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\begin{document}

\title{QCD Corrections to Single Top Quark and W~Boson Production at HERA}

\author{Kai-Peer~O.~Diener}

\address{Paul Scherrer Institut, \\
CH-5232 Villigen, \\ 
Switzerland\\ 
E-mail: kai.diener@psi.ch}

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\maketitle

\abstracts{
We review theoretical results on QCD corrections to flavour changing 
neutral current single top as well as single $W$ boson production at
HERA.
In both cases the inclusion of QCD corrections has a moderate effect on the 
size of the cross sections but reduces the QCD
scale dependence considerably.
This writeup summarizes a talk given at the Ringberg Workshop
``New Trends in HERA Physics 2003''.
}

%**************************************************************
%**************************************************************
\section{Introduction} \label{sec:intro}

At HERA, $27.5\mrm{GeV}$ electrons 
collide with $920\mrm{GeV}$ protons allowing on the one hand
for precise studies of the proton structure and associated non-perturbative
aspects of QCD, on the other hand for tests of perturbative QCD and the electroweak
(EW) standard model (SM) in general.

We review the production of a single top quark in a 
FCNC reaction and the production of a single $W$ boson
in $e^\pm p$ collisions at HERA energies.
This work is, thus, concerned with phenomenological 
aspects of the gauge sector 
(non-abelian contribution to $W$ boson production) 
and flavour structure (single top production) of 
the SM 
%with a phenomenological effective 
%Lagrangian allowing for flavour changing
%neutral current (FCNC)
as well as aspects of perturbative QCD. 
Typical momentum transfers of the reactions we consider 
define scales at 
which a perturbative treatment of QCD corrections to 
the leading-order (LO) EW processes is justified.
In fact, the inclusion of such corrections is necessary
to obtain theoretical predictions for the relevant 
observables which are of comparable accuracy as the 
corresponding experimental measurements at HERA.

In a recent search~\cite{Andreev:2003pm} for
events with a high-energy isolated electron or muon and
missing transverse momentum
the H1 collaboration claimed to find an excess of events with
transverse momentum of the hadronic system greater than $25\mrm{GeV}$.
Within the SM, such events are expected to be mainly due to $W$
boson production with subsequent leptonic decay.
It is, thus, important to control the effects 
of radiative QCD corrections to the LO EW $W$ boson 
production mechanism within and beyond the SM.
A possible explanation of an excess of $W$ boson production 
events beyond the SM would be the FCNC production of 
a short-lived top quark decaying into a bottom quark and 
a $W$ boson.

In section~\ref{sec:topprod}, we review a calculation of QCD
corrections to FCNC top quark production at HERA in the eikonal
approximation as carried out by Belyaev and Kidonakis 
in~\citeref{Belyaev:2001hf}.

We proceed in section~\ref{sec:Wprod} by describing QCD corrections to 
the $W$ production cross sections at HERA.
Section~\ref{subsec:tot} is concerned with corrections 
to the total cross section. 
We review the corrections to resolved photoproduction
as discussed in~\citerefs{Nason:1999xs} by Nason, R\"uckl and Spira.
Section~\ref{subsec:diff} is devoted to corrections to 
the $W$ production cross sections differential in the 
transverse momentum of the $W$ boson $p_T$ and 
its rapidity $y$.
We discuss next-to-leading order (NLO) QCD corrections to
the direct photoproduction part which 
is the dominant contribution to these distributions.
The results of section~\ref{subsec:diff} are quoted from
\citeref{Diener:2002if}.

Section~\ref{sec:conclusions} contains a short summary and 
concluding remarks.

%**************************************************************
%**************************************************************
\section{Soft Gluon Corrections to FCNC Top Production}
\label{sec:topprod}

In \citeref{Belyaev:2001hf}, FCNC couplings between a top quark $t$,
an up or charm quark $q$ and a photon ($F^{\mu \nu}$ is 
the corresponding field strength tensor) are
introduced through the effective Lagrangian
\begin{equation}
\Delta {\mathcal L}^\mrm{eff} = \frac{1}{\Lambda} 
        \kappa_{tq\gamma} e   \bar t \sigma_{\mu\nu} q
F^{\mu\nu} + \mrm{h.c.}, 
\end{equation}
where $\Lambda$ is a dimensionful scale parameter.
Clearly, HERA should be quite sensitive to $\kappa_{tu\gamma}$
through the partonic process \mbox{$e^\pm u \to e^\pm t$}.

To reduce the theoretical error in the determination of bounds
for $\kappa_{tu\gamma}$, the authors of \citeref{Belyaev:2001hf}
consider gluonic quantum corrections to the 
LO EW process.
They argue, that since the top quark is very heavy 
top production at HERA occurs --- if at all --- mostly close to
threshold and that in this regime the eikonal approximation
suffices to describe first-order QCD effects.

\begin{figure}[ht]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
\centerline{\epsfxsize=6cm\epsfbox{diener_fig1.eps}}   
\caption{\label{fig:BKscale} Born, one loop, and Born plus one loop cross section 
as a function of the QCD scale $\mu=\mu_R=\mu_F$ 
for FCNC single top quark production at HERA with \mbox{$m_t=175\mrm{GeV}$}
and \mbox{$\kappa_{tu\gamma}=0.1$} for \mbox{$\sqrt{S}=318\mrm{GeV}$}.}
\end{figure}
Fig.~\ref{fig:BKscale} is reproduced from \citeref{Belyaev:2001hf}
and shows the dependence of the Born, one loop and Born plus one loop
cross section on the renormalization and factorization scale 
\mbox{$\mu=\mu_R=\mu_F$}.
The parton densities were implemented in the CTEQ5M~\cite{Lai:1999wy} 
parameterization and the numerical input parameters were set to
\[
\sqrt{S}=318\mrm{GeV}, \quad \Lambda=m_t=175\mrm{GeV}, \quad \kappa_{tu\gamma}=0.1.
\]
Fig.~\ref{fig:BKscale} shows that above \mbox{$\mu=10\mrm{GeV}$} the inclusion
of NLO QCD corrections in the eikonal approximation leads to a significant
stabilization of the total cross section with respect to simultaneous
variation of the renormalization and factorization scale. 
It can also be inferred that \mbox{$\mu=m_t=175\mrm{GeV}$} as 
central value for the QCD scale is a sensible choice, unlike
\mbox{$\mu=Q=\sqrt{-(p_e-p_{e'})^2}$}, which would lead to significant
contributions from very low $Q$ values of the order of $m_e$ 
where the parton density functions are not well-defined.

\begin{figure}[ht]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
\centerline{\epsfxsize=6cm\epsfbox{diener_fig2.eps}\epsfxsize=6cm\epsfbox{diener_fig3.eps}}   
\caption{\label{fig:BKtmcm} Born, one loop,  and Born plus one loop 
cross section for the FCNC
single top quark production at HERA with \mbox{$\mu=\mu_R=\mu_F=m_t$}
and $\kappa_{tu\gamma}=0.1$ versus 
(a) the top quark mass with $\sqrt{S}=318\mrm{GeV}$  and~(b) the CM energy
with $m_t=175\mrm{GeV}$.}
\end{figure}
In Fig.~\ref{fig:BKtmcm}, the Born, one loop and Born plus 
one loop cross sections are plotted as a function of a) the top mass and 
b) the centre-of-mass energy.
The QCD scale $\mu$ is set to its central value $m_t$ and $\kappa_{tu\gamma}$ 
is chosen as above. 
A strong dependence of the cross sections on the top mass 
and CM energy is observed.
Indeed, a variation of $m_t$ by $\pm5\%$ leads to roughly $20\%$ uncertainty
in the cross section, a $6\%$ increase in CM energy causes an
increase of the cross section by $40\%$. 

Analyses of FCNC single top production from
an initial state $u$ quark at HERA have been carried out 
taking into account the results of \citeref{Belyaev:2001hf} discussed 
above.
The ZEUS collaboration found 
\mbox{$\kappa_{tu\gamma} < 0.174$}~\cite{Chekanov:2003yt},
the H1 collaboration established  
\mbox{$\kappa_{tu\gamma} < 0.27$}~\cite{Aktas:2003yd}, both at the 
$95\%$ confidence level.


%**************************************************************
%**************************************************************
\section{NLO QCD Corrections to $W$ Boson Production}
\label{sec:Wprod}

There are three different mechanisms of $W$ boson production 
in $e^\pm p$ collisions:
\begin{enumerate}
\item The deep-inelastic regime defined by the kinematical
        condition that \mbox{$Q^2 =-(p_e - p_{e'})^2 > Q^2_\mrm{cut}$} and 
        that the hadronic cross section can be written 
        \[      \sigma_{ep} = \hat \sigma_{eq} \otimes f_p^q,  \] with
        the quark density in the proton $f_p^q$ and ``$\otimes$'' indicating a
        convolution.
\item The photoproduction region with \mbox{$Q^2 =-(p_e - p_{e'})^2 < Q^2_\mrm{cut}$} 
        where a further distinction is made between
\begin{enumerate}
\item direct photoproduction with an initial state photon instead of 
        the electron in the partonic subprocess, i.e.,
        \[      \sigma_{ep} = \hat \sigma_{\gamma q} \otimes f_p^q \otimes f_e^\gamma,\]
        where $f_e^\gamma$ is given by the Weizs\"acker--Williams spectrum and 
\item resolved photoproduction where the initial state photon is split
        into a $q,\bar q$ pair and the photon in the initial state of the 
        partonic subprocess
        is thus replaced by a (anti) quark:
        \[      \sigma_{ep} = \hat \sigma_{\bar q q} \otimes f_p^q 
                \otimes f_e^\gamma \otimes f_\gamma^{\bar q}.\]
\end{enumerate}
\end{enumerate}
Which of the production 
mechanisms is dominant depends to a large extent on whether we
consider total or differential cross sections.
At lowest order there is no 
contribution from resolved photoproduction to the $p_T$ 
distribution of the $W$
as the partonic process in this case is $2 \to 1$ with vanishing $p_T$
in the final state. 
The resolved contributions to the total cross section are, however, 
quite significant and we shall thus proceed 
by reviewing the total and differential 
$W$ boson production cross sections separately.

%**************************************************************
\subsection{Corrections to the Total Cross Section} 
\label{subsec:tot}

In \citerefs{Nason:1999xs}, Nason, R\"uckl and Spira showed
that resolved photoproduction is the dominant contribution
to the LO total $W$ boson production cross section in
$e^\pm p$ collisions.
It contributes about $60\%$, while direct photoproduction and
deep-inelastic scattering account for roughly $15\%$ and $25\%$, 
respectively.
Their analysis of QCD quantum corrections was, thus, confined
to the resolved part.

\begin{figure}[ht]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
%\centerline{\epsfxsize=4.2cm\begin{turn}{-90}\epsfbox{nrsWplus.eps}\end{turn}
%\epsfxsize=4.2cm\begin{turn}{-90}\epsfbox{nrsWmin.eps}\end{turn}}   
\centerline{\epsfxsize=5cm\begin{turn}{-90}\epsfbox{diener_fig4.eps}\end{turn}}
\vspace*{.5cm}
\centerline{\epsfxsize=5cm\begin{turn}{-90}\epsfbox{diener_fig5.eps}\end{turn}}   
\vspace*{.5cm}
\caption{\label{fig:nrs}
Dependence of the individual
contributions to $W^+$ (up) and $W^-$ (down) production
with subsequent muonic decay on the renormalization and 
factorization scale $\mu = \mu_R = \mu_F = \xi M_W$.
The full curves represent the final predictions for the
total cross section of $W^\pm$ production in $e^+p$ collisions. 
}
\end{figure}

Fig.~\ref{fig:nrs} shows the dependence of
the total cross section for $W^\pm$ production 
with subsequent muonic decay on the QCD renormalization 
and factorization scale $\mu=\mu_R=\mu_F$.
The plots are reproduced from \citerefs{Nason:1999xs}
with CTEQ4M~\cite{Lai:1996mg}
and ACFGP~\cite{Aurenche:1992sb} parton densities for
the proton and the photon,
respectively. 
The calculation was carried out in the $\overline{\mrm{MS}}$ 
scheme using an NLO strong coupling constant with 
$\Lambda_5 = 202\mrm{MeV}$ and an angular cut of 
$5^\circ$ (instead of a fixed $Q^2_\mrm{cut}$) 
to separate photoproduction 
from deep inelastic scattering (DIS).
%The strong coupling constant is taken at NLO with
%$\Lambda_5 = 202\mrm{MeV}$ and an angular cut of 
%$5^\circ$ (instead of a fixed $Q^2_\mrm{cut}$) 
%is introduced to separate photoproduction 
%from deep inelastic scattering (DIS).
At $\mu = M_W$, the QCD corrections enhance 
the resolved contribution by about $40\%$. 
It can be observed that the scale
dependence in the sum of direct and resolved contributions is
significantly reduced, once the NLO corrections to the resolved part are
included. 
The full curves show  the total sum of NLO resolved, LO direct
and LO DIS contribution, which is the prediction of \citeref{Nason:1999xs} 
for the total $W^\pm$ production cross
sections. 
The residual scale dependence
is about $5\%$. 
Since the remaining dependence on $Q^2_\mrm{cut}$ is of similar
size, Nason, R\"uckl and Spira estimate 
the total theoretical uncertainty
to be less than about $10\%$. 

%**************************************************************
\subsection{Corrections to the Differential Cross Section} 
\label{subsec:diff}

We proceed with the discussion of QCD quantum corrections 
to the doubly and singly differential $W$ boson 
production cross sections 
\mbox{$\mrm{d^2}\sigma/\mrm{d}y \mrm{d}p_T$}
and \mbox{$\mrm{d} \sigma/ \mrm{d} p_T$} as discussed in 
\citerefs{Diener:2002if}.
As pointed out there, the resolved contribution
to the $p_T$ distribution is negligible for 
\mbox{$p_T > 15\mrm{GeV}$}.
Instead, direct photoproduction is the dominant mechanism whereas 
deep-inelastic scattering is smaller but comparable.
Thus, the authors of \citerefs{Diener:2002if} confined the
discussion of QCD corrected 
differential cross section distributions 
to the case of direct photoproduction.
Events with one jet in the final state in addition
to the $W$ boson are distinguished from events with two 
final state jets by requiring that their cones of
size $R=1$ defined 
by an inclusive $k_T$ algorithm do not overlap
and that both partons have a transverse energy of more
than $5\mrm{GeV}$.
All other configurations (especially with LO kinematics)
define one-jet events.

\begin{figure}[ht]
%\centerline{\epsfxsize=4.2cm\begin{turn}{-90}\epsfbox{scale1.ps}\end{turn}
%\epsfxsize=4.2cm\begin{turn}{-90}\epsfbox{scale2.ps}\end{turn}}   
\centerline{\epsfxsize=5cm\begin{turn}{-90}\epsfbox{diener_fig6.eps}\end{turn}}
\vspace*{.5cm}
\centerline{\epsfxsize=5cm\begin{turn}{-90}\epsfbox{diener_fig7.eps}\end{turn}} 
\vspace*{.5cm}
\caption{\label{fig:dss1} Dependence of the direct contribution
to the $W$ production cross section on the
renormalization and factorization scale $\mu = \mu_R = \mu_F = \xi M_W$
for \mbox{$p_T=40\mrm{GeV}$}.
The full curves represent the NLO predictions and the broken curves the
LO scale dependences.
The upper plot presents $W+1\mrm{jet}$ production, the lower one presents
the sum of $W+1\mrm{jet}$ plus $W+2\mrm{jets}$.}
\end{figure}
To estimate the theoretical uncertainties, the
QCD scale dependence of the direct
contributions to the processes $e^+ p \to W^\pm + X$
is presented in Fig.~\ref{fig:dss1}.
The scale dependence is significantly smaller, 
once the NLO corrections are included. 
The residual scale dependence is reduced from about $20\%$ down
to around $5\%$. 
The plots of Fig.~\ref{fig:dss1} clearly indicate that the NLO QCD
corrections are small at the central value of the QCD scale $m_W$.
Since the uncertainties of the parton densities are of similar size, the
total theoretical uncertainty can be estimated to be less than 
about $10\%$.

\begin{figure}[ht]
%\centerline{\epsfxsize=4.2cm\begin{turn}{-90}\epsfbox{ptfull+.ps}\end{turn}
%\epsfxsize=4.2cm\begin{turn}{-90}\epsfbox{y+.ps}\end{turn}}   
\centerline{\epsfxsize=5cm\begin{turn}{-90}\epsfbox{diener_fig8.eps}\end{turn}}
\vspace*{.5cm}
\centerline{\epsfxsize=5cm\begin{turn}{-90}\epsfbox{diener_fig9.eps}\end{turn}}   
\vspace*{.5cm}
\caption{\label{fig:dss2}  
Shown are the transverse momentum distributions (up),
and rapidity distribution (down) 
at $p_T=40\mrm{GeV}$ of $W^+$ bosons at HERA. 
The full curves show the
total distributions including the NLO corrections to the direct contribution,
while the broken lines exhibit the individual parts at LO and the direct
part at NLO.}
\end{figure}
Fig.~\ref{fig:dss2} depicts the transverse momentum (up)
as well as rapidity (down) distribution of 
$W^+$ bosons produced at HERA with \mbox{$\sqrt{S}=318\mrm{GeV}$}. 

The upper plot clearly demonstrates the dominance of the
direct photoproduction part, while the DIS part is smaller but still
relevant. The resolved photoproduction part can safely be neglected for
$p_T$ values larger than about $10$--$15\mrm{GeV}$. The rising total sum at $p_T$
values below about $15\mrm{GeV}$ signals the breakdown of the purely perturbative
results and underlines the necessity of a soft gluon resummation
to describe $W$ boson production for smaller transverse momenta
which, however, is beyond the scope of \citeref{Diener:2002if}.

The rapidity distribution of $W^+$ boson production at HERA for
\mbox{$p_T=40\mrm{GeV}$} is presented in the lower plot of
Fig.~\ref{fig:dss2} for the individual production mechanisms. 
The direct photoproduction part is shown at LO and NLO.
As in the case of the transverse momentum
distribution, the QCD corrections to the direct contribution amount to less
than about $10$--$15\%$ and are, thus, of moderate size. 
The figure shows that
$W$ bosons are preferably produced at larger rapidity values.



%**************************************************************
%**************************************************************
\section{Conclusions}
\label{sec:conclusions}

We have reviewed FCNC single top quark as well as single $W$ boson
production at HERA.

As the authors of \citeref{Belyaev:2001hf} could show, 
the inclusion of lowest-order QCD quantum effect in the eikonal
approximation leads to a stabilization of the 
FCNC single top production cross section with respect to 
variation of the renormalization and factorization scales.
The stabilization is most pronounced for 
\mbox{$10\mrm{GeV} < \mu_R = \mu_F < 75\mrm{GeV}$}.
For \mbox{$\mu_R = \mu_F = m_t$} QCD quantum effects 
lead to a $25\%$ increase in cross section,
rendering reliable control of the radiative 
corrections an indispensable
prerequisite for a precise determination of FCNC couplings.

There are several competing mechanisms to produce 
$W$ bosons at HERA.
The most important contributions to the total
cross section stem from resolved photons, whereas
the transverse momentum and rapidity distributions 
(for finite $p_T$) 
are dominated by direct photoproduction.

In their analysis of the dominant resolved part
the authors of \citeref{Nason:1999xs} found 
that QCD corrections enhance the total $W$ production
cross section by about $40\%$ while reducing the 
dependence on the renormalization and factorization 
scale to $5\%$. The total remaining theoretical uncertainty
is estimated to be smaller than $10\%$.

Direct photoproduction is the dominant contribution
to distributions of the $W$ production cross section in
($W$) transverse momentum and rapidity. 
The lowest-order QCD radiative corrections constitute 
a \mbox{$\pm 10$--$15\%$} effect and reduce the 
renormalization and factorization scale dependence to 
about $5\%$.
The total theoretical uncertainty is estimated to 
be roughly $10\%$.


We remark
that neither anomalous top quark 
production (with subsequent decay into a 
$W$ boson and a $b$ quark) 
nor higher-order QCD corrections to 
$W$ boson production within the SM 
can explain the lepton flavour asymmetry in the 
excess of isolated single lepton events as observed in 
\citeref{Andreev:2003pm}.


%**************************************************************
%**************************************************************
\section*{Acknowledgments}
I would like to thank M.~Spira and C.~Schwanenberger for the 
fruitful collaboration and useful discussions.
I appreciated the hospitality at Castle Ringberg and I would like 
to thank the organizers for their excellent work
and the invitation to participate in this meeting.

%**************************************************************
%**************************************************************
\section*{Note Added in Proof}
Meanwhile, the soft-gluonic corrections to FCNC single top production
at HERA (and the Tevatron)
have been evaluated up to next-to-next-to-leading 
order level~\cite{Kidonakis:2003sc}.
The authors observe a further stabilization of the cross section
with respect to variations of the QCD scale in comparison to the 
NLO order soft gluon-corrected result.



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\begin{thebibliography}{0}

%\cite{Andreev:2003pm}
\bibitem{Andreev:2003pm}
V.~Andreev {\it et al.}  [H1 Collaboration],
%``Isolated electrons and muons in events with missing transverse momentum  at HERA,''
Phys.\ Lett.\ B {\bf 561}, 241 (2003).
%[arXiv:hep-ex/0301030].
%%CITATION = HEP-EX 0301030;%%
%
%\cite{Belyaev:2001hf}
\bibitem{Belyaev:2001hf}
A.~Belyaev and N.~Kidonakis,
%``QCD corrections to FCNC single top production at HERA,''
Phys.\ Rev.\ D {\bf 65}, 037501 (2002).
%[arXiv:hep-ph/0102072].
%%CITATION = HEP-PH 0102072;%%
%
%\cite{Nason:1999xs}
\bibitem{Nason:1999xs}
P.~Nason, R.~R\"uckl and M.~Spira,
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J.\ Phys.\ G {\bf 25}, 1434 (1999);\\
%[arXiv:hep-ph/9902296].
%%CITATION = HEP-PH 9902296;%%
%\cite{Spira:1999ja}
%\bibitem{Spira:1999ja}
M.~Spira,
%``W boson production at NLO,''
arXiv:hep-ph/9905469.
%%CITATION = HEP-PH 9905469;%%
%
%\cite{Diener:2002if}
\bibitem{Diener:2002if}
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%``Photoproduction of W bosons at HERA: QCD corrections,''
Eur.\ Phys.\ J.\ C {\bf 25}, 405 (2002);\\
%[arXiv:hep-ph/0203269].
%%CITATION = HEP-PH 0203269;%%
%
%\cite{Diener:2003df}
%\bibitem{Diener:2003df}
K.~P.~Diener, C.~Schwanenberger and M.~Spira,
%``Photoproduction of W bosons at HERA: Reweighting method for  implementing QCD corrections in Monte Carlo programs,''
hep-ex/0302040.
%%CITATION = HEP-EX 0302040;%%
%
%\cite{Lai:1999wy}
\bibitem{Lai:1999wy}
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%``Global {QCD} analysis of parton structure of the nucleon: CTEQ5 parton  distributions,''
Eur.\ Phys.\ J.\ C {\bf 12}, 375 (2000).
%[arXiv:hep-ph/9903282].
%%CITATION = HEP-PH 9903282;%%
%
%\cite{Chekanov:2003yt} 
\bibitem{Chekanov:2003yt}
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Phys.\ Lett.\ B {\bf 559}, 153 (2003).
%[arXiv:hep-ex/0302010].
%%CITATION = HEP-EX 0302010;%%
%
%\cite{Aktas:2003yd}
\bibitem{Aktas:2003yd}
A.~Aktas {\it et al.}  [H1 Collaboration],
%``Search for single top quark production in e p collisions at HERA,''
arXiv:hep-ex/0310032.
%%CITATION = HEP-EX 0310032;%%
%
%\cite{Lai:1996mg}
\bibitem{Lai:1996mg}
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%``Improved parton distributions from global analysis of recent deep  inelastic scattering and inclusive jet data,''
Phys.\ Rev.\ D {\bf 55}, 1280 (1997).
%[arXiv:hep-ph/9606399].
%%CITATION = HEP-PH 9606399;%%
%
%\cite{Aurenche:1992sb}
\bibitem{Aurenche:1992sb}
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%%CITATION = ZEPYA,C56,589;%%
%
%\cite{Kidonakis:2003sc}
\bibitem{Kidonakis:2003sc}
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%``FCNC top quark production via anomalous t q V couplings beyond leading order,''
arXiv:hep-ph/0310299.
%%CITATION = HEP-PH 0310299;%%


\end{thebibliography}

\end{document}
