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\begin{document}
\vspace*{4cm}
\title{Electroweak physics at HERA}
\author{ N. OKAZAKI }
\address{High Energy Accelerator Research Organization, IPNS, \\
1-1 Oho, Tsukuba, Ibaraki, Japan}
\maketitle\abstracts{
Precise tests of electroweak (EW) physics have been performed at
the $e^{\pm}p$ collider HERA by making use of the longitudinally
polarised lepton beams. In this paper, the parity violation in $e^{\pm}p$
neutral current interaction at distances down to $10^{-18}$ m,
the charged current cross section as a function of the lepton beam
polarisation, a QCD and EW parameter fit and
cross sections for the single W production are
presented.}
\section{Deep Inelastic Scattering at HERA}
HERA was the first lepton-proton collider. It accelerated an electron
(or positron) beam of energy 27.5 GeV and a proton beam of energy 920 GeV.
The two beams collided at a centre-of-mass energy of 318 GeV. There were
two experiments for the collision, H1 and ZEUS.
HERA underwent a major upgrade in the year 2000 (HERAII).
In HERAII, high statistics data with longitudinally polarised lepton beam
and the polarisation of typically 30 $\sim$ 40 $\%$ were provided
to the experiments.
\begin{wrapfigure}[17]{r}[15pt]{2.82in}
\begin{center}
\psfig{figure=dsdq2_herapdf.eps,height=2.32in}
\caption{The $Q^{2}$ dependences of the NC and CC cross sections $d\sigma/dQ^{2}$
shown for the $e^{+}p$ and $e^{-}p$ data.
\hskip176pt
\label{fig:dsdq2} }
\end{center}
\end{wrapfigure}
The ep scattering at a high invariant mass of a hadronic system and high
four-momentum transfer, $q$, is known as deep inelastic scattering, DIS. There are
two processes in DIS, neutral current (NC) and charged current (CC) interactions.
NC and CC proceed via the $\gamma/Z^{0}$ and $W^{\pm}$ boson exchanges,
respectively.
The kinematics of DIS are usually described by $Q^{2} \ (= -q^{2})$,
Bjorken-$x$ and inelastisity, $y$.
CC cross sections at low $Q^{2}$ are suppressed due to the large mass of
the W boson in the propagator term
while NC cross sections are much larger at low $Q^{2}$, due to $\gamma$
exchange dominance. However CC and NC cross sections become the same order for
$Q^{2} \sim M^{2}_{W}, M^{2}_{Z}$ where $M_{W}$ and $M_{Z}$ are the masses
of the $Z^{0}$ and $W^{\pm}$ bosons, respectively.
The measurements of the HERA experiments demonstrate the unification of
electromagnetic and weak forces as shown in Fig. \ref{fig:dsdq2}.
\section{Electroweak effects in neutral current interactions}
The unpolarised cross section for $e^{\pm}p$ NC interaction can be written as,
%--------------------%
\begin{eqnarray}
\frac{d^{2} \sigma(e^{\pm}p) }{dx dQ^{2}} =
\frac{2 \pi \alpha^{2} }{ x Q^{2} }
\left[ Y_{+} F_{2}(x,Q^{2}) \mp Y_{-}xF_{3}(x,Q^{2}) \right] ,
\end{eqnarray}
%--------------------%
where $Y_{\pm} = 1 \pm (1-y)^{2}$.
$F_{2}$ and $xF_{3}$ are the proton structure functions.
The $xF_{3}$ reflects distribution of valence quarks as follows,
%--------------------%
\begin{eqnarray}
xF_{3} \simeq \sum_{i} -a_{e} \chi_{Z} [ 2 e_{i} a_{i} ]
\times x(q_{i} - \bar{q}_{i}).
\label{eq:F3}
\end{eqnarray}
%--------------------%
where, $i$ runs over quark flavors and $q, \bar{q}$ are the quark,
anti-quark density functions (PDFs) respectively.
The quantity $a_{i}$ ($a_{e}$) is the axial-vector coupling to
the $Z^{0}$ boson of quark $i$ (electron)
\footnote{
The vector coupling of electron, $v_{e}$, is small and terms containing
it are ignored in Eqs. (\ref{eq:F3}) and (\ref{eq:F2}) }.
The quantity $\chi_{Z}$ is the relative contribution of $\gamma$ and $Z^{0}$ exchange,
and $\chi_{Z} = \frac{1}{\sin^{2}2\theta_{W}} \frac{Q^{2}}{M^{2}_{Z} + Q^{2}} $
where $\theta_{W}$ is the Weinberg mixing angle.
The terms with $\chi^{2}_{Z}$, which arose from the pure $Z^{0}$ exchange,
are very small and have been ignored in Eq (\ref{eq:F3}).
The $xF_{3}$ structure function
arises as a result of parity violation.
It was measured from the difference between the
$e^{-}p$ and $e^{+}p$ cross section
\cite{ncZ} and it was found to be in agreement with the standard model
(SM) prediction.
The parity violation effect with the longitudinally polarised beam
is also seen in the structure function $F_{2}$
which is approximately given by,
%--------------------%
\begin{eqnarray}
F_{2} \simeq \sum_{i} \left[ e_{i}^{2} + P_{e}(2 \chi_{Z} a_{e} e_{i} v_{i}) \right]
\times x(q_{i} + \bar{q}_{i}),
\label{eq:F2}
\end{eqnarray}
%--------------------%
where $v_{i}$ is the vector couplings to the $Z^{0}$ boson of quark $i$,
and $P_{e}$ is the electron/positron polarisation defined as,
%--------------------%
\begin{eqnarray}
P_{e} = \frac{ N_{R} - N_{L} }{ N_{R} + N_{L} }.
\end{eqnarray}
%--------------------%
$N_{R}$ and $N_{L}$ are the number of leptons of right- and left- handed helicity,
respectively. The polarisation asymmetry of the NC cross section, $A^{\pm}$, is defined as,
%--------------------%
\begin{eqnarray}
A^{\pm} = \frac{2}{P_{R} - P_{L}} \cdot
\frac{\sigma^{\pm}(P_{R}) - \sigma^{\pm}(P_{L}) }
{\sigma^{\pm}(P_{R}) + \sigma^{\pm}(P_{L}) },
\end{eqnarray}
%--------------------%
for $e^{+}p$ and $e^{-}p$ scattering respectively,
where $P_{R}$ and $P_{L}$ are the positive and negative polarisation.
The asymmetry may be approximated as
%--------------------%
\begin{eqnarray}
A^{\pm} \simeq \chi_{Z} a_{e} \frac{2 v_{i}}{e_{i}},
\end{eqnarray}
%--------------------%
so that it is sensitive to the quark vector couplings which are discussed
in the next section.
The measurement of the H1 and ZEUS data \cite{ncHZ}
are combined as shown in Fig. \ref{fig:Asym}.
The measured asymmetry shows a deviation of $A^{\pm}$ from zero at high $Q^{2}$
and it is well described by the SM prediction
It demonstrates the parity violation at very small distances,
down to about $10^{-18}$ m.
%--------------------%
\begin{figure}[htbp]
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\vskip 0.05cm
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%*** Figure 1
\begin{minipage}{0.45\hsize}
\begin{center}
\psfig{figure=H1ZeusPrel_06_022_asymComb.eps,height=2.3in}
\caption{The measured polarisation asymmetries $A^{\pm}$
by the H1 and ZEUS combined analysis.The lines describe
the theoretical predictions of NLO QCD evaluated with H12000 PDF
and ZEUS-JETS PDF. \hskip178pt
\label{fig:Asym} }
\end{center}
\end{minipage}
%***
\hskip0.8 cm
%*** Figure 2
\begin{minipage}{0.45\hsize}
\begin{center}
\psfig{figure=cc_tot_H1ZEUS_hera2.eps,height=2.3in}
\caption{The total cross sections for $e^{-}p$ and $e^{+}p$
CC DIS as a function of the longitudinally polarisation of the
lepton beam.
The bands show SM predictions with HERAPDF0.1 NLO PDFs \hskip30pt
\label{fig:CcTot} }
\end{center}
\end{minipage}
\vskip 0.10 cm
\end{figure}
%--------------------%
\section{Charged current cross sections}
CC ep DIS is a pure weak process so that the parity is 100 $\%$
violated in the SM.
The cross section for CC DIS is given as,
%--------------------%
\begin{eqnarray}
\frac{d^{2} \sigma(e^{\pm}p) }{dx dQ^{2}} =
(1 \pm P_{e})
\frac{G^{2}_{F} }{ 4 \pi x }
\left( \frac{M^{2}_{W} }{ M^{2}_{W} + Q^{2} } \right)^{2}
\left[ Y_{+} F^{CC}_{2}(x,Q^{2}) \mp Y_{-}xF^{CC}_{3}(x,Q^{2}) \right],
\label{eq:CC}
\end{eqnarray}
%--------------------%
where $F^{CC}_{2}$, $xF^{CC}_{3}$ are the structure functions
and $G_{F}$ is Fermi coupling constant.
As Eq. (\ref{eq:CC}) shows, the CC cross section has a linear
dependence on polarisation of incoming lepton.
The measured CC total cross sections are shown as a function of the polarisation
in Fig. \ref{fig:CcTot}, including previous measurements with unpolarised beams
\cite{ccHp,ccZp}.
The data are compared to the SM predictions evaluated using HERAPDF0.1 \cite{herapdf}
which are the NLO QCD PDFs extracted from fits to the combined H1 and ZEUS
data. The SM prediction provides a reasonable description of the data.
Only left-handed particles and right-handed antiparticles take part in
weak interactions.
Lower limit on the mass of the W boson which couples to right-handed
chirality can be set by extrapolating the $e^{+}p$ and $e^{-}p$
cross section to $P_{e} = -1, +1$, respectively.
The limit by H1 $e^{+}p$ data analysis is 208 GeV at 95 $\%$ CL
\cite{ccHpub}.
\section{Electroweak parameter and QCD fit}
Precise measurement of structure functions at low $Q^{2}$ and
hence precise determination of PDFs have been performed.
Moreover high luminosity of HERAII data allows
high statistics measurements up to higher $Q^{2}$ region.
EW effects appear more clearly in high $Q^{2}$ DIS.
Therefore PDFs and EW parameters can be fitted simultaneously.
As Eqs. (\ref{eq:F3}) and (\ref{eq:F2}) show, the NC DIS cross section
has the sensitivity for quark couplings to the $Z^{0}$ boson.
Fig. \ref{fig:CoupleZ} show the 68 $\%$ CL contour plots from
the determination of $a_{i}$ and $v_{i}$ for u and d-quarks.
\cite{fit}
The fit of HERA data is consistent with the SM and has significantly improved
the determination of the u-quark coupling compared
with the LEP and Tevatron results.
%--------------------%
\begin{figure}[htbp]
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\vskip 0.05cm
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%*** Figure 3
\begin{center}
%\psfig{figure=auvu4ZEUS.eps,height=2.1in}
\psfig{figure=coupleZ.eps,height=4.76in,angle=270}
\caption{Contour plots of the 68 $\%$ limits on the electroweak
couplings of the quarks to the $Z^{0}$ compared to those from CDF and
LEP-II. Left side: $a_{u}$ v.s $v_{u}$, right side: $a_{d}$ v.s $v_{d}$.
\hskip158pt
\label{fig:CoupleZ} }
\end{center}
\vskip 0.1cm
\end{figure}
%--------------------%
\section{Single W production}
The single W production with a subsequent leptonic decay,
$W \rightarrow l \nu$, has been studied at HERA.
The signature of this process is a high transverse momentum ($P_{T}$) lepton
and large missing $P_{T}$ due to the decay neutrino.
The cross section predicted by the SM is quite small.
Therefore this channel is important in searches for the new physics
which have a similar event topology as this process.
The cross section measured by H1 is,
%--------------------%
\begin{eqnarray}
\sigma_{W} = 1.14 \pm 0.25 (stat.) \pm 0.14 (syst.) \ pb, \nonumber
\end{eqnarray}
%--------------------%
with $\sigma^{SM}_{W} = 1.27 \pm 0.19 $ of the SM expectation \cite{wH},
ZEUS obtains,
%--------------------%
\begin{eqnarray}
\sigma_{W} = 0.89^{+0.25}_{-0.22} (stat.) \pm 0.10 (syst.) \ pb, \nonumber
\end{eqnarray}
%--------------------%
with the SM prediction is $\sigma^{SM}_{W} = 1.2 \pm 0.18 $. \cite{wZ}
The results from the both experiments, H1 and ZEUS have a good agreement
with the SM.
\section{Summary}
Precise measurements for electroweak effects in the space-like scattering
have been presented.
The operation of HERA ended in June 2007 and the collected data
corresponds to about 1 fb$^{-1}$, after combining the data of the two experiments.
More analysis exploiting the full HERA data are expected to come, thus
further improving the statistical precision.
\section*{References}
\begin{thebibliography}{99}
\bibitem{ncZ} ZEUS Coll., \ S.Chekanov et al., arXiv:hep-ex/0901.2385. %NC-ZEUS
\bibitem{ncHZ} H1 Coll. and ZEUS Coll., H1prelim-06-142 Sent to
\textit{ ICHEP 06 } %the XXXIII International Conference on High Energy Physiscs,
\bibitem{ccHp} H1 Coll. H1prelim-06-041 Sent to
\textit{ ICHEP 06 } %the XXXIII International Conference on High Energy Physiscs,
\bibitem{ccZp} ZEUS Coll., ZEUS-prel-07-023 Sent to \textit{The 2007 EPS HEP}
% The 2009 Europhysics Conference on High Energy Physiscs
\bibitem{herapdf} H1 Coll. and ZEUS Coll., H1prelim-08-045 Sent to
\textit{ ICHEP 08 } %the XXXIV International Conference on High Energy Physiscs,
\bibitem{ccHpub} H1 Coll., \ A.Aktas et al., Coll., \Journal{\PLB}{634}{173}{2006}. %CC 03-04 H1
\bibitem{fit}ZEUS Coll., ZEUS-prel-07-027
Sent to \textit{Lepton Photon 2007 }
%XXIII International Symposium on Lepton and Photon Interactions at High Energy 2007}%QCD fit
\bibitem{wH}H1 Coll., \ F.D.Aaron et al., arXiv:hep-ex/0901.0488. %W-H1
\bibitem{wZ}ZEUS Coll., \ S.Chekanov et al., \Journal{\PLB}{672}{106}{2009}. %W-ZEUS
\end{thebibliography}
\end{document}
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