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\begin{titlepage}

\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em] 
\begin{small}
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%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
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\multicolumn{4}{l}{{\bf
                33rd International Conference on High Energy Physics, ICHEP06},
                July.~26-Aug.2,~2006,~Moscow} \\
                 & Abstract:        & {\bf }    &\\
                 & Parallel Session & {\bf Electroweak}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
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\vspace*{2cm}

\begin{center}
  \Large
  {\bf 
Neutral Current Interactions in $e^{\pm}p$ Scattering with Longitudinally Polarised Leptons}

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

\begin{abstract}

\noindent

The inclusive single differential cross section ${\rm d}\sigma/{\rm
d}Q^2$ and the reduced double differential cross section
$\tilde{\sigma}(x,Q^2)$ are presented for the neutral current process,
$e^{\pm}p\rightarrow e^{\pm} X$, in interactions with longitudinally
polarised lepton beams. The cross sections are measured in the region
of large negative four-momentum transfer squared, $Q^2 \geq 200\,{\rm
GeV}^2$ and inelasticity, $y<0.9$. The data are consistent with the
expected $Q^2$ dependence of polarised cross sections, albeit with the
limited statistical precision of the data. The reduced cross section
data are combined together with previously published unpolarised data
to determine the structure function $x\tilde{F}_3$ with
improved precision. The data are compared to predictions of the
Standard Model which is able to provide a good description of the data.

\noindent
\end{abstract}


\end{titlepage}

\pagestyle{plain}

\section{Introduction} % 

HERA measurements of proton structure in neutral current (NC) deep
inelastic scattering (DIS) with polarised lepton beams are crucial to
the understanding of the detailed dynamics of QCD as well as allowing
the chiral structure of electroweak interactions to be simultaneously
probed at the highest energies. Previously published
measurements~\cite{h1lowq2,h19497,h19899,h1hiq2} have already provided
unique constraints~\cite{h1lowq2,h1hiq2,cteq,mrst} on the parton
densities functions (PDFs) of the proton as well as the axial and
vector couplings of the light quarks to the $Z^0$
boson~\cite{ewpaper}.

First neutral current (NC) cross section measurements of data taken in
the second phase of HERA operation with longitudinally polarised
leptons beams have recently been reported~\cite{h1dis06} for both
electron and positron scattering data and are presented here in
detail. 

In this paper first measurements of the inclusive NC single
differential cross sections, ${\rm d}\sigma/{\rm d}Q^2$ and the
reduced cross section $\tilde{\sigma}(x,Q^2)$ are presented for $e^+p$
and $e^-p$ scattering. The data were taken with an incident lepton
beam energy of $27.5\,{\rm GeV}$, whilst the unpolarised proton beam
energy was $920\,{\rm GeV}$, yielding a centre-of-mass energy of
$\sqrt{s}=318\,{\rm GeV}$. Both data sets are further subdivided into
samples of left handed and right handed longitudinal polarisation,
$P_e=(N_R-N_L)/(N_R+N_L)$, where $N_R$ ($N_L$) is the number of right
(left) handed leptons in the beam. The corresponding data sets are
termed the $R$ and $L$ data sets respectively. The corresponding
luminosity and longitudinal lepton beam polarisation is given in
Tab.~\ref{lumi} below.
\begin{table}[h]
  \begin{center}
    \begin{tabular}{|r|c|c|}
\hline
 & $R$ & $L$\\
\hline
$e^+p$ & $\mathcal{L}=26.9$~pb$^{-1}$ & $\mathcal{L}=20.7$~pb$^{-1}$ \\
       & $P_e=(+33.6\pm 0.6)\%$ & $P_e=(-40.2\pm 1.1)\%$ \\
\hline
$e^-p$ & $\mathcal{L}=29.6$~pb$^{-1}$ & $\mathcal{L}=68.6$~pb$^{-1}$ \\
       & $P_e=(+37.0\pm 1.8)\%$ & $P_e=(-27.0\pm 1.3)\%$ \\
\hline
\end{tabular} 
\caption{ 
Table of luminosities, $\mathcal{L}$, and luminosity weighted
longitudinal polarisation, $P_e$ for the data sets presented here.}
\label{lumi}
\end{center}
\end{table}

\section{The Neutral Current Cross Section}
%
%The measured double differential CC cross section for collisions of polarised 
%electrons with unpolarised protons, corrected for QED radiative
%effects, may be expressed as
%
%\vspace{-6mm}

The generalised proton structure functions, $\tilde{F}_{2,3}$, may be
written as linear combinations of the hadronic structure functions
$F_{2}$, $F_{2,3}^{\gamma Z}$, and $F_{2,3}^{Z}$ containing
information on QCD parton dynamics as well as the EW couplings of the
quarks to the neutral vector bosons. The function $F_{2}$ is
associated to pure photon exchange terms, $F_{2,3}^{\gamma Z}$
correspond to photon-$Z^0$ interference and $F_{2,3}^{Z}$ correspond
to the pure $Z^0$ exchange terms. In addition the longitudinal
structure function $\tilde{F}_L$ may be similarly decomposed, however
this is important only at high $y$ and is expected to be negligible at
large $x$ (the familiar Bjorken $x$ variable) and $Q^2$. Neglecting
$\tilde{F}_{L}$, the linear combinations for arbitrarily polarised
$e^{\pm}p$ scattering are given by
\begin{eqnarray}
 \tilde{F}^{\pm}_2 = F_2 - (v_e \pm P_e a_e) \kappa  \frac{Q^2}{Q^2+M_Z^2}    F_2^{\gamma Z} 
            + (v_e^2+a_e^2 \pm P_e 2v_e a_e) \kappa^2 \left[\frac{Q^2}{Q^2+M_Z^2}\right]^2 F_2^Z \\
 x\tilde{F}^{\pm}_3 = -(a_e \pm P_e v_e)   \kappa    \frac{Q^2}{Q^2+M_Z^2}    xF_3^{\gamma Z} 
       + (2a_ev_e \pm P_e [v_e^2 + a_e^2] ) \kappa^2 \left[\frac{Q^2}{Q^2+M_Z^2}\right]^2 xF_3^Z 
\label{SF1}
\end{eqnarray}

Here $\kappa^{-1}=4\frac{M_W^2}{M_Z^2}(1-\frac{M_W^2}{M_Z^2})$
in the on-mass-shell scheme~\cite{oms} with $M_W$ and $M_Z$ the usual
weak vector boson masses. 
%The above expressions are related to the
%corresponding expressions for polarised positron scattering under the
%transformations $\tilde{F}_2^{L,R}(e^-p) \rightarrow
%\tilde{F}_2^{R,L}(e^+p)$ and 
%$x\tilde{F}_3^{L,R}(e^-p) \rightarrow x\tilde{F}_3^{R,L}(e^+p)$. For
%lepton beams with any polarisation the appropriate average of
%purely $L$ and $R$ structure functions is taken {\bf JOBBY!!!!}. 
For the first time
HERA is sensitive to the parity violating terms of the Standard Model,
namely those proportional to $P_e$ in eqs.~\ref{SF1} above.

In the quark parton model, the hadronic structure functions are
related to linear combinations of sums and differences of the quark
and anti-quark momentum distributions $xq(x,Q^2)$ and
$x\bar{q}(x,Q^2)$
\begin{eqnarray}
  \left[F_2,F_2^{\gamma Z},F_2^Z\right]  = x\sum_q [e_q^2,2e_qv_q,v_q^2+a_q^2](q+\bar{q}) \\
  \left[xF_3^{\gamma Z},xF3^Z \right] = 2x \sum_q[e_qa_q,v_qa_q](q-\bar{q})\,\,\,,
\label{SF2} 
\end{eqnarray} 
where $v_q$ and $a_q$ are the vector and axial-vector couplings of the light
quarks and $e_q$ is the charge of the quark of flavour $q$.

The differential cross section for $e^{\pm}$ scattering after
correction for QED radiative effects is then given by
\begin{equation}
\frac{\rm{d}^2\sigma^{\pm}_{NC}}{{\rm d}x{\rm d}Q^2}=
\frac{2\pi\alpha^2}{xQ^4}(Y_+\tilde{F}_2{\mp}Y_-x\tilde{F}_3-y^2\tilde{F}_L) \,\,\,,
\label{ncxsec} 
\end{equation} 
and the reduced cross section is defined by
\begin{equation}
\tilde{\sigma}^{\pm}(x,Q^2)\equiv
\frac{\rm{d}^2\sigma^{\pm}_{NC}}{{\rm d}x{\rm d}Q^2}\frac{xQ^4}{2\pi\alpha^2}\frac{1}{Y_+}\equiv
\tilde{F}_2{\mp}\frac{Y_-}{Y_+}x\tilde{F}_3-\frac{y^2}{Y_+}\tilde{F}_L \,\,\,.
\label{rxsec} 
\end{equation} 
The helicity dependence of the electroweak interactions is contained
within the terms $Y_{\pm}=1\pm(1-y^2)$ where $y$ characterises the
inelasticity of the interaction.

\section{Experimental Technique}
 
The H1 detector components most relevant to this analysis are the
liquid argon (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 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}.

The $ep$ luminosity is determined by measuring the QED bremsstrahlung
($ep\rightarrow ep\gamma$) event rate by tagging the low angle
scattered lepton in a detector located at $z=-44$m adjacent to the
beam pipe.

At HERA transverse polarisation of the lepton beam arises naturally
through synchrotron radiation via the Sokolov-Ternov
effect~\cite{spin}. In $2000$ a pair of spin rotators was installed in
the beamline on either side of the H1 detector, allowing transversely
polarised leptons to be rotated into longitudinally polarised states
and back again. The degree of polarisation is constant around the HERA
ring and is continuously measured using two independent polarimeters
LPOL~\cite{lpol} and TPOL~\cite{tpol}.  The polarimeters are situated
in beamline sections in which the beam leptons have longitudinal and
transverse polarisations respectively.  Both measurements rely on an
asymmetry in the energy spectrum of left and right handed circularly
polarised photons undergoing Compton scattering with the lepton
beam. The TPOL measurement uses in addition a spatial asymmetry. The
LPOL polarimeter measurements are used when available and TPOL
measurements otherwise. The polarisation profile weighted by the
luminosity values is shown in Fig.~\ref{profile_pol}(a) and (b) for the
$e^+$ and $e^-$ samples respectively.

Simulated DIS events are used in order to determine acceptance corrections.
DIS processes are generated using the DJANGO~\cite{django} Monte Carlo (MC)
simulation program, which is based on LEPTO~\cite{lepto} for 
the hard interaction and HERACLES~\cite{heracles} for single photon
emission and virtual EW corrections.
LEPTO combines ${\cal O}(\alpha_s)$ matrix elements with higher order QCD
effects using the colour dipole model as implemented in ARIADNE~\cite{cdm}.
The JETSET program~\cite{jetset} is used to simulate 
the hadronisation process.  
In the event generation the DIS cross section is calculated using the 
H1 PDF $2000$~\cite{h1hiq2} parametrisation for the proton PDFs.

The dominant $ep$ background contribution arises from photoproduction
processes. These are simulated using the PYTHIA~\cite{pythia} MC with
leading order PDFs for the proton taken from CTEQ~\cite{cteq5}
and for the photon from GRV~\cite{ggrv}. Further backgrounds from NC
DIS, QED-Compton scattering, lepton pair production, prompt photon
production and heavy gauge boson ($W^{\pm},Z^0$) production are also
simulated; their final contribution to the analysis sample is small. Further
details are given in~\cite{h1hiq2}.

The detector response to events produced by the generation
programs is simulated in detail using a program based on
GEANT~\cite{GEANT}. These simulated events are then subjected to the
same reconstruction and analysis chain as the data.

The selection of candidate NC events follows closely that of the
previously published analysis of unpolarised data from
H1~\cite{h1hiq2} and is briefly described below.

NC events are characterised by an isolated high transverse momentum
lepton and a hadronic system opposite in azimuth to the scattered
lepton. Such events are triggered mainly using information from the
LAr calorimeter. The calorimeter has finely segmented geometry
allowing the selection of localised energy deposits in the
electromagnetic sections of the calorimeter. This selection is found
to be $100$\% efficient for leptons above $11$~GeV.

The hadronic final state particles are reconstructed using a
combination of tracks and calorimeter deposits in an energy flow
algorithm that avoids double counting~\cite{bpthesis,andrei}.

The NC sample is used to carry out an {\it in-situ} calibration of the
electromagnetic and hadronic energy scales of the LAr calorimeter
using the method described in~\cite{h1hiq2, bpthesis, andrei}. The
hadronic calibration procedure is based on the double angle
reconstruction method~\cite{damethod} which uses the polar angles of
the scattered lepton and the hadronic final state to determine their
respective energies. The calibration procedure gives good agreement
between data and simulation within an estimated uncertainty of $2\%$
on the hadronic scale and between $1-3$\% on the electromagnetic
scale.

The NC kinematic quantities are determined using the $e\Sigma$
method~\cite{esigma} which uses information from the scattered lepton
and the hadronic final state optimising kinematic resolution and
minimising QED radiative corrections across the measured kinematic
plane.

\section{Measurement Procedure}

Candidate NC interactions are selected by requiring the scattered
lepton energy $E^{\prime}_e>11\,{\rm GeV}$, $Q^2_e > 133$~GeV$^2$ and
a reconstructed vertex within $35\,{\rm cm}$ in $z$ of the nominal
interaction point. Background from misidentified leptons is suppressed
by requiring a charged track to be associated to the lepton
candidate. Longitudinal energy-momentum conservation requires that
$E-P_z=2E_e$ where $E_e$ is the lepton beam energy and $E$ and $P_z$
are the usual components of the hadronic final state four-vector. By
requiring $E-P_z>35$~GeV the influence of radiative corrections is
considerably reduced which arises from initial state
bremmsstrahlung. This requirement further suppresses the contamination
from photoproduction background in which the the scattered lepton is
undetected in the backward beam pipe and a hadron is misidentified as a
lepton candidate.

The comparison of the data and the simulation (normalised to the
luminosity of the data) is shown in Fig.~\ref{nc} for the distribution
of scattered lepton energy $E_e^{\prime}$, the lepton polar angle
$\theta_e$, the reconstructed $Q^2_e$, and the transverse momentum
balance between the lepton and the hadronic final state,
$P_T^h/P_T^e$. In all cases the simulation provides an accurate
description of the data and is used to correct for selection and
resolution effects.

The systematic uncertainties on the cross section measurements are
discussed briefly below (see \cite{bpthesis,andrei} and
references therein for more details). Positive and negative
variations of one standard deviation of each error source are found to
yield errors which are symmetric to a good approximation. The
systematic uncertainties of each source are taken to be fully
uncorrelated between the cross section measurements unless stated
otherwise.

\begin{itemize}

\item An uncertainty of $1\%~(z_{imp}<+20{\rm cm})$, $2\%~(+20{\rm
  cm}<z_{imp}<+100{\rm cm})$ and $3\%$ for $z_{imp}>+100{\rm cm})$ is
  assigned to the scale of the electromagnetic energy measured in the
  LAr calorimeter, where $z_{imp}$ is the z co-ordinate of the impact
  position of the scattered lepton in the LAr calorimeter. $1\%$ of
  this uncertainty is considered as a correlated component.
    
\item An uncertainty of $0.5\%$ for $\theta_e<100^{\circ}$ and $2\%$ for
  $\theta_e>100^{\circ}$ in the identification of the scattered
  electron or positron.

\item An uncertainty in the polar angle measurement of the scattered
  lepton is taken to be $3$~mrad.

\item An uncertainty of $2\%$ is assigned to the scale of the hadronic energy
  measured in the detector, of which $1\%$ is considered as a
  correlated component to the uncertainty. 
    
\item A $10\%$ correlated uncertainty is assigned to the amount of energy
  in the LAr calorimeter attributed to noise.
  
\item A $1$\% uncertainty on the trigger efficiency.
  
\item A $30\%$ correlated uncertainty 
  on the subtracted $ep$ background is determined from a comparison of
  data and simulation after relaxing the anti-photoproduction cuts,
  such that the sample is dominated by photoproduction events. 
  
\item The combined uncertainty of finding an interaction vertex and
  associating a track to the scattered lepton is estimated to be
  $3\%$ for the $e^+p$ data and $2\%$ for the $e^-p$ data uncorrelated
  from point to point.

\item An uncertainty of $0.5\%$ accounts for the
  dependence of the acceptance correction on the PDFs used in the 
  MC simulation.

\item An error of $1\%$ is estimated for
  the QED radiative corrections. This accounts for missing
  contributions in the simulation of the lowest order QED effects and
  for the uncertainty on the higher order QED and EW corrections.

\item An uncertainty of $z$ dependence of the proton bunch
  structure is estimated to be $0.5\%$.

\item For the $e^+p$ data the relative uncertainty in the measurement of the lepton
  beam polarisation is taken to be $1.6\%$ for the LPOL and $3.5\%$
  for the TPOL~\cite{lorenzon}, yielding a total relative polarisation uncertainty of
  $1.8\%$ for the $R$ data set and $2.7\%$ for the $L$ data set. For
  the $e^-p$ data a global relative uncertainty of $5\%$ is considered. These
  uncertainties are not included in the error bars of the data points.

\item In addition, there is a global uncertainty of $2\%$
  on the luminosity measurement for both the $R$ and $L$ $e^-p$ data
  samples, and $1.3\%$ for both $e^+p$ data samples. A correlated
  component of this uncertainty is estimated to be $0.5\%$.

\end{itemize}
%
The total systematic error is formed by adding the individual
uncertainties in quadrature.

\section{Results}

The measured cross sections $\frac{{\rm d}\sigma}{{\rm d}Q^2}$ for
$y<0.9$ are shown in Figs.~\ref{dsdq2pos} and~\ref{dsdq2ele} for $e^+p$ and $e^-p$
scattering respectively. The data are measured in the range $Q^2\geq
200$~GeV$^2$ up to $Q^2\simeq 20\,000$~GeV$^2$ over which the cross sections
fall by almost six orders of magnitude with increasing $Q^2$. The
cross sections are found to be similar at low $Q^2$ for both lepton
charges and polarisations, however at the highest $Q^2$ differences
are observed between the $e^+p$ and $e^-p$ cross sections arising from
the structure function $x{\tilde F}_3$.

The measured data are compared to the Standard Model expectation from
the H1 PDF 2000 fit~\cite{h1hiq2} which provides an excellent
description of the $Q^2$ dependence of the data including
the difference between the $e^+p$ and $e^-p$ cross sections.

The Standard Model also predicts a difference in the cross section for
leptons with different helicity states arising from the chiral
structure of the neutral electroweak exchange. In
Fig.~\ref{dsdq2ratio} the ratio of cross sections for right polarised
to left polarised lepton beams is shown separately for $e^+p$ and
$e^-p$ scattering data. In both cases the ratio is found to be
consistent, within experimental uncertainty, with unity at low $Q^2$,
indicating little dependence of the cross section on beam
polarisation. Note the normalisation uncertainties of the measurements
are not included in the errors bars, but are indicated by the dashed
lines on the figure. At higher $Q^2$ the data have a tendency to
deviate from unity. For positron scattering the data indicate that
right handed positrons yield a larger cross section than left handed
positrons, whereas for electron scattering the data indicate the
opposite behaviour. This behaviour is consistent with the Standard
Model expectation shown as the solid curve in
Fig.~\ref{dsdq2ratio}.

The influence of lepton beam polarisation on the measured cross
sections can be enhanced by combining the $e^+p$ and $e^-p$ scattering
data in the ratio ${\rm R}$ defined as
\begin{equation}
{\rm R}=\frac
{{\rm d}\sigma/{\rm d}Q^2(e^+p,P_e>0) + {\rm d}\sigma/{\rm d}Q^2(e^-p,P_e<0)}
{{\rm d}\sigma/{\rm d}Q^2(e^+p,P_e<0) + {\rm d}\sigma/{\rm d}Q^2(e^-p,P_e>0)} \,\,\,.
\label{Rratio} 
\end{equation} 
For fixed polarisation and at fixed $x$ and $Q^2$ this is approximately given by
\begin{equation}
{\rm R} \simeq \frac
%{F_2-P_e^{\prime} a_e\kappa\frac{Q^2}{Q^2+M_Z^2}F_2^{\gamma Z}}
%{F_2-P_e^{\prime\prime} a_e\kappa\frac{Q^2}{Q^2+M_Z^2}F_2^{\gamma Z}} \,\,\,,
{F_2-P_e a_e\kappa\frac{Q^2}{Q^2+M_Z^2}F_2^{\gamma Z}}
{F_2-P_e a_e\kappa\frac{Q^2}{Q^2+M_Z^2}F_2^{\gamma Z}} \,\,\,,
\label{asymmetry} 
\end{equation} 
%where $P_e^{\prime}$ and $P_e^{\prime\prime}$ are the different
%polarisations given in Tab.~\ref{lumi} for the data sets considered.
The ratio ${\rm R}$ is shown in Fig.~\ref{Rratio}. At low $Q^2$ ${\rm R}$ is found
to be consistent with unity and deviations from this behaviour are
observed with increasing $Q^2$. The data are consistent with the
Standard Model expectation shown as the solid curve albeit within
the moderate precision of this data.

The NC polarised data maybe combined into an effective unpolarised
$e^+p$ data set and an effective unpolarised $e^-p$ data set after
correcting for the small residual polarisations. The unpolarised
reduced cross sections are shown for $e^{\pm}p$ scattering in
Fig.~\ref{xf3}(a) and demonstrate a clear suppression of the $e^+p$
cross section with respect to the $e^-p$ data. The data compare well
to the Standard Model prediction from the H1 PDF 2000 fit in which the
observed difference arises from the generalised structure function
$x{\tilde F}_3$ (see eq.~\ref{rxsec}). Thus $x{\tilde F}_3$ may be
obtained from
\begin{eqnarray}
x{\tilde F}_3 = \frac{Y_+}{2Y_-}\left[\tilde{\sigma}^-(x,Q^2)-\tilde{\sigma}^+(x,Q^2)\right]\,\,\,.
\label{extractxf3} 
\end{eqnarray} 
In order to improve the statistical precision the cross section data
presented here are combined with the previously
published~\cite{h1hiq2} unpolarised NC reduced cross sections in order
to determine $x{\tilde F}_3$. The combined HERA-I and HERA-II data
are shown in Fig.~\ref{xf3}(b) for the region of $x$ and $Q^2$ where
the expected sensitivity is larger than the normalisation uncertainty
of the data. In the context of the Standard Model it can be seen from
eq.~\ref{SF1} that the dominant contribution to $x{\tilde F}_3$ arises
from the $\gamma Z$ interference term, since the pure $Z$ exchange
term is suppresed by an additional factor of $\kappa Q^2/(Q^2+M_Z^2)$
and, in the case of unpolarised scattering, the small vector coupling
$v_e$. Thus $x{\tilde F}_3^{\gamma Z}$ may be determined by
\begin{equation}
xF^{\gamma Z}_3 \simeq x{\tilde F}_3 \frac{(Q^2+M_Z^2)}{a_e \kappa Q^2}
\label{extractxf3gz} 
\end{equation} 
neglecting terms proportional to $v_e$. The data are shown in
Fig.~\ref{xf3gz}(a) for three $Q^2$ values and compared to the
expectation. The weak $Q^2$ dependence of this non-singlet structure
function is also shown from the H1 PDF 2000 fit. Since the dependence
is weak, the data are transformed to one $Q^2$ value at $1\,500$
GeV$^2$ and then averaged. The combination of extracted $xF^{\gamma
Z}_3$ data for all $Q^2$ values is shown in Fig.~\ref{xf3gz}(b) and is
directly sensitive to the valence quark distributions. The Standard
Model prediction is also shown and found to be in excellent agreement
in both shape and magnitude with the data.

In leading order QCD a sum rule~\cite{rizvi} predicts that
\begin{equation}
\int_0^1{xF_3^{\gamma Z} \frac{dx}{x}} = \frac{1}{3} \int_0^1{(2u_v + d_v)dx} = 
\frac{5}{3}.
\end{equation}
Higher order corrections to this are expected to be of order
$\alpha_S/\pi$.  The structure function is measured over the $x$ range
$0.02$ to $0.65$ and the integral of $F_3^{\gamma Z}$ is measured to
be
\begin{equation}
\int_{0.02}^{0.65}{F_3^{\gamma Z}dx}=1.31 \pm 0.15 (stat) \pm 0.11 (syst)
\end{equation}
which is consistent with the results of the H1 PDF 2000 fit, $1.12 \pm
0.02$ and for the same $x$ interval at $Q^2 = 1500$\,GeV$^2$.

\section{Summary}
The NC cross sections ${\rm d}\sigma/{\rm d}Q^2$ have been measured
for $e^{\pm}p$ scattering with longitudinally polarised lepton beams
at HERA for the first time. The data are consistent with the predicted
behaviour of the polarised $ep$ scattering in the Standard Model with
the current limited statistical precision of the measurements.

The double differential NC cross sections ${\tilde
\sigma}^{\pm}(x,Q^2)$ have been measured at high $Q^2$ and an
extraction of the structure functions $x{\tilde F}_3$ and
$xF_3^{\gamma Z}$ has been performed. The data are in good agreement
with expectation of the Standard Model.

\section*{Acknowledgements}

We are grateful to the HERA machine group whose outstanding
efforts have made this experiment possible.
We thank the engineers and technicians for their work in constructing 
and maintaining the H1 detector, our funding agencies for
financial support, the DESY technical staff for continual assistance
and the DESY directorate for support and for the
hospitality which they extend to the non DESY
members of the collaboration.

\newpage
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\newpage

%%%%%%%%%%%%%%%%%%%% Fig.1: Polarisation profile %%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\begin{picture}(90,220)(0,0)
\setlength{\unitlength}{1 mm}
\put(-30,110){\epsfig{file=H1prelim-06-042.fig1a.eps,width=\textwidth}}
\put(-30,  0){\epsfig{file=H1prelim-06-042.fig1b.eps,width=\textwidth}}
\put( -5,200){\bf (a)}
\put( -5,85){\bf (b)}
\end{picture}
\end{center}
\caption{
  The polarisation profile weighted by the luminosity values for the
  (a) $e^+p$ data and (b) $e^-p$ data. The $L (R)$ sample corresponds to distribution having
  negative (positive) polarisation values.}
\label{profile_pol}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%% Fig.2: NC control plot %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\begin{picture}(90,200)(0,0)
\setlength{\unitlength}{1 mm}
\put(-25,100){\epsfig{file=H1prelim-06-042.fig2a.eps,width=13.5cm}}
\put(-25, -5){\epsfig{file=H1prelim-06-042.fig2b.eps,width=13.5cm}}
\put( 35,199){\bf (a)}
\put(106,199){\bf (b)}
\put( 35,148){\bf (c)}
\put(106,148){\bf (d)}
%
\put( 35, 94){\bf (e)}
\put(106, 94){\bf (f)}
\put( 35,43){\bf (g)}
\put(106,43){\bf (h)}
%
\end{picture}
\end{center}
\caption{
  Distributions of (a,e)~$E_e^{\prime}$, (b,f)~$\theta_e$, (c,g)~$Q_e^2$
  and (d,h)~$P_{T,h}/P_{T,e}$ shown in upper and lower plots for the
  selected events in the (a-d) $e^+p$ and (e-h) $e^-p$ data sets. The
  Monte Carlo (MC) contributions from the neutral current (NC) process
  and the $ep$ background (bkg) processes are shown as open histograms
  with the latter contribution alone being shown as shaded
  histograms.}
\label{nc}
\end{figure}

%%%%%%%%%%%%%%%%%%%% Fig.3: cross sections vs.Q2 e+  %%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\begin{picture}(90,200)(0,0)
\setlength{\unitlength}{1 mm}
\put(-30,100){\epsfig{file=H1prelim-06-042.fig3a.eps,width=\textwidth}}
\put(-30, 0){\epsfig{file=H1prelim-06-042.fig3b.eps,width=\textwidth}}
\put( 20,190){\bf (a)}
\put( 20, 90){\bf (b)}
\end{picture}
\end{center}
\caption{
The $Q^2$ dependence of the NC cross section ${\rm d}\sigma/{\rm
d}Q^2$ for $e^+p$ scattering with the (a) right handed $(R)$ and (b)
left handed $(L)$ polarisation data sets. The data (solid points) are
compared to the Standard Model prediction.
}
\label{dsdq2pos}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%% Fig.4: cross sections vs.Q2 e- %%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\begin{picture}(90,200)(0,0)
\setlength{\unitlength}{1 mm}
\put(-30,100){\epsfig{file=H1prelim-06-042.fig4a.eps,width=\textwidth}}
\put(-30, 0 ){\epsfig{file=H1prelim-06-042.fig4b.eps,width=\textwidth}}
\put( 20,190){\bf (a)}
\put( 20, 90){\bf (b)}
\end{picture}
\end{center}
\caption{
The $Q^2$ dependence of the NC cross section ${\rm d}\sigma/{\rm
d}Q^2$ for $e^-p$ scattering with the (a) right handed $(R)$ and (b)
left handed $(L)$ polarisation data sets. The data (solid points) are
compared to the Standard Model prediction.
}
\label{dsdq2ele}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%% Fig.5: ds/dQ2 ratios e+ and e-  %%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\begin{picture}(90,200)(0,0)
\setlength{\unitlength}{1 mm}
\put(-30,100){\epsfig{file=H1prelim-06-042.fig3c.eps,width=\textwidth}}
\put(-30, 0){\epsfig{file=H1prelim-06-042.fig4c.eps,width=\textwidth}}
\put( -3,190){\bf (a)}
\put( -3, 90){\bf (b)}
\end{picture}
\end{center}
\caption{
  The $Q^2$ dependences of the NC cross section ratio ${\rm
  d}\sigma/{\rm d}Q^2$ $R$/$L$ for the (a) $e^+p$ scattering and (b)
  $e^-p$ scattering. The data (solid points) are compared to the
  Standard Model prediction (solid curve).  The inner error bars
  represent the statistical uncertainties and the outer error bars
  represent the total errors. The normalisation uncertainty is not
  included in the error bars and is instead shown as the dashed lines.
  }
\label{dsdq2ratio} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%% Fig.6: ds/dQ2 ratio combined %%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\begin{picture}(90,110)(0,0)
\setlength{\unitlength}{1 mm}
\put(-35, -10){\epsfig{file=H1prelim-06-042.fig5.eps,width=\textwidth}}
\end{picture}
\end{center}
\caption{
  The $Q^2$ dependence of the combined $e^{\pm}p$ $L$ and $R$ NC cross
  section ratio ${\rm R}$.  The data (solid points) are compared to
  the Standard Model prediction. The inner error bars represent the
  statistical uncertainties and the outer error bars represent the
  total errors. The normalisation uncertainty is not included in the
  error bars.}
\label{Rratio} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%% Fig.7. xF3-tilde  %%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\begin{picture}(90,150)(0,0)
\setlength{\unitlength}{1 mm}
\put(-35, -10){\epsfig{file=H1prelim-06-042.fig6.eps,width=\textwidth}}
\put( -20,150){\bf (a)}
\put( -20, 70){\bf (b)}
\end{picture}
\end{center}
\caption{
  The unpolarised reduced cross sections $\tilde{\sigma}^{\pm}(x,Q^2)$
  are shown in (a) for HERA-II data (open/solid points) compared to
  the Standard Model (solid/dashed curves). The structure function
  $x\tilde{F}_3$ evaluated using HERA-I and HERA-II data is shown in
  (b) (solid points) compared to the Standard Model (solid curve). The inner
  error bars represent the statistical uncertainties and the outer
  error bars represent the total errors. The normalisation uncertainty
  is included in the error bars for (b) only.}
\label{xf3} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%% Fig.8. xF3-gZ  %%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\begin{picture}(90,150)(0,0)
\setlength{\unitlength}{1 mm}
\put(-35, -10){\epsfig{file=H1prelim-06-042.fig7.eps,width=\textwidth}}
\put( 20,126){\bf (a)}
\put( 20, 65){\bf (b)}
\end{picture}
\end{center}
\caption{
  The structure function $xF_3^{\gamma Z}$ extracted from all HERA-I
  and HERA-II data (solid and open points) is shown in (a) for three
  $Q^2$ bins together with the Standard Model expectation (full,
  dashed and dotted curves). In (b) the data are transformed to
  $Q^2=1\,500$~GeV$^2$ and combined in each $x$ bin. The inner
  error bars represent the statistical uncertainties and the outer
  error bars represent the total errors. The normalisation uncertainty
  is included in the error bars.}
\label{xf3gz} 
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}


