have a
linear dependence on polarisation, and furthermore, the cross section
for fully left handed positrons should be zero (similarly for fully
right handed electrons). This follows from the nonexistence of right
handed currents within the framework of the Standard Model.

In this paper we report on new measurements of the total CC cross
section for two values of lepton beam polarisation. The measurements
are compared to Standard Model expectations and a linear fit to the
data is performed. The fit is extrapolated to that of a fully left
handed polarised positron beam. 


For the analysis presented here, two independent data samples with
right and left handed polarised positrons are used. For convenience,
they are named hereafter as the $R$ and $L$ data sets, respectively.
The $R$ data set has a mean polarisation of $33.0\pm2.0\,\%$ and an
integrated luminosity of $15.3\pm0.4$\,pb$^{-1}$. The corresponding
numbers for the $L$ data set are $-40.2\pm1.5\,\%$ and
$21.7\pm0.6$\,pb$^{-1}$. In both data sets the incident positron beam energy
is $27.6$\,GeV, whilst the unpolarised proton beam energy is
$920$\,GeV. This yields a center-of-mass energy of $\sqrt{s}=319$\,GeV.
The measured cross sections are defined in terms of two of the kinematic
variables $Q^2$, $x$, and $y$, where $y$ quantifies the inelasticity
of the interaction.

The leading order double differential CC cross section
for $e^+p \rightarrow \bar{\nu} X$ can be written as
\begin{equation}
\frac{{\rm d}^2\sigma_{CC}}{{\rm d}x\;{\rm d}\QQ} = (1+P_e)
 \frac{G_F^2}{2\pi x } \left(\frac{M_W^2}{M_W^2+Q^2} \right)^2
x \left [ (\bar{u}+\bar{c})+(1-y)^2(d+s) \right ]\;\;\;,
\label{Scc}
\end{equation}
where $G_F$ is the Fermi coupling constant, $M_W$ is the mass of the
$W$ boson, and $\bar{u}$, $\bar{c}$, $d$, $s$ are quark distribution
functions of the proton.  $P_e$ is the degree of longitudinal
polarisation and is defined as $P_e=(N_R-N_L)/(N_R+N_L)$ with $N_R$,
and $N_L$ the number of right and left handed positrons in the beam.
It can be seen that the cross section has a linear dependence on
the polarisation of the lepton beam. The cross section for right
handed ($+$) positrons is enhanced, whilst the cross section for left
handed ($-$) positrons is suppressed. For $P_e\,=\,-1$ the cross
section is identically zero in the Standard Model.

\section{Experimental Technique}
 
The H1 detector components most relevant to this analysis are the LAr
calorimeter, which measures the positions and energies of charged and
neutral particles over the polar\footnote{The polar angle $\theta$ is
  defined with respect to the positive $z$ axis. The forward direction
  is the region of increasing $z$ and the direction of the incident
  proton beam.} angular range $4^\circ<\theta<154^\circ$, and the
inner tracking detectors which measure the angles and momenta of
charged particles over the range $7^\circ<\theta<165^\circ$. A full
description of the detector can be found in~\cite{h1det}.

At HERA transverse polarisation of the positron beam arises naturally
through synchrotron radiation via the Sokolov-Ternov
effect~\cite{spin}. In 2000 pairs of spin rotators were installed in
the beamline around the H1 detector allowing transversely polarised
positrons to become longitudinally polarised. The polarisation is
continuously measured using two independent polarimeters measuring the
transverse polarisation: TPOL~\cite{tpol} and the longitudinal
polarisation LPOL~\cite{lpol}. The luminosity weighted
polarisation distribution is shown in figure~\ref{pol} for the data
and is reproduced in the Monte Carlo simulation.

In order to determine acceptance corrections and background
contributions for the DIS cross section measurements, the detector
response to events produced by various Monte Carlo generation programs
is simulated in detail, and is described in ~\cite{h1hiq2}.

The selection and analysis of deep inelastic scattering (DIS)
processes with charged current (CC) interactions follows closely that
of published unpolarised data~\cite{h1hiq2} and is briefly described
below.  The CC events are identified as having missing
transverse momentum, $P_{T,h}$, where
$P_{T,h}=\sqrt{(\sum_i{p_{x,i}})^2+(\sum_i{p_{y,i}})^2}$ and is summed
over all particles of the hadronic final state. 

The CC kinematic quantities can only be determined with the hadron method
($h$ method)~\cite{jb} using the relations
\begin{equation}
   y_{h} = \frac{\Sigma}{ 2 \ E_e }
   \hspace*{2cm}
   Q^2_{h} = \frac{P_{T,h}^2}{ 1-y_{h}}
   \hspace*{2cm}
    x_h=\frac{Q^2_h} {s \ y_h}\;\;\;,
\end{equation}
This method is influenced by particle losses in the beam pipe and
fluctuations of the detector response to hadronic final state
particles, and therefore has moderate precision.  

NC interactions are also studied and provide an accurate and high
statistics data sample with which to check the detector response.  The
selection of NC interactions is based mainly on the requirement of an
identified scattered positron in the liquid argon (LAr) calorimeter
with an energy $E^\prime_e>11$\,GeV.  The NC sample is used to carry
out an {\it in-situ} calibration of the energy scale of hadronic final
state particles using the analysis described in~\cite{h1hiq2}. The
calibration procedure is based on the balance of the transverse energy
of the positrons with that of the hadronic final state.  The procedure
is found to give good agreement between data and simulation. 

In addition NC events are used for studies of systematic uncertainties
in the charged current analysis.  The data are processed such that all
information from the scattered positron is suppressed, the so-called
{\it pseudo-CC} sample. This sample then mimics CC interactions
allowing all efficiencies to be checked with high precision.

\section{Measurement Procedure}

Candidate CC interactions are selected by requiring
\mbox{$P_{T,h}>12$\,GeV}. In order to ensure high efficiency of the
trigger and kinematic resolution the analysis is futher restricted to
the domain of $Q^2_h\gtrsim 200$\,GeV$^2$ and $0.03<y_h<0.85$.
Finally, non-$ep$ background is rejected by searching for typical
cosmic ray and beam-induced background event topologies.

The good understanding of the detector can be seen in figures
\ref{rhnc},\ref{rhcc},\ref{lhnc} and~\ref{lhcc}. In figure~\ref{rhnc}
the NC $R$ sample is compared to simulation and demonstrates the
electromagnetic calibration of the detector. Figure~\ref{rhcc} shows
relevant distributions of the CC $R$ sample compared to simulation.
The contribution of background photoproduction processes is also shown
and can be seen to have a large influence at low $Q^2$.
Figure~\ref{lhnc} shows the same NC distributions for the $L$ sample.
The simulation provides a good description of the data. In particular
fig.~\ref{lhnc}d shows the ratio of transverse momentum of the
scattered positron to that of the hadronic final state. This
distribution is described by the simulation within the systematic
uncertainty assigned to the hadronic calibration. The details of the
analysis are not identical for the measurements of the right and left
handed cross sections, in particular the supression of photoproduction
background.

\section{Results}

The integrated polarised CC cross sections are measured in the range $Q^2
> 400$\,GeV$^2$ and $y<0.9$. The values are:

\begin{eqnarray*}
\label{result}
\sigma_{CC} (P_e=         +0.33) & = & 34.67\,{\rm pb} \pm 1.94\,{\rm pb \,\,(stat)}  \pm 1.66\,{\rm pb \,\,(sys)}\\
\sigma_{CC} (P_e=         -0.40) & = & 13.80\,{\rm pb} \pm 1.04\,{\rm pb \,\,(stat)}  \pm 0.94\,{\rm pb \,\,(sys)}
\end{eqnarray*}

The measurement of the unpolarised CC cross section based on the
HERA-I data set~\cite{h1hiq2} with a luminosity of $65.2$~pb$^{-1}$ and
measured in the same phase space region is:

\begin{eqnarray*}
\label{resultunpol}
\sigma_{CC} (P_e=\,\,\,\,\,0.00) & = & 28.44\,{\rm pb} \pm 0.77\,{\rm pb \,\,(stat)}  \pm 1.22\,{\rm pb \,\,(sys)}
\end{eqnarray*}

These are to be compared to Standard Model expectations using the
H1PDF 2000 fit which yields \mbox{$\sigma_{CC} = 34.91$\,pb} for
$P_e=+0.33$ , \mbox{$\sigma_{CC} = 26.25$\,pb} for $P_e=0.00$, and
\mbox{$\sigma_{CC} = 15.70$\,pb} for $P_e=-0.40$. The measurements are
shown in fig.~\ref{xsec} compared to the Standard Model expectations.

A linear fit to the polarisation dependence of the measured cross
sections was performed. The fit took into account the correlated
systematics between the measurements and is also shown in
fig.~\ref{xsec}. The fit provides a reasonable description of the data with
a $\chi^2=2.45$. The fit extrapolated to the point $P_e=-1.0$ yields a cross section of:

\begin{eqnarray*}
\label{result}
\sigma_{CC} (P_e=-1.00) & = & -3.7\,{\rm pb} \pm 2.4\,{\rm pb \,\,(stat)}  \pm 2.7\,{\rm pb \,\,(sys)}
\end{eqnarray*}

This extrapolated cross section expectation is consistent with the
Standard Model prediction of a zero cross section.

\newpage
\begin{thebibliography}{99}

\bibitem {klein}
%\cite{Klein:vs}
M.~Klein and T.~Riemann,
%``Electroweak Interactions Probing The Nucleon Structure,''
Z.\ Phys.\ C {\bf 24} (1984) 151.
%%CITATION = ZEPYA,C24,151;%%

\bibitem{h1hiq2}
C.~Adloff {\it et al.}  [H1 Collaboration],
%``Measurement and QCD Analysis of Neutral and Charged Current Cross Sections 
% at HERA''
Eur.\ Phys.\ J.\ C {\bf 30} (2003) 1-32
[hep-ex/0304003].
%%CITATION = HEP-EX 0304003;%%
\vspace{-2mm}

\bibitem{h1det}
% H1 Collaboration, I.~Abt {\it et al.}, Nucl.\ Instr.\ Meth.\ A386 (1997)
% 310 and 348.  -> No, 310 & 397
I.~Abt {\it et al.}  [H1 Collaboration],
%``The H1 detector at HERA,''
Nucl.\ Instrum.\ Meth.\ A {\bf 386} (1997) 310 and 348; \\
%%CITATION = NUIMA,A386,310;%%
%
R.~D.~Appuhn {\it et al.}  [H1 SPACAL Group Collaboration],
%``The H1 lead/scintillating-fibre calorimeter,''
Nucl.\ Instrum.\ Meth.\ A {\bf 386} (1997) 397.
%%CITATION = NUIMA,A386,397;%%

\bibitem{spin}
 A.A.~Sokolov and I.M.~Ternov, Sov.\ Phys.\ Dokl.\ {\bf 8} No. 12 (1964) 1203.;
 
\bibitem{tpol}
 D.B.~Barber {\it et al.}, Nucl.\ Instrum.\ Meth.\ A {\bf 329} (1993) 79.;
 
\bibitem{lpol}
 M.~Beckmann {\it et al.}, Nucl.\ Instrum.\ Meth.\ A {\bf 479} (2002) 334.;

\bibitem{jb}
A.~Blondel and F.~Jacquet, Proceedings of the Study of an $ep$ Facility for 
Europe, ed.\ U.~Amaldi, DESY 79/48 (1979) 391.


\end{thebibliography}

\newpage
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\begin{figure}[htb]
\begin{center}
\begin{picture}(90,90)(0,0)
\setlength{\unitlength}{1 mm}
\put(-30,  -5){\epsfig{file=PolHist.eps,width=\textwidth}}
\end{picture}
\end{center}
\caption{Distribution of luminosity versus polarisation $P_e$}
\label{pol}
\end{figure}

\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\begin{picture}(90,90)(0,0)
\setlength{\unitlength}{1 mm}
\put(-20,  -5){\epsfig{file=rhnc.ps,bbllx=547,bblly=47,bburx=767,bbury= 17,width= 4.1cm,angle=-90}}
\put(-05,92){\bf (a)}
\put( 63,92){\bf (b)}
\put(-05,41){\bf (c)}
\put( 63,41){\bf (d)}
\end{picture}
\end{center}
\caption{
  Distributions of $E_e^{\prime}$ (a), $\theta_e$ (b), $Q^2$ (c) and 
  $E-p_z$ (d) of the NC $R$ data set.}
\label{rhnc}
\end{figure}

\begin{figure}[htb]
\begin{center}
\begin{picture}(90,90)(0,0)
\setlength{\unitlength}{1 mm}
\put(-20,-5){\epsfig{file=rhcc.ps,bbllx=547,bblly=47,bburx=767,bbury= 17,width=4.1cm,angle=-90}}
\put(-05,92){\bf (a)}
\put( 63,92){\bf (b)}
\put(-05,41){\bf (c)}
\put( 63,41){\bf (d)}
\end{picture}
\end{center}
\caption{
  Distributions of $Q^2_h$ (a), $P_{T,h}$ (b), $E_h-p_{z,h}$ (c) and 
  $x_h$ (d) of the CC $R$ data set.}
\label{rhcc}
\end{figure}
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\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\begin{picture}(90,90)(0,0)
\setlength{\unitlength}{1 mm}
\put(-20,  -5){\epsfig{file=lhnc.ps,bbllx=547,bblly=47,bburx=767,bbury= 17,width= 4.1cm,angle=-90}}
\put(-05,92){\bf (a)}
\put( 63,92){\bf (b)}
\put(-05,41){\bf (c)}
\put( 63,41){\bf (d)}
\end{picture}
\end{center}
\caption{
  Distributions of $E_e^{\prime}$ (a), $\theta_e$ (b), $Q^2$ (c) and 
  $P_{T,h}/P_{T,e}$ (d) of the NC $L$ data set.}
\label{lhnc}
\end{figure}

\begin{figure}[htb]
\begin{center}
\begin{picture}(90,90)(0,0)
\setlength{\unitlength}{1 mm}
\put(-20,-5){\epsfig{file=lhcc.ps,bbllx=547,bblly=47,bburx=767,bbury= 17,width=4.1cm,angle=-90}}
\put(-05,92){\bf (a)}
\put( 63,92){\bf (b)}
\put(-05,41){\bf (c)}
\put( 63,41){\bf (d)}
\end{picture}
\end{center}
\caption{
  Distributions of $Q^2_h$ (a), $P_{T,h}$ (b), $E_h-p_{z,h}$ (c) and 
  $x_h$ (d) of the CC $L$ data set.}
\label{lhcc}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newpage
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\begin{figure}[htb]
\begin{center}
\begin{picture}(90,150)(0,0)
\setlength{\unitlength}{1 mm}
\put(-40,  0){\epsfig{file=TotXsection_DataFitH1PDF.eps,width=\textwidth}}
\end{picture}
\end{center}
\caption{
  The dependence of the $e^+p$ CC cross section with the lepton beam
  polarisation $P_e$ is shown. The data are compared to the prediction
  from the H1 PDF 2000 fit (full line). The dashed line shows the
  result of a linear fit to the data with $\chi^2 / n.d.f = 2.45$.  }
\label{xsec}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{document}
