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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 
%Submitted to & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
Submitted to & & &
\epsfig{file=/h1/www/images/H1logo_bw_small.epsi,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                XXII International Symposium on Lepton-Photon Interactions
                at High Energy},
                June~30,~2005,~Uppsala} \\
                 & Abstract:        & {\bf 386}    &\\
                 & Session: & {\bf Electroweak physics and beyond}   &\\
\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 
    Determination of Electroweak Parameters at HERA}

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

\begin{abstract}

\noindent
Using the $e^+p$ and $e^-p$ charged and neutral current cross sections
previously measured by the H1 experiment, a combined electroweak and QCD
analysis is performed to determine electroweak parameters accounting
for their correlation with parton distributions.
An improved measurement is obtained of the $W$ propagator mass
in charged current $ep$ scattering. The weak mixing angle $\sin^2\!\theta_W$
is determined in the on-mass-shell renormalisation scheme. 
A first measurement at HERA is made of the light quark weak couplings to
the $Z^0$ boson.
% and
%a possible contribution of right-handed isospin components to the weak
%couplings is investigated.
\end{abstract}


\end{titlepage}

\pagestyle{plain}

\section{Introduction}

The deep inelastic scattering (DIS) of leptons off nucleons has played 
an important role in revealing the structure of matter, in the discovery
of weak neutral current interactions and 
in the foundation of the Standard Model (SM) as the theory of strong and 
electroweak (EW) interactions.
At HERA, the first lepton-proton collider ever built, the study of DIS 
has been pursued since 1992 over a wide kinematic range. 
In terms of $Q^2$, the negative four-momentum transfer squared,
the kinematic coverage can extend up to 
the centre-of-mass energy squared ($s$), thus covering the region
where the electromagnetic and weak interactions become
of comparable strength.
Both charged current (CC) and neutral current (NC) interactions 
occur in $ep$ collisions and are studied by the two collider 
experiments H1 and ZEUS.
The data taken in a first phase of the HERA operation (HERA-I) until 2000
have been used in many QCD analyses to determine 
the strong interaction coupling constant 
$\alpha_s$~\cite{h1alphas,zeusalphas,zeusalphasnew}
and parton distribution functions (PDFs)~\cite{zeusalphas,h1ep300,h1ep320}.
%with increasing precision.
Based on the HERA-I data EW analyses have been also
performed~\cite{h1ep300,h1cc94,h1nccc96,h1em320,zeusccep300,zeusccem320,zeusccep320} 
to extract the $W$ boson mass value
from the CC data at high $Q^2$. Previously the QCD and EW sectors were
analysed independently.

The increased precision of the H1 data recently 
published~\cite{h1alphas,h1ep300,h1em320,h1ep320} permits 
a combined QCD and EW
analysis which is performed here for the first time.
A measurement is made of the CC propagator mass ($M_{\rm prop}$).
The masses of the $W$ boson ($M_W$) and top quark ($m_t$) and 
the weak mixing angle ($\sin^2\!\theta_W$) are determined 
within the electroweak $SU(2)_L \times U(1)$ Standard Model.
The vector and axial-vector weak couplings of
the light ($u$ and $d$) quarks to the $Z^0$ boson are measured
for the first time at HERA.
These results are complementary to determinations of EW parameters at LEP,
the Tevatron and low energy experiments~\cite{pdg04}.

\section{Charged and Neutral Current Cross Sections}\label{sec:smnccc}

\subsection{Charged Current Cross Section}

The charged current interactions, 
$e^\pm p \rightarrow \overline{\nu}_e^{\mbox{\tiny
\hspace{-3mm}\raisebox{0.3mm}{(}\hspace{2.5mm}\raisebox{0.3mm}{)}}}X$,
are mediated by the exchange of a $W$ boson in the $t$ channel.  
The measured cross section for unpolarised beams after correction for QED
radiative effects~\cite{ccrad} can be expressed as
%
%\vspace*{-6mm}
\begin{eqnarray}
 \frac{d^2\sigma^{\rm CC}(e^\pm p)}{dxdQ^2}&=&
\frac{G^2_F}{2\pi x}\left[\frac{M^2_W}{M^2_W+Q^2}\right]^2\phi^\pm_{CC}(x,Q^2)
\left(1+\Delta^{\pm, weak}_{CC}\right)\,,\label{eqn:xscc}\\
{\rm with} \hspace{5mm} \phi^\pm_{CC}(x,Q^2)&=&
\frac{1}{2}\left[Y_+W_2^\pm (x,Q^2)\mp Y_-xW_3^\pm (x,Q^2)-y^2W_L^\pm (x,Q^2)\right]\,.
\end{eqnarray}
Here $G_F$ is the Fermi constant accounting for radiative corrections to 
the $W$ propagator as measured in muon decays and $\Delta^{\pm,weak}_{CC}$ 
represents the other weak vertex and box corrections, which amount to 
a few per mil~\cite{wh} and are neglected.
%have very little dependence on the top quark mass $m_t$ and 
%the Higgs boson mass $M_H$~\cite{wh}.
The term $\phi^\pm_{CC}$~\cite{h1ep300}
contains the structure functions $W^\pm_2$, $xW^\pm_3$ and
$W^\pm_L$. The factors $Y_\pm$ are defined as $Y_\pm\equiv 1\pm (1-y)^2$ and
$y$ is the inelasticity variable which is related to Bjorken $x$ and 
$Q^2$ by $y=Q^2/xs$.

Within the SM, the CC cross section in Eqn.(\ref{eqn:xscc}) can
be expressed in the so-called on-mass-shell (OMS) scheme~\cite{oms}
by replacing the Fermi constant $G_F$ by
the following relation:
%
%\vspace{-6mm}
\begin{equation}
G_F=\frac{\pi\alpha}{\sqrt{2} M^2_W\left(1-\frac{M^2_W}{M^2_Z}\right)}\frac{1}{1-\Delta r}\,,
\label{eqn:gf}
\end{equation}
where $\alpha\equiv \alpha(Q^2=0)$ is the fine structure constant and 
$M_Z$ is the mass of the $Z^0$ boson.
%Contrary to $\Delta^{\pm,weak}_{CC}$, 
The term $\Delta r$ contains EW radiative corrections which can be expressed 
at one loop as~\cite{wh}
%
%\vspace{-4mm}
\begin{equation}
\Delta r=\Delta \alpha -\frac{\cos^2\!\theta_W}{\sin^2\!\theta_W}\Delta \rho
+\Delta r_{\rm rem}\label{eqn:dr}\,.
\end{equation}
The first term $\Delta \alpha$ is the fermionic part of 
the photon vacuum polarisation. It has a calculable leptonic contribution 
and an uncalculable hadronic component
which can however be estimated using $e^+e^-$ data~\cite{dalpha}. 
Numerically these two contributions are of similar size and
have a total value of $0.059$~\cite{eprc} when evaluated at $M^2_Z$. 
The quantity $\Delta \rho$ 
arises from the large mass difference between the top and bottom quarks 
in the vector boson self-energy loop:
%
%\vspace{-4mm}
\begin{equation}
\Delta \rho=\frac{3\alpha}{16\pi \sin^2\!\theta_W\cos^2\!\theta_W}\frac{m^2_t}{M^2_Z}\,.\label{eqn:drho}
\end{equation}
The second term in Eqn.(\ref{eqn:dr}) has a numerical value of about $0.03$. 
The last term $\Delta r_{\rm rem}$ is numerically smaller ($\sim 0.01$). 
It contains the remaining contributions including those with logarithmic 
dependence on $m_t$ and the Higgs boson mass $M_H$. 
In Eqns.(\ref{eqn:dr},\ref{eqn:drho}) and the OMS scheme, it is understood 
that 
\begin{equation}
\sin^2\!\theta_W=1-M^2_W/M^2_Z \hspace{5mm} {\rm and}\hspace{5mm}
 \cos^2\!\theta_W=M^2_W/M^2_Z\,.
\label{eqn:s2theta}
\end{equation}

In the quark parton model (QPM), 
the structure functions $W^\pm_2$ and $xW^\pm_3$ may be interpreted as
lepton charge dependent sums and differences of quark and anti-quark 
distributions and are given by
%
%\vspace{-6mm}
\begin{equation}
W^+_2=x(\overline{U}+D),\hspace{3mm} xW^+_3=x(D-\overline{U}),\hspace{3mm}
W^-_2=x(U+\overline{D}),\hspace{3mm} xW^-_3=x(U-\overline{D})\,,
\end{equation}
whereas $W^\pm_L=0$. 
The terms $xU$, $xD$, $x\overline{U}$ and $x\overline{D}$
are defined as the sum of up-type, of down-type and of their anti-quark-type 
distributions, i.e., below the $b$ quark mass threshold:
%
%\vspace{-6mm}
\begin{equation}
xU=x(u+c) , \hspace{3mm}
xD=x(d+s),\hspace{3mm}
x\overline{U}= x(\overline{u}+\overline{c}),\hspace{3mm}
x\overline{D}= x(\overline{d}+\overline{s})\,.
\end{equation}
In next-to-leading-order (NLO) QCD and the $\overline{{\rm MS}}$ 
normalisation scheme~\cite{msbar},
these simple relations do not hold any longer
and $W^\pm_L$ becomes non-zero.
Nevertheless the capability of the CC cross sections to probe up- and 
down-type quarks remains.

\subsection{Neutral Current Cross Section}

The NC interactions, $e^\pm p\rightarrow e^\pm X$, are mediated by
photon ($\gamma$) or $Z^0$ exchange in the $t$ channel.
The measured NC cross section with unpolarised beams 
after correction for QED radiative effects~\cite{ncrad} is given by
%
%\vspace*{-6mm}
\begin{eqnarray}
\frac{d^2\sigma^{\rm NC}(e^\pm p)}{dxdQ^2}&=&
\frac{2\pi\alpha^2}{xQ^4}\phi^\pm_{NC}(x,Q^2)
\left(1+\Delta^{\pm,weak}_{NC}\right)\,,\label{eqn:xsnc}\\
{\rm with}\hspace{5mm} \phi^\pm_{NC}(x,Q^2)&=&
Y_+\tilde{F}_2(x,Q^2)\mp Y_-x\tilde{F}_3(x,Q^2)-y^2\tilde{F}_L(x,Q^2)\,.
\end{eqnarray}
Here $\Delta^{\pm,weak}_{NC}$ represents weak radiative corrections 
which are typically less than $1\%$ and never more than $3\%$.
The NC structure function term $\phi^\pm_{NC}$~\cite{h1ep300} is
expressed in terms of the generalised structure functions 
$\tilde{F}_2$, $x\tilde{F}_3$ and $\tilde{F}_L$.
The first two can be further decomposed as~\cite{klein}
%
%\vspace{-6mm}
\begin{eqnarray}
 \label{f2p}
 \tilde{F}_2  \equiv & F_2 & - \ v_e  \ \frac{\kappa  Q^2}{(Q^2 + M_Z^2)}
  F^{\gamma Z}_2  \,\,\, + (v_e^2+a_e^2)  
 \left(\frac{\kappa  Q^2}{Q^2 + M_Z^2}\right)^2 F^Z_2\,, \\
 \label{f3p}
 x\tilde{F}_3    \equiv &      & - \ a_e  \ \frac{\kappa  Q^2}{(Q^2 + M_Z^2)} 
 xF^{\gamma Z}_3 + \,\, (2 v_e a_e) \,\,
 \left(\frac{\kappa  Q^2}{Q^2 + M_Z^2}\right)^2  xF^Z_3\,,
\end{eqnarray} 
with 
%
%\vspace{-6mm}
\begin{equation}
\kappa^{-1}=\frac{2\sqrt{2}\pi\alpha}{G_FM^2_Z}\,,
\end{equation}
in the modified on-mass-shell (MOMS) scheme~\cite{moms} in which
all EW parameters can be defined in terms of $\alpha$, $G_F$ and $M_Z$ 
(besides fermion masses and quark mixing angles) or
%
%\vspace{-6mm}
\begin{equation}
\kappa^{-1}
%\equiv 4\sin^2\!\theta_W\cos^2\!\theta_W
=4\frac{M_W^2}{M_Z^2}\left(1-\frac{M_W^2}{M_Z^2}\right)(1-\Delta r)
\label{eqn:kappa}
\end{equation}
in the OMS scheme.
The quantities $v_e$ and $a_e$ are the
vector and axial-vector weak couplings of the electron
to the $Z^{0}$~\cite{pdg04}.
In the bulk of the HERA phase space, $\tilde{F}_2$ is dominated by 
the electromagnetic structure function $F_2$
originating from photon exchange only. The functions $F^Z_2$ and $xF^Z_3$
are the contributions to $\tilde{F}_2$ and $x\tilde{F}_3$ from $Z^0$ 
exchange and the functions $F_2^{\gamma Z}$ and $xF_3^{\gamma Z}$ are 
the contributions from $\gamma Z$ interference. 
These contributions only become important at large values of $Q^2$.
%The longitudinal structure function $\tilde{F}_L$ may be
%decomposed in a manner similar to $\tilde{F}_2$. Its contribution is
%significant only at high $y$.

In the QPM, the longitudinal structure function $\tilde{F}_L=0$,
the structure functions $F_2$,
$F_2^{\gamma Z}$ and $F_2^Z$ are related to the sum of the quark
and anti-quark momentum distributions, $xq$ and $x\overline{q}$, 
%
%\vspace{-6mm}
\begin{equation}
 \label{eq:f2}
 [F_2,F_2^{\gamma Z},F_2^{Z}] = x \sum_q 
 [e_q^2, 2 e_q v_q, v_q^2+a_q^2] 
 \{q+\overline{q}\} 
\end{equation}
and the structure functions $xF_3^{\gamma Z}$ and $xF_3^Z$ to their
difference,% which determines the valence quark distributions, $xq_v(x,Q^2)$,
%
%\vspace{-4mm}
\begin{equation}
 \label{eq:xf3}
 [ x F_3^{\gamma Z},x F_3^{Z} ] = 2x \sum_q 
 [e_q a_q, v_q a_q]
 \{q -\overline{q} \}\,.% = 2 x \sum_{q=u,d} [e_q a_q, v_q a_q] q_v\,.
\end{equation}
In Eqns.(\ref{eq:f2},\ref{eq:xf3}), $e_q$ is the electric charge of 
quark $q$ ,
%in units of the positron charge
and $v_q$ and $a_q$ are, respectively, the vector and
axial-vector weak coupling constants of the quarks to the $Z^0$:
%
%\vspace*{-6mm}
\begin{eqnarray}
v_q&=&I^3_{q,L}-2e_q\sin^2\!\theta_W\label{eqn:vf}\\
a_q&=&I^3_{q,L}\label{eqn:af}
\end{eqnarray}
where $I^3_{q,L}$
%=1/2 (-1/2)$ 
is the third component of the weak isospin.

The weak radiative corrections $\Delta^{\pm,weak}_{NC}$ in 
Eqn.(\ref{eqn:xsnc}) correspond effectively to modifications of 
the weak neutral current couplings to so-called dressed couplings
 by four weak form factors
$\rho_{eq}$, $\kappa_e$, $\kappa_q$ and $\kappa_{eq}$~\cite{wh}.
The form factor $\rho_{eq}$ has a numerical value very close to $1$
for $Q^2\lesssim 10\,000\,{\rm GeV}^2$ and only at very high $Q^2$ is 
a deviation of a few percent reached~\cite{wh}. The form factors $\kappa_{e,q,eq}$ fall strongly with $Q^2$~\cite{wh} and
approach $1$ where the $\gamma Z$ and $Z^0$ contributions become significant.
Given the current precision of the data used 
(Section~\ref{sec:datafit}), in the following
analysis $\rho_{eq}=1$ is assumed and the weak mixing angle in Eqn.(\ref{eqn:vf}) is replaced by an effective one $\sin^2\!\theta_W^{eff}=\kappa_q(1-M_W^2/M_Z^2)$ where $\kappa_q$ is assumed to be flavour independent.


%In the Born approximation, $\rho=\kappa=1$, and 
%$\overline{v}_{eq}=\overline{v}_e\overline{v}_q$. 
%The coupling $\overline{v}_{eq}$ has no equivalent in the Born approximation.
%The form factors $\kappa_{e,q,eq}$ combined with $\sin^2\!\theta_W$
%give rise to effective mixing angles. 
%The form factors at one-loop level may be written as
%
%\vspace{-6mm}
%\begin{eqnarray}
%\rho_{eq}&=&1+\Delta \rho + \Delta \rho^{\rm rem}_{eq}\\
%\kappa_{e,q,eq}&=&1+\frac{\cos^2\!\theta_W}{\sin^2\!\theta_W}\Delta\rho
%  +\Delta \kappa^{\rm rem}_{e,q,eq}
%\end{eqnarray}
%where $\Delta \rho$ is the leading quadratic correction of 
%Eqn.(\ref{eqn:drho})
%and $\Delta \rho^{\rm rem}_{eq}$ and $\Delta \kappa^{\rm rem}_{e,q,eq}$
%are the remaining contributions with the leading $m_t$ terms subtracted.
%Numerically, $\rho_{eq}$ is very close to $1$ for 
%$Q^2\lesssim 10\,000\,{\rm GeV}^2$
%and only at very high $Q^2$ is a deviation of a few percent reached~\cite{wh}.


\section{Data Sets and Fit Strategies}\label{sec:datafit}

The analysis performed here uses (as in~\cite{h1ep320}) the following
H1 data sets:
two low $Q^2$ data sets ($1.5\leq Q^2\leq 150\,{\rm GeV}^2$)~\cite{h1alphas}, 
three high $Q^2$ NC data sets 
($100\leq Q^2\leq 30\,000\,{\rm GeV}^2$)~\cite{h1ep300,
h1ep320,h1em320} and three high $Q^2$ CC data
sets ($300\leq Q^2\leq 15\,000\,{\rm GeV}^2$)~\cite{h1ep300,h1ep320,h1em320}.
These data cover a Bjorken $x$ range from $3\cdot 10^{-5}$ to $0.65$ 
depending on $Q^2$.
% same data sets as 
%those in~\cite{h1ep320}.
%These are three NC $e^\pm p$ cross section data sets
%at high $Q^2$, three corresponding CC cross sections data also at high $Q^2$
%and two precision data at low $Q^2$.

The low $Q^2$ data are dominated by systematic uncertainties which have
a precision down to $2\%$ in most of the covered region. The high $Q^2$
data on the other hand are mostly limited by the statistical
precision which is up to $30\%$ or larger for 
$Q^2\gtrsim 10\,000\,{\rm GeV}^2$.

The combined EW-QCD analysis follows the same fit procedure used 
in~\cite{h1ep320}.
The QCD analysis is performed using 
the DGLAP evolution equations~\cite{dglap} at NLO~\cite{furmanski} 
in the $\overline{{\rm MS}}$ renormalisation scheme.
All quarks are taken to be massless.
%including the charm and bottom quarks.
%The bottom quark distribution, $xb$, is assumed to be zero for $Q^2<m_b^2$ 
%where $m_b$ is the bottom quark mass.
%Its contribution for $Q^2$ above the mass threshold is included in the $xD$
%and $x\overline{D}$ distributions.

Fits are performed to the measured cross sections 
%calculating the longitudinal structure functions to order $\alpha_s^2$ 
assuming the strong coupling constant to be equal to 
$\alpha_s(M_Z)=0.1185$~\cite{pdg00}.
The analysis uses an $x$-space program developed within
the H1 Collaboration~\cite{qcdfit}.
In the fit procedure, a $\chi^2$ function which is defined
in~\cite{h1alphas} is minimised. 
The minimisation takes into account correlations between
data points caused by systematic uncertainties~\cite{h1ep320}.
%and allow the error parameters, 
%including the relative normalisation of the various data sets, 
%to be determined by the fit. 

In the fits, five PDFs -- gluon, $xU$, $xD$, $x\overline{U}$ and 
$x\overline{D}$ -- are defined by 10 free parameters as in~\cite{h1ep320}. 
The PDFs obtained here are consistent with the H1 PDF 2000~\cite{h1ep320}.
For more details refer to~\cite{bpthesis}.
%in the analysis of the fits on the QCD sector are thus comparable to 
%H1 PDF 2000~\cite{h1ep320}, i.e., a simultaneous 
%determination of five sets of PDFs with 10 free parameters.
%These are the gluon distribution and the sum of up-type $xU$, 
%of down-type $xD$ and of their antiquark-type $x\overline{U}$ and 
%$x\overline{D}$ distributions. 
%Since EW parameters are fitted together,
%some correlation between the PDFs and EW parameters is thus expected.
Table~\ref{tab:fits} shows an overview of various fits that are
performed in subsequent section to determine different EW parameters.

% ------------------ TABLE : fits  -------------------------
\begin{table*}[htb]
  \renewcommand{\doublerulesep}{0.4pt}
  \renewcommand{\arraystretch}{1.2}
 \vspace{-0.1cm}

\begin{center}
    \begin{tabular}{|l|c|c|c|}
      \hline
 Fit & Scheme & \multicolumn{2}{|c|}{Fixed parameters} \\ \cline{3-4}
     &        & CC & NC \\\hline
$G$-$M_{\rm prop}$-${\rm PDF}$ & $-$ & $-$ & 
  $\alpha, M_Z, M_W$ \\
$M_{\rm prop}$-${\rm PDF}$ & MOMS & 
  $G_F$ & $\alpha, G_F, M_Z$ \\ \hline
$M_W$-${\rm PDF}$ & OMS & 
  \multicolumn{2}{|c|}{$\alpha, M_Z, m_t, M_H$} \\
$m_t$-${\rm PDF}$ & OMS & 
  \multicolumn{2}{|c|}{$\alpha, M_Z, M_W, M_H$} \\\hline
$v_u$-$a_u$-$v_d$-$a_d$-${\rm PDF}$ & $-$ &
 $G_F, M_W$ & $\alpha, M_Z, M_W$ \\
$v_u$-$a_u$-${\rm PDF}$ & $-$ &
 $G_F, M_W$ & $\alpha, M_Z, M_W, v_d, a_d$ \\
$v_d$-$a_d$-${\rm PDF}$ & $-$ &
 $G_F, M_W$ & $\alpha, M_Z, M_W, v_u, a_u$ \\
$I^3_{u,R}$-$I^3_{d,R}$-${\rm PDF}$ & $-$ &
 $G_F, M_W$ & $\alpha, M_Z, M_W, v_{q,L}, a_{q,L}$ \\
 \hline
    \end{tabular}
    \caption {\small \label{tab:fits}
               Summary of the fit assumptions. In the fits, in addition
               to the free parameters listed in the first column, 
               the systematic
               correlation uncertainty parameters are allowed to 
               vary (see Table 2 in~\cite{h1ep320}).
               The fixed parameters are
               set to values taken from~\cite{pdg04} and $M_H$ is set to 
               $120\,{\rm GeV}$.
               }
\end{center}
\end{table*}
% ------------------------------------------------------------------------


\section{Results}\label{sec:results}

\subsection{\boldmath Determination of Masses and $\sin^2\!\theta_W$}
\label{sec:mw}

The CC cross section data allow a simultaneous determination of 
$G_F$ and $M_W$ and of the PDFs as independent parameters
(fit $G$-$M_{\rm prop}$-${\rm PDF}$ in Table~\ref{tab:fits}). 
In this fit, the parameters $G_F$ and $M_W$ in Eqn.(\ref{eqn:xscc}) 
are considered to be a normalisation variable $G$ and a propagator mass 
$M_{\rm prop}$, respectively, independent of the SM.
The sensitivity to $G$ according to Eqn.(\ref{eqn:xscc}) results from
the normalisation of the CC cross section whereas the sensitivity
to $M_{\rm prop}$ arises from the $Q^2$ dependence.
The fit is performed including 
the NC cross section data 
in order to constrain the PDFs. 
The result of the fit to $G$ and $M_{\rm prop}$ 
is shown in Fig.~\ref{fig:gf} as the shaded area.
The $\chi^2$ value per degree of freedom (dof) is $533.0/610=0.87$.
%The small $\chi^2$ value hints that some of the systematic errors may have
%been overestimated.
The correlation between $G$, $M_{\rm prop}$ and the fitted PDF parameters
may be found in~\cite{gfmw_cor}.
The determination is consistent with the SM values
demonstrating the universality of the CC interaction over a large range of
$Q^2$ values.

Fixing $G$ to $G_F$ at the value of $1.16637\cdot 10^{-5}\,{\rm GeV}^2$ obtained 
from the muon lifetime measurement~\cite{pdg04}, one may fit the
CC propagator mass $M_{\rm prop}$ only.
For this fit ($M_{\rm prop}$-${\rm PDF}$), the EW parameters 
are defined in the MOMS scheme
and the propagator mass $M_{\rm prop}$ is considered to be 
independent of any other EW parameters.
Note that in the MOMS scheme, the use of $G_F$ makes the dependency of the CC 
and NC cross sections on $m_t$ and $M_H$ negligibly small.

The result of the fit, also shown 
in Fig.~\ref{fig:gf}, is
%
%\vspace{-6mm}
\begin{equation}
M_{\rm prop}=82.87\pm 1.82_{\rm exp}\left.^{+0.30}_{-0.16}\right|_{\rm model}\,{\rm GeV}\,.
\end{equation}
Here the first error is experimental and the second corresponds to
uncertainties due to input parameters and model assumptions as introduced 
in Table~5 in~\cite{h1ep320}
(e.g., the variation of $\alpha_s=0.1185\pm 0.0020$). 
The $\chi^2$ value per dof is $533.3/611$~\cite{bpthesis}.
%With respect to the strong correlation
%between the PDF parameters (unshown here), the correction between the EW and
%QCD sector is relatively weak but by far negligible.
%
% ------------------ TABLE : chi2/dof  -------------------------
%\begin{table*}[htb]
%  \renewcommand{\doublerulesep}{0.4pt}
%  \renewcommand{\arraystretch}{1.2}
% \vspace{-0.1cm}
%
%\begin{center}
%    \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
%      \hline
% $\rho_{M,B_g}$ & $\rho_{M,C_g}$ & $\rho_{M,D_g}$ & 
% $\rho_{M,B_U}$ & $\rho_{M,C_U}$ & $\rho_{M,F_U}$ & 
% $\rho_{M,A_D}$ & $\rho_{M,C_D}$ & 
% $\rho_{M,C_{\overline{U}}}$ & $\rho_{M,C_{\overline{D}}}$ \\ \hline
% $-0.13$ & $-0.21$ & $0.10$ & $0.07$ & $0.06$ & $0.06$ & $0.13$ & $0.08$ & 
% $0.08$ & $0.22$ \\ \hline
%    \end{tabular}
%    \caption {\small \label{tab:correlation}
%               The correlation coefficients between $M_{\rm prop}$ and
%               PDF parameters are shown.
%               }
%\end{center}
%\end{table*}
% ------------------------------------------------------------------------

The propagator mass $M_{\rm prop}$ measured here in the space-like region
can be compared with direct $W$ boson mass measurements obtained in 
the time-like region by the Tevatron and LEP experiments~\cite{pdg04}.
The value is consistent with 
%A comparison with 
the world average of 
$M_W=80.425\pm 0.038\,{\rm GeV}$~\cite{pdg04} within 1.3 standard deviations.

In comparison with previous measurements of $M_{\rm prop}$ at 
HERA~\cite{h1nccc96,h1ep300,h1em320,zeusccep300,zeusccem320,zeusccep320}, 
the precision is improved. 
This is due to the higher statistics data and 
the different treatment of 
the correlation between the propagator mass and the PDFs. 

Within the SM, the CC and NC cross sections can be expressed in the OMS
scheme in which  
all EW parameters are determined by $\alpha$, $M_Z$ and $M_W$ together with
$m_t$ and $M_H$ in the loop corrections.
In this scheme, the CC cross section normalisation
depends on $M_W$ via the $G_F-M_W$ relation (Eqn.(\ref{eqn:gf})).
Some additional sensitivity to $M_W$ comes through the $M_W$ 
dependent terms (e.g., Eqn.(\ref{eqn:kappa})) in the NC cross section.
Fixing $m_t$ to its world average value of $178\,{\rm GeV}$~\cite{pdg04} 
and assuming $M_H=120\,{\rm GeV}$, the fit $M_W$-${\rm PDF}$ leads to
%
%\vspace{-6mm}
\begin{equation}
\label{eqn:mw}
M_W=80.786\pm 0.205_{\rm exp}\left.^{+0.048}_{-0.029}\right|_{\rm model}\pm0.025_{\delta m_t}-0.084_{\delta M_H}\pm 0.033_{\delta(\Delta r)}\,{\rm GeV}\,.
\end{equation}
Here, in addition to the experimental and model uncertainties, three other
error sources are considered: the uncertainty on the top quark
mass $\delta m_t=4.3\,{\rm GeV}$~\cite{pdg04}, 
a variation of the Higgs mass from $120\,{\rm GeV}$
to $300\,{\rm GeV}$ and the uncertainty of higher order terms in 
$\Delta r$~\cite{bpthesis}. 
It should be pointed out that the result (\ref{eqn:mw}) on $M_W$
is not a direct measurement but a determination within the SM framework.

Together with the world average value of 
$M_Z=91.1876\pm 0.0021\,{\rm GeV}$~\cite{pdg04}, 
the result obtained on $M_W$ from Eqn.(\ref{eqn:mw}) represents
an indirect determination of $\sin^2\!\theta_W$ 
in the OMS scheme (Eqn.({\ref{eqn:s2theta}))
%
%\vspace{-6mm}
\begin{equation}
\sin^2\!\theta_W=0.2151
\pm 0.0040_{\rm exp}\left.^{+0.0019}_{-0.0011}\right|_{\rm th}
\end{equation}
where the first error is experimental and the second is theoretical covering
all remaining uncertainties in Eqn.(\ref{eqn:mw}). The uncertainty due to
$\delta M_Z$ is negligible.

Figure~\ref{fig:mw-mt} shows the correlation between $M_W$ and $m_t$ in 
a simultaneous fit to these parameters.
Fixing $M_W$ to the world average value and assuming $M_H=120\,{\rm GeV}$,
the fit $m_t$-${\rm PDF}$ gives $m_t=108\pm 44\,{\rm GeV}$
where the uncertainty is experimental.
The result represents the first determination of the top quark mass through
loop effects in the $ep$ data at HERA.

\subsection{\boldmath Determination of $v_{u,d}$ and $a_{u,d}$}
\label{sec:coupling}

At HERA, the NC interactions at high $Q^2$ receive contributions from 
$\gamma Z$ interference and $Z^0$ exchange (Eqns.(\ref{eq:f2},\ref{eq:xf3})). 
Thus the NC data
can be used to extract the weak couplings of up- and down-type quarks
to the $Z^0$ boson.
At high $Q^2$ and high $x$, where the NC $e^\pm p$ cross sections are 
sensitive to these couplings, the up- and down-type quark distributions 
are dominated by the light $u$ and $d$ quarks.
Therefore, this measurement can be considered to determine the light quark
couplings.
The CC cross section data help
disentangle the up and down quark distributions. 

In this analysis (fit $v_u$-$a_u$-$v_d$-$a_d$-${\rm PDF}$), the vector and 
axial-vector dressed couplings of $u$ and $d$ quarks are treated as 
free parameters.
The results of the fit 
are shown 
in Fig.~\ref{fig:coupling}
and are given in Table~\ref{tab:coupling}.
The effect of the $u$ and $d$ correlation is illustrated in 
Fig.~\ref{fig:coupling}
by fixing either $u$ or $d$ quark couplings to their SM values 
(fits $v_d$-$a_d$-${\rm PDF}$ and $v_u$-$a_u$-${\rm PDF}$). 
The precision is better for the $u$ quark as expected.
The superior precision for $a_u$ comes from 
the $\gamma Z$ interference contribution $xF_3^{\gamma Z}$ 
(Eqn.(\ref{eq:xf3})). The $d$-quark couplings $v_{d}$ and $a_d$
are mainly constrained by the $Z^0$ exchange term $F_2^Z$ 
(Eqn.(\ref{eq:f2})). These differences in sensitivity result in different
contour shapes shown in Fig.~\ref{fig:coupling}.

% ------------------ TABLE : coupling  -------------------------
\begin{table*}[htb]
  \renewcommand{\doublerulesep}{0.4pt}
  \renewcommand{\arraystretch}{1.2}
 \vspace{-0.1cm}

\begin{center}
    \begin{tabular}{|c|c|c|c|c|c|}
      \hline
 Fit & $a_u$ & $v_u$ & $a_d$ & $v_d$ & $\chi^2/{\rm dof}$ \\ \hline
 $v_u$-$a_u$-$v_d$-$a_d$-${\rm PDF}$ & $0.56\pm 0.10$ & $0.05\pm 0.19$ &
  $-0.77\pm 0.37$ & $-0.50\pm 0.37$ & $531.7/608$ \\
 $v_u$-$a_u$-${\rm PDF}$ & $0.57\pm 0.08$ & $0.27\pm 0.13$ & $-$ & $-$ & 
  $534.1/610$ \\
 $v_d$-$a_d$-${\rm PDF}$ & $-$ & $-$ & $-0.80\pm 0.24$ & $-0.33\pm 0.33$ &
  $532.6/610$ \\\hline
 SM value & $0.5$ & $0.196$ & $-0.5$ & $-0.346$ & $-$ \\\hline
    \end{tabular}
    \caption {\small \label{tab:coupling}
               The results of the fits to the weak neutral current couplings 
               in comparison with their SM values. The correlation between the
               the fit parameters may be
               found in~\cite{cor-coupling}.
               }
\end{center}
\end{table*}
% ------------------------------------------------------------------------

The results from this analysis are also
compared in Fig.~\ref{fig:coupling} with 
%similar results obtained recently by the CDF experiment~\cite{cdf}.
%The HERA determination has comparable precision as that from the Tevatron.
the combined LEP data where all possible solutions are explicitly 
shown~\cite{lepew}.
The HERA determination is sensitive to $u$ and $d$ quarks separately, 
contrary to other precise measurement of light quark-$Z^0$ couplings in 
$\nu N$ scattering~\cite{nuN} and atomic partity violation~\cite{apv} on
heavy nuclei.
The HERA determination also resolves the ambiguities
present in the LEP results.


\section{Conclusion} \label{sec:conclusion}

Using the neutral and charged current cross section data recently published
by H1, combined electroweak and QCD fits have been performed.
This analysis is the first in which the correlation between the electroweak
and parton distribution parameters is considered and a set of
electroweak theory parameters is determined in the space-like region based
on the HERA data.

Exploiting the $Q^2$ dependence of the charged current data, 
the propagator mass has been measured with the result
$M_{\rm prop}=82.87\pm 1.82_{\rm exp}\left.^{+0.30}_{-0.16}\right|_{\rm model}$.
Within the Standard Model framework,
the $W$ mass has been determined to be
$M_W=80.786\pm0.205_{\rm exp}\left.^{+0.063}_{-0.098}\right|_{\rm th}\,{\rm GeV}$
in the on-mass-shell scheme.
%These mass values represent a significant improvement over those earlier
%determinations which were obtained based on one of the samples available then
%by decoupling the electroweak parameters from 
%the parton distribution functions.
This mass value has also been used to derive an indirect
determination of $\sin^2\!\theta_W$ yielding 
$0.2151\pm 0.0040_{\rm exp}\left.^{+0.0019}_{-0.0011}\right|_{\rm th}$.
Furthermore, a result on the top quark mass via electroweak 
effects in $ep$ data has been obtained.
%Furthermore, the first result has been obtained on the top quark mass 
% via electroweak radiative effects in the $ep$ data.

The vector and axial-vector weak neutral current couplings of 
$u$ and $d$ quarks to the $Z^0$ boson have been determined at HERA for the
first time.
%A possible contribution to the weak neutral current couplings
%from right-handed current couplings has also been studied.
%All results are consistent with the electroweak Standard Model.

% relying on
%the neutral current data at high $Q^2$. 
%The precision of the up-type quark couplings is found to be
%comparable with that obtained by the combined analysis at LEP. 
%The HERA determination also resolves the sign and mirror ambiguities 
%in the LEP measurements. 

\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. It is our pleasure to thank H.~Spiesberger
for helpful discussions.

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% ---------- FIGURE: G_F vs. M_W  ----------------
%
\begin{figure}[htb] 
\begin{center}
\begin{picture}(50,125)
% \put(-40,-10){\epsfig{file=/afs/desy.de/user/z/zhang/h1raid8/paper/ew/lp05/fig1.eps,width=14cm}}
\put(-40,-10){\epsfig{file=H1prelim-05-041.fig1.eps,width=14cm}}
\end{picture}
\end{center}
  \caption{\label{fig:gf}
  The result of the fit to $G$ and $M_{\rm prop}$ at $68\%$ confidence 
  level (CL) shown as the shaded area. The world average values are indicated 
  with the star symbol. Fixing $G$ to $G_F$, the fit in the MOMS scheme
  results in a measurement of the propagator mass $M_{\rm prop}$ shown
  as the square with the horizontal error bars.
%  Within the SM and choosing $M_Z=91.1876\,{\rm GeV}$,
%  $m_t=178\,{\rm GeV}$ and $m_H=120\,{\rm GeV}$, the fit gives a constrained
%  determination of $M_W$ corresponding to the horizontal projection
%  of the result represented by the circle with the error bars.
  }
\end{figure} 
%---------------------------------------------------------------------------

% ---------- FIGURE: M_W vs. m_t  ----------------
%
\begin{figure}[htb] 
\begin{center}
\begin{picture}(50,125)
\put(-47.5,-35){\epsfig{file=H1prelim-05-041.fig2.ps,bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=15cm}}

\end{picture}
\end{center}
  \caption{\label{fig:mw-mt}
    The fitted $M_W$ as a function of $m_t$ shown for $M_H=120\,{\rm GeV}$ 
   (the large shaded band) and $M_H=300\,{\rm GeV}$ (the area between the two
  dashed lines). The vertical shaded band and the horizontal hatched band
  show the world average values of $m_t$ and $M_W$ respectively.}
\end{figure} 
%---------------------------------------------------------------------------

% ---------- FIGURE: light quark couplings  ----------------
%
\begin{figure}[htb] 
\begin{center}
\begin{picture}(50,160)
\put(-35,45){\epsfig{file=H1prelim-05-041.fig3a.ps,bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=12cm}}
\put(-35,-45){\epsfig{file=H1prelim-05-041.fig3b.ps,bbllx=0pt,bblly=0pt,bburx=594pt,bbury=842pt,width=12cm}}
\end{picture}
\end{center}
  \caption{\label{fig:coupling}
  Results at $68\%$ confidence level (CL) on the weak neutral current
  couplings of 
  $u$ (upper plot) and $d$ (lower plot) quarks to the $Z^0$ boson 
  determined in this analysis 
  (the shaded contours) in comparison with those determined by 
  the combined LEP data which have both sign and $v-a$ exchange ambiguities.
%  the CDF experiment (the open contours). 
  The dark-shaded contours correspond to results of a simultaneous fit of
  all four couplings whereas the lighted-shaded contours correspond to 
  results of fits where either $d$ or $u$ quark couplings are fixed 
  to their SM values.
  The stars show the expected SM values.}
\end{figure} 
%---------------------------------------------------------------------------

\end{document}

